%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP216^2 : TPTP v9.2.1. Bugfixed v9.2.1.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Feb 28 09:05:03 AM UTC 2026
% Result : Theorem 24.80s 8.34s
% Output : Refutation 24.80s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : ITP216^2 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.00/0.10 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.13/0.30 % Computer : n019.cluster.edu
% 0.13/0.30 % Model : x86_64 x86_64
% 0.13/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.30 % Memory : 8042.1875MB
% 0.13/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.30 % CPULimit : 300
% 0.13/0.30 % WCLimit : 300
% 0.13/0.30 % DateTime : Thu Feb 26 05:04:09 EST 2026
% 0.13/0.30 % CPUTime :
% 0.13/0.30 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.40 ---- Original DTF file ---
% 0.19/0.40 thf(spec,logic,$$dhol).
% 0.19/0.40 %------------------------------------------------------------------------------
% 0.19/0.40 % File : ITP216^2 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.19/0.40 % Domain : Interactive Theorem Proving
% 0.19/0.40 % Problem : Sledgehammer problem Time_Reasoning 00074_002979
% 0.19/0.40 % Version : [Des22] axioms.
% 0.19/0.40 % English :
% 0.19/0.40
% 0.19/0.40 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.40 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.19/0.40 % Source : [Des22]
% 0.19/0.40 % Names : 0029_Time_Reasoning_00074_002979 [Des22]
% 0.19/0.40
% 0.19/0.40 % Status : Theorem
% 0.19/0.40 % Rating : ? v9.2.1
% 0.19/0.40 % Syntax : Number of formulae : 7612 (2659 unt; 913 typ; 0 def)
% 0.19/0.40 % Number of atoms : 19157 (7619 equ; 6 cnn)
% 0.19/0.40 % Maximal formula atoms : 48 ( 2 avg)
% 0.19/0.40 % Number of connectives : 155299 (1816 ~; 175 |;1143 &;144024 @)
% 0.19/0.40 % ( 0 <=>;8141 =>; 0 <=; 0 <~>)
% 0.19/0.40 % Maximal formula depth : 38 ( 8 avg)
% 0.19/0.40 % Number of types : 13 ( 12 usr)
% 0.19/0.40 % Number of type conns : 6488 (6488 >; 0 *; 0 +; 0 <<)
% 0.19/0.41 % Number of symbols : 905 ( 901 usr; 20 con; 0-10 aty)
% 0.19/0.41 % Number of variables : 25679 (2885 ^;21417 !; 384 ?;25679 :)
% 0.19/0.41 % ( 993 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.41 % SPC : TH1_THM_EQU_NAR
% 0.19/0.41
% 0.19/0.41 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.19/0.41 % from the van Emde Boas Trees session in the Archive of Formal
% 0.19/0.41 % proofs -
% 0.19/0.41 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.19/0.41 % 2022-02-17 16:14:04.213
% 0.19/0.41 % Bugfixes : Added ()s around negated sides of equations.
% 0.19/0.41 %------------------------------------------------------------------------------
% 0.19/0.41 % Could-be-implicit typings (26)
% 0.19/0.41 thf(ty_t_Heap__Time__Monad_OHeap,type,
% 0.19/0.41 heap_Time_Heap: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Code__Numeral_Onatural,type,
% 0.19/0.41 code_natural: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Code__Numeral_Ointeger,type,
% 0.19/0.41 code_integer: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Code__Evaluation_Oterm,type,
% 0.19/0.41 code_term: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Heap_Oheap_Oheap__ext,type,
% 0.19/0.41 heap_ext: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Product__Type_Ounit,type,
% 0.19/0.41 product_unit: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Product__Type_Oprod,type,
% 0.19/0.41 product_prod: $tType > $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Old__Datatype_Onode,type,
% 0.19/0.41 old_node: $tType > $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Multiset_Omultiset,type,
% 0.19/0.41 multiset: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Typerep_Otyperep,type,
% 0.19/0.41 typerep: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Assertions_Oassn,type,
% 0.19/0.41 assn: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Predicate_Opred,type,
% 0.19/0.41 pred: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Sum__Type_Osum,type,
% 0.19/0.41 sum_sum: $tType > $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Predicate_Oseq,type,
% 0.19/0.41 seq: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Option_Ooption,type,
% 0.19/0.41 option: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_String_Ochar,type,
% 0.19/0.41 char: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Heap_Oarray,type,
% 0.19/0.41 array: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_List_Olist,type,
% 0.19/0.41 list: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Heap_Oref,type,
% 0.19/0.41 ref: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Set_Oset,type,
% 0.19/0.41 set: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Rat_Orat,type,
% 0.19/0.41 rat: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Num_Onum,type,
% 0.19/0.41 num: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Nat_Onat,type,
% 0.19/0.41 nat: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_Int_Oint,type,
% 0.19/0.41 int: $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_t_itself,type,
% 0.19/0.41 itself: $tType > $tType ).
% 0.19/0.41
% 0.19/0.41 thf(ty_tf_a,type,
% 0.19/0.41 a: $tType ).
% 0.19/0.41
% 0.19/0.41 % Explicit typings (887)
% 0.19/0.41 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.19/0.41 bounded_lattice_top:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Typerep_Otyperep,type,
% 0.19/0.41 typerep2:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Enum_Oenum,type,
% 0.19/0.41 enum:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Code__Evaluation_Oterm__of,type,
% 0.19/0.41 code_term_of:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_HOL_Otype,type,
% 0.19/0.41 type:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_HOL_Oequal,type,
% 0.19/0.41 cl_HOL_Oequal:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Heap_Oheap,type,
% 0.19/0.41 heap:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Odvd,type,
% 0.19/0.41 dvd:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Oone,type,
% 0.19/0.41 one:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Oring,type,
% 0.19/0.41 ring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Oplus,type,
% 0.19/0.41 plus:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Ozero,type,
% 0.19/0.41 zero:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Num_Onumeral,type,
% 0.19/0.41 numeral:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Power_Opower,type,
% 0.19/0.41 power:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Fields_Ofield,type,
% 0.19/0.41 field:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_GCD_Oring__gcd,type,
% 0.19/0.41 ring_gcd:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Lattices_Oinf,type,
% 0.19/0.41 inf:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Lattices_Osup,type,
% 0.19/0.41 sup:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Oring__1,type,
% 0.19/0.41 ring_1:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Ouminus,type,
% 0.19/0.41 uminus:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.41 ord:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Fields_Oinverse,type,
% 0.19/0.41 inverse:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Osemiring,type,
% 0.19/0.41 semiring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Nat_Oring__char__0,type,
% 0.19/0.41 ring_char_0:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Num_Oneg__numeral,type,
% 0.19/0.41 neg_numeral:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.41 order:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ocomm__ring,type,
% 0.19/0.41 comm_ring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_GCD_Osemiring__Gcd,type,
% 0.19/0.41 semiring_Gcd:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_GCD_Osemiring__gcd,type,
% 0.19/0.41 semiring_gcd:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.41 group_add:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Ono__bot,type,
% 0.19/0.41 no_bot:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Ono__top,type,
% 0.19/0.41 no_top:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Osemiring__0,type,
% 0.19/0.41 semiring_0:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Osemiring__1,type,
% 0.19/0.41 semiring_1:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.19/0.41 finite_finite:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Omonoid__add,type,
% 0.19/0.41 monoid_add:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 0.19/0.41 comm_ring_1:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Omonoid__mult,type,
% 0.19/0.41 monoid_mult:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.41 linorder:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.41 preorder:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Parity_Oring__parity,type,
% 0.19/0.41 ring_parity:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 0.19/0.41 idom_abs_sgn:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Oordered__ring,type,
% 0.19/0.41 ordered_ring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ozero__neq__one,type,
% 0.19/0.41 zero_neq_one:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Countable_Ocountable,type,
% 0.19/0.41 countable:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Fields_Ofield__char__0,type,
% 0.19/0.41 field_char_0:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Oab__group__add,type,
% 0.19/0.41 ab_group_add:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.19/0.41 semiring_char_0:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.19/0.41 order_bot:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Orderings_Oorder__top,type,
% 0.19/0.41 order_top:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ocomm__semiring,type,
% 0.19/0.41 comm_semiring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ozero__less__one,type,
% 0.19/0.41 zero_less_one:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Fields_Odivision__ring,type,
% 0.19/0.41 division_ring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Osemigroup__add,type,
% 0.19/0.41 semigroup_add:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Num_Osemiring__numeral,type,
% 0.19/0.41 semiring_numeral:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Osemidom__divide,type,
% 0.19/0.41 semidom_divide:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Osemidom__modulo,type,
% 0.19/0.41 semidom_modulo:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Complete__Lattices_OInf,type,
% 0.19/0.41 complete_Inf:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Complete__Lattices_OSup,type,
% 0.19/0.41 complete_Sup:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Osemigroup__mult,type,
% 0.19/0.41 semigroup_mult:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 0.19/0.41 comm_semiring_0:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 0.19/0.41 comm_semiring_1:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Olinordered__idom,type,
% 0.19/0.41 linordered_idom:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Olinordered__ring,type,
% 0.19/0.41 linordered_ring:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Rings_Osemiring__modulo,type,
% 0.19/0.41 semiring_modulo:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 0.19/0.41 comm_monoid_add:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Parity_Osemiring__parity,type,
% 0.19/0.41 semiring_parity:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Fields_Olinordered__field,type,
% 0.19/0.41 linordered_field:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 0.19/0.41 ab_semigroup_add:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 0.19/0.41 comm_monoid_diff:
% 0.19/0.41 !>[A: $tType] : $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 0.19/0.41 comm_monoid_mult:
% 0.19/0.41 !>[A: $tType] : $o ).
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% 0.19/0.41 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
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% 0.19/0.41 thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
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% 0.19/0.41 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
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% 0.19/0.41 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
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% 0.19/0.41 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
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% 0.19/0.41 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
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% 0.19/0.41 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
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% 0.19/0.41 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
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% 0.19/0.41 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
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% 0.19/0.41 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
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% 0.19/0.41 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
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% 0.19/0.41 thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
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% 0.19/0.41 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
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% 0.19/0.41 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Ofreeze,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Oget,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Olen_H,type,
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% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( array @ B ) > ( heap_Time_Heap @ A ) ) ).
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% 0.19/0.41 thf(sy_c_Array__Time_Olength,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Omake,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Omake_H,type,
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% 0.19/0.41 !>[B: $tType,A: $tType] : ( code_integer > ( B > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
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% 0.19/0.41 thf(sy_c_Array__Time_Omap__entry,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Onew,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Onew_H,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Onoteq,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Onth,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Onth_H,type,
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% 0.19/0.41 !>[A: $tType] : ( ( array @ A ) > code_integer > ( heap_Time_Heap @ A ) ) ).
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% 0.19/0.41 thf(sy_c_Array__Time_Oof__list,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Opresent,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Oswap,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Oupd,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Oupd_H,type,
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% 0.19/0.41 thf(sy_c_Array__Time_Oupdate,type,
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% 0.19/0.41 thf(sy_c_Assertions_Oassn_ORep__assn,type,
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% 0.19/0.41 thf(sy_c_Assertions_Oex__assn,type,
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% 0.19/0.41 thf(sy_c_Assertions_Oin__range,type,
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% 0.19/0.41
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Oone__assn__raw,type,
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Oone__assn__raw__rel,type,
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Oproper,type,
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Opure__assn,type,
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Opure__assn__raw,type,
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Opure__assn__raw__rel,type,
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% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_OrelH,type,
% 0.19/0.41 relH: ( set @ nat ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Osnga__assn,type,
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% 0.19/0.41 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > assn ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Osnga__assn__raw,type,
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% 0.19/0.41 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Osnga__assn__raw__rel,type,
% 0.19/0.41 snga_assn_raw_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Osngr__assn,type,
% 0.19/0.41 sngr_assn:
% 0.19/0.41 !>[A: $tType] : ( ( ref @ A ) > A > assn ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Osngr__assn__raw,type,
% 0.19/0.41 sngr_assn_raw:
% 0.19/0.41 !>[A: $tType] : ( ( ref @ A ) > A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Osngr__assn__raw__rel,type,
% 0.19/0.41 sngr_assn_raw_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Otimes__assn__raw,type,
% 0.19/0.41 times_assn_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
% 0.19/0.41 times_assn_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Owand__assn,type,
% 0.19/0.41 wand_assn: assn > assn > assn ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Owand__raw,type,
% 0.19/0.41 wand_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Assertions_Owand__raw__rel,type,
% 0.19/0.41 wand_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
% 0.19/0.41 bNF_Cardinal_Csum:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
% 0.19/0.41 bNF_Cardinal_cexp:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
% 0.19/0.41 bNF_Cardinal_cfinite:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
% 0.19/0.41 bNF_Ca4139267488887388095finite:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
% 0.19/0.41 bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
% 0.19/0.41 bNF_Cardinal_cprod:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
% 0.19/0.41 bNF_Cardinal_csum:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
% 0.19/0.41 bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
% 0.19/0.41 bNF_Cardinal_czero:
% 0.19/0.41 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
% 0.19/0.41 bNF_Ca8387033319878233205ardSuc:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
% 0.19/0.41 bNF_Ca6860139660246222851ard_of:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
% 0.19/0.41 bNF_Ca8970107618336181345der_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
% 0.19/0.41 bNF_Ca7293521722713021262ofinal:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
% 0.19/0.41 bNF_Ca6246979054910435723ardSuc:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
% 0.19/0.41 bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
% 0.19/0.41 bNF_Ca7133664381575040944arCard:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 0.19/0.41 bNF_Ca3754400796208372196lChain:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_OGr,type,
% 0.19/0.41 bNF_Gr:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_OGrp,type,
% 0.19/0.41 bNF_Grp:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_Ocollect,type,
% 0.19/0.41 bNF_collect:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_Oconvol,type,
% 0.19/0.41 bNF_convol:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_Oeq__onp,type,
% 0.19/0.41 bNF_eq_onp:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_OfstOp,type,
% 0.19/0.41 bNF_fstOp:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_Opick__middlep,type,
% 0.19/0.41 bNF_pick_middlep:
% 0.19/0.41 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_Orel__fun,type,
% 0.19/0.41 bNF_rel_fun:
% 0.19/0.41 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_OsndOp,type,
% 0.19/0.41 bNF_sndOp:
% 0.19/0.41 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Def_Ovimage2p,type,
% 0.19/0.41 bNF_vimage2p:
% 0.19/0.41 !>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
% 0.19/0.41 bNF_Gr5436034075474128252omCard:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 0.19/0.41 bNF_Greatest_image2:
% 0.19/0.41 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
% 0.19/0.41 bNF_Gr4221423524335903396lImage:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
% 0.19/0.41 bNF_Gr7122648621184425601vImage:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
% 0.19/0.41 bNF_Greatest_toCard:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
% 0.19/0.41 bNF_Gr1419584066657907630d_pred:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
% 0.19/0.41 bNF_Wellorder_Func:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
% 0.19/0.41 bNF_Wellorder_bsqr:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
% 0.19/0.41 bNF_Wellorder_curr:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
% 0.19/0.41 bNF_We2720479622203943262_image:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
% 0.19/0.41 bNF_We413866401316099525erIncl:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
% 0.19/0.41 bNF_Wellorder_ordIso:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
% 0.19/0.41 bNF_Wellorder_ordLeq:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
% 0.19/0.41 bNF_We4044943003108391690rdLess:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
% 0.19/0.41 bNF_Wellorder_compat:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
% 0.19/0.41 bNF_Wellorder_embed:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
% 0.19/0.41 bNF_Wellorder_embedS:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
% 0.19/0.41 bNF_Wellorder_iso:
% 0.19/0.41 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
% 0.19/0.41 bNF_Wellorder_wo_rel:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
% 0.19/0.41 bNF_We4791949203932849705sMinim:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
% 0.19/0.41 bNF_We1388413361240627857o_max2:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
% 0.19/0.41 bNF_We6954850376910717587_minim:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
% 0.19/0.41 bNF_Wellorder_wo_suc:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
% 0.19/0.41 basic_BNF_size_prod:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
% 0.19/0.41 basic_BNF_size_sum:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNFs_Ofsts,type,
% 0.19/0.41 basic_fsts:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNFs_Opred__fun,type,
% 0.19/0.41 basic_pred_fun:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNFs_Opred__prod,type,
% 0.19/0.41 basic_pred_prod:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNFs_Orel__prod,type,
% 0.19/0.41 basic_rel_prod:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Basic__BNFs_Osnds,type,
% 0.19/0.41 basic_snds:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Binomial_Obinomial,type,
% 0.19/0.41 binomial: nat > nat > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Binomial_Ogbinomial,type,
% 0.19/0.41 gbinomial:
% 0.19/0.41 !>[A: $tType] : ( A > nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 0.19/0.41 bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 0.19/0.41 bit_and_not_num: num > num > ( option @ num ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 0.19/0.41 bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 0.19/0.41 bit_concat_bit: nat > int > int > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 0.19/0.41 bit_or_not_num_neg: num > num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 0.19/0.41 bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
% 0.19/0.41 bit_ri4277139882892585799ns_not:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
% 0.19/0.41 bit_ri4674362597316999326ke_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
% 0.19/0.41 bit_se5824344872417868541ns_and:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
% 0.19/0.41 bit_se4197421643247451524op_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
% 0.19/0.41 bit_se8732182000553998342ip_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
% 0.19/0.41 bit_se2239418461657761734s_mask:
% 0.19/0.41 !>[A: $tType] : ( nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
% 0.19/0.41 bit_se1065995026697491101ons_or:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
% 0.19/0.41 bit_se4730199178511100633sh_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
% 0.19/0.41 bit_se5668285175392031749et_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
% 0.19/0.41 bit_se2584673776208193580ke_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
% 0.19/0.41 bit_se2638667681897837118et_bit:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
% 0.19/0.41 bit_se5824344971392196577ns_xor:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
% 0.19/0.41 bit_se5641148757651400278ts_bit:
% 0.19/0.41 !>[A: $tType] : ( A > nat > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
% 0.19/0.41 bit_se6407376104438227557le_bit:
% 0.19/0.41 !>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 0.19/0.41 bit_take_bit_num: nat > num > ( option @ num ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 0.19/0.41 bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 0.19/0.41 bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
% 0.19/0.41 bit_un6697907153464112080or_num: num > num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
% 0.19/0.41 bit_un4773296044027857193um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 0.19/0.41 bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 0.19/0.41 bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
% 0.19/0.41 boolea2506097494486148201lgebra:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
% 0.19/0.41 boolea3799213064322606851m_diff:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Cancellation_Oiterate__add,type,
% 0.19/0.41 iterate_add:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_ONat,type,
% 0.19/0.41 code_Nat: code_integer > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_ONeg,type,
% 0.19/0.41 code_Neg: num > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_OPos,type,
% 0.19/0.41 code_Pos: num > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_OSuc,type,
% 0.19/0.41 code_Suc: code_natural > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 0.19/0.41 code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ocr__integer,type,
% 0.19/0.41 code_cr_integer: int > code_integer > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ocr__natural,type,
% 0.19/0.41 code_cr_natural: nat > code_natural > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 0.19/0.41 code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 0.19/0.41 code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Odup,type,
% 0.19/0.41 code_dup: code_integer > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 0.19/0.41 code_int_of_integer: code_integer > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 0.19/0.41 code_integer_of_int: int > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
% 0.19/0.41 code_integer_of_nat: nat > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
% 0.19/0.41 code_i5400310926305786745atural: code_natural > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 0.19/0.41 code_integer_of_num: num > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 0.19/0.41 code_nat_of_integer: code_integer > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 0.19/0.41 code_case_natural:
% 0.19/0.41 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
% 0.19/0.41 code_nat_of_natural: code_natural > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
% 0.19/0.41 code_natural_of_nat: nat > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 0.19/0.41 code_rec_natural:
% 0.19/0.41 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
% 0.19/0.41 code_rec_set_natural:
% 0.19/0.41 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onatural__of__integer,type,
% 0.19/0.41 code_n4118661773612635043nteger: code_integer > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onegative,type,
% 0.19/0.41 code_negative: num > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 0.19/0.41 code_num_of_integer: code_integer > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Opcr__integer,type,
% 0.19/0.41 code_pcr_integer: int > code_integer > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Opcr__natural,type,
% 0.19/0.41 code_pcr_natural: nat > code_natural > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Opositive,type,
% 0.19/0.41 code_positive: num > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Osize__nat,type,
% 0.19/0.41 code_size_nat: code_natural > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Code__Numeral_Osub,type,
% 0.19/0.41 code_sub: num > num > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
% 0.19/0.41 complete_Inf_Inf:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 0.19/0.41 complete_Sup_Sup:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
% 0.19/0.41 comple1908693960933563346ssible:
% 0.19/0.41 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
% 0.19/0.41 comple187402453842119260l_fixp:
% 0.19/0.41 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
% 0.19/0.41 comple115746919287870866o_fixp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Partial__Order_Ochain,type,
% 0.19/0.41 comple1602240252501008431_chain:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 0.19/0.41 comple7038119648293358887notone:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Countable_Ofrom__nat,type,
% 0.19/0.41 from_nat:
% 0.19/0.41 !>[A: $tType] : ( nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Countable_Onth__item,type,
% 0.19/0.41 nth_item:
% 0.19/0.41 !>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Countable_Onth__item__rel,type,
% 0.19/0.41 nth_item_rel: nat > nat > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Countable_Oto__nat,type,
% 0.19/0.41 to_nat:
% 0.19/0.41 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Divides_Oadjust__div,type,
% 0.19/0.41 adjust_div: ( product_prod @ int @ int ) > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Divides_Odivmod__nat,type,
% 0.19/0.41 divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Divides_Oeucl__rel__int,type,
% 0.19/0.41 eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 0.19/0.41 unique5940410009612947441es_aux:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
% 0.19/0.41 unique8689654367752047608divmod:
% 0.19/0.41 !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
% 0.19/0.41 unique1321980374590559556d_step:
% 0.19/0.41 !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 0.19/0.41 equiv_congruent:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Equiv__Relations_Ocongruent2,type,
% 0.19/0.41 equiv_congruent2:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Equiv__Relations_Oequiv,type,
% 0.19/0.41 equiv_equiv:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Equiv__Relations_Oequivp,type,
% 0.19/0.41 equiv_equivp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Equiv__Relations_Oproj,type,
% 0.19/0.41 equiv_proj:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Equiv__Relations_Oquotient,type,
% 0.19/0.41 equiv_quotient:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
% 0.19/0.41 euclid6346220572633701492n_size:
% 0.19/0.41 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
% 0.19/0.41 euclid7384307370059645450egment:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
% 0.19/0.41 comm_s3205402744901411588hammer:
% 0.19/0.41 !>[A: $tType] : ( A > nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
% 0.19/0.41 semiring_char_0_fact:
% 0.19/0.41 !>[A: $tType] : ( nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 0.19/0.41 inverse_inverse:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ocard,type,
% 0.19/0.41 finite_card:
% 0.19/0.41 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
% 0.19/0.41 finite6289374366891150609ommute:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
% 0.19/0.41 finite1981172056906289455axioms:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
% 0.19/0.41 finite4980608107308702382axioms:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofinite,type,
% 0.19/0.41 finite_finite2:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofold,type,
% 0.19/0.41 finite_fold:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofolding,type,
% 0.19/0.41 finite_folding:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofolding__idem,type,
% 0.19/0.41 finite_folding_idem:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofolding__idem__axioms,type,
% 0.19/0.41 finite7837460588564673216axioms:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofolding__idem__on,type,
% 0.19/0.41 finite1890593828518410140dem_on:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
% 0.19/0.41 finite6916993218817215295axioms:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Obij__betw,type,
% 0.19/0.41 bij_betw:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Ocomp,type,
% 0.19/0.41 comp:
% 0.19/0.41 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Ofcomp,type,
% 0.19/0.41 fcomp:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( B > C ) > A > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Ofun__upd,type,
% 0.19/0.41 fun_upd:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Oid,type,
% 0.19/0.41 id:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Oinj__on,type,
% 0.19/0.41 inj_on:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Omap__fun,type,
% 0.19/0.41 map_fun:
% 0.19/0.41 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Omono__on,type,
% 0.19/0.41 mono_on:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Ooverride__on,type,
% 0.19/0.41 override_on:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Ostrict__mono__on,type,
% 0.19/0.41 strict_mono_on:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun_Othe__inv__into,type,
% 0.19/0.41 the_inv_into:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Omax__strict,type,
% 0.19/0.41 fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Omax__weak,type,
% 0.19/0.41 fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Omin__strict,type,
% 0.19/0.41 fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Omin__weak,type,
% 0.19/0.41 fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Opair__leq,type,
% 0.19/0.41 fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Opair__less,type,
% 0.19/0.41 fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 0.19/0.41 fun_reduction_pair:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 0.19/0.41 fun_rp_inv_image:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_OGcd__class_OGcd,type,
% 0.19/0.41 gcd_Gcd:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_OGcd__class_OLcm,type,
% 0.19/0.41 gcd_Lcm:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Obezw,type,
% 0.19/0.41 bezw: nat > nat > ( product_prod @ int @ int ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Obezw__rel,type,
% 0.19/0.41 bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Obounded__quasi__semilattice,type,
% 0.19/0.41 bounde8507323023520639062attice:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
% 0.19/0.41 bounde6485984586167503788ce_set:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
% 0.19/0.41 bounde2362111253966948842tice_F:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Ogcd__class_Ogcd,type,
% 0.19/0.41 gcd_gcd:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Ogcd__class_Olcm,type,
% 0.19/0.41 gcd_lcm:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 0.19/0.41 gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
% 0.19/0.41 semiring_gcd_Gcd_fin:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
% 0.19/0.41 semiring_gcd_Lcm_fin:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Oabel__semigroup,type,
% 0.19/0.41 abel_semigroup:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Oabel__semigroup__axioms,type,
% 0.19/0.41 abel_s757365448890700780axioms:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 0.19/0.41 abs_abs:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ocomm__monoid,type,
% 0.19/0.41 comm_monoid:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ocomm__monoid__axioms,type,
% 0.19/0.41 comm_monoid_axioms:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ogroup,type,
% 0.19/0.41 group:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ogroup__axioms,type,
% 0.19/0.41 group_axioms:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.19/0.41 minus_minus:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Omonoid,type,
% 0.19/0.41 monoid:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Omonoid__axioms,type,
% 0.19/0.41 monoid_axioms:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Oone__class_Oone,type,
% 0.19/0.41 one_one:
% 0.19/0.41 !>[A: $tType] : A ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 0.19/0.41 plus_plus:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Osemigroup,type,
% 0.19/0.41 semigroup:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 0.19/0.41 sgn_sgn:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 0.19/0.41 times_times:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.19/0.41 uminus_uminus:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 0.19/0.41 zero_zero:
% 0.19/0.41 !>[A: $tType] : A ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
% 0.19/0.41 groups3894954378712506084id_sum:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 0.19/0.41 groups7311177749621191930dd_sum:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
% 0.19/0.41 groups1027152243600224163dd_sum:
% 0.19/0.41 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
% 0.19/0.41 groups7121269368397514597t_prod:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
% 0.19/0.41 groups1962203154675924110t_prod:
% 0.19/0.41 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
% 0.19/0.41 groups778175481326437816id_set:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
% 0.19/0.41 groups_comm_monoid_F:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
% 0.19/0.41 groups_comm_monoid_G:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Ocomm__monoid__list,type,
% 0.19/0.41 groups1828464146339083142d_list:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
% 0.19/0.41 groups4802862169904069756st_set:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
% 0.19/0.41 groups4207007520872428315er_sum:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Omonoid__add_Osum__list,type,
% 0.19/0.41 groups4543113879258116180m_list:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
% 0.19/0.41 groups8242544230860333062m_list:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Omonoid__list,type,
% 0.19/0.41 groups_monoid_list:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Omonoid__list_OF,type,
% 0.19/0.41 groups_monoid_F:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
% 0.19/0.41 groups5270119922927024881d_list:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_HOL_ONO__MATCH,type,
% 0.19/0.41 nO_MATCH:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( A > B > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_HOL_OThe,type,
% 0.19/0.41 the:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_HOL_OUniq,type,
% 0.19/0.41 uniq:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_HOL_Odefault__class_Odefault,type,
% 0.19/0.41 default_default:
% 0.19/0.41 !>[A: $tType] : A ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_HOL_Oundefined,type,
% 0.19/0.41 undefined:
% 0.19/0.41 !>[A: $tType] : A ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oaddr__of__array,type,
% 0.19/0.41 addr_of_array:
% 0.19/0.41 !>[A: $tType] : ( ( array @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oaddr__of__ref,type,
% 0.19/0.41 addr_of_ref:
% 0.19/0.41 !>[A: $tType] : ( ( ref @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oarray_OArray,type,
% 0.19/0.41 array2:
% 0.19/0.41 !>[A: $tType] : ( nat > ( array @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oheap_Oarrays,type,
% 0.19/0.41 arrays:
% 0.19/0.41 !>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oheap_Oarrays__update,type,
% 0.19/0.41 arrays_update:
% 0.19/0.41 !>[Z: $tType] : ( ( ( typerep > nat > ( list @ nat ) ) > typerep > nat > ( list @ nat ) ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oheap_Olim,type,
% 0.19/0.41 lim:
% 0.19/0.41 !>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oheap_Olim__update,type,
% 0.19/0.41 lim_update:
% 0.19/0.41 !>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oheap_Orefs,type,
% 0.19/0.41 refs:
% 0.19/0.41 !>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap_Oref_ORef,type,
% 0.19/0.41 ref2:
% 0.19/0.41 !>[A: $tType] : ( nat > ( ref @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
% 0.19/0.41 heap_Time_Heap2:
% 0.19/0.41 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap,type,
% 0.19/0.41 heap_Time_case_Heap:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B ) > ( heap_Time_Heap @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
% 0.19/0.41 heap_Time_map_Heap:
% 0.19/0.41 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ Aa ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
% 0.19/0.41 heap_Time_pred_Heap:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( heap_Time_Heap @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
% 0.19/0.41 heap_Time_rec_Heap:
% 0.19/0.41 !>[A: $tType,C: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) > ( heap_Time_Heap @ A ) > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
% 0.19/0.41 heap_Time_rel_Heap:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
% 0.19/0.41 heap_Time_set_Heap:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
% 0.19/0.41 heap_Time_size_Heap:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
% 0.19/0.41 heap_Time_Heap_lub:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
% 0.19/0.41 heap_Time_Heap_ord:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oassert,type,
% 0.19/0.41 heap_Time_assert:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Obind,type,
% 0.19/0.41 heap_Time_bind:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oeffect,type,
% 0.19/0.41 heap_Time_effect:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oexecute,type,
% 0.19/0.41 heap_Time_execute:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oguard,type,
% 0.19/0.41 heap_Time_guard:
% 0.19/0.41 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oheap,type,
% 0.19/0.41 heap_Time_heap:
% 0.19/0.41 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Olift,type,
% 0.19/0.41 heap_Time_lift:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oraise,type,
% 0.19/0.41 heap_Time_raise:
% 0.19/0.41 !>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oreturn,type,
% 0.19/0.41 heap_Time_return:
% 0.19/0.41 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Osuccess,type,
% 0.19/0.41 heap_Time_success:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Otap,type,
% 0.19/0.41 heap_Time_tap:
% 0.19/0.41 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
% 0.19/0.41 heap_Time_timeFrame:
% 0.19/0.41 !>[A: $tType] : ( nat > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
% 0.19/0.41 heap_T5500966940807335491me_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Oureturn,type,
% 0.19/0.41 heap_Time_ureturn:
% 0.19/0.41 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Heap__Time__Monad_Owait,type,
% 0.19/0.41 heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Hilbert__Choice_Obijection,type,
% 0.19/0.41 hilbert_bijection:
% 0.19/0.41 !>[A: $tType] : ( ( A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 0.19/0.41 hilbert_inv_into:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Hoare__Triple_Ohoare__triple,type,
% 0.19/0.41 hoare_hoare_triple:
% 0.19/0.41 !>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Hoare__Triple_Onew__addrs,type,
% 0.19/0.41 hoare_new_addrs: ( heap_ext @ product_unit ) > ( set @ nat ) > ( heap_ext @ product_unit ) > ( set @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_If,type,
% 0.19/0.41 if:
% 0.19/0.41 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
% 0.19/0.41 complete_lattice_lfp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_OAbs__Integ,type,
% 0.19/0.41 abs_Integ: ( product_prod @ nat @ nat ) > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_ONeg,type,
% 0.19/0.41 neg: num > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_OPos,type,
% 0.19/0.41 pos: num > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_ORep__Integ,type,
% 0.19/0.41 rep_Integ: int > ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Odup,type,
% 0.19/0.41 dup: int > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Ointrel,type,
% 0.19/0.41 intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Onat,type,
% 0.19/0.41 nat2: int > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Opcr__int,type,
% 0.19/0.41 pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Opower__int,type,
% 0.19/0.41 power_int:
% 0.19/0.41 !>[A: $tType] : ( A > int > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Oring__1__class_OInts,type,
% 0.19/0.41 ring_1_Ints:
% 0.19/0.41 !>[A: $tType] : ( set @ A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 0.19/0.41 ring_1_of_int:
% 0.19/0.41 !>[A: $tType] : ( int > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Int_Osub,type,
% 0.19/0.41 sub: num > num > int ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 0.19/0.41 inf_inf:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 0.19/0.41 semilattice_neutr:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.19/0.41 sup_sup:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 0.19/0.41 lattic643756798349783984er_Max:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
% 0.19/0.41 lattic643756798350308766er_Min:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
% 0.19/0.41 lattic7752659483105999362nf_fin:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
% 0.19/0.41 lattic5214292709420241887eutr_F:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 0.19/0.41 lattic149705377957585745ce_set:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
% 0.19/0.41 lattic1715443433743089157tice_F:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 0.19/0.41 lattic5882676163264333800up_fin:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Lifting_OQuotient,type,
% 0.19/0.41 quotient:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oappend,type,
% 0.19/0.41 append:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oarg__min__list,type,
% 0.19/0.41 arg_min_list:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oarg__min__list__rel,type,
% 0.19/0.41 arg_min_list_rel:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Obutlast,type,
% 0.19/0.41 butlast:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oconcat,type,
% 0.19/0.41 concat:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ocount__list,type,
% 0.19/0.41 count_list:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Odistinct,type,
% 0.19/0.41 distinct:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Odrop,type,
% 0.19/0.41 drop:
% 0.19/0.41 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_OdropWhile,type,
% 0.19/0.41 dropWhile:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oenumerate,type,
% 0.19/0.41 enumerate:
% 0.19/0.41 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oextract,type,
% 0.19/0.41 extract:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ofilter,type,
% 0.19/0.41 filter:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ofind,type,
% 0.19/0.41 find:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ofold,type,
% 0.19/0.41 fold:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ofoldl,type,
% 0.19/0.41 foldl:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ofoldr,type,
% 0.19/0.41 foldr:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ogen__length,type,
% 0.19/0.41 gen_length:
% 0.19/0.41 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olast,type,
% 0.19/0.41 last:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olenlex,type,
% 0.19/0.41 lenlex:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olex,type,
% 0.19/0.41 lex:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olexn,type,
% 0.19/0.41 lexn:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olexord,type,
% 0.19/0.41 lexord:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olexordp,type,
% 0.19/0.41 lexordp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olinorder__class_Osort__key,type,
% 0.19/0.41 linorder_sort_key:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 0.19/0.41 linord4507533701916653071of_set:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_OCons,type,
% 0.19/0.41 cons:
% 0.19/0.41 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_ONil,type,
% 0.19/0.41 nil:
% 0.19/0.41 !>[A: $tType] : ( list @ A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Ocase__list,type,
% 0.19/0.41 case_list:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Ohd,type,
% 0.19/0.41 hd:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Olist__all,type,
% 0.19/0.41 list_all:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Olist__all2,type,
% 0.19/0.41 list_all2:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Omap,type,
% 0.19/0.41 map:
% 0.19/0.41 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Oset,type,
% 0.19/0.41 set2:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Osize__list,type,
% 0.19/0.41 size_list:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist_Otl,type,
% 0.19/0.41 tl:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olist__update,type,
% 0.19/0.41 list_update:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olistrel,type,
% 0.19/0.41 listrel:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olistrel1,type,
% 0.19/0.41 listrel1:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olistrel1p,type,
% 0.19/0.41 listrel1p:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olistrelp,type,
% 0.19/0.41 listrelp:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Olists,type,
% 0.19/0.41 lists:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omap__filter,type,
% 0.19/0.41 map_filter:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omap__project,type,
% 0.19/0.41 map_project:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omap__tailrec__rev,type,
% 0.19/0.41 map_tailrec_rev:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omap__tailrec__rev__rel,type,
% 0.19/0.41 map_tailrec_rev_rel:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omeasures,type,
% 0.19/0.41 measures:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omin__list,type,
% 0.19/0.41 min_list:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Omin__list__rel,type,
% 0.19/0.41 min_list_rel:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Onth,type,
% 0.19/0.41 nth:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Onths,type,
% 0.19/0.41 nths:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oord__class_Olexordp,type,
% 0.19/0.41 ord_lexordp:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Opartition,type,
% 0.19/0.41 partition:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oproduct,type,
% 0.19/0.41 product:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oproduct__lists,type,
% 0.19/0.41 product_lists:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oremdups,type,
% 0.19/0.41 remdups:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oremdups__adj,type,
% 0.19/0.41 remdups_adj:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oremdups__adj__rel,type,
% 0.19/0.41 remdups_adj_rel:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oremove1,type,
% 0.19/0.41 remove1:
% 0.19/0.41 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oreplicate,type,
% 0.19/0.41 replicate:
% 0.19/0.41 !>[A: $tType] : ( nat > A > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Orev,type,
% 0.19/0.41 rev:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Orotate,type,
% 0.19/0.41 rotate:
% 0.19/0.41 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Orotate1,type,
% 0.19/0.41 rotate1:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oshuffles,type,
% 0.19/0.41 shuffles:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oshuffles__rel,type,
% 0.19/0.41 shuffles_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Osorted__wrt,type,
% 0.19/0.41 sorted_wrt:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Osorted__wrt__rel,type,
% 0.19/0.41 sorted_wrt_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Osplice,type,
% 0.19/0.41 splice:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Osplice__rel,type,
% 0.19/0.41 splice_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Osuccessively,type,
% 0.19/0.41 successively:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Osuccessively__rel,type,
% 0.19/0.41 successively_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Otake,type,
% 0.19/0.41 take:
% 0.19/0.41 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_OtakeWhile,type,
% 0.19/0.41 takeWhile:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Othose,type,
% 0.19/0.41 those:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Otranspose,type,
% 0.19/0.41 transpose:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Otranspose__rel,type,
% 0.19/0.41 transpose_rel:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oupt,type,
% 0.19/0.41 upt: nat > nat > ( list @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oupto,type,
% 0.19/0.41 upto: int > int > ( list @ int ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Oupto__rel,type,
% 0.19/0.41 upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_List_Ozip,type,
% 0.19/0.41 zip:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Odom,type,
% 0.19/0.41 dom:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Ograph,type,
% 0.19/0.41 graph:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Omap__add,type,
% 0.19/0.41 map_add:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Omap__comp,type,
% 0.19/0.41 map_comp:
% 0.19/0.41 !>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Omap__le,type,
% 0.19/0.41 map_le:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Omap__of,type,
% 0.19/0.41 map_of:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Omap__upds,type,
% 0.19/0.41 map_upds:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Oran,type,
% 0.19/0.41 ran:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Map_Orestrict__map,type,
% 0.19/0.41 restrict_map:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_OCODE__ABORT,type,
% 0.19/0.41 cODE_ABORT:
% 0.19/0.41 !>[A: $tType] : ( ( product_unit > A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_OEps__Opt,type,
% 0.19/0.41 eps_Opt:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( option @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Obijective,type,
% 0.19/0.41 bijective:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Obrk__rel,type,
% 0.19/0.41 brk_rel:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Odflt__None__set,type,
% 0.19/0.41 dflt_None_set:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( option @ ( set @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Ofilter__rev,type,
% 0.19/0.41 filter_rev:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Ofilter__rev__aux,type,
% 0.19/0.41 filter_rev_aux:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Ofun__of__rel,type,
% 0.19/0.41 fun_of_rel:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Oinv__on,type,
% 0.19/0.41 inv_on:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Olist__all__zip,type,
% 0.19/0.41 list_all_zip:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Olist__all__zip__rel,type,
% 0.19/0.41 list_all_zip_rel:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Olist__collect__set,type,
% 0.19/0.41 list_collect_set:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omap__mmupd,type,
% 0.19/0.41 map_mmupd:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > ( option @ A ) ) > ( set @ B ) > A > B > ( option @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omap__to__set,type,
% 0.19/0.41 map_to_set:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omerge,type,
% 0.19/0.41 merge:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omerge__list,type,
% 0.19/0.41 merge_list:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omerge__list__rel,type,
% 0.19/0.41 merge_list_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omerge__rel,type,
% 0.19/0.41 merge_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort,type,
% 0.19/0.41 mergesort:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__by__rel,type,
% 0.19/0.41 mergesort_by_rel:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__by__rel__merge,type,
% 0.19/0.41 merges9089515139780605204_merge:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
% 0.19/0.41 merges2244889521215249637ge_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__by__rel__rel,type,
% 0.19/0.41 mergesort_by_rel_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__by__rel__split,type,
% 0.19/0.41 merges295452479951948502_split:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
% 0.19/0.41 merges7066485432131860899it_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Omergesort__remdups,type,
% 0.19/0.41 mergesort_remdups:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Opairself,type,
% 0.19/0.41 pairself:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Opairself__rel,type,
% 0.19/0.41 pairself_rel:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Opartition__rev,type,
% 0.19/0.41 partition_rev:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Opartition__rev__rel,type,
% 0.19/0.41 partition_rev_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Oquicksort__by__rel,type,
% 0.19/0.41 quicksort_by_rel:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Oquicksort__by__rel__rel,type,
% 0.19/0.41 quicksort_by_rel_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Orel__of,type,
% 0.19/0.41 rel_of:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Orel__restrict,type,
% 0.19/0.41 rel_restrict:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Oremove__rev,type,
% 0.19/0.41 remove_rev:
% 0.19/0.41 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Orevg,type,
% 0.19/0.41 revg:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Orevg__rel,type,
% 0.19/0.41 revg_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Oset__to__map,type,
% 0.19/0.41 set_to_map:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( option @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Oslice,type,
% 0.19/0.41 slice:
% 0.19/0.41 !>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Osu__rel__fun,type,
% 0.19/0.41 su_rel_fun:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Othe__default,type,
% 0.19/0.41 the_default:
% 0.19/0.41 !>[A: $tType] : ( A > ( option @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Ouncurry,type,
% 0.19/0.41 uncurry:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Ozipf,type,
% 0.19/0.41 zipf:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Misc_Ozipf__rel,type,
% 0.19/0.41 zipf_rel:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Oadd__mset,type,
% 0.19/0.41 add_mset:
% 0.19/0.41 !>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset,type,
% 0.19/0.41 comm_monoid_sum_mset:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
% 0.19/0.41 comm_m7189776963980413722m_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ocomm__monoid__mset,type,
% 0.19/0.41 comm_monoid_mset:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
% 0.19/0.41 comm_monoid_F:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
% 0.19/0.41 comm_m9189036328036947845d_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ofilter__mset,type,
% 0.19/0.41 filter_mset:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( multiset @ A ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ofold__mset,type,
% 0.19/0.41 fold_mset:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Oimage__mset,type,
% 0.19/0.41 image_mset:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ointer__mset,type,
% 0.19/0.41 inter_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Olinorder__class_Opart,type,
% 0.19/0.41 linorder_part:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Oms__strict,type,
% 0.19/0.41 ms_strict: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Oms__weak,type,
% 0.19/0.41 ms_weak: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omset,type,
% 0.19/0.41 mset:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omset__set,type,
% 0.19/0.41 mset_set:
% 0.19/0.41 !>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omult,type,
% 0.19/0.41 mult:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omult1,type,
% 0.19/0.41 mult1:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omulteqp__code,type,
% 0.19/0.41 multeqp_code:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omultiset_OAbs__multiset,type,
% 0.19/0.41 abs_multiset:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omultiset_Ocount,type,
% 0.19/0.41 count:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omultp,type,
% 0.19/0.41 multp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Omultp__code,type,
% 0.19/0.41 multp_code:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Opcr__multiset,type,
% 0.19/0.41 pcr_multiset:
% 0.19/0.41 !>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( C > nat ) > ( multiset @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Opw__leq,type,
% 0.19/0.41 pw_leq: ( multiset @ ( product_prod @ nat @ nat ) ) > ( multiset @ ( product_prod @ nat @ nat ) ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Orepeat__mset,type,
% 0.19/0.41 repeat_mset:
% 0.19/0.41 !>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Oreplicate__mset,type,
% 0.19/0.41 replicate_mset:
% 0.19/0.41 !>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Oset__mset,type,
% 0.19/0.41 set_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Osize__multiset,type,
% 0.19/0.41 size_multiset:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
% 0.19/0.41 subset_eq_mset_impl:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
% 0.19/0.41 subset751672762298770561pl_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Osubset__mset,type,
% 0.19/0.41 subset_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Osubseteq__mset,type,
% 0.19/0.41 subseteq_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Ounion__mset,type,
% 0.19/0.41 union_mset:
% 0.19/0.41 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Multiset_Owcount,type,
% 0.19/0.41 wcount:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_OSuc,type,
% 0.19/0.41 suc: nat > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_Ocompow,type,
% 0.19/0.41 compow:
% 0.19/0.41 !>[A: $tType] : ( nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 0.19/0.41 case_nat:
% 0.19/0.41 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 0.19/0.41 rec_nat:
% 0.19/0.41 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 0.19/0.41 semiring_1_of_nat:
% 0.19/0.41 !>[A: $tType] : ( nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 0.19/0.41 semiri8178284476397505188at_aux:
% 0.19/0.41 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat_Osize__class_Osize,type,
% 0.19/0.41 size_size:
% 0.19/0.41 !>[A: $tType] : ( A > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 0.19/0.41 nat_list_decode: nat > ( list @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 0.19/0.41 nat_list_decode_rel: nat > nat > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.19/0.41 nat_list_encode: ( list @ nat ) > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.19/0.41 nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 0.19/0.41 nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.19/0.41 nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.19/0.41 nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.19/0.41 nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.19/0.41 nat_set_decode: nat > ( set @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.19/0.41 nat_set_encode: ( set @ nat ) > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Osum__decode,type,
% 0.19/0.41 nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Nat__Bijection_Otriangle,type,
% 0.19/0.41 nat_triangle: nat > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_OBitM,type,
% 0.19/0.41 bitM: num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oinc,type,
% 0.19/0.41 inc: num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onat__of__num,type,
% 0.19/0.41 nat_of_num: num > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 0.19/0.41 neg_numeral_dbl:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
% 0.19/0.41 neg_numeral_dbl_dec:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
% 0.19/0.41 neg_numeral_dbl_inc:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
% 0.19/0.41 neg_numeral_is_num:
% 0.19/0.41 !>[A: $tType] : ( A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oneg__numeral__class_Osub,type,
% 0.19/0.41 neg_numeral_sub:
% 0.19/0.41 !>[A: $tType] : ( num > num > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onum_OBit0,type,
% 0.19/0.41 bit0: num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onum_OBit1,type,
% 0.19/0.41 bit1: num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onum_OOne,type,
% 0.19/0.41 one2: num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onum_Ocase__num,type,
% 0.19/0.41 case_num:
% 0.19/0.41 !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onum_Osize__num,type,
% 0.19/0.41 size_num: num > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onum__of__nat,type,
% 0.19/0.41 num_of_nat: nat > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 0.19/0.41 numeral_numeral:
% 0.19/0.41 !>[A: $tType] : ( num > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Opred__numeral,type,
% 0.19/0.41 pred_numeral: num > nat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Oring__1__class_Oiszero,type,
% 0.19/0.41 ring_1_iszero:
% 0.19/0.41 !>[A: $tType] : ( A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Num_Osqr,type,
% 0.19/0.41 sqr: num > num ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_OAtom,type,
% 0.19/0.41 old_Atom:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_OIn0,type,
% 0.19/0.41 old_In0:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_OIn1,type,
% 0.19/0.41 old_In1:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_OLeaf,type,
% 0.19/0.41 old_Leaf:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_ONode,type,
% 0.19/0.41 old_Node:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_ONumb,type,
% 0.19/0.41 old_Numb:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_OPush__Node,type,
% 0.19/0.41 old_Push_Node:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_OScons,type,
% 0.19/0.41 old_Scons:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_Odprod,type,
% 0.19/0.41 old_dprod:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_Odsum,type,
% 0.19/0.41 old_dsum:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_Ondepth,type,
% 0.19/0.41 old_ndepth:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
% 0.19/0.41 old_Abs_Node:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Obind,type,
% 0.19/0.41 bind:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ocombine__options,type,
% 0.19/0.41 combine_options:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ois__none,type,
% 0.19/0.41 is_none:
% 0.19/0.41 !>[A: $tType] : ( ( option @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_ONone,type,
% 0.19/0.41 none:
% 0.19/0.41 !>[A: $tType] : ( option @ A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_OSome,type,
% 0.19/0.41 some:
% 0.19/0.41 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Ocase__option,type,
% 0.19/0.41 case_option:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Omap__option,type,
% 0.19/0.41 map_option:
% 0.19/0.41 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Opred__option,type,
% 0.19/0.41 pred_option:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Orec__option,type,
% 0.19/0.41 rec_option:
% 0.19/0.41 !>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Orel__option,type,
% 0.19/0.41 rel_option:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Oset__option,type,
% 0.19/0.41 set_option:
% 0.19/0.41 !>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Osize__option,type,
% 0.19/0.41 size_option:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Ooption_Othe,type,
% 0.19/0.41 the2:
% 0.19/0.41 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Option_Othese,type,
% 0.19/0.41 these:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_OAbove,type,
% 0.19/0.41 order_Above:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_OAboveS,type,
% 0.19/0.41 order_AboveS:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_OUnder,type,
% 0.19/0.41 order_Under:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_OUnderS,type,
% 0.19/0.41 order_UnderS:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Oabove,type,
% 0.19/0.41 order_above:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_OaboveS,type,
% 0.19/0.41 order_aboveS:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 0.19/0.41 order_679001287576687338der_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Oofilter,type,
% 0.19/0.41 order_ofilter:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Opartial__order__on,type,
% 0.19/0.41 order_7125193373082350890der_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Opreorder__on,type,
% 0.19/0.41 order_preorder_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Orelation__of,type,
% 0.19/0.41 order_relation_of:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Ounder,type,
% 0.19/0.41 order_under:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_OunderS,type,
% 0.19/0.41 order_underS:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Order__Relation_Owell__order__on,type,
% 0.19/0.41 order_well_order_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.19/0.41 bot_bot:
% 0.19/0.41 !>[A: $tType] : A ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oord_Omax,type,
% 0.19/0.41 max:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oord_Omin,type,
% 0.19/0.41 min:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.19/0.41 ord_less:
% 0.19/0.41 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.41 ord_less_eq:
% 0.19/0.41 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oord__class_Omax,type,
% 0.19/0.41 ord_max:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oord__class_Omin,type,
% 0.19/0.41 ord_min:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 0.19/0.41 order_mono:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 0.19/0.41 order_strict_mono:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.19/0.41 top_top:
% 0.19/0.41 !>[A: $tType] : A ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Oflat__lub,type,
% 0.19/0.41 partial_flat_lub:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Oflat__ord,type,
% 0.19/0.41 partial_flat_ord:
% 0.19/0.41 !>[A: $tType] : ( A > A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Ofun__lub,type,
% 0.19/0.41 partial_fun_lub:
% 0.19/0.41 !>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Ofun__ord,type,
% 0.19/0.41 partial_fun_ord:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Oimg__lub,type,
% 0.19/0.41 partial_img_lub:
% 0.19/0.41 !>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( A > C ) > ( D > B ) > ( ( set @ C ) > D ) > ( set @ A ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Oimg__ord,type,
% 0.19/0.41 partial_img_ord:
% 0.19/0.41 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Partial__Function_Opartial__function__definitions,type,
% 0.19/0.41 partia7178651479351089652itions:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Power_Opower_Opower,type,
% 0.19/0.41 power2:
% 0.19/0.41 !>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Power_Opower__class_Opower,type,
% 0.19/0.41 power_power:
% 0.19/0.41 !>[A: $tType] : ( A > nat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Predicate_OSeq,type,
% 0.19/0.41 seq2:
% 0.19/0.41 !>[A: $tType] : ( ( product_unit > ( seq @ A ) ) > ( pred @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Predicate_Obind,type,
% 0.19/0.41 bind2:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Predicate_Oiterate__upto,type,
% 0.19/0.41 iterate_upto:
% 0.19/0.41 !>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( pred @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Predicate_Oiterate__upto__rel,type,
% 0.19/0.41 iterate_upto_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Predicate_Oseq_OEmpty,type,
% 0.19/0.41 empty:
% 0.19/0.41 !>[A: $tType] : ( seq @ A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Predicate_Oseq_OInsert,type,
% 0.19/0.41 insert:
% 0.19/0.41 !>[A: $tType] : ( A > ( pred @ A ) > ( seq @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_OPair,type,
% 0.19/0.41 product_Pair:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_OSigma,type,
% 0.19/0.41 product_Sigma:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_OUnity,type,
% 0.19/0.41 product_Unity: product_unit ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oapfst,type,
% 0.19/0.41 product_apfst:
% 0.19/0.41 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oapsnd,type,
% 0.19/0.41 product_apsnd:
% 0.19/0.41 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Ocurry,type,
% 0.19/0.41 product_curry:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.19/0.41 produc5280177257484947105e_prod:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Omap__prod,type,
% 0.19/0.41 product_map_prod:
% 0.19/0.41 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 0.19/0.41 product_rec_bool:
% 0.19/0.41 !>[T: $tType] : ( T > T > $o > T ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 0.19/0.41 product_rec_set_bool:
% 0.19/0.41 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.19/0.41 product_rec_prod:
% 0.19/0.41 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 0.19/0.41 product_rec_set_prod:
% 0.19/0.41 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 0.19/0.41 product_rec_set_unit:
% 0.19/0.41 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 0.19/0.41 product_rec_unit:
% 0.19/0.41 !>[T: $tType] : ( T > product_unit > T ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 0.19/0.41 product_case_prod:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 0.19/0.41 product_fst:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 0.19/0.41 product_snd:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 0.19/0.41 product_swap:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oproduct,type,
% 0.19/0.41 product_product:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Oscomp,type,
% 0.19/0.41 product_scomp:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
% 0.19/0.41 product_Abs_unit: $o > product_unit ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Ounit_ORep__unit,type,
% 0.19/0.41 product_Rep_unit: product_unit > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Product__Type_Ounit_Ocase__unit,type,
% 0.19/0.41 product_case_unit:
% 0.19/0.41 !>[A: $tType] : ( A > product_unit > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Pure_Otype,type,
% 0.19/0.41 type2:
% 0.19/0.41 !>[A: $tType] : ( itself @ A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
% 0.19/0.41 linorder_quicksort:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
% 0.19/0.41 linord6200660962353139674rt_rel:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Quotient_OQuotient3,type,
% 0.19/0.41 quotient3:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Oinc__shift,type,
% 0.19/0.41 inc_shift: code_natural > code_natural > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Oiterate,type,
% 0.19/0.41 iterate:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Oiterate__rel,type,
% 0.19/0.41 iterate_rel:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Olog,type,
% 0.19/0.41 log: code_natural > code_natural > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Olog__rel,type,
% 0.19/0.41 log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Ominus__shift,type,
% 0.19/0.41 minus_shift: code_natural > code_natural > code_natural > code_natural ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Onext,type,
% 0.19/0.41 next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Opick,type,
% 0.19/0.41 pick:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Orange,type,
% 0.19/0.41 range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Oselect,type,
% 0.19/0.41 select:
% 0.19/0.41 !>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Oselect__weight,type,
% 0.19/0.41 select_weight:
% 0.19/0.41 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random_Osplit__seed,type,
% 0.19/0.41 split_seed: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random__Pred_Obind,type,
% 0.19/0.41 random_bind:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random__Pred_Oempty,type,
% 0.19/0.41 random_empty:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random__Pred_Oiterate__upto,type,
% 0.19/0.41 random_iterate_upto:
% 0.19/0.41 !>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Random__Pred_Ounion,type,
% 0.19/0.41 random_union:
% 0.19/0.41 !>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_OAbs__Rat,type,
% 0.19/0.41 abs_Rat: ( product_prod @ int @ int ) > rat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_OFract,type,
% 0.19/0.41 fract: int > int > rat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_OFrct,type,
% 0.19/0.41 frct: ( product_prod @ int @ int ) > rat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_ORep__Rat,type,
% 0.19/0.41 rep_Rat: rat > ( product_prod @ int @ int ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
% 0.19/0.41 field_char_0_of_rat:
% 0.19/0.41 !>[A: $tType] : ( rat > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Onormalize,type,
% 0.19/0.41 normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Oof__int,type,
% 0.19/0.41 of_int: int > rat ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Opcr__rat,type,
% 0.19/0.41 pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Opositive,type,
% 0.19/0.41 positive: rat > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Oquotient__of,type,
% 0.19/0.41 quotient_of: rat > ( product_prod @ int @ int ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rat_Oratrel,type,
% 0.19/0.41 ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Oalloc,type,
% 0.19/0.41 ref_alloc:
% 0.19/0.41 !>[A: $tType] : ( A > ( heap_ext @ product_unit ) > ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Ochange,type,
% 0.19/0.41 ref_change:
% 0.19/0.41 !>[A: $tType] : ( ( A > A ) > ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Oget,type,
% 0.19/0.41 ref_get:
% 0.19/0.41 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Olookup,type,
% 0.19/0.41 ref_lookup:
% 0.19/0.41 !>[A: $tType] : ( ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Opresent,type,
% 0.19/0.41 ref_present:
% 0.19/0.41 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Oref,type,
% 0.19/0.41 ref_ref:
% 0.19/0.41 !>[A: $tType] : ( A > ( heap_Time_Heap @ ( ref @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Oset,type,
% 0.19/0.41 ref_set:
% 0.19/0.41 !>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Ref__Time_Oupdate,type,
% 0.19/0.41 ref_update:
% 0.19/0.41 !>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
% 0.19/0.41 refine_Imp_assert: $o > ( heap_Time_Heap @ product_unit ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Refine__Imp__Hol_Orefines,type,
% 0.19/0.41 refine_Imp_refines:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_ODomain,type,
% 0.19/0.41 domain:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_ODomainp,type,
% 0.19/0.41 domainp:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_OField,type,
% 0.19/0.41 field2:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_OId,type,
% 0.19/0.41 id2:
% 0.19/0.41 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_OId__on,type,
% 0.19/0.41 id_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_OImage,type,
% 0.19/0.41 image:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_ORange,type,
% 0.19/0.41 range2:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_ORangep,type,
% 0.19/0.41 rangep:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oantisym,type,
% 0.19/0.41 antisym:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oantisymp,type,
% 0.19/0.41 antisymp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oasym,type,
% 0.19/0.41 asym:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oasymp,type,
% 0.19/0.41 asymp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oconverse,type,
% 0.19/0.41 converse:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oconversep,type,
% 0.19/0.41 conversep:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oinv__image,type,
% 0.19/0.41 inv_image:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oirrefl,type,
% 0.19/0.41 irrefl:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oirreflp,type,
% 0.19/0.41 irreflp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Orefl__on,type,
% 0.19/0.41 refl_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Oreflp,type,
% 0.19/0.41 reflp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Orelcomp,type,
% 0.19/0.41 relcomp:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Orelcompp,type,
% 0.19/0.41 relcompp:
% 0.19/0.41 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Osingle__valued,type,
% 0.19/0.41 single_valued:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Osingle__valuedp,type,
% 0.19/0.41 single_valuedp:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Ototal__on,type,
% 0.19/0.41 total_on:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Otrans,type,
% 0.19/0.41 trans:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Relation_Otransp,type,
% 0.19/0.41 transp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
% 0.19/0.41 algebr8660921524188924756oprime:
% 0.19/0.41 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 0.19/0.41 divide_divide:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 0.19/0.41 dvd_dvd:
% 0.19/0.41 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Omodulo__class_Omodulo,type,
% 0.19/0.41 modulo_modulo:
% 0.19/0.41 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
% 0.19/0.41 normal6383669964737779283malize:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
% 0.19/0.41 unit_f5069060285200089521factor:
% 0.19/0.41 !>[A: $tType] : ( A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 0.19/0.41 zero_neq_one_of_bool:
% 0.19/0.41 !>[A: $tType] : ( $o > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_OCollect,type,
% 0.19/0.41 collect:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_OPow,type,
% 0.19/0.41 pow:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_Obind,type,
% 0.19/0.41 bind3:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_Odisjnt,type,
% 0.19/0.41 disjnt:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_Oimage,type,
% 0.19/0.41 image2:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_Oinsert,type,
% 0.19/0.41 insert2:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_Ois__singleton,type,
% 0.19/0.41 is_singleton:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set_Ovimage,type,
% 0.19/0.41 vimage:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
% 0.19/0.41 set_fo6178422350223883121st_nat:
% 0.19/0.41 !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
% 0.19/0.41 set_fo1817059534552279752at_rel:
% 0.19/0.41 !>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 0.19/0.41 set_ord_atLeast:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 0.19/0.41 set_or1337092689740270186AtMost:
% 0.19/0.41 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 0.19/0.41 set_or7035219750837199246ssThan:
% 0.19/0.41 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 0.19/0.41 set_ord_atMost:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
% 0.19/0.41 set_ord_greaterThan:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
% 0.19/0.41 set_or3652927894154168847AtMost:
% 0.19/0.41 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
% 0.19/0.41 set_or5935395276787703475ssThan:
% 0.19/0.41 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 0.19/0.41 set_ord_lessThan:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_String_Ochar_OChar,type,
% 0.19/0.41 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_String_Ointeger__of__char,type,
% 0.19/0.41 integer_of_char: char > code_integer ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
% 0.19/0.41 unique5772411509450598832har_of:
% 0.19/0.41 !>[A: $tType] : ( A > char ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Sum__Type_OInl,type,
% 0.19/0.41 sum_Inl:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Sum__Type_OInr,type,
% 0.19/0.41 sum_Inr:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Sum__Type_OPlus,type,
% 0.19/0.41 sum_Plus:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Sum__Type_Omap__sum,type,
% 0.19/0.41 sum_map_sum:
% 0.19/0.41 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 0.19/0.41 sum_case_sum:
% 0.19/0.41 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
% 0.19/0.41 syntax7388354845996824322omatch:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( A > B > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Time__Reasoning_Ofails,type,
% 0.19/0.41 time_fails:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Time__Reasoning_Othe__heap,type,
% 0.19/0.41 time_the_heap:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Time__Reasoning_Othe__res,type,
% 0.19/0.41 time_the_res:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Time__Reasoning_Otime,type,
% 0.19/0.41 time_time:
% 0.19/0.41 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Transitive__Closure_Oacyclic,type,
% 0.19/0.41 transitive_acyclic:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Transitive__Closure_Ontrancl,type,
% 0.19/0.41 transitive_ntrancl:
% 0.19/0.41 !>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 0.19/0.41 transitive_rtrancl:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Transitive__Closure_Ortranclp,type,
% 0.19/0.41 transitive_rtranclp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Transitive__Closure_Otrancl,type,
% 0.19/0.41 transitive_trancl:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Transitive__Closure_Otranclp,type,
% 0.19/0.41 transitive_tranclp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Typedef_Otype__definition,type,
% 0.19/0.41 type_definition:
% 0.19/0.41 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Typerep_Otyperep__class_Otyperep,type,
% 0.19/0.41 typerep_typerep:
% 0.19/0.41 !>[A: $tType] : ( ( itself @ A ) > typerep ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Oacc,type,
% 0.19/0.41 acc:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Oaccp,type,
% 0.19/0.41 accp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Ofinite__psubset,type,
% 0.19/0.41 finite_psubset:
% 0.19/0.41 !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Oless__than,type,
% 0.19/0.41 less_than: set @ ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Olex__prod,type,
% 0.19/0.41 lex_prod:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Omax__ext,type,
% 0.19/0.41 max_ext:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Omax__extp,type,
% 0.19/0.41 max_extp:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Omeasure,type,
% 0.19/0.41 measure:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Omin__ext,type,
% 0.19/0.41 min_ext:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Omlex__prod,type,
% 0.19/0.41 mlex_prod:
% 0.19/0.41 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Opred__nat,type,
% 0.19/0.41 pred_nat: set @ ( product_prod @ nat @ nat ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_Owf,type,
% 0.19/0.41 wf:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wellfounded_OwfP,type,
% 0.19/0.41 wfP:
% 0.19/0.41 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wfrec_Oadm__wf,type,
% 0.19/0.41 adm_wf:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wfrec_Ocut,type,
% 0.19/0.41 cut:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Wfrec_Osame__fst,type,
% 0.19/0.41 same_fst:
% 0.19/0.41 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Zorn_OChains,type,
% 0.19/0.41 chains:
% 0.19/0.41 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Zorn_Oinit__seg__of,type,
% 0.19/0.41 init_seg_of:
% 0.19/0.41 !>[A: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_Zorn_Opred__on_Ochain,type,
% 0.19/0.41 pred_chain:
% 0.19/0.41 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_fChoice,type,
% 0.19/0.41 fChoice:
% 0.19/0.41 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_c_member,type,
% 0.19/0.41 member:
% 0.19/0.41 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.41
% 0.19/0.41 thf(sy_v_h,type,
% 0.19/0.41 h: heap_ext @ product_unit ).
% 0.19/0.41
% 0.19/0.41 thf(sy_v_xs,type,
% 0.19/0.41 xs: list @ a ).
% 0.19/0.41
% 0.19/0.41 % Relevant facts (5993)
% 0.19/0.41 thf(fact_0_fails__def,axiom,
% 0.19/0.41 ! [A: $tType] :
% 0.19/0.41 ( ( time_fails @ A )
% 0.19/0.41 = ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 0.19/0.41 ( ( heap_Time_execute @ A @ M @ H )
% 0.19/0.41 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % fails_def
% 0.19/0.41 thf(fact_1_Heap__eqI,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
% 0.19/0.41 ( ! [H2: heap_ext @ product_unit] :
% 0.19/0.41 ( ( heap_Time_execute @ A @ F @ H2 )
% 0.19/0.41 = ( heap_Time_execute @ A @ G @ H2 ) )
% 0.19/0.41 => ( F = G ) ) ).
% 0.19/0.41
% 0.19/0.41 % Heap_eqI
% 0.19/0.41 thf(fact_2_execute__raise,axiom,
% 0.19/0.41 ! [A: $tType,S: list @ char] :
% 0.19/0.41 ( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S ) )
% 0.19/0.41 = ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % execute_raise
% 0.19/0.41 thf(fact_3_execute__guard_I1_J,axiom,
% 0.19/0.41 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.41 ( ~ ( P @ H3 )
% 0.19/0.41 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 )
% 0.19/0.41 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % execute_guard(1)
% 0.19/0.41 thf(fact_4_execute__assert_I2_J,axiom,
% 0.19/0.41 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ~ ( P @ X )
% 0.19/0.41 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 )
% 0.19/0.41 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % execute_assert(2)
% 0.19/0.41 thf(fact_5_successI,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.41 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.41 => ( heap_Time_success @ A @ F @ H3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % successI
% 0.19/0.41 thf(fact_6_success__def,axiom,
% 0.19/0.41 ! [A: $tType] :
% 0.19/0.41 ( ( heap_Time_success @ A )
% 0.19/0.41 = ( ^ [F2: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 0.19/0.41 ( ( heap_Time_execute @ A @ F2 @ H )
% 0.19/0.41 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_def
% 0.19/0.41 thf(fact_7_Heap__ordE,axiom,
% 0.19/0.41 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ( heap_Time_Heap_ord @ A @ X @ Y )
% 0.19/0.41 => ( ( ( heap_Time_execute @ A @ X @ H3 )
% 0.19/0.41 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.41 => ( ( heap_Time_execute @ A @ X @ H3 )
% 0.19/0.41 = ( heap_Time_execute @ A @ Y @ H3 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % Heap_ordE
% 0.19/0.41 thf(fact_8_Heap__ordI,axiom,
% 0.19/0.41 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A] :
% 0.19/0.41 ( ! [H2: heap_ext @ product_unit] :
% 0.19/0.41 ( ( ( heap_Time_execute @ A @ X @ H2 )
% 0.19/0.41 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.41 | ( ( heap_Time_execute @ A @ X @ H2 )
% 0.19/0.41 = ( heap_Time_execute @ A @ Y @ H2 ) ) )
% 0.19/0.41 => ( heap_Time_Heap_ord @ A @ X @ Y ) ) ).
% 0.19/0.41
% 0.19/0.41 % Heap_ordI
% 0.19/0.41 thf(fact_9_execute__bind_I2_J,axiom,
% 0.19/0.41 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 0.19/0.41 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.41 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.41 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 0.19/0.41 = ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % execute_bind(2)
% 0.19/0.41 thf(fact_10_the__default_Osimps_I2_J,axiom,
% 0.19/0.41 ! [A: $tType,X: A] :
% 0.19/0.41 ( ( the_default @ A @ X @ ( none @ A ) )
% 0.19/0.41 = X ) ).
% 0.19/0.41
% 0.19/0.41 % the_default.simps(2)
% 0.19/0.41 thf(fact_11_raise__bind,axiom,
% 0.19/0.41 ! [B: $tType,A: $tType,E2: list @ char,F: B > ( heap_Time_Heap @ A )] :
% 0.19/0.41 ( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E2 ) @ F )
% 0.19/0.41 = ( heap_Time_raise @ A @ E2 ) ) ).
% 0.19/0.41
% 0.19/0.41 % raise_bind
% 0.19/0.41 thf(fact_12_heap_Oleq__refl,axiom,
% 0.19/0.41 ! [A: $tType,X: heap_Time_Heap @ A] : ( heap_Time_Heap_ord @ A @ X @ X ) ).
% 0.19/0.41
% 0.19/0.41 % heap.leq_refl
% 0.19/0.41 thf(fact_13_heap_Oleq__trans,axiom,
% 0.19/0.41 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] :
% 0.19/0.41 ( ( heap_Time_Heap_ord @ A @ X @ Y )
% 0.19/0.41 => ( ( heap_Time_Heap_ord @ A @ Y @ Z2 )
% 0.19/0.41 => ( heap_Time_Heap_ord @ A @ X @ Z2 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % heap.leq_trans
% 0.19/0.41 thf(fact_14_heap_Oleq__antisym,axiom,
% 0.19/0.41 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A] :
% 0.19/0.41 ( ( heap_Time_Heap_ord @ A @ X @ Y )
% 0.19/0.41 => ( ( heap_Time_Heap_ord @ A @ Y @ X )
% 0.19/0.41 => ( X = Y ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % heap.leq_antisym
% 0.19/0.41 thf(fact_15_assert__cong,axiom,
% 0.19/0.41 ! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B ),X: A] :
% 0.19/0.41 ( ( P = P2 )
% 0.19/0.41 => ( ! [X2: A] :
% 0.19/0.41 ( ( P2 @ X2 )
% 0.19/0.41 => ( ( F @ X2 )
% 0.19/0.41 = ( F3 @ X2 ) ) )
% 0.19/0.41 => ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X ) @ F )
% 0.19/0.41 = ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X ) @ F3 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % assert_cong
% 0.19/0.41 thf(fact_16_success__ifI,axiom,
% 0.19/0.41 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,E2: heap_Time_Heap @ A] :
% 0.19/0.41 ( ( C2
% 0.19/0.41 => ( heap_Time_success @ A @ T2 @ H3 ) )
% 0.19/0.41 => ( ( ~ C2
% 0.19/0.41 => ( heap_Time_success @ A @ E2 @ H3 ) )
% 0.19/0.41 => ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_ifI
% 0.19/0.41 thf(fact_17_success__guardI,axiom,
% 0.19/0.41 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.41 ( ( P @ H3 )
% 0.19/0.41 => ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_guardI
% 0.19/0.41 thf(fact_18_success__assertI,axiom,
% 0.19/0.41 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ( P @ X )
% 0.19/0.41 => ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_assertI
% 0.19/0.41 thf(fact_19_fails__bind,axiom,
% 0.19/0.41 ! [A: $tType,B: $tType,M2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ( time_fails @ A @ ( heap_Time_bind @ B @ A @ M2 @ F ) @ H3 )
% 0.19/0.41 = ( ~ ( time_fails @ B @ M2 @ H3 )
% 0.19/0.41 => ( time_fails @ A @ ( F @ ( time_the_res @ B @ M2 @ H3 ) ) @ ( time_the_heap @ B @ M2 @ H3 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % fails_bind
% 0.19/0.41 thf(fact_20_success__of__listI,axiom,
% 0.19/0.41 ! [A: $tType] :
% 0.19/0.41 ( ( heap @ A )
% 0.19/0.41 => ! [Xs: list @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_of_listI
% 0.19/0.41 thf(fact_21_success__tapI,axiom,
% 0.19/0.41 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H3 ) ).
% 0.19/0.41
% 0.19/0.41 % success_tapI
% 0.19/0.41 thf(fact_22_success__heapI,axiom,
% 0.19/0.41 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H3 ) ).
% 0.19/0.41
% 0.19/0.41 % success_heapI
% 0.19/0.41 thf(fact_23_success__ureturnI,axiom,
% 0.19/0.41 ! [A: $tType,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 ) ).
% 0.19/0.41
% 0.19/0.41 % success_ureturnI
% 0.19/0.41 thf(fact_24_bind__ureturn,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A] :
% 0.19/0.41 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
% 0.19/0.41 = F ) ).
% 0.19/0.41
% 0.19/0.41 % bind_ureturn
% 0.19/0.41 thf(fact_25_ureturn__bind,axiom,
% 0.19/0.41 ! [A: $tType,B: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 0.19/0.41 ( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X ) @ F )
% 0.19/0.41 = ( F @ X ) ) ).
% 0.19/0.41
% 0.19/0.41 % ureturn_bind
% 0.19/0.41 thf(fact_26_option_Osimps_I6_J,axiom,
% 0.19/0.41 ! [A: $tType,C: $tType,F1: C,F22: A > C] :
% 0.19/0.41 ( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
% 0.19/0.41 = F1 ) ).
% 0.19/0.41
% 0.19/0.41 % option.simps(6)
% 0.19/0.41 thf(fact_27_success__bind__effectI,axiom,
% 0.19/0.41 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 0.19/0.41 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ X @ N )
% 0.19/0.41 => ( ( heap_Time_success @ B @ ( G @ X ) @ H4 )
% 0.19/0.41 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_bind_effectI
% 0.19/0.41 thf(fact_28_execute__guard_I2_J,axiom,
% 0.19/0.41 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.41 ( ( P @ H3 )
% 0.19/0.41 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 )
% 0.19/0.41 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H3 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % execute_guard(2)
% 0.19/0.41 thf(fact_29_combine__options__simps_I1_J,axiom,
% 0.19/0.41 ! [A: $tType,F: A > A > A,Y: option @ A] :
% 0.19/0.41 ( ( combine_options @ A @ F @ ( none @ A ) @ Y )
% 0.19/0.41 = Y ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_simps(1)
% 0.19/0.41 thf(fact_30_combine__options__simps_I2_J,axiom,
% 0.19/0.41 ! [A: $tType,F: A > A > A,X: option @ A] :
% 0.19/0.41 ( ( combine_options @ A @ F @ X @ ( none @ A ) )
% 0.19/0.41 = X ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_simps(2)
% 0.19/0.41 thf(fact_31_option_Oinject,axiom,
% 0.19/0.41 ! [A: $tType,X22: A,Y2: A] :
% 0.19/0.41 ( ( ( some @ A @ X22 )
% 0.19/0.41 = ( some @ A @ Y2 ) )
% 0.19/0.41 = ( X22 = Y2 ) ) ).
% 0.19/0.41
% 0.19/0.41 % option.inject
% 0.19/0.41 thf(fact_32_not__None__eq,axiom,
% 0.19/0.41 ! [A: $tType,X: option @ A] :
% 0.19/0.41 ( ( X
% 0.19/0.41 != ( none @ A ) )
% 0.19/0.41 = ( ? [Y3: A] :
% 0.19/0.41 ( X
% 0.19/0.41 = ( some @ A @ Y3 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % not_None_eq
% 0.19/0.41 thf(fact_33_not__Some__eq,axiom,
% 0.19/0.41 ! [A: $tType,X: option @ A] :
% 0.19/0.41 ( ( ! [Y3: A] :
% 0.19/0.41 ( X
% 0.19/0.41 != ( some @ A @ Y3 ) ) )
% 0.19/0.41 = ( X
% 0.19/0.41 = ( none @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % not_Some_eq
% 0.19/0.41 thf(fact_34_combine__options__simps_I3_J,axiom,
% 0.19/0.41 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 0.19/0.41 ( ( combine_options @ A @ F @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
% 0.19/0.41 = ( some @ A @ ( F @ A3 @ B2 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_simps(3)
% 0.19/0.41 thf(fact_35_option_Osimps_I7_J,axiom,
% 0.19/0.41 ! [C: $tType,A: $tType,F1: C,F22: A > C,X22: A] :
% 0.19/0.41 ( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
% 0.19/0.41 = ( F22 @ X22 ) ) ).
% 0.19/0.41
% 0.19/0.41 % option.simps(7)
% 0.19/0.41 thf(fact_36_effect__deterministic_I3_J,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
% 0.19/0.41 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
% 0.19/0.41 => ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
% 0.19/0.41 => ( N = N2 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % effect_deterministic(3)
% 0.19/0.41 thf(fact_37_effect__deterministic_I2_J,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
% 0.19/0.41 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
% 0.19/0.41 => ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
% 0.19/0.41 => ( H4 = H5 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % effect_deterministic(2)
% 0.19/0.41 thf(fact_38_effect__deterministic_I1_J,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
% 0.19/0.41 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
% 0.19/0.41 => ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
% 0.19/0.41 => ( A3 = B2 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % effect_deterministic(1)
% 0.19/0.41 thf(fact_39_effect__ifE,axiom,
% 0.19/0.41 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.41 ( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 @ H4 @ R @ N )
% 0.19/0.41 => ( ( C2
% 0.19/0.41 => ~ ( heap_Time_effect @ A @ T2 @ H3 @ H4 @ R @ N ) )
% 0.19/0.41 => ~ ( ~ C2
% 0.19/0.41 => ~ ( heap_Time_effect @ A @ E2 @ H3 @ H4 @ R @ N ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % effect_ifE
% 0.19/0.41 thf(fact_40_effect__ifI,axiom,
% 0.19/0.41 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,E2: heap_Time_Heap @ A] :
% 0.19/0.41 ( ( C2
% 0.19/0.41 => ( heap_Time_effect @ A @ T2 @ H3 @ H4 @ R @ N ) )
% 0.19/0.41 => ( ( ~ C2
% 0.19/0.41 => ( heap_Time_effect @ A @ E2 @ H3 @ H4 @ R @ N ) )
% 0.19/0.41 => ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 @ H4 @ R @ N ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % effect_ifI
% 0.19/0.41 thf(fact_41_combine__options__assoc,axiom,
% 0.19/0.41 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A,Z2: option @ A] :
% 0.19/0.41 ( ! [X2: A,Y4: A,Z3: A] :
% 0.19/0.41 ( ( F @ ( F @ X2 @ Y4 ) @ Z3 )
% 0.19/0.41 = ( F @ X2 @ ( F @ Y4 @ Z3 ) ) )
% 0.19/0.41 => ( ( combine_options @ A @ F @ ( combine_options @ A @ F @ X @ Y ) @ Z2 )
% 0.19/0.41 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z2 ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_assoc
% 0.19/0.41 thf(fact_42_combine__options__commute,axiom,
% 0.19/0.41 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A] :
% 0.19/0.41 ( ! [X2: A,Y4: A] :
% 0.19/0.41 ( ( F @ X2 @ Y4 )
% 0.19/0.41 = ( F @ Y4 @ X2 ) )
% 0.19/0.41 => ( ( combine_options @ A @ F @ X @ Y )
% 0.19/0.41 = ( combine_options @ A @ F @ Y @ X ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_commute
% 0.19/0.41 thf(fact_43_combine__options__left__commute,axiom,
% 0.19/0.41 ! [A: $tType,F: A > A > A,Y: option @ A,X: option @ A,Z2: option @ A] :
% 0.19/0.41 ( ! [X2: A,Y4: A] :
% 0.19/0.41 ( ( F @ X2 @ Y4 )
% 0.19/0.41 = ( F @ Y4 @ X2 ) )
% 0.19/0.41 => ( ! [X2: A,Y4: A,Z3: A] :
% 0.19/0.41 ( ( F @ ( F @ X2 @ Y4 ) @ Z3 )
% 0.19/0.41 = ( F @ X2 @ ( F @ Y4 @ Z3 ) ) )
% 0.19/0.41 => ( ( combine_options @ A @ F @ Y @ ( combine_options @ A @ F @ X @ Z2 ) )
% 0.19/0.41 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z2 ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_left_commute
% 0.19/0.41 thf(fact_44_mem__Collect__eq,axiom,
% 0.19/0.41 ! [A: $tType,A3: A,P: A > $o] :
% 0.19/0.41 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.19/0.41 = ( P @ A3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % mem_Collect_eq
% 0.19/0.41 thf(fact_45_Collect__mem__eq,axiom,
% 0.19/0.41 ! [A: $tType,A4: set @ A] :
% 0.19/0.41 ( ( collect @ A
% 0.19/0.41 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 0.19/0.41 = A4 ) ).
% 0.19/0.41
% 0.19/0.41 % Collect_mem_eq
% 0.19/0.41 thf(fact_46_Collect__cong,axiom,
% 0.19/0.41 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.41 ( ! [X2: A] :
% 0.19/0.41 ( ( P @ X2 )
% 0.19/0.41 = ( Q @ X2 ) )
% 0.19/0.41 => ( ( collect @ A @ P )
% 0.19/0.41 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % Collect_cong
% 0.19/0.41 thf(fact_47_ext,axiom,
% 0.19/0.41 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.19/0.41 ( ! [X2: A] :
% 0.19/0.41 ( ( F @ X2 )
% 0.19/0.41 = ( G @ X2 ) )
% 0.19/0.41 => ( F = G ) ) ).
% 0.19/0.41
% 0.19/0.41 % ext
% 0.19/0.41 thf(fact_48_option_Odistinct_I1_J,axiom,
% 0.19/0.41 ! [A: $tType,X22: A] :
% 0.19/0.41 ( ( none @ A )
% 0.19/0.41 != ( some @ A @ X22 ) ) ).
% 0.19/0.41
% 0.19/0.41 % option.distinct(1)
% 0.19/0.41 thf(fact_49_option_OdiscI,axiom,
% 0.19/0.41 ! [A: $tType,Option: option @ A,X22: A] :
% 0.19/0.41 ( ( Option
% 0.19/0.41 = ( some @ A @ X22 ) )
% 0.19/0.41 => ( Option
% 0.19/0.41 != ( none @ A ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % option.discI
% 0.19/0.41 thf(fact_50_option_Oexhaust,axiom,
% 0.19/0.41 ! [A: $tType,Y: option @ A] :
% 0.19/0.41 ( ( Y
% 0.19/0.41 != ( none @ A ) )
% 0.19/0.41 => ~ ! [X23: A] :
% 0.19/0.41 ( Y
% 0.19/0.41 != ( some @ A @ X23 ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % option.exhaust
% 0.19/0.41 thf(fact_51_split__option__ex,axiom,
% 0.19/0.41 ! [A: $tType] :
% 0.19/0.41 ( ( ^ [P3: ( option @ A ) > $o] :
% 0.19/0.41 ? [X4: option @ A] : ( P3 @ X4 ) )
% 0.19/0.41 = ( ^ [P4: ( option @ A ) > $o] :
% 0.19/0.41 ( ( P4 @ ( none @ A ) )
% 0.19/0.41 | ? [X3: A] : ( P4 @ ( some @ A @ X3 ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % split_option_ex
% 0.19/0.41 thf(fact_52_split__option__all,axiom,
% 0.19/0.41 ! [A: $tType] :
% 0.19/0.41 ( ( ^ [P3: ( option @ A ) > $o] :
% 0.19/0.41 ! [X4: option @ A] : ( P3 @ X4 ) )
% 0.19/0.41 = ( ^ [P4: ( option @ A ) > $o] :
% 0.19/0.41 ( ( P4 @ ( none @ A ) )
% 0.19/0.41 & ! [X3: A] : ( P4 @ ( some @ A @ X3 ) ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % split_option_all
% 0.19/0.41 thf(fact_53_combine__options__cases,axiom,
% 0.19/0.41 ! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
% 0.19/0.41 ( ( ( X
% 0.19/0.41 = ( none @ A ) )
% 0.19/0.41 => ( P @ X @ Y ) )
% 0.19/0.41 => ( ( ( Y
% 0.19/0.41 = ( none @ B ) )
% 0.19/0.41 => ( P @ X @ Y ) )
% 0.19/0.41 => ( ! [A5: A,B3: B] :
% 0.19/0.41 ( ( X
% 0.19/0.41 = ( some @ A @ A5 ) )
% 0.19/0.41 => ( ( Y
% 0.19/0.41 = ( some @ B @ B3 ) )
% 0.19/0.41 => ( P @ X @ Y ) ) )
% 0.19/0.41 => ( P @ X @ Y ) ) ) ) ).
% 0.19/0.41
% 0.19/0.41 % combine_options_cases
% 0.19/0.41 thf(fact_54_success__effectE,axiom,
% 0.19/0.41 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ( heap_Time_success @ A @ C2 @ H3 )
% 0.19/0.41 => ~ ! [R2: A,H6: heap_ext @ product_unit,N3: nat] :
% 0.19/0.41 ~ ( heap_Time_effect @ A @ C2 @ H3 @ H6 @ R2 @ N3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % success_effectE
% 0.19/0.41 thf(fact_55_effect__success,axiom,
% 0.19/0.41 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.41 ( ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N )
% 0.19/0.41 => ( heap_Time_success @ A @ C2 @ H3 ) ) ).
% 0.19/0.41
% 0.19/0.41 % effect_success
% 0.19/0.41 thf(fact_56_effect__raiseE,axiom,
% 0.19/0.41 ! [A: $tType,X: list @ char,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.41 ~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H3 @ H4 @ R @ N ) ).
% 0.19/0.41
% 0.19/0.41 % effect_raiseE
% 0.19/0.41 thf(fact_57_the__default_Osimps_I1_J,axiom,
% 0.19/0.41 ! [A: $tType,Uu2: A,X: A] :
% 0.19/0.41 ( ( the_default @ A @ Uu2 @ ( some @ A @ X ) )
% 0.19/0.41 = X ) ).
% 0.19/0.41
% 0.19/0.41 % the_default.simps(1)
% 0.19/0.41 thf(fact_58_Heap__cases,axiom,
% 0.19/0.41 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.19/0.41 ( ! [X2: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 0.19/0.41 ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.41 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ H6 ) ) )
% 0.19/0.41 => ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.42 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Heap_cases
% 0.19/0.42 thf(fact_59_successE,axiom,
% 0.19/0.42 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.19/0.42 ( ( heap_Time_success @ A @ F @ H3 )
% 0.19/0.42 => ~ ! [R2: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.42 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ H6 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % successE
% 0.19/0.42 thf(fact_60_execute__heap,axiom,
% 0.19/0.42 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
% 0.19/0.42 = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ F ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_heap
% 0.19/0.42 thf(fact_61_some__opt__sym__eq__trivial,axiom,
% 0.19/0.42 ! [A: $tType,X: A] :
% 0.19/0.42 ( ( eps_Opt @ A
% 0.19/0.42 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.19/0.42 @ X ) )
% 0.19/0.42 = ( some @ A @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % some_opt_sym_eq_trivial
% 0.19/0.42 thf(fact_62_success__newI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [N: nat,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % success_newI
% 0.19/0.42 thf(fact_63_success__makeI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [N: nat,F: nat > A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % success_makeI
% 0.19/0.42 thf(fact_64_mmupd__in__upd,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,K: A,K2: set @ A,M2: A > ( option @ B ),V: B] :
% 0.19/0.42 ( ( member @ A @ K @ K2 )
% 0.19/0.42 => ( ( map_mmupd @ A @ B @ M2 @ K2 @ V @ K )
% 0.19/0.42 = ( some @ B @ V ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % mmupd_in_upd
% 0.19/0.42 thf(fact_65_option_Ocollapse,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( some @ A @ ( the2 @ A @ Option ) )
% 0.19/0.42 = Option ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.collapse
% 0.19/0.42 thf(fact_66_not__Some__eq2,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
% 0.19/0.42 ( ( ! [X3: A,Y3: B] :
% 0.19/0.42 ( V
% 0.19/0.42 != ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) )
% 0.19/0.42 = ( V
% 0.19/0.42 = ( none @ ( product_prod @ A @ B ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_Some_eq2
% 0.19/0.42 thf(fact_67_effect__tapE,axiom,
% 0.19/0.42 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H3 @ H4 @ R @ N )
% 0.19/0.42 => ~ ( ( H4 = H3 )
% 0.19/0.42 => ( ( R
% 0.19/0.42 = ( F @ H3 ) )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_tapE
% 0.19/0.42 thf(fact_68_effect__tapI,axiom,
% 0.19/0.42 ! [A: $tType,H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
% 0.19/0.42 ( ( H4 = H3 )
% 0.19/0.42 => ( ( R
% 0.19/0.42 = ( F @ H3 ) )
% 0.19/0.42 => ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H3 @ H4 @ R @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_tapI
% 0.19/0.42 thf(fact_69_effect__assertI,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: A,H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ( H4 = H3 )
% 0.19/0.42 => ( ( R = X )
% 0.19/0.42 => ( ( N
% 0.19/0.42 = ( one_one @ nat ) )
% 0.19/0.42 => ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_assertI
% 0.19/0.42 thf(fact_70_mmupd__notin__upd,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,K: A,K2: set @ A,M2: A > ( option @ B ),V: B] :
% 0.19/0.42 ( ~ ( member @ A @ K @ K2 )
% 0.19/0.42 => ( ( map_mmupd @ A @ B @ M2 @ K2 @ V @ K )
% 0.19/0.42 = ( M2 @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % mmupd_notin_upd
% 0.19/0.42 thf(fact_71_fun__comp__eq__conv,axiom,
% 0.19/0.42 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,Fg: A > B] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.19/0.42 = Fg )
% 0.19/0.42 = ( ! [X3: A] :
% 0.19/0.42 ( ( F @ ( G @ X3 ) )
% 0.19/0.42 = ( Fg @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fun_comp_eq_conv
% 0.19/0.42 thf(fact_72_comp__cong__right,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ( comp @ B @ C @ A @ F @ X )
% 0.19/0.42 = ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_cong_right
% 0.19/0.42 thf(fact_73_comp__cong__left,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ( comp @ A @ B @ C @ X @ F )
% 0.19/0.42 = ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_cong_left
% 0.19/0.42 thf(fact_74_bex2I,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 0.19/0.42 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 0.19/0.42 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 0.19/0.42 => ( P @ A3 @ B2 ) )
% 0.19/0.42 => ? [A5: A,B3: B] :
% 0.19/0.42 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S2 )
% 0.19/0.42 & ( P @ A5 @ B3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bex2I
% 0.19/0.42 thf(fact_75_pairself_Ocases,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 0.19/0.42 ~ ! [F4: A > B,A5: A,B3: A] :
% 0.19/0.42 ( X
% 0.19/0.42 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F4 @ ( product_Pair @ A @ A @ A5 @ B3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % pairself.cases
% 0.19/0.42 thf(fact_76_fails__array__new,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [N: nat,X: A,H3: heap_ext @ product_unit] :
% 0.19/0.42 ~ ( time_fails @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % fails_array_new
% 0.19/0.42 thf(fact_77_timeFrame_Ocases,axiom,
% 0.19/0.42 ! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.19/0.42 ( ! [N3: nat,R2: A,H2: heap_ext @ product_unit,N4: nat] :
% 0.19/0.42 ( X
% 0.19/0.42 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N3 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) ) )
% 0.19/0.42 => ~ ! [N3: nat] :
% 0.19/0.42 ( X
% 0.19/0.42 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N3 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame.cases
% 0.19/0.42 thf(fact_78_execute__assert_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_assert(1)
% 0.19/0.42 thf(fact_79_execute__tap,axiom,
% 0.19/0.42 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_tap
% 0.19/0.42 thf(fact_80_option_Osel,axiom,
% 0.19/0.42 ! [A: $tType,X22: A] :
% 0.19/0.42 ( ( the2 @ A @ ( some @ A @ X22 ) )
% 0.19/0.42 = X22 ) ).
% 0.19/0.42
% 0.19/0.42 % option.sel
% 0.19/0.42 thf(fact_81_option_Oexpand,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 0.19/0.42 ( ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( Option2
% 0.19/0.42 = ( none @ A ) ) )
% 0.19/0.42 => ( ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( Option2
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( the2 @ A @ Option )
% 0.19/0.42 = ( the2 @ A @ Option2 ) ) ) )
% 0.19/0.42 => ( Option = Option2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.expand
% 0.19/0.42 thf(fact_82_effect__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_effect @ A )
% 0.19/0.42 = ( ^ [C3: heap_Time_Heap @ A,H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R3: A,N5: nat] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ C3 @ H )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ N5 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_def
% 0.19/0.42 thf(fact_83_effectI,axiom,
% 0.19/0.42 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,R: A,H4: heap_ext @ product_unit,N: nat] :
% 0.19/0.42 ( ( ( heap_Time_execute @ A @ C2 @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 0.19/0.42 => ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % effectI
% 0.19/0.42 thf(fact_84_map__mmupd__def,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( map_mmupd @ B @ A )
% 0.19/0.42 = ( ^ [M: B > ( option @ A ),K3: set @ B,V2: A,K4: B] : ( if @ ( option @ A ) @ ( member @ B @ K4 @ K3 ) @ ( some @ A @ V2 ) @ ( M @ K4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_mmupd_def
% 0.19/0.42 thf(fact_85_map__mmupdE,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,M2: B > ( option @ A ),K2: set @ B,V: A,K: B,X: A] :
% 0.19/0.42 ( ( ( map_mmupd @ B @ A @ M2 @ K2 @ V @ K )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 => ( ( ~ ( member @ B @ K @ K2 )
% 0.19/0.42 => ( ( M2 @ K )
% 0.19/0.42 != ( some @ A @ X ) ) )
% 0.19/0.42 => ~ ( ( member @ B @ K @ K2 )
% 0.19/0.42 => ( X != V ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_mmupdE
% 0.19/0.42 thf(fact_86_success__bind__executeI,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 0.19/0.42 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 0.19/0.42 => ( ( heap_Time_success @ B @ ( G @ X ) @ H4 )
% 0.19/0.42 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % success_bind_executeI
% 0.19/0.42 thf(fact_87_Eps__Opt__eq__Some__implies,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: A] :
% 0.19/0.42 ( ( ( eps_Opt @ A @ P )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 => ( P @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % Eps_Opt_eq_Some_implies
% 0.19/0.42 thf(fact_88_Eps__Opt__eq__Some,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: A] :
% 0.19/0.42 ( ! [X5: A] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ( P @ X5 )
% 0.19/0.42 => ( X5 = X ) ) )
% 0.19/0.42 => ( ( ( eps_Opt @ A @ P )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 = ( P @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Eps_Opt_eq_Some
% 0.19/0.42 thf(fact_89_option_Oexhaust__sel,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( Option
% 0.19/0.42 = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.exhaust_sel
% 0.19/0.42 thf(fact_90_effect__assertE,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 @ H4 @ R @ N )
% 0.19/0.42 => ~ ( ( P @ X )
% 0.19/0.42 => ( ( R = X )
% 0.19/0.42 => ( ( H4 = H3 )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_assertE
% 0.19/0.42 thf(fact_91_comp__apply,axiom,
% 0.19/0.42 ! [C: $tType,A: $tType,B: $tType] :
% 0.19/0.42 ( ( comp @ B @ A @ C )
% 0.19/0.42 = ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_apply
% 0.19/0.42 thf(fact_92_prod_Oinject,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 0.19/0.42 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 0.19/0.42 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 0.19/0.42 = ( ( X1 = Y1 )
% 0.19/0.42 & ( X22 = Y2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod.inject
% 0.19/0.42 thf(fact_93_old_Oprod_Oinject,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
% 0.19/0.42 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 0.19/0.42 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.19/0.42 = ( ( A3 = A6 )
% 0.19/0.42 & ( B2 = B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.prod.inject
% 0.19/0.42 thf(fact_94_execute__bind_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 0.19/0.42 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 0.19/0.42 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 0.19/0.42 = ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_bind(1)
% 0.19/0.42 thf(fact_95_prod__cases3,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.42 ~ ! [A5: A,B3: B,C4: C] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_cases3
% 0.19/0.42 thf(fact_96_prod__cases4,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.19/0.42 ~ ! [A5: A,B3: B,C4: C,D2: D] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_cases4
% 0.19/0.42 thf(fact_97_prod__cases5,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.19/0.42 ~ ! [A5: A,B3: B,C4: C,D2: D,E3: E] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_cases5
% 0.19/0.42 thf(fact_98_prod__cases6,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.19/0.42 ~ ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E3 @ F4 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_cases6
% 0.19/0.42 thf(fact_99_prod__cases7,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 0.19/0.42 ~ ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5,G4: G3] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_cases7
% 0.19/0.42 thf(fact_100_prod__induct3,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.42 ( ! [A5: A,B3: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) )
% 0.19/0.42 => ( P @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_induct3
% 0.19/0.42 thf(fact_101_prod__induct4,axiom,
% 0.19/0.42 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 0.19/0.42 ( ! [A5: A,B3: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
% 0.19/0.42 => ( P @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_induct4
% 0.19/0.42 thf(fact_102_timeFrame_Osimps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType,N: nat] :
% 0.19/0.42 ( ( heap_Time_timeFrame @ A @ N @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.42 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame.simps(2)
% 0.19/0.42 thf(fact_103_Pair__inject,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
% 0.19/0.42 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 0.19/0.42 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 0.19/0.42 => ~ ( ( A3 = A6 )
% 0.19/0.42 => ( B2 != B4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Pair_inject
% 0.19/0.42 thf(fact_104_prod__cases,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P5: product_prod @ A @ B] :
% 0.19/0.42 ( ! [A5: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.19/0.42 => ( P @ P5 ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_cases
% 0.19/0.42 thf(fact_105_surj__pair,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,P5: product_prod @ A @ B] :
% 0.19/0.42 ? [X2: A,Y4: B] :
% 0.19/0.42 ( P5
% 0.19/0.42 = ( product_Pair @ A @ B @ X2 @ Y4 ) ) ).
% 0.19/0.42
% 0.19/0.42 % surj_pair
% 0.19/0.42 thf(fact_106_old_Oprod_Oexhaust,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 0.19/0.42 ~ ! [A5: A,B3: B] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.prod.exhaust
% 0.19/0.42 thf(fact_107_one__reorient,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( one @ A )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ( ( ( one_one @ A )
% 0.19/0.42 = X )
% 0.19/0.42 = ( X
% 0.19/0.42 = ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % one_reorient
% 0.19/0.42 thf(fact_108_comp__eq__dest__lhs,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.19/0.42 = C2 )
% 0.19/0.42 => ( ( A3 @ ( B2 @ V ) )
% 0.19/0.42 = ( C2 @ V ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_eq_dest_lhs
% 0.19/0.42 thf(fact_109_comp__eq__elim,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.19/0.42 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.19/0.42 => ! [V3: A] :
% 0.19/0.42 ( ( A3 @ ( B2 @ V3 ) )
% 0.19/0.42 = ( C2 @ ( D3 @ V3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_eq_elim
% 0.19/0.42 thf(fact_110_comp__eq__dest,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.19/0.42 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 0.19/0.42 => ( ( A3 @ ( B2 @ V ) )
% 0.19/0.42 = ( C2 @ ( D3 @ V ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_eq_dest
% 0.19/0.42 thf(fact_111_comp__assoc,axiom,
% 0.19/0.42 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H3: A > C] :
% 0.19/0.42 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H3 )
% 0.19/0.42 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_assoc
% 0.19/0.42 thf(fact_112_comp__def,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType,B: $tType] :
% 0.19/0.42 ( ( comp @ B @ C @ A )
% 0.19/0.42 = ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_def
% 0.19/0.42 thf(fact_113_prod__induct7,axiom,
% 0.19/0.42 ! [G3: $tType,F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 0.19/0.42 ( ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.19/0.42 => ( P @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_induct7
% 0.19/0.42 thf(fact_114_prod__induct6,axiom,
% 0.19/0.42 ! [F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
% 0.19/0.42 ( ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E3 @ F4 ) ) ) ) ) )
% 0.19/0.42 => ( P @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_induct6
% 0.19/0.42 thf(fact_115_prod__induct5,axiom,
% 0.19/0.42 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 0.19/0.42 ( ! [A5: A,B3: B,C4: C,D2: D,E3: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) )
% 0.19/0.42 => ( P @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % prod_induct5
% 0.19/0.42 thf(fact_116_old_Oprod_Orec,axiom,
% 0.19/0.42 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
% 0.19/0.42 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.19/0.42 = ( F1 @ A3 @ B2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % old.prod.rec
% 0.19/0.42 thf(fact_117_timeFrame_Oelims,axiom,
% 0.19/0.42 ! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.42 ( ( ( heap_Time_timeFrame @ A @ X @ Xa )
% 0.19/0.42 = Y )
% 0.19/0.42 => ( ! [R2: A,H2: heap_ext @ product_unit,N4: nat] :
% 0.19/0.42 ( ( Xa
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) )
% 0.19/0.42 => ( Y
% 0.19/0.42 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) ) )
% 0.19/0.42 => ~ ( ( Xa
% 0.19/0.42 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.42 => ( Y
% 0.19/0.42 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame.elims
% 0.19/0.42 thf(fact_118_execute__bind__eq__SomeI,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y: B,H5: heap_ext @ product_unit,N2: nat] :
% 0.19/0.42 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 0.19/0.42 => ( ( ( heap_Time_execute @ B @ ( G @ X ) @ H4 )
% 0.19/0.42 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N2 ) ) ) )
% 0.19/0.42 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_bind_eq_SomeI
% 0.19/0.42 thf(fact_119_uncurry__apply,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
% 0.19/0.42 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 0.19/0.42 = ( F @ A3 @ B2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % uncurry_apply
% 0.19/0.42 thf(fact_120_timeFrame_Osimps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,N: nat,R: A,H3: heap_ext @ product_unit,N2: nat] :
% 0.19/0.42 ( ( heap_Time_timeFrame @ A @ N @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame.simps(1)
% 0.19/0.42 thf(fact_121_heap__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_heap @ A )
% 0.19/0.42 = ( ^ [F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( heap_Time_Heap2 @ A @ ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ F2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % heap_def
% 0.19/0.42 thf(fact_122_internal__case__prod__conv,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
% 0.19/0.42 ( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 0.19/0.42 = ( C2 @ A3 @ B2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % internal_case_prod_conv
% 0.19/0.42 thf(fact_123_option_Osplit__sel,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 0.19/0.42 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 0.19/0.42 = ( ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 => ( P @ F1 ) )
% 0.19/0.42 & ( ( Option
% 0.19/0.42 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 0.19/0.42 => ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.split_sel
% 0.19/0.42 thf(fact_124_option_Osplit__sel__asm,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 0.19/0.42 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 0.19/0.42 = ( ~ ( ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 & ~ ( P @ F1 ) )
% 0.19/0.42 | ( ( Option
% 0.19/0.42 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 0.19/0.42 & ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.split_sel_asm
% 0.19/0.42 thf(fact_125_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.19/0.42 ! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
% 0.19/0.42 ~ ! [F4: nat > A > A,A5: nat,B3: nat,Acc: A] :
% 0.19/0.42 ( X
% 0.19/0.42 != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_atLeastAtMost_nat.cases
% 0.19/0.42 thf(fact_126_one__natural_Orsp,axiom,
% 0.19/0.42 ( ( one_one @ nat )
% 0.19/0.42 = ( one_one @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % one_natural.rsp
% 0.19/0.42 thf(fact_127_timeFrame_Opelims,axiom,
% 0.19/0.42 ! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.42 ( ( ( heap_Time_timeFrame @ A @ X @ Xa )
% 0.19/0.42 = Y )
% 0.19/0.42 => ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ Xa ) )
% 0.19/0.42 => ( ! [R2: A,H2: heap_ext @ product_unit,N4: nat] :
% 0.19/0.42 ( ( Xa
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) )
% 0.19/0.42 => ( ( Y
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) )
% 0.19/0.42 => ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) ) ) ) )
% 0.19/0.42 => ~ ( ( Xa
% 0.19/0.42 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.42 => ( ( Y
% 0.19/0.42 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 0.19/0.42 => ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame.pelims
% 0.19/0.42 thf(fact_128_add__right__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel_semigroup_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.19/0.42 = ( plus_plus @ A @ C2 @ A3 ) )
% 0.19/0.42 = ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_right_cancel
% 0.19/0.42 thf(fact_129_add__left__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel_semigroup_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ C2 ) )
% 0.19/0.42 = ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_left_cancel
% 0.19/0.42 thf(fact_130_Heap_Oinject,axiom,
% 0.19/0.42 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.19/0.42 ( ( ( heap_Time_Heap2 @ A @ X )
% 0.19/0.42 = ( heap_Time_Heap2 @ A @ Ya ) )
% 0.19/0.42 = ( X = Ya ) ) ).
% 0.19/0.42
% 0.19/0.42 % Heap.inject
% 0.19/0.42 thf(fact_131_Heap__execute,axiom,
% 0.19/0.42 ! [A: $tType,F: heap_Time_Heap @ A] :
% 0.19/0.42 ( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
% 0.19/0.42 = F ) ).
% 0.19/0.42
% 0.19/0.42 % Heap_execute
% 0.19/0.42 thf(fact_132_timeFrame__assoc,axiom,
% 0.19/0.42 ! [A: $tType,N: nat,N2: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.42 ( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N2 @ F ) )
% 0.19/0.42 = ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N2 ) @ F ) ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame_assoc
% 0.19/0.42 thf(fact_133_add__right__imp__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel_semigroup_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.19/0.42 = ( plus_plus @ A @ C2 @ A3 ) )
% 0.19/0.42 => ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_right_imp_eq
% 0.19/0.42 thf(fact_134_add__left__imp__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel_semigroup_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ C2 ) )
% 0.19/0.42 => ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_left_imp_eq
% 0.19/0.42 thf(fact_135_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_semigroup_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ab_semigroup_add_class.add.left_commute
% 0.19/0.42 thf(fact_136_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_semigroup_add @ A )
% 0.19/0.42 => ( ( plus_plus @ A )
% 0.19/0.42 = ( ^ [A7: A,B5: A] : ( plus_plus @ A @ B5 @ A7 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ab_semigroup_add_class.add.commute
% 0.19/0.42 thf(fact_137_add_Oright__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.19/0.42 = ( plus_plus @ A @ C2 @ A3 ) )
% 0.19/0.42 = ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.right_cancel
% 0.19/0.42 thf(fact_138_add_Oleft__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ C2 ) )
% 0.19/0.42 = ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.left_cancel
% 0.19/0.42 thf(fact_139_add_Oassoc,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semigroup_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.assoc
% 0.19/0.42 thf(fact_140_group__cancel_Oadd2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comm_monoid_add @ A )
% 0.19/0.42 => ! [B6: A,K: A,B2: A,A3: A] :
% 0.19/0.42 ( ( B6
% 0.19/0.42 = ( plus_plus @ A @ K @ B2 ) )
% 0.19/0.42 => ( ( plus_plus @ A @ A3 @ B6 )
% 0.19/0.42 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group_cancel.add2
% 0.19/0.42 thf(fact_141_group__cancel_Oadd1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comm_monoid_add @ A )
% 0.19/0.42 => ! [A4: A,K: A,A3: A,B2: A] :
% 0.19/0.42 ( ( A4
% 0.19/0.42 = ( plus_plus @ A @ K @ A3 ) )
% 0.19/0.42 => ( ( plus_plus @ A @ A4 @ B2 )
% 0.19/0.42 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group_cancel.add1
% 0.19/0.42 thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( I = J )
% 0.19/0.42 & ( K = L ) )
% 0.19/0.42 => ( ( plus_plus @ A @ I @ K )
% 0.19/0.42 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_semiring(4)
% 0.19/0.42 thf(fact_143_Heap_Oexhaust,axiom,
% 0.19/0.42 ! [A: $tType,Y: heap_Time_Heap @ A] :
% 0.19/0.42 ~ ! [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.19/0.42 ( Y
% 0.19/0.42 != ( heap_Time_Heap2 @ A @ X2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % Heap.exhaust
% 0.19/0.42 thf(fact_144_execute_Osimps,axiom,
% 0.19/0.42 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ ( heap_Time_Heap2 @ A @ F ) )
% 0.19/0.42 = F ) ).
% 0.19/0.42
% 0.19/0.42 % execute.simps
% 0.19/0.42 thf(fact_145_option_Osimps_I5_J,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] :
% 0.19/0.42 ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
% 0.19/0.42 = ( F22 @ X22 ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(5)
% 0.19/0.42 thf(fact_146_option_Osimps_I4_J,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F1: B,F22: A > B] :
% 0.19/0.42 ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
% 0.19/0.42 = F1 ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(4)
% 0.19/0.42 thf(fact_147_effect__bindE,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit,H5: heap_ext @ product_unit,R4: A,N: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H3 @ H5 @ R4 @ N )
% 0.19/0.42 => ~ ! [H6: heap_ext @ product_unit,R2: B,N1: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ B @ F @ H3 @ H6 @ R2 @ N1 )
% 0.19/0.42 => ! [N22: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( G @ R2 ) @ H6 @ H5 @ R4 @ N22 )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_bindE
% 0.19/0.42 thf(fact_148_effect__bindI,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H5: heap_ext @ product_unit,R4: B,N2: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ R @ N )
% 0.19/0.42 => ( ( heap_Time_effect @ B @ ( G @ R ) @ H4 @ H5 @ R4 @ N2 )
% 0.19/0.42 => ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 @ H5 @ R4 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_bindI
% 0.19/0.42 thf(fact_149_option_Ocase__eq__if,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( case_option @ B @ A )
% 0.19/0.42 = ( ^ [F12: B,F23: A > B,Option3: option @ A] :
% 0.19/0.42 ( if @ B
% 0.19/0.42 @ ( Option3
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 @ F12
% 0.19/0.42 @ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.case_eq_if
% 0.19/0.42 thf(fact_150_uncurry__curry__id,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.19/0.42 ( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.19/0.42 = F ) ).
% 0.19/0.42
% 0.19/0.42 % uncurry_curry_id
% 0.19/0.42 thf(fact_151_curry__uncurry__id,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.19/0.42 ( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
% 0.19/0.42 = F ) ).
% 0.19/0.42
% 0.19/0.42 % curry_uncurry_id
% 0.19/0.42 thf(fact_152_accp_Ocases,axiom,
% 0.19/0.42 ! [A: $tType,R: A > A > $o,A3: A] :
% 0.19/0.42 ( ( accp @ A @ R @ A3 )
% 0.19/0.42 => ! [Y6: A] :
% 0.19/0.42 ( ( R @ Y6 @ A3 )
% 0.19/0.42 => ( accp @ A @ R @ Y6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp.cases
% 0.19/0.42 thf(fact_153_accp_Osimps,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( accp @ A )
% 0.19/0.42 = ( ^ [R3: A > A > $o,A7: A] :
% 0.19/0.42 ? [X3: A] :
% 0.19/0.42 ( ( A7 = X3 )
% 0.19/0.42 & ! [Y3: A] :
% 0.19/0.42 ( ( R3 @ Y3 @ X3 )
% 0.19/0.42 => ( accp @ A @ R3 @ Y3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp.simps
% 0.19/0.42 thf(fact_154_accpI,axiom,
% 0.19/0.42 ! [A: $tType,R: A > A > $o,X: A] :
% 0.19/0.42 ( ! [Y4: A] :
% 0.19/0.42 ( ( R @ Y4 @ X )
% 0.19/0.42 => ( accp @ A @ R @ Y4 ) )
% 0.19/0.42 => ( accp @ A @ R @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % accpI
% 0.19/0.42 thf(fact_155_accp__induct,axiom,
% 0.19/0.42 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 0.19/0.42 ( ( accp @ A @ R @ A3 )
% 0.19/0.42 => ( ! [X2: A] :
% 0.19/0.42 ( ( accp @ A @ R @ X2 )
% 0.19/0.42 => ( ! [Y6: A] :
% 0.19/0.42 ( ( R @ Y6 @ X2 )
% 0.19/0.42 => ( P @ Y6 ) )
% 0.19/0.42 => ( P @ X2 ) ) )
% 0.19/0.42 => ( P @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp_induct
% 0.19/0.42 thf(fact_156_accp__downward,axiom,
% 0.19/0.42 ! [A: $tType,R: A > A > $o,B2: A,A3: A] :
% 0.19/0.42 ( ( accp @ A @ R @ B2 )
% 0.19/0.42 => ( ( R @ A3 @ B2 )
% 0.19/0.42 => ( accp @ A @ R @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp_downward
% 0.19/0.42 thf(fact_157_not__accp__down,axiom,
% 0.19/0.42 ! [A: $tType,R5: A > A > $o,X: A] :
% 0.19/0.42 ( ~ ( accp @ A @ R5 @ X )
% 0.19/0.42 => ~ ! [Z3: A] :
% 0.19/0.42 ( ( R5 @ Z3 @ X )
% 0.19/0.42 => ( accp @ A @ R5 @ Z3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_accp_down
% 0.19/0.42 thf(fact_158_accp__induct__rule,axiom,
% 0.19/0.42 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 0.19/0.42 ( ( accp @ A @ R @ A3 )
% 0.19/0.42 => ( ! [X2: A] :
% 0.19/0.42 ( ( accp @ A @ R @ X2 )
% 0.19/0.42 => ( ! [Y6: A] :
% 0.19/0.42 ( ( R @ Y6 @ X2 )
% 0.19/0.42 => ( P @ Y6 ) )
% 0.19/0.42 => ( P @ X2 ) ) )
% 0.19/0.42 => ( P @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp_induct_rule
% 0.19/0.42 thf(fact_159_fun_Omap__comp,axiom,
% 0.19/0.42 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 0.19/0.42 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 0.19/0.42 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.19/0.42
% 0.19/0.42 % fun.map_comp
% 0.19/0.42 thf(fact_160_comp__cong,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F3: D > A,G5: E > D,X6: E] :
% 0.19/0.42 ( ( ( F @ ( G @ X ) )
% 0.19/0.42 = ( F3 @ ( G5 @ X6 ) ) )
% 0.19/0.42 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.19/0.42 = ( comp @ D @ A @ E @ F3 @ G5 @ X6 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_cong
% 0.19/0.42 thf(fact_161_curryI,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
% 0.19/0.42 ( ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.19/0.42 => ( product_curry @ A @ B @ $o @ F @ A3 @ B2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % curryI
% 0.19/0.42 thf(fact_162_curry__conv,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType,B: $tType] :
% 0.19/0.42 ( ( product_curry @ B @ C @ A )
% 0.19/0.42 = ( ^ [F2: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F2 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % curry_conv
% 0.19/0.42 thf(fact_163_curryE,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
% 0.19/0.42 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
% 0.19/0.42 => ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % curryE
% 0.19/0.42 thf(fact_164_curryD,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
% 0.19/0.42 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
% 0.19/0.42 => ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % curryD
% 0.19/0.42 thf(fact_165_fold__atLeastAtMost__nat_Opinduct,axiom,
% 0.19/0.42 ! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
% 0.19/0.42 ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
% 0.19/0.42 => ( ! [F4: nat > A > A,A5: nat,B3: nat,Acc: A] :
% 0.19/0.42 ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) )
% 0.19/0.42 => ( ( ~ ( ord_less @ nat @ B3 @ A5 )
% 0.19/0.42 => ( P @ F4 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F4 @ A5 @ Acc ) ) )
% 0.19/0.42 => ( P @ F4 @ A5 @ B3 @ Acc ) ) )
% 0.19/0.42 => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_atLeastAtMost_nat.pinduct
% 0.19/0.42 thf(fact_166_in__lex__prod,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.19/0.42 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 0.19/0.42 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A6 ) @ R )
% 0.19/0.42 | ( ( A3 = A6 )
% 0.19/0.42 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % in_lex_prod
% 0.19/0.42 thf(fact_167_rewriteR__comp__comp2,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H3: A > C,R1: D > B,R22: A > D,F: B > E,L: D > E] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ G @ H3 )
% 0.19/0.42 = ( comp @ D @ B @ A @ R1 @ R22 ) )
% 0.19/0.42 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 0.19/0.42 = L )
% 0.19/0.42 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H3 )
% 0.19/0.42 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % rewriteR_comp_comp2
% 0.19/0.42 thf(fact_168_rewriteL__comp__comp2,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H3: E > A,R: E > D] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.19/0.42 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.19/0.42 => ( ( ( comp @ A @ D @ E @ L2 @ H3 )
% 0.19/0.42 = R )
% 0.19/0.42 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H3 ) )
% 0.19/0.42 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % rewriteL_comp_comp2
% 0.19/0.42 thf(fact_169_rewriteR__comp__comp,axiom,
% 0.19/0.42 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H3: A > C,R: A > B,F: B > D] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ G @ H3 )
% 0.19/0.42 = R )
% 0.19/0.42 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H3 )
% 0.19/0.42 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % rewriteR_comp_comp
% 0.19/0.42 thf(fact_170_rewriteL__comp__comp,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H3: D > A] :
% 0.19/0.42 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.19/0.42 = L )
% 0.19/0.42 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H3 ) )
% 0.19/0.42 = ( comp @ A @ B @ D @ L @ H3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % rewriteL_comp_comp
% 0.19/0.42 thf(fact_171_type__copy__map__cong0,axiom,
% 0.19/0.42 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M3: B > A,G: C > B,X: C,N6: D > A,H3: C > D,F: A > E] :
% 0.19/0.42 ( ( ( M3 @ ( G @ X ) )
% 0.19/0.42 = ( N6 @ ( H3 @ X ) ) )
% 0.19/0.42 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M3 ) @ G @ X )
% 0.19/0.42 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N6 ) @ H3 @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % type_copy_map_cong0
% 0.19/0.42 thf(fact_172_comp__apply__eq,axiom,
% 0.19/0.42 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H3: D > A,K: C > D] :
% 0.19/0.42 ( ( ( F @ ( G @ X ) )
% 0.19/0.42 = ( H3 @ ( K @ X ) ) )
% 0.19/0.42 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.19/0.42 = ( comp @ D @ A @ C @ H3 @ K @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % comp_apply_eq
% 0.19/0.42 thf(fact_173_effect__returnE,axiom,
% 0.19/0.42 ! [A: $tType,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H3 @ H4 @ R @ N )
% 0.19/0.42 => ~ ( ( R = X )
% 0.19/0.42 => ( ( H4 = H3 )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_returnE
% 0.19/0.42 thf(fact_174_effect__returnI,axiom,
% 0.19/0.42 ! [A: $tType,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A] :
% 0.19/0.42 ( ( H3 = H4 )
% 0.19/0.42 => ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H3 @ H4 @ X @ ( one_one @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_returnI
% 0.19/0.42 thf(fact_175_add_Oright__commute,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_semigroup_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.42 = ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.right_commute
% 0.19/0.42 thf(fact_176_add__less__cancel__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_cancel_right
% 0.19/0.42 thf(fact_177_add__less__cancel__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_cancel_left
% 0.19/0.42 thf(fact_178_fold__if__return,axiom,
% 0.19/0.42 ! [A: $tType,B2: $o,C2: A,D3: A] :
% 0.19/0.42 ( ( B2
% 0.19/0.42 => ( ( heap_Time_return @ A @ C2 )
% 0.19/0.42 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D3 ) ) ) )
% 0.19/0.42 & ( ~ B2
% 0.19/0.42 => ( ( heap_Time_return @ A @ D3 )
% 0.19/0.42 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_if_return
% 0.19/0.42 thf(fact_179_add__less__imp__less__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.42 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_imp_less_right
% 0.19/0.42 thf(fact_180_add__less__imp__less__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.19/0.42 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_imp_less_left
% 0.19/0.42 thf(fact_181_add__strict__right__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_strict_right_mono
% 0.19/0.42 thf(fact_182_add__strict__left__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_strict_left_mono
% 0.19/0.42 thf(fact_183_add__strict__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( strict9044650504122735259up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.42 => ( ( ord_less @ A @ C2 @ D3 )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_strict_mono
% 0.19/0.42 thf(fact_184_add__mono__thms__linordered__field_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( ord_less @ A @ I @ J )
% 0.19/0.42 & ( K = L ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_field(1)
% 0.19/0.42 thf(fact_185_add__mono__thms__linordered__field_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( I = J )
% 0.19/0.42 & ( ord_less @ A @ K @ L ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_field(2)
% 0.19/0.42 thf(fact_186_add__mono__thms__linordered__field_I5_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( ord_less @ A @ I @ J )
% 0.19/0.42 & ( ord_less @ A @ K @ L ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_field(5)
% 0.19/0.42 thf(fact_187_success__returnI,axiom,
% 0.19/0.42 ! [A: $tType,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H3 ) ).
% 0.19/0.42
% 0.19/0.42 % success_returnI
% 0.19/0.42 thf(fact_188_fails__return,axiom,
% 0.19/0.42 ! [A: $tType,X: A,H3: heap_ext @ product_unit] :
% 0.19/0.42 ~ ( time_fails @ A @ ( heap_Time_return @ A @ X ) @ H3 ) ).
% 0.19/0.42
% 0.19/0.42 % fails_return
% 0.19/0.42 thf(fact_189_add_Oright__assoc,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_semigroup_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.right_assoc
% 0.19/0.42 thf(fact_190_nat__add__left__cancel__less,axiom,
% 0.19/0.42 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.42 = ( ord_less @ nat @ M2 @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_add_left_cancel_less
% 0.19/0.42 thf(fact_191_less__option__Some,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 0.19/0.42 = ( ord_less @ A @ X @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_option_Some
% 0.19/0.42 thf(fact_192_fold__atLeastAtMost__nat_Opelims,axiom,
% 0.19/0.42 ! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
% 0.19/0.42 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
% 0.19/0.42 = Y )
% 0.19/0.42 => ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
% 0.19/0.42 => ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
% 0.19/0.42 => ( Y = Xc ) )
% 0.19/0.42 & ( ~ ( ord_less @ nat @ Xb @ Xa )
% 0.19/0.42 => ( Y
% 0.19/0.42 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
% 0.19/0.42 => ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_atLeastAtMost_nat.pelims
% 0.19/0.42 thf(fact_193_fold__atLeastAtMost__nat_Opsimps,axiom,
% 0.19/0.42 ! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
% 0.19/0.42 ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
% 0.19/0.42 => ( ( ( ord_less @ nat @ B2 @ A3 )
% 0.19/0.42 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 0.19/0.42 = Acc2 ) )
% 0.19/0.42 & ( ~ ( ord_less @ nat @ B2 @ A3 )
% 0.19/0.42 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 0.19/0.42 = ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_atLeastAtMost_nat.psimps
% 0.19/0.42 thf(fact_194_in__measure,axiom,
% 0.19/0.42 ! [A: $tType,X: A,Y: A,F: A > nat] :
% 0.19/0.42 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F ) )
% 0.19/0.42 = ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % in_measure
% 0.19/0.42 thf(fact_195_less__add__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_semidom @ A )
% 0.19/0.42 => ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_add_one
% 0.19/0.42 thf(fact_196_add__mono1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono1
% 0.19/0.42 thf(fact_197_lift__collapse,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( heap_Time_lift @ B @ A )
% 0.19/0.42 = ( ^ [F2: B > A,X3: B] : ( heap_Time_return @ A @ ( F2 @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lift_collapse
% 0.19/0.42 thf(fact_198_add__lessD1,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.19/0.42 => ( ord_less @ nat @ I @ K ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_lessD1
% 0.19/0.42 thf(fact_199_add__less__mono,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ I @ J )
% 0.19/0.42 => ( ( ord_less @ nat @ K @ L )
% 0.19/0.42 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_mono
% 0.19/0.42 thf(fact_200_not__add__less1,axiom,
% 0.19/0.42 ! [I: nat,J: nat] :
% 0.19/0.42 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 0.19/0.42
% 0.19/0.42 % not_add_less1
% 0.19/0.42 thf(fact_201_less__option__None,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: option @ A] :
% 0.19/0.42 ~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_option_None
% 0.19/0.42 thf(fact_202_less__option__None__Some__code,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_option_None_Some_code
% 0.19/0.42 thf(fact_203_less__option__None__Some,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_option_None_Some
% 0.19/0.42 thf(fact_204_less__option__None__is__Some,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: option @ A] :
% 0.19/0.42 ( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
% 0.19/0.42 => ? [Z3: A] :
% 0.19/0.42 ( X
% 0.19/0.42 = ( some @ A @ Z3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_option_None_is_Some
% 0.19/0.42 thf(fact_205_lift__def,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( heap_Time_lift @ A @ B )
% 0.19/0.42 = ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lift_def
% 0.19/0.42 thf(fact_206_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( set_fo6178422350223883121st_nat @ A )
% 0.19/0.42 = ( ^ [F2: nat > A > A,A7: nat,B5: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A7 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A7 @ Acc3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_atLeastAtMost_nat.simps
% 0.19/0.42 thf(fact_207_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.19/0.42 ! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
% 0.19/0.42 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
% 0.19/0.42 = Y )
% 0.19/0.42 => ( ( ( ord_less @ nat @ Xb @ Xa )
% 0.19/0.42 => ( Y = Xc ) )
% 0.19/0.42 & ( ~ ( ord_less @ nat @ Xb @ Xa )
% 0.19/0.42 => ( Y
% 0.19/0.42 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % fold_atLeastAtMost_nat.elims
% 0.19/0.42 thf(fact_208_less__add__eq__less,axiom,
% 0.19/0.42 ! [K: nat,L: nat,M2: nat,N: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ K @ L )
% 0.19/0.42 => ( ( ( plus_plus @ nat @ M2 @ L )
% 0.19/0.42 = ( plus_plus @ nat @ K @ N ) )
% 0.19/0.42 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_add_eq_less
% 0.19/0.42 thf(fact_209_trans__less__add2,axiom,
% 0.19/0.42 ! [I: nat,J: nat,M2: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ I @ J )
% 0.19/0.42 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % trans_less_add2
% 0.19/0.42 thf(fact_210_trans__less__add1,axiom,
% 0.19/0.42 ! [I: nat,J: nat,M2: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ I @ J )
% 0.19/0.42 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % trans_less_add1
% 0.19/0.42 thf(fact_211_add__less__mono1,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ I @ J )
% 0.19/0.42 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_mono1
% 0.19/0.42 thf(fact_212_not__add__less2,axiom,
% 0.19/0.42 ! [J: nat,I: nat] :
% 0.19/0.42 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 0.19/0.42
% 0.19/0.42 % not_add_less2
% 0.19/0.42 thf(fact_213_less__numeral__extra_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_numeral_extra(4)
% 0.19/0.42 thf(fact_214_mlex__iff,axiom,
% 0.19/0.42 ! [A: $tType,X: A,Y: A,F: A > nat,R5: set @ ( product_prod @ A @ A )] :
% 0.19/0.42 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) )
% 0.19/0.42 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 0.19/0.42 | ( ( ( F @ X )
% 0.19/0.42 = ( F @ Y ) )
% 0.19/0.42 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % mlex_iff
% 0.19/0.42 thf(fact_215_mlex__less,axiom,
% 0.19/0.42 ! [A: $tType,F: A > nat,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 0.19/0.42 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 0.19/0.42 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % mlex_less
% 0.19/0.42 thf(fact_216_pairself_Opelims,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
% 0.19/0.42 ( ( ( pairself @ A @ B @ X @ Xa )
% 0.19/0.42 = Y )
% 0.19/0.42 => ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ Xa ) )
% 0.19/0.42 => ~ ! [A5: A,B3: A] :
% 0.19/0.42 ( ( Xa
% 0.19/0.42 = ( product_Pair @ A @ A @ A5 @ B3 ) )
% 0.19/0.42 => ( ( Y
% 0.19/0.42 = ( product_Pair @ B @ B @ ( X @ A5 ) @ ( X @ B3 ) ) )
% 0.19/0.42 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ ( product_Pair @ A @ A @ A5 @ B3 ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % pairself.pelims
% 0.19/0.42 thf(fact_217_execute__return,axiom,
% 0.19/0.42 ! [A: $tType,X: A] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
% 0.19/0.42 = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_return
% 0.19/0.42 thf(fact_218_same__fstI,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: A > $o,X: A,Y7: B,Y: B,R5: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.19/0.42 ( ( P @ X )
% 0.19/0.42 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R5 @ X ) )
% 0.19/0.42 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % same_fstI
% 0.19/0.42 thf(fact_219_zero__less__two,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_less_two
% 0.19/0.42 thf(fact_220_is__num__normalize_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_num_normalize(1)
% 0.19/0.42 thf(fact_221_add_Osafe__commute,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_semigroup_add @ A )
% 0.19/0.42 => ! [X: A,Y: A,A3: A,B2: A] :
% 0.19/0.42 ( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A3 )
% 0.19/0.42 => ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.safe_commute
% 0.19/0.42 thf(fact_222_bijective__def,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( bijective @ A @ B )
% 0.19/0.42 = ( ^ [R6: set @ ( product_prod @ A @ B )] :
% 0.19/0.42 ( ! [X3: A,Y3: B,Z5: B] :
% 0.19/0.42 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 )
% 0.19/0.42 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R6 ) )
% 0.19/0.42 => ( Y3 = Z5 ) )
% 0.19/0.42 & ! [X3: A,Y3: A,Z5: B] :
% 0.19/0.42 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R6 )
% 0.19/0.42 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R6 ) )
% 0.19/0.42 => ( X3 = Y3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bijective_def
% 0.19/0.42 thf(fact_223_ssubst__Pair__rhs,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,R: A,S: B,R5: set @ ( product_prod @ A @ B ),S3: B] :
% 0.19/0.42 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R5 )
% 0.19/0.42 => ( ( S3 = S )
% 0.19/0.42 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R5 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ssubst_Pair_rhs
% 0.19/0.42 thf(fact_224_add__is__0,axiom,
% 0.19/0.42 ! [M2: nat,N: nat] :
% 0.19/0.42 ( ( ( plus_plus @ nat @ M2 @ N )
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 = ( ( M2
% 0.19/0.42 = ( zero_zero @ nat ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_is_0
% 0.19/0.42 thf(fact_225_Nat_Oadd__0__right,axiom,
% 0.19/0.42 ! [M2: nat] :
% 0.19/0.42 ( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
% 0.19/0.42 = M2 ) ).
% 0.19/0.42
% 0.19/0.42 % Nat.add_0_right
% 0.19/0.42 thf(fact_226_timeFrame__zero,axiom,
% 0.19/0.42 ! [A: $tType,H3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.42 ( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H3 )
% 0.19/0.42 = H3 ) ).
% 0.19/0.42
% 0.19/0.42 % timeFrame_zero
% 0.19/0.42 thf(fact_227_Heap__Time__Monad_Obind__bind,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
% 0.19/0.42 ( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
% 0.19/0.42 = ( heap_Time_bind @ A @ B @ F
% 0.19/0.42 @ ^ [X3: A] : ( heap_Time_bind @ C @ B @ ( G @ X3 ) @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Heap_Time_Monad.bind_bind
% 0.19/0.42 thf(fact_228_not__gr__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [N: A] :
% 0.19/0.42 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 0.19/0.42 = ( N
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_gr_zero
% 0.19/0.42 thf(fact_229_add__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( monoid_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_0
% 0.19/0.42 thf(fact_230_zero__eq__add__iff__both__eq__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ( zero_zero @ A )
% 0.19/0.42 = ( plus_plus @ A @ X @ Y ) )
% 0.19/0.42 = ( ( X
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 & ( Y
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_eq_add_iff_both_eq_0
% 0.19/0.42 thf(fact_231_add__eq__0__iff__both__eq__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ X @ Y )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( ( X
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 & ( Y
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_eq_0_iff_both_eq_0
% 0.19/0.42 thf(fact_232_add__cancel__right__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 = ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( B2
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_cancel_right_right
% 0.19/0.42 thf(fact_233_add__cancel__right__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 = ( plus_plus @ A @ B2 @ A3 ) )
% 0.19/0.42 = ( B2
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_cancel_right_left
% 0.19/0.42 thf(fact_234_add__cancel__left__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = A3 )
% 0.19/0.42 = ( B2
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_cancel_left_right
% 0.19/0.42 thf(fact_235_add__cancel__left__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ B2 @ A3 )
% 0.19/0.42 = A3 )
% 0.19/0.42 = ( B2
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_cancel_left_left
% 0.19/0.42 thf(fact_236_double__zero__sym,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( zero_zero @ A )
% 0.19/0.42 = ( plus_plus @ A @ A3 @ A3 ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % double_zero_sym
% 0.19/0.42 thf(fact_237_add_Oright__neutral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( monoid_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.right_neutral
% 0.19/0.42 thf(fact_238_add__gr__0,axiom,
% 0.19/0.42 ! [M2: nat,N: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.42 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.19/0.42 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_gr_0
% 0.19/0.42 thf(fact_239_less__one,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 0.19/0.42 = ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_one
% 0.19/0.42 thf(fact_240_some__opt__eq__trivial,axiom,
% 0.19/0.42 ! [A: $tType,X: A] :
% 0.19/0.42 ( ( eps_Opt @ A
% 0.19/0.42 @ ^ [Y3: A] : ( Y3 = X ) )
% 0.19/0.42 = ( some @ A @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % some_opt_eq_trivial
% 0.19/0.42 thf(fact_241_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 0.19/0.42 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_less_double_add_iff_zero_less_single_add
% 0.19/0.42 thf(fact_242_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % double_add_less_zero_iff_single_add_less_zero
% 0.19/0.42 thf(fact_243_less__add__same__cancel2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 0.19/0.42 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_add_same_cancel2
% 0.19/0.42 thf(fact_244_less__add__same__cancel1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_add_same_cancel1
% 0.19/0.42 thf(fact_245_add__less__same__cancel2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_same_cancel2
% 0.19/0.42 thf(fact_246_add__less__same__cancel1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_same_cancel1
% 0.19/0.42 thf(fact_247_some__opt__false__trivial,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( eps_Opt @ A
% 0.19/0.42 @ ^ [Uu: A] : $false )
% 0.19/0.42 = ( none @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % some_opt_false_trivial
% 0.19/0.42 thf(fact_248_zero__reorient,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( zero @ A )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ( ( ( zero_zero @ A )
% 0.19/0.42 = X )
% 0.19/0.42 = ( X
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_reorient
% 0.19/0.42 thf(fact_249_fun_Omap__ident,axiom,
% 0.19/0.42 ! [A: $tType,D: $tType,T2: D > A] :
% 0.19/0.42 ( ( comp @ A @ A @ D
% 0.19/0.42 @ ^ [X3: A] : X3
% 0.19/0.42 @ T2 )
% 0.19/0.42 = T2 ) ).
% 0.19/0.42
% 0.19/0.42 % fun.map_ident
% 0.19/0.42 thf(fact_250_option_Odisc__eq__case_I2_J,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 = ( case_option @ $o @ A @ $false
% 0.19/0.42 @ ^ [Uu: A] : $true
% 0.19/0.42 @ Option ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.disc_eq_case(2)
% 0.19/0.42 thf(fact_251_option_Odisc__eq__case_I1_J,axiom,
% 0.19/0.42 ! [A: $tType,Option: option @ A] :
% 0.19/0.42 ( ( Option
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( case_option @ $o @ A @ $true
% 0.19/0.42 @ ^ [Uu: A] : $false
% 0.19/0.42 @ Option ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.disc_eq_case(1)
% 0.19/0.42 thf(fact_252_option_Ocase__distrib,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,H3: B > C,F1: B,F22: A > B,Option: option @ A] :
% 0.19/0.42 ( ( H3 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 0.19/0.42 = ( case_option @ C @ A @ ( H3 @ F1 )
% 0.19/0.42 @ ^ [X3: A] : ( H3 @ ( F22 @ X3 ) )
% 0.19/0.42 @ Option ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.case_distrib
% 0.19/0.42 thf(fact_253_effect__LetI,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: B,N: nat] :
% 0.19/0.42 ( ( X = T2 )
% 0.19/0.42 => ( ( heap_Time_effect @ B @ ( F @ X ) @ H3 @ H4 @ R @ N )
% 0.19/0.42 => ( heap_Time_effect @ B @ ( F @ T2 ) @ H3 @ H4 @ R @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_LetI
% 0.19/0.42 thf(fact_254_distrib__if__bind,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D3: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
% 0.19/0.42 ( ( B2
% 0.19/0.42 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F )
% 0.19/0.42 = ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
% 0.19/0.42 & ( ~ B2
% 0.19/0.42 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F )
% 0.19/0.42 = ( heap_Time_bind @ B @ A @ D3 @ F ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % distrib_if_bind
% 0.19/0.42 thf(fact_255_success__LetI,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit] :
% 0.19/0.42 ( ( X = T2 )
% 0.19/0.42 => ( ( heap_Time_success @ B @ ( F @ X ) @ H3 )
% 0.19/0.42 => ( heap_Time_success @ B @ ( F @ T2 ) @ H3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % success_LetI
% 0.19/0.42 thf(fact_256_curry__K,axiom,
% 0.19/0.42 ! [B: $tType,C: $tType,A: $tType,C2: C] :
% 0.19/0.42 ( ( product_curry @ A @ B @ C
% 0.19/0.42 @ ^ [X3: product_prod @ A @ B] : C2 )
% 0.19/0.42 = ( ^ [X3: A,Y3: B] : C2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % curry_K
% 0.19/0.42 thf(fact_257_case__optionE,axiom,
% 0.19/0.42 ! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
% 0.19/0.42 ( ( case_option @ $o @ A @ P @ Q @ X )
% 0.19/0.42 => ( ( ( X
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 => ~ P )
% 0.19/0.42 => ~ ! [Y4: A] :
% 0.19/0.42 ( ( X
% 0.19/0.42 = ( some @ A @ Y4 ) )
% 0.19/0.42 => ~ ( Q @ Y4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % case_optionE
% 0.19/0.42 thf(fact_258_curry__def,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType] :
% 0.19/0.42 ( ( product_curry @ A @ B @ C )
% 0.19/0.42 = ( ^ [C3: ( product_prod @ A @ B ) > C,X3: A,Y3: B] : ( C3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % curry_def
% 0.19/0.42 thf(fact_259_combine__options__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( combine_options @ A )
% 0.19/0.42 = ( ^ [F2: A > A > A,X3: option @ A,Y3: option @ A] :
% 0.19/0.42 ( case_option @ ( option @ A ) @ A @ Y3
% 0.19/0.42 @ ^ [Z5: A] :
% 0.19/0.42 ( case_option @ ( option @ A ) @ A @ ( some @ A @ Z5 )
% 0.19/0.42 @ ^ [Aa2: A] : ( some @ A @ ( F2 @ Z5 @ Aa2 ) )
% 0.19/0.42 @ Y3 )
% 0.19/0.42 @ X3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % combine_options_def
% 0.19/0.42 thf(fact_260_less__option__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ( ( ord_less @ ( option @ A ) )
% 0.19/0.42 = ( ^ [X3: option @ A] :
% 0.19/0.42 ( case_option @ $o @ A @ $false
% 0.19/0.42 @ ^ [Y3: A] :
% 0.19/0.42 ( case_option @ $o @ A @ $true
% 0.19/0.42 @ ^ [Z5: A] : ( ord_less @ A @ Z5 @ Y3 )
% 0.19/0.42 @ X3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_option_def
% 0.19/0.42 thf(fact_261_zero__less__iff__neq__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [N: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 0.19/0.42 = ( N
% 0.19/0.42 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_less_iff_neq_zero
% 0.19/0.42 thf(fact_262_gr__implies__not__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [M2: A,N: A] :
% 0.19/0.42 ( ( ord_less @ A @ M2 @ N )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % gr_implies_not_zero
% 0.19/0.42 thf(fact_263_not__less__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [N: A] :
% 0.19/0.42 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_less_zero
% 0.19/0.42 thf(fact_264_gr__zeroI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [N: A] :
% 0.19/0.42 ( ( N
% 0.19/0.42 != ( zero_zero @ A ) )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % gr_zeroI
% 0.19/0.42 thf(fact_265_add_Ogroup__left__neutral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.group_left_neutral
% 0.19/0.42 thf(fact_266_add_Ocomm__neutral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comm_monoid_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.comm_neutral
% 0.19/0.42 thf(fact_267_comm__monoid__add__class_Oadd__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( comm_monoid_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % comm_monoid_add_class.add_0
% 0.19/0.42 thf(fact_268_zero__neq__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( zero_neq_one @ A )
% 0.19/0.42 => ( ( zero_zero @ A )
% 0.19/0.42 != ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_neq_one
% 0.19/0.42 thf(fact_269_array__make,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ( ( array_new @ A )
% 0.19/0.42 = ( ^ [N5: nat,X3: A] :
% 0.19/0.42 ( array_make @ A @ N5
% 0.19/0.42 @ ^ [Uu: nat] : X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % array_make
% 0.19/0.42 thf(fact_270_plus__nat_Oadd__0,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.42 = N ) ).
% 0.19/0.42
% 0.19/0.42 % plus_nat.add_0
% 0.19/0.42 thf(fact_271_add__eq__self__zero,axiom,
% 0.19/0.42 ! [M2: nat,N: nat] :
% 0.19/0.42 ( ( ( plus_plus @ nat @ M2 @ N )
% 0.19/0.42 = M2 )
% 0.19/0.42 => ( N
% 0.19/0.42 = ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_eq_self_zero
% 0.19/0.42 thf(fact_272_less__numeral__extra_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_numeral_extra(1)
% 0.19/0.42 thf(fact_273_ureturn__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_ureturn @ A )
% 0.19/0.42 = ( ^ [X3: A] :
% 0.19/0.42 ( heap_Time_heap @ A
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ureturn_def
% 0.19/0.42 thf(fact_274_raise__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_raise @ A )
% 0.19/0.42 = ( ^ [S4: list @ char] :
% 0.19/0.42 ( heap_Time_Heap2 @ A
% 0.19/0.42 @ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % raise_def
% 0.19/0.42 thf(fact_275_bind__lift,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
% 0.19/0.42 ( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
% 0.19/0.42 = ( heap_Time_bind @ B @ A @ F
% 0.19/0.42 @ ^ [X3: B] : ( heap_Time_return @ A @ ( G @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_lift
% 0.19/0.42 thf(fact_276_guard__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_guard @ A )
% 0.19/0.42 = ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.19/0.42 ( heap_Time_Heap2 @ A
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % guard_def
% 0.19/0.42 thf(fact_277_pos__add__strict,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( strict7427464778891057005id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less @ A @ B2 @ C2 )
% 0.19/0.42 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % pos_add_strict
% 0.19/0.42 thf(fact_278_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.42 => ~ ! [C4: A] :
% 0.19/0.42 ( ( B2
% 0.19/0.42 = ( plus_plus @ A @ A3 @ C4 ) )
% 0.19/0.42 => ( C4
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % canonically_ordered_monoid_add_class.lessE
% 0.19/0.42 thf(fact_279_add__pos__pos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_pos_pos
% 0.19/0.42 thf(fact_280_add__neg__neg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_neg_neg
% 0.19/0.42 thf(fact_281_add__less__zeroD,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 0.19/0.42 | ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_zeroD
% 0.19/0.42 thf(fact_282_zero__less__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( zero_less_one @ A )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_less_one
% 0.19/0.42 thf(fact_283_not__one__less__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_one_less_zero
% 0.19/0.42 thf(fact_284_execute__ureturn,axiom,
% 0.19/0.42 ! [A: $tType,X: A] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
% 0.19/0.42 = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_ureturn
% 0.19/0.42 thf(fact_285_less__imp__add__positive,axiom,
% 0.19/0.42 ! [I: nat,J: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ I @ J )
% 0.19/0.42 => ? [K5: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K5 )
% 0.19/0.42 & ( ( plus_plus @ nat @ I @ K5 )
% 0.19/0.42 = J ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_imp_add_positive
% 0.19/0.42 thf(fact_286_pairself_Osimps,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: A > B,A3: A,B2: A] :
% 0.19/0.42 ( ( pairself @ A @ B @ F @ ( product_Pair @ A @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % pairself.simps
% 0.19/0.42 thf(fact_287_pairself_Oelims,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
% 0.19/0.42 ( ( ( pairself @ A @ B @ X @ Xa )
% 0.19/0.42 = Y )
% 0.19/0.42 => ~ ! [A5: A,B3: A] :
% 0.19/0.42 ( ( Xa
% 0.19/0.42 = ( product_Pair @ A @ A @ A5 @ B3 ) )
% 0.19/0.42 => ( Y
% 0.19/0.42 != ( product_Pair @ B @ B @ ( X @ A5 ) @ ( X @ B3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % pairself.elims
% 0.19/0.42 thf(fact_288_return__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_return @ A )
% 0.19/0.42 = ( ^ [X3: A] :
% 0.19/0.42 ( heap_Time_heap @ A
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % return_def
% 0.19/0.42 thf(fact_289_effect__ureturnI,axiom,
% 0.19/0.42 ! [A: $tType,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A] :
% 0.19/0.42 ( ( H3 = H4 )
% 0.19/0.42 => ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 @ H4 @ X @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_ureturnI
% 0.19/0.42 thf(fact_290_effect__ureturnE,axiom,
% 0.19/0.42 ! [A: $tType,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 @ H4 @ R @ N )
% 0.19/0.42 => ~ ( ( R = X )
% 0.19/0.42 => ( ( H4 = H3 )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_ureturnE
% 0.19/0.42 thf(fact_291_tap__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap_Time_tap @ A )
% 0.19/0.42 = ( ^ [F2: ( heap_ext @ product_unit ) > A] :
% 0.19/0.42 ( heap_Time_Heap2 @ A
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F2 @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % tap_def
% 0.19/0.42 thf(fact_292_double__eq__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % double_eq_0_iff
% 0.19/0.42 thf(fact_293_return__bind,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 0.19/0.42 ( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F )
% 0.19/0.42 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 0.19/0.42 @ ^ [Uu: product_unit] : ( F @ X ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % return_bind
% 0.19/0.42 thf(fact_294_bind__return,axiom,
% 0.19/0.42 ! [A: $tType,F: heap_Time_Heap @ A] :
% 0.19/0.42 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
% 0.19/0.42 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 0.19/0.42 @ ^ [Uu: product_unit] : F ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_return
% 0.19/0.42 thf(fact_295_add__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( semiri1453513574482234551roduct @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( B2
% 0.19/0.42 = ( plus_plus @ A @ B2 @ A3 ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_0_iff
% 0.19/0.42 thf(fact_296_verit__sum__simplify,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % verit_sum_simplify
% 0.19/0.42 thf(fact_297_dbl__inc__simps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dbl_inc_simps(2)
% 0.19/0.42 thf(fact_298_disjE__realizer2,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,P: $o,Q: A > $o,X: option @ A,R5: B > $o,F: B,G: A > B] :
% 0.19/0.42 ( ( case_option @ $o @ A @ P @ Q @ X )
% 0.19/0.42 => ( ( P
% 0.19/0.42 => ( R5 @ F ) )
% 0.19/0.42 => ( ! [Q2: A] :
% 0.19/0.42 ( ( Q @ Q2 )
% 0.19/0.42 => ( R5 @ ( G @ Q2 ) ) )
% 0.19/0.42 => ( R5 @ ( case_option @ B @ A @ F @ G @ X ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % disjE_realizer2
% 0.19/0.42 thf(fact_299_set__to__map__def,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( set_to_map @ B @ A )
% 0.19/0.42 = ( ^ [S5: set @ ( product_prod @ B @ A ),K4: B] :
% 0.19/0.42 ( eps_Opt @ A
% 0.19/0.42 @ ^ [V2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K4 @ V2 ) @ S5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % set_to_map_def
% 0.19/0.42 thf(fact_300_K__record__comp,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
% 0.19/0.42 ( ( comp @ C @ B @ A
% 0.19/0.42 @ ^ [X3: C] : C2
% 0.19/0.42 @ F )
% 0.19/0.42 = ( ^ [X3: A] : C2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % K_record_comp
% 0.19/0.42 thf(fact_301_pred__equals__eq2,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.19/0.42 ( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 ) )
% 0.19/0.42 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) ) )
% 0.19/0.42 = ( R5 = S2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % pred_equals_eq2
% 0.19/0.42 thf(fact_302_dbl__inc__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( neg_numeral_dbl_inc @ A )
% 0.19/0.42 = ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dbl_inc_def
% 0.19/0.42 thf(fact_303_zero__natural_Orsp,axiom,
% 0.19/0.42 ( ( zero_zero @ nat )
% 0.19/0.42 = ( zero_zero @ nat ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_natural.rsp
% 0.19/0.42 thf(fact_304_fails__wait,axiom,
% 0.19/0.42 ! [X: nat,H3: heap_ext @ product_unit] :
% 0.19/0.42 ~ ( time_fails @ product_unit @ ( heap_Time_wait @ X ) @ H3 ) ).
% 0.19/0.42
% 0.19/0.42 % fails_wait
% 0.19/0.42 thf(fact_305_Euclid__induct,axiom,
% 0.19/0.42 ! [P: nat > nat > $o,A3: nat,B2: nat] :
% 0.19/0.42 ( ! [A5: nat,B3: nat] :
% 0.19/0.42 ( ( P @ A5 @ B3 )
% 0.19/0.42 = ( P @ B3 @ A5 ) )
% 0.19/0.42 => ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
% 0.19/0.42 => ( ! [A5: nat,B3: nat] :
% 0.19/0.42 ( ( P @ A5 @ B3 )
% 0.19/0.42 => ( P @ A5 @ ( plus_plus @ nat @ A5 @ B3 ) ) )
% 0.19/0.42 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Euclid_induct
% 0.19/0.42 thf(fact_306_lookup__chain,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType] :
% 0.19/0.42 ( ( heap @ B )
% 0.19/0.42 => ! [R: ref @ B,F: heap_Time_Heap @ A] :
% 0.19/0.42 ( ( heap_Time_bind @ B @ A @ ( ref_lookup @ B @ R )
% 0.19/0.42 @ ^ [Uu: B] : F )
% 0.19/0.42 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 0.19/0.42 @ ^ [Uu: product_unit] : F ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lookup_chain
% 0.19/0.42 thf(fact_307_Ints__odd__less__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_odd_less_0
% 0.19/0.42 thf(fact_308_num__of__nat__plus__distrib,axiom,
% 0.19/0.42 ! [M2: nat,N: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.19/0.42 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.42 => ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.42 = ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % num_of_nat_plus_distrib
% 0.19/0.42 thf(fact_309_divides__aux__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.42 => ! [Q3: A,R: A] :
% 0.19/0.42 ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R ) )
% 0.19/0.42 = ( R
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % divides_aux_eq
% 0.19/0.42 thf(fact_310_sgn__pos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( sgn_sgn @ A @ A3 )
% 0.19/0.42 = ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_pos
% 0.19/0.42 thf(fact_311_abs__add__one__gt__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_add_one_gt_zero
% 0.19/0.42 thf(fact_312_map__option__case,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( map_option @ B @ A )
% 0.19/0.42 = ( ^ [F2: B > A] :
% 0.19/0.42 ( case_option @ ( option @ A ) @ B @ ( none @ A )
% 0.19/0.42 @ ^ [X3: B] : ( some @ A @ ( F2 @ X3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option_case
% 0.19/0.42 thf(fact_313_add__neg__numeral__special_I7_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_neg_numeral_special(7)
% 0.19/0.42 thf(fact_314_add__neg__numeral__special_I8_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_neg_numeral_special(8)
% 0.19/0.42 thf(fact_315_dbl__dec__simps_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dbl_dec_simps(3)
% 0.19/0.42 thf(fact_316_neg__equal__iff__equal,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ( uminus_uminus @ A @ A3 )
% 0.19/0.42 = ( uminus_uminus @ A @ B2 ) )
% 0.19/0.42 = ( A3 = B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_equal_iff_equal
% 0.19/0.42 thf(fact_317_add_Oinverse__inverse,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.inverse_inverse
% 0.19/0.42 thf(fact_318_abs__idempotent,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
% 0.19/0.42 = ( abs_abs @ A @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_idempotent
% 0.19/0.42 thf(fact_319_neg__equal__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( uminus_uminus @ A @ A3 )
% 0.19/0.42 = A3 )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_equal_zero
% 0.19/0.42 thf(fact_320_equal__neg__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 = ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % equal_neg_zero
% 0.19/0.42 thf(fact_321_neg__equal__0__iff__equal,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( uminus_uminus @ A @ A3 )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_equal_0_iff_equal
% 0.19/0.42 thf(fact_322_neg__0__equal__iff__equal,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( zero_zero @ A )
% 0.19/0.42 = ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ( zero_zero @ A )
% 0.19/0.42 = A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_0_equal_iff_equal
% 0.19/0.42 thf(fact_323_add_Oinverse__neutral,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.inverse_neutral
% 0.19/0.42 thf(fact_324_neg__less__iff__less,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_less_iff_less
% 0.19/0.42 thf(fact_325_minus__add__distrib,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % minus_add_distrib
% 0.19/0.42 thf(fact_326_minus__add__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = B2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % minus_add_cancel
% 0.19/0.42 thf(fact_327_add__minus__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
% 0.19/0.42 = B2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_minus_cancel
% 0.19/0.42 thf(fact_328_abs__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_zero
% 0.19/0.42 thf(fact_329_abs__eq__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( abs_abs @ A @ A3 )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_eq_0
% 0.19/0.42 thf(fact_330_abs__0__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( zero_zero @ A )
% 0.19/0.42 = ( abs_abs @ A @ A3 ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_0_eq
% 0.19/0.42 thf(fact_331_abs__add__abs,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
% 0.19/0.42 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_add_abs
% 0.19/0.42 thf(fact_332_abs__1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( idom_abs_sgn @ A )
% 0.19/0.42 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_1
% 0.19/0.42 thf(fact_333_abs__minus__cancel,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( abs_abs @ A @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_minus_cancel
% 0.19/0.42 thf(fact_334_sgn__1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( idom_abs_sgn @ A )
% 0.19/0.42 => ( ( sgn_sgn @ A @ ( one_one @ A ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_1
% 0.19/0.42 thf(fact_335_map__option__eq__Some,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: B > A,Xo: option @ B,Y: A] :
% 0.19/0.42 ( ( ( map_option @ B @ A @ F @ Xo )
% 0.19/0.42 = ( some @ A @ Y ) )
% 0.19/0.42 = ( ? [Z5: B] :
% 0.19/0.42 ( ( Xo
% 0.19/0.42 = ( some @ B @ Z5 ) )
% 0.19/0.42 & ( ( F @ Z5 )
% 0.19/0.42 = Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option_eq_Some
% 0.19/0.42 thf(fact_336_None__eq__map__option__iff,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: B > A,X: option @ B] :
% 0.19/0.42 ( ( ( none @ A )
% 0.19/0.42 = ( map_option @ B @ A @ F @ X ) )
% 0.19/0.42 = ( X
% 0.19/0.42 = ( none @ B ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % None_eq_map_option_iff
% 0.19/0.42 thf(fact_337_map__option__is__None,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: B > A,Opt: option @ B] :
% 0.19/0.42 ( ( ( map_option @ B @ A @ F @ Opt )
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( Opt
% 0.19/0.42 = ( none @ B ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option_is_None
% 0.19/0.42 thf(fact_338_option_Omap__disc__iff,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: A > B,A3: option @ A] :
% 0.19/0.42 ( ( ( map_option @ A @ B @ F @ A3 )
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( none @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.map_disc_iff
% 0.19/0.42 thf(fact_339_neg__less__0__iff__less,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_less_0_iff_less
% 0.19/0.42 thf(fact_340_neg__0__less__iff__less,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_0_less_iff_less
% 0.19/0.42 thf(fact_341_neg__less__pos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.19/0.42 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_less_pos
% 0.19/0.42 thf(fact_342_less__neg__neg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_neg_neg
% 0.19/0.42 thf(fact_343_ab__left__minus,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ab_left_minus
% 0.19/0.42 thf(fact_344_add_Oright__inverse,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.right_inverse
% 0.19/0.42 thf(fact_345_zero__less__abs__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_less_abs_iff
% 0.19/0.42 thf(fact_346_abs__neg__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_neg_one
% 0.19/0.42 thf(fact_347_case__map__option,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,C: $tType,G: A,H3: B > A,F: C > B,X: option @ C] :
% 0.19/0.42 ( ( case_option @ A @ B @ G @ H3 @ ( map_option @ C @ B @ F @ X ) )
% 0.19/0.42 = ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H3 @ F ) @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % case_map_option
% 0.19/0.42 thf(fact_348_dbl__inc__simps_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.42 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dbl_inc_simps(4)
% 0.19/0.42 thf(fact_349_abs__sgn__eq__1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 != ( zero_zero @ A ) )
% 0.19/0.42 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_sgn_eq_1
% 0.19/0.42 thf(fact_350_dbl__dec__simps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
% 0.19/0.42 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dbl_dec_simps(2)
% 0.19/0.42 thf(fact_351_sgn__neg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( sgn_sgn @ A @ A3 )
% 0.19/0.42 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_neg
% 0.19/0.42 thf(fact_352_minus__equation__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ( uminus_uminus @ A @ A3 )
% 0.19/0.42 = B2 )
% 0.19/0.42 = ( ( uminus_uminus @ A @ B2 )
% 0.19/0.42 = A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % minus_equation_iff
% 0.19/0.42 thf(fact_353_equation__minus__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 = ( uminus_uminus @ A @ B2 ) )
% 0.19/0.42 = ( B2
% 0.19/0.42 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % equation_minus_iff
% 0.19/0.42 thf(fact_354_same__sgn__abs__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ( sgn_sgn @ A @ B2 )
% 0.19/0.42 = ( sgn_sgn @ A @ A3 ) )
% 0.19/0.42 => ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % same_sgn_abs_add
% 0.19/0.42 thf(fact_355_sgn__minus__1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( idom_abs_sgn @ A )
% 0.19/0.42 => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.42 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_minus_1
% 0.19/0.42 thf(fact_356_option_Omap__ident,axiom,
% 0.19/0.42 ! [A: $tType,T2: option @ A] :
% 0.19/0.42 ( ( map_option @ A @ A
% 0.19/0.42 @ ^ [X3: A] : X3
% 0.19/0.42 @ T2 )
% 0.19/0.42 = T2 ) ).
% 0.19/0.42
% 0.19/0.42 % option.map_ident
% 0.19/0.42 thf(fact_357_abs__of__neg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( abs_abs @ A @ A3 )
% 0.19/0.42 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_of_neg
% 0.19/0.42 thf(fact_358_abs__sgn__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( A3
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.19/0.42 = ( zero_zero @ A ) ) )
% 0.19/0.42 & ( ( A3
% 0.19/0.42 != ( zero_zero @ A ) )
% 0.19/0.42 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_sgn_eq
% 0.19/0.42 thf(fact_359_abs__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 0.19/0.42 = ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_one
% 0.19/0.42 thf(fact_360_map__option_Ocompositionality,axiom,
% 0.19/0.42 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Option: option @ A] :
% 0.19/0.42 ( ( map_option @ B @ C @ F @ ( map_option @ A @ B @ G @ Option ) )
% 0.19/0.42 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Option ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option.compositionality
% 0.19/0.42 thf(fact_361_option_Omap__comp,axiom,
% 0.19/0.42 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: option @ A] :
% 0.19/0.42 ( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F @ V ) )
% 0.19/0.42 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.map_comp
% 0.19/0.42 thf(fact_362_map__option_Ocomp,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 0.19/0.42 ( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F ) @ ( map_option @ A @ B @ G ) )
% 0.19/0.42 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option.comp
% 0.19/0.42 thf(fact_363_minus__less__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.19/0.42 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % minus_less_iff
% 0.19/0.42 thf(fact_364_less__minus__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.19/0.42 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_minus_iff
% 0.19/0.42 thf(fact_365_is__num__normalize_I8_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( neg_numeral @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % is_num_normalize(8)
% 0.19/0.42 thf(fact_366_add_Oinverse__distrib__swap,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.inverse_distrib_swap
% 0.19/0.42 thf(fact_367_group__cancel_Oneg1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_group_add @ A )
% 0.19/0.42 => ! [A4: A,K: A,A3: A] :
% 0.19/0.42 ( ( A4
% 0.19/0.42 = ( plus_plus @ A @ K @ A3 ) )
% 0.19/0.42 => ( ( uminus_uminus @ A @ A4 )
% 0.19/0.42 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group_cancel.neg1
% 0.19/0.42 thf(fact_368_one__neq__neg__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ring_char_0 @ A )
% 0.19/0.42 => ( ( one_one @ A )
% 0.19/0.42 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % one_neq_neg_one
% 0.19/0.42 thf(fact_369_map__option__cong,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > B,G: A > B] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ! [A5: A] :
% 0.19/0.42 ( ( Y
% 0.19/0.42 = ( some @ A @ A5 ) )
% 0.19/0.42 => ( ( F @ A5 )
% 0.19/0.42 = ( G @ A5 ) ) )
% 0.19/0.42 => ( ( map_option @ A @ B @ F @ X )
% 0.19/0.42 = ( map_option @ A @ B @ G @ Y ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option_cong
% 0.19/0.42 thf(fact_370_option_Osimps_I9_J,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,F: A > B,X22: A] :
% 0.19/0.42 ( ( map_option @ A @ B @ F @ ( some @ A @ X22 ) )
% 0.19/0.42 = ( some @ B @ ( F @ X22 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(9)
% 0.19/0.42 thf(fact_371_option_Osimps_I8_J,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: A > B] :
% 0.19/0.42 ( ( map_option @ A @ B @ F @ ( none @ A ) )
% 0.19/0.42 = ( none @ B ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.simps(8)
% 0.19/0.42 thf(fact_372_same__sgn__sgn__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ( sgn_sgn @ A @ B2 )
% 0.19/0.42 = ( sgn_sgn @ A @ A3 ) )
% 0.19/0.42 => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % same_sgn_sgn_add
% 0.19/0.42 thf(fact_373_Ints__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ring_1 @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_add
% 0.19/0.42 thf(fact_374_success__lookupI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [R: ref @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( ref_lookup @ A @ R ) @ H3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % success_lookupI
% 0.19/0.42 thf(fact_375_Ints__1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ring_1 @ A )
% 0.19/0.42 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_1
% 0.19/0.42 thf(fact_376_Ints__nonzero__abs__less1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
% 0.19/0.42 => ( X
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_nonzero_abs_less1
% 0.19/0.42 thf(fact_377_sgn__if,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ( sgn_sgn @ A )
% 0.19/0.42 = ( ^ [X3: A] :
% 0.19/0.42 ( if @ A
% 0.19/0.42 @ ( X3
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 @ ( zero_zero @ A )
% 0.19/0.42 @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_if
% 0.19/0.42 thf(fact_378_sgn__1__neg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( sgn_sgn @ A @ A3 )
% 0.19/0.42 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.42 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_1_neg
% 0.19/0.42 thf(fact_379_abs__not__less__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_not_less_zero
% 0.19/0.42 thf(fact_380_abs__of__pos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( abs_abs @ A @ A3 )
% 0.19/0.42 = A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_of_pos
% 0.19/0.42 thf(fact_381_option_Orec__o__map,axiom,
% 0.19/0.42 ! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F: A > B] :
% 0.19/0.42 ( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F ) )
% 0.19/0.42 = ( rec_option @ C @ A @ G
% 0.19/0.42 @ ^ [X3: A] : ( Ga @ ( F @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.rec_o_map
% 0.19/0.42 thf(fact_382_neg__eq__iff__add__eq__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ( uminus_uminus @ A @ A3 )
% 0.19/0.42 = B2 )
% 0.19/0.42 = ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_eq_iff_add_eq_0
% 0.19/0.42 thf(fact_383_eq__neg__iff__add__eq__0,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 = ( uminus_uminus @ A @ B2 ) )
% 0.19/0.42 = ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % eq_neg_iff_add_eq_0
% 0.19/0.42 thf(fact_384_add_Oinverse__unique,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 => ( ( uminus_uminus @ A @ A3 )
% 0.19/0.42 = B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.inverse_unique
% 0.19/0.42 thf(fact_385_ab__group__add__class_Oab__left__minus,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ab_group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.19/0.42 = ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ab_group_add_class.ab_left_minus
% 0.19/0.42 thf(fact_386_add__eq__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( B2
% 0.19/0.42 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_eq_0_iff
% 0.19/0.42 thf(fact_387_zero__neq__neg__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ring_char_0 @ A )
% 0.19/0.42 => ( ( zero_zero @ A )
% 0.19/0.42 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_neq_neg_one
% 0.19/0.42 thf(fact_388_less__minus__one__simps_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_minus_one_simps(4)
% 0.19/0.42 thf(fact_389_less__minus__one__simps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_minus_one_simps(2)
% 0.19/0.42 thf(fact_390_option_Omap__sel,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,A3: option @ A,F: A > B] :
% 0.19/0.42 ( ( A3
% 0.19/0.42 != ( none @ A ) )
% 0.19/0.42 => ( ( the2 @ B @ ( map_option @ A @ B @ F @ A3 ) )
% 0.19/0.42 = ( F @ ( the2 @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % option.map_sel
% 0.19/0.42 thf(fact_391_Ints__double__eq__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ring_char_0 @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( ( plus_plus @ A @ A3 @ A3 )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_double_eq_0_iff
% 0.19/0.42 thf(fact_392_less__minus__one__simps_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_minus_one_simps(3)
% 0.19/0.42 thf(fact_393_less__minus__one__simps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_minus_one_simps(1)
% 0.19/0.42 thf(fact_394_sgn__1__pos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ( sgn_sgn @ A @ A3 )
% 0.19/0.42 = ( one_one @ A ) )
% 0.19/0.42 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % sgn_1_pos
% 0.19/0.42 thf(fact_395_Ints__odd__nonzero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ring_char_0 @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
% 0.19/0.42 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_odd_nonzero
% 0.19/0.42 thf(fact_396_map__option__o__empty,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,F: C > B] :
% 0.19/0.42 ( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F )
% 0.19/0.42 @ ^ [X3: A] : ( none @ C ) )
% 0.19/0.42 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option_o_empty
% 0.19/0.42 thf(fact_397_execute__lookup,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [R: ref @ A,H3: heap_ext @ product_unit] :
% 0.19/0.42 ( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R ) @ H3 )
% 0.19/0.42 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( ref_get @ A @ H3 @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % execute_lookup
% 0.19/0.42 thf(fact_398_change__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ( ( ref_change @ A )
% 0.19/0.42 = ( ^ [F2: A > A,R3: ref @ A] :
% 0.19/0.42 ( heap_Time_bind @ A @ A @ ( ref_lookup @ A @ R3 )
% 0.19/0.42 @ ^ [X3: A] :
% 0.19/0.42 ( heap_Time_bind @ product_unit @ A @ ( ref_update @ A @ R3 @ ( F2 @ X3 ) )
% 0.19/0.42 @ ^ [Uu: product_unit] : ( heap_Time_return @ A @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % change_def
% 0.19/0.42 thf(fact_399_fails__assert_H,axiom,
% 0.19/0.42 ! [P: $o,H3: heap_ext @ product_unit] :
% 0.19/0.42 ( ( time_fails @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
% 0.19/0.42 = ( ~ P ) ) ).
% 0.19/0.42
% 0.19/0.42 % fails_assert'
% 0.19/0.42 thf(fact_400_effect__lookupI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,X: A,R: ref @ A,N: nat] :
% 0.19/0.42 ( ( H4 = H3 )
% 0.19/0.42 => ( ( X
% 0.19/0.42 = ( ref_get @ A @ H3 @ R ) )
% 0.19/0.42 => ( ( N
% 0.19/0.42 = ( one_one @ nat ) )
% 0.19/0.42 => ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H3 @ H4 @ X @ N ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_lookupI
% 0.19/0.42 thf(fact_401_effect__lookupE,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [R: ref @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A,N: nat] :
% 0.19/0.42 ( ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H3 @ H4 @ X @ N )
% 0.19/0.42 => ~ ( ( H4 = H3 )
% 0.19/0.42 => ( ( X
% 0.19/0.42 = ( ref_get @ A @ H3 @ R ) )
% 0.19/0.42 => ( N
% 0.19/0.42 != ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % effect_lookupE
% 0.19/0.42 thf(fact_402_add_Ogroup__axioms,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( group_add @ A )
% 0.19/0.42 => ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add.group_axioms
% 0.19/0.42 thf(fact_403_num__of__nat__double,axiom,
% 0.19/0.42 ! [N: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.42 => ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
% 0.19/0.42 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % num_of_nat_double
% 0.19/0.42 thf(fact_404_Ints__nonzero__abs__ge1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( X
% 0.19/0.42 != ( zero_zero @ A ) )
% 0.19/0.42 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_nonzero_abs_ge1
% 0.19/0.42 thf(fact_405_lookup__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ( ( ref_lookup @ A )
% 0.19/0.42 = ( ^ [R3: ref @ A] :
% 0.19/0.42 ( heap_Time_tap @ A
% 0.19/0.42 @ ^ [H: heap_ext @ product_unit] : ( ref_get @ A @ H @ R3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % lookup_def
% 0.19/0.42 thf(fact_406_map__conv__bind__option,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType] :
% 0.19/0.42 ( ( map_option @ B @ A )
% 0.19/0.42 = ( ^ [F2: B > A,X3: option @ B] : ( bind @ B @ A @ X3 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_conv_bind_option
% 0.19/0.42 thf(fact_407_nat__add__left__cancel__le,axiom,
% 0.19/0.42 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.42 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_add_left_cancel_le
% 0.19/0.42 thf(fact_408_less__eq__option__None__code,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_option_None_code
% 0.19/0.42 thf(fact_409_bind__assoc,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,C: $tType,X: option @ C,F: C > ( option @ B ),G: B > ( option @ A )] :
% 0.19/0.42 ( ( bind @ B @ A @ ( bind @ C @ B @ X @ F ) @ G )
% 0.19/0.42 = ( bind @ C @ A @ X
% 0.19/0.42 @ ^ [Y3: C] : ( bind @ B @ A @ ( F @ Y3 ) @ G ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_assoc
% 0.19/0.42 thf(fact_410_le__zero__eq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [N: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 0.19/0.42 = ( N
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_zero_eq
% 0.19/0.42 thf(fact_411_add__le__cancel__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_cancel_left
% 0.19/0.42 thf(fact_412_add__le__cancel__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_cancel_right
% 0.19/0.42 thf(fact_413_neg__le__iff__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_le_iff_le
% 0.19/0.42 thf(fact_414_less__eq__option__Some,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 0.19/0.42 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_option_Some
% 0.19/0.42 thf(fact_415_less__eq__option__Some__None,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( none @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_option_Some_None
% 0.19/0.42 thf(fact_416_bind__runit,axiom,
% 0.19/0.42 ! [A: $tType,X: option @ A] :
% 0.19/0.42 ( ( bind @ A @ A @ X @ ( some @ A ) )
% 0.19/0.42 = X ) ).
% 0.19/0.42
% 0.19/0.42 % bind_runit
% 0.19/0.42 thf(fact_417_bind__rzero,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ B] :
% 0.19/0.42 ( ( bind @ B @ A @ X
% 0.19/0.42 @ ^ [X3: B] : ( none @ A ) )
% 0.19/0.42 = ( none @ A ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_rzero
% 0.19/0.42 thf(fact_418_add__le__same__cancel1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [B2: A,A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_same_cancel1
% 0.19/0.42 thf(fact_419_add__le__same__cancel2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_same_cancel2
% 0.19/0.42 thf(fact_420_le__add__same__cancel1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_add_same_cancel1
% 0.19/0.42 thf(fact_421_le__add__same__cancel2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere1937475149494474687imp_le @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_add_same_cancel2
% 0.19/0.42 thf(fact_422_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % double_add_le_zero_iff_single_add_le_zero
% 0.19/0.42 thf(fact_423_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_le_double_add_iff_zero_le_single_add
% 0.19/0.42 thf(fact_424_neg__0__le__iff__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_0_le_iff_le
% 0.19/0.42 thf(fact_425_neg__le__0__iff__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_le_0_iff_le
% 0.19/0.42 thf(fact_426_less__eq__neg__nonpos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_neg_nonpos
% 0.19/0.42 thf(fact_427_neg__less__eq__nonneg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord5086331880401160121up_add @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % neg_less_eq_nonneg
% 0.19/0.42 thf(fact_428_abs__of__nonneg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( abs_abs @ A @ A3 )
% 0.19/0.42 = A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_of_nonneg
% 0.19/0.42 thf(fact_429_abs__le__self__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
% 0.19/0.42 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_le_self_iff
% 0.19/0.42 thf(fact_430_abs__le__zero__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.42 = ( A3
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_le_zero_iff
% 0.19/0.42 thf(fact_431_abs__of__nonpos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( abs_abs @ A @ A3 )
% 0.19/0.42 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_of_nonpos
% 0.19/0.42 thf(fact_432_ord__eq__le__eq__trans,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ord @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.42 ( ( A3 = B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.42 => ( ( C2 = D3 )
% 0.19/0.42 => ( ord_less_eq @ A @ A3 @ D3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ord_eq_le_eq_trans
% 0.19/0.42 thf(fact_433_bind__option__cong__code,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B )] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ( bind @ A @ B @ X @ F )
% 0.19/0.42 = ( bind @ A @ B @ Y @ F ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_option_cong_code
% 0.19/0.42 thf(fact_434_group_Oleft__cancel,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( ( F @ A3 @ B2 )
% 0.19/0.42 = ( F @ A3 @ C2 ) )
% 0.19/0.42 = ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.left_cancel
% 0.19/0.42 thf(fact_435_group_Oleft__inverse,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( F @ ( Inverse @ A3 ) @ A3 )
% 0.19/0.42 = Z2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.left_inverse
% 0.19/0.42 thf(fact_436_group_Oright__cancel,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,B2: A,A3: A,C2: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( ( F @ B2 @ A3 )
% 0.19/0.42 = ( F @ C2 @ A3 ) )
% 0.19/0.42 = ( B2 = C2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.right_cancel
% 0.19/0.42 thf(fact_437_group_Oright__inverse,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( F @ A3 @ ( Inverse @ A3 ) )
% 0.19/0.42 = Z2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.right_inverse
% 0.19/0.42 thf(fact_438_group_Oinverse__unique,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( ( F @ A3 @ B2 )
% 0.19/0.42 = Z2 )
% 0.19/0.42 => ( ( Inverse @ A3 )
% 0.19/0.42 = B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.inverse_unique
% 0.19/0.42 thf(fact_439_group_Oinverse__inverse,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( Inverse @ ( Inverse @ A3 ) )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.inverse_inverse
% 0.19/0.42 thf(fact_440_group_Oinverse__neutral,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( Inverse @ Z2 )
% 0.19/0.42 = Z2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.inverse_neutral
% 0.19/0.42 thf(fact_441_group_Ogroup__left__neutral,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( F @ Z2 @ A3 )
% 0.19/0.42 = A3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.group_left_neutral
% 0.19/0.42 thf(fact_442_group_Oinverse__distrib__swap,axiom,
% 0.19/0.42 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
% 0.19/0.42 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.19/0.42 => ( ( Inverse @ ( F @ A3 @ B2 ) )
% 0.19/0.42 = ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % group.inverse_distrib_swap
% 0.19/0.42 thf(fact_443_accp__subset,axiom,
% 0.19/0.42 ! [A: $tType,R12: A > A > $o,R23: A > A > $o] :
% 0.19/0.42 ( ( ord_less_eq @ ( A > A > $o ) @ R12 @ R23 )
% 0.19/0.42 => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R23 ) @ ( accp @ A @ R12 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp_subset
% 0.19/0.42 thf(fact_444_le__some__optE,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [M2: A,X: option @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M2 ) @ X )
% 0.19/0.42 => ~ ! [M4: A] :
% 0.19/0.42 ( ( X
% 0.19/0.42 = ( some @ A @ M4 ) )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ M2 @ M4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_some_optE
% 0.19/0.42 thf(fact_445_pred__subset__eq2,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.19/0.42 ( ( ord_less_eq @ ( A > B > $o )
% 0.19/0.42 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
% 0.19/0.42 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
% 0.19/0.42 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % pred_subset_eq2
% 0.19/0.42 thf(fact_446_less__eq__option__def,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ( ( ord_less_eq @ ( option @ A ) )
% 0.19/0.42 = ( ^ [X3: option @ A,Y3: option @ A] :
% 0.19/0.42 ( case_option @ $o @ A @ $true
% 0.19/0.42 @ ^ [Z5: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z5 ) @ Y3 )
% 0.19/0.42 @ X3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_option_def
% 0.19/0.42 thf(fact_447_zero__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_le
% 0.19/0.42 thf(fact_448_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( ord_less_eq @ A @ I @ J )
% 0.19/0.42 & ( K = L ) )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_semiring(3)
% 0.19/0.42 thf(fact_449_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( I = J )
% 0.19/0.42 & ( ord_less_eq @ A @ K @ L ) )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_semiring(2)
% 0.19/0.42 thf(fact_450_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( ord_less_eq @ A @ I @ J )
% 0.19/0.42 & ( ord_less_eq @ A @ K @ L ) )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_semiring(1)
% 0.19/0.42 thf(fact_451_add__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ C2 @ D3 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono
% 0.19/0.42 thf(fact_452_add__left__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_left_mono
% 0.19/0.42 thf(fact_453_less__eqE,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ~ ! [C4: A] :
% 0.19/0.42 ( B2
% 0.19/0.42 != ( plus_plus @ A @ A3 @ C4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eqE
% 0.19/0.42 thf(fact_454_add__right__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6658533253407199908up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_right_mono
% 0.19/0.42 thf(fact_455_le__iff__add,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( canoni5634975068530333245id_add @ A )
% 0.19/0.42 => ( ( ord_less_eq @ A )
% 0.19/0.42 = ( ^ [A7: A,B5: A] :
% 0.19/0.42 ? [C3: A] :
% 0.19/0.42 ( B5
% 0.19/0.42 = ( plus_plus @ A @ A7 @ C3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_iff_add
% 0.19/0.42 thf(fact_456_add__le__imp__le__left,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.19/0.42 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_imp_le_left
% 0.19/0.42 thf(fact_457_add__le__imp__le__right,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere2412721322843649153imp_le @ A )
% 0.19/0.42 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.42 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_imp_le_right
% 0.19/0.42 thf(fact_458_accp__subset__induct,axiom,
% 0.19/0.42 ! [A: $tType,D4: A > $o,R5: A > A > $o,X: A,P: A > $o] :
% 0.19/0.42 ( ( ord_less_eq @ ( A > $o ) @ D4 @ ( accp @ A @ R5 ) )
% 0.19/0.42 => ( ! [X2: A,Z3: A] :
% 0.19/0.42 ( ( D4 @ X2 )
% 0.19/0.42 => ( ( R5 @ Z3 @ X2 )
% 0.19/0.42 => ( D4 @ Z3 ) ) )
% 0.19/0.42 => ( ( D4 @ X )
% 0.19/0.42 => ( ! [X2: A] :
% 0.19/0.42 ( ( D4 @ X2 )
% 0.19/0.42 => ( ! [Z6: A] :
% 0.19/0.42 ( ( R5 @ Z6 @ X2 )
% 0.19/0.42 => ( P @ Z6 ) )
% 0.19/0.42 => ( P @ X2 ) ) )
% 0.19/0.42 => ( P @ X ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % accp_subset_induct
% 0.19/0.42 thf(fact_459_le__numeral__extra_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_numeral_extra(4)
% 0.19/0.42 thf(fact_460_subrelI,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 0.19/0.42 ( ! [X2: A,Y4: B] :
% 0.19/0.42 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 0.19/0.42 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) )
% 0.19/0.42 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 0.19/0.42
% 0.19/0.42 % subrelI
% 0.19/0.42 thf(fact_461_le__minus__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.19/0.42 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_minus_iff
% 0.19/0.42 thf(fact_462_minus__le__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.19/0.42 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % minus_le_iff
% 0.19/0.42 thf(fact_463_le__imp__neg__le,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordered_ab_group_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_imp_neg_le
% 0.19/0.42 thf(fact_464_exists__leI,axiom,
% 0.19/0.42 ! [N: nat,P: nat > $o] :
% 0.19/0.42 ( ( ! [N7: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ N7 @ N )
% 0.19/0.42 => ~ ( P @ N7 ) )
% 0.19/0.42 => ( P @ N ) )
% 0.19/0.42 => ? [N4: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ N4 @ N )
% 0.19/0.42 & ( P @ N4 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % exists_leI
% 0.19/0.42 thf(fact_465_nat__le__iff__add,axiom,
% 0.19/0.42 ( ( ord_less_eq @ nat )
% 0.19/0.42 = ( ^ [M: nat,N5: nat] :
% 0.19/0.42 ? [K4: nat] :
% 0.19/0.42 ( N5
% 0.19/0.42 = ( plus_plus @ nat @ M @ K4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_le_iff_add
% 0.19/0.42 thf(fact_466_trans__le__add2,axiom,
% 0.19/0.42 ! [I: nat,J: nat,M2: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.42 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % trans_le_add2
% 0.19/0.42 thf(fact_467_trans__le__add1,axiom,
% 0.19/0.42 ! [I: nat,J: nat,M2: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.42 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % trans_le_add1
% 0.19/0.42 thf(fact_468_add__le__mono1,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_mono1
% 0.19/0.42 thf(fact_469_add__le__mono,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.42 => ( ( ord_less_eq @ nat @ K @ L )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_mono
% 0.19/0.42 thf(fact_470_le__Suc__ex,axiom,
% 0.19/0.42 ! [K: nat,L: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ K @ L )
% 0.19/0.42 => ? [N3: nat] :
% 0.19/0.42 ( L
% 0.19/0.42 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_Suc_ex
% 0.19/0.42 thf(fact_471_add__leD2,axiom,
% 0.19/0.42 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 0.19/0.42 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_leD2
% 0.19/0.42 thf(fact_472_add__leD1,axiom,
% 0.19/0.42 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 0.19/0.42 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_leD1
% 0.19/0.42 thf(fact_473_le__add2,axiom,
% 0.19/0.42 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_add2
% 0.19/0.42 thf(fact_474_le__add1,axiom,
% 0.19/0.42 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_add1
% 0.19/0.42 thf(fact_475_add__leE,axiom,
% 0.19/0.42 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 0.19/0.42 => ~ ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.42 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_leE
% 0.19/0.42 thf(fact_476_abs__le__D1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_le_D1
% 0.19/0.42 thf(fact_477_abs__ge__self,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_ge_self
% 0.19/0.42 thf(fact_478_less__eq__option__None,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_option_None
% 0.19/0.42 thf(fact_479_less__eq__option__None__is__None,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( preorder @ A )
% 0.19/0.42 => ! [X: option @ A] :
% 0.19/0.42 ( ( ord_less_eq @ ( option @ A ) @ X @ ( none @ A ) )
% 0.19/0.42 => ( X
% 0.19/0.42 = ( none @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % less_eq_option_None_is_None
% 0.19/0.42 thf(fact_480_bind_Obind__lunit,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: A,F: A > ( option @ B )] :
% 0.19/0.42 ( ( bind @ A @ B @ ( some @ A @ X ) @ F )
% 0.19/0.42 = ( F @ X ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind.bind_lunit
% 0.19/0.42 thf(fact_481_Option_Obind__cong,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.19/0.42 ( ( X = Y )
% 0.19/0.42 => ( ! [A5: A] :
% 0.19/0.42 ( ( Y
% 0.19/0.42 = ( some @ A @ A5 ) )
% 0.19/0.42 => ( ( F @ A5 )
% 0.19/0.42 = ( G @ A5 ) ) )
% 0.19/0.42 => ( ( bind @ A @ B @ X @ F )
% 0.19/0.42 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Option.bind_cong
% 0.19/0.42 thf(fact_482_bind__eq__Some__conv,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A ),X: A] :
% 0.19/0.42 ( ( ( bind @ B @ A @ F @ G )
% 0.19/0.42 = ( some @ A @ X ) )
% 0.19/0.42 = ( ? [Y3: B] :
% 0.19/0.42 ( ( F
% 0.19/0.42 = ( some @ B @ Y3 ) )
% 0.19/0.42 & ( ( G @ Y3 )
% 0.19/0.42 = ( some @ A @ X ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_eq_Some_conv
% 0.19/0.42 thf(fact_483_bind_Obind__lzero,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,F: A > ( option @ B )] :
% 0.19/0.42 ( ( bind @ A @ B @ ( none @ A ) @ F )
% 0.19/0.42 = ( none @ B ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind.bind_lzero
% 0.19/0.42 thf(fact_484_map__option__bind,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,F: B > A,X: option @ C,G: C > ( option @ B )] :
% 0.19/0.42 ( ( map_option @ B @ A @ F @ ( bind @ C @ B @ X @ G ) )
% 0.19/0.42 = ( bind @ C @ A @ X @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F ) @ G ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_option_bind
% 0.19/0.42 thf(fact_485_add__decreasing,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_decreasing
% 0.19/0.42 thf(fact_486_add__increasing,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.42 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_increasing
% 0.19/0.42 thf(fact_487_add__decreasing2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_decreasing2
% 0.19/0.42 thf(fact_488_add__increasing2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [C2: A,B2: A,A3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.42 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_increasing2
% 0.19/0.42 thf(fact_489_add__nonneg__nonneg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_nonneg_nonneg
% 0.19/0.42 thf(fact_490_add__nonpos__nonpos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_nonpos_nonpos
% 0.19/0.42 thf(fact_491_add__nonneg__eq__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.19/0.42 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.19/0.42 => ( ( ( plus_plus @ A @ X @ Y )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( ( X
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 & ( Y
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_nonneg_eq_0_iff
% 0.19/0.42 thf(fact_492_add__nonpos__eq__0__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ( plus_plus @ A @ X @ Y )
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 = ( ( X
% 0.19/0.42 = ( zero_zero @ A ) )
% 0.19/0.42 & ( Y
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_nonpos_eq_0_iff
% 0.19/0.42 thf(fact_493_not__one__le__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % not_one_le_zero
% 0.19/0.42 thf(fact_494_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord181362715937106298miring @ A )
% 0.19/0.42 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % linordered_nonzero_semiring_class.zero_le_one
% 0.19/0.42 thf(fact_495_zero__less__one__class_Ozero__le__one,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( zero_less_one @ A )
% 0.19/0.42 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % zero_less_one_class.zero_le_one
% 0.19/0.42 thf(fact_496_add__mono__thms__linordered__field_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( ord_less_eq @ A @ I @ J )
% 0.19/0.42 & ( ord_less @ A @ K @ L ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_field(4)
% 0.19/0.42 thf(fact_497_add__mono__thms__linordered__field_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [I: A,J: A,K: A,L: A] :
% 0.19/0.42 ( ( ( ord_less @ A @ I @ J )
% 0.19/0.42 & ( ord_less_eq @ A @ K @ L ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_mono_thms_linordered_field(3)
% 0.19/0.42 thf(fact_498_add__le__less__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ( ord_less @ A @ C2 @ D3 )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_le_less_mono
% 0.19/0.42 thf(fact_499_add__less__le__mono,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere580206878836729694up_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ C2 @ D3 )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_less_le_mono
% 0.19/0.42 thf(fact_500_le__minus__one__simps_I2_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_minus_one_simps(2)
% 0.19/0.42 thf(fact_501_le__minus__one__simps_I4_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_minus_one_simps(4)
% 0.19/0.42 thf(fact_502_abs__ge__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_ge_zero
% 0.19/0.42 thf(fact_503_abs__triangle__ineq,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_triangle_ineq
% 0.19/0.42 thf(fact_504_abs__leI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_leI
% 0.19/0.42 thf(fact_505_abs__le__D2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_le_D2
% 0.19/0.42 thf(fact_506_abs__le__iff,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 0.19/0.42 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.42 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_le_iff
% 0.19/0.42 thf(fact_507_abs__ge__minus__self,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_ge_minus_self
% 0.19/0.42 thf(fact_508_mono__nat__linear__lb,axiom,
% 0.19/0.42 ! [F: nat > nat,M2: nat,K: nat] :
% 0.19/0.42 ( ! [M5: nat,N3: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ M5 @ N3 )
% 0.19/0.42 => ( ord_less @ nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
% 0.19/0.42 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % mono_nat_linear_lb
% 0.19/0.42 thf(fact_509_nat__geq__1__eq__neqz,axiom,
% 0.19/0.42 ! [X: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
% 0.19/0.42 = ( X
% 0.19/0.42 != ( zero_zero @ nat ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_geq_1_eq_neqz
% 0.19/0.42 thf(fact_510_bind__split__asm,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F: B > ( option @ A )] :
% 0.19/0.42 ( ( P @ ( bind @ B @ A @ M2 @ F ) )
% 0.19/0.42 = ( ~ ( ( ( M2
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 & ~ ( P @ ( none @ A ) ) )
% 0.19/0.42 | ? [X3: B] :
% 0.19/0.42 ( ( M2
% 0.19/0.42 = ( some @ B @ X3 ) )
% 0.19/0.42 & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_split_asm
% 0.19/0.42 thf(fact_511_bind__split,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F: B > ( option @ A )] :
% 0.19/0.42 ( ( P @ ( bind @ B @ A @ M2 @ F ) )
% 0.19/0.42 = ( ( ( M2
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 => ( P @ ( none @ A ) ) )
% 0.19/0.42 & ! [V2: B] :
% 0.19/0.42 ( ( M2
% 0.19/0.42 = ( some @ B @ V2 ) )
% 0.19/0.42 => ( P @ ( F @ V2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_split
% 0.19/0.42 thf(fact_512_mlex__leq,axiom,
% 0.19/0.42 ! [A: $tType,F: A > nat,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
% 0.19/0.42 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
% 0.19/0.42 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % mlex_leq
% 0.19/0.42 thf(fact_513_bind__eq__None__conv,axiom,
% 0.19/0.42 ! [B: $tType,A: $tType,A3: option @ B,F: B > ( option @ A )] :
% 0.19/0.42 ( ( ( bind @ B @ A @ A3 @ F )
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( ( A3
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 | ( ( F @ ( the2 @ B @ A3 ) )
% 0.19/0.42 = ( none @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_eq_None_conv
% 0.19/0.42 thf(fact_514_bind__map__option,axiom,
% 0.19/0.42 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: option @ C,G: B > ( option @ A )] :
% 0.19/0.42 ( ( bind @ B @ A @ ( map_option @ C @ B @ F @ X ) @ G )
% 0.19/0.42 = ( bind @ C @ A @ X @ ( comp @ B @ ( option @ A ) @ C @ G @ F ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % bind_map_option
% 0.19/0.42 thf(fact_515_add__neg__nonpos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_neg_nonpos
% 0.19/0.42 thf(fact_516_add__nonneg__pos,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_nonneg_pos
% 0.19/0.42 thf(fact_517_add__nonpos__neg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_nonpos_neg
% 0.19/0.42 thf(fact_518_add__pos__nonneg,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere6911136660526730532id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.42 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_pos_nonneg
% 0.19/0.42 thf(fact_519_add__strict__increasing,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere8940638589300402666id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 0.19/0.42 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_strict_increasing
% 0.19/0.42 thf(fact_520_add__strict__increasing2,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere8940638589300402666id_add @ A )
% 0.19/0.42 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.42 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.42 => ( ( ord_less @ A @ B2 @ C2 )
% 0.19/0.42 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % add_strict_increasing2
% 0.19/0.42 thf(fact_521_discrete,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.42 => ( ( ord_less @ A )
% 0.19/0.42 = ( ^ [A7: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % discrete
% 0.19/0.42 thf(fact_522_le__minus__one__simps_I1_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_minus_one_simps(1)
% 0.19/0.42 thf(fact_523_le__minus__one__simps_I3_J,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % le_minus_one_simps(3)
% 0.19/0.42 thf(fact_524_dense__eq0__I,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 & ( dense_linorder @ A ) )
% 0.19/0.42 => ! [X: A] :
% 0.19/0.42 ( ! [E3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
% 0.19/0.42 => ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E3 ) )
% 0.19/0.42 => ( X
% 0.19/0.42 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % dense_eq0_I
% 0.19/0.42 thf(fact_525_abs__minus__le__zero,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.42 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % abs_minus_le_zero
% 0.19/0.42 thf(fact_526_success__changeI,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( heap @ A )
% 0.19/0.42 => ! [F: A > A,R: ref @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( ref_change @ A @ F @ R ) @ H3 ) ) ).
% 0.19/0.42
% 0.19/0.42 % success_changeI
% 0.19/0.42 thf(fact_527_semiring__norm_I6_J,axiom,
% 0.19/0.42 ! [M2: num,N: num] :
% 0.19/0.42 ( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
% 0.19/0.42 = ( bit0 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % semiring_norm(6)
% 0.19/0.42 thf(fact_528_field__le__epsilon,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_field @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ! [E3: A] :
% 0.19/0.42 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
% 0.19/0.42 => ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E3 ) ) )
% 0.19/0.42 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % field_le_epsilon
% 0.19/0.42 thf(fact_529_ex__has__greatest__nat__lemma,axiom,
% 0.19/0.42 ! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
% 0.19/0.42 ( ( P @ K )
% 0.19/0.42 => ( ! [X2: A] :
% 0.19/0.42 ( ( P @ X2 )
% 0.19/0.42 => ? [Y6: A] :
% 0.19/0.42 ( ( P @ Y6 )
% 0.19/0.42 & ~ ( ord_less_eq @ nat @ ( F @ Y6 ) @ ( F @ X2 ) ) ) )
% 0.19/0.42 => ? [Y4: A] :
% 0.19/0.42 ( ( P @ Y4 )
% 0.19/0.42 & ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % ex_has_greatest_nat_lemma
% 0.19/0.42 thf(fact_530_map__comp__def,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType,B: $tType] :
% 0.19/0.42 ( ( map_comp @ B @ C @ A )
% 0.19/0.42 = ( ^ [F2: B > ( option @ C ),G2: A > ( option @ B ),K4: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F2 @ ( G2 @ K4 ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_comp_def
% 0.19/0.42 thf(fact_531_convex__bound__lt,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linord715952674999750819strict @ A )
% 0.19/0.42 => ! [X: A,A3: A,Y: A,U: A,V: A] :
% 0.19/0.42 ( ( ord_less @ A @ X @ A3 )
% 0.19/0.42 => ( ( ord_less @ A @ Y @ A3 )
% 0.19/0.42 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 0.19/0.42 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 0.19/0.42 => ( ( ( plus_plus @ A @ U @ V )
% 0.19/0.42 = ( one_one @ A ) )
% 0.19/0.42 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % convex_bound_lt
% 0.19/0.42 thf(fact_532_map__comp__empty_I1_J,axiom,
% 0.19/0.42 ! [C: $tType,B: $tType,A: $tType,M2: C > ( option @ B )] :
% 0.19/0.42 ( ( map_comp @ C @ B @ A @ M2
% 0.19/0.42 @ ^ [X3: A] : ( none @ C ) )
% 0.19/0.42 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_comp_empty(1)
% 0.19/0.42 thf(fact_533_map__comp__empty_I2_J,axiom,
% 0.19/0.42 ! [B: $tType,D: $tType,C: $tType,M2: C > ( option @ B )] :
% 0.19/0.42 ( ( map_comp @ B @ D @ C
% 0.19/0.42 @ ^ [X3: B] : ( none @ D )
% 0.19/0.42 @ M2 )
% 0.19/0.42 = ( ^ [X3: C] : ( none @ D ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_comp_empty(2)
% 0.19/0.42 thf(fact_534_map__comp__None__iff,axiom,
% 0.19/0.42 ! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K: C] :
% 0.19/0.42 ( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K )
% 0.19/0.42 = ( none @ A ) )
% 0.19/0.42 = ( ( ( M22 @ K )
% 0.19/0.42 = ( none @ B ) )
% 0.19/0.42 | ? [K6: B] :
% 0.19/0.42 ( ( ( M22 @ K )
% 0.19/0.42 = ( some @ B @ K6 ) )
% 0.19/0.42 & ( ( M1 @ K6 )
% 0.19/0.42 = ( none @ A ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % map_comp_None_iff
% 0.19/0.42 thf(fact_535_Ints__eq__abs__less1,axiom,
% 0.19/0.42 ! [A: $tType] :
% 0.19/0.42 ( ( linordered_idom @ A )
% 0.19/0.42 => ! [X: A,Y: A] :
% 0.19/0.42 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
% 0.19/0.42 => ( ( X = Y )
% 0.19/0.42 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % Ints_eq_abs_less1
% 0.19/0.42 thf(fact_536_nat0__intermed__int__val,axiom,
% 0.19/0.42 ! [N: nat,F: nat > int,K: int] :
% 0.19/0.42 ( ! [I2: nat] :
% 0.19/0.42 ( ( ord_less @ nat @ I2 @ N )
% 0.19/0.42 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F @ I2 ) ) ) @ ( one_one @ int ) ) )
% 0.19/0.42 => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
% 0.19/0.42 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 0.19/0.42 => ? [I2: nat] :
% 0.19/0.42 ( ( ord_less_eq @ nat @ I2 @ N )
% 0.19/0.42 & ( ( F @ I2 )
% 0.19/0.42 = K ) ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat0_intermed_int_val
% 0.19/0.42 thf(fact_537_diff__diff__left,axiom,
% 0.19/0.42 ! [I: nat,J: nat,K: nat] :
% 0.19/0.42 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 0.19/0.42 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % diff_diff_left
% 0.19/0.42 thf(fact_538_nat__1__eq__mult__iff,axiom,
% 0.19/0.42 ! [M2: nat,N: nat] :
% 0.19/0.42 ( ( ( one_one @ nat )
% 0.19/0.42 = ( times_times @ nat @ M2 @ N ) )
% 0.19/0.42 = ( ( M2
% 0.19/0.42 = ( one_one @ nat ) )
% 0.19/0.42 & ( N
% 0.19/0.42 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.42
% 0.19/0.42 % nat_1_eq_mult_iff
% 0.19/0.42 thf(fact_539_nat__mult__eq__1__iff,axiom,
% 0.19/0.42 ! [M2: nat,N: nat] :
% 0.19/0.42 ( ( ( times_times @ nat @ M2 @ N )
% 0.19/0.43 = ( one_one @ nat ) )
% 0.19/0.43 = ( ( M2
% 0.19/0.43 = ( one_one @ nat ) )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_mult_eq_1_iff
% 0.19/0.43 thf(fact_540_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ A3 )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % cancel_comm_monoid_add_class.diff_cancel
% 0.19/0.43 thf(fact_541_diff__zero,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_zero
% 0.19/0.43 thf(fact_542_zero__diff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_monoid_diff @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_diff
% 0.19/0.43 thf(fact_543_diff__0__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_0_right
% 0.19/0.43 thf(fact_544_diff__self,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ A3 )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_self
% 0.19/0.43 thf(fact_545_mult_Oright__neutral,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( monoid_mult @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult.right_neutral
% 0.19/0.43 thf(fact_546_mult__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( monoid_mult @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_1
% 0.19/0.43 thf(fact_547_add__diff__cancel__right_H,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel2418104881723323429up_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_cancel_right'
% 0.19/0.43 thf(fact_548_add__diff__cancel__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel2418104881723323429up_add @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_cancel_right
% 0.19/0.43 thf(fact_549_add__diff__cancel__left_H,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel2418104881723323429up_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
% 0.19/0.43 = B2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_cancel_left'
% 0.19/0.43 thf(fact_550_add__diff__cancel__left,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel2418104881723323429up_add @ A )
% 0.19/0.43 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_cancel_left
% 0.19/0.43 thf(fact_551_diff__add__cancel,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_cancel
% 0.19/0.43 thf(fact_552_add__diff__cancel,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_cancel
% 0.19/0.43 thf(fact_553_minus__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % minus_diff_eq
% 0.19/0.43 thf(fact_554_Nat_Oadd__diff__assoc,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.43 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.add_diff_assoc
% 0.19/0.43 thf(fact_555_Nat_Oadd__diff__assoc2,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.19/0.43 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.add_diff_assoc2
% 0.19/0.43 thf(fact_556_Nat_Odiff__diff__right,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.43 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.diff_diff_right
% 0.19/0.43 thf(fact_557_zle__add1__eq__le,axiom,
% 0.19/0.43 ! [W: int,Z2: int] :
% 0.19/0.43 ( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
% 0.19/0.43 = ( ord_less_eq @ int @ W @ Z2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % zle_add1_eq_le
% 0.19/0.43 thf(fact_558_map__comp__simps_I2_J,axiom,
% 0.19/0.43 ! [B: $tType,C: $tType,A: $tType,M22: B > ( option @ A ),K: B,K7: A,M1: A > ( option @ C )] :
% 0.19/0.43 ( ( ( M22 @ K )
% 0.19/0.43 = ( some @ A @ K7 ) )
% 0.19/0.43 => ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K )
% 0.19/0.43 = ( M1 @ K7 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % map_comp_simps(2)
% 0.19/0.43 thf(fact_559_map__comp__simps_I1_J,axiom,
% 0.19/0.43 ! [B: $tType,A: $tType,C: $tType,M22: B > ( option @ A ),K: B,M1: A > ( option @ C )] :
% 0.19/0.43 ( ( ( M22 @ K )
% 0.19/0.43 = ( none @ A ) )
% 0.19/0.43 => ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K )
% 0.19/0.43 = ( none @ C ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % map_comp_simps(1)
% 0.19/0.43 thf(fact_560_diff__ge__0__iff__ge,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_ge_0_iff_ge
% 0.19/0.43 thf(fact_561_zero__comp__diff__simps_I1_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_comp_diff_simps(1)
% 0.19/0.43 thf(fact_562_diff__gt__0__iff__gt,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_gt_0_iff_gt
% 0.19/0.43 thf(fact_563_zero__comp__diff__simps_I2_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_comp_diff_simps(2)
% 0.19/0.43 thf(fact_564_le__add__diff__inverse,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.43 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_add_diff_inverse
% 0.19/0.43 thf(fact_565_le__add__diff__inverse2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.43 => ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_add_diff_inverse2
% 0.19/0.43 thf(fact_566_mult__cancel__left1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_15535105094025558882visors @ A )
% 0.19/0.43 => ! [C2: A,B2: A] :
% 0.19/0.43 ( ( C2
% 0.19/0.43 = ( times_times @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( ( C2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 | ( B2
% 0.19/0.43 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_cancel_left1
% 0.19/0.43 thf(fact_567_mult__cancel__left2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_15535105094025558882visors @ A )
% 0.19/0.43 => ! [C2: A,A3: A] :
% 0.19/0.43 ( ( ( times_times @ A @ C2 @ A3 )
% 0.19/0.43 = C2 )
% 0.19/0.43 = ( ( C2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 | ( A3
% 0.19/0.43 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_cancel_left2
% 0.19/0.43 thf(fact_568_mult__cancel__right1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_15535105094025558882visors @ A )
% 0.19/0.43 => ! [C2: A,B2: A] :
% 0.19/0.43 ( ( C2
% 0.19/0.43 = ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( ( C2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 | ( B2
% 0.19/0.43 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_cancel_right1
% 0.19/0.43 thf(fact_569_mult__cancel__right2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_15535105094025558882visors @ A )
% 0.19/0.43 => ! [A3: A,C2: A] :
% 0.19/0.43 ( ( ( times_times @ A @ A3 @ C2 )
% 0.19/0.43 = C2 )
% 0.19/0.43 = ( ( C2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 | ( A3
% 0.19/0.43 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_cancel_right2
% 0.19/0.43 thf(fact_570_diff__add__zero,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_monoid_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_zero
% 0.19/0.43 thf(fact_571_diff__numeral__special_I9_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_numeral_special(9)
% 0.19/0.43 thf(fact_572_diff__0,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_0
% 0.19/0.43 thf(fact_573_mult__minus1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_1 @ A )
% 0.19/0.43 => ! [Z2: A] :
% 0.19/0.43 ( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z2 )
% 0.19/0.43 = ( uminus_uminus @ A @ Z2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_minus1
% 0.19/0.43 thf(fact_574_mult__minus1__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_1 @ A )
% 0.19/0.43 => ! [Z2: A] :
% 0.19/0.43 ( ( times_times @ A @ Z2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.43 = ( uminus_uminus @ A @ Z2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_minus1_right
% 0.19/0.43 thf(fact_575_uminus__add__conv__diff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 0.19/0.43 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % uminus_add_conv_diff
% 0.19/0.43 thf(fact_576_diff__minus__eq__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 0.19/0.43 = ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_minus_eq_add
% 0.19/0.43 thf(fact_577_diff__numeral__special_I12_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_numeral_special(12)
% 0.19/0.43 thf(fact_578_le__imp__0__less,axiom,
% 0.19/0.43 ! [Z2: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 0.19/0.43 => ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_imp_0_less
% 0.19/0.43 thf(fact_579_q__pos__lemma,axiom,
% 0.19/0.43 ! [B4: int,Q4: int,R4: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 0.19/0.43 => ( ( ord_less @ int @ R4 @ B4 )
% 0.19/0.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.19/0.43 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % q_pos_lemma
% 0.19/0.43 thf(fact_580_zdiv__mono2__lemma,axiom,
% 0.19/0.43 ! [B2: int,Q3: int,R: int,B4: int,Q4: int,R4: int] :
% 0.19/0.43 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R )
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 0.19/0.43 => ( ( ord_less @ int @ R4 @ B4 )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.19/0.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.19/0.43 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 0.19/0.43 => ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zdiv_mono2_lemma
% 0.19/0.43 thf(fact_581_zdiv__mono2__neg__lemma,axiom,
% 0.19/0.43 ! [B2: int,Q3: int,R: int,B4: int,Q4: int,R4: int] :
% 0.19/0.43 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R )
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 0.19/0.43 => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( ord_less @ int @ R @ B2 )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 0.19/0.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 0.19/0.43 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 0.19/0.43 => ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zdiv_mono2_neg_lemma
% 0.19/0.43 thf(fact_582_unique__quotient__lemma,axiom,
% 0.19/0.43 ! [B2: int,Q4: int,R4: int,Q3: int,R: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 0.19/0.43 => ( ( ord_less @ int @ R4 @ B2 )
% 0.19/0.43 => ( ( ord_less @ int @ R @ B2 )
% 0.19/0.43 => ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % unique_quotient_lemma
% 0.19/0.43 thf(fact_583_unique__quotient__lemma__neg,axiom,
% 0.19/0.43 ! [B2: int,Q4: int,R4: int,Q3: int,R: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( ord_less @ int @ B2 @ R )
% 0.19/0.43 => ( ( ord_less @ int @ B2 @ R4 )
% 0.19/0.43 => ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % unique_quotient_lemma_neg
% 0.19/0.43 thf(fact_584_int__ge__induct,axiom,
% 0.19/0.43 ! [K: int,I: int,P: int > $o] :
% 0.19/0.43 ( ( ord_less_eq @ int @ K @ I )
% 0.19/0.43 => ( ( P @ K )
% 0.19/0.43 => ( ! [I2: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ K @ I2 )
% 0.19/0.43 => ( ( P @ I2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 0.19/0.43 => ( P @ I ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_ge_induct
% 0.19/0.43 thf(fact_585_int__induct,axiom,
% 0.19/0.43 ! [P: int > $o,K: int,I: int] :
% 0.19/0.43 ( ( P @ K )
% 0.19/0.43 => ( ! [I2: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ K @ I2 )
% 0.19/0.43 => ( ( P @ I2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 0.19/0.43 => ( ! [I2: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ I2 @ K )
% 0.19/0.43 => ( ( P @ I2 )
% 0.19/0.43 => ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 0.19/0.43 => ( P @ I ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_induct
% 0.19/0.43 thf(fact_586_zless__imp__add1__zle,axiom,
% 0.19/0.43 ! [W: int,Z2: int] :
% 0.19/0.43 ( ( ord_less @ int @ W @ Z2 )
% 0.19/0.43 => ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % zless_imp_add1_zle
% 0.19/0.43 thf(fact_587_add1__zle__eq,axiom,
% 0.19/0.43 ! [W: int,Z2: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 )
% 0.19/0.43 = ( ord_less @ int @ W @ Z2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % add1_zle_eq
% 0.19/0.43 thf(fact_588_nat__eq__add__iff1,axiom,
% 0.19/0.43 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ J @ I )
% 0.19/0.43 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 )
% 0.19/0.43 = N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_eq_add_iff1
% 0.19/0.43 thf(fact_589_nat__eq__add__iff2,axiom,
% 0.19/0.43 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.43 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( M2
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_eq_add_iff2
% 0.19/0.43 thf(fact_590_nat__le__add__iff1,axiom,
% 0.19/0.43 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ J @ I )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_le_add_iff1
% 0.19/0.43 thf(fact_591_nat__le__add__iff2,axiom,
% 0.19/0.43 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_le_add_iff2
% 0.19/0.43 thf(fact_592_nat__diff__add__eq1,axiom,
% 0.19/0.43 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ J @ I )
% 0.19/0.43 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_diff_add_eq1
% 0.19/0.43 thf(fact_593_nat__diff__add__eq2,axiom,
% 0.19/0.43 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.43 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_diff_add_eq2
% 0.19/0.43 thf(fact_594_bounded__Max__nat,axiom,
% 0.19/0.43 ! [P: nat > $o,X: nat,M3: nat] :
% 0.19/0.43 ( ( P @ X )
% 0.19/0.43 => ( ! [X2: nat] :
% 0.19/0.43 ( ( P @ X2 )
% 0.19/0.43 => ( ord_less_eq @ nat @ X2 @ M3 ) )
% 0.19/0.43 => ~ ! [M5: nat] :
% 0.19/0.43 ( ( P @ M5 )
% 0.19/0.43 => ~ ! [X7: nat] :
% 0.19/0.43 ( ( P @ X7 )
% 0.19/0.43 => ( ord_less_eq @ nat @ X7 @ M5 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bounded_Max_nat
% 0.19/0.43 thf(fact_595_subset__Collect__conv,axiom,
% 0.19/0.43 ! [A: $tType,S2: set @ A,P: A > $o] :
% 0.19/0.43 ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( collect @ A @ P ) )
% 0.19/0.43 = ( ! [X3: A] :
% 0.19/0.43 ( ( member @ A @ X3 @ S2 )
% 0.19/0.43 => ( P @ X3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % subset_Collect_conv
% 0.19/0.43 thf(fact_596_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel2418104881723323429up_add @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
% 0.19/0.43 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % cancel_ab_semigroup_add_class.diff_right_commute
% 0.19/0.43 thf(fact_597_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_semigroup_mult @ A )
% 0.19/0.43 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
% 0.19/0.43 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ab_semigroup_mult_class.mult.left_commute
% 0.19/0.43 thf(fact_598_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_semigroup_mult @ A )
% 0.19/0.43 => ( ( times_times @ A )
% 0.19/0.43 = ( ^ [A7: A,B5: A] : ( times_times @ A @ B5 @ A7 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ab_semigroup_mult_class.mult.commute
% 0.19/0.43 thf(fact_599_mult_Oassoc,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semigroup_mult @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult.assoc
% 0.19/0.43 thf(fact_600_diff__eq__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.19/0.43 = ( minus_minus @ A @ C2 @ D3 ) )
% 0.19/0.43 => ( ( A3 = B2 )
% 0.19/0.43 = ( C2 = D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_eq_diff_eq
% 0.19/0.43 thf(fact_601_one__integer_Orsp,axiom,
% 0.19/0.43 ( ( one_one @ int )
% 0.19/0.43 = ( one_one @ int ) ) ).
% 0.19/0.43
% 0.19/0.43 % one_integer.rsp
% 0.19/0.43 thf(fact_602_int__gr__induct,axiom,
% 0.19/0.43 ! [K: int,I: int,P: int > $o] :
% 0.19/0.43 ( ( ord_less @ int @ K @ I )
% 0.19/0.43 => ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 0.19/0.43 => ( ! [I2: int] :
% 0.19/0.43 ( ( ord_less @ int @ K @ I2 )
% 0.19/0.43 => ( ( P @ I2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 0.19/0.43 => ( P @ I ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_gr_induct
% 0.19/0.43 thf(fact_603_zless__add1__eq,axiom,
% 0.19/0.43 ! [W: int,Z2: int] :
% 0.19/0.43 ( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
% 0.19/0.43 = ( ( ord_less @ int @ W @ Z2 )
% 0.19/0.43 | ( W = Z2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zless_add1_eq
% 0.19/0.43 thf(fact_604_odd__nonzero,axiom,
% 0.19/0.43 ! [Z2: int] :
% 0.19/0.43 ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
% 0.19/0.43 != ( zero_zero @ int ) ) ).
% 0.19/0.43
% 0.19/0.43 % odd_nonzero
% 0.19/0.43 thf(fact_605_odd__less__0__iff,axiom,
% 0.19/0.43 ! [Z2: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
% 0.19/0.43 = ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % odd_less_0_iff
% 0.19/0.43 thf(fact_606_eq__add__iff1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 0.19/0.43 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 0.19/0.43 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 )
% 0.19/0.43 = D3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eq_add_iff1
% 0.19/0.43 thf(fact_607_eq__add__iff2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 0.19/0.43 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 0.19/0.43 = ( C2
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eq_add_iff2
% 0.19/0.43 thf(fact_608_square__diff__square__factored,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_ring @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
% 0.19/0.43 = ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % square_diff_square_factored
% 0.19/0.43 thf(fact_609_left__add__mult__distrib,axiom,
% 0.19/0.43 ! [I: nat,U: nat,J: nat,K: nat] :
% 0.19/0.43 ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% 0.19/0.43
% 0.19/0.43 % left_add_mult_distrib
% 0.19/0.43 thf(fact_610_nat__less__add__iff1,axiom,
% 0.19/0.43 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ J @ I )
% 0.19/0.43 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_less_add_iff1
% 0.19/0.43 thf(fact_611_nat__less__add__iff2,axiom,
% 0.19/0.43 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.43 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 0.19/0.43 = ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_less_add_iff2
% 0.19/0.43 thf(fact_612_le__add__iff1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_add_iff1
% 0.19/0.43 thf(fact_613_le__add__iff2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_add_iff2
% 0.19/0.43 thf(fact_614_less__add__iff1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 0.19/0.43 = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_add_iff1
% 0.19/0.43 thf(fact_615_less__add__iff2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 0.19/0.43 = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_add_iff2
% 0.19/0.43 thf(fact_616_square__diff__one__factored,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_1 @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
% 0.19/0.43 = ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % square_diff_one_factored
% 0.19/0.43 thf(fact_617_diff__eq__diff__less__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.19/0.43 = ( minus_minus @ A @ C2 @ D3 ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 = ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_eq_diff_less_eq
% 0.19/0.43 thf(fact_618_diff__right__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_right_mono
% 0.19/0.43 thf(fact_619_diff__left__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.43 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_left_mono
% 0.19/0.43 thf(fact_620_diff__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,D3: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ D3 @ C2 )
% 0.19/0.43 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_mono
% 0.19/0.43 thf(fact_621_eq__iff__diff__eq__0,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.19/0.43 = ( ^ [A7: A,B5: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A7 @ B5 )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eq_iff_diff_eq_0
% 0.19/0.43 thf(fact_622_diff__strict__right__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.43 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_strict_right_mono
% 0.19/0.43 thf(fact_623_diff__strict__left__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.43 ( ( ord_less @ A @ B2 @ A3 )
% 0.19/0.43 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_strict_left_mono
% 0.19/0.43 thf(fact_624_diff__eq__diff__less,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.19/0.43 = ( minus_minus @ A @ C2 @ D3 ) )
% 0.19/0.43 => ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.43 = ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_eq_diff_less
% 0.19/0.43 thf(fact_625_diff__strict__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,D3: A,C2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( ord_less @ A @ D3 @ C2 )
% 0.19/0.43 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_strict_mono
% 0.19/0.43 thf(fact_626_diff__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel2418104881723323429up_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_diff_eq
% 0.19/0.43 thf(fact_627_add__implies__diff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( cancel1802427076303600483id_add @ A )
% 0.19/0.43 => ! [C2: A,B2: A,A3: A] :
% 0.19/0.43 ( ( ( plus_plus @ A @ C2 @ B2 )
% 0.19/0.43 = A3 )
% 0.19/0.43 => ( C2
% 0.19/0.43 = ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_implies_diff
% 0.19/0.43 thf(fact_628_diff__add__eq__diff__diff__swap,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_eq_diff_diff_swap
% 0.19/0.43 thf(fact_629_diff__add__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_eq
% 0.19/0.43 thf(fact_630_diff__diff__eq2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_diff_eq2
% 0.19/0.43 thf(fact_631_add__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_eq
% 0.19/0.43 thf(fact_632_eq__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( minus_minus @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( ( plus_plus @ A @ A3 @ B2 )
% 0.19/0.43 = C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eq_diff_eq
% 0.19/0.43 thf(fact_633_diff__eq__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ( minus_minus @ A @ A3 @ B2 )
% 0.19/0.43 = C2 )
% 0.19/0.43 = ( A3
% 0.19/0.43 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_eq_eq
% 0.19/0.43 thf(fact_634_group__cancel_Osub1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_group_add @ A )
% 0.19/0.43 => ! [A4: A,K: A,A3: A,B2: A] :
% 0.19/0.43 ( ( A4
% 0.19/0.43 = ( plus_plus @ A @ K @ A3 ) )
% 0.19/0.43 => ( ( minus_minus @ A @ A4 @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % group_cancel.sub1
% 0.19/0.43 thf(fact_635_minus__diff__commute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_group_add @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
% 0.19/0.43 = ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % minus_diff_commute
% 0.19/0.43 thf(fact_636_ring__class_Oring__distribs_I2_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ring_class.ring_distribs(2)
% 0.19/0.43 thf(fact_637_ring__class_Oring__distribs_I1_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ring_class.ring_distribs(1)
% 0.19/0.43 thf(fact_638_comm__semiring__class_Odistrib,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_semiring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % comm_semiring_class.distrib
% 0.19/0.43 thf(fact_639_distrib__left,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % distrib_left
% 0.19/0.43 thf(fact_640_distrib__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % distrib_right
% 0.19/0.43 thf(fact_641_combine__common__factor,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring @ A )
% 0.19/0.43 => ! [A3: A,E2: A,B2: A,C2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % combine_common_factor
% 0.19/0.43 thf(fact_642_crossproduct__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiri1453513574482234551roduct @ A )
% 0.19/0.43 => ! [W: A,Y: A,X: A,Z2: A] :
% 0.19/0.43 ( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ W @ Z2 ) @ ( times_times @ A @ X @ Y ) ) )
% 0.19/0.43 = ( ( W = X )
% 0.19/0.43 | ( Y = Z2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % crossproduct_eq
% 0.19/0.43 thf(fact_643_crossproduct__noteq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiri1453513574482234551roduct @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( ( A3 != B2 )
% 0.19/0.43 & ( C2 != D3 ) )
% 0.19/0.43 = ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) )
% 0.19/0.43 != ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % crossproduct_noteq
% 0.19/0.43 thf(fact_644_comm__monoid__mult__class_Omult__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_monoid_mult @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % comm_monoid_mult_class.mult_1
% 0.19/0.43 thf(fact_645_mult_Ocomm__neutral,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_monoid_mult @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult.comm_neutral
% 0.19/0.43 thf(fact_646_abs__minus__commute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_minus_commute
% 0.19/0.43 thf(fact_647_Nat_Odiff__cancel,axiom,
% 0.19/0.43 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 0.19/0.43 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.diff_cancel
% 0.19/0.43 thf(fact_648_diff__cancel2,axiom,
% 0.19/0.43 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.43 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 0.19/0.43 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_cancel2
% 0.19/0.43 thf(fact_649_diff__add__inverse,axiom,
% 0.19/0.43 ! [N: nat,M2: nat] :
% 0.19/0.43 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
% 0.19/0.43 = M2 ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_inverse
% 0.19/0.43 thf(fact_650_diff__add__inverse2,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
% 0.19/0.43 = M2 ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_inverse2
% 0.19/0.43 thf(fact_651_le__diff__conv,axiom,
% 0.19/0.43 ! [J: nat,K: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.19/0.43 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_diff_conv
% 0.19/0.43 thf(fact_652_Nat_Ole__diff__conv2,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.43 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.le_diff_conv2
% 0.19/0.43 thf(fact_653_Nat_Odiff__add__assoc,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 0.19/0.43 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.diff_add_assoc
% 0.19/0.43 thf(fact_654_Nat_Odiff__add__assoc2,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
% 0.19/0.43 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.diff_add_assoc2
% 0.19/0.43 thf(fact_655_Nat_Ole__imp__diff__is__add,axiom,
% 0.19/0.43 ! [I: nat,J: nat,K: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ I @ J )
% 0.19/0.43 => ( ( ( minus_minus @ nat @ J @ I )
% 0.19/0.43 = K )
% 0.19/0.43 = ( J
% 0.19/0.43 = ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Nat.le_imp_diff_is_add
% 0.19/0.43 thf(fact_656_add__mult__distrib,axiom,
% 0.19/0.43 ! [M2: nat,N: nat,K: nat] :
% 0.19/0.43 ( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N ) @ K )
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_mult_distrib
% 0.19/0.43 thf(fact_657_add__mult__distrib2,axiom,
% 0.19/0.43 ! [K: nat,M2: nat,N: nat] :
% 0.19/0.43 ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_mult_distrib2
% 0.19/0.43 thf(fact_658_mlex__leI,axiom,
% 0.19/0.43 ! [A3: nat,A6: nat,B2: nat,B4: nat,N6: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ A3 @ A6 )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ B2 @ B4 )
% 0.19/0.43 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mlex_leI
% 0.19/0.43 thf(fact_659_nat__mult__1,axiom,
% 0.19/0.43 ! [N: nat] :
% 0.19/0.43 ( ( times_times @ nat @ ( one_one @ nat ) @ N )
% 0.19/0.43 = N ) ).
% 0.19/0.43
% 0.19/0.43 % nat_mult_1
% 0.19/0.43 thf(fact_660_nat__mult__1__right,axiom,
% 0.19/0.43 ! [N: nat] :
% 0.19/0.43 ( ( times_times @ nat @ N @ ( one_one @ nat ) )
% 0.19/0.43 = N ) ).
% 0.19/0.43
% 0.19/0.43 % nat_mult_1_right
% 0.19/0.43 thf(fact_661_mult__eq__if,axiom,
% 0.19/0.43 ( ( times_times @ nat )
% 0.19/0.43 = ( ^ [M: nat,N5: nat] :
% 0.19/0.43 ( if @ nat
% 0.19/0.43 @ ( M
% 0.19/0.43 = ( zero_zero @ nat ) )
% 0.19/0.43 @ ( zero_zero @ nat )
% 0.19/0.43 @ ( plus_plus @ nat @ N5 @ ( times_times @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_eq_if
% 0.19/0.43 thf(fact_662_lambda__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( monoid_mult @ A )
% 0.19/0.43 => ( ( ^ [X3: A] : X3 )
% 0.19/0.43 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % lambda_one
% 0.19/0.43 thf(fact_663_is__num__normalize_I6_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( neg_numeral_is_num @ A @ X )
% 0.19/0.43 => ( ( neg_numeral_is_num @ A @ Y )
% 0.19/0.43 => ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % is_num_normalize(6)
% 0.19/0.43 thf(fact_664_is__num__add__commute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( neg_numeral_is_num @ A @ X )
% 0.19/0.43 => ( ( neg_numeral_is_num @ A @ Y )
% 0.19/0.43 => ( ( plus_plus @ A @ X @ Y )
% 0.19/0.43 = ( plus_plus @ A @ Y @ X ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % is_num_add_commute
% 0.19/0.43 thf(fact_665_is__num__add__left__commute,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ! [X: A,Y: A,Z2: A] :
% 0.19/0.43 ( ( neg_numeral_is_num @ A @ X )
% 0.19/0.43 => ( ( neg_numeral_is_num @ A @ Y )
% 0.19/0.43 => ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y @ Z2 ) )
% 0.19/0.43 = ( plus_plus @ A @ Y @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % is_num_add_left_commute
% 0.19/0.43 thf(fact_666_is__num__normalize_I4_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % is_num_normalize(4)
% 0.19/0.43 thf(fact_667_le__iff__diff__le__0,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ( ( ord_less_eq @ A )
% 0.19/0.43 = ( ^ [A7: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_iff_diff_le_0
% 0.19/0.43 thf(fact_668_less__iff__diff__less__0,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ( ( ord_less @ A )
% 0.19/0.43 = ( ^ [A7: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_iff_diff_less_0
% 0.19/0.43 thf(fact_669_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( ( minus_minus @ A @ B2 @ A3 )
% 0.19/0.43 = C2 )
% 0.19/0.43 = ( B2
% 0.19/0.43 = ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.19/0.43 thf(fact_670_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.19/0.43 = B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.19/0.43 thf(fact_671_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.19/0.43 thf(fact_672_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
% 0.19/0.43 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.19/0.43 thf(fact_673_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.19/0.43 thf(fact_674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
% 0.19/0.43 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.19/0.43 thf(fact_675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.19/0.43 thf(fact_676_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.19/0.43 thf(fact_677_le__add__diff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_add_diff
% 0.19/0.43 thf(fact_678_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere1170586879665033532d_diff @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
% 0.19/0.43 = B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ordered_cancel_comm_monoid_diff_class.diff_add
% 0.19/0.43 thf(fact_679_le__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_diff_eq
% 0.19/0.43 thf(fact_680_diff__le__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_le_eq
% 0.19/0.43 thf(fact_681_add__le__imp__le__diff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [I: A,K: A,N: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.19/0.43 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_le_imp_le_diff
% 0.19/0.43 thf(fact_682_add__le__add__imp__diff__le,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [I: A,K: A,N: A,J: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.19/0.43 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 0.19/0.43 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_le_add_imp_diff_le
% 0.19/0.43 thf(fact_683_less__diff__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_diff_eq
% 0.19/0.43 thf(fact_684_diff__less__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordered_ab_group_add @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_less_eq
% 0.19/0.43 thf(fact_685_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ~ ( ord_less @ A @ A3 @ B2 )
% 0.19/0.43 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.43 = A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % linordered_semidom_class.add_diff_inverse
% 0.19/0.43 thf(fact_686_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_group_add @ A )
% 0.19/0.43 => ( ( minus_minus @ A )
% 0.19/0.43 = ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.19/0.43 thf(fact_687_diff__conv__add__uminus,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( group_add @ A )
% 0.19/0.43 => ( ( minus_minus @ A )
% 0.19/0.43 = ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_conv_add_uminus
% 0.19/0.43 thf(fact_688_group__cancel_Osub2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ab_group_add @ A )
% 0.19/0.43 => ! [B6: A,K: A,B2: A,A3: A] :
% 0.19/0.43 ( ( B6
% 0.19/0.43 = ( plus_plus @ A @ K @ B2 ) )
% 0.19/0.43 => ( ( minus_minus @ A @ A3 @ B6 )
% 0.19/0.43 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % group_cancel.sub2
% 0.19/0.43 thf(fact_689_abs__triangle__ineq2__sym,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.43 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_triangle_ineq2_sym
% 0.19/0.43 thf(fact_690_abs__triangle__ineq3,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.43 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_triangle_ineq3
% 0.19/0.43 thf(fact_691_abs__triangle__ineq2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.43 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_triangle_ineq2
% 0.19/0.43 thf(fact_692_add__scale__eq__noteq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiri1453513574482234551roduct @ A )
% 0.19/0.43 => ! [R: A,A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( R
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( ( A3 = B2 )
% 0.19/0.43 & ( C2 != D3 ) )
% 0.19/0.43 => ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
% 0.19/0.43 != ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D3 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_scale_eq_noteq
% 0.19/0.43 thf(fact_693_less__1__mult,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [M2: A,N: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( one_one @ A ) @ M2 )
% 0.19/0.43 => ( ( ord_less @ A @ ( one_one @ A ) @ N )
% 0.19/0.43 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_1_mult
% 0.19/0.43 thf(fact_694_square__eq__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_15535105094025558882visors @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ( times_times @ A @ X @ X )
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 = ( ( X
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 | ( X
% 0.19/0.43 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % square_eq_1_iff
% 0.19/0.43 thf(fact_695_diff__add__0,axiom,
% 0.19/0.43 ! [N: nat,M2: nat] :
% 0.19/0.43 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) )
% 0.19/0.43 = ( zero_zero @ nat ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_add_0
% 0.19/0.43 thf(fact_696_less__diff__conv,axiom,
% 0.19/0.43 ! [I: nat,J: nat,K: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 0.19/0.43 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_diff_conv
% 0.19/0.43 thf(fact_697_add__diff__inverse__nat,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ~ ( ord_less @ nat @ M2 @ N )
% 0.19/0.43 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M2 @ N ) )
% 0.19/0.43 = M2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_diff_inverse_nat
% 0.19/0.43 thf(fact_698_less__diff__conv2,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 0.19/0.43 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_diff_conv2
% 0.19/0.43 thf(fact_699_mlex__snd__decrI,axiom,
% 0.19/0.43 ! [A3: nat,A6: nat,B2: nat,B4: nat,N6: nat] :
% 0.19/0.43 ( ( A3 = A6 )
% 0.19/0.43 => ( ( ord_less @ nat @ B2 @ B4 )
% 0.19/0.43 => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mlex_snd_decrI
% 0.19/0.43 thf(fact_700_mlex__fst__decrI,axiom,
% 0.19/0.43 ! [A3: nat,A6: nat,B2: nat,N6: nat,B4: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ A3 @ A6 )
% 0.19/0.43 => ( ( ord_less @ nat @ B2 @ N6 )
% 0.19/0.43 => ( ( ord_less @ nat @ B4 @ N6 )
% 0.19/0.43 => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mlex_fst_decrI
% 0.19/0.43 thf(fact_701_mlex__bound,axiom,
% 0.19/0.43 ! [A3: nat,A4: nat,B2: nat,N6: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ A3 @ A4 )
% 0.19/0.43 => ( ( ord_less @ nat @ B2 @ N6 )
% 0.19/0.43 => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( times_times @ nat @ A4 @ N6 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mlex_bound
% 0.19/0.43 thf(fact_702_mult__eq__self__implies__10,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( M2
% 0.19/0.43 = ( times_times @ nat @ M2 @ N ) )
% 0.19/0.43 => ( ( N
% 0.19/0.43 = ( one_one @ nat ) )
% 0.19/0.43 | ( M2
% 0.19/0.43 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_eq_self_implies_10
% 0.19/0.43 thf(fact_703_map__comp__Some__iff,axiom,
% 0.19/0.43 ! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K: C,V: A] :
% 0.19/0.43 ( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K )
% 0.19/0.43 = ( some @ A @ V ) )
% 0.19/0.43 = ( ? [K6: B] :
% 0.19/0.43 ( ( ( M22 @ K )
% 0.19/0.43 = ( some @ B @ K6 ) )
% 0.19/0.43 & ( ( M1 @ K6 )
% 0.19/0.43 = ( some @ A @ V ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % map_comp_Some_iff
% 0.19/0.43 thf(fact_704_field__le__mult__one__interval,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ! [Z3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 0.19/0.43 => ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
% 0.19/0.43 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % field_le_mult_one_interval
% 0.19/0.43 thf(fact_705_abs__diff__le__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [X: A,A3: A,R: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 0.19/0.43 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 0.19/0.43 & ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_diff_le_iff
% 0.19/0.43 thf(fact_706_abs__diff__triangle__ineq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_diff_triangle_ineq
% 0.19/0.43 thf(fact_707_abs__triangle__ineq4,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ordere166539214618696060dd_abs @ A )
% 0.19/0.43 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_triangle_ineq4
% 0.19/0.43 thf(fact_708_abs__diff__less__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [X: A,A3: A,R: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 0.19/0.43 = ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 0.19/0.43 & ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % abs_diff_less_iff
% 0.19/0.43 thf(fact_709_sum__squares__ge__zero,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_ring @ A )
% 0.19/0.43 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % sum_squares_ge_zero
% 0.19/0.43 thf(fact_710_mult__left__le,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linord181362715937106298miring @ A )
% 0.19/0.43 => ! [C2: A,A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_left_le
% 0.19/0.43 thf(fact_711_mult__le__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linord181362715937106298miring @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_le_one
% 0.19/0.43 thf(fact_712_mult__right__le__one__le,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.19/0.43 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_right_le_one_le
% 0.19/0.43 thf(fact_713_mult__left__le__one__le,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 0.19/0.43 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_left_le_one_le
% 0.19/0.43 thf(fact_714_not__sum__squares__lt__zero,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_ring @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % not_sum_squares_lt_zero
% 0.19/0.43 thf(fact_715_nat__diff__split,axiom,
% 0.19/0.43 ! [P: nat > $o,A3: nat,B2: nat] :
% 0.19/0.43 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 0.19/0.43 = ( ( ( ord_less @ nat @ A3 @ B2 )
% 0.19/0.43 => ( P @ ( zero_zero @ nat ) ) )
% 0.19/0.43 & ! [D5: nat] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ nat @ B2 @ D5 ) )
% 0.19/0.43 => ( P @ D5 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_diff_split
% 0.19/0.43 thf(fact_716_nat__diff__split__asm,axiom,
% 0.19/0.43 ! [P: nat > $o,A3: nat,B2: nat] :
% 0.19/0.43 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 0.19/0.43 = ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
% 0.19/0.43 & ~ ( P @ ( zero_zero @ nat ) ) )
% 0.19/0.43 | ? [D5: nat] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ nat @ B2 @ D5 ) )
% 0.19/0.43 & ~ ( P @ D5 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_diff_split_asm
% 0.19/0.43 thf(fact_717_dbl__dec__def,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ( ( neg_numeral_dbl_dec @ A )
% 0.19/0.43 = ( ^ [X3: A] : ( minus_minus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % dbl_dec_def
% 0.19/0.43 thf(fact_718_mult__le__cancel__left1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [C2: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 0.19/0.43 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_le_cancel_left1
% 0.19/0.43 thf(fact_719_mult__le__cancel__left2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [C2: A,A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 0.19/0.43 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_le_cancel_left2
% 0.19/0.43 thf(fact_720_mult__le__cancel__right1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [C2: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 0.19/0.43 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_le_cancel_right1
% 0.19/0.43 thf(fact_721_mult__le__cancel__right2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [A3: A,C2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 0.19/0.43 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_le_cancel_right2
% 0.19/0.43 thf(fact_722_mult__less__cancel__left1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [C2: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 0.19/0.43 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 0.19/0.43 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_less_cancel_left1
% 0.19/0.43 thf(fact_723_mult__less__cancel__left2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [C2: A,A3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 0.19/0.43 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 0.19/0.43 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_less_cancel_left2
% 0.19/0.43 thf(fact_724_mult__less__cancel__right1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [C2: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 0.19/0.43 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_less_cancel_right1
% 0.19/0.43 thf(fact_725_mult__less__cancel__right2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_idom @ A )
% 0.19/0.43 => ! [A3: A,C2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 0.19/0.43 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 0.19/0.43 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_less_cancel_right2
% 0.19/0.43 thf(fact_726_convex__bound__le,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linord6961819062388156250ring_1 @ A )
% 0.19/0.43 => ! [X: A,A3: A,Y: A,U: A,V: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ X @ A3 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ Y @ A3 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 0.19/0.43 => ( ( ( plus_plus @ A @ U @ V )
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % convex_bound_le
% 0.19/0.43 thf(fact_727_is__num_Ocases,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( neg_numeral_is_num @ A @ A3 )
% 0.19/0.43 => ( ( A3
% 0.19/0.43 != ( one_one @ A ) )
% 0.19/0.43 => ( ! [X2: A] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( uminus_uminus @ A @ X2 ) )
% 0.19/0.43 => ~ ( neg_numeral_is_num @ A @ X2 ) )
% 0.19/0.43 => ~ ! [X2: A,Y4: A] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ A @ X2 @ Y4 ) )
% 0.19/0.43 => ( ( neg_numeral_is_num @ A @ X2 )
% 0.19/0.43 => ~ ( neg_numeral_is_num @ A @ Y4 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % is_num.cases
% 0.19/0.43 thf(fact_728_is__num_Osimps,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( neg_numeral @ A )
% 0.19/0.43 => ( ( neg_numeral_is_num @ A )
% 0.19/0.43 = ( ^ [A7: A] :
% 0.19/0.43 ( ( A7
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 | ? [X3: A] :
% 0.19/0.43 ( ( A7
% 0.19/0.43 = ( uminus_uminus @ A @ X3 ) )
% 0.19/0.43 & ( neg_numeral_is_num @ A @ X3 ) )
% 0.19/0.43 | ? [X3: A,Y3: A] :
% 0.19/0.43 ( ( A7
% 0.19/0.43 = ( plus_plus @ A @ X3 @ Y3 ) )
% 0.19/0.43 & ( neg_numeral_is_num @ A @ X3 )
% 0.19/0.43 & ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % is_num.simps
% 0.19/0.43 thf(fact_729_sum__squares__gt__zero__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linord4710134922213307826strict @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
% 0.19/0.43 = ( ( X
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 | ( Y
% 0.19/0.43 != ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % sum_squares_gt_zero_iff
% 0.19/0.43 thf(fact_730_sum__squares__le__zero__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linord4710134922213307826strict @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
% 0.19/0.43 = ( ( X
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 & ( Y
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % sum_squares_le_zero_iff
% 0.19/0.43 thf(fact_731_incr__lemma,axiom,
% 0.19/0.43 ! [D3: int,Z2: int,X: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
% 0.19/0.43 => ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % incr_lemma
% 0.19/0.43 thf(fact_732_decr__lemma,axiom,
% 0.19/0.43 ! [D3: int,X: int,Z2: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
% 0.19/0.43 => ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % decr_lemma
% 0.19/0.43 thf(fact_733_sum__squares__eq__zero__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linord4710134922213307826strict @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 = ( ( X
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 & ( Y
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % sum_squares_eq_zero_iff
% 0.19/0.43 thf(fact_734_frac__unique__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,A3: A] :
% 0.19/0.43 ( ( ( archimedean_frac @ A @ X )
% 0.19/0.43 = A3 )
% 0.19/0.43 = ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
% 0.19/0.43 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 & ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % frac_unique_iff
% 0.19/0.43 thf(fact_735_scaling__mono,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [U: A,V: A,R: A,S: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ U @ V )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
% 0.19/0.43 => ( ( ord_less_eq @ A @ R @ S )
% 0.19/0.43 => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % scaling_mono
% 0.19/0.43 thf(fact_736_divide__le__eq__1__neg,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.19/0.43 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_le_eq_1_neg
% 0.19/0.43 thf(fact_737_divide__le__eq__1__pos,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.19/0.43 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_le_eq_1_pos
% 0.19/0.43 thf(fact_738_div__by__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semidom_divide @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_by_1
% 0.19/0.43 thf(fact_739_div__self,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semidom_divide @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.19/0.43 = ( one_one @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_self
% 0.19/0.43 thf(fact_740_zero__eq__1__divide__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ( zero_zero @ A )
% 0.19/0.43 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.19/0.43 = ( A3
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_eq_1_divide_iff
% 0.19/0.43 thf(fact_741_one__divide__eq__0__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 = ( A3
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % one_divide_eq_0_iff
% 0.19/0.43 thf(fact_742_eq__divide__eq__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ( one_one @ A )
% 0.19/0.43 = ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ( A3
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 & ( A3 = B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eq_divide_eq_1
% 0.19/0.43 thf(fact_743_divide__eq__eq__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ( divide_divide @ A @ B2 @ A3 )
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 = ( ( A3
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 & ( A3 = B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_eq_eq_1
% 0.19/0.43 thf(fact_744_divide__self__if,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ( A3
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.19/0.43 = ( zero_zero @ A ) ) )
% 0.19/0.43 & ( ( A3
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.19/0.43 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_self_if
% 0.19/0.43 thf(fact_745_divide__self,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ A3 @ A3 )
% 0.19/0.43 = ( one_one @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_self
% 0.19/0.43 thf(fact_746_one__eq__divide__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ( one_one @ A )
% 0.19/0.43 = ( divide_divide @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 & ( A3 = B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % one_eq_divide_iff
% 0.19/0.43 thf(fact_747_divide__eq__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 = ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 & ( A3 = B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_eq_1_iff
% 0.19/0.43 thf(fact_748_divide__minus1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.43 = ( uminus_uminus @ A @ X ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_minus1
% 0.19/0.43 thf(fact_749_zero__le__divide__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_le_divide_1_iff
% 0.19/0.43 thf(fact_750_divide__le__0__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.43 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_le_0_1_iff
% 0.19/0.43 thf(fact_751_zero__less__divide__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.19/0.43 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_less_divide_1_iff
% 0.19/0.43 thf(fact_752_less__divide__eq__1__pos,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_divide_eq_1_pos
% 0.19/0.43 thf(fact_753_less__divide__eq__1__neg,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_divide_eq_1_neg
% 0.19/0.43 thf(fact_754_divide__less__eq__1__pos,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.19/0.43 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_less_eq_1_pos
% 0.19/0.43 thf(fact_755_divide__less__eq__1__neg,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.19/0.43 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_less_eq_1_neg
% 0.19/0.43 thf(fact_756_divide__less__0__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 0.19/0.43 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_less_0_1_iff
% 0.19/0.43 thf(fact_757_nonzero__divide__mult__cancel__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nonzero_divide_mult_cancel_right
% 0.19/0.43 thf(fact_758_nonzero__divide__mult__cancel__left,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 0.19/0.43 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nonzero_divide_mult_cancel_left
% 0.19/0.43 thf(fact_759_le__divide__eq__1__pos,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_divide_eq_1_pos
% 0.19/0.43 thf(fact_760_le__divide__eq__1__neg,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_divide_eq_1_neg
% 0.19/0.43 thf(fact_761_int__distrib_I2_J,axiom,
% 0.19/0.43 ! [W: int,Z1: int,Z22: int] :
% 0.19/0.43 ( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_distrib(2)
% 0.19/0.43 thf(fact_762_int__distrib_I1_J,axiom,
% 0.19/0.43 ! [Z1: int,Z22: int,W: int] :
% 0.19/0.43 ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_distrib(1)
% 0.19/0.43 thf(fact_763_plus__int__code_I2_J,axiom,
% 0.19/0.43 ! [L: int] :
% 0.19/0.43 ( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
% 0.19/0.43 = L ) ).
% 0.19/0.43
% 0.19/0.43 % plus_int_code(2)
% 0.19/0.43 thf(fact_764_plus__int__code_I1_J,axiom,
% 0.19/0.43 ! [K: int] :
% 0.19/0.43 ( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
% 0.19/0.43 = K ) ).
% 0.19/0.43
% 0.19/0.43 % plus_int_code(1)
% 0.19/0.43 thf(fact_765_zero__integer_Orsp,axiom,
% 0.19/0.43 ( ( zero_zero @ int )
% 0.19/0.43 = ( zero_zero @ int ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_integer.rsp
% 0.19/0.43 thf(fact_766_add__divide__distrib,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_divide_distrib
% 0.19/0.43 thf(fact_767_int__div__pos__eq,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.19/0.43 => ( ( ord_less @ int @ R @ B2 )
% 0.19/0.43 => ( ( divide_divide @ int @ A3 @ B2 )
% 0.19/0.43 = Q3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_div_pos_eq
% 0.19/0.43 thf(fact_768_int__div__neg__eq,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( ord_less @ int @ B2 @ R )
% 0.19/0.43 => ( ( divide_divide @ int @ A3 @ B2 )
% 0.19/0.43 = Q3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_div_neg_eq
% 0.19/0.43 thf(fact_769_split__zdiv,axiom,
% 0.19/0.43 ! [P: int > $o,N: int,K: int] :
% 0.19/0.43 ( ( P @ ( divide_divide @ int @ N @ K ) )
% 0.19/0.43 = ( ( ( K
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 => ( P @ ( zero_zero @ int ) ) )
% 0.19/0.43 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.43 => ! [I3: int,J2: int] :
% 0.19/0.43 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
% 0.19/0.43 & ( ord_less @ int @ J2 @ K )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 0.19/0.43 => ( P @ I3 ) ) )
% 0.19/0.43 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.19/0.43 => ! [I3: int,J2: int] :
% 0.19/0.43 ( ( ( ord_less @ int @ K @ J2 )
% 0.19/0.43 & ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 0.19/0.43 => ( P @ I3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % split_zdiv
% 0.19/0.43 thf(fact_770_frac__lt__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % frac_lt_1
% 0.19/0.43 thf(fact_771_frac__1__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 0.19/0.43 = ( archimedean_frac @ A @ X ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % frac_1_eq
% 0.19/0.43 thf(fact_772_right__inverse__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 = ( A3 = B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % right_inverse_eq
% 0.19/0.43 thf(fact_773_less__divide__eq__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 & ( ord_less @ A @ A3 @ B2 ) )
% 0.19/0.43 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_divide_eq_1
% 0.19/0.43 thf(fact_774_divide__less__eq__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 & ( ord_less @ A @ B2 @ A3 ) )
% 0.19/0.43 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 & ( ord_less @ A @ A3 @ B2 ) )
% 0.19/0.43 | ( A3
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_less_eq_1
% 0.19/0.43 thf(fact_775_divide__add__eq__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [Z2: A,X: A,Y: A] :
% 0.19/0.43 ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_add_eq_iff
% 0.19/0.43 thf(fact_776_add__divide__eq__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [Z2: A,X: A,Y: A] :
% 0.19/0.43 ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_divide_eq_iff
% 0.19/0.43 thf(fact_777_add__num__frac,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [Y: A,Z2: A,X: A] :
% 0.19/0.43 ( ( Y
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_num_frac
% 0.19/0.43 thf(fact_778_add__frac__num,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [Y: A,X: A,Z2: A] :
% 0.19/0.43 ( ( Y
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_frac_num
% 0.19/0.43 thf(fact_779_add__frac__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [Y: A,Z2: A,X: A,W: A] :
% 0.19/0.43 ( ( Y
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_frac_eq
% 0.19/0.43 thf(fact_780_add__divide__eq__if__simps_I1_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [Z2: A,A3: A,B2: A] :
% 0.19/0.43 ( ( ( Z2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
% 0.19/0.43 = A3 ) )
% 0.19/0.43 & ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_divide_eq_if_simps(1)
% 0.19/0.43 thf(fact_781_add__divide__eq__if__simps_I2_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [Z2: A,A3: A,B2: A] :
% 0.19/0.43 ( ( ( Z2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
% 0.19/0.43 = B2 ) )
% 0.19/0.43 & ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_divide_eq_if_simps(2)
% 0.19/0.43 thf(fact_782_less__half__sum,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.43 => ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % less_half_sum
% 0.19/0.43 thf(fact_783_gt__half__sum,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.43 => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % gt_half_sum
% 0.19/0.43 thf(fact_784_divide__eq__minus__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( field @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.19/0.43 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.43 = ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 & ( A3
% 0.19/0.43 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_eq_minus_1_iff
% 0.19/0.43 thf(fact_785_div__pos__geq,axiom,
% 0.19/0.43 ! [L: int,K: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.19/0.43 => ( ( ord_less_eq @ int @ L @ K )
% 0.19/0.43 => ( ( divide_divide @ int @ K @ L )
% 0.19/0.43 = ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_pos_geq
% 0.19/0.43 thf(fact_786_frac__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ( archimedean_frac @ A @ X )
% 0.19/0.43 = X )
% 0.19/0.43 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.19/0.43 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % frac_eq
% 0.19/0.43 thf(fact_787_frac__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 0.19/0.43 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
% 0.19/0.43 = ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
% 0.19/0.43 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 0.19/0.43 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
% 0.19/0.43 = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % frac_add
% 0.19/0.43 thf(fact_788_incr__mult__lemma,axiom,
% 0.19/0.43 ! [D3: int,P: int > $o,K: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ X2 @ D3 ) ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ( P @ X7 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ X7 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % incr_mult_lemma
% 0.19/0.43 thf(fact_789_divide__le__eq__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 & ( ord_less_eq @ A @ B2 @ A3 ) )
% 0.19/0.43 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.19/0.43 | ( A3
% 0.19/0.43 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divide_le_eq_1
% 0.19/0.43 thf(fact_790_le__divide__eq__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_field @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 0.19/0.43 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.43 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 0.19/0.43 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 0.19/0.43 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_divide_eq_1
% 0.19/0.43 thf(fact_791_minus__divide__add__eq__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [Z2: A,X: A,Y: A] :
% 0.19/0.43 ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % minus_divide_add_eq_iff
% 0.19/0.43 thf(fact_792_add__divide__eq__if__simps_I3_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( division_ring @ A )
% 0.19/0.43 => ! [Z2: A,A3: A,B2: A] :
% 0.19/0.43 ( ( ( Z2
% 0.19/0.43 = ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
% 0.19/0.43 = B2 ) )
% 0.19/0.43 & ( ( Z2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
% 0.19/0.43 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_divide_eq_if_simps(3)
% 0.19/0.43 thf(fact_793_frac__neg,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.19/0.43 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 0.19/0.43 = ( zero_zero @ A ) ) )
% 0.19/0.43 & ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.19/0.43 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 0.19/0.43 = ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % frac_neg
% 0.19/0.43 thf(fact_794_div__mult__self4,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,C2: A,A3: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mult_self4
% 0.19/0.43 thf(fact_795_div__mult__self3,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,C2: A,A3: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mult_self3
% 0.19/0.43 thf(fact_796_div__mult__self2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mult_self2
% 0.19/0.43 thf(fact_797_div__mult__self1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mult_self1
% 0.19/0.43 thf(fact_798_div__minus1__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid8851590272496341667cancel @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.43 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_minus1_right
% 0.19/0.43 thf(fact_799_div__pos__neg__trivial,axiom,
% 0.19/0.43 ! [K: int,L: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( divide_divide @ int @ K @ L )
% 0.19/0.43 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_pos_neg_trivial
% 0.19/0.43 thf(fact_800_bits__div__by__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( bit_semiring_bits @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % bits_div_by_1
% 0.19/0.43 thf(fact_801_dividend__less__times__div,axiom,
% 0.19/0.43 ! [N: nat,M2: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.43 => ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % dividend_less_times_div
% 0.19/0.43 thf(fact_802_dividend__less__div__times,axiom,
% 0.19/0.43 ! [N: nat,M2: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.43 => ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % dividend_less_div_times
% 0.19/0.43 thf(fact_803_split__div,axiom,
% 0.19/0.43 ! [P: nat > $o,M2: nat,N: nat] :
% 0.19/0.43 ( ( P @ ( divide_divide @ nat @ M2 @ N ) )
% 0.19/0.43 = ( ( ( N
% 0.19/0.43 = ( zero_zero @ nat ) )
% 0.19/0.43 => ( P @ ( zero_zero @ nat ) ) )
% 0.19/0.43 & ( ( N
% 0.19/0.43 != ( zero_zero @ nat ) )
% 0.19/0.43 => ! [I3: nat,J2: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ J2 @ N )
% 0.19/0.43 => ( ( M2
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J2 ) )
% 0.19/0.43 => ( P @ I3 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % split_div
% 0.19/0.43 thf(fact_804_div__less__dividend,axiom,
% 0.19/0.43 ! [N: nat,M2: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 0.19/0.43 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.19/0.43 => ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_less_dividend
% 0.19/0.43 thf(fact_805_div__eq__dividend__iff,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.19/0.43 => ( ( ( divide_divide @ nat @ M2 @ N )
% 0.19/0.43 = M2 )
% 0.19/0.43 = ( N
% 0.19/0.43 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_eq_dividend_iff
% 0.19/0.43 thf(fact_806_div__add__self1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_add_self1
% 0.19/0.43 thf(fact_807_div__add__self2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_add_self2
% 0.19/0.43 thf(fact_808_upto_Opinduct,axiom,
% 0.19/0.43 ! [A0: int,A1: int,P: int > int > $o] :
% 0.19/0.43 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 0.19/0.43 => ( ! [I2: int,J3: int] :
% 0.19/0.43 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J3 ) )
% 0.19/0.43 => ( ( ( ord_less_eq @ int @ I2 @ J3 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) )
% 0.19/0.43 => ( P @ I2 @ J3 ) ) )
% 0.19/0.43 => ( P @ A0 @ A1 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % upto.pinduct
% 0.19/0.43 thf(fact_809_eucl__rel__int_Osimps,axiom,
% 0.19/0.43 ( eucl_rel_int
% 0.19/0.43 = ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
% 0.19/0.43 ( ? [K4: int] :
% 0.19/0.43 ( ( A12 = K4 )
% 0.19/0.43 & ( A23
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 & ( A33
% 0.19/0.43 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K4 ) ) )
% 0.19/0.43 | ? [L3: int,K4: int,Q5: int] :
% 0.19/0.43 ( ( A12 = K4 )
% 0.19/0.43 & ( A23 = L3 )
% 0.19/0.43 & ( A33
% 0.19/0.43 = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
% 0.19/0.43 & ( L3
% 0.19/0.43 != ( zero_zero @ int ) )
% 0.19/0.43 & ( K4
% 0.19/0.43 = ( times_times @ int @ Q5 @ L3 ) ) )
% 0.19/0.43 | ? [R3: int,L3: int,K4: int,Q5: int] :
% 0.19/0.43 ( ( A12 = K4 )
% 0.19/0.43 & ( A23 = L3 )
% 0.19/0.43 & ( A33
% 0.19/0.43 = ( product_Pair @ int @ int @ Q5 @ R3 ) )
% 0.19/0.43 & ( ( sgn_sgn @ int @ R3 )
% 0.19/0.43 = ( sgn_sgn @ int @ L3 ) )
% 0.19/0.43 & ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L3 ) )
% 0.19/0.43 & ( K4
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ Q5 @ L3 ) @ R3 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int.simps
% 0.19/0.43 thf(fact_810_eucl__rel__int_Ocases,axiom,
% 0.19/0.43 ! [A1: int,A22: int,A32: product_prod @ int @ int] :
% 0.19/0.43 ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 0.19/0.43 => ( ( ( A22
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 => ( A32
% 0.19/0.43 != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
% 0.19/0.43 => ( ! [Q2: int] :
% 0.19/0.43 ( ( A32
% 0.19/0.43 = ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) )
% 0.19/0.43 => ( ( A22
% 0.19/0.43 != ( zero_zero @ int ) )
% 0.19/0.43 => ( A1
% 0.19/0.43 != ( times_times @ int @ Q2 @ A22 ) ) ) )
% 0.19/0.43 => ~ ! [R2: int,Q2: int] :
% 0.19/0.43 ( ( A32
% 0.19/0.43 = ( product_Pair @ int @ int @ Q2 @ R2 ) )
% 0.19/0.43 => ( ( ( sgn_sgn @ int @ R2 )
% 0.19/0.43 = ( sgn_sgn @ int @ A22 ) )
% 0.19/0.43 => ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ A22 ) )
% 0.19/0.43 => ( A1
% 0.19/0.43 != ( plus_plus @ int @ ( times_times @ int @ Q2 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int.cases
% 0.19/0.43 thf(fact_811_eucl__rel__int__iff,axiom,
% 0.19/0.43 ! [K: int,L: int,Q3: int,R: int] :
% 0.19/0.43 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.43 = ( ( K
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ R ) )
% 0.19/0.43 & ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.19/0.43 & ( ord_less @ int @ R @ L ) ) )
% 0.19/0.43 & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 0.19/0.43 => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( ord_less @ int @ L @ R )
% 0.19/0.43 & ( ord_less_eq @ int @ R @ ( zero_zero @ int ) ) ) )
% 0.19/0.43 & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 0.19/0.43 => ( Q3
% 0.19/0.43 = ( zero_zero @ int ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int_iff
% 0.19/0.43 thf(fact_812_power_Opower__eq__if,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( power2 @ A )
% 0.19/0.43 = ( ^ [One: A,Times: A > A > A,P6: A,M: nat] :
% 0.19/0.43 ( if @ A
% 0.19/0.43 @ ( M
% 0.19/0.43 = ( zero_zero @ nat ) )
% 0.19/0.43 @ One
% 0.19/0.43 @ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % power.power_eq_if
% 0.19/0.43 thf(fact_813_split__neg__lemma,axiom,
% 0.19/0.43 ! [K: int,P: int > int > $o,N: int] :
% 0.19/0.43 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 0.19/0.43 = ( ! [I3: int,J2: int] :
% 0.19/0.43 ( ( ( ord_less @ int @ K @ J2 )
% 0.19/0.43 & ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 0.19/0.43 => ( P @ I3 @ J2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % split_neg_lemma
% 0.19/0.43 thf(fact_814_split__pos__lemma,axiom,
% 0.19/0.43 ! [K: int,P: int > int > $o,N: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.43 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 0.19/0.43 = ( ! [I3: int,J2: int] :
% 0.19/0.43 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
% 0.19/0.43 & ( ord_less @ int @ J2 @ K )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 0.19/0.43 => ( P @ I3 @ J2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % split_pos_lemma
% 0.19/0.43 thf(fact_815_mod__add__self2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_add_self2
% 0.19/0.43 thf(fact_816_mod__add__self1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_add_self1
% 0.19/0.43 thf(fact_817_mod__by__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semidom_modulo @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_by_1
% 0.19/0.43 thf(fact_818_bits__mod__by__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( bit_semiring_bits @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bits_mod_by_1
% 0.19/0.43 thf(fact_819_mod__mult__self1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_mult_self1
% 0.19/0.43 thf(fact_820_mod__mult__self2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_mult_self2
% 0.19/0.43 thf(fact_821_mod__mult__self3,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [C2: A,B2: A,A3: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_mult_self3
% 0.19/0.43 thf(fact_822_mod__mult__self4,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [B2: A,C2: A,A3: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_mult_self4
% 0.19/0.43 thf(fact_823_mod__minus1__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid8851590272496341667cancel @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.43 = ( zero_zero @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_minus1_right
% 0.19/0.43 thf(fact_824_mod__int__unique,axiom,
% 0.19/0.43 ! [K: int,L: int,Q3: int,R: int] :
% 0.19/0.43 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.43 => ( ( modulo_modulo @ int @ K @ L )
% 0.19/0.43 = R ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_int_unique
% 0.19/0.43 thf(fact_825_unique__quotient,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int,Q4: int,R4: int] :
% 0.19/0.43 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.43 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R4 ) )
% 0.19/0.43 => ( Q3 = Q4 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % unique_quotient
% 0.19/0.43 thf(fact_826_unique__remainder,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int,Q4: int,R4: int] :
% 0.19/0.43 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.43 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R4 ) )
% 0.19/0.43 => ( R = R4 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % unique_remainder
% 0.19/0.43 thf(fact_827_full__exhaustive__int_H_Ocases,axiom,
% 0.19/0.43 ! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
% 0.19/0.43 ~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
% 0.19/0.43 ( X
% 0.19/0.43 != ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % full_exhaustive_int'.cases
% 0.19/0.43 thf(fact_828_exhaustive__int_H_Ocases,axiom,
% 0.19/0.43 ! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
% 0.19/0.43 ~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
% 0.19/0.43 ( X
% 0.19/0.43 != ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % exhaustive_int'.cases
% 0.19/0.43 thf(fact_829_small__lazy_H_Ocases,axiom,
% 0.19/0.43 ! [X: product_prod @ int @ int] :
% 0.19/0.43 ~ ! [D2: int,I2: int] :
% 0.19/0.43 ( X
% 0.19/0.43 != ( product_Pair @ int @ int @ D2 @ I2 ) ) ).
% 0.19/0.43
% 0.19/0.43 % small_lazy'.cases
% 0.19/0.43 thf(fact_830_eucl__rel__int,axiom,
% 0.19/0.43 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int
% 0.19/0.43 thf(fact_831_mod__add__right__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_add_right_eq
% 0.19/0.43 thf(fact_832_mod__add__left__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_add_left_eq
% 0.19/0.43 thf(fact_833_mod__add__cong,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,C2: A,A6: A,B2: A,B4: A] :
% 0.19/0.43 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ A6 @ C2 ) )
% 0.19/0.43 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 0.19/0.43 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ ( plus_plus @ A @ A6 @ B4 ) @ C2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_add_cong
% 0.19/0.43 thf(fact_834_mod__add__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_add_eq
% 0.19/0.43 thf(fact_835_eucl__rel__int__by0,axiom,
% 0.19/0.43 ! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int_by0
% 0.19/0.43 thf(fact_836_div__int__unique,axiom,
% 0.19/0.43 ! [K: int,L: int,Q3: int,R: int] :
% 0.19/0.43 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.43 => ( ( divide_divide @ int @ K @ L )
% 0.19/0.43 = Q3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_int_unique
% 0.19/0.43 thf(fact_837_mod__eqE,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid8851590272496341667cancel @ A )
% 0.19/0.43 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.43 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 0.19/0.43 = ( modulo_modulo @ A @ B2 @ C2 ) )
% 0.19/0.43 => ~ ! [D2: A] :
% 0.19/0.43 ( B2
% 0.19/0.43 != ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_eqE
% 0.19/0.43 thf(fact_838_div__add1__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid3128863361964157862miring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_add1_eq
% 0.19/0.43 thf(fact_839_mult__div__mod__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring_modulo @ A )
% 0.19/0.43 => ! [B2: A,A3: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_div_mod_eq
% 0.19/0.43 thf(fact_840_mod__mult__div__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring_modulo @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_mult_div_eq
% 0.19/0.43 thf(fact_841_mod__div__mult__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring_modulo @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_div_mult_eq
% 0.19/0.43 thf(fact_842_div__mult__mod__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring_modulo @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mult_mod_eq
% 0.19/0.43 thf(fact_843_mod__div__decomp,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semiring_modulo @ A )
% 0.19/0.43 => ! [A3: A,B2: A] :
% 0.19/0.43 ( A3
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_div_decomp
% 0.19/0.43 thf(fact_844_cancel__div__mod__rules_I1_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semidom_modulo @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % cancel_div_mod_rules(1)
% 0.19/0.43 thf(fact_845_cancel__div__mod__rules_I2_J,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( semidom_modulo @ A )
% 0.19/0.43 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.43 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % cancel_div_mod_rules(2)
% 0.19/0.43 thf(fact_846_div__mult1__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( euclid3128863361964157862miring @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mult1_eq
% 0.19/0.43 thf(fact_847_eucl__rel__int__dividesI,axiom,
% 0.19/0.43 ! [L: int,K: int,Q3: int] :
% 0.19/0.43 ( ( L
% 0.19/0.43 != ( zero_zero @ int ) )
% 0.19/0.43 => ( ( K
% 0.19/0.43 = ( times_times @ int @ Q3 @ L ) )
% 0.19/0.43 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int_dividesI
% 0.19/0.43 thf(fact_848_div__mod__decomp__int,axiom,
% 0.19/0.43 ! [A4: int,N: int] :
% 0.19/0.43 ( A4
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A4 @ N ) @ N ) @ ( modulo_modulo @ int @ A4 @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mod_decomp_int
% 0.19/0.43 thf(fact_849_mod__pos__neg__trivial,axiom,
% 0.19/0.43 ! [K: int,L: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( modulo_modulo @ int @ K @ L )
% 0.19/0.43 = ( plus_plus @ int @ K @ L ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_pos_neg_trivial
% 0.19/0.43 thf(fact_850_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.43 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.43 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 0.19/0.43 => ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.19/0.43 thf(fact_851_int__mod__pos__eq,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 0.19/0.43 => ( ( ord_less @ int @ R @ B2 )
% 0.19/0.43 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.19/0.43 = R ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_mod_pos_eq
% 0.19/0.43 thf(fact_852_int__mod__neg__eq,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int] :
% 0.19/0.43 ( ( A3
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 0.19/0.43 => ( ( ord_less @ int @ B2 @ R )
% 0.19/0.43 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 0.19/0.43 = R ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % int_mod_neg_eq
% 0.19/0.43 thf(fact_853_split__zmod,axiom,
% 0.19/0.43 ! [P: int > $o,N: int,K: int] :
% 0.19/0.43 ( ( P @ ( modulo_modulo @ int @ N @ K ) )
% 0.19/0.43 = ( ( ( K
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 => ( P @ N ) )
% 0.19/0.43 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 0.19/0.43 => ! [I3: int,J2: int] :
% 0.19/0.43 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
% 0.19/0.43 & ( ord_less @ int @ J2 @ K )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 0.19/0.43 => ( P @ J2 ) ) )
% 0.19/0.43 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.19/0.43 => ! [I3: int,J2: int] :
% 0.19/0.43 ( ( ( ord_less @ int @ K @ J2 )
% 0.19/0.43 & ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
% 0.19/0.43 & ( N
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 0.19/0.43 => ( P @ J2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % split_zmod
% 0.19/0.43 thf(fact_854_zmod__zmult2__eq,axiom,
% 0.19/0.43 ! [C2: int,A3: int,B2: int] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
% 0.19/0.43 => ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zmod_zmult2_eq
% 0.19/0.43 thf(fact_855_zminus1__lemma,axiom,
% 0.19/0.43 ! [A3: int,B2: int,Q3: int,R: int] :
% 0.19/0.43 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.43 => ( ( B2
% 0.19/0.43 != ( zero_zero @ int ) )
% 0.19/0.43 => ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
% 0.19/0.43 @ ( product_Pair @ int @ int
% 0.19/0.43 @ ( if @ int
% 0.19/0.43 @ ( R
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 @ ( uminus_uminus @ int @ Q3 )
% 0.19/0.43 @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
% 0.19/0.43 @ ( if @ int
% 0.19/0.43 @ ( R
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 @ ( zero_zero @ int )
% 0.19/0.43 @ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zminus1_lemma
% 0.19/0.43 thf(fact_856_eucl__rel__int__remainderI,axiom,
% 0.19/0.43 ! [R: int,L: int,K: int,Q3: int] :
% 0.19/0.43 ( ( ( sgn_sgn @ int @ R )
% 0.19/0.43 = ( sgn_sgn @ int @ L ) )
% 0.19/0.43 => ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
% 0.19/0.43 => ( ( K
% 0.19/0.43 = ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R ) )
% 0.19/0.43 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % eucl_rel_int_remainderI
% 0.19/0.43 thf(fact_857_verit__le__mono__div__int,axiom,
% 0.19/0.43 ! [A4: int,B6: int,N: int] :
% 0.19/0.43 ( ( ord_less @ int @ A4 @ B6 )
% 0.19/0.43 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.19/0.43 => ( ord_less_eq @ int
% 0.19/0.43 @ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
% 0.19/0.43 @ ( if @ int
% 0.19/0.43 @ ( ( modulo_modulo @ int @ B6 @ N )
% 0.19/0.43 = ( zero_zero @ int ) )
% 0.19/0.43 @ ( one_one @ int )
% 0.19/0.43 @ ( zero_zero @ int ) ) )
% 0.19/0.43 @ ( divide_divide @ int @ B6 @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % verit_le_mono_div_int
% 0.19/0.43 thf(fact_858_bezw__0,axiom,
% 0.19/0.43 ! [X: nat] :
% 0.19/0.43 ( ( bezw @ X @ ( zero_zero @ nat ) )
% 0.19/0.43 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bezw_0
% 0.19/0.43 thf(fact_859_verit__le__mono__div,axiom,
% 0.19/0.43 ! [A4: nat,B6: nat,N: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ A4 @ B6 )
% 0.19/0.43 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.43 => ( ord_less_eq @ nat
% 0.19/0.43 @ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
% 0.19/0.43 @ ( if @ nat
% 0.19/0.43 @ ( ( modulo_modulo @ nat @ B6 @ N )
% 0.19/0.43 = ( zero_zero @ nat ) )
% 0.19/0.43 @ ( one_one @ nat )
% 0.19/0.43 @ ( zero_zero @ nat ) ) )
% 0.19/0.43 @ ( divide_divide @ nat @ B6 @ N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % verit_le_mono_div
% 0.19/0.43 thf(fact_860_zero__le__ceiling,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 0.19/0.43 = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_le_ceiling
% 0.19/0.43 thf(fact_861_ceiling__less__zero,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 0.19/0.43 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_less_zero
% 0.19/0.43 thf(fact_862_floor__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 0.19/0.43 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
% 0.19/0.43 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
% 0.19/0.43 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 0.19/0.43 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
% 0.19/0.43 = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_add
% 0.19/0.43 thf(fact_863_cpmi,axiom,
% 0.19/0.43 ! [D4: int,P: int > $o,P2: int > $o,B6: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ? [Z6: int] :
% 0.19/0.43 ! [X2: int] :
% 0.19/0.43 ( ( ord_less @ int @ X2 @ Z6 )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 = ( P2 @ X2 ) ) )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ B6 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ( ! [X2: int,K5: int] :
% 0.19/0.43 ( ( P2 @ X2 )
% 0.19/0.43 = ( P2 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K5 @ D4 ) ) ) )
% 0.19/0.43 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.19/0.43 = ( ? [X3: int] :
% 0.19/0.43 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 & ( P2 @ X3 ) )
% 0.19/0.43 | ? [X3: int] :
% 0.19/0.43 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 & ? [Y3: int] :
% 0.19/0.43 ( ( member @ int @ Y3 @ B6 )
% 0.19/0.43 & ( P @ ( plus_plus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % cpmi
% 0.19/0.43 thf(fact_864_cppi,axiom,
% 0.19/0.43 ! [D4: int,P: int > $o,P2: int > $o,A4: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ? [Z6: int] :
% 0.19/0.43 ! [X2: int] :
% 0.19/0.43 ( ( ord_less @ int @ Z6 @ X2 )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 = ( P2 @ X2 ) ) )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ A4 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ( ! [X2: int,K5: int] :
% 0.19/0.43 ( ( P2 @ X2 )
% 0.19/0.43 = ( P2 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K5 @ D4 ) ) ) )
% 0.19/0.43 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.19/0.43 = ( ? [X3: int] :
% 0.19/0.43 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 & ( P2 @ X3 ) )
% 0.19/0.43 | ? [X3: int] :
% 0.19/0.43 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 & ? [Y3: int] :
% 0.19/0.43 ( ( member @ int @ Y3 @ A4 )
% 0.19/0.43 & ( P @ ( minus_minus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % cppi
% 0.19/0.43 thf(fact_865_bset_I6_J,axiom,
% 0.19/0.43 ! [D4: int,B6: set @ int,T2: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ X7 @ T2 )
% 0.19/0.43 => ( ord_less_eq @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(6)
% 0.19/0.43 thf(fact_866_bset_I8_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,B6: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ T2 @ X7 )
% 0.19/0.43 => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(8)
% 0.19/0.43 thf(fact_867_aset_I6_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,A4: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ X7 @ T2 )
% 0.19/0.43 => ( ord_less_eq @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(6)
% 0.19/0.43 thf(fact_868_atLeastAtMost__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ord @ A )
% 0.19/0.43 => ! [I: A,L: A,U: A] :
% 0.19/0.43 ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 0.19/0.43 = ( ( ord_less_eq @ A @ L @ I )
% 0.19/0.43 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % atLeastAtMost_iff
% 0.19/0.43 thf(fact_869_Icc__eq__Icc,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( order @ A )
% 0.19/0.43 => ! [L: A,H3: A,L4: A,H4: A] :
% 0.19/0.43 ( ( ( set_or1337092689740270186AtMost @ A @ L @ H3 )
% 0.19/0.43 = ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) )
% 0.19/0.43 = ( ( ( L = L4 )
% 0.19/0.43 & ( H3 = H4 ) )
% 0.19/0.43 | ( ~ ( ord_less_eq @ A @ L @ H3 )
% 0.19/0.43 & ~ ( ord_less_eq @ A @ L4 @ H4 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Icc_eq_Icc
% 0.19/0.43 thf(fact_870_atLeastatMost__subset__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( preorder @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.19/0.43 = ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.19/0.43 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % atLeastatMost_subset_iff
% 0.19/0.43 thf(fact_871_floor__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
% 0.19/0.43 = ( one_one @ int ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_one
% 0.19/0.43 thf(fact_872_ceiling__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
% 0.19/0.43 = ( one_one @ int ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_one
% 0.19/0.43 thf(fact_873_floor__add2,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,Y: A] :
% 0.19/0.43 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 0.19/0.43 | ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
% 0.19/0.43 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
% 0.19/0.43 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_add2
% 0.19/0.43 thf(fact_874_zero__less__floor,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.19/0.43 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % zero_less_floor
% 0.19/0.43 thf(fact_875_floor__le__zero,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
% 0.19/0.43 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_le_zero
% 0.19/0.43 thf(fact_876_one__le__floor,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.19/0.43 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % one_le_floor
% 0.19/0.43 thf(fact_877_floor__less__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 0.19/0.43 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_less_one
% 0.19/0.43 thf(fact_878_ceiling__le__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 0.19/0.43 = ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_le_one
% 0.19/0.43 thf(fact_879_one__less__ceiling,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 0.19/0.43 = ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % one_less_ceiling
% 0.19/0.43 thf(fact_880_ceiling__add__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 0.19/0.43 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_add_one
% 0.19/0.43 thf(fact_881_floor__diff__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 0.19/0.43 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_diff_one
% 0.19/0.43 thf(fact_882_ceiling__diff__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 0.19/0.43 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_diff_one
% 0.19/0.43 thf(fact_883_atLeastatMost__psubset__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( preorder @ A )
% 0.19/0.43 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.43 ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.19/0.43 = ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 0.19/0.43 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.19/0.43 & ( ord_less_eq @ A @ B2 @ D3 )
% 0.19/0.43 & ( ( ord_less @ A @ C2 @ A3 )
% 0.19/0.43 | ( ord_less @ A @ B2 @ D3 ) ) ) )
% 0.19/0.43 & ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % atLeastatMost_psubset_iff
% 0.19/0.43 thf(fact_884_nat__mod__eq__iff,axiom,
% 0.19/0.43 ! [X: nat,N: nat,Y: nat] :
% 0.19/0.43 ( ( ( modulo_modulo @ nat @ X @ N )
% 0.19/0.43 = ( modulo_modulo @ nat @ Y @ N ) )
% 0.19/0.43 = ( ? [Q1: nat,Q22: nat] :
% 0.19/0.43 ( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
% 0.19/0.43 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_mod_eq_iff
% 0.19/0.43 thf(fact_885_le__floor__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % le_floor_add
% 0.19/0.43 thf(fact_886_bset_I1_J,axiom,
% 0.19/0.43 ! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
% 0.19/0.43 ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ B6 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ B6 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( Q @ X2 )
% 0.19/0.43 => ( Q @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ( P @ X7 )
% 0.19/0.43 & ( Q @ X7 ) )
% 0.19/0.43 => ( ( P @ ( minus_minus @ int @ X7 @ D4 ) )
% 0.19/0.43 & ( Q @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(1)
% 0.19/0.43 thf(fact_887_bset_I2_J,axiom,
% 0.19/0.43 ! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
% 0.19/0.43 ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ B6 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ B6 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( Q @ X2 )
% 0.19/0.43 => ( Q @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ( P @ X7 )
% 0.19/0.43 | ( Q @ X7 ) )
% 0.19/0.43 => ( ( P @ ( minus_minus @ int @ X7 @ D4 ) )
% 0.19/0.43 | ( Q @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(2)
% 0.19/0.43 thf(fact_888_aset_I1_J,axiom,
% 0.19/0.43 ! [D4: int,A4: set @ int,P: int > $o,Q: int > $o] :
% 0.19/0.43 ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ A4 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ A4 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( Q @ X2 )
% 0.19/0.43 => ( Q @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ( P @ X7 )
% 0.19/0.43 & ( Q @ X7 ) )
% 0.19/0.43 => ( ( P @ ( plus_plus @ int @ X7 @ D4 ) )
% 0.19/0.43 & ( Q @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(1)
% 0.19/0.43 thf(fact_889_aset_I2_J,axiom,
% 0.19/0.43 ! [D4: int,A4: set @ int,P: int > $o,Q: int > $o] :
% 0.19/0.43 ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ A4 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( P @ X2 )
% 0.19/0.43 => ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ( ! [X2: int] :
% 0.19/0.43 ( ! [Xa2: int] :
% 0.19/0.43 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb2: int] :
% 0.19/0.43 ( ( member @ int @ Xb2 @ A4 )
% 0.19/0.43 => ( X2
% 0.19/0.43 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 0.19/0.43 => ( ( Q @ X2 )
% 0.19/0.43 => ( Q @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ( P @ X7 )
% 0.19/0.43 | ( Q @ X7 ) )
% 0.19/0.43 => ( ( P @ ( plus_plus @ int @ X7 @ D4 ) )
% 0.19/0.43 | ( Q @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(2)
% 0.19/0.43 thf(fact_890_nat__mod__eq__lemma,axiom,
% 0.19/0.43 ! [X: nat,N: nat,Y: nat] :
% 0.19/0.43 ( ( ( modulo_modulo @ nat @ X @ N )
% 0.19/0.43 = ( modulo_modulo @ nat @ Y @ N ) )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ Y @ X )
% 0.19/0.43 => ? [Q2: nat] :
% 0.19/0.43 ( X
% 0.19/0.43 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % nat_mod_eq_lemma
% 0.19/0.43 thf(fact_891_mod__eq__nat2E,axiom,
% 0.19/0.43 ! [M2: nat,Q3: nat,N: nat] :
% 0.19/0.43 ( ( ( modulo_modulo @ nat @ M2 @ Q3 )
% 0.19/0.43 = ( modulo_modulo @ nat @ N @ Q3 ) )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.43 => ~ ! [S6: nat] :
% 0.19/0.43 ( N
% 0.19/0.43 != ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q3 @ S6 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_eq_nat2E
% 0.19/0.43 thf(fact_892_mod__eq__nat1E,axiom,
% 0.19/0.43 ! [M2: nat,Q3: nat,N: nat] :
% 0.19/0.43 ( ( ( modulo_modulo @ nat @ M2 @ Q3 )
% 0.19/0.43 = ( modulo_modulo @ nat @ N @ Q3 ) )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ N @ M2 )
% 0.19/0.43 => ~ ! [S6: nat] :
% 0.19/0.43 ( M2
% 0.19/0.43 != ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S6 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_eq_nat1E
% 0.19/0.43 thf(fact_893_ceiling__add__le,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_add_le
% 0.19/0.43 thf(fact_894_mod__mult2__eq,axiom,
% 0.19/0.43 ! [M2: nat,N: nat,Q3: nat] :
% 0.19/0.43 ( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q3 ) )
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mod_mult2_eq
% 0.19/0.43 thf(fact_895_div__mod__decomp,axiom,
% 0.19/0.43 ! [A4: nat,N: nat] :
% 0.19/0.43 ( A4
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A4 @ N ) @ N ) @ ( modulo_modulo @ nat @ A4 @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_mod_decomp
% 0.19/0.43 thf(fact_896_one__add__floor,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A] :
% 0.19/0.43 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 0.19/0.43 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % one_add_floor
% 0.19/0.43 thf(fact_897_split__mod,axiom,
% 0.19/0.43 ! [P: nat > $o,M2: nat,N: nat] :
% 0.19/0.43 ( ( P @ ( modulo_modulo @ nat @ M2 @ N ) )
% 0.19/0.43 = ( ( ( N
% 0.19/0.43 = ( zero_zero @ nat ) )
% 0.19/0.43 => ( P @ M2 ) )
% 0.19/0.43 & ( ( N
% 0.19/0.43 != ( zero_zero @ nat ) )
% 0.19/0.43 => ! [I3: nat,J2: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ J2 @ N )
% 0.19/0.43 => ( ( M2
% 0.19/0.43 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J2 ) )
% 0.19/0.43 => ( P @ J2 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % split_mod
% 0.19/0.43 thf(fact_898_bset_I3_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,B6: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( X7 = T2 )
% 0.19/0.43 => ( ( minus_minus @ int @ X7 @ D4 )
% 0.19/0.43 = T2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(3)
% 0.19/0.43 thf(fact_899_bset_I4_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,B6: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ T2 @ B6 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( X7 != T2 )
% 0.19/0.43 => ( ( minus_minus @ int @ X7 @ D4 )
% 0.19/0.43 != T2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(4)
% 0.19/0.43 thf(fact_900_bset_I5_J,axiom,
% 0.19/0.43 ! [D4: int,B6: set @ int,T2: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less @ int @ X7 @ T2 )
% 0.19/0.43 => ( ord_less @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(5)
% 0.19/0.43 thf(fact_901_bset_I7_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,B6: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ T2 @ B6 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less @ int @ T2 @ X7 )
% 0.19/0.43 => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % bset(7)
% 0.19/0.43 thf(fact_902_aset_I3_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,A4: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( X7 = T2 )
% 0.19/0.43 => ( ( plus_plus @ int @ X7 @ D4 )
% 0.19/0.43 = T2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(3)
% 0.19/0.43 thf(fact_903_aset_I4_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,A4: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ T2 @ A4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( X7 != T2 )
% 0.19/0.43 => ( ( plus_plus @ int @ X7 @ D4 )
% 0.19/0.43 != T2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(4)
% 0.19/0.43 thf(fact_904_aset_I5_J,axiom,
% 0.19/0.43 ! [D4: int,T2: int,A4: set @ int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ( ( member @ int @ T2 @ A4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less @ int @ X7 @ T2 )
% 0.19/0.43 => ( ord_less @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(5)
% 0.19/0.43 thf(fact_905_aset_I7_J,axiom,
% 0.19/0.43 ! [D4: int,A4: set @ int,T2: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less @ int @ T2 @ X7 )
% 0.19/0.43 => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(7)
% 0.19/0.43 thf(fact_906_aset_I8_J,axiom,
% 0.19/0.43 ! [D4: int,A4: set @ int,T2: int] :
% 0.19/0.43 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 0.19/0.43 => ! [X7: int] :
% 0.19/0.43 ( ! [Xa3: int] :
% 0.19/0.43 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.43 => ! [Xb3: int] :
% 0.19/0.43 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.43 => ( X7
% 0.19/0.43 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.43 => ( ( ord_less_eq @ int @ T2 @ X7 )
% 0.19/0.43 => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % aset(8)
% 0.19/0.43 thf(fact_907_ceiling__divide__lower,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [Q3: A,P5: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
% 0.19/0.43 => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P5 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_divide_lower
% 0.19/0.43 thf(fact_908_floor__divide__upper,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [Q3: A,P5: A] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
% 0.19/0.43 => ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % floor_divide_upper
% 0.19/0.43 thf(fact_909_power__decreasing__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [B2: A,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.43 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.43 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
% 0.19/0.43 = ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % power_decreasing_iff
% 0.19/0.43 thf(fact_910_divmod__nat__def,axiom,
% 0.19/0.43 ( divmod_nat
% 0.19/0.43 = ( ^ [M: nat,N5: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M @ N5 ) @ ( modulo_modulo @ nat @ M @ N5 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % divmod_nat_def
% 0.19/0.43 thf(fact_911_distrib__left__NO__MATCH,axiom,
% 0.19/0.43 ! [B: $tType,A: $tType] :
% 0.19/0.43 ( ( semiring @ A )
% 0.19/0.43 => ! [X: B,Y: B,A3: A,B2: A,C2: A] :
% 0.19/0.43 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
% 0.19/0.43 => ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % distrib_left_NO_MATCH
% 0.19/0.43 thf(fact_912_distrib__right__NO__MATCH,axiom,
% 0.19/0.43 ! [B: $tType,A: $tType] :
% 0.19/0.43 ( ( semiring @ A )
% 0.19/0.43 => ! [X: B,Y: B,C2: A,A3: A,B2: A] :
% 0.19/0.43 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
% 0.19/0.43 => ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.43 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % distrib_right_NO_MATCH
% 0.19/0.43 thf(fact_913_Suc__times__mod__eq,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
% 0.19/0.43 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N ) ) @ M2 )
% 0.19/0.43 = ( one_one @ nat ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Suc_times_mod_eq
% 0.19/0.43 thf(fact_914_div__add__self1__no__field,axiom,
% 0.19/0.43 ! [B: $tType,A: $tType] :
% 0.19/0.43 ( ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 & ( field @ B ) )
% 0.19/0.43 => ! [X: B,B2: A,A3: A] :
% 0.19/0.43 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 0.19/0.43 => ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_add_self1_no_field
% 0.19/0.43 thf(fact_915_div__add__self2__no__field,axiom,
% 0.19/0.43 ! [B: $tType,A: $tType] :
% 0.19/0.43 ( ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.43 & ( field @ B ) )
% 0.19/0.43 => ! [X: B,B2: A,A3: A] :
% 0.19/0.43 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 0.19/0.43 => ( ( B2
% 0.19/0.43 != ( zero_zero @ A ) )
% 0.19/0.43 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 0.19/0.43 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % div_add_self2_no_field
% 0.19/0.43 thf(fact_916_power__one,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( monoid_mult @ A )
% 0.19/0.43 => ! [N: nat] :
% 0.19/0.43 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 0.19/0.43 = ( one_one @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % power_one
% 0.19/0.43 thf(fact_917_add__Suc__right,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.43 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % add_Suc_right
% 0.19/0.43 thf(fact_918_power__one__right,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( monoid_mult @ A )
% 0.19/0.43 => ! [A3: A] :
% 0.19/0.43 ( ( power_power @ A @ A3 @ ( one_one @ nat ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % power_one_right
% 0.19/0.43 thf(fact_919_power__inject__exp,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [A3: A,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.43 => ( ( ( power_power @ A @ A3 @ M2 )
% 0.19/0.43 = ( power_power @ A @ A3 @ N ) )
% 0.19/0.43 = ( M2 = N ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % power_inject_exp
% 0.19/0.43 thf(fact_920_mult__Suc__right,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( times_times @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.43 = ( plus_plus @ nat @ M2 @ ( times_times @ nat @ M2 @ N ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % mult_Suc_right
% 0.19/0.43 thf(fact_921_diff__Suc__1,axiom,
% 0.19/0.43 ! [N: nat] :
% 0.19/0.43 ( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
% 0.19/0.43 = N ) ).
% 0.19/0.43
% 0.19/0.43 % diff_Suc_1
% 0.19/0.43 thf(fact_922_of__int__eq__1__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_char_0 @ A )
% 0.19/0.43 => ! [Z2: int] :
% 0.19/0.43 ( ( ( ring_1_of_int @ A @ Z2 )
% 0.19/0.43 = ( one_one @ A ) )
% 0.19/0.43 = ( Z2
% 0.19/0.43 = ( one_one @ int ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % of_int_eq_1_iff
% 0.19/0.43 thf(fact_923_of__int__1,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_1 @ A )
% 0.19/0.43 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 0.19/0.43 = ( one_one @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % of_int_1
% 0.19/0.43 thf(fact_924_of__int__add,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( ring_1 @ A )
% 0.19/0.43 => ! [W: int,Z2: int] :
% 0.19/0.43 ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z2 ) )
% 0.19/0.43 = ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % of_int_add
% 0.19/0.43 thf(fact_925_power__strict__increasing__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [B2: A,X: nat,Y: nat] :
% 0.19/0.43 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 0.19/0.43 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
% 0.19/0.43 = ( ord_less @ nat @ X @ Y ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % power_strict_increasing_iff
% 0.19/0.43 thf(fact_926_left__minus__one__mult__self,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_ring_1 @ A )
% 0.19/0.43 => ! [N: nat,A3: A] :
% 0.19/0.43 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
% 0.19/0.43 = A3 ) ) ).
% 0.19/0.43
% 0.19/0.43 % left_minus_one_mult_self
% 0.19/0.43 thf(fact_927_minus__one__mult__self,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( comm_ring_1 @ A )
% 0.19/0.43 => ! [N: nat] :
% 0.19/0.43 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
% 0.19/0.43 = ( one_one @ A ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % minus_one_mult_self
% 0.19/0.43 thf(fact_928_diff__Suc__diff__eq2,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
% 0.19/0.43 = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_Suc_diff_eq2
% 0.19/0.43 thf(fact_929_diff__Suc__diff__eq1,axiom,
% 0.19/0.43 ! [K: nat,J: nat,I: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ K @ J )
% 0.19/0.43 => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
% 0.19/0.43 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % diff_Suc_diff_eq1
% 0.19/0.43 thf(fact_930_Suc__diff,axiom,
% 0.19/0.43 ! [M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.43 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
% 0.19/0.43 => ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
% 0.19/0.43 = ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % Suc_diff
% 0.19/0.43 thf(fact_931_Suc__mod__mult__self4,axiom,
% 0.19/0.43 ! [N: nat,K: nat,M2: nat] :
% 0.19/0.43 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M2 ) ) @ N )
% 0.19/0.43 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 0.19/0.43
% 0.19/0.43 % Suc_mod_mult_self4
% 0.19/0.43 thf(fact_932_Suc__mod__mult__self3,axiom,
% 0.19/0.43 ! [K: nat,N: nat,M2: nat] :
% 0.19/0.43 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M2 ) ) @ N )
% 0.19/0.43 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 0.19/0.43
% 0.19/0.43 % Suc_mod_mult_self3
% 0.19/0.43 thf(fact_933_Suc__mod__mult__self2,axiom,
% 0.19/0.43 ! [M2: nat,N: nat,K: nat] :
% 0.19/0.43 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N @ K ) ) ) @ N )
% 0.19/0.43 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 0.19/0.43
% 0.19/0.43 % Suc_mod_mult_self2
% 0.19/0.43 thf(fact_934_Suc__mod__mult__self1,axiom,
% 0.19/0.43 ! [M2: nat,K: nat,N: nat] :
% 0.19/0.43 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K @ N ) ) ) @ N )
% 0.19/0.43 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 0.19/0.43
% 0.19/0.43 % Suc_mod_mult_self1
% 0.19/0.43 thf(fact_935_ceiling__add__of__int,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.43 => ! [X: A,Z2: int] :
% 0.19/0.43 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) )
% 0.19/0.43 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 ) ) ) ).
% 0.19/0.43
% 0.19/0.43 % ceiling_add_of_int
% 0.19/0.43 thf(fact_936_power__strict__decreasing__iff,axiom,
% 0.19/0.43 ! [A: $tType] :
% 0.19/0.43 ( ( linordered_semidom @ A )
% 0.19/0.43 => ! [B2: A,M2: nat,N: nat] :
% 0.19/0.43 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.44 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
% 0.19/0.44 = ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_strict_decreasing_iff
% 0.19/0.44 thf(fact_937_power__increasing__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [B2: A,X: nat,Y: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
% 0.19/0.44 = ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_increasing_iff
% 0.19/0.44 thf(fact_938_Suc__diff__1,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.19/0.44 = N ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_diff_1
% 0.19/0.44 thf(fact_939_of__int__1__le__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [Z2: int] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
% 0.19/0.44 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_1_le_iff
% 0.19/0.44 thf(fact_940_of__int__le__1__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [Z2: int] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
% 0.19/0.44 = ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_le_1_iff
% 0.19/0.44 thf(fact_941_of__int__less__1__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [Z2: int] :
% 0.19/0.44 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
% 0.19/0.44 = ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_less_1_iff
% 0.19/0.44 thf(fact_942_of__int__1__less__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [Z2: int] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
% 0.19/0.44 = ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_1_less_iff
% 0.19/0.44 thf(fact_943_power__gt1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_gt1
% 0.19/0.44 thf(fact_944_add__Suc__shift,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.44 = ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_Suc_shift
% 0.19/0.44 thf(fact_945_add__Suc,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.44 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_Suc
% 0.19/0.44 thf(fact_946_nat__arith_Osuc1,axiom,
% 0.19/0.44 ! [A4: nat,K: nat,A3: nat] :
% 0.19/0.44 ( ( A4
% 0.19/0.44 = ( plus_plus @ nat @ K @ A3 ) )
% 0.19/0.44 => ( ( suc @ A4 )
% 0.19/0.44 = ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_arith.suc1
% 0.19/0.44 thf(fact_947_power__Suc__le__self,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_Suc_le_self
% 0.19/0.44 thf(fact_948_power__Suc__less__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_Suc_less_one
% 0.19/0.44 thf(fact_949_power__minus_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [X: A,N: nat] :
% 0.19/0.44 ( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
% 0.19/0.44 => ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
% 0.19/0.44 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_minus'
% 0.19/0.44 thf(fact_950_one__le__power,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_le_power
% 0.19/0.44 thf(fact_951_left__right__inverse__power,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( monoid_mult @ A )
% 0.19/0.44 => ! [X: A,Y: A,N: nat] :
% 0.19/0.44 ( ( ( times_times @ A @ X @ Y )
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 => ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % left_right_inverse_power
% 0.19/0.44 thf(fact_952_power__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( power @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_0
% 0.19/0.44 thf(fact_953_power__one__over,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( division_ring @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
% 0.19/0.44 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_one_over
% 0.19/0.44 thf(fact_954_power__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( monoid_mult @ A )
% 0.19/0.44 => ! [A3: A,M2: nat,N: nat] :
% 0.19/0.44 ( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.44 = ( times_times @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_add
% 0.19/0.44 thf(fact_955_nat__compl__induct_H,axiom,
% 0.19/0.44 ! [P: nat > $o,N: nat] :
% 0.19/0.44 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.44 => ( ! [N3: nat] :
% 0.19/0.44 ( ! [Nn: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ Nn @ N3 )
% 0.19/0.44 => ( P @ Nn ) )
% 0.19/0.44 => ( P @ ( suc @ N3 ) ) )
% 0.19/0.44 => ( P @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_compl_induct'
% 0.19/0.44 thf(fact_956_nat__compl__induct,axiom,
% 0.19/0.44 ! [P: nat > $o,N: nat] :
% 0.19/0.44 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.44 => ( ! [N3: nat] :
% 0.19/0.44 ( ! [Nn: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ Nn @ N3 )
% 0.19/0.44 => ( P @ Nn ) )
% 0.19/0.44 => ( P @ ( suc @ N3 ) ) )
% 0.19/0.44 => ( P @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_compl_induct
% 0.19/0.44 thf(fact_957_one__is__add,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ( suc @ ( zero_zero @ nat ) )
% 0.19/0.44 = ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.44 = ( ( ( M2
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.44 & ( N
% 0.19/0.44 = ( zero_zero @ nat ) ) )
% 0.19/0.44 | ( ( M2
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 & ( N
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_is_add
% 0.19/0.44 thf(fact_958_add__is__1,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ( plus_plus @ nat @ M2 @ N )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.44 = ( ( ( M2
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.44 & ( N
% 0.19/0.44 = ( zero_zero @ nat ) ) )
% 0.19/0.44 | ( ( M2
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 & ( N
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_is_1
% 0.19/0.44 thf(fact_959_nat__in__between__eq_I2_J,axiom,
% 0.19/0.44 ! [A3: nat,B2: nat] :
% 0.19/0.44 ( ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.19/0.44 & ( ord_less @ nat @ B2 @ ( suc @ A3 ) ) )
% 0.19/0.44 = ( B2 = A3 ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_in_between_eq(2)
% 0.19/0.44 thf(fact_960_nat__in__between__eq_I1_J,axiom,
% 0.19/0.44 ! [A3: nat,B2: nat] :
% 0.19/0.44 ( ( ( ord_less @ nat @ A3 @ B2 )
% 0.19/0.44 & ( ord_less_eq @ nat @ B2 @ ( suc @ A3 ) ) )
% 0.19/0.44 = ( B2
% 0.19/0.44 = ( suc @ A3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_in_between_eq(1)
% 0.19/0.44 thf(fact_961_less__imp__Suc__add,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.44 => ? [K5: nat] :
% 0.19/0.44 ( N
% 0.19/0.44 = ( suc @ ( plus_plus @ nat @ M2 @ K5 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_imp_Suc_add
% 0.19/0.44 thf(fact_962_less__iff__Suc__add,axiom,
% 0.19/0.44 ( ( ord_less @ nat )
% 0.19/0.44 = ( ^ [M: nat,N5: nat] :
% 0.19/0.44 ? [K4: nat] :
% 0.19/0.44 ( N5
% 0.19/0.44 = ( suc @ ( plus_plus @ nat @ M @ K4 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_iff_Suc_add
% 0.19/0.44 thf(fact_963_less__add__Suc2,axiom,
% 0.19/0.44 ! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_add_Suc2
% 0.19/0.44 thf(fact_964_less__add__Suc1,axiom,
% 0.19/0.44 ! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_add_Suc1
% 0.19/0.44 thf(fact_965_less__natE,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.44 => ~ ! [Q2: nat] :
% 0.19/0.44 ( N
% 0.19/0.44 != ( suc @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_natE
% 0.19/0.44 thf(fact_966_Suc__to__right,axiom,
% 0.19/0.44 ! [N: nat,M2: nat] :
% 0.19/0.44 ( ( ( suc @ N )
% 0.19/0.44 = M2 )
% 0.19/0.44 => ( N
% 0.19/0.44 = ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_to_right
% 0.19/0.44 thf(fact_967_One__nat__def,axiom,
% 0.19/0.44 ( ( one_one @ nat )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % One_nat_def
% 0.19/0.44 thf(fact_968_mult__Suc,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( times_times @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.44 = ( plus_plus @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mult_Suc
% 0.19/0.44 thf(fact_969_Suc__eq__plus1__left,axiom,
% 0.19/0.44 ( suc
% 0.19/0.44 = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_eq_plus1_left
% 0.19/0.44 thf(fact_970_plus__1__eq__Suc,axiom,
% 0.19/0.44 ( ( plus_plus @ nat @ ( one_one @ nat ) )
% 0.19/0.44 = suc ) ).
% 0.19/0.44
% 0.19/0.44 % plus_1_eq_Suc
% 0.19/0.44 thf(fact_971_Suc__eq__plus1,axiom,
% 0.19/0.44 ( suc
% 0.19/0.44 = ( ^ [N5: nat] : ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_eq_plus1
% 0.19/0.44 thf(fact_972_diff__Suc__eq__diff__pred,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.44 = ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_Suc_eq_diff_pred
% 0.19/0.44 thf(fact_973_power__le__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_le_one
% 0.19/0.44 thf(fact_974_power__less__power__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_less_power_Suc
% 0.19/0.44 thf(fact_975_power__gt1__lemma,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_gt1_lemma
% 0.19/0.44 thf(fact_976_power__0__left,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( ( N
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.19/0.44 = ( one_one @ A ) ) )
% 0.19/0.44 & ( ( N
% 0.19/0.44 != ( zero_zero @ nat ) )
% 0.19/0.44 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 0.19/0.44 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_0_left
% 0.19/0.44 thf(fact_977_power__strict__increasing,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [N: nat,N6: nat,A3: A] :
% 0.19/0.44 ( ( ord_less @ nat @ N @ N6 )
% 0.19/0.44 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_strict_increasing
% 0.19/0.44 thf(fact_978_power__less__imp__less__exp,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,M2: nat,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 0.19/0.44 => ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_less_imp_less_exp
% 0.19/0.44 thf(fact_979_power__minus,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.19/0.44 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_minus
% 0.19/0.44 thf(fact_980_power__increasing,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [N: nat,N6: nat,A3: A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ N @ N6 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_increasing
% 0.19/0.44 thf(fact_981_floor__add__int,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,Z2: int] :
% 0.19/0.44 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
% 0.19/0.44 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_add_int
% 0.19/0.44 thf(fact_982_int__add__floor,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [Z2: int,X: A] :
% 0.19/0.44 ( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.19/0.44 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_add_floor
% 0.19/0.44 thf(fact_983_nat__induct__non__zero,axiom,
% 0.19/0.44 ! [N: nat,P: nat > $o] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ( P @ ( one_one @ nat ) )
% 0.19/0.44 => ( ! [N3: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.19/0.44 => ( ( P @ N3 )
% 0.19/0.44 => ( P @ ( suc @ N3 ) ) ) )
% 0.19/0.44 => ( P @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_induct_non_zero
% 0.19/0.44 thf(fact_984_of__int__leD,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [N: int,X: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 0.19/0.44 => ( ( N
% 0.19/0.44 = ( zero_zero @ int ) )
% 0.19/0.44 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_leD
% 0.19/0.44 thf(fact_985_of__int__lessD,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [N: int,X: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 0.19/0.44 => ( ( N
% 0.19/0.44 = ( zero_zero @ int ) )
% 0.19/0.44 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_lessD
% 0.19/0.44 thf(fact_986_floor__exists,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim462609752435547400_field @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ? [Z3: int] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
% 0.19/0.44 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_exists
% 0.19/0.44 thf(fact_987_floor__exists1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim462609752435547400_field @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ? [X2: int] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ X )
% 0.19/0.44 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) )
% 0.19/0.44 & ! [Y6: int] :
% 0.19/0.44 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X )
% 0.19/0.44 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
% 0.19/0.44 => ( Y6 = X2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_exists1
% 0.19/0.44 thf(fact_988_power__Suc__less,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_Suc_less
% 0.19/0.44 thf(fact_989_power__strict__decreasing,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [N: nat,N6: nat,A3: A] :
% 0.19/0.44 ( ( ord_less @ nat @ N @ N6 )
% 0.19/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_strict_decreasing
% 0.19/0.44 thf(fact_990_power__decreasing,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [N: nat,N6: nat,A3: A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ N @ N6 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_decreasing
% 0.19/0.44 thf(fact_991_power__le__imp__le__exp,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,M2: nat,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 0.19/0.44 => ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_le_imp_le_exp
% 0.19/0.44 thf(fact_992_self__le__power,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % self_le_power
% 0.19/0.44 thf(fact_993_one__less__power,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_semidom @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_less_power
% 0.19/0.44 thf(fact_994_nz__le__conv__less,axiom,
% 0.19/0.44 ! [K: nat,M2: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.19/0.44 => ( ( ord_less_eq @ nat @ K @ M2 )
% 0.19/0.44 => ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nz_le_conv_less
% 0.19/0.44 thf(fact_995_ceiling__altdef,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ( ( archimedean_ceiling @ A )
% 0.19/0.44 = ( ^ [X3: A] :
% 0.19/0.44 ( if @ int
% 0.19/0.44 @ ( X3
% 0.19/0.44 = ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
% 0.19/0.44 @ ( archim6421214686448440834_floor @ A @ X3 )
% 0.19/0.44 @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_altdef
% 0.19/0.44 thf(fact_996_Suc__diff__eq__diff__pred,axiom,
% 0.19/0.44 ! [N: nat,M2: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
% 0.19/0.44 = ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_diff_eq_diff_pred
% 0.19/0.44 thf(fact_997_Suc__pred_H,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( N
% 0.19/0.44 = ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_pred'
% 0.19/0.44 thf(fact_998_add__eq__if,axiom,
% 0.19/0.44 ( ( plus_plus @ nat )
% 0.19/0.44 = ( ^ [M: nat,N5: nat] :
% 0.19/0.44 ( if @ nat
% 0.19/0.44 @ ( M
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 @ N5
% 0.19/0.44 @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_eq_if
% 0.19/0.44 thf(fact_999_Suc__n__minus__m__eq,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( ord_less @ nat @ ( one_one @ nat ) @ M2 )
% 0.19/0.44 => ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
% 0.19/0.44 = ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_n_minus_m_eq
% 0.19/0.44 thf(fact_1000_floor__unique,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [Z2: int,X: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
% 0.19/0.44 => ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
% 0.19/0.44 => ( ( archim6421214686448440834_floor @ A @ X )
% 0.19/0.44 = Z2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_unique
% 0.19/0.44 thf(fact_1001_floor__eq__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,A3: int] :
% 0.19/0.44 ( ( ( archim6421214686448440834_floor @ A @ X )
% 0.19/0.44 = A3 )
% 0.19/0.44 = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
% 0.19/0.44 & ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_eq_iff
% 0.19/0.44 thf(fact_1002_floor__split,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [P: int > $o,T2: A] :
% 0.19/0.44 ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
% 0.19/0.44 = ( ! [I3: int] :
% 0.19/0.44 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
% 0.19/0.44 & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
% 0.19/0.44 => ( P @ I3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_split
% 0.19/0.44 thf(fact_1003_ceiling__correct,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
% 0.19/0.44 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_correct
% 0.19/0.44 thf(fact_1004_ceiling__unique,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [Z2: int,X: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X )
% 0.19/0.44 => ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) )
% 0.19/0.44 => ( ( archimedean_ceiling @ A @ X )
% 0.19/0.44 = Z2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_unique
% 0.19/0.44 thf(fact_1005_ceiling__eq__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,A3: int] :
% 0.19/0.44 ( ( ( archimedean_ceiling @ A @ X )
% 0.19/0.44 = A3 )
% 0.19/0.44 = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
% 0.19/0.44 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_eq_iff
% 0.19/0.44 thf(fact_1006_ceiling__split,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [P: int > $o,T2: A] :
% 0.19/0.44 ( ( P @ ( archimedean_ceiling @ A @ T2 ) )
% 0.19/0.44 = ( ! [I3: int] :
% 0.19/0.44 ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
% 0.19/0.44 & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
% 0.19/0.44 => ( P @ I3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_split
% 0.19/0.44 thf(fact_1007_less__floor__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [Z2: int,X: A] :
% 0.19/0.44 ( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.19/0.44 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_floor_iff
% 0.19/0.44 thf(fact_1008_floor__le__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,Z2: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
% 0.19/0.44 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_le_iff
% 0.19/0.44 thf(fact_1009_ceiling__less__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,Z2: int] :
% 0.19/0.44 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
% 0.19/0.44 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_less_iff
% 0.19/0.44 thf(fact_1010_le__ceiling__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [Z2: int,X: A] :
% 0.19/0.44 ( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
% 0.19/0.44 = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % le_ceiling_iff
% 0.19/0.44 thf(fact_1011_floor__correct,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
% 0.19/0.44 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_correct
% 0.19/0.44 thf(fact_1012_power__diff__power__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.44 => ! [A3: A,N: nat,M2: nat] :
% 0.19/0.44 ( ( A3
% 0.19/0.44 != ( zero_zero @ A ) )
% 0.19/0.44 => ( ( ( ord_less_eq @ nat @ N @ M2 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 0.19/0.44 = ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
% 0.19/0.44 & ( ~ ( ord_less_eq @ nat @ N @ M2 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_diff_power_eq
% 0.19/0.44 thf(fact_1013_power__eq__if,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( power @ A )
% 0.19/0.44 => ( ( power_power @ A )
% 0.19/0.44 = ( ^ [P6: A,M: nat] :
% 0.19/0.44 ( if @ A
% 0.19/0.44 @ ( M
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 @ ( one_one @ A )
% 0.19/0.44 @ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_eq_if
% 0.19/0.44 thf(fact_1014_power__minus__mult,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( monoid_mult @ A )
% 0.19/0.44 => ! [N: nat,A3: A] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
% 0.19/0.44 = ( power_power @ A @ A3 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_minus_mult
% 0.19/0.44 thf(fact_1015_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [K: nat,N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ K @ N )
% 0.19/0.44 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
% 0.19/0.44 = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_one_power_add_eq_neg_one_power_diff
% 0.19/0.44 thf(fact_1016_option_Osize__gen_I2_J,axiom,
% 0.19/0.44 ! [A: $tType,X: A > nat,X22: A] :
% 0.19/0.44 ( ( size_option @ A @ X @ ( some @ A @ X22 ) )
% 0.19/0.44 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % option.size_gen(2)
% 0.19/0.44 thf(fact_1017_Heap_Osize__gen,axiom,
% 0.19/0.44 ! [A: $tType,Xa: A > nat,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.19/0.44 ( ( heap_Time_size_Heap @ A @ Xa @ ( heap_Time_Heap2 @ A @ X ) )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Heap.size_gen
% 0.19/0.44 thf(fact_1018_num_Osize__gen_I2_J,axiom,
% 0.19/0.44 ! [X22: num] :
% 0.19/0.44 ( ( size_num @ ( bit0 @ X22 ) )
% 0.19/0.44 = ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % num.size_gen(2)
% 0.19/0.44 thf(fact_1019_normalize__negative,axiom,
% 0.19/0.44 ! [Q3: int,P5: int] :
% 0.19/0.44 ( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
% 0.19/0.44 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.19/0.44 = ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % normalize_negative
% 0.19/0.44 thf(fact_1020_option_Osize__gen_I1_J,axiom,
% 0.19/0.44 ! [A: $tType,X: A > nat] :
% 0.19/0.44 ( ( size_option @ A @ X @ ( none @ A ) )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % option.size_gen(1)
% 0.19/0.44 thf(fact_1021_Heap_Osize_I2_J,axiom,
% 0.19/0.44 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.19/0.44 ( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X ) )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Heap.size(2)
% 0.19/0.44 thf(fact_1022_num_Osize_I5_J,axiom,
% 0.19/0.44 ! [X22: num] :
% 0.19/0.44 ( ( size_size @ num @ ( bit0 @ X22 ) )
% 0.19/0.44 = ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % num.size(5)
% 0.19/0.44 thf(fact_1023_option_Osize_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( size_size @ ( option @ A ) @ ( none @ A ) )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % option.size(3)
% 0.19/0.44 thf(fact_1024_normalize__denom__zero,axiom,
% 0.19/0.44 ! [P5: int] :
% 0.19/0.44 ( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
% 0.19/0.44 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % normalize_denom_zero
% 0.19/0.44 thf(fact_1025_option_Osize__neq,axiom,
% 0.19/0.44 ! [A: $tType,X: option @ A] :
% 0.19/0.44 ( ( size_size @ ( option @ A ) @ X )
% 0.19/0.44 != ( zero_zero @ nat ) ) ).
% 0.19/0.44
% 0.19/0.44 % option.size_neq
% 0.19/0.44 thf(fact_1026_Heap_Osize__neq,axiom,
% 0.19/0.44 ! [A: $tType,X: heap_Time_Heap @ A] :
% 0.19/0.44 ( ( size_size @ ( heap_Time_Heap @ A ) @ X )
% 0.19/0.44 != ( zero_zero @ nat ) ) ).
% 0.19/0.44
% 0.19/0.44 % Heap.size_neq
% 0.19/0.44 thf(fact_1027_option_Osize__gen__o__map,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,F: B > nat,G: A > B] :
% 0.19/0.44 ( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F ) @ ( map_option @ A @ B @ G ) )
% 0.19/0.44 = ( size_option @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % option.size_gen_o_map
% 0.19/0.44 thf(fact_1028_normalize__denom__pos,axiom,
% 0.19/0.44 ! [R: product_prod @ int @ int,P5: int,Q3: int] :
% 0.19/0.44 ( ( ( normalize @ R )
% 0.19/0.44 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.19/0.44 => ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% 0.19/0.44
% 0.19/0.44 % normalize_denom_pos
% 0.19/0.44 thf(fact_1029_normalize__crossproduct,axiom,
% 0.19/0.44 ! [Q3: int,S: int,P5: int,R: int] :
% 0.19/0.44 ( ( Q3
% 0.19/0.44 != ( zero_zero @ int ) )
% 0.19/0.44 => ( ( S
% 0.19/0.44 != ( zero_zero @ int ) )
% 0.19/0.44 => ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.19/0.44 = ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
% 0.19/0.44 => ( ( times_times @ int @ P5 @ S )
% 0.19/0.44 = ( times_times @ int @ R @ Q3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % normalize_crossproduct
% 0.19/0.44 thf(fact_1030_option_Osize_I4_J,axiom,
% 0.19/0.44 ! [A: $tType,X22: A] :
% 0.19/0.44 ( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
% 0.19/0.44 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % option.size(4)
% 0.19/0.44 thf(fact_1031_divmod__nat__if,axiom,
% 0.19/0.44 ( divmod_nat
% 0.19/0.44 = ( ^ [M: nat,N5: nat] :
% 0.19/0.44 ( if @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ( ( N5
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 | ( ord_less @ nat @ M @ N5 ) )
% 0.19/0.44 @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M )
% 0.19/0.44 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
% 0.19/0.44 @ ( divmod_nat @ ( minus_minus @ nat @ M @ N5 ) @ N5 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_nat_if
% 0.19/0.44 thf(fact_1032_sum__gp__multiplied,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ( monoid_mult @ A )
% 0.19/0.44 & ( comm_ring @ A ) )
% 0.19/0.44 => ! [M2: nat,N: nat,X: A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
% 0.19/0.44 = ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_gp_multiplied
% 0.19/0.44 thf(fact_1033_pair__lessI2,axiom,
% 0.19/0.44 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.19/0.44 => ( ( ord_less @ nat @ S @ T2 )
% 0.19/0.44 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % pair_lessI2
% 0.19/0.44 thf(fact_1034_pair__less__iff1,axiom,
% 0.19/0.44 ! [X: nat,Y: nat,Z2: nat] :
% 0.19/0.44 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z2 ) ) @ fun_pair_less )
% 0.19/0.44 = ( ord_less @ nat @ Y @ Z2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % pair_less_iff1
% 0.19/0.44 thf(fact_1035_Suc__times__numeral__mod__eq,axiom,
% 0.19/0.44 ! [K: num,N: nat] :
% 0.19/0.44 ( ( ( numeral_numeral @ nat @ K )
% 0.19/0.44 != ( one_one @ nat ) )
% 0.19/0.44 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
% 0.19/0.44 = ( one_one @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_times_numeral_mod_eq
% 0.19/0.44 thf(fact_1036_one__int_Otransfer,axiom,
% 0.19/0.44 pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% 0.19/0.44
% 0.19/0.44 % one_int.transfer
% 0.19/0.44 thf(fact_1037_less__than__iff,axiom,
% 0.19/0.44 ! [X: nat,Y: nat] :
% 0.19/0.44 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
% 0.19/0.44 = ( ord_less @ nat @ X @ Y ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_than_iff
% 0.19/0.44 thf(fact_1038_one__int__def,axiom,
% 0.19/0.44 ( ( one_one @ int )
% 0.19/0.44 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_int_def
% 0.19/0.44 thf(fact_1039_sum__telescope_H_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ab_group_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,F: nat > A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) ) ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
% 0.19/0.44 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_telescope''
% 0.19/0.44 thf(fact_1040_case__prod__conv,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
% 0.19/0.44 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 0.19/0.44 = ( F @ A3 @ B2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prod_conv
% 0.19/0.44 thf(fact_1041_case__prod__curry,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 0.19/0.44 = F ) ).
% 0.19/0.44
% 0.19/0.44 % case_prod_curry
% 0.19/0.44 thf(fact_1042_curry__case__prod,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 0.19/0.44 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 0.19/0.44 = F ) ).
% 0.19/0.44
% 0.19/0.44 % curry_case_prod
% 0.19/0.44 thf(fact_1043_distrib__right__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ( numeral @ A )
% 0.19/0.44 & ( semiring @ A ) )
% 0.19/0.44 => ! [A3: A,B2: A,V: num] :
% 0.19/0.44 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
% 0.19/0.44 = ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % distrib_right_numeral
% 0.19/0.44 thf(fact_1044_distrib__left__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ( numeral @ A )
% 0.19/0.44 & ( semiring @ A ) )
% 0.19/0.44 => ! [V: num,B2: A,C2: A] :
% 0.19/0.44 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.44 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % distrib_left_numeral
% 0.19/0.44 thf(fact_1045_add__neg__numeral__simps_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_neg_numeral_simps(3)
% 0.19/0.44 thf(fact_1046_add__numeral__left,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [V: num,W: num,Z2: A] :
% 0.19/0.44 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
% 0.19/0.44 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_numeral_left
% 0.19/0.44 thf(fact_1047_numeral__plus__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_plus_numeral
% 0.19/0.44 thf(fact_1048_semiring__norm_I168_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ! [V: num,W: num,Y: A] :
% 0.19/0.44 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
% 0.19/0.44 = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(168)
% 0.19/0.44 thf(fact_1049_diff__numeral__simps_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_numeral_simps(3)
% 0.19/0.44 thf(fact_1050_diff__numeral__simps_I2_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_numeral_simps(2)
% 0.19/0.44 thf(fact_1051_ceiling__add__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,V: num] :
% 0.19/0.44 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 0.19/0.44 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_add_numeral
% 0.19/0.44 thf(fact_1052_power__add__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( monoid_mult @ A )
% 0.19/0.44 => ! [A3: A,M2: num,N: num] :
% 0.19/0.44 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
% 0.19/0.44 = ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_add_numeral
% 0.19/0.44 thf(fact_1053_power__add__numeral2,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( monoid_mult @ A )
% 0.19/0.44 => ! [A3: A,M2: num,N: num,B2: A] :
% 0.19/0.44 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
% 0.19/0.44 = ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_add_numeral2
% 0.19/0.44 thf(fact_1054_numeral__less__floor,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [V: num,X: A] :
% 0.19/0.44 ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.19/0.44 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_less_floor
% 0.19/0.44 thf(fact_1055_floor__le__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,V: num] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.19/0.44 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_le_numeral
% 0.19/0.44 thf(fact_1056_sum_Ocl__ivl__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [N: nat,M2: nat,G: nat > A] :
% 0.19/0.44 ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.19/0.44 = ( zero_zero @ A ) ) )
% 0.19/0.44 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.cl_ivl_Suc
% 0.19/0.44 thf(fact_1057_ceiling__less__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,V: num] :
% 0.19/0.44 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 0.19/0.44 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_less_numeral
% 0.19/0.44 thf(fact_1058_numeral__le__ceiling,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [V: num,X: A] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
% 0.19/0.44 = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_le_ceiling
% 0.19/0.44 thf(fact_1059_neg__numeral__less__floor,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [V: num,X: A] :
% 0.19/0.44 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 0.19/0.44 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_numeral_less_floor
% 0.19/0.44 thf(fact_1060_floor__le__neg__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,V: num] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.19/0.44 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % floor_le_neg_numeral
% 0.19/0.44 thf(fact_1061_ceiling__less__neg__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,V: num] :
% 0.19/0.44 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 0.19/0.44 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ceiling_less_neg_numeral
% 0.19/0.44 thf(fact_1062_neg__numeral__le__ceiling,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [V: num,X: A] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
% 0.19/0.44 = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_numeral_le_ceiling
% 0.19/0.44 thf(fact_1063_nested__case__prod__simp,axiom,
% 0.19/0.44 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.19/0.44 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 0.19/0.44 = ( ^ [F2: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
% 0.19/0.44 ( product_case_prod @ B @ C @ A
% 0.19/0.44 @ ^ [A7: B,B5: C] : ( F2 @ A7 @ B5 @ Y3 )
% 0.19/0.44 @ X3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nested_case_prod_simp
% 0.19/0.44 thf(fact_1064_prod_Ocase__distrib,axiom,
% 0.19/0.44 ! [C: $tType,D: $tType,B: $tType,A: $tType,H3: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.19/0.44 ( ( H3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.19/0.44 = ( product_case_prod @ A @ B @ D
% 0.19/0.44 @ ^ [X12: A,X24: B] : ( H3 @ ( F @ X12 @ X24 ) )
% 0.19/0.44 @ Prod ) ) ).
% 0.19/0.44
% 0.19/0.44 % prod.case_distrib
% 0.19/0.44 thf(fact_1065_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,M2: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeast_Suc_atMost_Suc_shift
% 0.19/0.44 thf(fact_1066_sum_OatLeastAtMost__shift__bounds,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeastAtMost_shift_bounds
% 0.19/0.44 thf(fact_1067_split__cong,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
% 0.19/0.44 ( ! [X2: A,Y4: B] :
% 0.19/0.44 ( ( ( product_Pair @ A @ B @ X2 @ Y4 )
% 0.19/0.44 = Q3 )
% 0.19/0.44 => ( ( F @ X2 @ Y4 )
% 0.19/0.44 = ( G @ X2 @ Y4 ) ) )
% 0.19/0.44 => ( ( P5 = Q3 )
% 0.19/0.44 => ( ( product_case_prod @ A @ B @ C @ F @ P5 )
% 0.19/0.44 = ( product_case_prod @ A @ B @ C @ G @ Q3 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % split_cong
% 0.19/0.44 thf(fact_1068_old_Oprod_Ocase,axiom,
% 0.19/0.44 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.19/0.44 = ( F @ X1 @ X22 ) ) ).
% 0.19/0.44
% 0.19/0.44 % old.prod.case
% 0.19/0.44 thf(fact_1069_uminus__int_Oabs__eq,axiom,
% 0.19/0.44 ! [X: product_prod @ nat @ nat] :
% 0.19/0.44 ( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
% 0.19/0.44 = ( abs_Integ
% 0.19/0.44 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 )
% 0.19/0.44 @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % uminus_int.abs_eq
% 0.19/0.44 thf(fact_1070_case__prod__Pair__iden,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
% 0.19/0.44 = P5 ) ).
% 0.19/0.44
% 0.19/0.44 % case_prod_Pair_iden
% 0.19/0.44 thf(fact_1071_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,M2: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.shift_bounds_cl_Suc_ivl
% 0.19/0.44 thf(fact_1072_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.shift_bounds_cl_nat_ivl
% 0.19/0.44 thf(fact_1073_cond__case__prod__eta,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 0.19/0.44 ( ! [X2: A,Y4: B] :
% 0.19/0.44 ( ( F @ X2 @ Y4 )
% 0.19/0.44 = ( G @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
% 0.19/0.44 => ( ( product_case_prod @ A @ B @ C @ F )
% 0.19/0.44 = G ) ) ).
% 0.19/0.44
% 0.19/0.44 % cond_case_prod_eta
% 0.19/0.44 thf(fact_1074_case__prod__eta,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ C
% 0.19/0.44 @ ^ [X3: A,Y3: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 0.19/0.44 = F ) ).
% 0.19/0.44
% 0.19/0.44 % case_prod_eta
% 0.19/0.44 thf(fact_1075_case__prodE2,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
% 0.19/0.44 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
% 0.19/0.44 => ~ ! [X2: B,Y4: C] :
% 0.19/0.44 ( ( Z2
% 0.19/0.44 = ( product_Pair @ B @ C @ X2 @ Y4 ) )
% 0.19/0.44 => ~ ( Q @ ( P @ X2 @ Y4 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodE2
% 0.19/0.44 thf(fact_1076_eq__Abs__Integ,axiom,
% 0.19/0.44 ! [Z2: int] :
% 0.19/0.44 ~ ! [X2: nat,Y4: nat] :
% 0.19/0.44 ( Z2
% 0.19/0.44 != ( abs_Integ @ ( product_Pair @ nat @ nat @ X2 @ Y4 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % eq_Abs_Integ
% 0.19/0.44 thf(fact_1077_one__le__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linord181362715937106298miring @ A )
% 0.19/0.44 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_le_numeral
% 0.19/0.44 thf(fact_1078_not__numeral__less__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linord181362715937106298miring @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_numeral_less_one
% 0.19/0.44 thf(fact_1079_one__plus__numeral__commute,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [X: num] :
% 0.19/0.44 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.19/0.44 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_plus_numeral_commute
% 0.19/0.44 thf(fact_1080_numeral__Bit0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 0.19/0.44 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_Bit0
% 0.19/0.44 thf(fact_1081_one__neq__neg__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_char_0 @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( one_one @ A )
% 0.19/0.44 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_neq_neg_numeral
% 0.19/0.44 thf(fact_1082_numeral__neq__neg__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_char_0 @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( numeral_numeral @ A @ N )
% 0.19/0.44 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_neq_neg_one
% 0.19/0.44 thf(fact_1083_uncurry__def,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.19/0.44 ( ( uncurry @ A @ B @ C )
% 0.19/0.44 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.19/0.44
% 0.19/0.44 % uncurry_def
% 0.19/0.44 thf(fact_1084_internal__case__prod__def,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType] :
% 0.19/0.44 ( ( produc5280177257484947105e_prod @ A @ B @ C )
% 0.19/0.44 = ( product_case_prod @ A @ B @ C ) ) ).
% 0.19/0.44
% 0.19/0.44 % internal_case_prod_def
% 0.19/0.44 thf(fact_1085_sum_OatLeastAtMost__rev,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,N: nat,M2: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeastAtMost_rev
% 0.19/0.44 thf(fact_1086_numeral__code_I2_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 0.19/0.44 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_code(2)
% 0.19/0.44 thf(fact_1087_sum__shift__lb__Suc0__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [F: nat > A,K: nat] :
% 0.19/0.44 ( ( ( F @ ( zero_zero @ nat ) )
% 0.19/0.44 = ( zero_zero @ A ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_shift_lb_Suc0_0
% 0.19/0.44 thf(fact_1088_sum_OatLeast0__atMost__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeast0_atMost_Suc
% 0.19/0.44 thf(fact_1089_sum_OatLeast0__atMost__Suc__shift,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeast0_atMost_Suc_shift
% 0.19/0.44 thf(fact_1090_sum_OatLeast__Suc__atMost,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,G: nat > A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeast_Suc_atMost
% 0.19/0.44 thf(fact_1091_sum_Onat__ivl__Suc_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,G: nat > A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.nat_ivl_Suc'
% 0.19/0.44 thf(fact_1092_neg__numeral__le__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_numeral_le_one
% 0.19/0.44 thf(fact_1093_neg__one__le__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_one_le_numeral
% 0.19/0.44 thf(fact_1094_neg__numeral__le__neg__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_numeral_le_neg_one
% 0.19/0.44 thf(fact_1095_not__numeral__le__neg__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_numeral_le_neg_one
% 0.19/0.44 thf(fact_1096_not__one__le__neg__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_one_le_neg_numeral
% 0.19/0.44 thf(fact_1097_not__neg__one__less__neg__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_neg_one_less_neg_numeral
% 0.19/0.44 thf(fact_1098_not__one__less__neg__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_one_less_neg_numeral
% 0.19/0.44 thf(fact_1099_not__numeral__less__neg__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_numeral_less_neg_one
% 0.19/0.44 thf(fact_1100_neg__one__less__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_one_less_numeral
% 0.19/0.44 thf(fact_1101_neg__numeral__less__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_numeral_less_one
% 0.19/0.44 thf(fact_1102_sum_OSuc__reindex__ivl,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,G: nat > A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( G @ M2 )
% 0.19/0.44 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.Suc_reindex_ivl
% 0.19/0.44 thf(fact_1103_sum__Suc__diff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ab_group_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,F: nat > A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( minus_minus @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_Suc_diff
% 0.19/0.44 thf(fact_1104_sum__power__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ( monoid_mult @ A )
% 0.19/0.44 & ( comm_ring @ A ) )
% 0.19/0.44 => ! [X: A,M2: nat,I4: set @ nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I3 ) )
% 0.19/0.44 @ I4 )
% 0.19/0.44 = ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I4 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_power_add
% 0.19/0.44 thf(fact_1105_sum_OatLeast__atMost__pred__shift,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,M2: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ( comp @ nat @ A @ nat @ G
% 0.19/0.44 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeast_atMost_pred_shift
% 0.19/0.44 thf(fact_1106_sum__atLeastAtMost__code,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [F: nat > A,A3: nat,B2: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 0.19/0.44 = ( set_fo6178422350223883121st_nat @ A
% 0.19/0.44 @ ^ [A7: nat] : ( plus_plus @ A @ ( F @ A7 ) )
% 0.19/0.44 @ A3
% 0.19/0.44 @ B2
% 0.19/0.44 @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_atLeastAtMost_code
% 0.19/0.44 thf(fact_1107_sum_Oub__add__nat,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,G: nat > A,P5: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.ub_add_nat
% 0.19/0.44 thf(fact_1108_sum_OatLeastAtMost__shift__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,G: nat > A] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeastAtMost_shift_0
% 0.19/0.44 thf(fact_1109_zero__int__def,axiom,
% 0.19/0.44 ( ( zero_zero @ int )
% 0.19/0.44 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % zero_int_def
% 0.19/0.44 thf(fact_1110_pair__lessI1,axiom,
% 0.19/0.44 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ A3 @ B2 )
% 0.19/0.44 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% 0.19/0.44
% 0.19/0.44 % pair_lessI1
% 0.19/0.44 thf(fact_1111_zero__int_Otransfer,axiom,
% 0.19/0.44 pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% 0.19/0.44
% 0.19/0.44 % zero_int.transfer
% 0.19/0.44 thf(fact_1112_sum__natinterval__diff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ab_group_add @ A )
% 0.19/0.44 => ! [M2: nat,N: nat,F: nat > A] :
% 0.19/0.44 ( ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( minus_minus @ A @ ( F @ M2 ) @ ( F @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
% 0.19/0.44 & ( ~ ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_natinterval_diff
% 0.19/0.44 thf(fact_1113_convex__sum__bound__le,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType] :
% 0.19/0.44 ( ( linordered_idom @ B )
% 0.19/0.44 => ! [I4: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
% 0.19/0.44 ( ! [I2: A] :
% 0.19/0.44 ( ( member @ A @ I2 @ I4 )
% 0.19/0.44 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
% 0.19/0.44 => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I4 )
% 0.19/0.44 = ( one_one @ B ) )
% 0.19/0.44 => ( ! [I2: A] :
% 0.19/0.44 ( ( member @ A @ I2 @ I4 )
% 0.19/0.44 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I2 ) @ B2 ) ) @ Delta ) )
% 0.19/0.44 => ( ord_less_eq @ B
% 0.19/0.44 @ ( abs_abs @ B
% 0.19/0.44 @ ( minus_minus @ B
% 0.19/0.44 @ ( groups7311177749621191930dd_sum @ A @ B
% 0.19/0.44 @ ^ [I3: A] : ( times_times @ B @ ( A3 @ I3 ) @ ( X @ I3 ) )
% 0.19/0.44 @ I4 )
% 0.19/0.44 @ B2 ) )
% 0.19/0.44 @ Delta ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % convex_sum_bound_le
% 0.19/0.44 thf(fact_1114_sum__comp__morphism,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,C: $tType] :
% 0.19/0.44 ( ( ( comm_monoid_add @ B )
% 0.19/0.44 & ( comm_monoid_add @ A ) )
% 0.19/0.44 => ! [H3: B > A,G: C > B,A4: set @ C] :
% 0.19/0.44 ( ( ( H3 @ ( zero_zero @ B ) )
% 0.19/0.44 = ( zero_zero @ A ) )
% 0.19/0.44 => ( ! [X2: B,Y4: B] :
% 0.19/0.44 ( ( H3 @ ( plus_plus @ B @ X2 @ Y4 ) )
% 0.19/0.44 = ( plus_plus @ A @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H3 @ G ) @ A4 )
% 0.19/0.44 = ( H3 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A4 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_comp_morphism
% 0.19/0.44 thf(fact_1115_sum__gp,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ( division_ring @ A )
% 0.19/0.44 & ( comm_ring @ A ) )
% 0.19/0.44 => ! [N: nat,M2: nat,X: A] :
% 0.19/0.44 ( ( ( ord_less @ nat @ N @ M2 )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( zero_zero @ A ) ) )
% 0.19/0.44 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.19/0.44 => ( ( ( X
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
% 0.19/0.44 & ( ( X
% 0.19/0.44 != ( one_one @ A ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_gp
% 0.19/0.44 thf(fact_1116_divmod__digit__1_I1_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 0.19/0.44 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
% 0.19/0.44 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_digit_1(1)
% 0.19/0.44 thf(fact_1117_sum__gp__offset,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ( division_ring @ A )
% 0.19/0.44 & ( comm_ring @ A ) )
% 0.19/0.44 => ! [X: A,M2: nat,N: nat] :
% 0.19/0.44 ( ( ( X
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
% 0.19/0.44 & ( ( X
% 0.19/0.44 != ( one_one @ A ) )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
% 0.19/0.44 = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_gp_offset
% 0.19/0.44 thf(fact_1118_Sum__Icc__nat,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.44 @ ^ [X3: nat] : X3
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.44 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Sum_Icc_nat
% 0.19/0.44 thf(fact_1119_arith__series__nat,axiom,
% 0.19/0.44 ! [A3: nat,D3: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.44 @ ^ [I3: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I3 @ D3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.19/0.44 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % arith_series_nat
% 0.19/0.44 thf(fact_1120_mod__double__modulus,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [M2: A,X: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
% 0.19/0.44 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.19/0.44 => ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 0.19/0.44 = ( modulo_modulo @ A @ X @ M2 ) )
% 0.19/0.44 | ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 0.19/0.44 = ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mod_double_modulus
% 0.19/0.44 thf(fact_1121_sum_Odistrib,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: B > A,H3: B > A,A4: set @ B] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ B @ A
% 0.19/0.44 @ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H3 @ X3 ) )
% 0.19/0.44 @ A4 )
% 0.19/0.44 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ A4 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.distrib
% 0.19/0.44 thf(fact_1122_case__prodI,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
% 0.19/0.44 ( ( F @ A3 @ B2 )
% 0.19/0.44 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodI
% 0.19/0.44 thf(fact_1123_case__prodI2,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
% 0.19/0.44 ( ! [A5: A,B3: B] :
% 0.19/0.44 ( ( P5
% 0.19/0.44 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.19/0.44 => ( C2 @ A5 @ B3 ) )
% 0.19/0.44 => ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodI2
% 0.19/0.44 thf(fact_1124_mem__case__prodI,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B2: C] :
% 0.19/0.44 ( ( member @ A @ Z2 @ ( C2 @ A3 @ B2 ) )
% 0.19/0.44 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mem_case_prodI
% 0.19/0.44 thf(fact_1125_mem__case__prodI2,axiom,
% 0.19/0.44 ! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
% 0.19/0.44 ( ! [A5: A,B3: B] :
% 0.19/0.44 ( ( P5
% 0.19/0.44 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.19/0.44 => ( member @ C @ Z2 @ ( C2 @ A5 @ B3 ) ) )
% 0.19/0.44 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mem_case_prodI2
% 0.19/0.44 thf(fact_1126_case__prodI2_H,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 0.19/0.44 ( ! [A5: A,B3: B] :
% 0.19/0.44 ( ( ( product_Pair @ A @ B @ A5 @ B3 )
% 0.19/0.44 = P5 )
% 0.19/0.44 => ( C2 @ A5 @ B3 @ X ) )
% 0.19/0.44 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodI2'
% 0.19/0.44 thf(fact_1127_one__eq__numeral__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_char_0 @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( ( one_one @ A )
% 0.19/0.44 = ( numeral_numeral @ A @ N ) )
% 0.19/0.44 = ( one2 = N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_eq_numeral_iff
% 0.19/0.44 thf(fact_1128_numeral__eq__one__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_char_0 @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( ( numeral_numeral @ A @ N )
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 = ( N = one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_eq_one_iff
% 0.19/0.44 thf(fact_1129_of__nat__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.44 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_nat_add
% 0.19/0.44 thf(fact_1130_of__nat__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_nat_1
% 0.19/0.44 thf(fact_1131_of__nat__1__eq__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_char_0 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( ( one_one @ A )
% 0.19/0.44 = ( semiring_1_of_nat @ A @ N ) )
% 0.19/0.44 = ( N
% 0.19/0.44 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_nat_1_eq_iff
% 0.19/0.44 thf(fact_1132_of__nat__eq__1__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_char_0 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( ( semiring_1_of_nat @ A @ N )
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 = ( N
% 0.19/0.44 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_nat_eq_1_iff
% 0.19/0.44 thf(fact_1133_semiring__norm_I2_J,axiom,
% 0.19/0.44 ( ( plus_plus @ num @ one2 @ one2 )
% 0.19/0.44 = ( bit0 @ one2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(2)
% 0.19/0.44 thf(fact_1134_numeral__le__one__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linord181362715937106298miring @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.19/0.44 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_le_one_iff
% 0.19/0.44 thf(fact_1135_one__less__numeral__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linord181362715937106298miring @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.44 = ( ord_less @ num @ one2 @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_less_numeral_iff
% 0.19/0.44 thf(fact_1136_numeral__eq__neg__one__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_char_0 @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( N = one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_eq_neg_one_iff
% 0.19/0.44 thf(fact_1137_neg__one__eq__numeral__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_char_0 @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( ( uminus_uminus @ A @ ( one_one @ A ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.19/0.44 = ( N = one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_one_eq_numeral_iff
% 0.19/0.44 thf(fact_1138_of__nat__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ! [M2: nat] :
% 0.19/0.44 ( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
% 0.19/0.44 = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_nat_Suc
% 0.19/0.44 thf(fact_1139_Suc__numeral,axiom,
% 0.19/0.44 ! [N: num] :
% 0.19/0.44 ( ( suc @ ( numeral_numeral @ nat @ N ) )
% 0.19/0.44 = ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_numeral
% 0.19/0.44 thf(fact_1140_not__neg__one__le__neg__numeral__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
% 0.19/0.44 = ( M2 != one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_neg_one_le_neg_numeral_iff
% 0.19/0.44 thf(fact_1141_one__add__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_add_one
% 0.19/0.44 thf(fact_1142_neg__numeral__less__neg__one__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( M2 != one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_numeral_less_neg_one_iff
% 0.19/0.44 thf(fact_1143_bits__one__mod__two__eq__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % bits_one_mod_two_eq_one
% 0.19/0.44 thf(fact_1144_one__mod__two__eq__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_mod_two_eq_one
% 0.19/0.44 thf(fact_1145_add__2__eq__Suc,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.44 = ( suc @ ( suc @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_2_eq_Suc
% 0.19/0.44 thf(fact_1146_add__2__eq__Suc_H,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( suc @ ( suc @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_2_eq_Suc'
% 0.19/0.44 thf(fact_1147_Suc__1,axiom,
% 0.19/0.44 ( ( suc @ ( one_one @ nat ) )
% 0.19/0.44 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_1
% 0.19/0.44 thf(fact_1148_numeral__plus__one,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_plus_one
% 0.19/0.44 thf(fact_1149_one__plus__numeral,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_plus_numeral
% 0.19/0.44 thf(fact_1150_add__self__div__2,axiom,
% 0.19/0.44 ! [M2: nat] :
% 0.19/0.44 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = M2 ) ).
% 0.19/0.44
% 0.19/0.44 % add_self_div_2
% 0.19/0.44 thf(fact_1151_bits__1__div__2,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( zero_zero @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % bits_1_div_2
% 0.19/0.44 thf(fact_1152_one__div__two__eq__zero,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( zero_zero @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_div_two_eq_zero
% 0.19/0.44 thf(fact_1153_add__neg__numeral__special_I9_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_neg_numeral_special(9)
% 0.19/0.44 thf(fact_1154_diff__numeral__special_I11_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_numeral_special(11)
% 0.19/0.44 thf(fact_1155_diff__numeral__special_I10_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_numeral_special(10)
% 0.19/0.44 thf(fact_1156_not__mod__2__eq__0__eq__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 != ( zero_zero @ A ) )
% 0.19/0.44 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_mod_2_eq_0_eq_1
% 0.19/0.44 thf(fact_1157_not__mod__2__eq__1__eq__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 != ( one_one @ A ) )
% 0.19/0.44 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_mod_2_eq_1_eq_0
% 0.19/0.44 thf(fact_1158_minus__1__div__2__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid8789492081693882211th_nat @ A )
% 0.19/0.44 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % minus_1_div_2_eq
% 0.19/0.44 thf(fact_1159_bits__minus__1__mod__2__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.44 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % bits_minus_1_mod_2_eq
% 0.19/0.44 thf(fact_1160_minus__1__mod__2__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid8789492081693882211th_nat @ A )
% 0.19/0.44 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % minus_1_mod_2_eq
% 0.19/0.44 thf(fact_1161_diff__numeral__special_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_numeral_special(3)
% 0.19/0.44 thf(fact_1162_diff__numeral__special_I4_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % diff_numeral_special(4)
% 0.19/0.44 thf(fact_1163_add__self__mod__2,axiom,
% 0.19/0.44 ! [M2: nat] :
% 0.19/0.44 ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( zero_zero @ nat ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_self_mod_2
% 0.19/0.44 thf(fact_1164_power__minus1__even,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_minus1_even
% 0.19/0.44 thf(fact_1165_mod2__gr__0,axiom,
% 0.19/0.44 ! [M2: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.19/0.44 = ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mod2_gr_0
% 0.19/0.44 thf(fact_1166_lex__prod__def,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType] :
% 0.19/0.44 ( ( lex_prod @ A @ B )
% 0.19/0.44 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 0.19/0.44 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.19/0.44 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.19/0.44 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.19/0.44 @ ^ [A7: A,B5: B] :
% 0.19/0.44 ( product_case_prod @ A @ B @ $o
% 0.19/0.44 @ ^ [A8: A,B7: B] :
% 0.19/0.44 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A8 ) @ Ra )
% 0.19/0.44 | ( ( A7 = A8 )
% 0.19/0.44 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % lex_prod_def
% 0.19/0.44 thf(fact_1167_same__fst__def,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType] :
% 0.19/0.44 ( ( same_fst @ A @ B )
% 0.19/0.44 = ( ^ [P4: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.19/0.44 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 0.19/0.44 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 0.19/0.44 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 0.19/0.44 @ ^ [X9: A,Y8: B] :
% 0.19/0.44 ( product_case_prod @ A @ B @ $o
% 0.19/0.44 @ ^ [X3: A,Y3: B] :
% 0.19/0.44 ( ( X9 = X3 )
% 0.19/0.44 & ( P4 @ X3 )
% 0.19/0.44 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y3 ) @ ( R6 @ X3 ) ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % same_fst_def
% 0.19/0.44 thf(fact_1168_mem__case__prodE,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P5: product_prod @ B @ C] :
% 0.19/0.44 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P5 ) )
% 0.19/0.44 => ~ ! [X2: B,Y4: C] :
% 0.19/0.44 ( ( P5
% 0.19/0.44 = ( product_Pair @ B @ C @ X2 @ Y4 ) )
% 0.19/0.44 => ~ ( member @ A @ Z2 @ ( C2 @ X2 @ Y4 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mem_case_prodE
% 0.19/0.44 thf(fact_1169_case__prodD,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.19/0.44 => ( F @ A3 @ B2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodD
% 0.19/0.44 thf(fact_1170_case__prodE,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
% 0.19/0.44 => ~ ! [X2: A,Y4: B] :
% 0.19/0.44 ( ( P5
% 0.19/0.44 = ( product_Pair @ A @ B @ X2 @ Y4 ) )
% 0.19/0.44 => ~ ( C2 @ X2 @ Y4 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodE
% 0.19/0.44 thf(fact_1171_add__One__commute,axiom,
% 0.19/0.44 ! [N: num] :
% 0.19/0.44 ( ( plus_plus @ num @ one2 @ N )
% 0.19/0.44 = ( plus_plus @ num @ N @ one2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % add_One_commute
% 0.19/0.44 thf(fact_1172_case__prodD_H,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,C: $tType,R5: A > B > C > $o,A3: A,B2: B,C2: C] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R5 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C2 )
% 0.19/0.44 => ( R5 @ A3 @ B2 @ C2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodD'
% 0.19/0.44 thf(fact_1173_case__prodE_H,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z2 )
% 0.19/0.44 => ~ ! [X2: A,Y4: B] :
% 0.19/0.44 ( ( P5
% 0.19/0.44 = ( product_Pair @ A @ B @ X2 @ Y4 ) )
% 0.19/0.44 => ~ ( C2 @ X2 @ Y4 @ Z2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % case_prodE'
% 0.19/0.44 thf(fact_1174_sum_OatLeast__int__atMost__int__shift,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: int > A,M2: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.atLeast_int_atMost_int_shift
% 0.19/0.44 thf(fact_1175_less__int_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.19/0.44 ( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.19/0.44 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.19/0.44 @ ^ [X3: nat,Y3: nat] :
% 0.19/0.44 ( product_case_prod @ nat @ nat @ $o
% 0.19/0.44 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
% 0.19/0.44 @ Xa
% 0.19/0.44 @ X ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_int.abs_eq
% 0.19/0.44 thf(fact_1176_less__eq__int_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.19/0.44 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.19/0.44 @ ^ [X3: nat,Y3: nat] :
% 0.19/0.44 ( product_case_prod @ nat @ nat @ $o
% 0.19/0.44 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
% 0.19/0.44 @ Xa
% 0.19/0.44 @ X ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_eq_int.abs_eq
% 0.19/0.44 thf(fact_1177_Collect__case__prod__mono,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
% 0.19/0.44 ( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B6 )
% 0.19/0.44 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Collect_case_prod_mono
% 0.19/0.44 thf(fact_1178_zadd__int__left,axiom,
% 0.19/0.44 ! [M2: nat,N: nat,Z2: int] :
% 0.19/0.44 ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z2 ) )
% 0.19/0.44 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N ) ) @ Z2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % zadd_int_left
% 0.19/0.44 thf(fact_1179_int__plus,axiom,
% 0.19/0.44 ! [N: nat,M2: nat] :
% 0.19/0.44 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M2 ) )
% 0.19/0.44 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_plus
% 0.19/0.44 thf(fact_1180_int__ops_I5_J,axiom,
% 0.19/0.44 ! [A3: nat,B2: nat] :
% 0.19/0.44 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
% 0.19/0.44 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_ops(5)
% 0.19/0.44 thf(fact_1181_int__ops_I2_J,axiom,
% 0.19/0.44 ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
% 0.19/0.44 = ( one_one @ int ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_ops(2)
% 0.19/0.44 thf(fact_1182_zle__iff__zadd,axiom,
% 0.19/0.44 ( ( ord_less_eq @ int )
% 0.19/0.44 = ( ^ [W2: int,Z5: int] :
% 0.19/0.44 ? [N5: nat] :
% 0.19/0.44 ( Z5
% 0.19/0.44 = ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % zle_iff_zadd
% 0.19/0.44 thf(fact_1183_numeral__One,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( numeral @ A )
% 0.19/0.44 => ( ( numeral_numeral @ A @ one2 )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_One
% 0.19/0.44 thf(fact_1184_numerals_I1_J,axiom,
% 0.19/0.44 ( ( numeral_numeral @ nat @ one2 )
% 0.19/0.44 = ( one_one @ nat ) ) ).
% 0.19/0.44
% 0.19/0.44 % numerals(1)
% 0.19/0.44 thf(fact_1185_execute__bind__case,axiom,
% 0.19/0.44 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit] :
% 0.19/0.44 ( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H3 )
% 0.19/0.44 = ( case_option @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.44 @ ( product_case_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.44 @ ^ [X3: B] :
% 0.19/0.44 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.44 @ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ A @ N5 @ ( heap_Time_execute @ A @ ( G @ X3 ) @ H7 ) ) ) )
% 0.19/0.44 @ ( heap_Time_execute @ B @ F @ H3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % execute_bind_case
% 0.19/0.44 thf(fact_1186_Sum__Icc__int,axiom,
% 0.19/0.44 ! [M2: int,N: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ M2 @ N )
% 0.19/0.44 => ( ( groups7311177749621191930dd_sum @ int @ int
% 0.19/0.44 @ ^ [X3: int] : X3
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) )
% 0.19/0.44 = ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M2 @ ( minus_minus @ int @ M2 @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Sum_Icc_int
% 0.19/0.44 thf(fact_1187_Heap__Time__Monad_Obind__def,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType] :
% 0.19/0.44 ( ( heap_Time_bind @ A @ B )
% 0.19/0.44 = ( ^ [F2: heap_Time_Heap @ A,G2: A > ( heap_Time_Heap @ B )] :
% 0.19/0.44 ( heap_Time_Heap2 @ B
% 0.19/0.44 @ ^ [H: heap_ext @ product_unit] :
% 0.19/0.44 ( case_option @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.44 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.44 @ ^ [R3: A] :
% 0.19/0.44 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.19/0.44 @ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ B @ N5 @ ( heap_Time_execute @ B @ ( G2 @ R3 ) @ H7 ) ) ) )
% 0.19/0.44 @ ( heap_Time_execute @ A @ F2 @ H ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Heap_Time_Monad.bind_def
% 0.19/0.44 thf(fact_1188_zless__iff__Suc__zadd,axiom,
% 0.19/0.44 ( ( ord_less @ int )
% 0.19/0.44 = ( ^ [W2: int,Z5: int] :
% 0.19/0.44 ? [N5: nat] :
% 0.19/0.44 ( Z5
% 0.19/0.44 = ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % zless_iff_Suc_zadd
% 0.19/0.44 thf(fact_1189_int__ops_I4_J,axiom,
% 0.19/0.44 ! [A3: nat] :
% 0.19/0.44 ( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
% 0.19/0.44 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_ops(4)
% 0.19/0.44 thf(fact_1190_int__Suc,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( semiring_1_of_nat @ int @ ( suc @ N ) )
% 0.19/0.44 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_Suc
% 0.19/0.44 thf(fact_1191_int__bit__induct,axiom,
% 0.19/0.44 ! [P: int > $o,K: int] :
% 0.19/0.44 ( ( P @ ( zero_zero @ int ) )
% 0.19/0.44 => ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.19/0.44 => ( ! [K5: int] :
% 0.19/0.44 ( ( P @ K5 )
% 0.19/0.44 => ( ( K5
% 0.19/0.44 != ( zero_zero @ int ) )
% 0.19/0.44 => ( P @ ( times_times @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
% 0.19/0.44 => ( ! [K5: int] :
% 0.19/0.44 ( ( P @ K5 )
% 0.19/0.44 => ( ( K5
% 0.19/0.44 != ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.19/0.44 => ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
% 0.19/0.44 => ( P @ K ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_bit_induct
% 0.19/0.44 thf(fact_1192_uminus__numeral__One,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % uminus_numeral_One
% 0.19/0.44 thf(fact_1193_int__def,axiom,
% 0.19/0.44 ( ( semiring_1_of_nat @ int )
% 0.19/0.44 = ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % int_def
% 0.19/0.44 thf(fact_1194_num__of__nat__One,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
% 0.19/0.44 => ( ( num_of_nat @ N )
% 0.19/0.44 = one2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % num_of_nat_One
% 0.19/0.44 thf(fact_1195_neg__zdiv__mult__2,axiom,
% 0.19/0.44 ! [A3: int,B2: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 0.19/0.44 => ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.44 = ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_zdiv_mult_2
% 0.19/0.44 thf(fact_1196_pos__zdiv__mult__2,axiom,
% 0.19/0.44 ! [A3: int,B2: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.19/0.44 => ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.44 = ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % pos_zdiv_mult_2
% 0.19/0.44 thf(fact_1197_pos__zmod__mult__2,axiom,
% 0.19/0.44 ! [A3: int,B2: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 0.19/0.44 => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.44 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % pos_zmod_mult_2
% 0.19/0.44 thf(fact_1198_mod__mult2__eq_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [A3: A,M2: nat,N: nat] :
% 0.19/0.44 ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mod_mult2_eq'
% 0.19/0.44 thf(fact_1199_numeral__num__of__nat__unfold,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( ( N
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 0.19/0.44 = ( one_one @ A ) ) )
% 0.19/0.44 & ( ( N
% 0.19/0.44 != ( zero_zero @ nat ) )
% 0.19/0.44 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 0.19/0.44 = ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % numeral_num_of_nat_unfold
% 0.19/0.44 thf(fact_1200_double__arith__series,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_semiring_1 @ A )
% 0.19/0.44 => ! [A3: A,D3: A,N: nat] :
% 0.19/0.44 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 0.19/0.44 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 0.19/0.44 = ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % double_arith_series
% 0.19/0.44 thf(fact_1201_double__gauss__sum,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_semiring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 0.19/0.44 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % double_gauss_sum
% 0.19/0.44 thf(fact_1202_neg__zmod__mult__2,axiom,
% 0.19/0.44 ! [A3: int,B2: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 0.19/0.44 => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.44 = ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_zmod_mult_2
% 0.19/0.44 thf(fact_1203_mult__2,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_numeral @ A )
% 0.19/0.44 => ! [Z2: A] :
% 0.19/0.44 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
% 0.19/0.44 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mult_2
% 0.19/0.44 thf(fact_1204_mult__2__right,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_numeral @ A )
% 0.19/0.44 => ! [Z2: A] :
% 0.19/0.44 ( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mult_2_right
% 0.19/0.44 thf(fact_1205_left__add__twice,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_numeral @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.44 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % left_add_twice
% 0.19/0.44 thf(fact_1206_pos__eucl__rel__int__mult__2,axiom,
% 0.19/0.44 ! [B2: int,A3: int,Q3: int,R: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
% 0.19/0.44 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.44 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % pos_eucl_rel_int_mult_2
% 0.19/0.44 thf(fact_1207_one__power2,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_power2
% 0.19/0.44 thf(fact_1208_nat__1__add__1,axiom,
% 0.19/0.44 ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
% 0.19/0.44 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_1_add_1
% 0.19/0.44 thf(fact_1209_Suc__nat__number__of__add,axiom,
% 0.19/0.44 ! [V: num,N: nat] :
% 0.19/0.44 ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 0.19/0.44 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_nat_number_of_add
% 0.19/0.44 thf(fact_1210_double__gauss__sum__from__Suc__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_semiring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
% 0.19/0.44 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % double_gauss_sum_from_Suc_0
% 0.19/0.44 thf(fact_1211_arith__series,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [A3: A,D3: A,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % arith_series
% 0.19/0.44 thf(fact_1212_gauss__sum,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % gauss_sum
% 0.19/0.44 thf(fact_1213_nat__approx__posE,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [E2: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
% 0.19/0.44 => ~ ! [N3: nat] :
% 0.19/0.44 ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_approx_posE
% 0.19/0.44 thf(fact_1214_neg__eucl__rel__int__mult__2,axiom,
% 0.19/0.44 ! [B2: int,A3: int,Q3: int,R: int] :
% 0.19/0.44 ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
% 0.19/0.44 => ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.44 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_eucl_rel_int_mult_2
% 0.19/0.44 thf(fact_1215_gauss__sum__from__Suc__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % gauss_sum_from_Suc_0
% 0.19/0.44 thf(fact_1216_exp__add__not__zero__imp__right,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.44 != ( zero_zero @ A ) )
% 0.19/0.44 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % exp_add_not_zero_imp_right
% 0.19/0.44 thf(fact_1217_exp__add__not__zero__imp__left,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
% 0.19/0.44 != ( zero_zero @ A ) )
% 0.19/0.44 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
% 0.19/0.44 != ( zero_zero @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % exp_add_not_zero_imp_left
% 0.19/0.44 thf(fact_1218_sum__power2__eq__zero__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A,Y: A] :
% 0.19/0.44 ( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.19/0.44 = ( zero_zero @ A ) )
% 0.19/0.44 = ( ( X
% 0.19/0.44 = ( zero_zero @ A ) )
% 0.19/0.44 & ( Y
% 0.19/0.44 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_power2_eq_zero_iff
% 0.19/0.44 thf(fact_1219_div__exp__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [A3: A,M2: nat,N: nat] :
% 0.19/0.44 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_exp_eq
% 0.19/0.44 thf(fact_1220_power2__eq__1__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_15535105094025558882visors @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 = ( ( A3
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 | ( A3
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power2_eq_1_iff
% 0.19/0.44 thf(fact_1221_plus__int_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.19/0.44 ( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.19/0.44 = ( abs_Integ
% 0.19/0.44 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.19/0.44 @ ^ [X3: nat,Y3: nat] :
% 0.19/0.44 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) )
% 0.19/0.44 @ Xa
% 0.19/0.44 @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % plus_int.abs_eq
% 0.19/0.44 thf(fact_1222_abs__square__eq__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( one_one @ A ) )
% 0.19/0.44 = ( ( abs_abs @ A @ X )
% 0.19/0.44 = ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % abs_square_eq_1
% 0.19/0.44 thf(fact_1223_minus__int_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.19/0.44 ( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.19/0.44 = ( abs_Integ
% 0.19/0.44 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.19/0.44 @ ^ [X3: nat,Y3: nat] :
% 0.19/0.44 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) )
% 0.19/0.44 @ Xa
% 0.19/0.44 @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % minus_int.abs_eq
% 0.19/0.44 thf(fact_1224_nat__induct2,axiom,
% 0.19/0.44 ! [P: nat > $o,N: nat] :
% 0.19/0.44 ( ( P @ ( zero_zero @ nat ) )
% 0.19/0.44 => ( ( P @ ( one_one @ nat ) )
% 0.19/0.44 => ( ! [N3: nat] :
% 0.19/0.44 ( ( P @ N3 )
% 0.19/0.44 => ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 0.19/0.44 => ( P @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_induct2
% 0.19/0.44 thf(fact_1225_times__int_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 0.19/0.44 ( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 0.19/0.44 = ( abs_Integ
% 0.19/0.44 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.19/0.44 @ ^ [X3: nat,Y3: nat] :
% 0.19/0.44 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) )
% 0.19/0.44 @ Xa
% 0.19/0.44 @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % times_int.abs_eq
% 0.19/0.44 thf(fact_1226_sum__power2__le__zero__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A,Y: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
% 0.19/0.44 = ( ( X
% 0.19/0.44 = ( zero_zero @ A ) )
% 0.19/0.44 & ( Y
% 0.19/0.44 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_power2_le_zero_iff
% 0.19/0.44 thf(fact_1227_sum__power2__ge__zero,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_power2_ge_zero
% 0.19/0.44 thf(fact_1228_bits__stable__imp__add__self,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = A3 )
% 0.19/0.44 => ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.19/0.44 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % bits_stable_imp_add_self
% 0.19/0.44 thf(fact_1229_not__sum__power2__lt__zero,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A,Y: A] :
% 0.19/0.44 ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_sum_power2_lt_zero
% 0.19/0.44 thf(fact_1230_sum__power2__gt__zero__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A,Y: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 0.19/0.44 = ( ( X
% 0.19/0.44 != ( zero_zero @ A ) )
% 0.19/0.44 | ( Y
% 0.19/0.44 != ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum_power2_gt_zero_iff
% 0.19/0.44 thf(fact_1231_power2__sum,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_semiring_1 @ A )
% 0.19/0.44 => ! [X: A,Y: A] :
% 0.19/0.44 ( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power2_sum
% 0.19/0.44 thf(fact_1232_square__le__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 0.19/0.44 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 0.19/0.44 => ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % square_le_1
% 0.19/0.44 thf(fact_1233_abs__square__le__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 0.19/0.44 = ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % abs_square_le_1
% 0.19/0.44 thf(fact_1234_abs__square__less__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( linordered_idom @ A )
% 0.19/0.44 => ! [X: A] :
% 0.19/0.44 ( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 0.19/0.44 = ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % abs_square_less_1
% 0.19/0.44 thf(fact_1235_div__exp__mod__exp__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [A3: A,N: nat,M2: nat] :
% 0.19/0.44 ( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 0.19/0.44 = ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_exp_mod_exp_eq
% 0.19/0.44 thf(fact_1236_ex__power__ivl2,axiom,
% 0.19/0.44 ! [B2: nat,K: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 0.19/0.44 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 0.19/0.44 => ? [N3: nat] :
% 0.19/0.44 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
% 0.19/0.44 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ex_power_ivl2
% 0.19/0.44 thf(fact_1237_ex__power__ivl1,axiom,
% 0.19/0.44 ! [B2: nat,K: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 0.19/0.44 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
% 0.19/0.44 => ? [N3: nat] :
% 0.19/0.44 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
% 0.19/0.44 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % ex_power_ivl1
% 0.19/0.44 thf(fact_1238_power2__diff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_ring_1 @ A )
% 0.19/0.44 => ! [X: A,Y: A] :
% 0.19/0.44 ( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power2_diff
% 0.19/0.44 thf(fact_1239_power__minus1__odd,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % power_minus1_odd
% 0.19/0.44 thf(fact_1240_mask__eq__sum__exp,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 @ ( collect @ nat
% 0.19/0.44 @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mask_eq_sum_exp
% 0.19/0.44 thf(fact_1241_sum_Oin__pairs,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_add @ A )
% 0.19/0.44 => ! [G: nat > A,M2: nat,N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.19/0.44 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.44 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sum.in_pairs
% 0.19/0.44 thf(fact_1242_gauss__sum__nat,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.44 @ ^ [X3: nat] : X3
% 0.19/0.44 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.19/0.44 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % gauss_sum_nat
% 0.19/0.44 thf(fact_1243_divmod__step__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [L: num,R: A,Q3: A] :
% 0.19/0.44 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 0.19/0.44 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R ) )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L ) ) ) ) )
% 0.19/0.44 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 0.19/0.44 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R ) )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_step_eq
% 0.19/0.44 thf(fact_1244_divmod__step__def,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ( ( unique1321980374590559556d_step @ A )
% 0.19/0.44 = ( ^ [L3: num] :
% 0.19/0.44 ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 0.19/0.44 @ ^ [Q5: A,R3: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R3 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_step_def
% 0.19/0.44 thf(fact_1245_divmod__step__nat__def,axiom,
% 0.19/0.44 ( ( unique1321980374590559556d_step @ nat )
% 0.19/0.44 = ( ^ [L3: num] :
% 0.19/0.44 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.19/0.44 @ ^ [Q5: nat,R3: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R3 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R3 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_step_nat_def
% 0.19/0.44 thf(fact_1246_divmod__step__int__def,axiom,
% 0.19/0.44 ( ( unique1321980374590559556d_step @ int )
% 0.19/0.44 = ( ^ [L3: num] :
% 0.19/0.44 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.19/0.44 @ ^ [Q5: int,R3: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R3 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R3 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_step_int_def
% 0.19/0.44 thf(fact_1247_set__bit__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.19/0.44 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % set_bit_0
% 0.19/0.44 thf(fact_1248_mlex__eq,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( mlex_prod @ A )
% 0.19/0.44 = ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
% 0.19/0.44 ( collect @ ( product_prod @ A @ A )
% 0.19/0.44 @ ( product_case_prod @ A @ A @ $o
% 0.19/0.44 @ ^ [X3: A,Y3: A] :
% 0.19/0.44 ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
% 0.19/0.44 | ( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
% 0.19/0.44 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mlex_eq
% 0.19/0.44 thf(fact_1249_unset__bit__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.44 => ! [N: nat,A3: A] :
% 0.19/0.44 ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
% 0.19/0.44 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unset_bit_Suc
% 0.19/0.44 thf(fact_1250_flip__bit__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.44 => ! [N: nat,A3: A] :
% 0.19/0.44 ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
% 0.19/0.44 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % flip_bit_Suc
% 0.19/0.44 thf(fact_1251_set__bit__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.44 => ! [N: nat,A3: A] :
% 0.19/0.44 ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
% 0.19/0.44 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % set_bit_Suc
% 0.19/0.44 thf(fact_1252_split__part,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 0.19/0.44 ( ( product_case_prod @ A @ B @ $o
% 0.19/0.44 @ ^ [A7: A,B5: B] :
% 0.19/0.44 ( P
% 0.19/0.44 & ( Q @ A7 @ B5 ) ) )
% 0.19/0.44 = ( ^ [Ab: product_prod @ A @ B] :
% 0.19/0.44 ( P
% 0.19/0.44 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % split_part
% 0.19/0.44 thf(fact_1253_prod_Odisc__eq__case,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.19/0.44 ( product_case_prod @ A @ B @ $o
% 0.19/0.44 @ ^ [Uu: A,Uv: B] : $true
% 0.19/0.44 @ Prod ) ).
% 0.19/0.44
% 0.19/0.44 % prod.disc_eq_case
% 0.19/0.44 thf(fact_1254_divmod__step__integer__def,axiom,
% 0.19/0.44 ( ( unique1321980374590559556d_step @ code_integer )
% 0.19/0.44 = ( ^ [L3: num] :
% 0.19/0.44 ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.19/0.44 @ ^ [Q5: code_integer,R3: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R3 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R3 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_step_integer_def
% 0.19/0.44 thf(fact_1255_signed__take__bit__rec,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.44 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 0.19/0.44 = ( ^ [N5: nat,A7: A] :
% 0.19/0.44 ( if @ A
% 0.19/0.44 @ ( N5
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.19/0.44 @ ( plus_plus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_rec
% 0.19/0.44 thf(fact_1256_round__unique,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,Y: int] :
% 0.19/0.44 ( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
% 0.19/0.44 => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
% 0.19/0.44 => ( ( archimedean_round @ A @ X )
% 0.19/0.44 = Y ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % round_unique
% 0.19/0.44 thf(fact_1257_rel__of__def,axiom,
% 0.19/0.44 ! [B: $tType,A: $tType] :
% 0.19/0.44 ( ( rel_of @ A @ B )
% 0.19/0.44 = ( ^ [M: A > ( option @ B ),P4: ( product_prod @ A @ B ) > $o] :
% 0.19/0.44 ( collect @ ( product_prod @ A @ B )
% 0.19/0.44 @ ( product_case_prod @ A @ B @ $o
% 0.19/0.44 @ ^ [K4: A,V2: B] :
% 0.19/0.44 ( ( ( M @ K4 )
% 0.19/0.44 = ( some @ B @ V2 ) )
% 0.19/0.44 & ( P4 @ ( product_Pair @ A @ B @ K4 @ V2 ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % rel_of_def
% 0.19/0.44 thf(fact_1258_dbl__simps_I4_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dbl_simps(4)
% 0.19/0.44 thf(fact_1259_round__altdef,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ( ( archimedean_round @ A )
% 0.19/0.44 = ( ^ [X3: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X3 ) ) @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % round_altdef
% 0.19/0.44 thf(fact_1260_round__unique_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A,N: int] :
% 0.19/0.44 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.19/0.44 => ( ( archimedean_round @ A @ X )
% 0.19/0.44 = N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % round_unique'
% 0.19/0.44 thf(fact_1261_of__int__round__abs__le,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_round_abs_le
% 0.19/0.44 thf(fact_1262_divmod__algorithm__code_I5_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
% 0.19/0.44 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 0.19/0.44 @ ^ [Q5: A,R3: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R3 ) )
% 0.19/0.44 @ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(5)
% 0.19/0.44 thf(fact_1263_signed__take__bit__of__minus__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_of_minus_1
% 0.19/0.44 thf(fact_1264_signed__take__bit__Suc__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_Suc_1
% 0.19/0.44 thf(fact_1265_signed__take__bit__numeral__of__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.44 => ! [K: num] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_numeral_of_1
% 0.19/0.44 thf(fact_1266_round__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ( ( archimedean_round @ A @ ( one_one @ A ) )
% 0.19/0.44 = ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % round_1
% 0.19/0.44 thf(fact_1267_divmod__algorithm__code_I2_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [M2: num] :
% 0.19/0.44 ( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(2)
% 0.19/0.44 thf(fact_1268_dbl__simps_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dbl_simps(3)
% 0.19/0.44 thf(fact_1269_divmod__algorithm__code_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(3)
% 0.19/0.44 thf(fact_1270_less__eq__integer__code_I1_J,axiom,
% 0.19/0.44 ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% 0.19/0.44
% 0.19/0.44 % less_eq_integer_code(1)
% 0.19/0.44 thf(fact_1271_sgn__integer__code,axiom,
% 0.19/0.44 ( ( sgn_sgn @ code_integer )
% 0.19/0.44 = ( ^ [K4: code_integer] :
% 0.19/0.44 ( if @ code_integer
% 0.19/0.44 @ ( K4
% 0.19/0.44 = ( zero_zero @ code_integer ) )
% 0.19/0.44 @ ( zero_zero @ code_integer )
% 0.19/0.44 @ ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sgn_integer_code
% 0.19/0.44 thf(fact_1272_exhaustive__integer_H_Ocases,axiom,
% 0.19/0.44 ! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
% 0.19/0.44 ~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % exhaustive_integer'.cases
% 0.19/0.44 thf(fact_1273_full__exhaustive__integer_H_Ocases,axiom,
% 0.19/0.44 ! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
% 0.19/0.44 ~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % full_exhaustive_integer'.cases
% 0.19/0.44 thf(fact_1274_times__integer__code_I1_J,axiom,
% 0.19/0.44 ! [K: code_integer] :
% 0.19/0.44 ( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.19/0.44 = ( zero_zero @ code_integer ) ) ).
% 0.19/0.44
% 0.19/0.44 % times_integer_code(1)
% 0.19/0.44 thf(fact_1275_times__integer__code_I2_J,axiom,
% 0.19/0.44 ! [L: code_integer] :
% 0.19/0.44 ( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 0.19/0.44 = ( zero_zero @ code_integer ) ) ).
% 0.19/0.44
% 0.19/0.44 % times_integer_code(2)
% 0.19/0.44 thf(fact_1276_minus__integer__code_I1_J,axiom,
% 0.19/0.44 ! [K: code_integer] :
% 0.19/0.44 ( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.19/0.44 = K ) ).
% 0.19/0.44
% 0.19/0.44 % minus_integer_code(1)
% 0.19/0.44 thf(fact_1277_plus__integer__code_I2_J,axiom,
% 0.19/0.44 ! [L: code_integer] :
% 0.19/0.44 ( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 0.19/0.44 = L ) ).
% 0.19/0.44
% 0.19/0.44 % plus_integer_code(2)
% 0.19/0.44 thf(fact_1278_plus__integer__code_I1_J,axiom,
% 0.19/0.44 ! [K: code_integer] :
% 0.19/0.44 ( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.19/0.44 = K ) ).
% 0.19/0.44
% 0.19/0.44 % plus_integer_code(1)
% 0.19/0.44 thf(fact_1279_minus__integer__code_I2_J,axiom,
% 0.19/0.44 ! [L: code_integer] :
% 0.19/0.44 ( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 0.19/0.44 = ( uminus_uminus @ code_integer @ L ) ) ).
% 0.19/0.44
% 0.19/0.44 % minus_integer_code(2)
% 0.19/0.44 thf(fact_1280_divmod__integer_H__def,axiom,
% 0.19/0.44 ( ( unique8689654367752047608divmod @ code_integer )
% 0.19/0.44 = ( ^ [M: num,N5: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N5 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N5 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_integer'_def
% 0.19/0.44 thf(fact_1281_signed__take__bit__add,axiom,
% 0.19/0.44 ! [N: nat,K: int,L: int] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
% 0.19/0.44 = ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_add
% 0.19/0.44 thf(fact_1282_dbl__def,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( neg_numeral_dbl @ A )
% 0.19/0.44 = ( ^ [X3: A] : ( plus_plus @ A @ X3 @ X3 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dbl_def
% 0.19/0.44 thf(fact_1283_divmod__int__def,axiom,
% 0.19/0.44 ( ( unique8689654367752047608divmod @ int )
% 0.19/0.44 = ( ^ [M: num,N5: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_int_def
% 0.19/0.44 thf(fact_1284_divmod__def,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ( ( unique8689654367752047608divmod @ A )
% 0.19/0.44 = ( ^ [M: num,N5: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N5 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_def
% 0.19/0.44 thf(fact_1285_divmod_H__nat__def,axiom,
% 0.19/0.44 ( ( unique8689654367752047608divmod @ nat )
% 0.19/0.44 = ( ^ [M: num,N5: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N5 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N5 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod'_nat_def
% 0.19/0.44 thf(fact_1286_signed__take__bit__int__greater__eq,axiom,
% 0.19/0.44 ! [K: int,N: nat] :
% 0.19/0.44 ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.19/0.44 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_int_greater_eq
% 0.19/0.44 thf(fact_1287_divmod__divmod__step,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ( ( unique8689654367752047608divmod @ A )
% 0.19/0.44 = ( ^ [M: num,N5: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M @ N5 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M ) ) @ ( unique1321980374590559556d_step @ A @ N5 @ ( unique8689654367752047608divmod @ A @ M @ ( bit0 @ N5 ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_divmod_step
% 0.19/0.44 thf(fact_1288_round__def,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ( ( archimedean_round @ A )
% 0.19/0.44 = ( ^ [X3: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % round_def
% 0.19/0.44 thf(fact_1289_signed__take__bit__Suc,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.44 => ! [N: nat,A3: A] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
% 0.19/0.44 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_Suc
% 0.19/0.44 thf(fact_1290_of__int__round__le,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_round_le
% 0.19/0.44 thf(fact_1291_of__int__round__ge,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_round_ge
% 0.19/0.44 thf(fact_1292_of__int__round__gt,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( archim2362893244070406136eiling @ A )
% 0.19/0.44 => ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_int_round_gt
% 0.19/0.44 thf(fact_1293_integer__of__int__code,axiom,
% 0.19/0.44 ( code_integer_of_int
% 0.19/0.44 = ( ^ [K4: int] :
% 0.19/0.44 ( if @ code_integer @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K4 ) ) )
% 0.19/0.44 @ ( if @ code_integer
% 0.19/0.44 @ ( K4
% 0.19/0.44 = ( zero_zero @ int ) )
% 0.19/0.44 @ ( zero_zero @ code_integer )
% 0.19/0.44 @ ( if @ code_integer
% 0.19/0.44 @ ( ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( zero_zero @ int ) )
% 0.19/0.44 @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 0.19/0.44 @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % integer_of_int_code
% 0.19/0.44 thf(fact_1294_divmod__algorithm__code_I6_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 0.19/0.44 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 0.19/0.44 @ ^ [Q5: A,R3: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ A ) ) )
% 0.19/0.44 @ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(6)
% 0.19/0.44 thf(fact_1295_of__nat__code,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_1 @ A )
% 0.19/0.44 => ( ( semiring_1_of_nat @ A )
% 0.19/0.44 = ( ^ [N5: nat] :
% 0.19/0.44 ( semiri8178284476397505188at_aux @ A
% 0.19/0.44 @ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
% 0.19/0.44 @ N5
% 0.19/0.44 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % of_nat_code
% 0.19/0.44 thf(fact_1296_signed__take__bit__Suc__minus__bit1,axiom,
% 0.19/0.44 ! [N: nat,K: num] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.19/0.44 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_Suc_minus_bit1
% 0.19/0.44 thf(fact_1297_concat__bit__Suc,axiom,
% 0.19/0.44 ! [N: nat,K: int,L: int] :
% 0.19/0.44 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 0.19/0.44 = ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % concat_bit_Suc
% 0.19/0.44 thf(fact_1298_even__succ__mod__exp,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.44 = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_succ_mod_exp
% 0.19/0.44 thf(fact_1299_even__succ__div__exp,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [A3: A,N: nat] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.44 = ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_succ_div_exp
% 0.19/0.44 thf(fact_1300_take__bit__rec,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.44 => ( ( bit_se2584673776208193580ke_bit @ A )
% 0.19/0.44 = ( ^ [N5: nat,A7: A] :
% 0.19/0.44 ( if @ A
% 0.19/0.44 @ ( N5
% 0.19/0.44 = ( zero_zero @ nat ) )
% 0.19/0.44 @ ( zero_zero @ A )
% 0.19/0.44 @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % take_bit_rec
% 0.19/0.44 thf(fact_1301_nat__dvd__1__iff__1,axiom,
% 0.19/0.44 ! [M2: nat] :
% 0.19/0.44 ( ( dvd_dvd @ nat @ M2 @ ( one_one @ nat ) )
% 0.19/0.44 = ( M2
% 0.19/0.44 = ( one_one @ nat ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % nat_dvd_1_iff_1
% 0.19/0.44 thf(fact_1302_integer__of__int__eq__of__int,axiom,
% 0.19/0.44 ( code_integer_of_int
% 0.19/0.44 = ( ring_1_of_int @ code_integer ) ) ).
% 0.19/0.44
% 0.19/0.44 % integer_of_int_eq_of_int
% 0.19/0.44 thf(fact_1303_dvd__add__triv__right__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_s4317794764714335236cancel @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add_triv_right_iff
% 0.19/0.44 thf(fact_1304_dvd__add__triv__left__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_s4317794764714335236cancel @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add_triv_left_iff
% 0.19/0.44 thf(fact_1305_dvd__add__times__triv__right__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_s4317794764714335236cancel @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add_times_triv_right_iff
% 0.19/0.44 thf(fact_1306_dvd__add__times__triv__left__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_s4317794764714335236cancel @ A )
% 0.19/0.44 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add_times_triv_left_iff
% 0.19/0.44 thf(fact_1307_unit__prod,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_prod
% 0.19/0.44 thf(fact_1308_div__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.19/0.44 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.44 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_add
% 0.19/0.44 thf(fact_1309_unit__div__1__div__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.19/0.44 = A3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_div_1_div_1
% 0.19/0.44 thf(fact_1310_unit__div__1__unit,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_div_1_unit
% 0.19/0.44 thf(fact_1311_unit__div,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_div
% 0.19/0.44 thf(fact_1312_take__bit__Suc__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % take_bit_Suc_1
% 0.19/0.44 thf(fact_1313_take__bit__numeral__1,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.44 => ! [L: num] :
% 0.19/0.44 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
% 0.19/0.44 = ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % take_bit_numeral_1
% 0.19/0.44 thf(fact_1314_semiring__norm_I9_J,axiom,
% 0.19/0.44 ! [M2: num,N: num] :
% 0.19/0.44 ( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 0.19/0.44 = ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(9)
% 0.19/0.44 thf(fact_1315_semiring__norm_I7_J,axiom,
% 0.19/0.44 ! [M2: num,N: num] :
% 0.19/0.44 ( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(7)
% 0.19/0.44 thf(fact_1316_semiring__norm_I16_J,axiom,
% 0.19/0.44 ! [M2: num,N: num] :
% 0.19/0.44 ( ( times_times @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ ( bit0 @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(16)
% 0.19/0.44 thf(fact_1317_unit__mult__div__div,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 0.19/0.44 = ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_mult_div_div
% 0.19/0.44 thf(fact_1318_unit__div__mult__self,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
% 0.19/0.44 = B2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_div_mult_self
% 0.19/0.44 thf(fact_1319_take__bit__of__1__eq__0__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 0.19/0.44 = ( zero_zero @ A ) )
% 0.19/0.44 = ( N
% 0.19/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % take_bit_of_1_eq_0_iff
% 0.19/0.44 thf(fact_1320_semiring__norm_I10_J,axiom,
% 0.19/0.44 ! [M2: num,N: num] :
% 0.19/0.44 ( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(10)
% 0.19/0.44 thf(fact_1321_semiring__norm_I8_J,axiom,
% 0.19/0.44 ! [M2: num] :
% 0.19/0.44 ( ( plus_plus @ num @ ( bit1 @ M2 ) @ one2 )
% 0.19/0.44 = ( bit0 @ ( plus_plus @ num @ M2 @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(8)
% 0.19/0.44 thf(fact_1322_semiring__norm_I5_J,axiom,
% 0.19/0.44 ! [M2: num] :
% 0.19/0.44 ( ( plus_plus @ num @ ( bit0 @ M2 ) @ one2 )
% 0.19/0.44 = ( bit1 @ M2 ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(5)
% 0.19/0.44 thf(fact_1323_semiring__norm_I4_J,axiom,
% 0.19/0.44 ! [N: num] :
% 0.19/0.44 ( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
% 0.19/0.44 = ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(4)
% 0.19/0.44 thf(fact_1324_semiring__norm_I3_J,axiom,
% 0.19/0.44 ! [N: num] :
% 0.19/0.44 ( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
% 0.19/0.44 = ( bit1 @ N ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_norm(3)
% 0.19/0.44 thf(fact_1325_odd__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) )
% 0.19/0.44 = ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.44 != ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % odd_add
% 0.19/0.44 thf(fact_1326_even__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.44 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_add
% 0.19/0.44 thf(fact_1327_dbl__inc__simps_I3_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
% 0.19/0.44 = ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dbl_inc_simps(3)
% 0.19/0.44 thf(fact_1328_even__plus__one__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
% 0.19/0.44 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_plus_one_iff
% 0.19/0.44 thf(fact_1329_even__diff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_parity @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_diff
% 0.19/0.44 thf(fact_1330_neg__one__even__power,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.44 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.19/0.44 = ( one_one @ A ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_one_even_power
% 0.19/0.44 thf(fact_1331_neg__one__odd__power,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( ring_1 @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.44 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.19/0.44 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % neg_one_odd_power
% 0.19/0.44 thf(fact_1332_even__diff__nat,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) )
% 0.19/0.44 = ( ( ord_less @ nat @ M2 @ N )
% 0.19/0.44 | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_diff_nat
% 0.19/0.44 thf(fact_1333_divmod__algorithm__code_I4_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [N: num] :
% 0.19/0.44 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(4)
% 0.19/0.44 thf(fact_1334_mod__Suc__eq__mod__add3,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( modulo_modulo @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.19/0.44 = ( modulo_modulo @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mod_Suc_eq_mod_add3
% 0.19/0.44 thf(fact_1335_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.19/0.44 ! [M2: nat,V: num] :
% 0.19/0.44 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 0.19/0.44 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_mod_eq_add3_mod_numeral
% 0.19/0.44 thf(fact_1336_div__Suc__eq__div__add3,axiom,
% 0.19/0.44 ! [M2: nat,N: nat] :
% 0.19/0.44 ( ( divide_divide @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 0.19/0.44 = ( divide_divide @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_Suc_eq_div_add3
% 0.19/0.44 thf(fact_1337_Suc__div__eq__add3__div__numeral,axiom,
% 0.19/0.44 ! [M2: nat,V: num] :
% 0.19/0.44 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 0.19/0.44 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % Suc_div_eq_add3_div_numeral
% 0.19/0.44 thf(fact_1338_dbl__dec__simps_I4_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( neg_numeral @ A )
% 0.19/0.44 => ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.44 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dbl_dec_simps(4)
% 0.19/0.44 thf(fact_1339_even__succ__div__2,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_semiring_bits @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_succ_div_2
% 0.19/0.44 thf(fact_1340_odd__succ__div__two,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % odd_succ_div_two
% 0.19/0.44 thf(fact_1341_even__succ__div__two,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.44 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % even_succ_div_two
% 0.19/0.44 thf(fact_1342_odd__two__times__div__two__nat,axiom,
% 0.19/0.44 ! [N: nat] :
% 0.19/0.44 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.44 => ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 0.19/0.44 = ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % odd_two_times_div_two_nat
% 0.19/0.44 thf(fact_1343_odd__two__times__div__two__succ,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.44 => ! [A3: A] :
% 0.19/0.44 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.44 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
% 0.19/0.44 = A3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % odd_two_times_div_two_succ
% 0.19/0.44 thf(fact_1344_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( semiring_parity @ A )
% 0.19/0.44 => ! [N: nat] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
% 0.19/0.44 = ( N
% 0.19/0.44 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % semiring_parity_class.even_mask_iff
% 0.19/0.44 thf(fact_1345_zmod__numeral__Bit1,axiom,
% 0.19/0.44 ! [V: num,W: num] :
% 0.19/0.44 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
% 0.19/0.44 = ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % zmod_numeral_Bit1
% 0.19/0.44 thf(fact_1346_divmod__algorithm__code_I8_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( ( ord_less @ num @ M2 @ N )
% 0.19/0.44 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
% 0.19/0.44 & ( ~ ( ord_less @ num @ M2 @ N )
% 0.19/0.44 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(8)
% 0.19/0.44 thf(fact_1347_divmod__algorithm__code_I7_J,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( unique1627219031080169319umeral @ A )
% 0.19/0.44 => ! [M2: num,N: num] :
% 0.19/0.44 ( ( ( ord_less_eq @ num @ M2 @ N )
% 0.19/0.44 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
% 0.19/0.44 & ( ~ ( ord_less_eq @ num @ M2 @ N )
% 0.19/0.44 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.44 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divmod_algorithm_code(7)
% 0.19/0.44 thf(fact_1348_signed__take__bit__Suc__bit1,axiom,
% 0.19/0.44 ! [N: nat,K: num] :
% 0.19/0.44 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 0.19/0.44 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % signed_take_bit_Suc_bit1
% 0.19/0.44 thf(fact_1349_uminus__integer__code_I1_J,axiom,
% 0.19/0.44 ( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
% 0.19/0.44 = ( zero_zero @ code_integer ) ) ).
% 0.19/0.44
% 0.19/0.44 % uminus_integer_code(1)
% 0.19/0.44 thf(fact_1350_take__bit__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: nat,X: int] :
% 0.19/0.44 ( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % take_bit_integer.abs_eq
% 0.19/0.44 thf(fact_1351_abs__integer__code,axiom,
% 0.19/0.44 ( ( abs_abs @ code_integer )
% 0.19/0.44 = ( ^ [K4: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K4 ) @ K4 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % abs_integer_code
% 0.19/0.44 thf(fact_1352_less__integer__code_I1_J,axiom,
% 0.19/0.44 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_integer_code(1)
% 0.19/0.44 thf(fact_1353_unset__bit__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: nat,X: int] :
% 0.19/0.44 ( ( bit_se2638667681897837118et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unset_bit_integer.abs_eq
% 0.19/0.44 thf(fact_1354_flip__bit__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: nat,X: int] :
% 0.19/0.44 ( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % flip_bit_integer.abs_eq
% 0.19/0.44 thf(fact_1355_set__bit__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: nat,X: int] :
% 0.19/0.44 ( ( bit_se5668285175392031749et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % set_bit_integer.abs_eq
% 0.19/0.44 thf(fact_1356_take__bit__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.44 => ! [N: nat,A3: A,B2: A] :
% 0.19/0.44 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
% 0.19/0.44 = ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % take_bit_add
% 0.19/0.44 thf(fact_1357_zero__integer__def,axiom,
% 0.19/0.44 ( ( zero_zero @ code_integer )
% 0.19/0.44 = ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % zero_integer_def
% 0.19/0.44 thf(fact_1358_less__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: int,X: int] :
% 0.19/0.44 ( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( ord_less @ int @ Xa @ X ) ) ).
% 0.19/0.44
% 0.19/0.44 % less_integer.abs_eq
% 0.19/0.44 thf(fact_1359_uminus__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [X: int] :
% 0.19/0.44 ( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % uminus_integer.abs_eq
% 0.19/0.44 thf(fact_1360_abs__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [X: int] :
% 0.19/0.44 ( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % abs_integer.abs_eq
% 0.19/0.44 thf(fact_1361_dvd__add__right__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_s4317794764714335236cancel @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.19/0.44 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add_right_iff
% 0.19/0.44 thf(fact_1362_dvd__add__left__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_s4317794764714335236cancel @ A )
% 0.19/0.44 => ! [A3: A,C2: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.19/0.44 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add_left_iff
% 0.19/0.44 thf(fact_1363_dvd__add,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_semiring_1 @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.19/0.44 => ( ( dvd_dvd @ A @ A3 @ C2 )
% 0.19/0.44 => ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_add
% 0.19/0.44 thf(fact_1364_divide__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: int,X: int] :
% 0.19/0.44 ( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % divide_integer.abs_eq
% 0.19/0.44 thf(fact_1365_dvd__unit__imp__unit,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ B2 )
% 0.19/0.44 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_unit_imp_unit
% 0.19/0.44 thf(fact_1366_unit__imp__dvd,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [B2: A,A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_imp_dvd
% 0.19/0.44 thf(fact_1367_one__dvd,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( comm_monoid_mult @ A )
% 0.19/0.44 => ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% 0.19/0.44
% 0.19/0.44 % one_dvd
% 0.19/0.44 thf(fact_1368_modulo__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: int,X: int] :
% 0.19/0.44 ( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( modulo_modulo @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % modulo_integer.abs_eq
% 0.19/0.44 thf(fact_1369_sgn__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [X: int] :
% 0.19/0.44 ( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % sgn_integer.abs_eq
% 0.19/0.44 thf(fact_1370_concat__bit__assoc,axiom,
% 0.19/0.44 ! [N: nat,K: int,M2: nat,L: int,R: int] :
% 0.19/0.44 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M2 @ L @ R ) )
% 0.19/0.44 = ( bit_concat_bit @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% 0.19/0.44
% 0.19/0.44 % concat_bit_assoc
% 0.19/0.44 thf(fact_1371_not__is__unit__0,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % not_is_unit_0
% 0.19/0.44 thf(fact_1372_pinf_I9_J,axiom,
% 0.19/0.44 ! [B: $tType] :
% 0.19/0.44 ( ( ( plus @ B )
% 0.19/0.44 & ( linorder @ B )
% 0.19/0.44 & ( dvd @ B ) )
% 0.19/0.44 => ! [D3: B,S: B] :
% 0.19/0.44 ? [Z3: B] :
% 0.19/0.44 ! [X7: B] :
% 0.19/0.44 ( ( ord_less @ B @ Z3 @ X7 )
% 0.19/0.44 => ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) )
% 0.19/0.44 = ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % pinf(9)
% 0.19/0.44 thf(fact_1373_pinf_I10_J,axiom,
% 0.19/0.44 ! [B: $tType] :
% 0.19/0.44 ( ( ( plus @ B )
% 0.19/0.44 & ( linorder @ B )
% 0.19/0.44 & ( dvd @ B ) )
% 0.19/0.44 => ! [D3: B,S: B] :
% 0.19/0.44 ? [Z3: B] :
% 0.19/0.44 ! [X7: B] :
% 0.19/0.44 ( ( ord_less @ B @ Z3 @ X7 )
% 0.19/0.44 => ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) )
% 0.19/0.44 = ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % pinf(10)
% 0.19/0.44 thf(fact_1374_minf_I9_J,axiom,
% 0.19/0.44 ! [B: $tType] :
% 0.19/0.44 ( ( ( plus @ B )
% 0.19/0.44 & ( linorder @ B )
% 0.19/0.44 & ( dvd @ B ) )
% 0.19/0.44 => ! [D3: B,S: B] :
% 0.19/0.44 ? [Z3: B] :
% 0.19/0.44 ! [X7: B] :
% 0.19/0.44 ( ( ord_less @ B @ X7 @ Z3 )
% 0.19/0.44 => ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) )
% 0.19/0.44 = ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % minf(9)
% 0.19/0.44 thf(fact_1375_minf_I10_J,axiom,
% 0.19/0.44 ! [B: $tType] :
% 0.19/0.44 ( ( ( plus @ B )
% 0.19/0.44 & ( linorder @ B )
% 0.19/0.44 & ( dvd @ B ) )
% 0.19/0.44 => ! [D3: B,S: B] :
% 0.19/0.44 ? [Z3: B] :
% 0.19/0.44 ! [X7: B] :
% 0.19/0.44 ( ( ord_less @ B @ X7 @ Z3 )
% 0.19/0.44 => ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) )
% 0.19/0.44 = ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % minf(10)
% 0.19/0.44 thf(fact_1376_unit__mult__right__cancel,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( ( times_times @ A @ B2 @ A3 )
% 0.19/0.44 = ( times_times @ A @ C2 @ A3 ) )
% 0.19/0.44 = ( B2 = C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_mult_right_cancel
% 0.19/0.44 thf(fact_1377_unit__mult__left__cancel,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( ( times_times @ A @ A3 @ B2 )
% 0.19/0.44 = ( times_times @ A @ A3 @ C2 ) )
% 0.19/0.44 = ( B2 = C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_mult_left_cancel
% 0.19/0.44 thf(fact_1378_mult__unit__dvd__iff_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.44 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mult_unit_dvd_iff'
% 0.19/0.44 thf(fact_1379_dvd__mult__unit__iff_H,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_mult_unit_iff'
% 0.19/0.44 thf(fact_1380_mult__unit__dvd__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % mult_unit_dvd_iff
% 0.19/0.44 thf(fact_1381_dvd__mult__unit__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_mult_unit_iff
% 0.19/0.44 thf(fact_1382_is__unit__mult__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 0.19/0.44 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % is_unit_mult_iff
% 0.19/0.44 thf(fact_1383_div__plus__div__distrib__dvd__left,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.44 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.44 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_plus_div_distrib_dvd_left
% 0.19/0.44 thf(fact_1384_div__plus__div__distrib__dvd__right,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( euclid4440199948858584721cancel @ A )
% 0.19/0.44 => ! [C2: A,B2: A,A3: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.19/0.44 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.44 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_plus_div_distrib_dvd_right
% 0.19/0.44 thf(fact_1385_unit__div__cancel,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [A3: A,B2: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 0.19/0.44 = ( divide_divide @ A @ C2 @ A3 ) )
% 0.19/0.44 = ( B2 = C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % unit_div_cancel
% 0.19/0.44 thf(fact_1386_div__unit__dvd__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % div_unit_dvd_iff
% 0.19/0.44 thf(fact_1387_dvd__div__unit__iff,axiom,
% 0.19/0.44 ! [A: $tType] :
% 0.19/0.44 ( ( algebraic_semidom @ A )
% 0.19/0.44 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.44 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.44 => ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
% 0.19/0.44 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % dvd_div_unit_iff
% 0.19/0.44 thf(fact_1388_xor__num_Ocases,axiom,
% 0.19/0.44 ! [X: product_prod @ num @ num] :
% 0.19/0.44 ( ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ one2 @ one2 ) )
% 0.19/0.44 => ( ! [N3: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
% 0.19/0.44 => ( ! [N3: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
% 0.19/0.44 => ( ! [M5: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) )
% 0.19/0.44 => ( ! [M5: num,N3: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
% 0.19/0.44 => ( ! [M5: num,N3: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
% 0.19/0.44 => ( ! [M5: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) )
% 0.19/0.44 => ( ! [M5: num,N3: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
% 0.19/0.44 => ~ ! [M5: num,N3: num] :
% 0.19/0.44 ( X
% 0.19/0.44 != ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % xor_num.cases
% 0.19/0.44 thf(fact_1389_plus__integer_Oabs__eq,axiom,
% 0.19/0.44 ! [Xa: int,X: int] :
% 0.19/0.44 ( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.44 = ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % plus_integer.abs_eq
% 0.19/0.44 thf(fact_1390_bezout__lemma__nat,axiom,
% 0.19/0.44 ! [D3: nat,A3: nat,B2: nat,X: nat,Y: nat] :
% 0.19/0.44 ( ( dvd_dvd @ nat @ D3 @ A3 )
% 0.19/0.44 => ( ( dvd_dvd @ nat @ D3 @ B2 )
% 0.19/0.44 => ( ( ( ( times_times @ nat @ A3 @ X )
% 0.19/0.44 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D3 ) )
% 0.19/0.44 | ( ( times_times @ nat @ B2 @ X )
% 0.19/0.44 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y ) @ D3 ) ) )
% 0.19/0.44 => ? [X2: nat,Y4: nat] :
% 0.19/0.44 ( ( dvd_dvd @ nat @ D3 @ A3 )
% 0.19/0.44 & ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A3 @ B2 ) )
% 0.19/0.44 & ( ( ( times_times @ nat @ A3 @ X2 )
% 0.19/0.44 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y4 ) @ D3 ) )
% 0.19/0.44 | ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X2 )
% 0.19/0.44 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D3 ) ) ) ) ) ) ) ).
% 0.19/0.44
% 0.19/0.44 % bezout_lemma_nat
% 0.19/0.44 thf(fact_1391_bezout__add__nat,axiom,
% 0.19/0.44 ! [A3: nat,B2: nat] :
% 0.19/0.44 ? [D2: nat,X2: nat,Y4: nat] :
% 0.19/0.44 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 0.19/0.44 & ( dvd_dvd @ nat @ D2 @ B2 )
% 0.19/0.44 & ( ( ( times_times @ nat @ A3 @ X2 )
% 0.19/0.45 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) )
% 0.19/0.45 | ( ( times_times @ nat @ B2 @ X2 )
% 0.19/0.45 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bezout_add_nat
% 0.19/0.45 thf(fact_1392_one__integer__def,axiom,
% 0.19/0.45 ( ( one_one @ code_integer )
% 0.19/0.45 = ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_integer_def
% 0.19/0.45 thf(fact_1393_less__eq__integer_Oabs__eq,axiom,
% 0.19/0.45 ! [Xa: int,X: int] :
% 0.19/0.45 ( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.45 = ( ord_less_eq @ int @ Xa @ X ) ) ).
% 0.19/0.45
% 0.19/0.45 % less_eq_integer.abs_eq
% 0.19/0.45 thf(fact_1394_times__integer_Oabs__eq,axiom,
% 0.19/0.45 ! [Xa: int,X: int] :
% 0.19/0.45 ( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.45 = ( code_integer_of_int @ ( times_times @ int @ Xa @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % times_integer.abs_eq
% 0.19/0.45 thf(fact_1395_zdvd__period,axiom,
% 0.19/0.45 ! [A3: int,D3: int,X: int,T2: int,C2: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ A3 @ D3 )
% 0.19/0.45 => ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
% 0.19/0.45 = ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D3 ) ) @ T2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % zdvd_period
% 0.19/0.45 thf(fact_1396_zdvd__reduce,axiom,
% 0.19/0.45 ! [K: int,N: int,M2: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M2 ) ) )
% 0.19/0.45 = ( dvd_dvd @ int @ K @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % zdvd_reduce
% 0.19/0.45 thf(fact_1397_minus__integer_Oabs__eq,axiom,
% 0.19/0.45 ! [Xa: int,X: int] :
% 0.19/0.45 ( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.45 = ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % minus_integer.abs_eq
% 0.19/0.45 thf(fact_1398_unity__coeff__ex,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( dvd @ A )
% 0.19/0.45 & ( semiring_0 @ A ) )
% 0.19/0.45 => ! [P: A > $o,L: A] :
% 0.19/0.45 ( ( ? [X3: A] : ( P @ ( times_times @ A @ L @ X3 ) ) )
% 0.19/0.45 = ( ? [X3: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X3 @ ( zero_zero @ A ) ) )
% 0.19/0.45 & ( P @ X3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unity_coeff_ex
% 0.19/0.45 thf(fact_1399_unit__dvdE,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [A3: A,B2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.45 => ~ ( ( A3
% 0.19/0.45 != ( zero_zero @ A ) )
% 0.19/0.45 => ! [C4: A] :
% 0.19/0.45 ( B2
% 0.19/0.45 != ( times_times @ A @ A3 @ C4 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_dvdE
% 0.19/0.45 thf(fact_1400_inf__period_I4_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( comm_ring @ A )
% 0.19/0.45 & ( dvd @ A ) )
% 0.19/0.45 => ! [D3: A,D4: A,T2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ D3 @ D4 )
% 0.19/0.45 => ! [X7: A,K8: A] :
% 0.19/0.45 ( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T2 ) ) )
% 0.19/0.45 = ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K8 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % inf_period(4)
% 0.19/0.45 thf(fact_1401_inf__period_I3_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( comm_ring @ A )
% 0.19/0.45 & ( dvd @ A ) )
% 0.19/0.45 => ! [D3: A,D4: A,T2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ D3 @ D4 )
% 0.19/0.45 => ! [X7: A,K8: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T2 ) )
% 0.19/0.45 = ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K8 @ D4 ) ) @ T2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % inf_period(3)
% 0.19/0.45 thf(fact_1402_unit__div__eq__0__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [B2: A,A3: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.19/0.45 = ( zero_zero @ A ) )
% 0.19/0.45 = ( A3
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_div_eq_0_iff
% 0.19/0.45 thf(fact_1403_numeral__Bit1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( numeral @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 0.19/0.45 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % numeral_Bit1
% 0.19/0.45 thf(fact_1404_unit__eq__div1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.19/0.45 = C2 )
% 0.19/0.45 = ( A3
% 0.19/0.45 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_eq_div1
% 0.19/0.45 thf(fact_1405_unit__eq__div2,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( A3
% 0.19/0.45 = ( divide_divide @ A @ C2 @ B2 ) )
% 0.19/0.45 = ( ( times_times @ A @ A3 @ B2 )
% 0.19/0.45 = C2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_eq_div2
% 0.19/0.45 thf(fact_1406_div__mult__unit2,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [C2: A,B2: A,A3: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 0.19/0.45 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.45 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % div_mult_unit2
% 0.19/0.45 thf(fact_1407_unit__div__commute,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [B2: A,A3: A,C2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 0.19/0.45 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_div_commute
% 0.19/0.45 thf(fact_1408_unit__div__mult__swap,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [C2: A,A3: A,B2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 0.19/0.45 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_div_mult_swap
% 0.19/0.45 thf(fact_1409_is__unit__div__mult2__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [B2: A,C2: A,A3: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 0.19/0.45 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % is_unit_div_mult2_eq
% 0.19/0.45 thf(fact_1410_unit__imp__mod__eq__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( euclid3725896446679973847miring @ A )
% 0.19/0.45 => ! [B2: A,A3: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % unit_imp_mod_eq_0
% 0.19/0.45 thf(fact_1411_is__unit__power__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [A3: A,N: nat] :
% 0.19/0.45 ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
% 0.19/0.45 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.45 | ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % is_unit_power_iff
% 0.19/0.45 thf(fact_1412_bezout__add__strong__nat,axiom,
% 0.19/0.45 ! [A3: nat,B2: nat] :
% 0.19/0.45 ( ( A3
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 => ? [D2: nat,X2: nat,Y4: nat] :
% 0.19/0.45 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 0.19/0.45 & ( dvd_dvd @ nat @ D2 @ B2 )
% 0.19/0.45 & ( ( times_times @ nat @ A3 @ X2 )
% 0.19/0.45 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bezout_add_strong_nat
% 0.19/0.45 thf(fact_1413_take__bit__Suc__bit1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [N: nat,K: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_Suc_bit1
% 0.19/0.45 thf(fact_1414_numeral__code_I3_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( numeral @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 0.19/0.45 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % numeral_code(3)
% 0.19/0.45 thf(fact_1415_stable__imp__take__bit__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [A3: A,N: nat] :
% 0.19/0.45 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = A3 )
% 0.19/0.45 => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.19/0.45 = ( zero_zero @ A ) ) )
% 0.19/0.45 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 0.19/0.45 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % stable_imp_take_bit_eq
% 0.19/0.45 thf(fact_1416_is__unitE,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [A3: A,C2: A] :
% 0.19/0.45 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.45 => ~ ( ( A3
% 0.19/0.45 != ( zero_zero @ A ) )
% 0.19/0.45 => ! [B3: A] :
% 0.19/0.45 ( ( B3
% 0.19/0.45 != ( zero_zero @ A ) )
% 0.19/0.45 => ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.45 = B3 )
% 0.19/0.45 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
% 0.19/0.45 = A3 )
% 0.19/0.45 => ( ( ( times_times @ A @ A3 @ B3 )
% 0.19/0.45 = ( one_one @ A ) )
% 0.19/0.45 => ( ( divide_divide @ A @ C2 @ A3 )
% 0.19/0.45 != ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % is_unitE
% 0.19/0.45 thf(fact_1417_is__unit__div__mult__cancel__left,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [A3: A,B2: A] :
% 0.19/0.45 ( ( A3
% 0.19/0.45 != ( zero_zero @ A ) )
% 0.19/0.45 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 0.19/0.45 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % is_unit_div_mult_cancel_left
% 0.19/0.45 thf(fact_1418_is__unit__div__mult__cancel__right,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [A3: A,B2: A] :
% 0.19/0.45 ( ( A3
% 0.19/0.45 != ( zero_zero @ A ) )
% 0.19/0.45 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.19/0.45 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
% 0.19/0.45 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % is_unit_div_mult_cancel_right
% 0.19/0.45 thf(fact_1419_odd__even__add,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_parity @ A )
% 0.19/0.45 => ! [A3: A,B2: A] :
% 0.19/0.45 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
% 0.19/0.45 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % odd_even_add
% 0.19/0.45 thf(fact_1420_odd__one,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_parity @ A )
% 0.19/0.45 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % odd_one
% 0.19/0.45 thf(fact_1421_dvd__power__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( algebraic_semidom @ A )
% 0.19/0.45 => ! [X: A,M2: nat,N: nat] :
% 0.19/0.45 ( ( X
% 0.19/0.45 != ( zero_zero @ A ) )
% 0.19/0.45 => ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
% 0.19/0.45 = ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
% 0.19/0.45 | ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dvd_power_iff
% 0.19/0.45 thf(fact_1422_dvd__power,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A )
% 0.19/0.45 => ! [N: nat,X: A] :
% 0.19/0.45 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 | ( X
% 0.19/0.45 = ( one_one @ A ) ) )
% 0.19/0.45 => ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dvd_power
% 0.19/0.45 thf(fact_1423_Suc3__eq__add__3,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 0.19/0.45 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc3_eq_add_3
% 0.19/0.45 thf(fact_1424_dvd__mult__cancel2,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.19/0.45 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M2 ) @ M2 )
% 0.19/0.45 = ( N
% 0.19/0.45 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dvd_mult_cancel2
% 0.19/0.45 thf(fact_1425_dvd__mult__cancel1,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 0.19/0.45 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N ) @ M2 )
% 0.19/0.45 = ( N
% 0.19/0.45 = ( one_one @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dvd_mult_cancel1
% 0.19/0.45 thf(fact_1426_dvd__minus__add,axiom,
% 0.19/0.45 ! [Q3: nat,N: nat,R: nat,M2: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ Q3 @ N )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R @ M2 ) )
% 0.19/0.45 => ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q3 ) )
% 0.19/0.45 = ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M2 ) @ Q3 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dvd_minus_add
% 0.19/0.45 thf(fact_1427_power__dvd__imp__le,axiom,
% 0.19/0.45 ! [I: nat,M2: nat,N: nat] :
% 0.19/0.45 ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
% 0.19/0.45 => ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
% 0.19/0.45 => ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % power_dvd_imp_le
% 0.19/0.45 thf(fact_1428_aset_I10_J,axiom,
% 0.19/0.45 ! [D3: int,D4: int,A4: set @ int,T2: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ D3 @ D4 )
% 0.19/0.45 => ! [X7: int] :
% 0.19/0.45 ( ! [Xa3: int] :
% 0.19/0.45 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.45 => ! [Xb3: int] :
% 0.19/0.45 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.45 => ( X7
% 0.19/0.45 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.45 => ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 0.19/0.45 => ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % aset(10)
% 0.19/0.45 thf(fact_1429_aset_I9_J,axiom,
% 0.19/0.45 ! [D3: int,D4: int,A4: set @ int,T2: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ D3 @ D4 )
% 0.19/0.45 => ! [X7: int] :
% 0.19/0.45 ( ! [Xa3: int] :
% 0.19/0.45 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.45 => ! [Xb3: int] :
% 0.19/0.45 ( ( member @ int @ Xb3 @ A4 )
% 0.19/0.45 => ( X7
% 0.19/0.45 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.45 => ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 0.19/0.45 => ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % aset(9)
% 0.19/0.45 thf(fact_1430_bset_I10_J,axiom,
% 0.19/0.45 ! [D3: int,D4: int,B6: set @ int,T2: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ D3 @ D4 )
% 0.19/0.45 => ! [X7: int] :
% 0.19/0.45 ( ! [Xa3: int] :
% 0.19/0.45 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.45 => ! [Xb3: int] :
% 0.19/0.45 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.45 => ( X7
% 0.19/0.45 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.45 => ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 0.19/0.45 => ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bset(10)
% 0.19/0.45 thf(fact_1431_bset_I9_J,axiom,
% 0.19/0.45 ! [D3: int,D4: int,B6: set @ int,T2: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ D3 @ D4 )
% 0.19/0.45 => ! [X7: int] :
% 0.19/0.45 ( ! [Xa3: int] :
% 0.19/0.45 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 0.19/0.45 => ! [Xb3: int] :
% 0.19/0.45 ( ( member @ int @ Xb3 @ B6 )
% 0.19/0.45 => ( X7
% 0.19/0.45 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 0.19/0.45 => ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 0.19/0.45 => ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bset(9)
% 0.19/0.45 thf(fact_1432_num_Osize__gen_I3_J,axiom,
% 0.19/0.45 ! [X32: num] :
% 0.19/0.45 ( ( size_num @ ( bit1 @ X32 ) )
% 0.19/0.45 = ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % num.size_gen(3)
% 0.19/0.45 thf(fact_1433_num_Osize_I6_J,axiom,
% 0.19/0.45 ! [X32: num] :
% 0.19/0.45 ( ( size_size @ num @ ( bit1 @ X32 ) )
% 0.19/0.45 = ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % num.size(6)
% 0.19/0.45 thf(fact_1434_odd__iff__mod__2__eq__one,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_parity @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.45 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % odd_iff_mod_2_eq_one
% 0.19/0.45 thf(fact_1435_Suc__mod__eq__add3__mod,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
% 0.19/0.45 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_mod_eq_add3_mod
% 0.19/0.45 thf(fact_1436_Suc__div__eq__add3__div,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
% 0.19/0.45 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_div_eq_add3_div
% 0.19/0.45 thf(fact_1437_even__diff__iff,axiom,
% 0.19/0.45 ! [K: int,L: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
% 0.19/0.45 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % even_diff_iff
% 0.19/0.45 thf(fact_1438_even__abs__add__iff,axiom,
% 0.19/0.45 ! [K: int,L: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
% 0.19/0.45 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % even_abs_add_iff
% 0.19/0.45 thf(fact_1439_even__add__abs__iff,axiom,
% 0.19/0.45 ! [K: int,L: int] :
% 0.19/0.45 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
% 0.19/0.45 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % even_add_abs_iff
% 0.19/0.45 thf(fact_1440_signed__take__bit__eq__take__bit__shift,axiom,
% 0.19/0.45 ( ( bit_ri4674362597316999326ke_bit @ int )
% 0.19/0.45 = ( ^ [N5: nat,K4: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ ( plus_plus @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % signed_take_bit_eq_take_bit_shift
% 0.19/0.45 thf(fact_1441_take__bit__incr__eq,axiom,
% 0.19/0.45 ! [N: nat,K: int] :
% 0.19/0.45 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.19/0.45 != ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
% 0.19/0.45 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 0.19/0.45 = ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_incr_eq
% 0.19/0.45 thf(fact_1442_oddE,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_parity @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ~ ! [B3: A] :
% 0.19/0.45 ( A3
% 0.19/0.45 != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % oddE
% 0.19/0.45 thf(fact_1443_parity__cases,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_parity @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 != ( zero_zero @ A ) ) )
% 0.19/0.45 => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 != ( one_one @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % parity_cases
% 0.19/0.45 thf(fact_1444_mod2__eq__if,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_parity @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = ( zero_zero @ A ) ) )
% 0.19/0.45 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mod2_eq_if
% 0.19/0.45 thf(fact_1445_minus__one__power__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ring_1 @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.45 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.19/0.45 = ( one_one @ A ) ) )
% 0.19/0.45 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.45 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.19/0.45 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % minus_one_power_iff
% 0.19/0.45 thf(fact_1446_take__bit__int__greater__eq,axiom,
% 0.19/0.45 ! [K: int,N: nat] :
% 0.19/0.45 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 0.19/0.45 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_int_greater_eq
% 0.19/0.45 thf(fact_1447_take__bit__Suc__minus__1__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_Suc_minus_1_eq
% 0.19/0.45 thf(fact_1448_take__bit__Suc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [N: nat,A3: A] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
% 0.19/0.45 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_Suc
% 0.19/0.45 thf(fact_1449_take__bit__numeral__minus__1__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [K: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_numeral_minus_1_eq
% 0.19/0.45 thf(fact_1450_even__mask__div__iff_H,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.45 => ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.19/0.45 = ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % even_mask_div_iff'
% 0.19/0.45 thf(fact_1451_even__mask__div__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.19/0.45 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.19/0.45 = ( zero_zero @ A ) )
% 0.19/0.45 | ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % even_mask_div_iff
% 0.19/0.45 thf(fact_1452_Code__Numeral_Onegative__def,axiom,
% 0.19/0.45 ( code_negative
% 0.19/0.45 = ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Code_Numeral.negative_def
% 0.19/0.45 thf(fact_1453_take__bit__numeral__minus__numeral__int,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.19/0.45 = ( case_option @ int @ num @ ( zero_zero @ int )
% 0.19/0.45 @ ^ [Q5: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ int @ Q5 ) ) )
% 0.19/0.45 @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_numeral_minus_numeral_int
% 0.19/0.45 thf(fact_1454_signed__take__bit__numeral__minus__bit1,axiom,
% 0.19/0.45 ! [L: num,K: num] :
% 0.19/0.45 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.19/0.45 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % signed_take_bit_numeral_minus_bit1
% 0.19/0.45 thf(fact_1455_execute__change,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( heap @ A )
% 0.19/0.45 => ! [F: A > A,R: ref @ A,H3: heap_ext @ product_unit] :
% 0.19/0.45 ( ( heap_Time_execute @ A @ ( ref_change @ A @ F @ R ) @ H3 )
% 0.19/0.45 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % execute_change
% 0.19/0.45 thf(fact_1456_signed__take__bit__numeral__bit1,axiom,
% 0.19/0.45 ! [L: num,K: num] :
% 0.19/0.45 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 0.19/0.45 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % signed_take_bit_numeral_bit1
% 0.19/0.45 thf(fact_1457_take__bit__Suc__minus__bit1,axiom,
% 0.19/0.45 ! [N: nat,K: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.19/0.45 = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_Suc_minus_bit1
% 0.19/0.45 thf(fact_1458_divmod__BitM__2__eq,axiom,
% 0.19/0.45 ! [M2: num] :
% 0.19/0.45 ( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
% 0.19/0.45 = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % divmod_BitM_2_eq
% 0.19/0.45 thf(fact_1459_flip__bit__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 0.19/0.45 = ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % flip_bit_0
% 0.19/0.45 thf(fact_1460_of__bool__eq_I2_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( zero_neq_one @ A )
% 0.19/0.45 => ( ( zero_neq_one_of_bool @ A @ $true )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % of_bool_eq(2)
% 0.19/0.45 thf(fact_1461_of__bool__eq__1__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( zero_neq_one @ A )
% 0.19/0.45 => ! [P: $o] :
% 0.19/0.45 ( ( ( zero_neq_one_of_bool @ A @ P )
% 0.19/0.45 = ( one_one @ A ) )
% 0.19/0.45 = P ) ) ).
% 0.19/0.45
% 0.19/0.45 % of_bool_eq_1_iff
% 0.19/0.45 thf(fact_1462_of__bool__less__one__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linordered_semidom @ A )
% 0.19/0.45 => ! [P: $o] :
% 0.19/0.45 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
% 0.19/0.45 = ( ~ P ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % of_bool_less_one_iff
% 0.19/0.45 thf(fact_1463_of__bool__not__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ring_1 @ A )
% 0.19/0.45 => ! [P: $o] :
% 0.19/0.45 ( ( zero_neq_one_of_bool @ A @ ~ P )
% 0.19/0.45 = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % of_bool_not_iff
% 0.19/0.45 thf(fact_1464_take__bit__num__simps_I3_J,axiom,
% 0.19/0.45 ! [N: nat,M2: num] :
% 0.19/0.45 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
% 0.19/0.45 = ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 0.19/0.45 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
% 0.19/0.45 @ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_num_simps(3)
% 0.19/0.45 thf(fact_1465_take__bit__num__simps_I4_J,axiom,
% 0.19/0.45 ! [N: nat,M2: num] :
% 0.19/0.45 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
% 0.19/0.45 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_num_simps(4)
% 0.19/0.45 thf(fact_1466_take__bit__num__simps_I6_J,axiom,
% 0.19/0.45 ! [R: num,M2: num] :
% 0.19/0.45 ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit0 @ M2 ) )
% 0.19/0.45 = ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 0.19/0.45 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
% 0.19/0.45 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_num_simps(6)
% 0.19/0.45 thf(fact_1467_take__bit__of__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 0.19/0.45 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_of_1
% 0.19/0.45 thf(fact_1468_add__neg__numeral__special_I6_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( neg_numeral @ A )
% 0.19/0.45 => ! [M2: num] :
% 0.19/0.45 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_neg_numeral_special(6)
% 0.19/0.45 thf(fact_1469_add__neg__numeral__special_I5_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( neg_numeral @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.19/0.45 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_neg_numeral_special(5)
% 0.19/0.45 thf(fact_1470_diff__numeral__special_I5_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( neg_numeral @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.45 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_numeral_special(5)
% 0.19/0.45 thf(fact_1471_diff__numeral__special_I6_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( neg_numeral @ A )
% 0.19/0.45 => ! [M2: num] :
% 0.19/0.45 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_numeral_special(6)
% 0.19/0.45 thf(fact_1472_take__bit__num__simps_I7_J,axiom,
% 0.19/0.45 ! [R: num,M2: num] :
% 0.19/0.45 ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit1 @ M2 ) )
% 0.19/0.45 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_num_simps(7)
% 0.19/0.45 thf(fact_1473_take__bit__numeral__numeral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.45 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_numeral_numeral
% 0.19/0.45 thf(fact_1474_bits__1__div__exp,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.45 = ( zero_neq_one_of_bool @ A
% 0.19/0.45 @ ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bits_1_div_exp
% 0.19/0.45 thf(fact_1475_one__div__2__pow__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.45 = ( zero_neq_one_of_bool @ A
% 0.19/0.45 @ ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_div_2_pow_eq
% 0.19/0.45 thf(fact_1476_one__mod__2__pow__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( euclid5411537665997757685th_nat @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.45 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_mod_2_pow_eq
% 0.19/0.45 thf(fact_1477_mult__inc,axiom,
% 0.19/0.45 ! [X: num,Y: num] :
% 0.19/0.45 ( ( times_times @ num @ X @ ( inc @ Y ) )
% 0.19/0.45 = ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% 0.19/0.45
% 0.19/0.45 % mult_inc
% 0.19/0.45 thf(fact_1478_add__inc,axiom,
% 0.19/0.45 ! [X: num,Y: num] :
% 0.19/0.45 ( ( plus_plus @ num @ X @ ( inc @ Y ) )
% 0.19/0.45 = ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_inc
% 0.19/0.45 thf(fact_1479_of__bool__less__eq__one,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linordered_semidom @ A )
% 0.19/0.45 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % of_bool_less_eq_one
% 0.19/0.45 thf(fact_1480_of__bool__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( zero_neq_one @ A )
% 0.19/0.45 => ( ( zero_neq_one_of_bool @ A )
% 0.19/0.45 = ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % of_bool_def
% 0.19/0.45 thf(fact_1481_split__of__bool,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( zero_neq_one @ A )
% 0.19/0.45 => ! [P: A > $o,P5: $o] :
% 0.19/0.45 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 0.19/0.45 = ( ( P5
% 0.19/0.45 => ( P @ ( one_one @ A ) ) )
% 0.19/0.45 & ( ~ P5
% 0.19/0.45 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % split_of_bool
% 0.19/0.45 thf(fact_1482_split__of__bool__asm,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( zero_neq_one @ A )
% 0.19/0.45 => ! [P: A > $o,P5: $o] :
% 0.19/0.45 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 0.19/0.45 = ( ~ ( ( P5
% 0.19/0.45 & ~ ( P @ ( one_one @ A ) ) )
% 0.19/0.45 | ( ~ P5
% 0.19/0.45 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % split_of_bool_asm
% 0.19/0.45 thf(fact_1483_add__One,axiom,
% 0.19/0.45 ! [X: num] :
% 0.19/0.45 ( ( plus_plus @ num @ X @ one2 )
% 0.19/0.45 = ( inc @ X ) ) ).
% 0.19/0.45
% 0.19/0.45 % add_One
% 0.19/0.45 thf(fact_1484_effect__updateI,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( heap @ A )
% 0.19/0.45 => ! [H4: heap_ext @ product_unit,R: ref @ A,V: A,H3: heap_ext @ product_unit,N: nat,X: product_unit] :
% 0.19/0.45 ( ( H4
% 0.19/0.45 = ( ref_set @ A @ R @ V @ H3 ) )
% 0.19/0.45 => ( ( N
% 0.19/0.45 = ( one_one @ nat ) )
% 0.19/0.45 => ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R @ V ) @ H3 @ H4 @ X @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % effect_updateI
% 0.19/0.45 thf(fact_1485_effect__updateE,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( heap @ A )
% 0.19/0.45 => ! [R4: ref @ A,V: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: product_unit,N: nat] :
% 0.19/0.45 ( ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R4 @ V ) @ H3 @ H4 @ R @ N )
% 0.19/0.45 => ~ ( ( H4
% 0.19/0.45 = ( ref_set @ A @ R4 @ V @ H3 ) )
% 0.19/0.45 => ( N
% 0.19/0.45 != ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % effect_updateE
% 0.19/0.45 thf(fact_1486_pred__numeral__def,axiom,
% 0.19/0.45 ( pred_numeral
% 0.19/0.45 = ( ^ [K4: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K4 ) @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % pred_numeral_def
% 0.19/0.45 thf(fact_1487_BitM__plus__one,axiom,
% 0.19/0.45 ! [N: num] :
% 0.19/0.45 ( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
% 0.19/0.45 = ( bit0 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % BitM_plus_one
% 0.19/0.45 thf(fact_1488_one__plus__BitM,axiom,
% 0.19/0.45 ! [N: num] :
% 0.19/0.45 ( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
% 0.19/0.45 = ( bit0 @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_plus_BitM
% 0.19/0.45 thf(fact_1489_numeral__inc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( numeral @ A )
% 0.19/0.45 => ! [X: num] :
% 0.19/0.45 ( ( numeral_numeral @ A @ ( inc @ X ) )
% 0.19/0.45 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % numeral_inc
% 0.19/0.45 thf(fact_1490_numeral__BitM,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( neg_numeral @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( numeral_numeral @ A @ ( bitM @ N ) )
% 0.19/0.45 = ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % numeral_BitM
% 0.19/0.45 thf(fact_1491_take__bit__numeral__minus__bit1,axiom,
% 0.19/0.45 ! [L: num,K: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 0.19/0.45 = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_numeral_minus_bit1
% 0.19/0.45 thf(fact_1492_bits__induct,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [P: A > $o,A3: A] :
% 0.19/0.45 ( ! [A5: A] :
% 0.19/0.45 ( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = A5 )
% 0.19/0.45 => ( P @ A5 ) )
% 0.19/0.45 => ( ! [A5: A,B3: $o] :
% 0.19/0.45 ( ( P @ A5 )
% 0.19/0.45 => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = A5 )
% 0.19/0.45 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
% 0.19/0.45 => ( P @ A3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bits_induct
% 0.19/0.45 thf(fact_1493_effect__changeE,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( heap @ A )
% 0.19/0.45 => ! [F: A > A,R4: ref @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.19/0.45 ( ( heap_Time_effect @ A @ ( ref_change @ A @ F @ R4 ) @ H3 @ H4 @ R @ N )
% 0.19/0.45 => ~ ( ( H4
% 0.19/0.45 = ( ref_set @ A @ R4 @ ( F @ ( ref_get @ A @ H3 @ R4 ) ) @ H3 ) )
% 0.19/0.45 => ( ( R
% 0.19/0.45 = ( F @ ( ref_get @ A @ H3 @ R4 ) ) )
% 0.19/0.45 => ( N
% 0.19/0.45 != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % effect_changeE
% 0.19/0.45 thf(fact_1494_effect__changeI,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( heap @ A )
% 0.19/0.45 => ! [H4: heap_ext @ product_unit,R: ref @ A,F: A > A,H3: heap_ext @ product_unit,X: A,N: nat] :
% 0.19/0.45 ( ( H4
% 0.19/0.45 = ( ref_set @ A @ R @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ H3 ) )
% 0.19/0.45 => ( ( X
% 0.19/0.45 = ( F @ ( ref_get @ A @ H3 @ R ) ) )
% 0.19/0.45 => ( ( N
% 0.19/0.45 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 0.19/0.45 => ( heap_Time_effect @ A @ ( ref_change @ A @ F @ R ) @ H3 @ H4 @ X @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % effect_changeI
% 0.19/0.45 thf(fact_1495_divide__int__unfold,axiom,
% 0.19/0.45 ! [L: int,K: int,N: nat,M2: nat] :
% 0.19/0.45 ( ( ( ( ( sgn_sgn @ int @ L )
% 0.19/0.45 = ( zero_zero @ int ) )
% 0.19/0.45 | ( ( sgn_sgn @ int @ K )
% 0.19/0.45 = ( zero_zero @ int ) )
% 0.19/0.45 | ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) )
% 0.19/0.45 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.19/0.45 = ( zero_zero @ int ) ) )
% 0.19/0.45 & ( ~ ( ( ( sgn_sgn @ int @ L )
% 0.19/0.45 = ( zero_zero @ int ) )
% 0.19/0.45 | ( ( sgn_sgn @ int @ K )
% 0.19/0.45 = ( zero_zero @ int ) )
% 0.19/0.45 | ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) )
% 0.19/0.45 => ( ( ( ( sgn_sgn @ int @ K )
% 0.19/0.45 = ( sgn_sgn @ int @ L ) )
% 0.19/0.45 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.19/0.45 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) )
% 0.19/0.45 & ( ( ( sgn_sgn @ int @ K )
% 0.19/0.45 != ( sgn_sgn @ int @ L ) )
% 0.19/0.45 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.19/0.45 = ( uminus_uminus @ int
% 0.19/0.45 @ ( semiring_1_of_nat @ int
% 0.19/0.45 @ ( plus_plus @ nat @ ( divide_divide @ nat @ M2 @ N )
% 0.19/0.45 @ ( zero_neq_one_of_bool @ nat
% 0.19/0.45 @ ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % divide_int_unfold
% 0.19/0.45 thf(fact_1496_take__bit__numeral__bit1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [L: num,K: num] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_numeral_bit1
% 0.19/0.45 thf(fact_1497_Divides_Oadjust__div__eq,axiom,
% 0.19/0.45 ! [Q3: int,R: int] :
% 0.19/0.45 ( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 0.19/0.45 = ( plus_plus @ int @ Q3
% 0.19/0.45 @ ( zero_neq_one_of_bool @ int
% 0.19/0.45 @ ( R
% 0.19/0.45 != ( zero_zero @ int ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Divides.adjust_div_eq
% 0.19/0.45 thf(fact_1498_mask__numeral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
% 0.19/0.45 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mask_numeral
% 0.19/0.45 thf(fact_1499_and__int__unfold,axiom,
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ int )
% 0.19/0.45 = ( ^ [K4: int,L3: int] :
% 0.19/0.45 ( if @ int
% 0.19/0.45 @ ( ( K4
% 0.19/0.45 = ( zero_zero @ int ) )
% 0.19/0.45 | ( L3
% 0.19/0.45 = ( zero_zero @ int ) ) )
% 0.19/0.45 @ ( zero_zero @ int )
% 0.19/0.45 @ ( if @ int
% 0.19/0.45 @ ( K4
% 0.19/0.45 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.19/0.45 @ L3
% 0.19/0.45 @ ( if @ int
% 0.19/0.45 @ ( L3
% 0.19/0.45 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.19/0.45 @ K4
% 0.19/0.45 @ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_int_unfold
% 0.19/0.45 thf(fact_1500_the__res__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( time_the_res @ A )
% 0.19/0.45 = ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 0.19/0.45 ( case_option @ A @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ A )
% 0.19/0.45 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A
% 0.19/0.45 @ ^ [R3: A] :
% 0.19/0.45 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ A
% 0.19/0.45 @ ^ [X3: heap_ext @ product_unit,Xa4: nat] : R3 ) )
% 0.19/0.45 @ ( heap_Time_execute @ A @ M @ H ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % the_res_def
% 0.19/0.45 thf(fact_1501_take__bit__eq__mask__iff__exp__dvd,axiom,
% 0.19/0.45 ! [N: nat,K: int] :
% 0.19/0.45 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.19/0.45 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 0.19/0.45 = ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_eq_mask_iff_exp_dvd
% 0.19/0.45 thf(fact_1502_Code__Numeral_Opositive__def,axiom,
% 0.19/0.45 ( code_positive
% 0.19/0.45 = ( numeral_numeral @ code_integer ) ) ).
% 0.19/0.45
% 0.19/0.45 % Code_Numeral.positive_def
% 0.19/0.45 thf(fact_1503_divide__int__def,axiom,
% 0.19/0.45 ( ( divide_divide @ int )
% 0.19/0.45 = ( ^ [K4: int,L3: int] :
% 0.19/0.45 ( if @ int
% 0.19/0.45 @ ( L3
% 0.19/0.45 = ( zero_zero @ int ) )
% 0.19/0.45 @ ( zero_zero @ int )
% 0.19/0.45 @ ( if @ int
% 0.19/0.45 @ ( ( sgn_sgn @ int @ K4 )
% 0.19/0.45 = ( sgn_sgn @ int @ L3 ) )
% 0.19/0.45 @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
% 0.19/0.45 @ ( uminus_uminus @ int
% 0.19/0.45 @ ( semiring_1_of_nat @ int
% 0.19/0.45 @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
% 0.19/0.45 @ ( zero_neq_one_of_bool @ nat
% 0.19/0.45 @ ~ ( dvd_dvd @ int @ L3 @ K4 ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % divide_int_def
% 0.19/0.45 thf(fact_1504_sum__gp0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( division_ring @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( ( X
% 0.19/0.45 = ( one_one @ A ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
% 0.19/0.45 & ( ( X
% 0.19/0.45 != ( one_one @ A ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_gp0
% 0.19/0.45 thf(fact_1505_atMost__eq__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( order @ A )
% 0.19/0.45 => ! [X: A,Y: A] :
% 0.19/0.45 ( ( ( set_ord_atMost @ A @ X )
% 0.19/0.45 = ( set_ord_atMost @ A @ Y ) )
% 0.19/0.45 = ( X = Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atMost_eq_iff
% 0.19/0.45 thf(fact_1506_atMost__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ! [I: A,K: A] :
% 0.19/0.45 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
% 0.19/0.45 = ( ord_less_eq @ A @ I @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atMost_iff
% 0.19/0.45 thf(fact_1507_atMost__subset__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( preorder @ A )
% 0.19/0.45 => ! [X: A,Y: A] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
% 0.19/0.45 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atMost_subset_iff
% 0.19/0.45 thf(fact_1508_bit_Oconj__one__right,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [X: A] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = X ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit.conj_one_right
% 0.19/0.45 thf(fact_1509_and_Oright__neutral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % and.right_neutral
% 0.19/0.45 thf(fact_1510_and_Oleft__neutral,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % and.left_neutral
% 0.19/0.45 thf(fact_1511_and__numerals_I2_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [Y: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_numerals(2)
% 0.19/0.45 thf(fact_1512_and__numerals_I8_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_numerals(8)
% 0.19/0.45 thf(fact_1513_Icc__subset__Iic__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( preorder @ A )
% 0.19/0.45 => ! [L: A,H3: A,H4: A] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) @ ( set_ord_atMost @ A @ H4 ) )
% 0.19/0.45 = ( ~ ( ord_less_eq @ A @ L @ H3 )
% 0.19/0.45 | ( ord_less_eq @ A @ H3 @ H4 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Icc_subset_Iic_iff
% 0.19/0.45 thf(fact_1514_mask__Suc__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mask_Suc_0
% 0.19/0.45 thf(fact_1515_sum_OatMost__Suc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.atMost_Suc
% 0.19/0.45 thf(fact_1516_take__bit__minus__one__eq__mask,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_minus_one_eq_mask
% 0.19/0.45 thf(fact_1517_and__numerals_I1_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [Y: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 0.19/0.45 = ( zero_zero @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_numerals(1)
% 0.19/0.45 thf(fact_1518_and__numerals_I5_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 0.19/0.45 = ( zero_zero @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_numerals(5)
% 0.19/0.45 thf(fact_1519_nat__numeral__diff__1,axiom,
% 0.19/0.45 ! [V: num] :
% 0.19/0.45 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
% 0.19/0.45 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_numeral_diff_1
% 0.19/0.45 thf(fact_1520_and__numerals_I7_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num,Y: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_numerals(7)
% 0.19/0.45 thf(fact_1521_mask__integer_Oabs__eq,axiom,
% 0.19/0.45 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 0.19/0.45 = ( ^ [X3: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mask_integer.abs_eq
% 0.19/0.45 thf(fact_1522_not__Iic__eq__Icc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( no_bot @ A )
% 0.19/0.45 => ! [H4: A,L: A,H3: A] :
% 0.19/0.45 ( ( set_ord_atMost @ A @ H4 )
% 0.19/0.45 != ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_Iic_eq_Icc
% 0.19/0.45 thf(fact_1523_atMost__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ( ( set_ord_atMost @ A )
% 0.19/0.45 = ( ^ [U2: A] :
% 0.19/0.45 ( collect @ A
% 0.19/0.45 @ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ U2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atMost_def
% 0.19/0.45 thf(fact_1524_and__eq__minus__1__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [A3: A,B2: A] :
% 0.19/0.45 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
% 0.19/0.45 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( ( A3
% 0.19/0.45 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 & ( B2
% 0.19/0.45 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_eq_minus_1_iff
% 0.19/0.45 thf(fact_1525_atMost__atLeast0,axiom,
% 0.19/0.45 ( ( set_ord_atMost @ nat )
% 0.19/0.45 = ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atMost_atLeast0
% 0.19/0.45 thf(fact_1526_nat__one__as__int,axiom,
% 0.19/0.45 ( ( one_one @ nat )
% 0.19/0.45 = ( nat2 @ ( one_one @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_one_as_int
% 0.19/0.45 thf(fact_1527_not__Iic__le__Icc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( no_bot @ A )
% 0.19/0.45 => ! [H3: A,L4: A,H4: A] :
% 0.19/0.45 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H3 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_Iic_le_Icc
% 0.19/0.45 thf(fact_1528_nat__int__add,axiom,
% 0.19/0.45 ! [A3: nat,B2: nat] :
% 0.19/0.45 ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
% 0.19/0.45 = ( plus_plus @ nat @ A3 @ B2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_int_add
% 0.19/0.45 thf(fact_1529_option_Othe__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( the2 @ A )
% 0.19/0.45 = ( case_option @ A @ A @ ( undefined @ A )
% 0.19/0.45 @ ^ [X24: A] : X24 ) ) ).
% 0.19/0.45
% 0.19/0.45 % option.the_def
% 0.19/0.45 thf(fact_1530_nat__plus__as__int,axiom,
% 0.19/0.45 ( ( plus_plus @ nat )
% 0.19/0.45 = ( ^ [A7: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_plus_as_int
% 0.19/0.45 thf(fact_1531_nat__add__distrib,axiom,
% 0.19/0.45 ! [Z2: int,Z7: int] :
% 0.19/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 0.19/0.45 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
% 0.19/0.45 => ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z7 ) )
% 0.19/0.45 = ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_add_distrib
% 0.19/0.45 thf(fact_1532_Suc__as__int,axiom,
% 0.19/0.45 ( suc
% 0.19/0.45 = ( ^ [A7: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( one_one @ int ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_as_int
% 0.19/0.45 thf(fact_1533_nat__abs__triangle__ineq,axiom,
% 0.19/0.45 ! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % nat_abs_triangle_ineq
% 0.19/0.45 thf(fact_1534_sum_OatMost__Suc__shift,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 0.19/0.45 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.atMost_Suc_shift
% 0.19/0.45 thf(fact_1535_sum__telescope,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_group_add @ A )
% 0.19/0.45 => ! [F: nat > A,I: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( minus_minus @ A @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ I ) )
% 0.19/0.45 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_telescope
% 0.19/0.45 thf(fact_1536_the__heap__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( time_the_heap @ A )
% 0.19/0.45 = ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 0.19/0.45 ( case_option @ ( heap_ext @ product_unit ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ ( heap_ext @ product_unit ) )
% 0.19/0.45 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( heap_ext @ product_unit )
% 0.19/0.45 @ ^ [X3: A] :
% 0.19/0.45 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( heap_ext @ product_unit )
% 0.19/0.45 @ ^ [H7: heap_ext @ product_unit,Xa4: nat] : H7 ) )
% 0.19/0.45 @ ( heap_Time_execute @ A @ M @ H ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % the_heap_def
% 0.19/0.45 thf(fact_1537_and__one__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
% 0.19/0.45 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_one_eq
% 0.19/0.45 thf(fact_1538_one__and__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
% 0.19/0.45 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_and_eq
% 0.19/0.45 thf(fact_1539_take__bit__eq__mask__iff,axiom,
% 0.19/0.45 ! [N: nat,K: int] :
% 0.19/0.45 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 0.19/0.45 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 0.19/0.45 = ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 0.19/0.45 = ( zero_zero @ int ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_eq_mask_iff
% 0.19/0.45 thf(fact_1540_Suc__nat__eq__nat__zadd1,axiom,
% 0.19/0.45 ! [Z2: int] :
% 0.19/0.45 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 0.19/0.45 => ( ( suc @ ( nat2 @ Z2 ) )
% 0.19/0.45 = ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_nat_eq_nat_zadd1
% 0.19/0.45 thf(fact_1541_Divides_Oadjust__div__def,axiom,
% 0.19/0.45 ( adjust_div
% 0.19/0.45 = ( product_case_prod @ int @ int @ int
% 0.19/0.45 @ ^ [Q5: int,R3: int] :
% 0.19/0.45 ( plus_plus @ int @ Q5
% 0.19/0.45 @ ( zero_neq_one_of_bool @ int
% 0.19/0.45 @ ( R3
% 0.19/0.45 != ( zero_zero @ int ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Divides.adjust_div_def
% 0.19/0.45 thf(fact_1542_sum__gp__basic,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
% 0.19/0.45 = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_gp_basic
% 0.19/0.45 thf(fact_1543_sum__power__shift,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [M2: nat,N: nat,X: A] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.19/0.45 = ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_power_shift
% 0.19/0.45 thf(fact_1544_sum_Otriangle__reindex__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.19/0.45 @ ( collect @ ( product_prod @ nat @ nat )
% 0.19/0.45 @ ( product_case_prod @ nat @ nat @ $o
% 0.19/0.45 @ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] :
% 0.19/0.45 ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ K4 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.triangle_reindex_eq
% 0.19/0.45 thf(fact_1545_and__int__rec,axiom,
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ int )
% 0.19/0.45 = ( ^ [K4: int,L3: int] :
% 0.19/0.45 ( plus_plus @ int
% 0.19/0.45 @ ( zero_neq_one_of_bool @ int
% 0.19/0.45 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 0.19/0.45 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 0.19/0.45 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_int_rec
% 0.19/0.45 thf(fact_1546_mask__half__int,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mask_half_int
% 0.19/0.45 thf(fact_1547_sum_Oin__pairs__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.in_pairs_0
% 0.19/0.45 thf(fact_1548_mask__nat__def,axiom,
% 0.19/0.45 ( ( bit_se2239418461657761734s_mask @ nat )
% 0.19/0.45 = ( ^ [N5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mask_nat_def
% 0.19/0.45 thf(fact_1549_mask__eq__exp__minus__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ( ( bit_se2239418461657761734s_mask @ A )
% 0.19/0.45 = ( ^ [N5: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % mask_eq_exp_minus_1
% 0.19/0.45 thf(fact_1550_sum_Ozero__middle,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [P5: nat,K: nat,G: nat > A,H3: nat > A] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ K @ P5 )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( zero_zero @ A ) @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ P5 ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.zero_middle
% 0.19/0.45 thf(fact_1551_and__minus__numerals_I3_J,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.19/0.45 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_minus_numerals(3)
% 0.19/0.45 thf(fact_1552_and__minus__numerals_I7_J,axiom,
% 0.19/0.45 ! [N: num,M2: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
% 0.19/0.45 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_minus_numerals(7)
% 0.19/0.45 thf(fact_1553_and__minus__numerals_I4_J,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.19/0.45 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_minus_numerals(4)
% 0.19/0.45 thf(fact_1554_and__minus__numerals_I8_J,axiom,
% 0.19/0.45 ! [N: num,M2: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
% 0.19/0.45 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_minus_numerals(8)
% 0.19/0.45 thf(fact_1555_gbinomial__partial__row__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_partial_row_sum
% 0.19/0.45 thf(fact_1556_gbinomial__r__part__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_r_part_sum
% 0.19/0.45 thf(fact_1557_and__num_Oelims,axiom,
% 0.19/0.45 ! [X: num,Xa: num,Y: option @ num] :
% 0.19/0.45 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.19/0.45 = Y )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ one2 ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ? [N3: num] :
% 0.19/0.45 ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( none @ num ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ? [N3: num] :
% 0.19/0.45 ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ one2 ) ) ) )
% 0.19/0.45 => ( ( ? [M5: num] :
% 0.19/0.45 ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( none @ num ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ( ? [M5: num] :
% 0.19/0.45 ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ one2 ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ~ ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 0.19/0.45 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 0.19/0.45 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_num.elims
% 0.19/0.45 thf(fact_1558_take__bit__Suc__from__most,axiom,
% 0.19/0.45 ! [N: nat,K: int] :
% 0.19/0.45 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
% 0.19/0.45 = ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % take_bit_Suc_from_most
% 0.19/0.45 thf(fact_1559_gbinomial__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( semiring_char_0 @ A )
% 0.19/0.45 & ( semidom_divide @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_1
% 0.19/0.45 thf(fact_1560_gbinomial__0_I1_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( semiring_char_0 @ A )
% 0.19/0.45 & ( semidom_divide @ A ) )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_0(1)
% 0.19/0.45 thf(fact_1561_and__nat__numerals_I2_J,axiom,
% 0.19/0.45 ! [Y: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
% 0.19/0.45 = ( one_one @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_nat_numerals(2)
% 0.19/0.45 thf(fact_1562_and__nat__numerals_I4_J,axiom,
% 0.19/0.45 ! [X: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.45 = ( one_one @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_nat_numerals(4)
% 0.19/0.45 thf(fact_1563_bit__disjunctive__add__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [A3: A,B2: A,N: nat] :
% 0.19/0.45 ( ! [N3: nat] :
% 0.19/0.45 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
% 0.19/0.45 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 0.19/0.45 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 0.19/0.45 | ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit_disjunctive_add_iff
% 0.19/0.45 thf(fact_1564_bit__1__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
% 0.19/0.45 = ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit_1_iff
% 0.19/0.45 thf(fact_1565_not__bit__1__Suc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % not_bit_1_Suc
% 0.19/0.45 thf(fact_1566_bit__numeral__simps_I1_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit_numeral_simps(1)
% 0.19/0.45 thf(fact_1567_gbinomial__Suc__Suc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 0.19/0.45 = ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_Suc_Suc
% 0.19/0.45 thf(fact_1568_and__integer_Oabs__eq,axiom,
% 0.19/0.45 ! [Xa: int,X: int] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.19/0.45 = ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_integer.abs_eq
% 0.19/0.45 thf(fact_1569_gbinomial__addition__formula,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( gbinomial @ A @ A3 @ ( suc @ K ) )
% 0.19/0.45 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_addition_formula
% 0.19/0.45 thf(fact_1570_gbinomial__absorb__comp,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
% 0.19/0.45 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_absorb_comp
% 0.19/0.45 thf(fact_1571_gbinomial__mult__1_H,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
% 0.19/0.45 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_mult_1'
% 0.19/0.45 thf(fact_1572_gbinomial__mult__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
% 0.19/0.45 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_mult_1
% 0.19/0.45 thf(fact_1573_Suc__times__gbinomial,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [K: nat,A3: A] :
% 0.19/0.45 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
% 0.19/0.45 = ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_times_gbinomial
% 0.19/0.45 thf(fact_1574_gbinomial__absorption,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [K: nat,A3: A] :
% 0.19/0.45 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
% 0.19/0.45 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_absorption
% 0.19/0.45 thf(fact_1575_gbinomial__parallel__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K4 ) ) @ K4 )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_parallel_sum
% 0.19/0.45 thf(fact_1576_gbinomial__rec,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 0.19/0.45 = ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_rec
% 0.19/0.45 thf(fact_1577_gbinomial__factors,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 0.19/0.45 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_factors
% 0.19/0.45 thf(fact_1578_gbinomial__negated__upper,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ( ( gbinomial @ A )
% 0.19/0.45 = ( ^ [A7: A,K4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A7 ) @ ( one_one @ A ) ) @ K4 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_negated_upper
% 0.19/0.45 thf(fact_1579_gbinomial__index__swap,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [K: nat,N: nat] :
% 0.19/0.45 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
% 0.19/0.45 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_index_swap
% 0.19/0.45 thf(fact_1580_int__bit__bound,axiom,
% 0.19/0.45 ! [K: int] :
% 0.19/0.45 ~ ! [N3: nat] :
% 0.19/0.45 ( ! [M6: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ N3 @ M6 )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M6 )
% 0.19/0.45 = ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
% 0.19/0.45 => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
% 0.19/0.45 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % int_bit_bound
% 0.19/0.45 thf(fact_1581_gbinomial__minus,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,K: nat] :
% 0.19/0.45 ( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
% 0.19/0.45 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_minus
% 0.19/0.45 thf(fact_1582_gbinomial__reduce__nat,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [K: nat,A3: A] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.19/0.45 => ( ( gbinomial @ A @ A3 @ K )
% 0.19/0.45 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_reduce_nat
% 0.19/0.45 thf(fact_1583_gbinomial__sum__lower__neg,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [A3: A,M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_sum_lower_neg
% 0.19/0.45 thf(fact_1584_numeral__and__num,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 0.19/0.45 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % numeral_and_num
% 0.19/0.45 thf(fact_1585_gbinomial__sum__up__index,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [K: nat,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [J2: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J2 ) @ K )
% 0.19/0.45 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.19/0.45 = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_sum_up_index
% 0.19/0.45 thf(fact_1586_gbinomial__partial__sum__poly,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [M2: nat,A3: A,X: A,Y: A] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_partial_sum_poly
% 0.19/0.45 thf(fact_1587_even__bit__succ__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [A3: A,N: nat] :
% 0.19/0.45 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
% 0.19/0.45 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 0.19/0.45 | ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % even_bit_succ_iff
% 0.19/0.45 thf(fact_1588_odd__bit__iff__bit__pred,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ! [A3: A,N: nat] :
% 0.19/0.45 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 0.19/0.45 = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
% 0.19/0.45 | ( N
% 0.19/0.45 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % odd_bit_iff_bit_pred
% 0.19/0.45 thf(fact_1589_and__num_Osimps_I9_J,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.45 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 0.19/0.45 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 0.19/0.45 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_num.simps(9)
% 0.19/0.45 thf(fact_1590_gbinomial__absorption_H,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [K: nat,A3: A] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.19/0.45 => ( ( gbinomial @ A @ A3 @ K )
% 0.19/0.45 = ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_absorption'
% 0.19/0.45 thf(fact_1591_gbinomial__partial__sum__poly__xpos,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [M2: nat,A3: A,X: A,Y: A] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A3 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_partial_sum_poly_xpos
% 0.19/0.45 thf(fact_1592_gbinomial__sum__nat__pow2,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ K4 ) ) @ K4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K4 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_sum_nat_pow2
% 0.19/0.45 thf(fact_1593_bit__sum__mult__2__cases,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [A3: A,B2: A,N: nat] :
% 0.19/0.45 ( ! [J3: nat] :
% 0.19/0.45 ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J3 ) )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
% 0.19/0.45 = ( ( ( N
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 0.19/0.45 & ( ( N
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit_sum_mult_2_cases
% 0.19/0.45 thf(fact_1594_and__not__num_Osimps_I8_J,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 0.19/0.45 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 0.19/0.45 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 0.19/0.45 @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_not_num.simps(8)
% 0.19/0.45 thf(fact_1595_bit__rec,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_semiring_bits @ A )
% 0.19/0.45 => ( ( bit_se5641148757651400278ts_bit @ A )
% 0.19/0.45 = ( ^ [A7: A,N5: nat] :
% 0.19/0.45 ( ( ( N5
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A7 ) )
% 0.19/0.45 & ( ( N5
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit_rec
% 0.19/0.45 thf(fact_1596_and__nat__unfold,axiom,
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ nat )
% 0.19/0.45 = ( ^ [M: nat,N5: nat] :
% 0.19/0.45 ( if @ nat
% 0.19/0.45 @ ( ( M
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 | ( N5
% 0.19/0.45 = ( zero_zero @ nat ) ) )
% 0.19/0.45 @ ( zero_zero @ nat )
% 0.19/0.45 @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_nat_unfold
% 0.19/0.45 thf(fact_1597_and__nat__rec,axiom,
% 0.19/0.45 ( ( bit_se5824344872417868541ns_and @ nat )
% 0.19/0.45 = ( ^ [M: nat,N5: nat] :
% 0.19/0.45 ( plus_plus @ nat
% 0.19/0.45 @ ( zero_neq_one_of_bool @ nat
% 0.19/0.45 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
% 0.19/0.45 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
% 0.19/0.45 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_nat_rec
% 0.19/0.45 thf(fact_1598_set__bit__eq,axiom,
% 0.19/0.45 ( ( bit_se5668285175392031749et_bit @ int )
% 0.19/0.45 = ( ^ [N5: nat,K4: int] :
% 0.19/0.45 ( plus_plus @ int @ K4
% 0.19/0.45 @ ( times_times @ int
% 0.19/0.45 @ ( zero_neq_one_of_bool @ int
% 0.19/0.45 @ ~ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N5 ) )
% 0.19/0.45 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % set_bit_eq
% 0.19/0.45 thf(fact_1599_and__not__num_Oelims,axiom,
% 0.19/0.45 ! [X: num,Xa: num,Y: option @ num] :
% 0.19/0.45 ( ( ( bit_and_not_num @ X @ Xa )
% 0.19/0.45 = Y )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( none @ num ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ? [N3: num] :
% 0.19/0.45 ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ one2 ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ? [N3: num] :
% 0.19/0.45 ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( none @ num ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( bit0 @ M5 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( bit0 @ M5 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 0.19/0.45 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 0.19/0.45 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ~ ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_not_num.elims
% 0.19/0.45 thf(fact_1600_and__not__num_Opelims,axiom,
% 0.19/0.45 ! [X: num,Xa: num,Y: option @ num] :
% 0.19/0.45 ( ( ( bit_and_not_num @ X @ Xa )
% 0.19/0.45 = Y )
% 0.19/0.45 => ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( none @ num ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ one2 ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( none @ num ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( bit0 @ M5 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( bit0 @ M5 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 0.19/0.45 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 0.19/0.45 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ~ ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_not_num.pelims
% 0.19/0.45 thf(fact_1601_and__num_Opelims,axiom,
% 0.19/0.45 ! [X: num,Xa: num,Y: option @ num] :
% 0.19/0.45 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 0.19/0.45 = Y )
% 0.19/0.45 => ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ one2 ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( none @ num ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ one2 ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( none @ num ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ one2 ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ~ ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 0.19/0.45 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 0.19/0.45 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % and_num.pelims
% 0.19/0.45 thf(fact_1602_gbinomial__code,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ( ( gbinomial @ A )
% 0.19/0.45 = ( ^ [A7: A,K4: nat] :
% 0.19/0.45 ( if @ A
% 0.19/0.45 @ ( K4
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 @ ( one_one @ A )
% 0.19/0.45 @ ( divide_divide @ A
% 0.19/0.45 @ ( set_fo6178422350223883121st_nat @ A
% 0.19/0.45 @ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ L3 ) ) )
% 0.19/0.45 @ ( zero_zero @ nat )
% 0.19/0.45 @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) )
% 0.19/0.45 @ ( one_one @ A ) )
% 0.19/0.45 @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % gbinomial_code
% 0.19/0.45 thf(fact_1603_xor__num_Oelims,axiom,
% 0.19/0.45 ! [X: num,Xa: num,Y: option @ num] :
% 0.19/0.45 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.19/0.45 = Y )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( none @ num ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( bit1 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( bit0 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( bit1 @ M5 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( bit0 @ M5 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ~ ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( Y
% 0.19/0.45 != ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_num.elims
% 0.19/0.45 thf(fact_1604_choose__alternating__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_ring_1 @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_alternating_sum
% 0.19/0.45 thf(fact_1605_choose__linear__sum,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [I3: nat] : ( times_times @ nat @ I3 @ ( binomial @ N @ I3 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_linear_sum
% 0.19/0.45 thf(fact_1606_binomial__r__part__sum,axiom,
% 0.19/0.45 ! [M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_r_part_sum
% 0.19/0.45 thf(fact_1607_sum_Otriangle__reindex,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.19/0.45 @ ( collect @ ( product_prod @ nat @ nat )
% 0.19/0.45 @ ( product_case_prod @ nat @ nat @ $o
% 0.19/0.45 @ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] :
% 0.19/0.45 ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ K4 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.triangle_reindex
% 0.19/0.45 thf(fact_1608_lessThan__eq__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [X: A,Y: A] :
% 0.19/0.45 ( ( ( set_ord_lessThan @ A @ X )
% 0.19/0.45 = ( set_ord_lessThan @ A @ Y ) )
% 0.19/0.45 = ( X = Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_eq_iff
% 0.19/0.45 thf(fact_1609_lessThan__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ! [I: A,K: A] :
% 0.19/0.45 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 0.19/0.45 = ( ord_less @ A @ I @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_iff
% 0.19/0.45 thf(fact_1610_binomial__n__n,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( binomial @ N @ N )
% 0.19/0.45 = ( one_one @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_n_n
% 0.19/0.45 thf(fact_1611_lessThan__subset__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [X: A,Y: A] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
% 0.19/0.45 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_subset_iff
% 0.19/0.45 thf(fact_1612_binomial__Suc__Suc,axiom,
% 0.19/0.45 ! [N: nat,K: nat] :
% 0.19/0.45 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 0.19/0.45 = ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_Suc_Suc
% 0.19/0.45 thf(fact_1613_binomial__n__0,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( binomial @ N @ ( zero_zero @ nat ) )
% 0.19/0.45 = ( one_one @ nat ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_n_0
% 0.19/0.45 thf(fact_1614_sum_OlessThan__Suc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.lessThan_Suc
% 0.19/0.45 thf(fact_1615_choose__one,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( binomial @ N @ ( one_one @ nat ) )
% 0.19/0.45 = N ) ).
% 0.19/0.45
% 0.19/0.45 % choose_one
% 0.19/0.45 thf(fact_1616_lessThan__def,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ( ( set_ord_lessThan @ A )
% 0.19/0.45 = ( ^ [U2: A] :
% 0.19/0.45 ( collect @ A
% 0.19/0.45 @ ^ [X3: A] : ( ord_less @ A @ X3 @ U2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_def
% 0.19/0.45 thf(fact_1617_choose__mult__lemma,axiom,
% 0.19/0.45 ! [M2: nat,R: nat,K: nat] :
% 0.19/0.45 ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R ) @ K ) @ ( plus_plus @ nat @ M2 @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K ) @ K ) )
% 0.19/0.45 = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R ) @ M2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_mult_lemma
% 0.19/0.45 thf(fact_1618_lessThan__Suc__atMost,axiom,
% 0.19/0.45 ! [K: nat] :
% 0.19/0.45 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 0.19/0.45 = ( set_ord_atMost @ nat @ K ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_Suc_atMost
% 0.19/0.45 thf(fact_1619_bit__integer_Oabs__eq,axiom,
% 0.19/0.45 ! [X: int] :
% 0.19/0.45 ( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
% 0.19/0.45 = ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% 0.19/0.45
% 0.19/0.45 % bit_integer.abs_eq
% 0.19/0.45 thf(fact_1620_lessThan__strict__subset__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [M2: A,N: A] :
% 0.19/0.45 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N ) )
% 0.19/0.45 = ( ord_less @ A @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_strict_subset_iff
% 0.19/0.45 thf(fact_1621_Suc__times__binomial__add,axiom,
% 0.19/0.45 ! [A3: nat,B2: nat] :
% 0.19/0.45 ( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
% 0.19/0.45 = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_times_binomial_add
% 0.19/0.45 thf(fact_1622_binomial__absorb__comp,axiom,
% 0.19/0.45 ! [N: nat,K: nat] :
% 0.19/0.45 ( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
% 0.19/0.45 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_absorb_comp
% 0.19/0.45 thf(fact_1623_fact__fact__dvd__fact,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linordered_semidom @ A )
% 0.19/0.45 => ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_fact_dvd_fact
% 0.19/0.45 thf(fact_1624_sum_Onat__diff__reindex,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.nat_diff_reindex
% 0.19/0.45 thf(fact_1625_Iic__subset__Iio__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [A3: A,B2: A] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
% 0.19/0.45 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Iic_subset_Iio_iff
% 0.19/0.45 thf(fact_1626_binomial__absorption,axiom,
% 0.19/0.45 ! [K: nat,N: nat] :
% 0.19/0.45 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 0.19/0.45 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_absorption
% 0.19/0.45 thf(fact_1627_sum__diff__distrib,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ! [Q: A > nat,P: A > nat,N: A] :
% 0.19/0.45 ( ! [X2: A] : ( ord_less_eq @ nat @ ( Q @ X2 ) @ ( P @ X2 ) )
% 0.19/0.45 => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ A @ nat
% 0.19/0.45 @ ^ [X3: A] : ( minus_minus @ nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_diff_distrib
% 0.19/0.45 thf(fact_1628_sum_OlessThan__Suc__shift,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 0.19/0.45 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.lessThan_Suc_shift
% 0.19/0.45 thf(fact_1629_sum__lessThan__telescope,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_group_add @ A )
% 0.19/0.45 => ! [F: nat > A,M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [N5: nat] : ( minus_minus @ A @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ M2 ) )
% 0.19/0.45 = ( minus_minus @ A @ ( F @ M2 ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_lessThan_telescope
% 0.19/0.45 thf(fact_1630_sum__lessThan__telescope_H,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_group_add @ A )
% 0.19/0.45 => ! [F: nat > A,M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [N5: nat] : ( minus_minus @ A @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ M2 ) )
% 0.19/0.45 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_lessThan_telescope'
% 0.19/0.45 thf(fact_1631_sum_OatLeast1__atMost__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.atLeast1_atMost_eq
% 0.19/0.45 thf(fact_1632_sum__bounds__lt__plus1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [F: nat > A,Mm: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ Mm ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_bounds_lt_plus1
% 0.19/0.45 thf(fact_1633_sum_Onested__swap_H,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [A3: nat > nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [J2: nat] :
% 0.19/0.45 ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 0.19/0.45 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.nested_swap'
% 0.19/0.45 thf(fact_1634_sum__choose__lower,axiom,
% 0.19/0.45 ! [R: nat,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [K4: nat] : ( binomial @ ( plus_plus @ nat @ R @ K4 ) @ K4 )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_choose_lower
% 0.19/0.45 thf(fact_1635_choose__rising__sum_I2_J,axiom,
% 0.19/0.45 ! [N: nat,M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ M2 ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_rising_sum(2)
% 0.19/0.45 thf(fact_1636_choose__rising__sum_I1_J,axiom,
% 0.19/0.45 ! [N: nat,M2: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ M2 ) )
% 0.19/0.45 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_rising_sum(1)
% 0.19/0.45 thf(fact_1637_binomial__code,axiom,
% 0.19/0.45 ( binomial
% 0.19/0.45 = ( ^ [N5: nat,K4: nat] : ( if @ nat @ ( ord_less @ nat @ N5 @ K4 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( binomial @ N5 @ ( minus_minus @ nat @ N5 @ K4 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N5 @ K4 ) @ ( one_one @ nat ) ) @ N5 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K4 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_code
% 0.19/0.45 thf(fact_1638_choose__reduce__nat,axiom,
% 0.19/0.45 ! [N: nat,K: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.19/0.45 => ( ( binomial @ N @ K )
% 0.19/0.45 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_reduce_nat
% 0.19/0.45 thf(fact_1639_times__binomial__minus1__eq,axiom,
% 0.19/0.45 ! [K: nat,N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 0.19/0.45 => ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
% 0.19/0.45 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % times_binomial_minus1_eq
% 0.19/0.45 thf(fact_1640_one__diff__power__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 0.19/0.45 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_diff_power_eq
% 0.19/0.45 thf(fact_1641_power__diff__1__eq,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
% 0.19/0.45 = ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % power_diff_1_eq
% 0.19/0.45 thf(fact_1642_geometric__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field @ A )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( X
% 0.19/0.45 != ( one_one @ A ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 0.19/0.45 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % geometric_sum
% 0.19/0.45 thf(fact_1643_sum_OatMost__shift,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 0.19/0.45 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.atMost_shift
% 0.19/0.45 thf(fact_1644_vandermonde,axiom,
% 0.19/0.45 ! [M2: nat,N: nat,R: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ nat @ ( binomial @ M2 @ K4 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ R ) )
% 0.19/0.45 = ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R ) ) ).
% 0.19/0.45
% 0.19/0.45 % vandermonde
% 0.19/0.45 thf(fact_1645_binomial__addition__formula,axiom,
% 0.19/0.45 ! [N: nat,K: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 => ( ( binomial @ N @ ( suc @ K ) )
% 0.19/0.45 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_addition_formula
% 0.19/0.45 thf(fact_1646_power__diff__sumr2,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat,Y: A] :
% 0.19/0.45 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
% 0.19/0.45 = ( times_times @ A @ ( minus_minus @ A @ X @ Y )
% 0.19/0.45 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % power_diff_sumr2
% 0.19/0.45 thf(fact_1647_diff__power__eq__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat,Y: A] :
% 0.19/0.45 ( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
% 0.19/0.45 = ( times_times @ A @ ( minus_minus @ A @ X @ Y )
% 0.19/0.45 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % diff_power_eq_sum
% 0.19/0.45 thf(fact_1648_sum__gp__strict,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( division_ring @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( ( X
% 0.19/0.45 = ( one_one @ A ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 0.19/0.45 = ( semiring_1_of_nat @ A @ N ) ) )
% 0.19/0.45 & ( ( X
% 0.19/0.45 != ( one_one @ A ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 0.19/0.45 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_gp_strict
% 0.19/0.45 thf(fact_1649_binomial,axiom,
% 0.19/0.45 ! [A3: nat,B2: nat,N: nat] :
% 0.19/0.45 ( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K4 ) ) @ ( power_power @ nat @ A3 @ K4 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial
% 0.19/0.45 thf(fact_1650_choose__two,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.19/0.45 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_two
% 0.19/0.45 thf(fact_1651_one__diff__power__eq_H,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( monoid_mult @ A )
% 0.19/0.45 & ( comm_ring @ A ) )
% 0.19/0.45 => ! [X: A,N: nat] :
% 0.19/0.45 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 0.19/0.45 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
% 0.19/0.45 @ ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 0.19/0.45 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % one_diff_power_eq'
% 0.19/0.45 thf(fact_1652_binomial__ring,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A )
% 0.19/0.45 => ! [A3: A,B2: A,N: nat] :
% 0.19/0.45 ( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( power_power @ A @ A3 @ K4 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % binomial_ring
% 0.19/0.45 thf(fact_1653_xor__num_Osimps_I6_J,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 0.19/0.45 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_num.simps(6)
% 0.19/0.45 thf(fact_1654_xor__num_Osimps_I8_J,axiom,
% 0.19/0.45 ! [M2: num,N: num] :
% 0.19/0.45 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 0.19/0.45 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_num.simps(8)
% 0.19/0.45 thf(fact_1655_choose__alternating__linear__sum,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_ring_1 @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( N
% 0.19/0.45 != ( one_one @ nat ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
% 0.19/0.45 @ ( set_ord_atMost @ nat @ N ) )
% 0.19/0.45 = ( zero_zero @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % choose_alternating_linear_sum
% 0.19/0.45 thf(fact_1656_fact__Suc__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_char_0 @ A )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_Suc_0
% 0.19/0.45 thf(fact_1657_fact__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_char_0 @ A )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_1
% 0.19/0.45 thf(fact_1658_fact__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_char_0 @ A )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_0
% 0.19/0.45 thf(fact_1659_fact__code,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_char_0 @ A )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A )
% 0.19/0.45 = ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_code
% 0.19/0.45 thf(fact_1660_fact__reduce,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_char_0 @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A @ N )
% 0.19/0.45 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_reduce
% 0.19/0.45 thf(fact_1661_fact__num__eq__if,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( semiring_char_0 @ A )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A )
% 0.19/0.45 = ( ^ [M: nat] :
% 0.19/0.45 ( if @ A
% 0.19/0.45 @ ( M
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 @ ( one_one @ A )
% 0.19/0.45 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_num_eq_if
% 0.19/0.45 thf(fact_1662_xor__num_Opelims,axiom,
% 0.19/0.45 ! [X: num,Xa: num,Y: option @ num] :
% 0.19/0.45 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 0.19/0.45 = Y )
% 0.19/0.45 => ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( none @ num ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( bit1 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ( ( X = one2 )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( bit0 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( bit1 @ M5 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit0 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ( ( Xa = one2 )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( bit0 @ M5 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 0.19/0.45 => ( ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit0 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.19/0.45 => ~ ! [M5: num] :
% 0.19/0.45 ( ( X
% 0.19/0.45 = ( bit1 @ M5 ) )
% 0.19/0.45 => ! [N3: num] :
% 0.19/0.45 ( ( Xa
% 0.19/0.45 = ( bit1 @ N3 ) )
% 0.19/0.45 => ( ( Y
% 0.19/0.45 = ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 0.19/0.45 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_num.pelims
% 0.19/0.45 thf(fact_1663_dvd__fact,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.45 => ( dvd_dvd @ nat @ M2 @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % dvd_fact
% 0.19/0.45 thf(fact_1664_fact__ge__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linordered_semidom @ A )
% 0.19/0.45 => ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_ge_1
% 0.19/0.45 thf(fact_1665_pochhammer__double,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [Z2: A,N: nat] :
% 0.19/0.45 ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.45 = ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % pochhammer_double
% 0.19/0.45 thf(fact_1666_fact__double,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( field_char_0 @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 0.19/0.45 = ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % fact_double
% 0.19/0.45 thf(fact_1667_pochhammer__code,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A )
% 0.19/0.45 => ( ( comm_s3205402744901411588hammer @ A )
% 0.19/0.45 = ( ^ [A7: A,N5: nat] :
% 0.19/0.45 ( if @ A
% 0.19/0.45 @ ( N5
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 @ ( one_one @ A )
% 0.19/0.45 @ ( set_fo6178422350223883121st_nat @ A
% 0.19/0.45 @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ O ) ) )
% 0.19/0.45 @ ( zero_zero @ nat )
% 0.19/0.45 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) )
% 0.19/0.45 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % pochhammer_code
% 0.19/0.45 thf(fact_1668_xor__Suc__0__eq,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.19/0.45 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.19/0.45 @ ( zero_neq_one_of_bool @ nat
% 0.19/0.45 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_Suc_0_eq
% 0.19/0.45 thf(fact_1669_Suc__0__xor__eq,axiom,
% 0.19/0.45 ! [N: nat] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.19/0.45 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 0.19/0.45 @ ( zero_neq_one_of_bool @ nat
% 0.19/0.45 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Suc_0_xor_eq
% 0.19/0.45 thf(fact_1670_push__bit__numeral__minus__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_ri3973907225187159222ations @ A )
% 0.19/0.45 => ! [N: num] :
% 0.19/0.45 ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.19/0.45 = ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % push_bit_numeral_minus_1
% 0.19/0.45 thf(fact_1671_num__of__integer__code,axiom,
% 0.19/0.45 ( code_num_of_integer
% 0.19/0.45 = ( ^ [K4: code_integer] :
% 0.19/0.45 ( if @ num @ ( ord_less_eq @ code_integer @ K4 @ ( one_one @ code_integer ) ) @ one2
% 0.19/0.45 @ ( product_case_prod @ code_integer @ code_integer @ num
% 0.19/0.45 @ ^ [L3: code_integer,J2: code_integer] :
% 0.19/0.45 ( if @ num
% 0.19/0.45 @ ( J2
% 0.19/0.45 = ( zero_zero @ code_integer ) )
% 0.19/0.45 @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) )
% 0.19/0.45 @ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one2 ) )
% 0.19/0.45 @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % num_of_integer_code
% 0.19/0.45 thf(fact_1672_Sum__Ico__nat,axiom,
% 0.19/0.45 ! [M2: nat,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 0.19/0.45 @ ^ [X3: nat] : X3
% 0.19/0.45 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.19/0.45 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % Sum_Ico_nat
% 0.19/0.45 thf(fact_1673_ivl__diff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [I: A,N: A,M2: A] :
% 0.19/0.45 ( ( ord_less_eq @ A @ I @ N )
% 0.19/0.45 => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
% 0.19/0.45 = ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ivl_diff
% 0.19/0.45 thf(fact_1674_push__bit__push__bit,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [M2: nat,N: nat,A3: A] :
% 0.19/0.45 ( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
% 0.19/0.45 = ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % push_bit_push_bit
% 0.19/0.45 thf(fact_1675_lessThan__minus__lessThan,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [N: A,M2: A] :
% 0.19/0.45 ( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M2 ) )
% 0.19/0.45 = ( set_or7035219750837199246ssThan @ A @ M2 @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_minus_lessThan
% 0.19/0.45 thf(fact_1676_pochhammer__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
% 0.19/0.45 = A3 ) ) ).
% 0.19/0.45
% 0.19/0.45 % pochhammer_1
% 0.19/0.45 thf(fact_1677_atLeastLessThan__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ord @ A )
% 0.19/0.45 => ! [I: A,L: A,U: A] :
% 0.19/0.45 ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 0.19/0.45 = ( ( ord_less_eq @ A @ L @ I )
% 0.19/0.45 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atLeastLessThan_iff
% 0.19/0.45 thf(fact_1678_ivl__subset,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [I: A,J: A,M2: A,N: A] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
% 0.19/0.45 = ( ( ord_less_eq @ A @ J @ I )
% 0.19/0.45 | ( ( ord_less_eq @ A @ M2 @ I )
% 0.19/0.45 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % ivl_subset
% 0.19/0.45 thf(fact_1679_pochhammer__0,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A )
% 0.19/0.45 => ! [A3: A] :
% 0.19/0.45 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
% 0.19/0.45 = ( one_one @ A ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % pochhammer_0
% 0.19/0.45 thf(fact_1680_xor__numerals_I1_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [Y: num] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 0.19/0.45 = ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_numerals(1)
% 0.19/0.45 thf(fact_1681_xor__numerals_I2_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [Y: num] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.19/0.45 = ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_numerals(2)
% 0.19/0.45 thf(fact_1682_xor__numerals_I5_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 0.19/0.45 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_numerals(5)
% 0.19/0.45 thf(fact_1683_xor__numerals_I8_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 0.19/0.45 = ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_numerals(8)
% 0.19/0.45 thf(fact_1684_push__bit__of__1,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
% 0.19/0.45 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % push_bit_of_1
% 0.19/0.45 thf(fact_1685_sum_Oop__ivl__Suc,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [N: nat,M2: nat,G: nat > A] :
% 0.19/0.45 ( ( ( ord_less @ nat @ N @ M2 )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 0.19/0.45 = ( zero_zero @ A ) ) )
% 0.19/0.45 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.op_ivl_Suc
% 0.19/0.45 thf(fact_1686_xor__numerals_I4_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num,Y: num] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_numerals(4)
% 0.19/0.45 thf(fact_1687_xor__numerals_I6_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.19/0.45 => ! [X: num,Y: num] :
% 0.19/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 0.19/0.45 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % xor_numerals(6)
% 0.19/0.45 thf(fact_1688_flip__bit__nat__def,axiom,
% 0.19/0.45 ( ( bit_se8732182000553998342ip_bit @ nat )
% 0.19/0.45 = ( ^ [M: nat,N5: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M @ ( one_one @ nat ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % flip_bit_nat_def
% 0.19/0.45 thf(fact_1689_push__bit__integer_Oabs__eq,axiom,
% 0.19/0.45 ! [Xa: nat,X: int] :
% 0.19/0.45 ( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 0.19/0.45 = ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa @ X ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % push_bit_integer.abs_eq
% 0.19/0.45 thf(fact_1690_atLeastLessThan__inj_I2_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.45 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.19/0.45 = ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.19/0.45 => ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.45 => ( ( ord_less @ A @ C2 @ D3 )
% 0.19/0.45 => ( B2 = D3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atLeastLessThan_inj(2)
% 0.19/0.45 thf(fact_1691_atLeastLessThan__inj_I1_J,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.45 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.19/0.45 = ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.19/0.45 => ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.45 => ( ( ord_less @ A @ C2 @ D3 )
% 0.19/0.45 => ( A3 = C2 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atLeastLessThan_inj(1)
% 0.19/0.45 thf(fact_1692_atLeastLessThan__eq__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.45 ( ( ord_less @ A @ A3 @ B2 )
% 0.19/0.45 => ( ( ord_less @ A @ C2 @ D3 )
% 0.19/0.45 => ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.19/0.45 = ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.19/0.45 = ( ( A3 = C2 )
% 0.19/0.45 & ( B2 = D3 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atLeastLessThan_eq_iff
% 0.19/0.45 thf(fact_1693_flip__bit__eq__xor,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ( ( bit_se8732182000553998342ip_bit @ A )
% 0.19/0.45 = ( ^ [N5: nat,A7: A] : ( bit_se5824344971392196577ns_xor @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % flip_bit_eq_xor
% 0.19/0.45 thf(fact_1694_push__bit__add,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [N: nat,A3: A,B2: A] :
% 0.19/0.45 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.19/0.45 = ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % push_bit_add
% 0.19/0.45 thf(fact_1695_atLeastLessThan__subset__iff,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( linorder @ A )
% 0.19/0.45 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.19/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.19/0.45 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.19/0.45 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.19/0.45 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % atLeastLessThan_subset_iff
% 0.19/0.45 thf(fact_1696_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.19/0.45 ! [L: nat,U: nat] :
% 0.19/0.45 ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
% 0.19/0.45 = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% 0.19/0.45
% 0.19/0.45 % atLeastLessThanSuc_atLeastAtMost
% 0.19/0.45 thf(fact_1697_lessThan__atLeast0,axiom,
% 0.19/0.45 ( ( set_ord_lessThan @ nat )
% 0.19/0.45 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % lessThan_atLeast0
% 0.19/0.45 thf(fact_1698_push__bit__take__bit,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.19/0.45 => ! [M2: nat,N: nat,A3: A] :
% 0.19/0.45 ( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 0.19/0.45 = ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % push_bit_take_bit
% 0.19/0.45 thf(fact_1699_pochhammer__fact,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ( semiring_char_0 @ A )
% 0.19/0.45 & ( comm_semiring_1 @ A ) )
% 0.19/0.45 => ( ( semiring_char_0_fact @ A )
% 0.19/0.45 = ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % pochhammer_fact
% 0.19/0.45 thf(fact_1700_sum_Oshift__bounds__Suc__ivl,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,M2: nat,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.19/0.45 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.shift_bounds_Suc_ivl
% 0.19/0.45 thf(fact_1701_sum_Oshift__bounds__nat__ivl,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.19/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.19/0.45 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 0.19/0.45 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.shift_bounds_nat_ivl
% 0.19/0.45 thf(fact_1702_sum_Oivl__cong,axiom,
% 0.19/0.45 ! [A: $tType,B: $tType] :
% 0.19/0.45 ( ( ( ord @ B )
% 0.19/0.45 & ( comm_monoid_add @ A ) )
% 0.19/0.45 => ! [A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
% 0.19/0.45 ( ( A3 = C2 )
% 0.19/0.45 => ( ( B2 = D3 )
% 0.19/0.45 => ( ! [X2: B] :
% 0.19/0.45 ( ( ord_less_eq @ B @ C2 @ X2 )
% 0.19/0.45 => ( ( ord_less @ B @ X2 @ D3 )
% 0.19/0.45 => ( ( G @ X2 )
% 0.19/0.45 = ( H3 @ X2 ) ) ) )
% 0.19/0.45 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.ivl_cong
% 0.19/0.45 thf(fact_1703_sum_OatLeastLessThan__concat,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_monoid_add @ A )
% 0.19/0.45 => ! [M2: nat,N: nat,P5: nat,G: nat > A] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.19/0.45 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum.atLeastLessThan_concat
% 0.19/0.45 thf(fact_1704_sum__diff__nat__ivl,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( ab_group_add @ A )
% 0.19/0.45 => ! [M2: nat,N: nat,P5: nat,F: nat > A] :
% 0.19/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.19/0.45 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.19/0.45 => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
% 0.19/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% 0.19/0.45
% 0.19/0.45 % sum_diff_nat_ivl
% 0.19/0.45 thf(fact_1705_pochhammer__0__left,axiom,
% 0.19/0.45 ! [A: $tType] :
% 0.19/0.45 ( ( comm_semiring_1 @ A )
% 0.19/0.45 => ! [N: nat] :
% 0.19/0.45 ( ( ( N
% 0.19/0.45 = ( zero_zero @ nat ) )
% 0.19/0.45 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 0.19/0.45 = ( one_one @ A ) ) )
% 0.19/0.45 & ( ( N
% 0.19/0.45 != ( zero_zero @ nat ) )
% 0.19/0.45 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 0.25/0.45 = ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_0_left
% 0.25/0.45 thf(fact_1706_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeast_Suc_lessThan_Suc_shift
% 0.25/0.45 thf(fact_1707_sum_OatLeastLessThan__shift__bounds,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeastLessThan_shift_bounds
% 0.25/0.45 thf(fact_1708_concat__bit__eq,axiom,
% 0.25/0.45 ( bit_concat_bit
% 0.25/0.45 = ( ^ [N5: nat,K4: int,L3: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N5 @ K4 ) @ ( bit_se4730199178511100633sh_bit @ int @ N5 @ L3 ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % concat_bit_eq
% 0.25/0.45 thf(fact_1709_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( preorder @ A )
% 0.25/0.45 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.25/0.45 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 0.25/0.45 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.45 & ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % atLeastAtMost_subseteq_atLeastLessThan_iff
% 0.25/0.45 thf(fact_1710_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( dense_linorder @ A )
% 0.25/0.45 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.45 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.25/0.45 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.45 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.45 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % atLeastLessThan_subseteq_atLeastAtMost_iff
% 0.25/0.45 thf(fact_1711_sum__shift__lb__Suc0__0__upt,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [F: nat > A,K: nat] :
% 0.25/0.45 ( ( ( F @ ( zero_zero @ nat ) )
% 0.25/0.45 = ( zero_zero @ A ) )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum_shift_lb_Suc0_0_upt
% 0.25/0.45 thf(fact_1712_sum_OatLeast0__lessThan__Suc,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.45 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeast0_lessThan_Suc
% 0.25/0.45 thf(fact_1713_sum_OatLeast__Suc__lessThan,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.45 ( ( ord_less @ nat @ M2 @ N )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.45 = ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeast_Suc_lessThan
% 0.25/0.45 thf(fact_1714_sum_OatLeastLessThan__Suc,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [A3: nat,B2: nat,G: nat > A] :
% 0.25/0.45 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 0.25/0.45 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeastLessThan_Suc
% 0.25/0.45 thf(fact_1715_sum_Olast__plus,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.45 = ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.last_plus
% 0.25/0.45 thf(fact_1716_pochhammer__rec,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_semiring_1 @ A )
% 0.25/0.45 => ! [A3: A,N: nat] :
% 0.25/0.45 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.25/0.45 = ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_rec
% 0.25/0.45 thf(fact_1717_pochhammer__Suc,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_semiring_1 @ A )
% 0.25/0.45 => ! [A3: A,N: nat] :
% 0.25/0.45 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.25/0.45 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_Suc
% 0.25/0.45 thf(fact_1718_pochhammer__rec_H,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_semiring_1 @ A )
% 0.25/0.45 => ! [Z2: A,N: nat] :
% 0.25/0.45 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
% 0.25/0.45 = ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_rec'
% 0.25/0.45 thf(fact_1719_pochhammer__product_H,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_semiring_1 @ A )
% 0.25/0.45 => ! [Z2: A,N: nat,M2: nat] :
% 0.25/0.45 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
% 0.25/0.45 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_product'
% 0.25/0.45 thf(fact_1720_bit__iff__and__push__bit__not__eq__0,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.45 => ( ( bit_se5641148757651400278ts_bit @ A )
% 0.25/0.45 = ( ^ [A7: A,N5: nat] :
% 0.25/0.45 ( ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) )
% 0.25/0.45 != ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % bit_iff_and_push_bit_not_eq_0
% 0.25/0.45 thf(fact_1721_sum__Suc__diff_H,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( ab_group_add @ A )
% 0.25/0.45 => ! [M2: nat,N: nat,F: nat > A] :
% 0.25/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [I3: nat] : ( minus_minus @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 0.25/0.45 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.45 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M2 ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum_Suc_diff'
% 0.25/0.45 thf(fact_1722_sum_OatLeastLessThan__rev,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,N: nat,M2: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
% 0.25/0.45 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeastLessThan_rev
% 0.25/0.45 thf(fact_1723_sum_Onested__swap,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [A3: nat > nat > A,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
% 0.25/0.45 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [J2: nat] :
% 0.25/0.45 ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 0.25/0.45 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 0.25/0.45 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.nested_swap
% 0.25/0.45 thf(fact_1724_sum_Onat__group,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,K: nat,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [M: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
% 0.25/0.45 @ ( set_ord_lessThan @ nat @ N ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.nat_group
% 0.25/0.45 thf(fact_1725_numeral__xor__num,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.45 => ! [M2: num,N: num] :
% 0.25/0.45 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 0.25/0.45 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % numeral_xor_num
% 0.25/0.45 thf(fact_1726_sum_Ohead__if,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [N: nat,M2: nat,G: nat > A] :
% 0.25/0.45 ( ( ( ord_less @ nat @ N @ M2 )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.45 = ( zero_zero @ A ) ) )
% 0.25/0.45 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.25/0.45 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.45 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.head_if
% 0.25/0.45 thf(fact_1727_sum_OatLeast0__lessThan__Suc__shift,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.45 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeast0_lessThan_Suc_shift
% 0.25/0.45 thf(fact_1728_sum_OatLeastLessThan__shift__0,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeastLessThan_shift_0
% 0.25/0.45 thf(fact_1729_pochhammer__product,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_semiring_1 @ A )
% 0.25/0.45 => ! [M2: nat,N: nat,Z2: A] :
% 0.25/0.45 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.45 => ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
% 0.25/0.45 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_product
% 0.25/0.45 thf(fact_1730_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,N: nat,M2: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 0.25/0.45 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeastLessThan_rev_at_least_Suc_atMost
% 0.25/0.45 thf(fact_1731_sum_OatLeast__lessThan__pred__shift,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_monoid_add @ A )
% 0.25/0.45 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.45 ( ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.45 @ ( comp @ nat @ A @ nat @ G
% 0.25/0.45 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.25/0.45 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.45 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % sum.atLeast_lessThan_pred_shift
% 0.25/0.45 thf(fact_1732_pochhammer__absorb__comp,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.45 ( ( comm_ring_1 @ A )
% 0.25/0.45 => ! [R: A,K: nat] :
% 0.25/0.45 ( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K ) )
% 0.25/0.45 = ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.25/0.45
% 0.25/0.45 % pochhammer_absorb_comp
% 0.25/0.45 thf(fact_1733_pochhammer__same,axiom,
% 0.25/0.45 ! [A: $tType] :
% 0.25/0.46 ( ( ( semiring_char_0 @ A )
% 0.25/0.46 & ( comm_ring_1 @ A )
% 0.25/0.46 & ( semiri3467727345109120633visors @ A ) )
% 0.25/0.46 => ! [N: nat] :
% 0.25/0.46 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
% 0.25/0.46 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_same
% 0.25/0.46 thf(fact_1734_pochhammer__minus,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_ring_1 @ A )
% 0.25/0.46 => ! [B2: A,K: nat] :
% 0.25/0.46 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
% 0.25/0.46 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_minus
% 0.25/0.46 thf(fact_1735_pochhammer__minus_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_ring_1 @ A )
% 0.25/0.46 => ! [B2: A,K: nat] :
% 0.25/0.46 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
% 0.25/0.46 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_minus'
% 0.25/0.46 thf(fact_1736_gbinomial__pochhammer_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( field_char_0 @ A )
% 0.25/0.46 => ( ( gbinomial @ A )
% 0.25/0.46 = ( ^ [A7: A,K4: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ K4 ) ) @ ( one_one @ A ) ) @ K4 ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % gbinomial_pochhammer'
% 0.25/0.46 thf(fact_1737_gbinomial__pochhammer,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( field_char_0 @ A )
% 0.25/0.46 => ( ( gbinomial @ A )
% 0.25/0.46 = ( ^ [A7: A,K4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A7 ) @ K4 ) ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % gbinomial_pochhammer
% 0.25/0.46 thf(fact_1738_xor__nat__unfold,axiom,
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ nat )
% 0.25/0.46 = ( ^ [M: nat,N5: nat] :
% 0.25/0.46 ( if @ nat
% 0.25/0.46 @ ( M
% 0.25/0.46 = ( zero_zero @ nat ) )
% 0.25/0.46 @ N5
% 0.25/0.46 @ ( if @ nat
% 0.25/0.46 @ ( N5
% 0.25/0.46 = ( zero_zero @ nat ) )
% 0.25/0.46 @ M
% 0.25/0.46 @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_nat_unfold
% 0.25/0.46 thf(fact_1739_num__of__integer_Oabs__eq,axiom,
% 0.25/0.46 ! [X: int] :
% 0.25/0.46 ( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( num_of_nat @ ( nat2 @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % num_of_integer.abs_eq
% 0.25/0.46 thf(fact_1740_signed__take__bit__code,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 0.25/0.46 = ( ^ [N5: nat,A7: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ N5 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N5 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % signed_take_bit_code
% 0.25/0.46 thf(fact_1741_xor__nat__rec,axiom,
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ nat )
% 0.25/0.46 = ( ^ [M: nat,N5: nat] :
% 0.25/0.46 ( plus_plus @ nat
% 0.25/0.46 @ ( zero_neq_one_of_bool @ nat
% 0.25/0.46 @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
% 0.25/0.46 != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
% 0.25/0.46 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_nat_rec
% 0.25/0.46 thf(fact_1742_xor__one__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 0.25/0.46 @ ( zero_neq_one_of_bool @ A
% 0.25/0.46 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_one_eq
% 0.25/0.46 thf(fact_1743_one__xor__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.46 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 0.25/0.46 @ ( zero_neq_one_of_bool @ A
% 0.25/0.46 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_xor_eq
% 0.25/0.46 thf(fact_1744_pochhammer__binomial__sum,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_ring_1 @ A )
% 0.25/0.46 => ! [A3: A,B2: A,N: nat] :
% 0.25/0.46 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 0.25/0.46 = ( groups7311177749621191930dd_sum @ nat @ A
% 0.25/0.46 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_binomial_sum
% 0.25/0.46 thf(fact_1745_divmod__integer__def,axiom,
% 0.25/0.46 ( code_divmod_integer
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K4 @ L3 ) @ ( modulo_modulo @ code_integer @ K4 @ L3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divmod_integer_def
% 0.25/0.46 thf(fact_1746_sum__power2,axiom,
% 0.25/0.46 ! [K: nat] :
% 0.25/0.46 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
% 0.25/0.46 = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % sum_power2
% 0.25/0.46 thf(fact_1747_nat__of__integer__code,axiom,
% 0.25/0.46 ( code_nat_of_integer
% 0.25/0.46 = ( ^ [K4: code_integer] :
% 0.25/0.46 ( if @ nat @ ( ord_less_eq @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
% 0.25/0.46 @ ( product_case_prod @ code_integer @ code_integer @ nat
% 0.25/0.46 @ ^ [L3: code_integer,J2: code_integer] :
% 0.25/0.46 ( if @ nat
% 0.25/0.46 @ ( J2
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) )
% 0.25/0.46 @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( one_one @ nat ) ) )
% 0.25/0.46 @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer_code
% 0.25/0.46 thf(fact_1748_pochhammer__times__pochhammer__half,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( field_char_0 @ A )
% 0.25/0.46 => ! [Z2: A,N: nat] :
% 0.25/0.46 ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [K4: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_times_pochhammer_half
% 0.25/0.46 thf(fact_1749_int__of__integer__code,axiom,
% 0.25/0.46 ( code_int_of_integer
% 0.25/0.46 = ( ^ [K4: code_integer] :
% 0.25/0.46 ( if @ int @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K4 ) ) )
% 0.25/0.46 @ ( if @ int
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( zero_zero @ int )
% 0.25/0.46 @ ( product_case_prod @ code_integer @ code_integer @ int
% 0.25/0.46 @ ^ [L3: code_integer,J2: code_integer] :
% 0.25/0.46 ( if @ int
% 0.25/0.46 @ ( J2
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) )
% 0.25/0.46 @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
% 0.25/0.46 @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_code
% 0.25/0.46 thf(fact_1750_divmod__integer__eq__cases,axiom,
% 0.25/0.46 ( code_divmod_integer
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] :
% 0.25/0.46 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( L3
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
% 0.25/0.46 @ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L3
% 0.25/0.46 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( ( sgn_sgn @ code_integer @ K4 )
% 0.25/0.46 = ( sgn_sgn @ code_integer @ L3 ) )
% 0.25/0.46 @ ( code_divmod_abs @ K4 @ L3 )
% 0.25/0.46 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ^ [R3: code_integer,S4: code_integer] :
% 0.25/0.46 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( S4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S4 ) ) )
% 0.25/0.46 @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divmod_integer_eq_cases
% 0.25/0.46 thf(fact_1751_time__def,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( time_time @ A )
% 0.25/0.46 = ( ^ [C3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 0.25/0.46 ( case_option @ nat @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( zero_zero @ nat )
% 0.25/0.46 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ nat
% 0.25/0.46 @ ^ [X3: A] :
% 0.25/0.46 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ nat
% 0.25/0.46 @ ^ [Xa4: heap_ext @ product_unit,T3: nat] : T3 ) )
% 0.25/0.46 @ ( heap_Time_execute @ A @ C3 @ H ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % time_def
% 0.25/0.46 thf(fact_1752_bit__cut__integer__def,axiom,
% 0.25/0.46 ( code_bit_cut_integer
% 0.25/0.46 = ( ^ [K4: code_integer] :
% 0.25/0.46 ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
% 0.25/0.46 @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit_cut_integer_def
% 0.25/0.46 thf(fact_1753_integer__of__num_I3_J,axiom,
% 0.25/0.46 ! [N: num] :
% 0.25/0.46 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 0.25/0.46 = ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_of_num(3)
% 0.25/0.46 thf(fact_1754_less__eq__int_Orep__eq,axiom,
% 0.25/0.46 ( ( ord_less_eq @ int )
% 0.25/0.46 = ( ^ [X3: int,Xa4: int] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.46 @ ^ [Y3: nat,Z5: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
% 0.25/0.46 @ ( rep_Integ @ X3 )
% 0.25/0.46 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_eq_int.rep_eq
% 0.25/0.46 thf(fact_1755_nat__of__integer__of__nat,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 0.25/0.46 = N ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer_of_nat
% 0.25/0.46 thf(fact_1756_int__of__integer__of__nat,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 0.25/0.46 = ( semiring_1_of_nat @ int @ N ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_of_nat
% 0.25/0.46 thf(fact_1757_integer__of__int__int__of__integer,axiom,
% 0.25/0.46 ! [K: code_integer] :
% 0.25/0.46 ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
% 0.25/0.46 = K ) ).
% 0.25/0.46
% 0.25/0.46 % integer_of_int_int_of_integer
% 0.25/0.46 thf(fact_1758_int__of__integer__integer__of__int,axiom,
% 0.25/0.46 ! [K: int] :
% 0.25/0.46 ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
% 0.25/0.46 = K ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_integer_of_int
% 0.25/0.46 thf(fact_1759_int__of__integer__inverse,axiom,
% 0.25/0.46 ! [X: code_integer] :
% 0.25/0.46 ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
% 0.25/0.46 = X ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_inverse
% 0.25/0.46 thf(fact_1760_apsnd__conv,axiom,
% 0.25/0.46 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
% 0.25/0.46 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
% 0.25/0.46 = ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % apsnd_conv
% 0.25/0.46 thf(fact_1761_prod_Oneutral__const,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [A4: set @ B] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.46 @ ^ [Uu: B] : ( one_one @ A )
% 0.25/0.46 @ A4 )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.neutral_const
% 0.25/0.46 thf(fact_1762_of__int__integer__of,axiom,
% 0.25/0.46 ! [K: code_integer] :
% 0.25/0.46 ( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
% 0.25/0.46 = K ) ).
% 0.25/0.46
% 0.25/0.46 % of_int_integer_of
% 0.25/0.46 thf(fact_1763_int__of__integer__of__int,axiom,
% 0.25/0.46 ! [K: int] :
% 0.25/0.46 ( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
% 0.25/0.46 = K ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_of_int
% 0.25/0.46 thf(fact_1764_zero__integer_Orep__eq,axiom,
% 0.25/0.46 ( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
% 0.25/0.46 = ( zero_zero @ int ) ) ).
% 0.25/0.46
% 0.25/0.46 % zero_integer.rep_eq
% 0.25/0.46 thf(fact_1765_int__of__integer__numeral,axiom,
% 0.25/0.46 ! [K: num] :
% 0.25/0.46 ( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
% 0.25/0.46 = ( numeral_numeral @ int @ K ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_numeral
% 0.25/0.46 thf(fact_1766_plus__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % plus_integer.rep_eq
% 0.25/0.46 thf(fact_1767_one__integer_Orep__eq,axiom,
% 0.25/0.46 ( ( code_int_of_integer @ ( one_one @ code_integer ) )
% 0.25/0.46 = ( one_one @ int ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_integer.rep_eq
% 0.25/0.46 thf(fact_1768_uminus__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
% 0.25/0.46 = ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % uminus_integer.rep_eq
% 0.25/0.46 thf(fact_1769_times__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % times_integer.rep_eq
% 0.25/0.46 thf(fact_1770_minus__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_integer.rep_eq
% 0.25/0.46 thf(fact_1771_abs__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
% 0.25/0.46 = ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % abs_integer.rep_eq
% 0.25/0.46 thf(fact_1772_divide__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divide_integer.rep_eq
% 0.25/0.46 thf(fact_1773_modulo__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % modulo_integer.rep_eq
% 0.25/0.46 thf(fact_1774_sgn__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
% 0.25/0.46 = ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % sgn_integer.rep_eq
% 0.25/0.46 thf(fact_1775_prod_OatMost__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atMost_Suc
% 0.25/0.46 thf(fact_1776_prod_OlessThan__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.lessThan_Suc
% 0.25/0.46 thf(fact_1777_nat__of__integer__non__positive,axiom,
% 0.25/0.46 ! [K: code_integer] :
% 0.25/0.46 ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 0.25/0.46 => ( ( code_nat_of_integer @ K )
% 0.25/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer_non_positive
% 0.25/0.46 thf(fact_1778_prod_Ocl__ivl__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [N: nat,M2: nat,G: nat > A] :
% 0.25/0.46 ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.46 = ( one_one @ A ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.cl_ivl_Suc
% 0.25/0.46 thf(fact_1779_prod_Oop__ivl__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [N: nat,M2: nat,G: nat > A] :
% 0.25/0.46 ( ( ( ord_less @ nat @ N @ M2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.46 = ( one_one @ A ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.op_ivl_Suc
% 0.25/0.46 thf(fact_1780_xor__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_integer.rep_eq
% 0.25/0.46 thf(fact_1781_push__bit__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % push_bit_integer.rep_eq
% 0.25/0.46 thf(fact_1782_xor__integer_Oabs__eq,axiom,
% 0.25/0.46 ! [Xa: int,X: int] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_integer.abs_eq
% 0.25/0.46 thf(fact_1783_nat__of__integer_Orep__eq,axiom,
% 0.25/0.46 ( code_nat_of_integer
% 0.25/0.46 = ( ^ [X3: code_integer] : ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer.rep_eq
% 0.25/0.46 thf(fact_1784_integer__eq__iff,axiom,
% 0.25/0.46 ( ( ^ [Y5: code_integer,Z4: code_integer] : ( Y5 = Z4 ) )
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ K4 )
% 0.25/0.46 = ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_eq_iff
% 0.25/0.46 thf(fact_1785_integer__eqI,axiom,
% 0.25/0.46 ! [K: code_integer,L: code_integer] :
% 0.25/0.46 ( ( ( code_int_of_integer @ K )
% 0.25/0.46 = ( code_int_of_integer @ L ) )
% 0.25/0.46 => ( K = L ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_eqI
% 0.25/0.46 thf(fact_1786_int__of__integer__inject,axiom,
% 0.25/0.46 ! [X: code_integer,Y: code_integer] :
% 0.25/0.46 ( ( ( code_int_of_integer @ X )
% 0.25/0.46 = ( code_int_of_integer @ Y ) )
% 0.25/0.46 = ( X = Y ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_inject
% 0.25/0.46 thf(fact_1787_prod_Onot__neutral__contains__not__neutral,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: B > A,A4: set @ B] :
% 0.25/0.46 ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 0.25/0.46 != ( one_one @ A ) )
% 0.25/0.46 => ~ ! [A5: B] :
% 0.25/0.46 ( ( member @ B @ A5 @ A4 )
% 0.25/0.46 => ( ( G @ A5 )
% 0.25/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.not_neutral_contains_not_neutral
% 0.25/0.46 thf(fact_1788_prod_Oneutral,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [A4: set @ B,G: B > A] :
% 0.25/0.46 ( ! [X2: B] :
% 0.25/0.46 ( ( member @ B @ X2 @ A4 )
% 0.25/0.46 => ( ( G @ X2 )
% 0.25/0.46 = ( one_one @ A ) ) )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 0.25/0.46 = ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.neutral
% 0.25/0.46 thf(fact_1789_apsnd__compose,axiom,
% 0.25/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
% 0.25/0.46 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
% 0.25/0.46 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% 0.25/0.46
% 0.25/0.46 % apsnd_compose
% 0.25/0.46 thf(fact_1790_prod__ge__1,axiom,
% 0.25/0.46 ! [A: $tType,B: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ! [A4: set @ B,F: B > A] :
% 0.25/0.46 ( ! [X2: B] :
% 0.25/0.46 ( ( member @ B @ X2 @ A4 )
% 0.25/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X2 ) ) )
% 0.25/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod_ge_1
% 0.25/0.46 thf(fact_1791_int__of__integer__less__iff,axiom,
% 0.25/0.46 ! [X: code_integer,Y: code_integer] :
% 0.25/0.46 ( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
% 0.25/0.46 = ( ord_less @ code_integer @ X @ Y ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_less_iff
% 0.25/0.46 thf(fact_1792_integer__less__iff,axiom,
% 0.25/0.46 ( ( ord_less @ code_integer )
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_less_iff
% 0.25/0.46 thf(fact_1793_less__integer_Orep__eq,axiom,
% 0.25/0.46 ( ( ord_less @ code_integer )
% 0.25/0.46 = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_integer.rep_eq
% 0.25/0.46 thf(fact_1794_less__eq__integer_Orep__eq,axiom,
% 0.25/0.46 ( ( ord_less_eq @ code_integer )
% 0.25/0.46 = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_eq_integer.rep_eq
% 0.25/0.46 thf(fact_1795_integer__less__eq__iff,axiom,
% 0.25/0.46 ( ( ord_less_eq @ code_integer )
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_less_eq_iff
% 0.25/0.46 thf(fact_1796_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.shift_bounds_cl_Suc_ivl
% 0.25/0.46 thf(fact_1797_prod_Oshift__bounds__Suc__ivl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.shift_bounds_Suc_ivl
% 0.25/0.46 thf(fact_1798_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.shift_bounds_cl_nat_ivl
% 0.25/0.46 thf(fact_1799_prod_Oshift__bounds__nat__ivl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.shift_bounds_nat_ivl
% 0.25/0.46 thf(fact_1800_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
% 0.25/0.46 ! [L: code_integer,U: code_integer] :
% 0.25/0.46 ( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThanPlusOne_atLeastAtMost_integer
% 0.25/0.46 thf(fact_1801_prod__le__1,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ! [A4: set @ B,F: B > A] :
% 0.25/0.46 ( ! [X2: B] :
% 0.25/0.46 ( ( member @ B @ X2 @ A4 )
% 0.25/0.46 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X2 ) )
% 0.25/0.46 & ( ord_less_eq @ A @ ( F @ X2 ) @ ( one_one @ A ) ) ) )
% 0.25/0.46 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod_le_1
% 0.25/0.46 thf(fact_1802_prod_Oivl__cong,axiom,
% 0.25/0.46 ! [A: $tType,B: $tType] :
% 0.25/0.46 ( ( ( ord @ B )
% 0.25/0.46 & ( comm_monoid_mult @ A ) )
% 0.25/0.46 => ! [A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
% 0.25/0.46 ( ( A3 = C2 )
% 0.25/0.46 => ( ( B2 = D3 )
% 0.25/0.46 => ( ! [X2: B] :
% 0.25/0.46 ( ( ord_less_eq @ B @ C2 @ X2 )
% 0.25/0.46 => ( ( ord_less @ B @ X2 @ D3 )
% 0.25/0.46 => ( ( G @ X2 )
% 0.25/0.46 = ( H3 @ X2 ) ) ) )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.ivl_cong
% 0.25/0.46 thf(fact_1803_take__bit__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % take_bit_integer.rep_eq
% 0.25/0.46 thf(fact_1804_and__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_integer.rep_eq
% 0.25/0.46 thf(fact_1805_bit__integer_Orep__eq,axiom,
% 0.25/0.46 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 0.25/0.46 = ( ^ [X3: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit_integer.rep_eq
% 0.25/0.46 thf(fact_1806_prod_OatLeastLessThan__concat,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,P5: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.46 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.25/0.46 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastLessThan_concat
% 0.25/0.46 thf(fact_1807_mask__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
% 0.25/0.46 = ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% 0.25/0.46
% 0.25/0.46 % mask_integer.rep_eq
% 0.25/0.46 thf(fact_1808_integer__of__num__def,axiom,
% 0.25/0.46 ( code_integer_of_num
% 0.25/0.46 = ( numeral_numeral @ code_integer ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_of_num_def
% 0.25/0.46 thf(fact_1809_unset__bit__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % unset_bit_integer.rep_eq
% 0.25/0.46 thf(fact_1810_set__bit__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_bit_integer.rep_eq
% 0.25/0.46 thf(fact_1811_flip__bit__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % flip_bit_integer.rep_eq
% 0.25/0.46 thf(fact_1812_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_Suc_atMost_Suc_shift
% 0.25/0.46 thf(fact_1813_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_Suc_lessThan_Suc_shift
% 0.25/0.46 thf(fact_1814_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.25/0.46 ! [L: int,U: int] :
% 0.25/0.46 ( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.25/0.46 thf(fact_1815_prod_OatLeastAtMost__shift__bounds,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastAtMost_shift_bounds
% 0.25/0.46 thf(fact_1816_prod_OatLeastLessThan__shift__bounds,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastLessThan_shift_bounds
% 0.25/0.46 thf(fact_1817_prod_OatLeastAtMost__rev,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat,M2: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastAtMost_rev
% 0.25/0.46 thf(fact_1818_prod_Onat__diff__reindex,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.nat_diff_reindex
% 0.25/0.46 thf(fact_1819_nat__of__integer_Oabs__eq,axiom,
% 0.25/0.46 ! [X: int] :
% 0.25/0.46 ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( nat2 @ X ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer.abs_eq
% 0.25/0.46 thf(fact_1820_num__of__integer_Orep__eq,axiom,
% 0.25/0.46 ( code_num_of_integer
% 0.25/0.46 = ( ^ [X3: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % num_of_integer.rep_eq
% 0.25/0.46 thf(fact_1821_prod_OatLeast0__atMost__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast0_atMost_Suc
% 0.25/0.46 thf(fact_1822_prod_OatLeast0__lessThan__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast0_lessThan_Suc
% 0.25/0.46 thf(fact_1823_prod_OatLeast__Suc__lessThan,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less @ nat @ M2 @ N )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_Suc_lessThan
% 0.25/0.46 thf(fact_1824_prod_Onat__ivl__Suc_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.nat_ivl_Suc'
% 0.25/0.46 thf(fact_1825_prod_OatLeast__Suc__atMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_Suc_atMost
% 0.25/0.46 thf(fact_1826_prod_OatLeastLessThan__Suc,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [A3: nat,B2: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastLessThan_Suc
% 0.25/0.46 thf(fact_1827_divmod__abs__code_I6_J,axiom,
% 0.25/0.46 ! [J: code_integer] :
% 0.25/0.46 ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
% 0.25/0.46 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divmod_abs_code(6)
% 0.25/0.46 thf(fact_1828_prod_Olast__plus,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.last_plus
% 0.25/0.46 thf(fact_1829_prod_OSuc__reindex__ivl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.46 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ M2 )
% 0.25/0.46 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.Suc_reindex_ivl
% 0.25/0.46 thf(fact_1830_prod_OatMost__Suc__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 0.25/0.46 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atMost_Suc_shift
% 0.25/0.46 thf(fact_1831_prod_OlessThan__Suc__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 0.25/0.46 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.lessThan_Suc_shift
% 0.25/0.46 thf(fact_1832_prod_OatLeastLessThan__rev,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat,M2: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastLessThan_rev
% 0.25/0.46 thf(fact_1833_prod_Onested__swap,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [A3: nat > nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [J2: nat] :
% 0.25/0.46 ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.nested_swap
% 0.25/0.46 thf(fact_1834_prod_OatLeast1__atMost__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast1_atMost_eq
% 0.25/0.46 thf(fact_1835_prod_Onat__group,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,K: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [M: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.nat_group
% 0.25/0.46 thf(fact_1836_fact__prod,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( semiring_char_0 @ A )
% 0.25/0.46 => ( ( semiring_char_0_fact @ A )
% 0.25/0.46 = ( ^ [N5: nat] :
% 0.25/0.46 ( semiring_1_of_nat @ A
% 0.25/0.46 @ ( groups7121269368397514597t_prod @ nat @ nat
% 0.25/0.46 @ ^ [X3: nat] : X3
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % fact_prod
% 0.25/0.46 thf(fact_1837_prod_Onested__swap_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [A3: nat > nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [J2: nat] :
% 0.25/0.46 ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.nested_swap'
% 0.25/0.46 thf(fact_1838_prod__atLeastAtMost__code,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [F: nat > A,A3: nat,B2: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 0.25/0.46 = ( set_fo6178422350223883121st_nat @ A
% 0.25/0.46 @ ^ [A7: nat] : ( times_times @ A @ ( F @ A7 ) )
% 0.25/0.46 @ A3
% 0.25/0.46 @ B2
% 0.25/0.46 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod_atLeastAtMost_code
% 0.25/0.46 thf(fact_1839_prod_Ohead__if,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [N: nat,M2: nat,G: nat > A] :
% 0.25/0.46 ( ( ( ord_less @ nat @ N @ M2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.46 = ( one_one @ A ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.head_if
% 0.25/0.46 thf(fact_1840_prod_Oub__add__nat,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A,P5: nat] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
% 0.25/0.46 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.ub_add_nat
% 0.25/0.46 thf(fact_1841_prod_OatLeast0__atMost__Suc__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast0_atMost_Suc_shift
% 0.25/0.46 thf(fact_1842_prod_OatLeast0__lessThan__Suc__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast0_lessThan_Suc_shift
% 0.25/0.46 thf(fact_1843_prod_OatLeastLessThan__shift__0,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastLessThan_shift_0
% 0.25/0.46 thf(fact_1844_integer__of__num__triv_I1_J,axiom,
% 0.25/0.46 ( ( code_integer_of_num @ one2 )
% 0.25/0.46 = ( one_one @ code_integer ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_of_num_triv(1)
% 0.25/0.46 thf(fact_1845_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat,M2: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastLessThan_rev_at_least_Suc_atMost
% 0.25/0.46 thf(fact_1846_prod_OatMost__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 0.25/0.46 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 0.25/0.46 @ ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atMost_shift
% 0.25/0.46 thf(fact_1847_pochhammer__prod,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_semiring_1 @ A )
% 0.25/0.46 => ( ( comm_s3205402744901411588hammer @ A )
% 0.25/0.46 = ( ^ [A7: A,N5: nat] :
% 0.25/0.46 ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ I3 ) )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_prod
% 0.25/0.46 thf(fact_1848_prod_OatLeast__atMost__pred__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ( comp @ nat @ A @ nat @ G
% 0.25/0.46 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_atMost_pred_shift
% 0.25/0.46 thf(fact_1849_prod_OatLeast__lessThan__pred__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ( comp @ nat @ A @ nat @ G
% 0.25/0.46 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_lessThan_pred_shift
% 0.25/0.46 thf(fact_1850_integer__of__num_I2_J,axiom,
% 0.25/0.46 ! [N: num] :
% 0.25/0.46 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 0.25/0.46 = ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_of_num(2)
% 0.25/0.46 thf(fact_1851_divmod__abs__code_I5_J,axiom,
% 0.25/0.46 ! [J: code_integer] :
% 0.25/0.46 ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
% 0.25/0.46 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divmod_abs_code(5)
% 0.25/0.46 thf(fact_1852_prod_OatLeastAtMost__shift__0,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeastAtMost_shift_0
% 0.25/0.46 thf(fact_1853_sum_OatLeast__int__lessThan__int__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_add @ A )
% 0.25/0.46 => ! [G: int > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.25/0.46 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % sum.atLeast_int_lessThan_int_shift
% 0.25/0.46 thf(fact_1854_pochhammer__Suc__prod,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_semiring_1 @ A )
% 0.25/0.46 => ! [A3: A,N: nat] :
% 0.25/0.46 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_Suc_prod
% 0.25/0.46 thf(fact_1855_pochhammer__prod__rev,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_semiring_1 @ A )
% 0.25/0.46 => ( ( comm_s3205402744901411588hammer @ A )
% 0.25/0.46 = ( ^ [A7: A,N5: nat] :
% 0.25/0.46 ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N5 @ I3 ) ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_prod_rev
% 0.25/0.46 thf(fact_1856_fact__div__fact,axiom,
% 0.25/0.46 ! [N: nat,M2: nat] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ N @ M2 )
% 0.25/0.46 => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ nat
% 0.25/0.46 @ ^ [X3: nat] : X3
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % fact_div_fact
% 0.25/0.46 thf(fact_1857_integer__of__num__triv_I2_J,axiom,
% 0.25/0.46 ( ( code_integer_of_num @ ( bit0 @ one2 ) )
% 0.25/0.46 = ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % integer_of_num_triv(2)
% 0.25/0.46 thf(fact_1858_prod_Oin__pairs,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.in_pairs
% 0.25/0.46 thf(fact_1859_prod_Oin__pairs__0,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.in_pairs_0
% 0.25/0.46 thf(fact_1860_nat__of__integer__less__iff,axiom,
% 0.25/0.46 ! [X: code_integer,Y: code_integer] :
% 0.25/0.46 ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
% 0.25/0.46 => ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
% 0.25/0.46 => ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
% 0.25/0.46 = ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer_less_iff
% 0.25/0.46 thf(fact_1861_pochhammer__Suc__prod__rev,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_semiring_1 @ A )
% 0.25/0.46 => ! [A3: A,N: nat] :
% 0.25/0.46 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pochhammer_Suc_prod_rev
% 0.25/0.46 thf(fact_1862_divmod__integer__code,axiom,
% 0.25/0.46 ( code_divmod_integer
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] :
% 0.25/0.46 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L3 )
% 0.25/0.46 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ ( code_divmod_abs @ K4 @ L3 )
% 0.25/0.46 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ^ [R3: code_integer,S4: code_integer] :
% 0.25/0.46 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( S4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S4 ) ) )
% 0.25/0.46 @ ( code_divmod_abs @ K4 @ L3 ) ) )
% 0.25/0.46 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( L3
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
% 0.25/0.46 @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
% 0.25/0.46 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K4 @ L3 )
% 0.25/0.46 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ^ [R3: code_integer,S4: code_integer] :
% 0.25/0.46 ( if @ ( product_prod @ code_integer @ code_integer )
% 0.25/0.46 @ ( S4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S4 ) ) )
% 0.25/0.46 @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divmod_integer_code
% 0.25/0.46 thf(fact_1863_bit__cut__integer__code,axiom,
% 0.25/0.46 ( code_bit_cut_integer
% 0.25/0.46 = ( ^ [K4: code_integer] :
% 0.25/0.46 ( if @ ( product_prod @ code_integer @ $o )
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( zero_zero @ code_integer ) )
% 0.25/0.46 @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
% 0.25/0.46 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
% 0.25/0.46 @ ^ [R3: code_integer,S4: code_integer] :
% 0.25/0.46 ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R3 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ S4 ) )
% 0.25/0.46 @ ( S4
% 0.25/0.46 = ( one_one @ code_integer ) ) )
% 0.25/0.46 @ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit_cut_integer_code
% 0.25/0.46 thf(fact_1864_prod_Ozero__middle,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [P5: nat,K: nat,G: nat > A,H3: nat > A] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 0.25/0.46 => ( ( ord_less_eq @ nat @ K @ P5 )
% 0.25/0.46 => ( ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( one_one @ A ) @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ P5 ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.zero_middle
% 0.25/0.46 thf(fact_1865_xor__int__rec,axiom,
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ int )
% 0.25/0.46 = ( ^ [K4: int,L3: int] :
% 0.25/0.46 ( plus_plus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) )
% 0.25/0.46 != ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_int_rec
% 0.25/0.46 thf(fact_1866_divmod__abs__def,axiom,
% 0.25/0.46 ( code_divmod_abs
% 0.25/0.46 = ( ^ [K4: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L3 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divmod_abs_def
% 0.25/0.46 thf(fact_1867_less__int_Orep__eq,axiom,
% 0.25/0.46 ( ( ord_less @ int )
% 0.25/0.46 = ( ^ [X3: int,Xa4: int] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.46 @ ^ [Y3: nat,Z5: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
% 0.25/0.46 @ ( rep_Integ @ X3 )
% 0.25/0.46 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_int.rep_eq
% 0.25/0.46 thf(fact_1868_make_H__def,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( ( heap @ A )
% 0.25/0.46 & ( semiring_1 @ B ) )
% 0.25/0.46 => ( ( array_make2 @ B @ A )
% 0.25/0.46 = ( ^ [I3: code_integer,F2: B > A] : ( array_make @ A @ ( code_nat_of_integer @ I3 ) @ ( comp @ B @ A @ nat @ F2 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % make'_def
% 0.25/0.46 thf(fact_1869_new_H__def,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( heap @ A )
% 0.25/0.46 => ( ( array_new2 @ A )
% 0.25/0.46 = ( comp @ nat @ ( A > ( heap_Time_Heap @ ( array @ A ) ) ) @ code_integer @ ( array_new @ A ) @ code_nat_of_integer ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % new'_def
% 0.25/0.46 thf(fact_1870_xor__int__unfold,axiom,
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ int )
% 0.25/0.46 = ( ^ [K4: int,L3: int] :
% 0.25/0.46 ( if @ int
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.25/0.46 @ ( bit_ri4277139882892585799ns_not @ int @ L3 )
% 0.25/0.46 @ ( if @ int
% 0.25/0.46 @ ( L3
% 0.25/0.46 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.25/0.46 @ ( bit_ri4277139882892585799ns_not @ int @ K4 )
% 0.25/0.46 @ ( if @ int
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( zero_zero @ int ) )
% 0.25/0.46 @ L3
% 0.25/0.46 @ ( if @ int
% 0.25/0.46 @ ( L3
% 0.25/0.46 = ( zero_zero @ int ) )
% 0.25/0.46 @ K4
% 0.25/0.46 @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_int_unfold
% 0.25/0.46 thf(fact_1871_take__bit__num__code,axiom,
% 0.25/0.46 ( bit_take_bit_num
% 0.25/0.46 = ( ^ [N5: nat,M: num] :
% 0.25/0.46 ( product_case_prod @ nat @ num @ ( option @ num )
% 0.25/0.46 @ ^ [A7: nat,X3: num] :
% 0.25/0.46 ( case_nat @ ( option @ num ) @ ( none @ num )
% 0.25/0.46 @ ^ [O: nat] :
% 0.25/0.46 ( case_num @ ( option @ num ) @ ( some @ num @ one2 )
% 0.25/0.46 @ ^ [P6: num] :
% 0.25/0.46 ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 0.25/0.46 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
% 0.25/0.46 @ ( bit_take_bit_num @ O @ P6 ) )
% 0.25/0.46 @ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 0.25/0.46 @ X3 )
% 0.25/0.46 @ A7 )
% 0.25/0.46 @ ( product_Pair @ nat @ num @ N5 @ M ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % take_bit_num_code
% 0.25/0.46 thf(fact_1872_prod_Otriangle__reindex,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.25/0.46 @ ( collect @ ( product_prod @ nat @ nat )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ $o
% 0.25/0.46 @ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [K4: nat] :
% 0.25/0.46 ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ K4 ) )
% 0.25/0.46 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.triangle_reindex
% 0.25/0.46 thf(fact_1873_case__nat__add__eq__if,axiom,
% 0.25/0.46 ! [A: $tType,A3: A,F: nat > A,V: num,N: nat] :
% 0.25/0.46 ( ( case_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 0.25/0.46 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % case_nat_add_eq_if
% 0.25/0.46 thf(fact_1874_rec__nat__add__eq__if,axiom,
% 0.25/0.46 ! [A: $tType,A3: A,F: nat > A > A,V: num,N: nat] :
% 0.25/0.46 ( ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 0.25/0.46 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rec_nat_add_eq_if
% 0.25/0.46 thf(fact_1875_bit_Ocompl__zero,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.compl_zero
% 0.25/0.46 thf(fact_1876_bit_Ocompl__one,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 = ( zero_zero @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.compl_one
% 0.25/0.46 thf(fact_1877_bit_Oxor__cancel__right,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.xor_cancel_right
% 0.25/0.46 thf(fact_1878_bit_Oxor__cancel__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.xor_cancel_left
% 0.25/0.46 thf(fact_1879_bit_Oxor__one__right,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.xor_one_right
% 0.25/0.46 thf(fact_1880_bit_Oxor__one__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 0.25/0.46 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.xor_one_left
% 0.25/0.46 thf(fact_1881_push__bit__minus__one__eq__not__mask,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [N: nat] :
% 0.25/0.46 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % push_bit_minus_one_eq_not_mask
% 0.25/0.46 thf(fact_1882_not__one__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
% 0.25/0.46 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_one_eq
% 0.25/0.46 thf(fact_1883_not__add__distrib,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_add_distrib
% 0.25/0.46 thf(fact_1884_not__diff__distrib,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_diff_distrib
% 0.25/0.46 thf(fact_1885_minus__eq__not__plus__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( uminus_uminus @ A )
% 0.25/0.46 = ( ^ [A7: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_eq_not_plus_1
% 0.25/0.46 thf(fact_1886_minus__eq__not__minus__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( uminus_uminus @ A )
% 0.25/0.46 = ( ^ [A7: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_eq_not_minus_1
% 0.25/0.46 thf(fact_1887_not__eq__complement,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( bit_ri4277139882892585799ns_not @ A )
% 0.25/0.46 = ( ^ [A7: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_eq_complement
% 0.25/0.46 thf(fact_1888_prod__int__eq,axiom,
% 0.25/0.46 ! [I: nat,J: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ int @ int
% 0.25/0.46 @ ^ [X3: int] : X3
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod_int_eq
% 0.25/0.46 thf(fact_1889_prod__int__plus__eq,axiom,
% 0.25/0.46 ! [I: nat,J: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ int @ int
% 0.25/0.46 @ ^ [X3: int] : X3
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod_int_plus_eq
% 0.25/0.46 thf(fact_1890_push__bit__mask__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [M2: nat,N: nat] :
% 0.25/0.46 ( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2239418461657761734s_mask @ A @ N ) )
% 0.25/0.46 = ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % push_bit_mask_eq
% 0.25/0.46 thf(fact_1891_prod_OatLeast__int__lessThan__int__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: int > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_int_lessThan_int_shift
% 0.25/0.46 thf(fact_1892_unset__bit__eq__and__not,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( ( bit_se2638667681897837118et_bit @ A )
% 0.25/0.46 = ( ^ [N5: nat,A7: A] : ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % unset_bit_eq_and_not
% 0.25/0.46 thf(fact_1893_prod_OatLeast__int__atMost__int__shift,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: int > A,M2: nat,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.atLeast_int_atMost_int_shift
% 0.25/0.46 thf(fact_1894_int__numeral__not__and__num,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.46 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_numeral_not_and_num
% 0.25/0.46 thf(fact_1895_int__numeral__and__not__num,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.46 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_numeral_and_not_num
% 0.25/0.46 thf(fact_1896_prod_Otriangle__reindex__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: nat > nat > A,N: nat] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.25/0.46 @ ( collect @ ( product_prod @ nat @ nat )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ $o
% 0.25/0.46 @ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 0.25/0.46 = ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [K4: nat] :
% 0.25/0.46 ( groups7121269368397514597t_prod @ nat @ A
% 0.25/0.46 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ K4 ) )
% 0.25/0.46 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.triangle_reindex_eq
% 0.25/0.46 thf(fact_1897_and__not__numerals_I8_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.25/0.46 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_not_numerals(8)
% 0.25/0.46 thf(fact_1898_not__int__rec,axiom,
% 0.25/0.46 ( ( bit_ri4277139882892585799ns_not @ int )
% 0.25/0.46 = ( ^ [K4: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_int_rec
% 0.25/0.46 thf(fact_1899_Nitpick_Ocase__nat__unfold,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( case_nat @ A )
% 0.25/0.46 = ( ^ [X3: A,F2: nat > A,N5: nat] :
% 0.25/0.46 ( if @ A
% 0.25/0.46 @ ( N5
% 0.25/0.46 = ( zero_zero @ nat ) )
% 0.25/0.46 @ X3
% 0.25/0.46 @ ( F2 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Nitpick.case_nat_unfold
% 0.25/0.46 thf(fact_1900_or__int__rec,axiom,
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ int )
% 0.25/0.46 = ( ^ [K4: int,L3: int] :
% 0.25/0.46 ( plus_plus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 0.25/0.46 | ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_int_rec
% 0.25/0.46 thf(fact_1901_or__numerals_I4_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [X: num,Y: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.25/0.46 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(4)
% 0.25/0.46 thf(fact_1902_or__numerals_I6_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [X: num,Y: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 0.25/0.46 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(6)
% 0.25/0.46 thf(fact_1903_or__numerals_I7_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [X: num,Y: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.25/0.46 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(7)
% 0.25/0.46 thf(fact_1904_or__nat__rec,axiom,
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ nat )
% 0.25/0.46 = ( ^ [M: nat,N5: nat] :
% 0.25/0.46 ( plus_plus @ nat
% 0.25/0.46 @ ( zero_neq_one_of_bool @ nat
% 0.25/0.46 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
% 0.25/0.46 | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
% 0.25/0.46 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_nat_rec
% 0.25/0.46 thf(fact_1905_or__not__numerals_I9_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 0.25/0.46 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_not_numerals(9)
% 0.25/0.46 thf(fact_1906_bit_Odisj__one__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.disj_one_left
% 0.25/0.46 thf(fact_1907_bit_Odisj__one__right,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.disj_one_right
% 0.25/0.46 thf(fact_1908_or__numerals_I8_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 0.25/0.46 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(8)
% 0.25/0.46 thf(fact_1909_or__numerals_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [Y: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 0.25/0.46 = ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(2)
% 0.25/0.46 thf(fact_1910_bit_Odisj__cancel__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.disj_cancel_left
% 0.25/0.46 thf(fact_1911_bit_Odisj__cancel__right,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.disj_cancel_right
% 0.25/0.46 thf(fact_1912_or__numerals_I5_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 0.25/0.46 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(5)
% 0.25/0.46 thf(fact_1913_or__numerals_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.46 => ! [Y: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 0.25/0.46 = ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_numerals(1)
% 0.25/0.46 thf(fact_1914_or__integer_Oabs__eq,axiom,
% 0.25/0.46 ! [Xa: int,X: int] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_integer.abs_eq
% 0.25/0.46 thf(fact_1915_or__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_integer.rep_eq
% 0.25/0.46 thf(fact_1916_disjunctive__add,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ! [N3: nat] :
% 0.25/0.46 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
% 0.25/0.46 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
% 0.25/0.46 => ( ( plus_plus @ A @ A3 @ B2 )
% 0.25/0.46 = ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % disjunctive_add
% 0.25/0.46 thf(fact_1917_plus__and__or,axiom,
% 0.25/0.46 ! [X: int,Y: int] :
% 0.25/0.46 ( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) )
% 0.25/0.46 = ( plus_plus @ int @ X @ Y ) ) ).
% 0.25/0.46
% 0.25/0.46 % plus_and_or
% 0.25/0.46 thf(fact_1918_not__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
% 0.25/0.46 = ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_integer.rep_eq
% 0.25/0.46 thf(fact_1919_set__bit__nat__def,axiom,
% 0.25/0.46 ( ( bit_se5668285175392031749et_bit @ nat )
% 0.25/0.46 = ( ^ [M: nat,N5: nat] : ( bit_se1065995026697491101ons_or @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M @ ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_bit_nat_def
% 0.25/0.46 thf(fact_1920_not__integer_Oabs__eq,axiom,
% 0.25/0.46 ! [X: int] :
% 0.25/0.46 ( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_integer.abs_eq
% 0.25/0.46 thf(fact_1921_set__bit__eq__or,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ( ( bit_se5668285175392031749et_bit @ A )
% 0.25/0.46 = ( ^ [N5: nat,A7: A] : ( bit_se1065995026697491101ons_or @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_bit_eq_or
% 0.25/0.46 thf(fact_1922_bit_Ocomplement__unique,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [A3: A,X: A,Y: A] :
% 0.25/0.46 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
% 0.25/0.46 = ( zero_zero @ A ) )
% 0.25/0.46 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 => ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
% 0.25/0.46 = ( zero_zero @ A ) )
% 0.25/0.46 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 => ( X = Y ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.complement_unique
% 0.25/0.46 thf(fact_1923_bit_Ocompl__unique,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [X: A,Y: A] :
% 0.25/0.46 ( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
% 0.25/0.46 = ( zero_zero @ A ) )
% 0.25/0.46 => ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
% 0.25/0.46 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 => ( ( bit_ri4277139882892585799ns_not @ A @ X )
% 0.25/0.46 = Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.compl_unique
% 0.25/0.46 thf(fact_1924_or__one__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.46 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_one_eq
% 0.25/0.46 thf(fact_1925_one__or__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.46 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_or_eq
% 0.25/0.46 thf(fact_1926_mask__Suc__double,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [N: nat] :
% 0.25/0.46 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
% 0.25/0.46 = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mask_Suc_double
% 0.25/0.46 thf(fact_1927_or__not__numerals_I5_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.25/0.46 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_not_numerals(5)
% 0.25/0.46 thf(fact_1928_Suc__0__or__eq,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.25/0.46 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Suc_0_or_eq
% 0.25/0.46 thf(fact_1929_or__Suc__0__eq,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 0.25/0.46 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_Suc_0_eq
% 0.25/0.46 thf(fact_1930_or__not__numerals_I8_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 0.25/0.46 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_not_numerals(8)
% 0.25/0.46 thf(fact_1931_or__int__unfold,axiom,
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ int )
% 0.25/0.46 = ( ^ [K4: int,L3: int] :
% 0.25/0.46 ( if @ int
% 0.25/0.46 @ ( ( K4
% 0.25/0.46 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 0.25/0.46 | ( L3
% 0.25/0.46 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.25/0.46 @ ( uminus_uminus @ int @ ( one_one @ int ) )
% 0.25/0.46 @ ( if @ int
% 0.25/0.46 @ ( K4
% 0.25/0.46 = ( zero_zero @ int ) )
% 0.25/0.46 @ L3
% 0.25/0.46 @ ( if @ int
% 0.25/0.46 @ ( L3
% 0.25/0.46 = ( zero_zero @ int ) )
% 0.25/0.46 @ K4
% 0.25/0.46 @ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_int_unfold
% 0.25/0.46 thf(fact_1932_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.abstract_boolean_algebra_sym_diff_axioms
% 0.25/0.46 thf(fact_1933_or__nat__unfold,axiom,
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ nat )
% 0.25/0.46 = ( ^ [M: nat,N5: nat] :
% 0.25/0.46 ( if @ nat
% 0.25/0.46 @ ( M
% 0.25/0.46 = ( zero_zero @ nat ) )
% 0.25/0.46 @ N5
% 0.25/0.46 @ ( if @ nat
% 0.25/0.46 @ ( N5
% 0.25/0.46 = ( zero_zero @ nat ) )
% 0.25/0.46 @ M
% 0.25/0.46 @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_nat_unfold
% 0.25/0.46 thf(fact_1934_drop__bit__rec,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ( ( bit_se4197421643247451524op_bit @ A )
% 0.25/0.46 = ( ^ [N5: nat,A7: A] :
% 0.25/0.46 ( if @ A
% 0.25/0.46 @ ( N5
% 0.25/0.46 = ( zero_zero @ nat ) )
% 0.25/0.46 @ A7
% 0.25/0.46 @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_rec
% 0.25/0.46 thf(fact_1935_image__atLeastZeroLessThan__integer,axiom,
% 0.25/0.46 ! [U: code_integer] :
% 0.25/0.46 ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
% 0.25/0.46 => ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
% 0.25/0.46 = ( image2 @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_atLeastZeroLessThan_integer
% 0.25/0.46 thf(fact_1936_uminus__int__def,axiom,
% 0.25/0.46 ( ( uminus_uminus @ int )
% 0.25/0.46 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.46 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % uminus_int_def
% 0.25/0.46 thf(fact_1937_bit_Oabstract__boolean__algebra__axioms,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit.abstract_boolean_algebra_axioms
% 0.25/0.46 thf(fact_1938_map__fun__apply,axiom,
% 0.25/0.46 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 0.25/0.46 ( ( map_fun @ B @ C @ D @ A )
% 0.25/0.46 = ( ^ [F2: B > C,G2: D > A,H: C > D,X3: B] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % map_fun_apply
% 0.25/0.46 thf(fact_1939_drop__bit__drop__bit,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,A3: A] :
% 0.25/0.46 ( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
% 0.25/0.46 = ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_drop_bit
% 0.25/0.46 thf(fact_1940_image__add__atLeastAtMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_semidom @ A )
% 0.25/0.46 => ! [K: A,I: A,J: A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_atLeastAtMost
% 0.25/0.46 thf(fact_1941_image__diff__atLeastAtMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_idom @ A )
% 0.25/0.46 => ! [D3: A,A3: A,B2: A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( minus_minus @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D3 @ B2 ) @ ( minus_minus @ A @ D3 @ A3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_diff_atLeastAtMost
% 0.25/0.46 thf(fact_1942_image__add__atLeastLessThan,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_semidom @ A )
% 0.25/0.46 => ! [K: A,I: A,J: A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_atLeastLessThan
% 0.25/0.46 thf(fact_1943_max__0__1_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_0_1(1)
% 0.25/0.46 thf(fact_1944_max__0__1_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_0_1(2)
% 0.25/0.46 thf(fact_1945_image__uminus__atLeastAtMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordered_ab_group_add @ A )
% 0.25/0.46 => ! [X: A,Y: A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_uminus_atLeastAtMost
% 0.25/0.46 thf(fact_1946_max__0__1_I6_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 0.25/0.46 = ( numeral_numeral @ A @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_0_1(6)
% 0.25/0.46 thf(fact_1947_max__0__1_I5_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.25/0.46 = ( numeral_numeral @ A @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_0_1(5)
% 0.25/0.46 thf(fact_1948_image__add__atMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordered_ab_group_add @ A )
% 0.25/0.46 => ! [C2: A,A3: A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A3 ) )
% 0.25/0.46 = ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_atMost
% 0.25/0.46 thf(fact_1949_image__add__atLeastAtMost_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_semidom @ A )
% 0.25/0.46 => ! [K: A,I: A,J: A] :
% 0.25/0.46 ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_atLeastAtMost'
% 0.25/0.46 thf(fact_1950_image__minus__const__atLeastAtMost_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_idom @ A )
% 0.25/0.46 => ! [D3: A,A3: A,B2: A] :
% 0.25/0.46 ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_minus_const_atLeastAtMost'
% 0.25/0.46 thf(fact_1951_image__add__atLeastLessThan_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_semidom @ A )
% 0.25/0.46 => ! [K: A,I: A,J: A] :
% 0.25/0.46 ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_atLeastLessThan'
% 0.25/0.46 thf(fact_1952_drop__bit__of__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [N: nat] :
% 0.25/0.46 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
% 0.25/0.46 = ( zero_neq_one_of_bool @ A
% 0.25/0.46 @ ( N
% 0.25/0.46 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_of_1
% 0.25/0.46 thf(fact_1953_image__mult__atLeastAtMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [D3: A,A3: A,B2: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
% 0.25/0.46 => ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_mult_atLeastAtMost
% 0.25/0.46 thf(fact_1954_image__divide__atLeastAtMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [D3: A,A3: A,B2: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [C3: A] : ( divide_divide @ A @ C3 @ D3 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_divide_atLeastAtMost
% 0.25/0.46 thf(fact_1955_drop__bit__integer_Orep__eq,axiom,
% 0.25/0.46 ! [X: nat,Xa: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa ) )
% 0.25/0.46 = ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_integer.rep_eq
% 0.25/0.46 thf(fact_1956_drop__bit__integer_Oabs__eq,axiom,
% 0.25/0.46 ! [Xa: nat,X: int] :
% 0.25/0.46 ( ( bit_se4197421643247451524op_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_integer.abs_eq
% 0.25/0.46 thf(fact_1957_max__add__distrib__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordere2412721322843649153imp_le @ A )
% 0.25/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.25/0.46 ( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
% 0.25/0.46 = ( ord_max @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_add_distrib_left
% 0.25/0.46 thf(fact_1958_max__add__distrib__right,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordere2412721322843649153imp_le @ A )
% 0.25/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.25/0.46 ( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z2 ) )
% 0.25/0.46 = ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_add_distrib_right
% 0.25/0.46 thf(fact_1959_max__diff__distrib__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordered_ab_group_add @ A )
% 0.25/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.25/0.46 ( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
% 0.25/0.46 = ( ord_max @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % max_diff_distrib_left
% 0.25/0.46 thf(fact_1960_translation__Compl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ab_group_add @ A )
% 0.25/0.46 => ! [A3: A,T2: set @ A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 0.25/0.46 = ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % translation_Compl
% 0.25/0.46 thf(fact_1961_nat__add__max__right,axiom,
% 0.25/0.46 ! [M2: nat,N: nat,Q3: nat] :
% 0.25/0.46 ( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q3 ) )
% 0.25/0.46 = ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_add_max_right
% 0.25/0.46 thf(fact_1962_nat__add__max__left,axiom,
% 0.25/0.46 ! [M2: nat,N: nat,Q3: nat] :
% 0.25/0.46 ( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q3 )
% 0.25/0.46 = ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_add_max_left
% 0.25/0.46 thf(fact_1963_image__eq__imp__comp,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A4: set @ B,G: C > A,B6: set @ C,H3: A > D] :
% 0.25/0.46 ( ( ( image2 @ B @ A @ F @ A4 )
% 0.25/0.46 = ( image2 @ C @ A @ G @ B6 ) )
% 0.25/0.46 => ( ( image2 @ B @ D @ ( comp @ A @ D @ B @ H3 @ F ) @ A4 )
% 0.25/0.46 = ( image2 @ C @ D @ ( comp @ A @ D @ C @ H3 @ G ) @ B6 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_eq_imp_comp
% 0.25/0.46 thf(fact_1964_image__comp,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
% 0.25/0.46 ( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ R ) )
% 0.25/0.46 = ( image2 @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_comp
% 0.25/0.46 thf(fact_1965_map__fun__def,axiom,
% 0.25/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType] :
% 0.25/0.46 ( ( map_fun @ C @ A @ B @ D )
% 0.25/0.46 = ( ^ [F2: C > A,G2: B > D,H: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G2 @ H ) @ F2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % map_fun_def
% 0.25/0.46 thf(fact_1966_map__fun_Ocompositionality,axiom,
% 0.25/0.46 ! [D: $tType,F5: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F5,H3: C > A,I: B > D,Fun: A > B] :
% 0.25/0.46 ( ( map_fun @ E @ C @ D @ F5 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H3 @ I @ Fun ) )
% 0.25/0.46 = ( map_fun @ E @ A @ B @ F5 @ ( comp @ C @ A @ E @ H3 @ F ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Fun ) ) ).
% 0.25/0.46
% 0.25/0.46 % map_fun.compositionality
% 0.25/0.46 thf(fact_1967_map__fun_Ocomp,axiom,
% 0.25/0.46 ! [E: $tType,C: $tType,A: $tType,F5: $tType,D: $tType,B: $tType,F: E > C,G: D > F5,H3: C > A,I: B > D] :
% 0.25/0.46 ( ( comp @ ( C > D ) @ ( E > F5 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F5 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H3 @ I ) )
% 0.25/0.46 = ( map_fun @ E @ A @ B @ F5 @ ( comp @ C @ A @ E @ H3 @ F ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % map_fun.comp
% 0.25/0.46 thf(fact_1968_translation__subtract__Compl,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ab_group_add @ A )
% 0.25/0.46 => ! [A3: A,T2: set @ A] :
% 0.25/0.46 ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.46 @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 0.25/0.46 = ( uminus_uminus @ ( set @ A )
% 0.25/0.46 @ ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.46 @ T2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % translation_subtract_Compl
% 0.25/0.46 thf(fact_1969_nat__minus__add__max,axiom,
% 0.25/0.46 ! [N: nat,M2: nat] :
% 0.25/0.46 ( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M2 ) @ M2 )
% 0.25/0.46 = ( ord_max @ nat @ N @ M2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_minus_add_max
% 0.25/0.46 thf(fact_1970_translation__diff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ab_group_add @ A )
% 0.25/0.46 => ! [A3: A,S: set @ A,T2: set @ A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
% 0.25/0.46 = ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % translation_diff
% 0.25/0.46 thf(fact_1971_translation__subtract__diff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ab_group_add @ A )
% 0.25/0.46 => ! [A3: A,S: set @ A,T2: set @ A] :
% 0.25/0.46 ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.46 @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
% 0.25/0.46 = ( minus_minus @ ( set @ A )
% 0.25/0.46 @ ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.46 @ S )
% 0.25/0.46 @ ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.46 @ T2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % translation_subtract_diff
% 0.25/0.46 thf(fact_1972_take__bit__drop__bit,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [M2: nat,N: nat,A3: A] :
% 0.25/0.46 ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
% 0.25/0.46 = ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % take_bit_drop_bit
% 0.25/0.46 thf(fact_1973_bit__drop__bit__eq,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [N: nat,A3: A] :
% 0.25/0.46 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
% 0.25/0.46 = ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit_drop_bit_eq
% 0.25/0.46 thf(fact_1974_div__push__bit__of__1__eq__drop__bit,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [A3: A,N: nat] :
% 0.25/0.46 ( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
% 0.25/0.46 = ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % div_push_bit_of_1_eq_drop_bit
% 0.25/0.46 thf(fact_1975_bit__iff__and__drop__bit__eq__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ( ( bit_se5641148757651400278ts_bit @ A )
% 0.25/0.46 = ( ^ [A7: A,N5: nat] :
% 0.25/0.46 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N5 @ A7 ) @ ( one_one @ A ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit_iff_and_drop_bit_eq_1
% 0.25/0.46 thf(fact_1976_bits__ident,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_se359711467146920520ations @ A )
% 0.25/0.46 => ! [N: nat,A3: A] :
% 0.25/0.46 ( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 0.25/0.46 = A3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % bits_ident
% 0.25/0.46 thf(fact_1977_slice__eq__mask,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.46 => ! [N: nat,M2: nat,A3: A] :
% 0.25/0.46 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) )
% 0.25/0.46 = ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M2 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % slice_eq_mask
% 0.25/0.46 thf(fact_1978_image__add__0,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comm_monoid_add @ A )
% 0.25/0.46 => ! [S2: set @ A] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
% 0.25/0.46 = S2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_0
% 0.25/0.46 thf(fact_1979_times__int__def,axiom,
% 0.25/0.46 ( ( times_times @ int )
% 0.25/0.46 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.46 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % times_int_def
% 0.25/0.46 thf(fact_1980_minus__int__def,axiom,
% 0.25/0.46 ( ( minus_minus @ int )
% 0.25/0.46 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.46 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_int_def
% 0.25/0.46 thf(fact_1981_plus__int__def,axiom,
% 0.25/0.46 ( ( plus_plus @ int )
% 0.25/0.46 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.46 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % plus_int_def
% 0.25/0.46 thf(fact_1982_mlex__prod__def,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( mlex_prod @ A )
% 0.25/0.46 = ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
% 0.25/0.46 ( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
% 0.25/0.46 @ ^ [X3: A] : ( product_Pair @ nat @ A @ ( F2 @ X3 ) @ X3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mlex_prod_def
% 0.25/0.46 thf(fact_1983_rat__minus__code,axiom,
% 0.25/0.46 ! [P5: rat,Q3: rat] :
% 0.25/0.46 ( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q3 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int,C3: int] :
% 0.25/0.46 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
% 0.25/0.46 @ ( quotient_of @ Q3 ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_minus_code
% 0.25/0.46 thf(fact_1984_rat__plus__code,axiom,
% 0.25/0.46 ! [P5: rat,Q3: rat] :
% 0.25/0.46 ( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q3 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int,C3: int] :
% 0.25/0.46 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
% 0.25/0.46 @ ( quotient_of @ Q3 ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_plus_code
% 0.25/0.46 thf(fact_1985_image__Suc__atLeastAtMost,axiom,
% 0.25/0.46 ! [I: nat,J: nat] :
% 0.25/0.46 ( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_Suc_atLeastAtMost
% 0.25/0.46 thf(fact_1986_image__Suc__atLeastLessThan,axiom,
% 0.25/0.46 ! [I: nat,J: nat] :
% 0.25/0.46 ( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_Suc_atLeastLessThan
% 0.25/0.46 thf(fact_1987_in__inv__image,axiom,
% 0.25/0.46 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.25/0.46 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
% 0.25/0.46 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).
% 0.25/0.46
% 0.25/0.46 % in_inv_image
% 0.25/0.46 thf(fact_1988_pair__imageI,axiom,
% 0.25/0.46 ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 0.25/0.46 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A4 )
% 0.25/0.46 => ( member @ C @ ( F @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pair_imageI
% 0.25/0.46 thf(fact_1989_int__of__integer__max,axiom,
% 0.25/0.46 ! [K: code_integer,L: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
% 0.25/0.46 = ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_max
% 0.25/0.46 thf(fact_1990_rat__one__code,axiom,
% 0.25/0.46 ( ( quotient_of @ ( one_one @ rat ) )
% 0.25/0.46 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_one_code
% 0.25/0.46 thf(fact_1991_of__nat__of__integer,axiom,
% 0.25/0.46 ! [K: code_integer] :
% 0.25/0.46 ( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
% 0.25/0.46 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 0.25/0.46
% 0.25/0.46 % of_nat_of_integer
% 0.25/0.46 thf(fact_1992_rat__zero__code,axiom,
% 0.25/0.46 ( ( quotient_of @ ( zero_zero @ rat ) )
% 0.25/0.46 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_zero_code
% 0.25/0.46 thf(fact_1993_quotient__of__number_I3_J,axiom,
% 0.25/0.46 ! [K: num] :
% 0.25/0.46 ( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
% 0.25/0.46 = ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % quotient_of_number(3)
% 0.25/0.46 thf(fact_1994_quotient__of__number_I4_J,axiom,
% 0.25/0.46 ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 0.25/0.46 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % quotient_of_number(4)
% 0.25/0.46 thf(fact_1995_quotient__of__number_I5_J,axiom,
% 0.25/0.46 ! [K: num] :
% 0.25/0.46 ( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
% 0.25/0.46 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % quotient_of_number(5)
% 0.25/0.46 thf(fact_1996_diff__rat__def,axiom,
% 0.25/0.46 ( ( minus_minus @ rat )
% 0.25/0.46 = ( ^ [Q5: rat,R3: rat] : ( plus_plus @ rat @ Q5 @ ( uminus_uminus @ rat @ R3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % diff_rat_def
% 0.25/0.46 thf(fact_1997_quotient__of__div,axiom,
% 0.25/0.46 ! [R: rat,N: int,D3: int] :
% 0.25/0.46 ( ( ( quotient_of @ R )
% 0.25/0.46 = ( product_Pair @ int @ int @ N @ D3 ) )
% 0.25/0.46 => ( R
% 0.25/0.46 = ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % quotient_of_div
% 0.25/0.46 thf(fact_1998_zero__notin__Suc__image,axiom,
% 0.25/0.46 ! [A4: set @ nat] :
% 0.25/0.46 ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % zero_notin_Suc_image
% 0.25/0.46 thf(fact_1999_None__notin__image__Some,axiom,
% 0.25/0.46 ! [A: $tType,A4: set @ A] :
% 0.25/0.46 ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% 0.25/0.46
% 0.25/0.46 % None_notin_image_Some
% 0.25/0.46 thf(fact_2000_plus__integer__def,axiom,
% 0.25/0.46 ( ( plus_plus @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( plus_plus @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % plus_integer_def
% 0.25/0.46 thf(fact_2001_times__integer__def,axiom,
% 0.25/0.46 ( ( times_times @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( times_times @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % times_integer_def
% 0.25/0.46 thf(fact_2002_minus__integer__def,axiom,
% 0.25/0.46 ( ( minus_minus @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( minus_minus @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_integer_def
% 0.25/0.46 thf(fact_2003_divide__integer__def,axiom,
% 0.25/0.46 ( ( divide_divide @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( divide_divide @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % divide_integer_def
% 0.25/0.46 thf(fact_2004_modulo__integer__def,axiom,
% 0.25/0.46 ( ( modulo_modulo @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( modulo_modulo @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % modulo_integer_def
% 0.25/0.46 thf(fact_2005_and__integer__def,axiom,
% 0.25/0.46 ( ( bit_se5824344872417868541ns_and @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344872417868541ns_and @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_integer_def
% 0.25/0.46 thf(fact_2006_or__integer__def,axiom,
% 0.25/0.46 ( ( bit_se1065995026697491101ons_or @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se1065995026697491101ons_or @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % or_integer_def
% 0.25/0.46 thf(fact_2007_xor__integer__def,axiom,
% 0.25/0.46 ( ( bit_se5824344971392196577ns_xor @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344971392196577ns_xor @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % xor_integer_def
% 0.25/0.46 thf(fact_2008_image__int__atLeastAtMost,axiom,
% 0.25/0.46 ! [A3: nat,B2: nat] :
% 0.25/0.46 ( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_int_atLeastAtMost
% 0.25/0.46 thf(fact_2009_image__int__atLeastLessThan,axiom,
% 0.25/0.46 ! [A3: nat,B2: nat] :
% 0.25/0.46 ( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_int_atLeastLessThan
% 0.25/0.46 thf(fact_2010_quotient__of__denom__pos,axiom,
% 0.25/0.46 ! [R: rat,P5: int,Q3: int] :
% 0.25/0.46 ( ( ( quotient_of @ R )
% 0.25/0.46 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.25/0.46 => ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % quotient_of_denom_pos
% 0.25/0.46 thf(fact_2011_image__Suc__atMost,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_Suc_atMost
% 0.25/0.46 thf(fact_2012_image__Suc__lessThan,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_Suc_lessThan
% 0.25/0.46 thf(fact_2013_inv__image__def,axiom,
% 0.25/0.46 ! [A: $tType,B: $tType] :
% 0.25/0.46 ( ( inv_image @ B @ A )
% 0.25/0.46 = ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
% 0.25/0.46 ( collect @ ( product_prod @ A @ A )
% 0.25/0.46 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.46 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inv_image_def
% 0.25/0.46 thf(fact_2014_rat__uminus__code,axiom,
% 0.25/0.46 ! [P5: rat] :
% 0.25/0.46 ( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A7 ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_uminus_code
% 0.25/0.46 thf(fact_2015_rat__abs__code,axiom,
% 0.25/0.46 ! [P5: rat] :
% 0.25/0.46 ( ( quotient_of @ ( abs_abs @ rat @ P5 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A7 ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_abs_code
% 0.25/0.46 thf(fact_2016_uminus__integer__def,axiom,
% 0.25/0.46 ( ( uminus_uminus @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % uminus_integer_def
% 0.25/0.46 thf(fact_2017_abs__integer__def,axiom,
% 0.25/0.46 ( ( abs_abs @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % abs_integer_def
% 0.25/0.46 thf(fact_2018_image__add__int__atLeastLessThan,axiom,
% 0.25/0.46 ! [L: int,U: int] :
% 0.25/0.46 ( ( image2 @ int @ int
% 0.25/0.46 @ ^ [X3: int] : ( plus_plus @ int @ X3 @ L )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_int_atLeastLessThan
% 0.25/0.46 thf(fact_2019_sgn__integer__def,axiom,
% 0.25/0.46 ( ( sgn_sgn @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % sgn_integer_def
% 0.25/0.46 thf(fact_2020_not__integer__def,axiom,
% 0.25/0.46 ( ( bit_ri4277139882892585799ns_not @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_integer_def
% 0.25/0.46 thf(fact_2021_image__add__integer__atLeastLessThan,axiom,
% 0.25/0.46 ! [L: code_integer,U: code_integer] :
% 0.25/0.46 ( ( image2 @ code_integer @ code_integer
% 0.25/0.46 @ ^ [X3: code_integer] : ( plus_plus @ code_integer @ X3 @ L )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_add_integer_atLeastLessThan
% 0.25/0.46 thf(fact_2022_rat__times__code,axiom,
% 0.25/0.46 ! [P5: rat,Q3: rat] :
% 0.25/0.46 ( ( quotient_of @ ( times_times @ rat @ P5 @ Q3 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int,C3: int] :
% 0.25/0.46 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ B5 ) @ ( times_times @ int @ C3 @ D5 ) ) )
% 0.25/0.46 @ ( quotient_of @ Q3 ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_times_code
% 0.25/0.46 thf(fact_2023_rat__divide__code,axiom,
% 0.25/0.46 ! [P5: rat,Q3: rat] :
% 0.25/0.46 ( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q3 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int,C3: int] :
% 0.25/0.46 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ C3 @ B5 ) ) )
% 0.25/0.46 @ ( quotient_of @ Q3 ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_divide_code
% 0.25/0.46 thf(fact_2024_image__atLeastZeroLessThan__int,axiom,
% 0.25/0.46 ! [U: int] :
% 0.25/0.46 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
% 0.25/0.46 => ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
% 0.25/0.46 = ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_atLeastZeroLessThan_int
% 0.25/0.46 thf(fact_2025_quotient__of__int,axiom,
% 0.25/0.46 ! [A3: int] :
% 0.25/0.46 ( ( quotient_of @ ( of_int @ A3 ) )
% 0.25/0.46 = ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % quotient_of_int
% 0.25/0.46 thf(fact_2026_rp__inv__image__def,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( fun_rp_inv_image @ A @ B )
% 0.25/0.46 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 0.25/0.46 @ ^ [R6: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F2 ) @ ( inv_image @ A @ B @ S5 @ F2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rp_inv_image_def
% 0.25/0.46 thf(fact_2027_rat__inverse__code,axiom,
% 0.25/0.46 ! [P5: rat] :
% 0.25/0.46 ( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
% 0.25/0.46 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 0.25/0.46 @ ^ [A7: int,B5: int] :
% 0.25/0.46 ( if @ ( product_prod @ int @ int )
% 0.25/0.46 @ ( A7
% 0.25/0.46 = ( zero_zero @ int ) )
% 0.25/0.46 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.46 @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A7 ) @ B5 ) @ ( abs_abs @ int @ A7 ) ) )
% 0.25/0.46 @ ( quotient_of @ P5 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rat_inverse_code
% 0.25/0.46 thf(fact_2028_diff__numeral__special_I8_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num] :
% 0.25/0.46 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % diff_numeral_special(8)
% 0.25/0.46 thf(fact_2029_diff__numeral__special_I7_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [N: num] :
% 0.25/0.46 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % diff_numeral_special(7)
% 0.25/0.46 thf(fact_2030_minus__sub__one__diff__one,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [M2: num] :
% 0.25/0.46 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
% 0.25/0.46 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_sub_one_diff_one
% 0.25/0.46 thf(fact_2031_add__neg__numeral__special_I4_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [N: num] :
% 0.25/0.46 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % add_neg_numeral_special(4)
% 0.25/0.46 thf(fact_2032_int__of__integer__sub,axiom,
% 0.25/0.46 ! [K: num,L: num] :
% 0.25/0.46 ( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
% 0.25/0.46 = ( neg_numeral_sub @ int @ K @ L ) ) ).
% 0.25/0.46
% 0.25/0.46 % int_of_integer_sub
% 0.25/0.46 thf(fact_2033_inverse__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_1
% 0.25/0.46 thf(fact_2034_inverse__eq__1__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( field @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( ( inverse_inverse @ A @ X )
% 0.25/0.46 = ( one_one @ A ) )
% 0.25/0.46 = ( X
% 0.25/0.46 = ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_eq_1_iff
% 0.25/0.46 thf(fact_2035_right__inverse,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( A3
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % right_inverse
% 0.25/0.46 thf(fact_2036_left__inverse,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( A3
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 0.25/0.46 = ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % left_inverse
% 0.25/0.46 thf(fact_2037_inverse__eq__divide__numeral,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [W: num] :
% 0.25/0.46 ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
% 0.25/0.46 = ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_eq_divide_numeral
% 0.25/0.46 thf(fact_2038_add__neg__numeral__simps_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num,N: num] :
% 0.25/0.46 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % add_neg_numeral_simps(2)
% 0.25/0.46 thf(fact_2039_add__neg__numeral__simps_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num,N: num] :
% 0.25/0.46 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % add_neg_numeral_simps(1)
% 0.25/0.46 thf(fact_2040_semiring__norm_I166_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [V: num,W: num,Y: A] :
% 0.25/0.46 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
% 0.25/0.46 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W ) @ Y ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % semiring_norm(166)
% 0.25/0.46 thf(fact_2041_semiring__norm_I167_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [V: num,W: num,Y: A] :
% 0.25/0.46 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
% 0.25/0.46 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V ) @ Y ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % semiring_norm(167)
% 0.25/0.46 thf(fact_2042_inverse__eq__divide__neg__numeral,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [W: num] :
% 0.25/0.46 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
% 0.25/0.46 = ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_eq_divide_neg_numeral
% 0.25/0.46 thf(fact_2043_diff__numeral__special_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num] :
% 0.25/0.46 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % diff_numeral_special(2)
% 0.25/0.46 thf(fact_2044_diff__numeral__special_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [N: num] :
% 0.25/0.46 ( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % diff_numeral_special(1)
% 0.25/0.46 thf(fact_2045_add__neg__numeral__special_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num] :
% 0.25/0.46 ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % add_neg_numeral_special(1)
% 0.25/0.46 thf(fact_2046_add__neg__numeral__special_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num] :
% 0.25/0.46 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % add_neg_numeral_special(2)
% 0.25/0.46 thf(fact_2047_add__neg__numeral__special_I3_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( neg_numeral @ A )
% 0.25/0.46 => ! [M2: num] :
% 0.25/0.46 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 0.25/0.46 = ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % add_neg_numeral_special(3)
% 0.25/0.46 thf(fact_2048_inverse__unique,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ( times_times @ A @ A3 @ B2 )
% 0.25/0.46 = ( one_one @ A ) )
% 0.25/0.46 => ( ( inverse_inverse @ A @ A3 )
% 0.25/0.46 = B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_unique
% 0.25/0.46 thf(fact_2049_inverse__eq__divide,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ( ( inverse_inverse @ A )
% 0.25/0.46 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_eq_divide
% 0.25/0.46 thf(fact_2050_prod__inversef,axiom,
% 0.25/0.46 ! [A: $tType,B: $tType] :
% 0.25/0.46 ( ( field @ A )
% 0.25/0.46 => ! [F: B > A,A4: set @ B] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ B @ A @ ( comp @ A @ A @ B @ ( inverse_inverse @ A ) @ F ) @ A4 )
% 0.25/0.46 = ( inverse_inverse @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod_inversef
% 0.25/0.46 thf(fact_2051_inverse__le__1__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 0.25/0.46 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.25/0.46 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_le_1_iff
% 0.25/0.46 thf(fact_2052_one__less__inverse__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 0.25/0.46 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.25/0.46 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_less_inverse_iff
% 0.25/0.46 thf(fact_2053_one__less__inverse,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.25/0.46 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.46 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_less_inverse
% 0.25/0.46 thf(fact_2054_division__ring__inverse__add,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( A3
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( B2
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.25/0.46 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % division_ring_inverse_add
% 0.25/0.46 thf(fact_2055_inverse__add,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( field @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( A3
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( B2
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 0.25/0.46 = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_add
% 0.25/0.46 thf(fact_2056_field__class_Ofield__inverse,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( field @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( A3
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 0.25/0.46 = ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % field_class.field_inverse
% 0.25/0.46 thf(fact_2057_nonzero__inverse__eq__divide,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( division_ring @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( A3
% 0.25/0.46 != ( zero_zero @ A ) )
% 0.25/0.46 => ( ( inverse_inverse @ A @ A3 )
% 0.25/0.46 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % nonzero_inverse_eq_divide
% 0.25/0.46 thf(fact_2058_one__le__inverse__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 0.25/0.46 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 0.25/0.46 & ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_le_inverse_iff
% 0.25/0.46 thf(fact_2059_inverse__less__1__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 0.25/0.46 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 0.25/0.46 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % inverse_less_1_iff
% 0.25/0.46 thf(fact_2060_one__le__inverse,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.25/0.46 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.46 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % one_le_inverse
% 0.25/0.46 thf(fact_2061_Frct__code__post_I5_J,axiom,
% 0.25/0.46 ! [K: num] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
% 0.25/0.46 = ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(5)
% 0.25/0.46 thf(fact_2062_Code__Numeral_Odup__def,axiom,
% 0.25/0.46 ( code_dup
% 0.25/0.46 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
% 0.25/0.46 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.dup_def
% 0.25/0.46 thf(fact_2063_Frct__code__post_I6_J,axiom,
% 0.25/0.46 ! [K: num,L: num] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
% 0.25/0.46 = ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(6)
% 0.25/0.46 thf(fact_2064_Frct__code__post_I4_J,axiom,
% 0.25/0.46 ! [K: num] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
% 0.25/0.46 = ( numeral_numeral @ rat @ K ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(4)
% 0.25/0.46 thf(fact_2065_eq__numeral__iff__iszero_I8_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [Y: num] :
% 0.25/0.46 ( ( ( one_one @ A )
% 0.25/0.46 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
% 0.25/0.46 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_numeral_iff_iszero(8)
% 0.25/0.46 thf(fact_2066_eq__numeral__iff__iszero_I7_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 0.25/0.46 = ( one_one @ A ) )
% 0.25/0.46 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_numeral_iff_iszero(7)
% 0.25/0.46 thf(fact_2067_Inf_OINF__image,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
% 0.25/0.46 ( ( Inf @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 0.25/0.46 = ( Inf @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Inf.INF_image
% 0.25/0.46 thf(fact_2068_not__iszero__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_iszero_1
% 0.25/0.46 thf(fact_2069_Code__Numeral_Odup__code_I1_J,axiom,
% 0.25/0.46 ( ( code_dup @ ( zero_zero @ code_integer ) )
% 0.25/0.46 = ( zero_zero @ code_integer ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.dup_code(1)
% 0.25/0.46 thf(fact_2070_not__iszero__neg__1,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_iszero_neg_1
% 0.25/0.46 thf(fact_2071_dup_Orep__eq,axiom,
% 0.25/0.46 ! [X: code_integer] :
% 0.25/0.46 ( ( code_int_of_integer @ ( code_dup @ X ) )
% 0.25/0.46 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % dup.rep_eq
% 0.25/0.46 thf(fact_2072_dup_Oabs__eq,axiom,
% 0.25/0.46 ! [X: int] :
% 0.25/0.46 ( ( code_dup @ ( code_integer_of_int @ X ) )
% 0.25/0.46 = ( code_integer_of_int @ ( plus_plus @ int @ X @ X ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % dup.abs_eq
% 0.25/0.46 thf(fact_2073_eq__numeral__iff__iszero_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [X: num,Y: num] :
% 0.25/0.46 ( ( ( numeral_numeral @ A @ X )
% 0.25/0.46 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
% 0.25/0.46 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_numeral_iff_iszero(2)
% 0.25/0.46 thf(fact_2074_eq__numeral__iff__iszero_I3_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [X: num,Y: num] :
% 0.25/0.46 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 0.25/0.46 = ( numeral_numeral @ A @ Y ) )
% 0.25/0.46 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_numeral_iff_iszero(3)
% 0.25/0.46 thf(fact_2075_Frct__code__post_I2_J,axiom,
% 0.25/0.46 ! [A3: int] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
% 0.25/0.46 = ( zero_zero @ rat ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(2)
% 0.25/0.46 thf(fact_2076_Frct__code__post_I1_J,axiom,
% 0.25/0.46 ! [A3: int] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
% 0.25/0.46 = ( zero_zero @ rat ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(1)
% 0.25/0.46 thf(fact_2077_Frct__code__post_I3_J,axiom,
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
% 0.25/0.46 = ( one_one @ rat ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(3)
% 0.25/0.46 thf(fact_2078_eq__numeral__iff__iszero_I6_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [Y: num] :
% 0.25/0.46 ( ( ( one_one @ A )
% 0.25/0.46 = ( numeral_numeral @ A @ Y ) )
% 0.25/0.46 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_numeral_iff_iszero(6)
% 0.25/0.46 thf(fact_2079_eq__numeral__iff__iszero_I5_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ring_1 @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( ( numeral_numeral @ A @ X )
% 0.25/0.46 = ( one_one @ A ) )
% 0.25/0.46 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_numeral_iff_iszero(5)
% 0.25/0.46 thf(fact_2080_Frct__code__post_I7_J,axiom,
% 0.25/0.46 ! [A3: int,B2: int] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
% 0.25/0.46 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(7)
% 0.25/0.46 thf(fact_2081_Frct__code__post_I8_J,axiom,
% 0.25/0.46 ! [A3: int,B2: int] :
% 0.25/0.46 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
% 0.25/0.46 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Frct_code_post(8)
% 0.25/0.46 thf(fact_2082_Sup_OSUP__image,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
% 0.25/0.46 ( ( Sup @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 0.25/0.46 = ( Sup @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Sup.SUP_image
% 0.25/0.46 thf(fact_2083_Code__Numeral_Osub__code_I9_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( code_sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 0.25/0.46 = ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.sub_code(9)
% 0.25/0.46 thf(fact_2084_Code__Numeral_Osub__code_I8_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( code_sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 0.25/0.46 = ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.sub_code(8)
% 0.25/0.46 thf(fact_2085_less__eq__int__def,axiom,
% 0.25/0.46 ( ( ord_less_eq @ int )
% 0.25/0.46 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.46 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_eq_int_def
% 0.25/0.46 thf(fact_2086_less__int__def,axiom,
% 0.25/0.46 ( ( ord_less @ int )
% 0.25/0.46 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
% 0.25/0.46 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.46 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.46 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.46 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_int_def
% 0.25/0.46 thf(fact_2087_image__affinity__atLeastAtMost__div__diff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [A3: A,B2: A,M2: A,C2: A] :
% 0.25/0.46 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_affinity_atLeastAtMost_div_diff
% 0.25/0.46 thf(fact_2088_image__affinity__atLeastAtMost__div,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [A3: A,B2: A,M2: A,C2: A] :
% 0.25/0.46 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_affinity_atLeastAtMost_div
% 0.25/0.46 thf(fact_2089_image__affinity__atLeastAtMost__diff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [A3: A,B2: A,M2: A,C2: A] :
% 0.25/0.46 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_affinity_atLeastAtMost_diff
% 0.25/0.46 thf(fact_2090_id__apply,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( id @ A )
% 0.25/0.46 = ( ^ [X3: A] : X3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % id_apply
% 0.25/0.46 thf(fact_2091_bijective__Empty,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bijective_Empty
% 0.25/0.46 thf(fact_2092_image__id,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( image2 @ A @ A @ ( id @ A ) )
% 0.25/0.46 = ( id @ ( set @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_id
% 0.25/0.46 thf(fact_2093_fun_Omap__id,axiom,
% 0.25/0.46 ! [A: $tType,D: $tType,T2: D > A] :
% 0.25/0.46 ( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
% 0.25/0.46 = T2 ) ).
% 0.25/0.46
% 0.25/0.46 % fun.map_id
% 0.25/0.46 thf(fact_2094_id__comp,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,G: A > B] :
% 0.25/0.46 ( ( comp @ B @ B @ A @ ( id @ B ) @ G )
% 0.25/0.46 = G ) ).
% 0.25/0.46
% 0.25/0.46 % id_comp
% 0.25/0.46 thf(fact_2095_comp__id,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.46 ( ( comp @ A @ B @ A @ F @ ( id @ A ) )
% 0.25/0.46 = F ) ).
% 0.25/0.46
% 0.25/0.46 % comp_id
% 0.25/0.46 thf(fact_2096_lessThan__0,axiom,
% 0.25/0.46 ( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
% 0.25/0.46 = ( bot_bot @ ( set @ nat ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % lessThan_0
% 0.25/0.46 thf(fact_2097_apsnd__id,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
% 0.25/0.46 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % apsnd_id
% 0.25/0.46 thf(fact_2098_map__mmupd__empty,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,M2: A > ( option @ B ),V: B] :
% 0.25/0.46 ( ( map_mmupd @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) @ V )
% 0.25/0.46 = M2 ) ).
% 0.25/0.46
% 0.25/0.46 % map_mmupd_empty
% 0.25/0.46 thf(fact_2099_set__to__map__empty,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.46 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_to_map_empty
% 0.25/0.46 thf(fact_2100_atLeastatMost__empty__iff2,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( preorder @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastatMost_empty_iff2
% 0.25/0.46 thf(fact_2101_atLeastatMost__empty__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( preorder @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastatMost_empty_iff
% 0.25/0.46 thf(fact_2102_atLeastatMost__empty,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [B2: A,A3: A] :
% 0.25/0.46 ( ( ord_less @ A @ B2 @ A3 )
% 0.25/0.46 => ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastatMost_empty
% 0.25/0.46 thf(fact_2103_atLeastLessThan__empty,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [B2: A,A3: A] :
% 0.25/0.46 ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.25/0.46 => ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThan_empty
% 0.25/0.46 thf(fact_2104_atLeastLessThan__empty__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( preorder @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThan_empty_iff
% 0.25/0.46 thf(fact_2105_atLeastLessThan__empty__iff2,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( preorder @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThan_empty_iff2
% 0.25/0.46 thf(fact_2106_prod_Oempty,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( comm_monoid_mult @ A )
% 0.25/0.46 => ! [G: B > A] :
% 0.25/0.46 ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % prod.empty
% 0.25/0.46 thf(fact_2107_group__add__class_Ominus__comp__minus,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( group_add @ A )
% 0.25/0.46 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 0.25/0.46 = ( id @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % group_add_class.minus_comp_minus
% 0.25/0.46 thf(fact_2108_boolean__algebra__class_Ominus__comp__minus,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( boolea8198339166811842893lgebra @ A )
% 0.25/0.46 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 0.25/0.46 = ( id @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % boolean_algebra_class.minus_comp_minus
% 0.25/0.46 thf(fact_2109_rel__of__empty,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
% 0.25/0.46 ( ( rel_of @ A @ B
% 0.25/0.46 @ ^ [X3: A] : ( none @ B )
% 0.25/0.46 @ P )
% 0.25/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % rel_of_empty
% 0.25/0.46 thf(fact_2110_comp__the__Some,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
% 0.25/0.46 = ( id @ A ) ) ).
% 0.25/0.46
% 0.25/0.46 % comp_the_Some
% 0.25/0.46 thf(fact_2111_fun_Omap__id0,axiom,
% 0.25/0.46 ! [A: $tType,D: $tType] :
% 0.25/0.46 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 0.25/0.46 = ( id @ ( D > A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % fun.map_id0
% 0.25/0.46 thf(fact_2112_Iio__eq__empty__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ( linorder @ A )
% 0.25/0.46 & ( order_bot @ A ) )
% 0.25/0.46 => ! [N: A] :
% 0.25/0.46 ( ( ( set_ord_lessThan @ A @ N )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 = ( N
% 0.25/0.46 = ( bot_bot @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Iio_eq_empty_iff
% 0.25/0.46 thf(fact_2113_option_Omap__id0,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( map_option @ A @ A @ ( id @ A ) )
% 0.25/0.46 = ( id @ ( option @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % option.map_id0
% 0.25/0.46 thf(fact_2114_set__notEmptyE,axiom,
% 0.25/0.46 ! [A: $tType,S2: set @ A] :
% 0.25/0.46 ( ( S2
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ~ ! [X2: A] :
% 0.25/0.46 ~ ( member @ A @ X2 @ S2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_notEmptyE
% 0.25/0.46 thf(fact_2115_memb__imp__not__empty,axiom,
% 0.25/0.46 ! [A: $tType,X: A,S2: set @ A] :
% 0.25/0.46 ( ( member @ A @ X @ S2 )
% 0.25/0.46 => ( S2
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % memb_imp_not_empty
% 0.25/0.46 thf(fact_2116_id__def,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( id @ A )
% 0.25/0.46 = ( ^ [X3: A] : X3 ) ) ).
% 0.25/0.46
% 0.25/0.46 % id_def
% 0.25/0.46 thf(fact_2117_eq__id__iff,axiom,
% 0.25/0.46 ! [A: $tType,F: A > A] :
% 0.25/0.46 ( ( ! [X3: A] :
% 0.25/0.46 ( ( F @ X3 )
% 0.25/0.46 = X3 ) )
% 0.25/0.46 = ( F
% 0.25/0.46 = ( id @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % eq_id_iff
% 0.25/0.46 thf(fact_2118_comp__eq__id__dest,axiom,
% 0.25/0.46 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
% 0.25/0.46 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 0.25/0.46 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 0.25/0.46 => ( ( A3 @ ( B2 @ V ) )
% 0.25/0.46 = ( C2 @ V ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % comp_eq_id_dest
% 0.25/0.46 thf(fact_2119_pointfree__idE,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
% 0.25/0.46 ( ( ( comp @ B @ A @ A @ F @ G )
% 0.25/0.46 = ( id @ A ) )
% 0.25/0.46 => ( ( F @ ( G @ X ) )
% 0.25/0.46 = X ) ) ).
% 0.25/0.46
% 0.25/0.46 % pointfree_idE
% 0.25/0.46 thf(fact_2120_option_Omap__id,axiom,
% 0.25/0.46 ! [A: $tType,T2: option @ A] :
% 0.25/0.46 ( ( map_option @ A @ A @ ( id @ A ) @ T2 )
% 0.25/0.46 = T2 ) ).
% 0.25/0.46
% 0.25/0.46 % option.map_id
% 0.25/0.46 thf(fact_2121_not__empty__eq__Iic__eq__empty,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( preorder @ A )
% 0.25/0.46 => ! [H3: A] :
% 0.25/0.46 ( ( bot_bot @ ( set @ A ) )
% 0.25/0.46 != ( set_ord_atMost @ A @ H3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % not_empty_eq_Iic_eq_empty
% 0.25/0.46 thf(fact_2122_lessThan__non__empty,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( no_bot @ A )
% 0.25/0.46 => ! [X: A] :
% 0.25/0.46 ( ( set_ord_lessThan @ A @ X )
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % lessThan_non_empty
% 0.25/0.46 thf(fact_2123_subset__minus__empty,axiom,
% 0.25/0.46 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.46 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.46 => ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % subset_minus_empty
% 0.25/0.46 thf(fact_2124_atLeastLessThan0,axiom,
% 0.25/0.46 ! [M2: nat] :
% 0.25/0.46 ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
% 0.25/0.46 = ( bot_bot @ ( set @ nat ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThan0
% 0.25/0.46 thf(fact_2125_lessThan__empty__iff,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( ( set_ord_lessThan @ nat @ N )
% 0.25/0.46 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.46 = ( N
% 0.25/0.46 = ( zero_zero @ nat ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % lessThan_empty_iff
% 0.25/0.46 thf(fact_2126_set__to__map__empty__iff_I1_J,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.46 ( ( ( set_to_map @ A @ B @ S2 )
% 0.25/0.46 = ( ^ [X3: A] : ( none @ B ) ) )
% 0.25/0.46 = ( S2
% 0.25/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_to_map_empty_iff(1)
% 0.25/0.46 thf(fact_2127_Code__Numeral_Osub__code_I1_J,axiom,
% 0.25/0.46 ( ( code_sub @ one2 @ one2 )
% 0.25/0.46 = ( zero_zero @ code_integer ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.sub_code(1)
% 0.25/0.46 thf(fact_2128_set__to__map__empty__iff_I2_J,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.46 ( ( ( ^ [X3: A] : ( none @ B ) )
% 0.25/0.46 = ( set_to_map @ A @ B @ S2 ) )
% 0.25/0.46 = ( S2
% 0.25/0.46 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_to_map_empty_iff(2)
% 0.25/0.46 thf(fact_2129_less__integer__def,axiom,
% 0.25/0.46 ( ( ord_less @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_integer_def
% 0.25/0.46 thf(fact_2130_less__eq__integer__def,axiom,
% 0.25/0.46 ( ( ord_less_eq @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less_eq @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % less_eq_integer_def
% 0.25/0.46 thf(fact_2131_Code__Numeral_Osub__code_I6_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( code_sub @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
% 0.25/0.46 = ( code_dup @ ( code_sub @ M2 @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.sub_code(6)
% 0.25/0.46 thf(fact_2132_Code__Numeral_Osub__code_I7_J,axiom,
% 0.25/0.46 ! [M2: num,N: num] :
% 0.25/0.46 ( ( code_sub @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 0.25/0.46 = ( code_dup @ ( code_sub @ M2 @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.sub_code(7)
% 0.25/0.46 thf(fact_2133_sub_Orep__eq,axiom,
% 0.25/0.46 ! [X: num,Xa: num] :
% 0.25/0.46 ( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
% 0.25/0.46 = ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % sub.rep_eq
% 0.25/0.46 thf(fact_2134_sub_Oabs__eq,axiom,
% 0.25/0.46 ( code_sub
% 0.25/0.46 = ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X3 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % sub.abs_eq
% 0.25/0.46 thf(fact_2135_image__mult__atLeastAtMost__if,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [C2: A,X: A,Y: A] :
% 0.25/0.46 ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.25/0.46 => ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.25/0.46 => ( ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.46 => ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.25/0.46 => ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_mult_atLeastAtMost_if
% 0.25/0.46 thf(fact_2136_image__mult__atLeastAtMost__if_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [X: A,Y: A,C2: A] :
% 0.25/0.46 ( ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.46 => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less_eq @ A @ X @ Y )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_mult_atLeastAtMost_if'
% 0.25/0.46 thf(fact_2137_image__affinity__atLeastAtMost,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linordered_field @ A )
% 0.25/0.46 => ! [A3: A,B2: A,M2: A,C2: A] :
% 0.25/0.46 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.46 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.46 => ( ( image2 @ A @ A
% 0.25/0.46 @ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 0.25/0.46 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 0.25/0.46 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_affinity_atLeastAtMost
% 0.25/0.46 thf(fact_2138_and__int_Osimps,axiom,
% 0.25/0.46 ( ( bit_se5824344872417868541ns_and @ int )
% 0.25/0.46 = ( ^ [K4: int,L3: int] :
% 0.25/0.46 ( if @ int
% 0.25/0.46 @ ( ( member @ int @ K4 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ L3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 @ ( uminus_uminus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 0.25/0.46 @ ( plus_plus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_int.simps
% 0.25/0.46 thf(fact_2139_and__int_Oelims,axiom,
% 0.25/0.46 ! [X: int,Xa: int,Y: int] :
% 0.25/0.46 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
% 0.25/0.46 = Y )
% 0.25/0.46 => ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( Y
% 0.25/0.46 = ( uminus_uminus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
% 0.25/0.46 & ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( Y
% 0.25/0.46 = ( plus_plus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_int.elims
% 0.25/0.46 thf(fact_2140_and__int_Opelims,axiom,
% 0.25/0.46 ! [X: int,Xa: int,Y: int] :
% 0.25/0.46 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
% 0.25/0.46 = Y )
% 0.25/0.46 => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
% 0.25/0.46 => ~ ( ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( Y
% 0.25/0.46 = ( uminus_uminus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
% 0.25/0.46 & ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( Y
% 0.25/0.46 = ( plus_plus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
% 0.25/0.46 => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_int.pelims
% 0.25/0.46 thf(fact_2141_and__int_Opsimps,axiom,
% 0.25/0.46 ! [K: int,L: int] :
% 0.25/0.46 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
% 0.25/0.46 => ( ( ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 0.25/0.46 = ( uminus_uminus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
% 0.25/0.46 & ( ~ ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 0.25/0.46 = ( plus_plus @ int
% 0.25/0.46 @ ( zero_neq_one_of_bool @ int
% 0.25/0.46 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 0.25/0.46 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
% 0.25/0.46 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_int.psimps
% 0.25/0.46 thf(fact_2142_and__int_Opinduct,axiom,
% 0.25/0.46 ! [A0: int,A1: int,P: int > int > $o] :
% 0.25/0.46 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 0.25/0.46 => ( ! [K5: int,L5: int] :
% 0.25/0.46 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K5 @ L5 ) )
% 0.25/0.46 => ( ( ~ ( ( member @ int @ K5 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 0.25/0.46 & ( member @ int @ L5 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 0.25/0.46 => ( P @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 0.25/0.46 => ( P @ K5 @ L5 ) ) )
% 0.25/0.46 => ( P @ A0 @ A1 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % and_int.pinduct
% 0.25/0.46 thf(fact_2143_mask__mod__exp,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( euclid5411537665997757685th_nat @ A )
% 0.25/0.46 => ! [N: nat,M2: nat] :
% 0.25/0.46 ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 0.25/0.46 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mask_mod_exp
% 0.25/0.46 thf(fact_2144_pair__leqI2,axiom,
% 0.25/0.46 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.25/0.46 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.25/0.46 => ( ( ord_less_eq @ nat @ S @ T2 )
% 0.25/0.46 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % pair_leqI2
% 0.25/0.46 thf(fact_2145_min__eq__arg_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linorder @ A )
% 0.25/0.46 => ! [M2: A,N: A] :
% 0.25/0.46 ( ( ( ord_min @ A @ M2 @ N )
% 0.25/0.46 = M2 )
% 0.25/0.46 = ( ord_less_eq @ A @ M2 @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_eq_arg(1)
% 0.25/0.46 thf(fact_2146_min__eq__arg_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linorder @ A )
% 0.25/0.46 => ! [M2: A,N: A] :
% 0.25/0.46 ( ( ( ord_min @ A @ M2 @ N )
% 0.25/0.46 = N )
% 0.25/0.46 = ( ord_less_eq @ A @ N @ M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_eq_arg(2)
% 0.25/0.46 thf(fact_2147_min__arg__le_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [N: A,M2: A] :
% 0.25/0.46 ( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M2 @ N ) )
% 0.25/0.46 = ( ( ord_min @ A @ M2 @ N )
% 0.25/0.46 = N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_arg_le(1)
% 0.25/0.46 thf(fact_2148_min__arg__le_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [M2: A,N: A] :
% 0.25/0.46 ( ( ord_less_eq @ A @ M2 @ ( ord_min @ A @ M2 @ N ) )
% 0.25/0.46 = ( ( ord_min @ A @ M2 @ N )
% 0.25/0.46 = M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_arg_le(2)
% 0.25/0.46 thf(fact_2149_min__arg__not__ge_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linorder @ A )
% 0.25/0.46 => ! [M2: A,N: A] :
% 0.25/0.46 ( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ M2 ) )
% 0.25/0.46 = ( ( ord_min @ A @ M2 @ N )
% 0.25/0.46 = M2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_arg_not_ge(1)
% 0.25/0.46 thf(fact_2150_min__arg__not__ge_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linorder @ A )
% 0.25/0.46 => ! [M2: A,N: A] :
% 0.25/0.46 ( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ N ) )
% 0.25/0.46 = ( ( ord_min @ A @ M2 @ N )
% 0.25/0.46 = N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_arg_not_ge(2)
% 0.25/0.46 thf(fact_2151_min__less__self__conv_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linorder @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 )
% 0.25/0.46 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_less_self_conv(1)
% 0.25/0.46 thf(fact_2152_min__less__self__conv_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linorder @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
% 0.25/0.46 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_less_self_conv(2)
% 0.25/0.46 thf(fact_2153_min__simps_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.46 => ( ( ord_min @ A @ A3 @ B2 )
% 0.25/0.46 = A3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_simps(1)
% 0.25/0.46 thf(fact_2154_min__simps_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [B2: A,A3: A] :
% 0.25/0.46 ( ( ord_less @ A @ B2 @ A3 )
% 0.25/0.46 => ( ( ord_min @ A @ A3 @ B2 )
% 0.25/0.46 = B2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_simps(2)
% 0.25/0.46 thf(fact_2155_min__0__1_I2_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 0.25/0.46 = ( zero_zero @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_0_1(2)
% 0.25/0.46 thf(fact_2156_min__0__1_I1_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 0.25/0.46 = ( zero_zero @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_0_1(1)
% 0.25/0.46 thf(fact_2157_min__0__1_I5_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_0_1(5)
% 0.25/0.46 thf(fact_2158_min__0__1_I6_J,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord181362715937106298miring @ A )
% 0.25/0.46 => ! [X: num] :
% 0.25/0.46 ( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 0.25/0.46 = ( one_one @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_0_1(6)
% 0.25/0.46 thf(fact_2159_atLeastAtMost__singleton__iff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.46 ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 = ( ( A3 = B2 )
% 0.25/0.46 & ( B2 = C2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastAtMost_singleton_iff
% 0.25/0.46 thf(fact_2160_atLeastAtMost__singleton,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [A3: A] :
% 0.25/0.46 ( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
% 0.25/0.46 = ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastAtMost_singleton
% 0.25/0.46 thf(fact_2161_min__Suc__gt_I2_J,axiom,
% 0.25/0.46 ! [A3: nat,B2: nat] :
% 0.25/0.46 ( ( ord_less @ nat @ A3 @ B2 )
% 0.25/0.46 => ( ( ord_min @ nat @ B2 @ ( suc @ A3 ) )
% 0.25/0.46 = ( suc @ A3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_Suc_gt(2)
% 0.25/0.46 thf(fact_2162_min__Suc__gt_I1_J,axiom,
% 0.25/0.46 ! [A3: nat,B2: nat] :
% 0.25/0.46 ( ( ord_less @ nat @ A3 @ B2 )
% 0.25/0.46 => ( ( ord_min @ nat @ ( suc @ A3 ) @ B2 )
% 0.25/0.46 = ( suc @ A3 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_Suc_gt(1)
% 0.25/0.46 thf(fact_2163_case__prod__Pair,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
% 0.25/0.46 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % case_prod_Pair
% 0.25/0.46 thf(fact_2164_single__Diff__lessThan,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( preorder @ A )
% 0.25/0.46 => ! [K: A] :
% 0.25/0.46 ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
% 0.25/0.46 = ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % single_Diff_lessThan
% 0.25/0.46 thf(fact_2165_min__add__distrib__right,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordere2412721322843649153imp_le @ A )
% 0.25/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.25/0.46 ( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z2 ) )
% 0.25/0.46 = ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_add_distrib_right
% 0.25/0.46 thf(fact_2166_min__add__distrib__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordere2412721322843649153imp_le @ A )
% 0.25/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.25/0.46 ( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
% 0.25/0.46 = ( ord_min @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_add_distrib_left
% 0.25/0.46 thf(fact_2167_min__diff__distrib__left,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( ordered_ab_group_add @ A )
% 0.25/0.46 => ! [X: A,Y: A,Z2: A] :
% 0.25/0.46 ( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
% 0.25/0.46 = ( ord_min @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % min_diff_distrib_left
% 0.25/0.46 thf(fact_2168_bot__option__def,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ( ( bot_bot @ ( option @ A ) )
% 0.25/0.46 = ( none @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_option_def
% 0.25/0.46 thf(fact_2169_bot__empty__eq2,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( bot_bot @ ( A > B > $o ) )
% 0.25/0.46 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bot_empty_eq2
% 0.25/0.46 thf(fact_2170_map__fun_Oidentity,axiom,
% 0.25/0.46 ! [B: $tType,A: $tType] :
% 0.25/0.46 ( ( map_fun @ A @ A @ B @ B
% 0.25/0.46 @ ^ [X3: A] : X3
% 0.25/0.46 @ ^ [X3: B] : X3 )
% 0.25/0.46 = ( id @ ( A > B ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % map_fun.identity
% 0.25/0.46 thf(fact_2171_map__option_Oidentity,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( map_option @ A @ A
% 0.25/0.46 @ ^ [X3: A] : X3 )
% 0.25/0.46 = ( id @ ( option @ A ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % map_option.identity
% 0.25/0.46 thf(fact_2172_minus__max__eq__min,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord5086331880401160121up_add @ A )
% 0.25/0.46 => ! [X: A,Y: A] :
% 0.25/0.46 ( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
% 0.25/0.46 = ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_max_eq_min
% 0.25/0.46 thf(fact_2173_minus__min__eq__max,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( linord5086331880401160121up_add @ A )
% 0.25/0.46 => ! [X: A,Y: A] :
% 0.25/0.46 ( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
% 0.25/0.46 = ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % minus_min_eq_max
% 0.25/0.46 thf(fact_2174_insert__minus__eq,axiom,
% 0.25/0.46 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.25/0.46 ( ( X != Y )
% 0.25/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 = ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % insert_minus_eq
% 0.25/0.46 thf(fact_2175_set__minus__singleton__eq,axiom,
% 0.25/0.46 ! [A: $tType,X: A,X10: set @ A] :
% 0.25/0.46 ( ~ ( member @ A @ X @ X10 )
% 0.25/0.46 => ( ( minus_minus @ ( set @ A ) @ X10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.46 = X10 ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_minus_singleton_eq
% 0.25/0.46 thf(fact_2176_atLeastAtMost__singleton_H,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ! [A3: A,B2: A] :
% 0.25/0.46 ( ( A3 = B2 )
% 0.25/0.46 => ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 0.25/0.46 = ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastAtMost_singleton'
% 0.25/0.46 thf(fact_2177_remove__subset,axiom,
% 0.25/0.46 ! [A: $tType,X: A,S2: set @ A] :
% 0.25/0.46 ( ( member @ A @ X @ S2 )
% 0.25/0.46 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % remove_subset
% 0.25/0.46 thf(fact_2178_nat__of__integer__def,axiom,
% 0.25/0.46 ( code_nat_of_integer
% 0.25/0.46 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% 0.25/0.46
% 0.25/0.46 % nat_of_integer_def
% 0.25/0.46 thf(fact_2179_bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % bit_integer_def
% 0.25/0.46 thf(fact_2180_mask__integer__def,axiom,
% 0.25/0.46 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % mask_integer_def
% 0.25/0.46 thf(fact_2181_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
% 0.25/0.46 ! [A: $tType] :
% 0.25/0.46 ( ( order @ A )
% 0.25/0.46 => ( ( set_or7035219750837199246ssThan @ A )
% 0.25/0.46 = ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastLessThan_eq_atLeastAtMost_diff
% 0.25/0.46 thf(fact_2182_atLeastAtMostPlus1__int__conv,axiom,
% 0.25/0.46 ! [M2: int,N: int] :
% 0.25/0.46 ( ( ord_less_eq @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 0.25/0.46 => ( ( set_or1337092689740270186AtMost @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 0.25/0.46 = ( insert2 @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % atLeastAtMostPlus1_int_conv
% 0.25/0.46 thf(fact_2183_simp__from__to,axiom,
% 0.25/0.46 ( ( set_or1337092689740270186AtMost @ int )
% 0.25/0.46 = ( ^ [I3: int,J2: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J2 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % simp_from_to
% 0.25/0.46 thf(fact_2184_Code__Numeral_Osub__def,axiom,
% 0.25/0.46 ( code_sub
% 0.25/0.46 = ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
% 0.25/0.46 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % Code_Numeral.sub_def
% 0.25/0.46 thf(fact_2185_num__of__integer__def,axiom,
% 0.25/0.46 ( code_num_of_integer
% 0.25/0.46 = ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % num_of_integer_def
% 0.25/0.46 thf(fact_2186_take__bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se2584673776208193580ke_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2584673776208193580ke_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % take_bit_integer_def
% 0.25/0.46 thf(fact_2187_drop__bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se4197421643247451524op_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4197421643247451524op_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % drop_bit_integer_def
% 0.25/0.46 thf(fact_2188_push__bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se4730199178511100633sh_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4730199178511100633sh_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % push_bit_integer_def
% 0.25/0.46 thf(fact_2189_unset__bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se2638667681897837118et_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2638667681897837118et_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % unset_bit_integer_def
% 0.25/0.46 thf(fact_2190_set__bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se5668285175392031749et_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5668285175392031749et_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % set_bit_integer_def
% 0.25/0.46 thf(fact_2191_flip__bit__integer__def,axiom,
% 0.25/0.46 ( ( bit_se8732182000553998342ip_bit @ code_integer )
% 0.25/0.46 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se8732182000553998342ip_bit @ int ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % flip_bit_integer_def
% 0.25/0.46 thf(fact_2192_pair__leqI1,axiom,
% 0.25/0.46 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 0.25/0.46 ( ( ord_less @ nat @ A3 @ B2 )
% 0.25/0.46 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% 0.25/0.46
% 0.25/0.46 % pair_leqI1
% 0.25/0.46 thf(fact_2193_image__minus__const__atLeastLessThan__nat,axiom,
% 0.25/0.46 ! [C2: nat,Y: nat,X: nat] :
% 0.25/0.46 ( ( ( ord_less @ nat @ C2 @ Y )
% 0.25/0.46 => ( ( image2 @ nat @ nat
% 0.25/0.46 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ nat @ C2 @ Y )
% 0.25/0.46 => ( ( ( ord_less @ nat @ X @ Y )
% 0.25/0.46 => ( ( image2 @ nat @ nat
% 0.25/0.46 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
% 0.25/0.46 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
% 0.25/0.46 & ( ~ ( ord_less @ nat @ X @ Y )
% 0.25/0.46 => ( ( image2 @ nat @ nat
% 0.25/0.46 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
% 0.25/0.46 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
% 0.25/0.46 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % image_minus_const_atLeastLessThan_nat
% 0.25/0.46 thf(fact_2194_card__atLeastAtMost__int,axiom,
% 0.25/0.46 ! [L: int,U: int] :
% 0.25/0.46 ( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
% 0.25/0.46 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % card_atLeastAtMost_int
% 0.25/0.46 thf(fact_2195_range__mod,axiom,
% 0.25/0.46 ! [N: nat] :
% 0.25/0.46 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.25/0.46 => ( ( image2 @ nat @ nat
% 0.25/0.46 @ ^ [M: nat] : ( modulo_modulo @ nat @ M @ N )
% 0.25/0.46 @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.46 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.25/0.46
% 0.25/0.46 % range_mod
% 0.25/0.46 thf(fact_2196_Some__image__these__eq,axiom,
% 0.25/0.46 ! [A: $tType,A4: set @ ( option @ A )] :
% 0.25/0.46 ( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
% 0.25/0.46 = ( collect @ ( option @ A )
% 0.25/0.46 @ ^ [X3: option @ A] :
% 0.25/0.46 ( ( member @ ( option @ A ) @ X3 @ A4 )
% 0.25/0.47 & ( X3
% 0.25/0.47 != ( none @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Some_image_these_eq
% 0.25/0.47 thf(fact_2197_prod__mono2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( linordered_idom @ B )
% 0.25/0.47 => ! [B6: set @ A,A4: set @ A,F: A > B] :
% 0.25/0.47 ( ( finite_finite2 @ A @ B6 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.47 => ( ! [B3: A] :
% 0.25/0.47 ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) )
% 0.25/0.47 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B3 ) ) )
% 0.25/0.47 => ( ! [A5: A] :
% 0.25/0.47 ( ( member @ A @ A5 @ A4 )
% 0.25/0.47 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A5 ) ) )
% 0.25/0.47 => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B6 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_mono2
% 0.25/0.47 thf(fact_2198_the__dflt__None__nonempty,axiom,
% 0.25/0.47 ! [A: $tType,S2: set @ A] :
% 0.25/0.47 ( ( S2
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 => ( ( dflt_None_set @ A @ S2 )
% 0.25/0.47 = ( some @ ( set @ A ) @ S2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % the_dflt_None_nonempty
% 0.25/0.47 thf(fact_2199_the__dflt__None__empty,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 = ( none @ ( set @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % the_dflt_None_empty
% 0.25/0.47 thf(fact_2200_atMost__UNIV__triv,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atMost_UNIV_triv
% 0.25/0.47 thf(fact_2201_finite__atLeastAtMost,axiom,
% 0.25/0.47 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastAtMost
% 0.25/0.47 thf(fact_2202_finite__atLeastLessThan,axiom,
% 0.25/0.47 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastLessThan
% 0.25/0.47 thf(fact_2203_int__of__integer__min,axiom,
% 0.25/0.47 ! [K: code_integer,L: code_integer] :
% 0.25/0.47 ( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
% 0.25/0.47 = ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % int_of_integer_min
% 0.25/0.47 thf(fact_2204_finite__atMost,axiom,
% 0.25/0.47 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atMost
% 0.25/0.47 thf(fact_2205_finite__lessThan,axiom,
% 0.25/0.47 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_lessThan
% 0.25/0.47 thf(fact_2206_finite__atLeastAtMost__int,axiom,
% 0.25/0.47 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastAtMost_int
% 0.25/0.47 thf(fact_2207_finite__atLeastLessThan__int,axiom,
% 0.25/0.47 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastLessThan_int
% 0.25/0.47 thf(fact_2208_finite__atLeastLessThan__integer,axiom,
% 0.25/0.47 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastLessThan_integer
% 0.25/0.47 thf(fact_2209_finite__atLeastAtMost__integer,axiom,
% 0.25/0.47 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastAtMost_integer
% 0.25/0.47 thf(fact_2210_finite__option__UNIV,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 0.25/0.47 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_option_UNIV
% 0.25/0.47 thf(fact_2211_card__eq__UNIV,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [S2: set @ A] :
% 0.25/0.47 ( ( ( finite_card @ A @ S2 )
% 0.25/0.47 = ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
% 0.25/0.47 = ( S2
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_eq_UNIV
% 0.25/0.47 thf(fact_2212_card__eq__UNIV2,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [S2: set @ A] :
% 0.25/0.47 ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( finite_card @ A @ S2 ) )
% 0.25/0.47 = ( S2
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_eq_UNIV2
% 0.25/0.47 thf(fact_2213_these__empty,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % these_empty
% 0.25/0.47 thf(fact_2214_these__insert__None,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ ( option @ A )] :
% 0.25/0.47 ( ( these @ A @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ A4 ) )
% 0.25/0.47 = ( these @ A @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % these_insert_None
% 0.25/0.47 thf(fact_2215_the__dflt__None__set,axiom,
% 0.25/0.47 ! [A: $tType,X: set @ A] :
% 0.25/0.47 ( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X ) )
% 0.25/0.47 = X ) ).
% 0.25/0.47
% 0.25/0.47 % the_dflt_None_set
% 0.25/0.47 thf(fact_2216_surj__plus,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ab_group_add @ A )
% 0.25/0.47 => ! [A3: A] :
% 0.25/0.47 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_plus
% 0.25/0.47 thf(fact_2217_infinite__Icc__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
% 0.25/0.47 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Icc_iff
% 0.25/0.47 thf(fact_2218_infinite__Ico__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
% 0.25/0.47 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Ico_iff
% 0.25/0.47 thf(fact_2219_prod_Oinfinite,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [A4: set @ B,G: B > A] :
% 0.25/0.47 ( ~ ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 0.25/0.47 = ( one_one @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.infinite
% 0.25/0.47 thf(fact_2220_card__ge__UNIV,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite @ A )
% 0.25/0.47 => ! [S2: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S2 ) )
% 0.25/0.47 = ( S2
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_ge_UNIV
% 0.25/0.47 thf(fact_2221_surj__diff__right,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ab_group_add @ A )
% 0.25/0.47 => ! [A3: A] :
% 0.25/0.47 ( ( image2 @ A @ A
% 0.25/0.47 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.47 @ ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_diff_right
% 0.25/0.47 thf(fact_2222_prod__eq__1__iff,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,F: A > nat] :
% 0.25/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.47 => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A4 )
% 0.25/0.47 = ( one_one @ nat ) )
% 0.25/0.47 = ( ! [X3: A] :
% 0.25/0.47 ( ( member @ A @ X3 @ A4 )
% 0.25/0.47 => ( ( F @ X3 )
% 0.25/0.47 = ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_eq_1_iff
% 0.25/0.47 thf(fact_2223_prod_Odelta,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,A3: B,B2: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ S2 )
% 0.25/0.47 => ( ( ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( B2 @ A3 ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.delta
% 0.25/0.47 thf(fact_2224_prod_Odelta_H,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,A3: B,B2: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ S2 )
% 0.25/0.47 => ( ( ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( B2 @ A3 ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.delta'
% 0.25/0.47 thf(fact_2225_these__insert__Some,axiom,
% 0.25/0.47 ! [A: $tType,X: A,A4: set @ ( option @ A )] :
% 0.25/0.47 ( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) )
% 0.25/0.47 = ( insert2 @ A @ X @ ( these @ A @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % these_insert_Some
% 0.25/0.47 thf(fact_2226_these__image__Some__eq,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A] :
% 0.25/0.47 ( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
% 0.25/0.47 = A4 ) ).
% 0.25/0.47
% 0.25/0.47 % these_image_Some_eq
% 0.25/0.47 thf(fact_2227_sum_Oinsert,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [A4: set @ B,X: B,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ~ ( member @ B @ X @ A4 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 0.25/0.47 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.insert
% 0.25/0.47 thf(fact_2228_atLeastLessThan__singleton,axiom,
% 0.25/0.47 ! [M2: nat] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
% 0.25/0.47 = ( insert2 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastLessThan_singleton
% 0.25/0.47 thf(fact_2229_atMost__0,axiom,
% 0.25/0.47 ( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
% 0.25/0.47 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atMost_0
% 0.25/0.47 thf(fact_2230_card__atLeastLessThan__int,axiom,
% 0.25/0.47 ! [L: int,U: int] :
% 0.25/0.47 ( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
% 0.25/0.47 = ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_atLeastLessThan_int
% 0.25/0.47 thf(fact_2231_card__doubleton__eq__2__iff,axiom,
% 0.25/0.47 ! [A: $tType,A3: A,B2: A] :
% 0.25/0.47 ( ( ( finite_card @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.47 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.47 = ( A3 != B2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_doubleton_eq_2_iff
% 0.25/0.47 thf(fact_2232_atMost__eq__UNIV__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( order_top @ A )
% 0.25/0.47 => ! [X: A] :
% 0.25/0.47 ( ( ( set_ord_atMost @ A @ X )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( X
% 0.25/0.47 = ( top_top @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atMost_eq_UNIV_iff
% 0.25/0.47 thf(fact_2233_card__le__if__inj__on__rel,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,B6: set @ A,A4: set @ B,R: B > A > $o] :
% 0.25/0.47 ( ( finite_finite2 @ A @ B6 )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ A4 )
% 0.25/0.47 => ? [B8: A] :
% 0.25/0.47 ( ( member @ A @ B8 @ B6 )
% 0.25/0.47 & ( R @ A5 @ B8 ) ) )
% 0.25/0.47 => ( ! [A13: B,A24: B,B3: A] :
% 0.25/0.47 ( ( member @ B @ A13 @ A4 )
% 0.25/0.47 => ( ( member @ B @ A24 @ A4 )
% 0.25/0.47 => ( ( member @ A @ B3 @ B6 )
% 0.25/0.47 => ( ( R @ A13 @ B3 )
% 0.25/0.47 => ( ( R @ A24 @ B3 )
% 0.25/0.47 => ( A13 = A24 ) ) ) ) ) )
% 0.25/0.47 => ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_le_if_inj_on_rel
% 0.25/0.47 thf(fact_2234_top__option__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( order_top @ A )
% 0.25/0.47 => ( ( top_top @ ( option @ A ) )
% 0.25/0.47 = ( some @ A @ ( top_top @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % top_option_def
% 0.25/0.47 thf(fact_2235_finite__if__eq__beyond__finite,axiom,
% 0.25/0.47 ! [A: $tType,S2: set @ A,S3: set @ A] :
% 0.25/0.47 ( ( finite_finite2 @ A @ S2 )
% 0.25/0.47 => ( finite_finite2 @ ( set @ A )
% 0.25/0.47 @ ( collect @ ( set @ A )
% 0.25/0.47 @ ^ [S4: set @ A] :
% 0.25/0.47 ( ( minus_minus @ ( set @ A ) @ S4 @ S2 )
% 0.25/0.47 = ( minus_minus @ ( set @ A ) @ S3 @ S2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_if_eq_beyond_finite
% 0.25/0.47 thf(fact_2236_finite__range__Some,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
% 0.25/0.47 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_range_Some
% 0.25/0.47 thf(fact_2237_obtain__subset__with__card__n,axiom,
% 0.25/0.47 ! [A: $tType,N: nat,S2: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
% 0.25/0.47 => ~ ! [T4: set @ A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
% 0.25/0.47 => ( ( ( finite_card @ A @ T4 )
% 0.25/0.47 = N )
% 0.25/0.47 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % obtain_subset_with_card_n
% 0.25/0.47 thf(fact_2238_atLeastAtMost__eq__UNIV__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( bounded_lattice @ A )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( ( X
% 0.25/0.47 = ( bot_bot @ A ) )
% 0.25/0.47 & ( Y
% 0.25/0.47 = ( top_top @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastAtMost_eq_UNIV_iff
% 0.25/0.47 thf(fact_2239_UNIV__option__conv,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( top_top @ ( set @ ( option @ A ) ) )
% 0.25/0.47 = ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UNIV_option_conv
% 0.25/0.47 thf(fact_2240_bounded__nat__set__is__finite,axiom,
% 0.25/0.47 ! [N6: set @ nat,N: nat] :
% 0.25/0.47 ( ! [X2: nat] :
% 0.25/0.47 ( ( member @ nat @ X2 @ N6 )
% 0.25/0.47 => ( ord_less @ nat @ X2 @ N ) )
% 0.25/0.47 => ( finite_finite2 @ nat @ N6 ) ) ).
% 0.25/0.47
% 0.25/0.47 % bounded_nat_set_is_finite
% 0.25/0.47 thf(fact_2241_finite__nat__set__iff__bounded,axiom,
% 0.25/0.47 ( ( finite_finite2 @ nat )
% 0.25/0.47 = ( ^ [N9: set @ nat] :
% 0.25/0.47 ? [M: nat] :
% 0.25/0.47 ! [X3: nat] :
% 0.25/0.47 ( ( member @ nat @ X3 @ N9 )
% 0.25/0.47 => ( ord_less @ nat @ X3 @ M ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_nat_set_iff_bounded
% 0.25/0.47 thf(fact_2242_finite__nat__set__iff__bounded__le,axiom,
% 0.25/0.47 ( ( finite_finite2 @ nat )
% 0.25/0.47 = ( ^ [N9: set @ nat] :
% 0.25/0.47 ? [M: nat] :
% 0.25/0.47 ! [X3: nat] :
% 0.25/0.47 ( ( member @ nat @ X3 @ N9 )
% 0.25/0.47 => ( ord_less_eq @ nat @ X3 @ M ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_nat_set_iff_bounded_le
% 0.25/0.47 thf(fact_2243_surjD,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.25/0.47 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 => ? [X2: B] :
% 0.25/0.47 ( Y
% 0.25/0.47 = ( F @ X2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surjD
% 0.25/0.47 thf(fact_2244_surjE,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 0.25/0.47 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 => ~ ! [X2: B] :
% 0.25/0.47 ( Y
% 0.25/0.47 != ( F @ X2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surjE
% 0.25/0.47 thf(fact_2245_surjI,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.25/0.47 ( ! [X2: A] :
% 0.25/0.47 ( ( G @ ( F @ X2 ) )
% 0.25/0.47 = X2 )
% 0.25/0.47 => ( ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surjI
% 0.25/0.47 thf(fact_2246_surj__def,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,F: B > A] :
% 0.25/0.47 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( ! [Y3: A] :
% 0.25/0.47 ? [X3: B] :
% 0.25/0.47 ( Y3
% 0.25/0.47 = ( F @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_def
% 0.25/0.47 thf(fact_2247_these__empty__eq,axiom,
% 0.25/0.47 ! [A: $tType,B6: set @ ( option @ A )] :
% 0.25/0.47 ( ( ( these @ A @ B6 )
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 = ( ( B6
% 0.25/0.47 = ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.25/0.47 | ( B6
% 0.25/0.47 = ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % these_empty_eq
% 0.25/0.47 thf(fact_2248_these__not__empty__eq,axiom,
% 0.25/0.47 ! [A: $tType,B6: set @ ( option @ A )] :
% 0.25/0.47 ( ( ( these @ A @ B6 )
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 = ( ( B6
% 0.25/0.47 != ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.25/0.47 & ( B6
% 0.25/0.47 != ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % these_not_empty_eq
% 0.25/0.47 thf(fact_2249_not__UNIV__eq__Icc,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( no_top @ A )
% 0.25/0.47 => ! [L4: A,H4: A] :
% 0.25/0.47 ( ( top_top @ ( set @ A ) )
% 0.25/0.47 != ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % not_UNIV_eq_Icc
% 0.25/0.47 thf(fact_2250_infinite__Iic,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ( linorder @ A )
% 0.25/0.47 & ( no_bot @ A ) )
% 0.25/0.47 => ! [A3: A] :
% 0.25/0.47 ~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Iic
% 0.25/0.47 thf(fact_2251_infinite__Iio,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ( linorder @ A )
% 0.25/0.47 & ( no_bot @ A ) )
% 0.25/0.47 => ! [A3: A] :
% 0.25/0.47 ~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Iio
% 0.25/0.47 thf(fact_2252_not__UNIV__eq__Iic,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( no_top @ A )
% 0.25/0.47 => ! [H4: A] :
% 0.25/0.47 ( ( top_top @ ( set @ A ) )
% 0.25/0.47 != ( set_ord_atMost @ A @ H4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % not_UNIV_eq_Iic
% 0.25/0.47 thf(fact_2253_card__1__singletonI,axiom,
% 0.25/0.47 ! [A: $tType,S2: set @ A,X: A] :
% 0.25/0.47 ( ( finite_finite2 @ A @ S2 )
% 0.25/0.47 => ( ( ( finite_card @ A @ S2 )
% 0.25/0.47 = ( one_one @ nat ) )
% 0.25/0.47 => ( ( member @ A @ X @ S2 )
% 0.25/0.47 => ( S2
% 0.25/0.47 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_1_singletonI
% 0.25/0.47 thf(fact_2254_finite__M__bounded__by__nat,axiom,
% 0.25/0.47 ! [P: nat > $o,I: nat] :
% 0.25/0.47 ( finite_finite2 @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [K4: nat] :
% 0.25/0.47 ( ( P @ K4 )
% 0.25/0.47 & ( ord_less @ nat @ K4 @ I ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_M_bounded_by_nat
% 0.25/0.47 thf(fact_2255_finite__less__ub,axiom,
% 0.25/0.47 ! [F: nat > nat,U: nat] :
% 0.25/0.47 ( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F @ N3 ) )
% 0.25/0.47 => ( finite_finite2 @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [N5: nat] : ( ord_less_eq @ nat @ ( F @ N5 ) @ U ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_less_ub
% 0.25/0.47 thf(fact_2256_card__insert__disjoint_H,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,X: A] :
% 0.25/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.47 => ( ~ ( member @ A @ X @ A4 )
% 0.25/0.47 => ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.25/0.47 = ( finite_card @ A @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_insert_disjoint'
% 0.25/0.47 thf(fact_2257_in__these__eq,axiom,
% 0.25/0.47 ! [A: $tType,X: A,A4: set @ ( option @ A )] :
% 0.25/0.47 ( ( member @ A @ X @ ( these @ A @ A4 ) )
% 0.25/0.47 = ( member @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % in_these_eq
% 0.25/0.47 thf(fact_2258_infinite__Icc,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.47 => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Icc
% 0.25/0.47 thf(fact_2259_infinite__Ico,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.47 => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Ico
% 0.25/0.47 thf(fact_2260_subset__eq__atLeast0__atMost__finite,axiom,
% 0.25/0.47 ! [N6: set @ nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.25/0.47 => ( finite_finite2 @ nat @ N6 ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_eq_atLeast0_atMost_finite
% 0.25/0.47 thf(fact_2261_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.25/0.47 ! [N6: set @ nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.25/0.47 => ( finite_finite2 @ nat @ N6 ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_eq_atLeast0_lessThan_finite
% 0.25/0.47 thf(fact_2262_atMost__Suc,axiom,
% 0.25/0.47 ! [K: nat] :
% 0.25/0.47 ( ( set_ord_atMost @ nat @ ( suc @ K ) )
% 0.25/0.47 = ( insert2 @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atMost_Suc
% 0.25/0.47 thf(fact_2263_surj__fun__eq,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,F: B > A,X10: set @ B,G1: A > C,G22: A > C] :
% 0.25/0.47 ( ( ( image2 @ B @ A @ F @ X10 )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ X10 )
% 0.25/0.47 => ( ( comp @ A @ C @ B @ G1 @ F @ X2 )
% 0.25/0.47 = ( comp @ A @ C @ B @ G22 @ F @ X2 ) ) )
% 0.25/0.47 => ( G1 = G22 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_fun_eq
% 0.25/0.47 thf(fact_2264_comp__surj,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
% 0.25/0.47 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 => ( ( ( image2 @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( top_top @ ( set @ C ) ) )
% 0.25/0.47 => ( ( image2 @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % comp_surj
% 0.25/0.47 thf(fact_2265_fun_Oset__map,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
% 0.25/0.47 ( ( image2 @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
% 0.25/0.47 = ( image2 @ A @ B @ F @ ( image2 @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.set_map
% 0.25/0.47 thf(fact_2266_fun_Omap__cong,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
% 0.25/0.47 ( ( X = Ya )
% 0.25/0.47 => ( ! [Z3: A] :
% 0.25/0.47 ( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 0.25/0.47 => ( ( F @ Z3 )
% 0.25/0.47 = ( G @ Z3 ) ) )
% 0.25/0.47 => ( ( comp @ A @ B @ D @ F @ X )
% 0.25/0.47 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.map_cong
% 0.25/0.47 thf(fact_2267_fun_Omap__cong0,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,D: $tType,X: D > A,F: A > B,G: A > B] :
% 0.25/0.47 ( ! [Z3: A] :
% 0.25/0.47 ( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 0.25/0.47 => ( ( F @ Z3 )
% 0.25/0.47 = ( G @ Z3 ) ) )
% 0.25/0.47 => ( ( comp @ A @ B @ D @ F @ X )
% 0.25/0.47 = ( comp @ A @ B @ D @ G @ X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.map_cong0
% 0.25/0.47 thf(fact_2268_fun_Oinj__map__strong,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F: A > B,Fa: A > B] :
% 0.25/0.47 ( ! [Z3: A,Za: A] :
% 0.25/0.47 ( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 0.25/0.47 => ( ( member @ A @ Za @ ( image2 @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
% 0.25/0.47 => ( ( ( F @ Z3 )
% 0.25/0.47 = ( Fa @ Za ) )
% 0.25/0.47 => ( Z3 = Za ) ) ) )
% 0.25/0.47 => ( ( ( comp @ A @ B @ D @ F @ X )
% 0.25/0.47 = ( comp @ A @ B @ D @ Fa @ Xa ) )
% 0.25/0.47 => ( X = Xa ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.inj_map_strong
% 0.25/0.47 thf(fact_2269_lessThan__Suc,axiom,
% 0.25/0.47 ! [K: nat] :
% 0.25/0.47 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 0.25/0.47 = ( insert2 @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % lessThan_Suc
% 0.25/0.47 thf(fact_2270_not__UNIV__le__Icc,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( no_top @ A )
% 0.25/0.47 => ! [L: A,H3: A] :
% 0.25/0.47 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % not_UNIV_le_Icc
% 0.25/0.47 thf(fact_2271_surj__id,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( image2 @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_id
% 0.25/0.47 thf(fact_2272_not__UNIV__le__Iic,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( no_top @ A )
% 0.25/0.47 => ! [H3: A] :
% 0.25/0.47 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % not_UNIV_le_Iic
% 0.25/0.47 thf(fact_2273_sum_Ofinite__Collect__op,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [I4: set @ B,X: B > A,Y: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [I3: B] :
% 0.25/0.47 ( ( member @ B @ I3 @ I4 )
% 0.25/0.47 & ( ( X @ I3 )
% 0.25/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [I3: B] :
% 0.25/0.47 ( ( member @ B @ I3 @ I4 )
% 0.25/0.47 & ( ( Y @ I3 )
% 0.25/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.25/0.47 => ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [I3: B] :
% 0.25/0.47 ( ( member @ B @ I3 @ I4 )
% 0.25/0.47 & ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.25/0.47 != ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.finite_Collect_op
% 0.25/0.47 thf(fact_2274_prod_Ofinite__Collect__op,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [I4: set @ B,X: B > A,Y: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [I3: B] :
% 0.25/0.47 ( ( member @ B @ I3 @ I4 )
% 0.25/0.47 & ( ( X @ I3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [I3: B] :
% 0.25/0.47 ( ( member @ B @ I3 @ I4 )
% 0.25/0.47 & ( ( Y @ I3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 => ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [I3: B] :
% 0.25/0.47 ( ( member @ B @ I3 @ I4 )
% 0.25/0.47 & ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
% 0.25/0.47 != ( one_one @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.finite_Collect_op
% 0.25/0.47 thf(fact_2275_finite__atLeastZeroLessThan__int,axiom,
% 0.25/0.47 ! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastZeroLessThan_int
% 0.25/0.47 thf(fact_2276_card__eq__sum,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( finite_card @ A )
% 0.25/0.47 = ( groups7311177749621191930dd_sum @ A @ nat
% 0.25/0.47 @ ^ [X3: A] : ( one_one @ nat ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_eq_sum
% 0.25/0.47 thf(fact_2277_prod_Ointer__filter,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [A4: set @ B,G: B > A,P: B > $o] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ A4 )
% 0.25/0.47 & ( P @ X3 ) ) ) )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G @ X3 ) @ ( one_one @ A ) )
% 0.25/0.47 @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.inter_filter
% 0.25/0.47 thf(fact_2278_prod__gen__delta,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,A3: B,B2: B > A,C2: A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ S2 )
% 0.25/0.47 => ( ( ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( power_power @ A @ C2 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_gen_delta
% 0.25/0.47 thf(fact_2279_finite__atLeastZeroLessThan__integer,axiom,
% 0.25/0.47 ! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_atLeastZeroLessThan_integer
% 0.25/0.47 thf(fact_2280_card__2__iff_H,axiom,
% 0.25/0.47 ! [A: $tType,S2: set @ A] :
% 0.25/0.47 ( ( ( finite_card @ A @ S2 )
% 0.25/0.47 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.47 = ( ? [X3: A] :
% 0.25/0.47 ( ( member @ A @ X3 @ S2 )
% 0.25/0.47 & ? [Y3: A] :
% 0.25/0.47 ( ( member @ A @ Y3 @ S2 )
% 0.25/0.47 & ( X3 != Y3 )
% 0.25/0.47 & ! [Z5: A] :
% 0.25/0.47 ( ( member @ A @ Z5 @ S2 )
% 0.25/0.47 => ( ( Z5 = X3 )
% 0.25/0.47 | ( Z5 = Y3 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_2_iff'
% 0.25/0.47 thf(fact_2281_sum_Orelated,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [R5: A > A > $o,S2: set @ B,H3: B > A,G: B > A] :
% 0.25/0.47 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
% 0.25/0.47 => ( ! [X13: A,Y12: A,X23: A,Y22: A] :
% 0.25/0.47 ( ( ( R5 @ X13 @ X23 )
% 0.25/0.47 & ( R5 @ Y12 @ Y22 ) )
% 0.25/0.47 => ( R5 @ ( plus_plus @ A @ X13 @ Y12 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ B @ S2 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ S2 )
% 0.25/0.47 => ( R5 @ ( H3 @ X2 ) @ ( G @ X2 ) ) )
% 0.25/0.47 => ( R5 @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.related
% 0.25/0.47 thf(fact_2282_prod_Orelated,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [R5: A > A > $o,S2: set @ B,H3: B > A,G: B > A] :
% 0.25/0.47 ( ( R5 @ ( one_one @ A ) @ ( one_one @ A ) )
% 0.25/0.47 => ( ! [X13: A,Y12: A,X23: A,Y22: A] :
% 0.25/0.47 ( ( ( R5 @ X13 @ X23 )
% 0.25/0.47 & ( R5 @ Y12 @ Y22 ) )
% 0.25/0.47 => ( R5 @ ( times_times @ A @ X13 @ Y12 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ B @ S2 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ S2 )
% 0.25/0.47 => ( R5 @ ( H3 @ X2 ) @ ( G @ X2 ) ) )
% 0.25/0.47 => ( R5 @ ( groups7121269368397514597t_prod @ B @ A @ H3 @ S2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.related
% 0.25/0.47 thf(fact_2283_sum_Oinsert__if,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [A4: set @ B,X: B,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( ( member @ B @ X @ A4 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 0.25/0.47 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ X @ A4 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 0.25/0.47 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.insert_if
% 0.25/0.47 thf(fact_2284_atLeast0__atMost__Suc,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeast0_atMost_Suc
% 0.25/0.47 thf(fact_2285_atLeast0__lessThan__Suc,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeast0_lessThan_Suc
% 0.25/0.47 thf(fact_2286_Icc__eq__insert__lb__nat,axiom,
% 0.25/0.47 ! [M2: nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.47 => ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
% 0.25/0.47 = ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Icc_eq_insert_lb_nat
% 0.25/0.47 thf(fact_2287_atLeastAtMostSuc__conv,axiom,
% 0.25/0.47 ! [M2: nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.25/0.47 => ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastAtMostSuc_conv
% 0.25/0.47 thf(fact_2288_atLeastAtMost__insertL,axiom,
% 0.25/0.47 ! [M2: nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.47 => ( ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
% 0.25/0.47 = ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastAtMost_insertL
% 0.25/0.47 thf(fact_2289_prod_Oreindex__bij__witness__not__neutral,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S7: set @ B,T5: set @ C,S2: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H3: C > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ S7 )
% 0.25/0.47 => ( ( finite_finite2 @ C @ T5 )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
% 0.25/0.47 => ( ( I @ ( J @ A5 ) )
% 0.25/0.47 = A5 ) )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
% 0.25/0.47 => ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
% 0.25/0.47 => ( ! [B3: C] :
% 0.25/0.47 ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.25/0.47 => ( ( J @ ( I @ B3 ) )
% 0.25/0.47 = B3 ) )
% 0.25/0.47 => ( ! [B3: C] :
% 0.25/0.47 ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.25/0.47 => ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) ) )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ S7 )
% 0.25/0.47 => ( ( G @ A5 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [B3: C] :
% 0.25/0.47 ( ( member @ C @ B3 @ T5 )
% 0.25/0.47 => ( ( H3 @ B3 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ S2 )
% 0.25/0.47 => ( ( H3 @ ( J @ A5 ) )
% 0.25/0.47 = ( G @ A5 ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ C @ A @ H3 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.reindex_bij_witness_not_neutral
% 0.25/0.47 thf(fact_2290_dflt__None__set__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dflt_None_set @ A )
% 0.25/0.47 = ( ^ [S5: set @ A] :
% 0.25/0.47 ( if @ ( option @ ( set @ A ) )
% 0.25/0.47 @ ( S5
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 @ ( none @ ( set @ A ) )
% 0.25/0.47 @ ( some @ ( set @ A ) @ S5 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % dflt_None_set_def
% 0.25/0.47 thf(fact_2291_sum__eq__1__iff,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,F: A > nat] :
% 0.25/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.47 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 )
% 0.25/0.47 = ( one_one @ nat ) )
% 0.25/0.47 = ( ? [X3: A] :
% 0.25/0.47 ( ( member @ A @ X3 @ A4 )
% 0.25/0.47 & ( ( F @ X3 )
% 0.25/0.47 = ( one_one @ nat ) )
% 0.25/0.47 & ! [Y3: A] :
% 0.25/0.47 ( ( member @ A @ Y3 @ A4 )
% 0.25/0.47 => ( ( X3 != Y3 )
% 0.25/0.47 => ( ( F @ Y3 )
% 0.25/0.47 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum_eq_1_iff
% 0.25/0.47 thf(fact_2292_surj__Compl__image__subset,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
% 0.25/0.47 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ A ) ) )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F @ A4 ) ) @ ( image2 @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surj_Compl_image_subset
% 0.25/0.47 thf(fact_2293_card__atLeastZeroLessThan__int,axiom,
% 0.25/0.47 ! [U: int] :
% 0.25/0.47 ( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
% 0.25/0.47 = ( nat2 @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_atLeastZeroLessThan_int
% 0.25/0.47 thf(fact_2294_atMost__nat__numeral,axiom,
% 0.25/0.47 ! [K: num] :
% 0.25/0.47 ( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
% 0.25/0.47 = ( insert2 @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atMost_nat_numeral
% 0.25/0.47 thf(fact_2295_lessThan__nat__numeral,axiom,
% 0.25/0.47 ! [K: num] :
% 0.25/0.47 ( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
% 0.25/0.47 = ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % lessThan_nat_numeral
% 0.25/0.47 thf(fact_2296_sum__Suc,axiom,
% 0.25/0.47 ! [A: $tType,F: A > nat,A4: set @ A] :
% 0.25/0.47 ( ( groups7311177749621191930dd_sum @ A @ nat
% 0.25/0.47 @ ^ [X3: A] : ( suc @ ( F @ X3 ) )
% 0.25/0.47 @ A4 )
% 0.25/0.47 = ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 ) @ ( finite_card @ A @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum_Suc
% 0.25/0.47 thf(fact_2297_prod_Osetdiff__irrelevant,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [A4: set @ B,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 0.25/0.47 @ ( minus_minus @ ( set @ B ) @ A4
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( G @ X3 )
% 0.25/0.47 = ( one_one @ A ) ) ) ) )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.setdiff_irrelevant
% 0.25/0.47 thf(fact_2298_notin__range__Some,axiom,
% 0.25/0.47 ! [A: $tType,X: option @ A] :
% 0.25/0.47 ( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
% 0.25/0.47 = ( X
% 0.25/0.47 = ( none @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % notin_range_Some
% 0.25/0.47 thf(fact_2299_less__1__prod2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( linordered_idom @ B )
% 0.25/0.47 => ! [I4: set @ A,I: A,F: A > B] :
% 0.25/0.47 ( ( finite_finite2 @ A @ I4 )
% 0.25/0.47 => ( ( member @ A @ I @ I4 )
% 0.25/0.47 => ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
% 0.25/0.47 => ( ! [I2: A] :
% 0.25/0.47 ( ( member @ A @ I2 @ I4 )
% 0.25/0.47 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I2 ) ) )
% 0.25/0.47 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I4 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % less_1_prod2
% 0.25/0.47 thf(fact_2300_less__1__prod,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( linordered_idom @ B )
% 0.25/0.47 => ! [I4: set @ A,F: A > B] :
% 0.25/0.47 ( ( finite_finite2 @ A @ I4 )
% 0.25/0.47 => ( ( I4
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 => ( ! [I2: A] :
% 0.25/0.47 ( ( member @ A @ I2 @ I4 )
% 0.25/0.47 => ( ord_less @ B @ ( one_one @ B ) @ ( F @ I2 ) ) )
% 0.25/0.47 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % less_1_prod
% 0.25/0.47 thf(fact_2301_sum_Oreindex__nontrivial,axiom,
% 0.25/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [A4: set @ B,H3: B > C,G: C > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ! [X2: B,Y4: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ A4 )
% 0.25/0.47 => ( ( member @ B @ Y4 @ A4 )
% 0.25/0.47 => ( ( X2 != Y4 )
% 0.25/0.47 => ( ( ( H3 @ X2 )
% 0.25/0.47 = ( H3 @ Y4 ) )
% 0.25/0.47 => ( ( G @ ( H3 @ X2 ) )
% 0.25/0.47 = ( zero_zero @ A ) ) ) ) ) )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 0.25/0.47 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.reindex_nontrivial
% 0.25/0.47 thf(fact_2302_prod_Oreindex__nontrivial,axiom,
% 0.25/0.47 ! [C: $tType,A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [A4: set @ B,H3: B > C,G: C > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ! [X2: B,Y4: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ A4 )
% 0.25/0.47 => ( ( member @ B @ Y4 @ A4 )
% 0.25/0.47 => ( ( X2 != Y4 )
% 0.25/0.47 => ( ( ( H3 @ X2 )
% 0.25/0.47 = ( H3 @ Y4 ) )
% 0.25/0.47 => ( ( G @ ( H3 @ X2 ) )
% 0.25/0.47 = ( one_one @ A ) ) ) ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.reindex_nontrivial
% 0.25/0.47 thf(fact_2303_sum_Osubset__diff,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [B6: set @ B,A4: set @ B,G: B > A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
% 0.25/0.47 => ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
% 0.25/0.47 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.subset_diff
% 0.25/0.47 thf(fact_2304_prod_Osame__carrier,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ C5 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
% 0.25/0.47 => ( ( G @ A5 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [B3: B] :
% 0.25/0.47 ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
% 0.25/0.47 => ( ( H3 @ B3 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ B6 ) )
% 0.25/0.47 = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ C5 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.same_carrier
% 0.25/0.47 thf(fact_2305_prod_Osame__carrierI,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ C5 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
% 0.25/0.47 => ( ! [A5: B] :
% 0.25/0.47 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
% 0.25/0.47 => ( ( G @ A5 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [B3: B] :
% 0.25/0.47 ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
% 0.25/0.47 => ( ( H3 @ B3 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ C5 ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ B6 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.same_carrierI
% 0.25/0.47 thf(fact_2306_prod_Omono__neutral__left,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [T6: set @ B,S2: set @ B,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ T6 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_left
% 0.25/0.47 thf(fact_2307_prod_Omono__neutral__right,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [T6: set @ B,S2: set @ B,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ T6 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_right
% 0.25/0.47 thf(fact_2308_prod_Omono__neutral__cong__left,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [T6: set @ B,S2: set @ B,H3: B > A,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ T6 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( H3 @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ S2 )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( H3 @ X2 ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_cong_left
% 0.25/0.47 thf(fact_2309_prod_Omono__neutral__cong__right,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [T6: set @ B,S2: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ T6 )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ S2 )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( H3 @ X2 ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 0.25/0.47 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ S2 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_cong_right
% 0.25/0.47 thf(fact_2310_atLeastLessThanSuc,axiom,
% 0.25/0.47 ! [M2: nat,N: nat] :
% 0.25/0.47 ( ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.47 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
% 0.25/0.47 & ( ~ ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.47 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
% 0.25/0.47 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastLessThanSuc
% 0.25/0.47 thf(fact_2311_Option_Othese__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( these @ A )
% 0.25/0.47 = ( ^ [A9: set @ ( option @ A )] :
% 0.25/0.47 ( image2 @ ( option @ A ) @ A @ ( the2 @ A )
% 0.25/0.47 @ ( collect @ ( option @ A )
% 0.25/0.47 @ ^ [X3: option @ A] :
% 0.25/0.47 ( ( member @ ( option @ A ) @ X3 @ A9 )
% 0.25/0.47 & ( X3
% 0.25/0.47 != ( none @ A ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Option.these_def
% 0.25/0.47 thf(fact_2312_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeast0_atMost_Suc_eq_insert_0
% 0.25/0.47 thf(fact_2313_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeast0_lessThan_Suc_eq_insert_0
% 0.25/0.47 thf(fact_2314_atMost__Suc__eq__insert__0,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_ord_atMost @ nat @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atMost_Suc_eq_insert_0
% 0.25/0.47 thf(fact_2315_lessThan__Suc__eq__insert__0,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_ord_lessThan @ nat @ ( suc @ N ) )
% 0.25/0.47 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % lessThan_Suc_eq_insert_0
% 0.25/0.47 thf(fact_2316_card__2__iff,axiom,
% 0.25/0.47 ! [A: $tType,S2: set @ A] :
% 0.25/0.47 ( ( ( finite_card @ A @ S2 )
% 0.25/0.47 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.47 = ( ? [X3: A,Y3: A] :
% 0.25/0.47 ( ( S2
% 0.25/0.47 = ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.47 & ( X3 != Y3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_2_iff
% 0.25/0.47 thf(fact_2317_card__3__iff,axiom,
% 0.25/0.47 ! [A: $tType,S2: set @ A] :
% 0.25/0.47 ( ( ( finite_card @ A @ S2 )
% 0.25/0.47 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 0.25/0.47 = ( ? [X3: A,Y3: A,Z5: A] :
% 0.25/0.47 ( ( S2
% 0.25/0.47 = ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.25/0.47 & ( X3 != Y3 )
% 0.25/0.47 & ( Y3 != Z5 )
% 0.25/0.47 & ( X3 != Z5 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_3_iff
% 0.25/0.47 thf(fact_2318_sum__image__le,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType] :
% 0.25/0.47 ( ( ordere6911136660526730532id_add @ B )
% 0.25/0.47 => ! [I4: set @ C,G: A > B,F: C > A] :
% 0.25/0.47 ( ( finite_finite2 @ C @ I4 )
% 0.25/0.47 => ( ! [I2: C] :
% 0.25/0.47 ( ( member @ C @ I2 @ I4 )
% 0.25/0.47 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I2 ) ) ) )
% 0.25/0.47 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image2 @ C @ A @ F @ I4 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum_image_le
% 0.25/0.47 thf(fact_2319_sum_Oremove,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [A4: set @ B,X: B,G: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( member @ B @ X @ A4 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
% 0.25/0.47 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.remove
% 0.25/0.47 thf(fact_2320_sum_Oinsert__remove,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [A4: set @ B,G: B > A,X: B] :
% 0.25/0.47 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 0.25/0.47 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.insert_remove
% 0.25/0.47 thf(fact_2321_atLeast1__atMost__eq__remove0,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.25/0.47 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeast1_atMost_eq_remove0
% 0.25/0.47 thf(fact_2322_atLeast1__lessThan__eq__remove0,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 0.25/0.47 = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeast1_lessThan_eq_remove0
% 0.25/0.47 thf(fact_2323_atLeastLessThan__nat__numeral,axiom,
% 0.25/0.47 ! [M2: nat,K: num] :
% 0.25/0.47 ( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
% 0.25/0.47 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
% 0.25/0.47 = ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
% 0.25/0.47 & ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
% 0.25/0.47 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
% 0.25/0.47 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastLessThan_nat_numeral
% 0.25/0.47 thf(fact_2324_sum_Odelta__remove,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [S2: set @ B,A3: B,B2: B > A,C2: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ S2 )
% 0.25/0.47 => ( ( ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ A3 @ S2 )
% 0.25/0.47 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.25/0.47 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
% 0.25/0.47 @ S2 )
% 0.25/0.47 = ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.delta_remove
% 0.25/0.47 thf(fact_2325_prod__le__power,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( linordered_semidom @ A )
% 0.25/0.47 => ! [A4: set @ B,F: B > A,N: A,K: nat] :
% 0.25/0.47 ( ! [I2: B] :
% 0.25/0.47 ( ( member @ B @ I2 @ A4 )
% 0.25/0.47 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I2 ) )
% 0.25/0.47 & ( ord_less_eq @ A @ ( F @ I2 ) @ N ) ) )
% 0.25/0.47 => ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
% 0.25/0.47 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
% 0.25/0.47 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_le_power
% 0.25/0.47 thf(fact_2326_card__Diff__insert,axiom,
% 0.25/0.47 ! [A: $tType,A3: A,A4: set @ A,B6: set @ A] :
% 0.25/0.47 ( ( member @ A @ A3 @ A4 )
% 0.25/0.47 => ( ~ ( member @ A @ A3 @ B6 )
% 0.25/0.47 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B6 ) ) )
% 0.25/0.47 = ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_Diff_insert
% 0.25/0.47 thf(fact_2327_card__insert__le__m1,axiom,
% 0.25/0.47 ! [A: $tType,N: nat,Y: set @ A,X: A] :
% 0.25/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.25/0.47 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.25/0.47 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_insert_le_m1
% 0.25/0.47 thf(fact_2328_card__Diff__singleton__if,axiom,
% 0.25/0.47 ! [A: $tType,X: A,A4: set @ A] :
% 0.25/0.47 ( ( ( member @ A @ X @ A4 )
% 0.25/0.47 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.47 = ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
% 0.25/0.47 & ( ~ ( member @ A @ X @ A4 )
% 0.25/0.47 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.47 = ( finite_card @ A @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_Diff_singleton_if
% 0.25/0.47 thf(fact_2329_card__Diff__singleton,axiom,
% 0.25/0.47 ! [A: $tType,X: A,A4: set @ A] :
% 0.25/0.47 ( ( member @ A @ X @ A4 )
% 0.25/0.47 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.47 = ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_Diff_singleton
% 0.25/0.47 thf(fact_2330_in__finite__psubset,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.47 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B6 ) @ ( finite_psubset @ A ) )
% 0.25/0.47 = ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.47 & ( finite_finite2 @ A @ B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % in_finite_psubset
% 0.25/0.47 thf(fact_2331_card__1__singletonE,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A] :
% 0.25/0.47 ( ( ( finite_card @ A @ A4 )
% 0.25/0.47 = ( one_one @ nat ) )
% 0.25/0.47 => ~ ! [X2: A] :
% 0.25/0.47 ( A4
% 0.25/0.47 != ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_1_singletonE
% 0.25/0.47 thf(fact_2332_integer__of__nat__def,axiom,
% 0.25/0.47 ( code_integer_of_nat
% 0.25/0.47 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat_def
% 0.25/0.47 thf(fact_2333_card__UNIV__unit,axiom,
% 0.25/0.47 ( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
% 0.25/0.47 = ( one_one @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_UNIV_unit
% 0.25/0.47 thf(fact_2334_card__lessThan,axiom,
% 0.25/0.47 ! [U: nat] :
% 0.25/0.47 ( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
% 0.25/0.47 = U ) ).
% 0.25/0.47
% 0.25/0.47 % card_lessThan
% 0.25/0.47 thf(fact_2335_card__atLeastLessThan,axiom,
% 0.25/0.47 ! [L: nat,U: nat] :
% 0.25/0.47 ( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
% 0.25/0.47 = ( minus_minus @ nat @ U @ L ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_atLeastLessThan
% 0.25/0.47 thf(fact_2336_card__atMost,axiom,
% 0.25/0.47 ! [U: nat] :
% 0.25/0.47 ( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
% 0.25/0.47 = ( suc @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_atMost
% 0.25/0.47 thf(fact_2337_Collect__const__case__prod,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: $o] :
% 0.25/0.47 ( ( P
% 0.25/0.47 => ( ( collect @ ( product_prod @ A @ B )
% 0.25/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.47 @ ^ [A7: A,B5: B] : P ) )
% 0.25/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
% 0.25/0.47 & ( ~ P
% 0.25/0.47 => ( ( collect @ ( product_prod @ A @ B )
% 0.25/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.47 @ ^ [A7: A,B5: B] : P ) )
% 0.25/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_const_case_prod
% 0.25/0.47 thf(fact_2338_nat__of__integer__integer__of__nat,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
% 0.25/0.47 = N ) ).
% 0.25/0.47
% 0.25/0.47 % nat_of_integer_integer_of_nat
% 0.25/0.47 thf(fact_2339_card__atLeastAtMost,axiom,
% 0.25/0.47 ! [L: nat,U: nat] :
% 0.25/0.47 ( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
% 0.25/0.47 = ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_atLeastAtMost
% 0.25/0.47 thf(fact_2340_integer__of__nat_Orep__eq,axiom,
% 0.25/0.47 ! [X: nat] :
% 0.25/0.47 ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
% 0.25/0.47 = ( semiring_1_of_nat @ int @ X ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat.rep_eq
% 0.25/0.47 thf(fact_2341_int__of__integer__integer__of__nat,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
% 0.25/0.47 = ( semiring_1_of_nat @ int @ N ) ) ).
% 0.25/0.47
% 0.25/0.47 % int_of_integer_integer_of_nat
% 0.25/0.47 thf(fact_2342_top__empty__eq2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( top_top @ ( A > B > $o ) )
% 0.25/0.47 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % top_empty_eq2
% 0.25/0.47 thf(fact_2343_int__of__integer__induct,axiom,
% 0.25/0.47 ! [Y: int,P: int > $o] :
% 0.25/0.47 ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 0.25/0.47 => ( ! [X2: code_integer] : ( P @ ( code_int_of_integer @ X2 ) )
% 0.25/0.47 => ( P @ Y ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % int_of_integer_induct
% 0.25/0.47 thf(fact_2344_int__of__integer__cases,axiom,
% 0.25/0.47 ! [Y: int] :
% 0.25/0.47 ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 0.25/0.47 => ~ ! [X2: code_integer] :
% 0.25/0.47 ( Y
% 0.25/0.47 != ( code_int_of_integer @ X2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % int_of_integer_cases
% 0.25/0.47 thf(fact_2345_int__of__integer,axiom,
% 0.25/0.47 ! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % int_of_integer
% 0.25/0.47 thf(fact_2346_integer__of__int__inject,axiom,
% 0.25/0.47 ! [X: int,Y: int] :
% 0.25/0.47 ( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
% 0.25/0.47 => ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 0.25/0.47 => ( ( ( code_integer_of_int @ X )
% 0.25/0.47 = ( code_integer_of_int @ Y ) )
% 0.25/0.47 = ( X = Y ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_int_inject
% 0.25/0.47 thf(fact_2347_integer__of__int__induct,axiom,
% 0.25/0.47 ! [P: code_integer > $o,X: code_integer] :
% 0.25/0.47 ( ! [Y4: int] :
% 0.25/0.47 ( ( member @ int @ Y4 @ ( top_top @ ( set @ int ) ) )
% 0.25/0.47 => ( P @ ( code_integer_of_int @ Y4 ) ) )
% 0.25/0.47 => ( P @ X ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_int_induct
% 0.25/0.47 thf(fact_2348_integer__of__int__cases,axiom,
% 0.25/0.47 ! [X: code_integer] :
% 0.25/0.47 ~ ! [Y4: int] :
% 0.25/0.47 ( ( X
% 0.25/0.47 = ( code_integer_of_int @ Y4 ) )
% 0.25/0.47 => ~ ( member @ int @ Y4 @ ( top_top @ ( set @ int ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_int_cases
% 0.25/0.47 thf(fact_2349_subset__card__intvl__is__intvl,axiom,
% 0.25/0.47 ! [A4: set @ nat,K: nat] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ nat ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) )
% 0.25/0.47 => ( A4
% 0.25/0.47 = ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_card_intvl_is_intvl
% 0.25/0.47 thf(fact_2350_integer__of__int__inverse,axiom,
% 0.25/0.47 ! [Y: int] :
% 0.25/0.47 ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 0.25/0.47 => ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
% 0.25/0.47 = Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_int_inverse
% 0.25/0.47 thf(fact_2351_subset__eq__atLeast0__lessThan__card,axiom,
% 0.25/0.47 ! [N6: set @ nat,N: nat] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.25/0.47 => ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% 0.25/0.47
% 0.25/0.47 % subset_eq_atLeast0_lessThan_card
% 0.25/0.47 thf(fact_2352_integer__of__nat__eq__of__nat,axiom,
% 0.25/0.47 ( code_integer_of_nat
% 0.25/0.47 = ( semiring_1_of_nat @ code_integer ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat_eq_of_nat
% 0.25/0.47 thf(fact_2353_card__less__Suc2,axiom,
% 0.25/0.47 ! [M3: set @ nat,I: nat] :
% 0.25/0.47 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M3 )
% 0.25/0.47 => ( ( finite_card @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [K4: nat] :
% 0.25/0.47 ( ( member @ nat @ ( suc @ K4 ) @ M3 )
% 0.25/0.47 & ( ord_less @ nat @ K4 @ I ) ) ) )
% 0.25/0.47 = ( finite_card @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [K4: nat] :
% 0.25/0.47 ( ( member @ nat @ K4 @ M3 )
% 0.25/0.47 & ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_less_Suc2
% 0.25/0.47 thf(fact_2354_card__less__Suc,axiom,
% 0.25/0.47 ! [M3: set @ nat,I: nat] :
% 0.25/0.47 ( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
% 0.25/0.47 => ( ( suc
% 0.25/0.47 @ ( finite_card @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [K4: nat] :
% 0.25/0.47 ( ( member @ nat @ ( suc @ K4 ) @ M3 )
% 0.25/0.47 & ( ord_less @ nat @ K4 @ I ) ) ) ) )
% 0.25/0.47 = ( finite_card @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [K4: nat] :
% 0.25/0.47 ( ( member @ nat @ K4 @ M3 )
% 0.25/0.47 & ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_less_Suc
% 0.25/0.47 thf(fact_2355_card__less,axiom,
% 0.25/0.47 ! [M3: set @ nat,I: nat] :
% 0.25/0.47 ( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
% 0.25/0.47 => ( ( finite_card @ nat
% 0.25/0.47 @ ( collect @ nat
% 0.25/0.47 @ ^ [K4: nat] :
% 0.25/0.47 ( ( member @ nat @ K4 @ M3 )
% 0.25/0.47 & ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) )
% 0.25/0.47 != ( zero_zero @ nat ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_less
% 0.25/0.47 thf(fact_2356_integer__of__nat__0,axiom,
% 0.25/0.47 ( ( code_integer_of_nat @ ( zero_zero @ nat ) )
% 0.25/0.47 = ( zero_zero @ code_integer ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat_0
% 0.25/0.47 thf(fact_2357_integer__of__nat_Oabs__eq,axiom,
% 0.25/0.47 ( code_integer_of_nat
% 0.25/0.47 = ( ^ [X3: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat.abs_eq
% 0.25/0.47 thf(fact_2358_integer__of__nat__numeral,axiom,
% 0.25/0.47 ! [N: num] :
% 0.25/0.47 ( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
% 0.25/0.47 = ( numeral_numeral @ code_integer @ N ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat_numeral
% 0.25/0.47 thf(fact_2359_integer__of__nat__1,axiom,
% 0.25/0.47 ( ( code_integer_of_nat @ ( one_one @ nat ) )
% 0.25/0.47 = ( one_one @ code_integer ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_nat_1
% 0.25/0.47 thf(fact_2360_card__sum__le__nat__sum,axiom,
% 0.25/0.47 ! [S2: set @ nat] :
% 0.25/0.47 ( ord_less_eq @ nat
% 0.25/0.47 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 0.25/0.47 @ ^ [X3: nat] : X3
% 0.25/0.47 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
% 0.25/0.47 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 0.25/0.47 @ ^ [X3: nat] : X3
% 0.25/0.47 @ S2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_sum_le_nat_sum
% 0.25/0.47 thf(fact_2361_set__encode__insert,axiom,
% 0.25/0.47 ! [A4: set @ nat,N: nat] :
% 0.25/0.47 ( ( finite_finite2 @ nat @ A4 )
% 0.25/0.47 => ( ~ ( member @ nat @ N @ A4 )
% 0.25/0.47 => ( ( nat_set_encode @ ( insert2 @ nat @ N @ A4 ) )
% 0.25/0.47 = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % set_encode_insert
% 0.25/0.47 thf(fact_2362_card__greaterThanLessThan__int,axiom,
% 0.25/0.47 ! [L: int,U: int] :
% 0.25/0.47 ( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
% 0.25/0.47 = ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_greaterThanLessThan_int
% 0.25/0.47 thf(fact_2363_wmax__insertI,axiom,
% 0.25/0.47 ! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % wmax_insertI
% 0.25/0.47 thf(fact_2364_wmin__insertI,axiom,
% 0.25/0.47 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % wmin_insertI
% 0.25/0.47 thf(fact_2365_smin__insertI,axiom,
% 0.25/0.47 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % smin_insertI
% 0.25/0.47 thf(fact_2366_prod_Oinsert_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [I4: set @ B,P5: B > A,I: B] :
% 0.25/0.47 ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( P5 @ X3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 => ( ( ( member @ B @ I @ I4 )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 0.25/0.47 = ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ I @ I4 )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 0.25/0.47 = ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.insert'
% 0.25/0.47 thf(fact_2367_greaterThanLessThan__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ord @ A )
% 0.25/0.47 => ! [I: A,L: A,U: A] :
% 0.25/0.47 ( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.25/0.47 = ( ( ord_less @ A @ L @ I )
% 0.25/0.47 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_iff
% 0.25/0.47 thf(fact_2368_finite__greaterThanLessThan__int,axiom,
% 0.25/0.47 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_greaterThanLessThan_int
% 0.25/0.47 thf(fact_2369_greaterThanLessThan__empty__iff2,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.25/0.47 = ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
% 0.25/0.47 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_empty_iff2
% 0.25/0.47 thf(fact_2370_greaterThanLessThan__empty__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_empty_iff
% 0.25/0.47 thf(fact_2371_greaterThanLessThan__empty,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( order @ A )
% 0.25/0.47 => ! [L: A,K: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ L @ K )
% 0.25/0.47 => ( ( set_or5935395276787703475ssThan @ A @ K @ L )
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_empty
% 0.25/0.47 thf(fact_2372_infinite__Ioo__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
% 0.25/0.47 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Ioo_iff
% 0.25/0.47 thf(fact_2373_image__uminus__greaterThanLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ordered_ab_group_add @ A )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
% 0.25/0.47 = ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % image_uminus_greaterThanLessThan
% 0.25/0.47 thf(fact_2374_prod_Oempty_H,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [P5: B > A] :
% 0.25/0.47 ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
% 0.25/0.47 = ( one_one @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.empty'
% 0.25/0.47 thf(fact_2375_smin__emptyI,axiom,
% 0.25/0.47 ! [X10: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( X10
% 0.25/0.47 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% 0.25/0.47
% 0.25/0.47 % smin_emptyI
% 0.25/0.47 thf(fact_2376_wmax__emptyI,axiom,
% 0.25/0.47 ! [X10: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X10 )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X10 ) @ fun_max_weak ) ) ).
% 0.25/0.47
% 0.25/0.47 % wmax_emptyI
% 0.25/0.47 thf(fact_2377_wmin__emptyI,axiom,
% 0.25/0.47 ! [X10: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% 0.25/0.47
% 0.25/0.47 % wmin_emptyI
% 0.25/0.47 thf(fact_2378_prod_Onon__neutral_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [G: B > A,I4: set @ B] :
% 0.25/0.47 ( ( groups1962203154675924110t_prod @ B @ A @ G
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( G @ X3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 = ( groups1962203154675924110t_prod @ B @ A @ G @ I4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.non_neutral'
% 0.25/0.47 thf(fact_2379_infinite__Ioo,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.47 => ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % infinite_Ioo
% 0.25/0.47 thf(fact_2380_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
% 0.25/0.47 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.47 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.47 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_subseteq_greaterThanLessThan
% 0.25/0.47 thf(fact_2381_prod__decode__aux_Ocases,axiom,
% 0.25/0.47 ! [X: product_prod @ nat @ nat] :
% 0.25/0.47 ~ ! [K5: nat,M5: nat] :
% 0.25/0.47 ( X
% 0.25/0.47 != ( product_Pair @ nat @ nat @ K5 @ M5 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_decode_aux.cases
% 0.25/0.47 thf(fact_2382_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.25/0.47 ! [L: int,U: int] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 0.25/0.47 = ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.25/0.47 thf(fact_2383_prod_Omono__neutral__left_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,T6: set @ B,G: B > A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
% 0.25/0.47 = ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_left'
% 0.25/0.47 thf(fact_2384_prod_Omono__neutral__right_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,T6: set @ B,G: B > A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 0.25/0.47 = ( groups1962203154675924110t_prod @ B @ A @ G @ S2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_right'
% 0.25/0.47 thf(fact_2385_prod_Omono__neutral__cong__left_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,T6: set @ B,H3: B > A,G: B > A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [I2: B] :
% 0.25/0.47 ( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( H3 @ I2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ S2 )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( H3 @ X2 ) ) )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
% 0.25/0.47 = ( groups1962203154675924110t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_cong_left'
% 0.25/0.47 thf(fact_2386_prod_Omono__neutral__cong__right_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [S2: set @ B,T6: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) )
% 0.25/0.47 => ( ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ S2 )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( H3 @ X2 ) ) )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 0.25/0.47 = ( groups1962203154675924110t_prod @ B @ A @ H3 @ S2 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.mono_neutral_cong_right'
% 0.25/0.47 thf(fact_2387_prod_Odistrib_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [I4: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( G @ X3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( H3 @ X3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A
% 0.25/0.47 @ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
% 0.25/0.47 @ I4 )
% 0.25/0.47 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.distrib'
% 0.25/0.47 thf(fact_2388_prod_OG__def,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ( ( groups1962203154675924110t_prod @ B @ A )
% 0.25/0.47 = ( ^ [P6: B > A,I5: set @ B] :
% 0.25/0.47 ( if @ A
% 0.25/0.47 @ ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I5 )
% 0.25/0.47 & ( ( P6 @ X3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 @ ( groups7121269368397514597t_prod @ B @ A @ P6
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I5 )
% 0.25/0.47 & ( ( P6 @ X3 )
% 0.25/0.47 != ( one_one @ A ) ) ) ) )
% 0.25/0.47 @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.G_def
% 0.25/0.47 thf(fact_2389_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.25/0.47 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.47 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.47 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_subseteq_atLeastAtMost_iff
% 0.25/0.47 thf(fact_2390_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( dense_linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.25/0.47 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.47 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.47 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % greaterThanLessThan_subseteq_atLeastLessThan_iff
% 0.25/0.47 thf(fact_2391_atLeastAtMost__diff__ends,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( linorder @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.47 = ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastAtMost_diff_ends
% 0.25/0.47 thf(fact_2392_smax__insertI,axiom,
% 0.25/0.47 ! [Y: product_prod @ nat @ nat,Y9: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X10: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y9 )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
% 0.25/0.47 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ Y9 ) @ fun_max_strict )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ X10 ) @ Y9 ) @ fun_max_strict ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % smax_insertI
% 0.25/0.47 thf(fact_2393_set__decode__plus__power__2,axiom,
% 0.25/0.47 ! [N: nat,Z2: nat] :
% 0.25/0.47 ( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
% 0.25/0.47 => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
% 0.25/0.47 = ( insert2 @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % set_decode_plus_power_2
% 0.25/0.47 thf(fact_2394_prod__decode__aux_Osimps,axiom,
% 0.25/0.47 ( nat_prod_decode_aux
% 0.25/0.47 = ( ^ [K4: nat,M: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M @ K4 ) @ ( product_Pair @ nat @ nat @ M @ ( minus_minus @ nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus @ nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_decode_aux.simps
% 0.25/0.47 thf(fact_2395_prod__decode__aux_Oelims,axiom,
% 0.25/0.47 ! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
% 0.25/0.47 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.25/0.47 = Y )
% 0.25/0.47 => ( ( ( ord_less_eq @ nat @ Xa @ X )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
% 0.25/0.47 & ( ~ ( ord_less_eq @ nat @ Xa @ X )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_decode_aux.elims
% 0.25/0.47 thf(fact_2396_min__rpair__set,axiom,
% 0.25/0.47 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% 0.25/0.47
% 0.25/0.47 % min_rpair_set
% 0.25/0.47 thf(fact_2397_triangle__Suc,axiom,
% 0.25/0.47 ! [N: nat] :
% 0.25/0.47 ( ( nat_triangle @ ( suc @ N ) )
% 0.25/0.47 = ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % triangle_Suc
% 0.25/0.47 thf(fact_2398_finite__greaterThanLessThan,axiom,
% 0.25/0.47 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_greaterThanLessThan
% 0.25/0.47 thf(fact_2399_finite__greaterThanLessThan__integer,axiom,
% 0.25/0.47 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_greaterThanLessThan_integer
% 0.25/0.47 thf(fact_2400_card__greaterThanLessThan,axiom,
% 0.25/0.47 ! [L: nat,U: nat] :
% 0.25/0.47 ( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
% 0.25/0.47 = ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_greaterThanLessThan
% 0.25/0.47 thf(fact_2401_max__rpair__set,axiom,
% 0.25/0.47 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% 0.25/0.47
% 0.25/0.47 % max_rpair_set
% 0.25/0.47 thf(fact_2402_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.25/0.47 ! [L: nat,U: nat] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
% 0.25/0.47 = ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastSucLessThan_greaterThanLessThan
% 0.25/0.47 thf(fact_2403_smax__emptyI,axiom,
% 0.25/0.47 ! [Y9: set @ ( product_prod @ nat @ nat )] :
% 0.25/0.47 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y9 )
% 0.25/0.47 => ( ( Y9
% 0.25/0.47 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y9 ) @ fun_max_strict ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % smax_emptyI
% 0.25/0.47 thf(fact_2404_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
% 0.25/0.47 ! [L: code_integer,U: code_integer] :
% 0.25/0.47 ( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
% 0.25/0.47 = ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% 0.25/0.47
% 0.25/0.47 % atLeastPlusOneLessThan_greaterThanLessThan_integer
% 0.25/0.47 thf(fact_2405_prod__decode__aux_Opelims,axiom,
% 0.25/0.47 ! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
% 0.25/0.47 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 0.25/0.47 = Y )
% 0.25/0.47 => ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
% 0.25/0.47 => ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
% 0.25/0.47 & ( ~ ( ord_less_eq @ nat @ Xa @ X )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 0.25/0.47 => ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_decode_aux.pelims
% 0.25/0.47 thf(fact_2406_total__on__singleton,axiom,
% 0.25/0.47 ! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % total_on_singleton
% 0.25/0.47 thf(fact_2407_prod__encode__prod__decode__aux,axiom,
% 0.25/0.47 ! [K: nat,M2: nat] :
% 0.25/0.47 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
% 0.25/0.47 = ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_encode_prod_decode_aux
% 0.25/0.47 thf(fact_2408_sum_Oinsert_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [I4: set @ B,P5: B > A,I: B] :
% 0.25/0.47 ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( P5 @ X3 )
% 0.25/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.25/0.47 => ( ( ( member @ B @ I @ I4 )
% 0.25/0.47 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 0.25/0.47 = ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I4 ) ) )
% 0.25/0.47 & ( ~ ( member @ B @ I @ I4 )
% 0.25/0.47 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 0.25/0.47 = ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.insert'
% 0.25/0.47 thf(fact_2409_max__ext_Omax__extI,axiom,
% 0.25/0.47 ! [A: $tType,X10: set @ A,Y9: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ( ( finite_finite2 @ A @ X10 )
% 0.25/0.47 => ( ( finite_finite2 @ A @ Y9 )
% 0.25/0.47 => ( ( Y9
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 => ( ! [X2: A] :
% 0.25/0.47 ( ( member @ A @ X2 @ X10 )
% 0.25/0.47 => ? [Xa2: A] :
% 0.25/0.47 ( ( member @ A @ Xa2 @ Y9 )
% 0.25/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R5 ) ) )
% 0.25/0.47 => ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X10 @ Y9 ) @ ( max_ext @ A @ R5 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % max_ext.max_extI
% 0.25/0.47 thf(fact_2410_total__on__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( total_on @ A )
% 0.25/0.47 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ! [X3: A] :
% 0.25/0.47 ( ( member @ A @ X3 @ A9 )
% 0.25/0.47 => ! [Y3: A] :
% 0.25/0.47 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.47 => ( ( X3 != Y3 )
% 0.25/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 0.25/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % total_on_def
% 0.25/0.47 thf(fact_2411_total__onI,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ( ! [X2: A,Y4: A] :
% 0.25/0.47 ( ( member @ A @ X2 @ A4 )
% 0.25/0.47 => ( ( member @ A @ Y4 @ A4 )
% 0.25/0.47 => ( ( X2 != Y4 )
% 0.25/0.47 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 0.25/0.47 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R ) ) ) ) )
% 0.25/0.47 => ( total_on @ A @ A4 @ R ) ) ).
% 0.25/0.47
% 0.25/0.47 % total_onI
% 0.25/0.47 thf(fact_2412_sum_Odistrib__triv_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [I4: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B @ I4 )
% 0.25/0.47 => ( ( groups1027152243600224163dd_sum @ B @ A
% 0.25/0.47 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
% 0.25/0.47 @ I4 )
% 0.25/0.47 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.distrib_triv'
% 0.25/0.47 thf(fact_2413_le__prod__encode__1,axiom,
% 0.25/0.47 ! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_prod_encode_1
% 0.25/0.47 thf(fact_2414_le__prod__encode__2,axiom,
% 0.25/0.47 ! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % le_prod_encode_2
% 0.25/0.47 thf(fact_2415_sum_Odistrib_H,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_add @ A )
% 0.25/0.47 => ! [I4: set @ B,G: B > A,H3: B > A] :
% 0.25/0.47 ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( G @ X3 )
% 0.25/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ B
% 0.25/0.47 @ ( collect @ B
% 0.25/0.47 @ ^ [X3: B] :
% 0.25/0.47 ( ( member @ B @ X3 @ I4 )
% 0.25/0.47 & ( ( H3 @ X3 )
% 0.25/0.47 != ( zero_zero @ A ) ) ) ) )
% 0.25/0.47 => ( ( groups1027152243600224163dd_sum @ B @ A
% 0.25/0.47 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
% 0.25/0.47 @ I4 )
% 0.25/0.47 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sum.distrib'
% 0.25/0.47 thf(fact_2416_prod__encode__def,axiom,
% 0.25/0.47 ( nat_prod_encode
% 0.25/0.47 = ( product_case_prod @ nat @ nat @ nat
% 0.25/0.47 @ ^ [M: nat,N5: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M @ N5 ) ) @ M ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_encode_def
% 0.25/0.47 thf(fact_2417_max__ext_Ocases,axiom,
% 0.25/0.47 ! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
% 0.25/0.47 => ~ ( ( finite_finite2 @ A @ A1 )
% 0.25/0.47 => ( ( finite_finite2 @ A @ A22 )
% 0.25/0.47 => ( ( A22
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 => ~ ! [X7: A] :
% 0.25/0.47 ( ( member @ A @ X7 @ A1 )
% 0.25/0.47 => ? [Xa3: A] :
% 0.25/0.47 ( ( member @ A @ Xa3 @ A22 )
% 0.25/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R5 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % max_ext.cases
% 0.25/0.47 thf(fact_2418_max__ext_Osimps,axiom,
% 0.25/0.47 ! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
% 0.25/0.47 = ( ( finite_finite2 @ A @ A1 )
% 0.25/0.47 & ( finite_finite2 @ A @ A22 )
% 0.25/0.47 & ( A22
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 & ! [X3: A] :
% 0.25/0.47 ( ( member @ A @ X3 @ A1 )
% 0.25/0.47 => ? [Y3: A] :
% 0.25/0.47 ( ( member @ A @ Y3 @ A22 )
% 0.25/0.47 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % max_ext.simps
% 0.25/0.47 thf(fact_2419_max__extp__eq,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( max_extp @ A )
% 0.25/0.47 = ( ^ [R3: A > A > $o,X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % max_extp_eq
% 0.25/0.47 thf(fact_2420_max__extp__max__ext__eq,axiom,
% 0.25/0.47 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ( ( max_extp @ A
% 0.25/0.47 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) )
% 0.25/0.47 = ( ^ [X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % max_extp_max_ext_eq
% 0.25/0.47 thf(fact_2421_one__div__numeral,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( unique1627219031080169319umeral @ A )
% 0.25/0.47 => ! [N: num] :
% 0.25/0.47 ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.25/0.47 = ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % one_div_numeral
% 0.25/0.47 thf(fact_2422_sqr_Osimps_I3_J,axiom,
% 0.25/0.47 ! [N: num] :
% 0.25/0.47 ( ( sqr @ ( bit1 @ N ) )
% 0.25/0.47 = ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sqr.simps(3)
% 0.25/0.47 thf(fact_2423_one__mod__numeral,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( unique1627219031080169319umeral @ A )
% 0.25/0.47 => ! [N: num] :
% 0.25/0.47 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 0.25/0.47 = ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % one_mod_numeral
% 0.25/0.47 thf(fact_2424_card__UNION,axiom,
% 0.25/0.47 ! [A: $tType,A4: set @ ( set @ A )] :
% 0.25/0.47 ( ( finite_finite2 @ ( set @ A ) @ A4 )
% 0.25/0.47 => ( ! [X2: set @ A] :
% 0.25/0.47 ( ( member @ ( set @ A ) @ X2 @ A4 )
% 0.25/0.47 => ( finite_finite2 @ A @ X2 ) )
% 0.25/0.47 => ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
% 0.25/0.47 = ( nat2
% 0.25/0.47 @ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
% 0.25/0.47 @ ^ [I5: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I5 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I5 ) ) ) )
% 0.25/0.47 @ ( collect @ ( set @ ( set @ A ) )
% 0.25/0.47 @ ^ [I5: set @ ( set @ A )] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I5 @ A4 )
% 0.25/0.47 & ( I5
% 0.25/0.47 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % card_UNION
% 0.25/0.47 thf(fact_2425_Sup__atMost,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [Y: A] :
% 0.25/0.47 ( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
% 0.25/0.47 = Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_atMost
% 0.25/0.47 thf(fact_2426_Sup__lessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 & ( dense_linorder @ A ) )
% 0.25/0.47 => ! [Y: A] :
% 0.25/0.47 ( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
% 0.25/0.47 = Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_lessThan
% 0.25/0.47 thf(fact_2427_fst__apsnd,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 0.25/0.47 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 0.25/0.47 = ( product_fst @ A @ C @ X ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_apsnd
% 0.25/0.47 thf(fact_2428_fst__comp__apsnd,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.25/0.47 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.25/0.47 = ( product_fst @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_comp_apsnd
% 0.25/0.47 thf(fact_2429_apsnd__eq__conv,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
% 0.25/0.47 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 0.25/0.47 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 0.25/0.47 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 0.25/0.47 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % apsnd_eq_conv
% 0.25/0.47 thf(fact_2430_snd__apsnd,axiom,
% 0.25/0.47 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 0.25/0.47 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 0.25/0.47 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_apsnd
% 0.25/0.47 thf(fact_2431_snd__comp__apsnd,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 0.25/0.47 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 0.25/0.47 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_comp_apsnd
% 0.25/0.47 thf(fact_2432_fst__divmod__integer,axiom,
% 0.25/0.47 ! [K: code_integer,L: code_integer] :
% 0.25/0.47 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 0.25/0.47 = ( divide_divide @ code_integer @ K @ L ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_divmod_integer
% 0.25/0.47 thf(fact_2433_snd__divmod__integer,axiom,
% 0.25/0.47 ! [K: code_integer,L: code_integer] :
% 0.25/0.47 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 0.25/0.47 = ( modulo_modulo @ code_integer @ K @ L ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_divmod_integer
% 0.25/0.47 thf(fact_2434_Sup__atLeastAtMost,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.47 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.47 = Y ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_atLeastAtMost
% 0.25/0.47 thf(fact_2435_Inf__atLeastAtMost,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.47 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 0.25/0.47 = X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_atLeastAtMost
% 0.25/0.47 thf(fact_2436_Sup__atLeastLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 & ( dense_linorder @ A ) )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ord_less @ A @ X @ Y )
% 0.25/0.47 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
% 0.25/0.47 = Y ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_atLeastLessThan
% 0.25/0.47 thf(fact_2437_Inf__atLeastLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ord_less @ A @ X @ Y )
% 0.25/0.47 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
% 0.25/0.47 = X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_atLeastLessThan
% 0.25/0.47 thf(fact_2438_prod_Ocollapse,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.25/0.47 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 0.25/0.47 = Prod ) ).
% 0.25/0.47
% 0.25/0.47 % prod.collapse
% 0.25/0.47 thf(fact_2439_Inf__atMost,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [X: A] :
% 0.25/0.47 ( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
% 0.25/0.47 = ( bot_bot @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_atMost
% 0.25/0.47 thf(fact_2440_Sup__greaterThanLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 & ( dense_linorder @ A ) )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ord_less @ A @ X @ Y )
% 0.25/0.47 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
% 0.25/0.47 = Y ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_greaterThanLessThan
% 0.25/0.47 thf(fact_2441_img__fst,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 0.25/0.47 => ( member @ A @ A3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % img_fst
% 0.25/0.47 thf(fact_2442_Inf__greaterThanLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 & ( dense_linorder @ A ) )
% 0.25/0.47 => ! [X: A,Y: A] :
% 0.25/0.47 ( ( ord_less @ A @ X @ Y )
% 0.25/0.47 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
% 0.25/0.47 = X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_greaterThanLessThan
% 0.25/0.47 thf(fact_2443_img__snd,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 0.25/0.47 => ( member @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % img_snd
% 0.25/0.47 thf(fact_2444_fst__divmod__abs,axiom,
% 0.25/0.47 ! [K: code_integer,L: code_integer] :
% 0.25/0.47 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 0.25/0.47 = ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_divmod_abs
% 0.25/0.47 thf(fact_2445_snd__divmod__abs,axiom,
% 0.25/0.47 ! [K: code_integer,L: code_integer] :
% 0.25/0.47 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 0.25/0.47 = ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_divmod_abs
% 0.25/0.47 thf(fact_2446_Inf__atMostLessThan,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [X: A] :
% 0.25/0.47 ( ( ord_less @ A @ ( top_top @ A ) @ X )
% 0.25/0.47 => ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
% 0.25/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_atMostLessThan
% 0.25/0.47 thf(fact_2447_split__comp__eq,axiom,
% 0.25/0.47 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
% 0.25/0.47 ( ( ^ [U2: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
% 0.25/0.47 = ( product_case_prod @ D @ B @ C
% 0.25/0.47 @ ^ [X3: D] : ( F @ ( G @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % split_comp_eq
% 0.25/0.47 thf(fact_2448_case__prod__beta_H,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_case_prod @ A @ B @ C )
% 0.25/0.47 = ( ^ [F2: A > B > C,X3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % case_prod_beta'
% 0.25/0.47 thf(fact_2449_case__prod__unfold,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_case_prod @ A @ B @ C )
% 0.25/0.47 = ( ^ [C3: A > B > C,P6: product_prod @ A @ B] : ( C3 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % case_prod_unfold
% 0.25/0.47 thf(fact_2450_Product__Type_OCollect__case__prodD,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: A > B > $o] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) )
% 0.25/0.47 => ( A4 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Product_Type.Collect_case_prodD
% 0.25/0.47 thf(fact_2451_case__prod__beta,axiom,
% 0.25/0.47 ! [A: $tType,C: $tType,B: $tType] :
% 0.25/0.47 ( ( product_case_prod @ B @ C @ A )
% 0.25/0.47 = ( ^ [F2: B > C > A,P6: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % case_prod_beta
% 0.25/0.47 thf(fact_2452_split__beta,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_case_prod @ A @ B @ C )
% 0.25/0.47 = ( ^ [F2: A > B > C,Prod2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % split_beta
% 0.25/0.47 thf(fact_2453_effect__heapI,axiom,
% 0.25/0.47 ! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A] :
% 0.25/0.47 ( ( N
% 0.25/0.47 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( H4
% 0.25/0.47 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( R
% 0.25/0.47 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
% 0.25/0.47 => ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H3 @ H4 @ R @ N ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % effect_heapI
% 0.25/0.47 thf(fact_2454_effect__heapE,axiom,
% 0.25/0.47 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.47 ( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H3 @ H4 @ R @ N )
% 0.25/0.47 => ~ ( ( H4
% 0.25/0.47 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( N
% 0.25/0.47 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( R
% 0.25/0.47 != ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % effect_heapE
% 0.25/0.47 thf(fact_2455_effect__guardI,axiom,
% 0.25/0.47 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
% 0.25/0.47 ( ( P @ H3 )
% 0.25/0.47 => ( ( H4
% 0.25/0.47 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( N
% 0.25/0.47 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( R
% 0.25/0.47 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
% 0.25/0.47 => ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % effect_guardI
% 0.25/0.47 thf(fact_2456_effect__guardE,axiom,
% 0.25/0.47 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.47 ( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 @ H4 @ R @ N )
% 0.25/0.47 => ~ ( ( H4
% 0.25/0.47 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( N
% 0.25/0.47 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 0.25/0.47 => ( ( R
% 0.25/0.47 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
% 0.25/0.47 => ~ ( P @ H3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % effect_guardE
% 0.25/0.47 thf(fact_2457_effectE,axiom,
% 0.25/0.47 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.47 ( ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N )
% 0.25/0.47 => ~ ( ( R
% 0.25/0.47 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) )
% 0.25/0.47 => ( ( H4
% 0.25/0.47 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) ) )
% 0.25/0.47 => ( ( N
% 0.25/0.47 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) ) )
% 0.25/0.47 => ~ ( heap_Time_success @ A @ C2 @ H3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % effectE
% 0.25/0.47 thf(fact_2458_prod__eq__iff,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( ^ [Y5: product_prod @ A @ B,Z4: product_prod @ A @ B] : ( Y5 = Z4 ) )
% 0.25/0.47 = ( ^ [S4: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 0.25/0.47 ( ( ( product_fst @ A @ B @ S4 )
% 0.25/0.47 = ( product_fst @ A @ B @ T3 ) )
% 0.25/0.47 & ( ( product_snd @ A @ B @ S4 )
% 0.25/0.47 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_eq_iff
% 0.25/0.47 thf(fact_2459_prod__eqI,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,Q3: product_prod @ A @ B] :
% 0.25/0.47 ( ( ( product_fst @ A @ B @ P5 )
% 0.25/0.47 = ( product_fst @ A @ B @ Q3 ) )
% 0.25/0.47 => ( ( ( product_snd @ A @ B @ P5 )
% 0.25/0.47 = ( product_snd @ A @ B @ Q3 ) )
% 0.25/0.47 => ( P5 = Q3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod_eqI
% 0.25/0.47 thf(fact_2460_prod_Oexpand,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
% 0.25/0.47 ( ( ( ( product_fst @ A @ B @ Prod )
% 0.25/0.47 = ( product_fst @ A @ B @ Prod3 ) )
% 0.25/0.47 & ( ( product_snd @ A @ B @ Prod )
% 0.25/0.47 = ( product_snd @ A @ B @ Prod3 ) ) )
% 0.25/0.47 => ( Prod = Prod3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.expand
% 0.25/0.47 thf(fact_2461_All__prod__contract,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.25/0.47 ( ( ! [A7: A,X8: B] : ( P @ A7 @ X8 ) )
% 0.25/0.47 = ( ! [Z5: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z5 ) @ ( product_snd @ A @ B @ Z5 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % All_prod_contract
% 0.25/0.47 thf(fact_2462_Ex__prod__contract,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.25/0.47 ( ( ? [A7: A,X8: B] : ( P @ A7 @ X8 ) )
% 0.25/0.47 = ( ? [Z5: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z5 ) @ ( product_snd @ A @ B @ Z5 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Ex_prod_contract
% 0.25/0.47 thf(fact_2463_prod_Osplit__sel,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.25/0.47 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.25/0.47 = ( ( Prod
% 0.25/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.25/0.47 => ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.split_sel
% 0.25/0.47 thf(fact_2464_prod_Osplit__sel__asm,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 0.25/0.47 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 0.25/0.47 = ( ~ ( ( Prod
% 0.25/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 0.25/0.47 & ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.split_sel_asm
% 0.25/0.47 thf(fact_2465_surjective__pairing,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 0.25/0.47 ( T2
% 0.25/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % surjective_pairing
% 0.25/0.47 thf(fact_2466_prod_Oexhaust__sel,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 0.25/0.47 ( Prod
% 0.25/0.47 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.exhaust_sel
% 0.25/0.47 thf(fact_2467_exI__realizer,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 0.25/0.47 ( ( P @ Y @ X )
% 0.25/0.47 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % exI_realizer
% 0.25/0.47 thf(fact_2468_conjI__realizer,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,P: A > $o,P5: A,Q: B > $o,Q3: B] :
% 0.25/0.47 ( ( P @ P5 )
% 0.25/0.47 => ( ( Q @ Q3 )
% 0.25/0.47 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q3 ) ) )
% 0.25/0.47 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q3 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % conjI_realizer
% 0.25/0.47 thf(fact_2469_execute__bind__success,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 0.25/0.47 ( ( heap_Time_success @ A @ F @ H3 )
% 0.25/0.47 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 0.25/0.47 = ( heap_Time_timeFrame @ B @ ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) @ ( heap_Time_execute @ B @ ( G @ ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) @ ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % execute_bind_success
% 0.25/0.47 thf(fact_2470_sndI,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 0.25/0.47 ( ( X
% 0.25/0.47 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.25/0.47 => ( ( product_snd @ A @ B @ X )
% 0.25/0.47 = Z2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % sndI
% 0.25/0.47 thf(fact_2471_sndE,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: B > $o] :
% 0.25/0.47 ( ( X
% 0.25/0.47 = ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.25/0.47 => ( ( P @ ( product_snd @ A @ B @ X ) )
% 0.25/0.47 => ( P @ B2 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sndE
% 0.25/0.47 thf(fact_2472_snd__eqD,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
% 0.25/0.47 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.25/0.47 = A3 )
% 0.25/0.47 => ( Y = A3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_eqD
% 0.25/0.47 thf(fact_2473_snd__conv,axiom,
% 0.25/0.47 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 0.25/0.47 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 0.25/0.47 = X22 ) ).
% 0.25/0.47
% 0.25/0.47 % snd_conv
% 0.25/0.47 thf(fact_2474_Some__Inf,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [A4: set @ A] :
% 0.25/0.47 ( ( some @ A @ ( complete_Inf_Inf @ A @ A4 ) )
% 0.25/0.47 = ( complete_Inf_Inf @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Some_Inf
% 0.25/0.47 thf(fact_2475_fstI,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 0.25/0.47 ( ( X
% 0.25/0.47 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 0.25/0.47 => ( ( product_fst @ A @ B @ X )
% 0.25/0.47 = Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % fstI
% 0.25/0.47 thf(fact_2476_fstE,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: A > $o] :
% 0.25/0.47 ( ( X
% 0.25/0.47 = ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.25/0.47 => ( ( P @ ( product_fst @ A @ B @ X ) )
% 0.25/0.47 => ( P @ A3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fstE
% 0.25/0.47 thf(fact_2477_fst__eqD,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
% 0.25/0.47 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.25/0.47 = A3 )
% 0.25/0.47 => ( X = A3 ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_eqD
% 0.25/0.47 thf(fact_2478_fst__conv,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 0.25/0.47 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 0.25/0.47 = X1 ) ).
% 0.25/0.47
% 0.25/0.47 % fst_conv
% 0.25/0.47 thf(fact_2479_snd__fst__flip,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( product_snd @ B @ A )
% 0.25/0.47 = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
% 0.25/0.47 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.25/0.47 @ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_fst_flip
% 0.25/0.47 thf(fact_2480_fst__snd__flip,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_fst @ A @ B )
% 0.25/0.47 = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
% 0.25/0.47 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 0.25/0.47 @ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_snd_flip
% 0.25/0.47 thf(fact_2481_finite__UNION__then__finite,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,B6: B > ( set @ A ),A4: set @ B,A3: B] :
% 0.25/0.47 ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A4 ) ) )
% 0.25/0.47 => ( ( member @ B @ A3 @ A4 )
% 0.25/0.47 => ( finite_finite2 @ A @ ( B6 @ A3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_UNION_then_finite
% 0.25/0.47 thf(fact_2482_case__prod__comp,axiom,
% 0.25/0.47 ! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
% 0.25/0.47 ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
% 0.25/0.47 = ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % case_prod_comp
% 0.25/0.47 thf(fact_2483_fn__snd__conv,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,F: B > C] :
% 0.25/0.47 ( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_snd @ A @ B @ X3 ) ) )
% 0.25/0.47 = ( product_case_prod @ A @ B @ C
% 0.25/0.47 @ ^ [Uu: A] : F ) ) ).
% 0.25/0.47
% 0.25/0.47 % fn_snd_conv
% 0.25/0.47 thf(fact_2484_snd__def,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_snd @ A @ B )
% 0.25/0.47 = ( product_case_prod @ A @ B @ B
% 0.25/0.47 @ ^ [X12: A,X24: B] : X24 ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_def
% 0.25/0.47 thf(fact_2485_fn__fst__conv,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 0.25/0.47 ( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X3 ) ) )
% 0.25/0.47 = ( product_case_prod @ A @ B @ C
% 0.25/0.47 @ ^ [A7: A,Uu: B] : ( F @ A7 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fn_fst_conv
% 0.25/0.47 thf(fact_2486_fst__def,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_fst @ A @ B )
% 0.25/0.47 = ( product_case_prod @ A @ B @ A
% 0.25/0.47 @ ^ [X12: A,X24: B] : X12 ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_def
% 0.25/0.47 thf(fact_2487_Collect__split__mono__strong,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X10: set @ A,A4: set @ ( product_prod @ A @ B ),Y9: set @ B,P: A > B > $o,Q: A > B > $o] :
% 0.25/0.47 ( ( X10
% 0.25/0.47 = ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) )
% 0.25/0.47 => ( ( Y9
% 0.25/0.47 = ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) )
% 0.25/0.47 => ( ! [X2: A] :
% 0.25/0.47 ( ( member @ A @ X2 @ X10 )
% 0.25/0.47 => ! [Xa3: B] :
% 0.25/0.47 ( ( member @ B @ Xa3 @ Y9 )
% 0.25/0.47 => ( ( P @ X2 @ Xa3 )
% 0.25/0.47 => ( Q @ X2 @ Xa3 ) ) ) )
% 0.25/0.47 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
% 0.25/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_split_mono_strong
% 0.25/0.47 thf(fact_2488_Some__Sup,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [A4: set @ A] :
% 0.25/0.47 ( ( A4
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 => ( ( some @ A @ ( complete_Sup_Sup @ A @ A4 ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Some_Sup
% 0.25/0.47 thf(fact_2489_Inf__option__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ( ( complete_Inf_Inf @ ( option @ A ) )
% 0.25/0.47 = ( ^ [A9: set @ ( option @ A )] : ( if @ ( option @ A ) @ ( member @ ( option @ A ) @ ( none @ A ) @ A9 ) @ ( none @ A ) @ ( some @ A @ ( complete_Inf_Inf @ A @ ( these @ A @ A9 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_option_def
% 0.25/0.47 thf(fact_2490_Some__INF,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [F: B > A,A4: set @ B] :
% 0.25/0.47 ( ( some @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A4 ) ) )
% 0.25/0.47 = ( complete_Inf_Inf @ ( option @ A )
% 0.25/0.47 @ ( image2 @ B @ ( option @ A )
% 0.25/0.47 @ ^ [X3: B] : ( some @ A @ ( F @ X3 ) )
% 0.25/0.47 @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Some_INF
% 0.25/0.47 thf(fact_2491_INF__image,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.25/0.47 ( ( complete_Inf @ A )
% 0.25/0.47 => ! [G: B > A,F: C > B,A4: set @ C] :
% 0.25/0.47 ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 0.25/0.47 = ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % INF_image
% 0.25/0.47 thf(fact_2492_SUP__image,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,C: $tType] :
% 0.25/0.47 ( ( complete_Sup @ A )
% 0.25/0.47 => ! [G: B > A,F: C > B,A4: set @ C] :
% 0.25/0.47 ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SUP_image
% 0.25/0.47 thf(fact_2493_in__snd__imageE,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,Y: A,S2: set @ ( product_prod @ B @ A )] :
% 0.25/0.47 ( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) )
% 0.25/0.47 => ~ ! [X2: B] :
% 0.25/0.47 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ S2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % in_snd_imageE
% 0.25/0.47 thf(fact_2494_in__fst__imageE,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X: A,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.47 ( ( member @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
% 0.25/0.47 => ~ ! [Y4: B] :
% 0.25/0.47 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % in_fst_imageE
% 0.25/0.47 thf(fact_2495_Some__SUP,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ B )
% 0.25/0.47 => ! [A4: set @ A,F: A > B] :
% 0.25/0.47 ( ( A4
% 0.25/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 => ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A4 ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( option @ B )
% 0.25/0.47 @ ( image2 @ A @ ( option @ B )
% 0.25/0.47 @ ^ [X3: A] : ( some @ B @ ( F @ X3 ) )
% 0.25/0.47 @ A4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Some_SUP
% 0.25/0.47 thf(fact_2496_finite__range__prod,axiom,
% 0.25/0.47 ! [A: $tType,C: $tType,B: $tType,F: B > ( product_prod @ A @ C )] :
% 0.25/0.47 ( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
% 0.25/0.47 => ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
% 0.25/0.47 => ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % finite_range_prod
% 0.25/0.47 thf(fact_2497_UN__atMost__UNIV,axiom,
% 0.25/0.47 ( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 0.25/0.47 = ( top_top @ ( set @ nat ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_atMost_UNIV
% 0.25/0.47 thf(fact_2498_UN__lessThan__UNIV,axiom,
% 0.25/0.47 ( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 0.25/0.47 = ( top_top @ ( set @ nat ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_lessThan_UNIV
% 0.25/0.47 thf(fact_2499_Sup__option__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ( ( complete_Sup_Sup @ ( option @ A ) )
% 0.25/0.47 = ( ^ [A9: set @ ( option @ A )] :
% 0.25/0.47 ( if @ ( option @ A )
% 0.25/0.47 @ ( ( A9
% 0.25/0.47 = ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.25/0.47 | ( A9
% 0.25/0.47 = ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
% 0.25/0.47 @ ( none @ A )
% 0.25/0.47 @ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A9 ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_option_def
% 0.25/0.47 thf(fact_2500_snd__image__mp,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ A,X: B,Y: A] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) @ B6 )
% 0.25/0.47 => ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 )
% 0.25/0.47 => ( member @ A @ Y @ B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_image_mp
% 0.25/0.47 thf(fact_2501_fst__image__mp,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B6: set @ A,X: A,Y: B] :
% 0.25/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) @ B6 )
% 0.25/0.47 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A4 )
% 0.25/0.47 => ( member @ A @ X @ B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_image_mp
% 0.25/0.47 thf(fact_2502_prod_OUnion__comp,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( comm_monoid_mult @ A )
% 0.25/0.47 => ! [B6: set @ ( set @ B ),G: B > A] :
% 0.25/0.47 ( ! [X2: set @ B] :
% 0.25/0.47 ( ( member @ ( set @ B ) @ X2 @ B6 )
% 0.25/0.47 => ( finite_finite2 @ B @ X2 ) )
% 0.25/0.47 => ( ! [A14: set @ B] :
% 0.25/0.47 ( ( member @ ( set @ B ) @ A14 @ B6 )
% 0.25/0.47 => ! [A25: set @ B] :
% 0.25/0.47 ( ( member @ ( set @ B ) @ A25 @ B6 )
% 0.25/0.47 => ( ( A14 != A25 )
% 0.25/0.47 => ! [X2: B] :
% 0.25/0.47 ( ( member @ B @ X2 @ A14 )
% 0.25/0.47 => ( ( member @ B @ X2 @ A25 )
% 0.25/0.47 => ( ( G @ X2 )
% 0.25/0.47 = ( one_one @ A ) ) ) ) ) ) )
% 0.25/0.47 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
% 0.25/0.47 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B6 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % prod.Union_comp
% 0.25/0.47 thf(fact_2503_UN__UN__finite__eq,axiom,
% 0.25/0.47 ! [A: $tType,A4: nat > ( set @ A )] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.47 @ ( image2 @ nat @ ( set @ A )
% 0.25/0.47 @ ^ [N5: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
% 0.25/0.47 @ ( top_top @ ( set @ nat ) ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_UN_finite_eq
% 0.25/0.47 thf(fact_2504_UN__le__add__shift,axiom,
% 0.25/0.47 ! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.47 @ ( image2 @ nat @ ( set @ A )
% 0.25/0.47 @ ^ [I3: nat] : ( M3 @ ( plus_plus @ nat @ I3 @ K ) )
% 0.25/0.47 @ ( set_ord_atMost @ nat @ N ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_le_add_shift
% 0.25/0.47 thf(fact_2505_UN__le__add__shift__strict,axiom,
% 0.25/0.47 ! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.47 @ ( image2 @ nat @ ( set @ A )
% 0.25/0.47 @ ^ [I3: nat] : ( M3 @ ( plus_plus @ nat @ I3 @ K ) )
% 0.25/0.47 @ ( set_ord_lessThan @ nat @ N ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_le_add_shift_strict
% 0.25/0.47 thf(fact_2506_rat__sgn__code,axiom,
% 0.25/0.47 ! [P5: rat] :
% 0.25/0.47 ( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
% 0.25/0.47 = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % rat_sgn_code
% 0.25/0.47 thf(fact_2507_bezw__non__0,axiom,
% 0.25/0.47 ! [Y: nat,X: nat] :
% 0.25/0.47 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
% 0.25/0.47 => ( ( bezw @ X @ Y )
% 0.25/0.47 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % bezw_non_0
% 0.25/0.47 thf(fact_2508_UN__finite__subset,axiom,
% 0.25/0.47 ! [A: $tType,A4: nat > ( set @ A ),C5: set @ A] :
% 0.25/0.47 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_finite_subset
% 0.25/0.47 thf(fact_2509_UN__finite2__eq,axiom,
% 0.25/0.47 ! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
% 0.25/0.47 ( ! [N3: nat] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
% 0.25/0.47 => ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
% 0.25/0.47 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_finite2_eq
% 0.25/0.47 thf(fact_2510_bezw_Oelims,axiom,
% 0.25/0.47 ! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
% 0.25/0.47 ( ( ( bezw @ X @ Xa )
% 0.25/0.47 = Y )
% 0.25/0.47 => ( ( ( Xa
% 0.25/0.47 = ( zero_zero @ nat ) )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 0.25/0.47 & ( ( Xa
% 0.25/0.47 != ( zero_zero @ nat ) )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % bezw.elims
% 0.25/0.47 thf(fact_2511_bezw_Osimps,axiom,
% 0.25/0.47 ( bezw
% 0.25/0.47 = ( ^ [X3: nat,Y3: nat] :
% 0.25/0.47 ( if @ ( product_prod @ int @ int )
% 0.25/0.47 @ ( Y3
% 0.25/0.47 = ( zero_zero @ nat ) )
% 0.25/0.47 @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
% 0.25/0.47 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % bezw.simps
% 0.25/0.47 thf(fact_2512_max__ext__def,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( max_ext @ A )
% 0.25/0.47 = ( ^ [R6: set @ ( product_prod @ A @ A )] :
% 0.25/0.47 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.25/0.47 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 0.25/0.47 @ ( max_extp @ A
% 0.25/0.47 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % max_ext_def
% 0.25/0.47 thf(fact_2513_UN__finite2__subset,axiom,
% 0.25/0.47 ! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
% 0.25/0.47 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
% 0.25/0.47 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UN_finite2_subset
% 0.25/0.47 thf(fact_2514_bezw_Opelims,axiom,
% 0.25/0.47 ! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
% 0.25/0.47 ( ( ( bezw @ X @ Xa )
% 0.25/0.47 = Y )
% 0.25/0.47 => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
% 0.25/0.47 => ~ ( ( ( ( Xa
% 0.25/0.47 = ( zero_zero @ nat ) )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 0.25/0.47 & ( ( Xa
% 0.25/0.47 != ( zero_zero @ nat ) )
% 0.25/0.47 => ( Y
% 0.25/0.47 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
% 0.25/0.47 => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % bezw.pelims
% 0.25/0.47 thf(fact_2515_snd__diag__id,axiom,
% 0.25/0.47 ! [A: $tType,Z2: A] :
% 0.25/0.47 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
% 0.25/0.47 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 0.25/0.47 @ Z2 )
% 0.25/0.47 = ( id @ A @ Z2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_diag_id
% 0.25/0.47 thf(fact_2516_fst__diag__id,axiom,
% 0.25/0.47 ! [A: $tType,Z2: A] :
% 0.25/0.47 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
% 0.25/0.47 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 0.25/0.47 @ Z2 )
% 0.25/0.47 = ( id @ A @ Z2 ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_diag_id
% 0.25/0.47 thf(fact_2517_size__prod__simp,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( basic_BNF_size_prod @ A @ B )
% 0.25/0.47 = ( ^ [F2: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % size_prod_simp
% 0.25/0.47 thf(fact_2518_fst__diag__snd,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
% 0.25/0.47 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 0.25/0.47 @ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
% 0.25/0.47 @ ( product_snd @ A @ B ) ) )
% 0.25/0.47 = ( product_snd @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_diag_snd
% 0.25/0.47 thf(fact_2519_snd__diag__fst,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
% 0.25/0.47 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 0.25/0.47 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 0.25/0.47 @ ( product_fst @ A @ B ) ) )
% 0.25/0.47 = ( product_fst @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_diag_fst
% 0.25/0.47 thf(fact_2520_None__in__Inf,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ! [A4: set @ ( option @ A )] :
% 0.25/0.47 ( ( member @ ( option @ A ) @ ( none @ A ) @ A4 )
% 0.25/0.47 => ( ( complete_Inf_Inf @ ( option @ A ) @ A4 )
% 0.25/0.47 = ( none @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % None_in_Inf
% 0.25/0.47 thf(fact_2521_empty__Sup,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 0.25/0.47 = ( none @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % empty_Sup
% 0.25/0.47 thf(fact_2522_singleton__None__Sup,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( comple6319245703460814977attice @ A )
% 0.25/0.47 => ( ( complete_Sup_Sup @ ( option @ A ) @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) )
% 0.25/0.47 = ( none @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % singleton_None_Sup
% 0.25/0.47 thf(fact_2523_INF__INT__eq2,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
% 0.25/0.47 ( ( complete_Inf_Inf @ ( A > B > $o )
% 0.25/0.47 @ ( image2 @ C @ ( A > B > $o )
% 0.25/0.47 @ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R @ I3 ) )
% 0.25/0.47 @ S2 ) )
% 0.25/0.47 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % INF_INT_eq2
% 0.25/0.47 thf(fact_2524_INF__Int__eq2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
% 0.25/0.47 ( ( complete_Inf_Inf @ ( A > B > $o )
% 0.25/0.47 @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
% 0.25/0.47 @ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
% 0.25/0.47 @ S2 ) )
% 0.25/0.47 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % INF_Int_eq2
% 0.25/0.47 thf(fact_2525_SUP__UN__eq2,axiom,
% 0.25/0.47 ! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( A > B > $o )
% 0.25/0.47 @ ( image2 @ C @ ( A > B > $o )
% 0.25/0.47 @ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R @ I3 ) )
% 0.25/0.47 @ S2 ) )
% 0.25/0.47 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SUP_UN_eq2
% 0.25/0.47 thf(fact_2526_SUP__Sup__eq2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( A > B > $o )
% 0.25/0.47 @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
% 0.25/0.47 @ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
% 0.25/0.47 @ S2 ) )
% 0.25/0.47 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SUP_Sup_eq2
% 0.25/0.47 thf(fact_2527_Inf__INT__eq2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( complete_Inf_Inf @ ( A > B > $o ) )
% 0.25/0.47 = ( ^ [S5: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S5 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Inf_INT_eq2
% 0.25/0.47 thf(fact_2528_Sup__SUP__eq2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( complete_Sup_Sup @ ( A > B > $o ) )
% 0.25/0.47 = ( ^ [S5: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S5 ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sup_SUP_eq2
% 0.25/0.47 thf(fact_2529_fst__diag__fst,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
% 0.25/0.47 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 0.25/0.47 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 0.25/0.47 @ ( product_fst @ A @ B ) ) )
% 0.25/0.47 = ( product_fst @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_diag_fst
% 0.25/0.47 thf(fact_2530_snd__diag__snd,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
% 0.25/0.47 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 0.25/0.47 @ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
% 0.25/0.47 @ ( product_snd @ A @ B ) ) )
% 0.25/0.47 = ( product_snd @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_diag_snd
% 0.25/0.47 thf(fact_2531_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
% 0.25/0.47 ( ( P @ X @ Y )
% 0.25/0.47 => ( ( A3
% 0.25/0.47 = ( product_Pair @ A @ B @ X @ Y ) )
% 0.25/0.47 => ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % BNF_Greatest_Fixpoint.subst_Pair
% 0.25/0.47 thf(fact_2532_normalize__def,axiom,
% 0.25/0.47 ( normalize
% 0.25/0.47 = ( ^ [P6: product_prod @ int @ int] :
% 0.25/0.47 ( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
% 0.25/0.47 @ ( if @ ( product_prod @ int @ int )
% 0.25/0.47 @ ( ( product_snd @ int @ int @ P6 )
% 0.25/0.47 = ( zero_zero @ int ) )
% 0.25/0.47 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.47 @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % normalize_def
% 0.25/0.47 thf(fact_2533_range__prod,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,F: C > ( product_prod @ A @ B )] :
% 0.25/0.47 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F @ ( top_top @ ( set @ C ) ) )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) )
% 0.25/0.47 @ ^ [Uu: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % range_prod
% 0.25/0.47 thf(fact_2534_fun_Oin__rel,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,D: $tType,R5: A > B > $o,A3: D > A,B2: D > B] :
% 0.25/0.47 ( ( bNF_rel_fun @ D @ D @ A @ B
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ R5
% 0.25/0.47 @ A3
% 0.25/0.47 @ B2 )
% 0.25/0.47 = ( ? [Z5: D > ( product_prod @ A @ B )] :
% 0.25/0.47 ( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z5
% 0.25/0.47 @ ( collect @ ( D > ( product_prod @ A @ B ) )
% 0.25/0.47 @ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) ) )
% 0.25/0.47 & ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z5 )
% 0.25/0.47 = A3 )
% 0.25/0.47 & ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z5 )
% 0.25/0.47 = B2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.in_rel
% 0.25/0.47 thf(fact_2535_apfst__apsnd,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.25/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 0.25/0.47 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % apfst_apsnd
% 0.25/0.47 thf(fact_2536_apsnd__apfst,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
% 0.25/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 0.25/0.47 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % apsnd_apfst
% 0.25/0.47 thf(fact_2537_eq__snd__iff,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,B2: A,P5: product_prod @ B @ A] :
% 0.25/0.47 ( ( B2
% 0.25/0.47 = ( product_snd @ B @ A @ P5 ) )
% 0.25/0.47 = ( ? [A7: B] :
% 0.25/0.47 ( P5
% 0.25/0.47 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % eq_snd_iff
% 0.25/0.47 thf(fact_2538_SigmaI,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( member @ A @ A3 @ A4 )
% 0.25/0.47 => ( ( member @ B @ B2 @ ( B6 @ A3 ) )
% 0.25/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SigmaI
% 0.25/0.47 thf(fact_2539_mem__Sigma__iff,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 0.25/0.47 = ( ( member @ A @ A3 @ A4 )
% 0.25/0.47 & ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % mem_Sigma_iff
% 0.25/0.47 thf(fact_2540_gcd__add1,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( semiring_gcd @ A )
% 0.25/0.47 => ! [M2: A,N: A] :
% 0.25/0.47 ( ( gcd_gcd @ A @ ( plus_plus @ A @ M2 @ N ) @ N )
% 0.25/0.47 = ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % gcd_add1
% 0.25/0.47 thf(fact_2541_gcd__add2,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( semiring_gcd @ A )
% 0.25/0.47 => ! [M2: A,N: A] :
% 0.25/0.47 ( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ M2 @ N ) )
% 0.25/0.47 = ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % gcd_add2
% 0.25/0.47 thf(fact_2542_gcd_Obottom__left__bottom,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( semiring_gcd @ A )
% 0.25/0.47 => ! [A3: A] :
% 0.25/0.47 ( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.47 = ( one_one @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % gcd.bottom_left_bottom
% 0.25/0.47 thf(fact_2543_gcd_Obottom__right__bottom,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( semiring_gcd @ A )
% 0.25/0.47 => ! [A3: A] :
% 0.25/0.47 ( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.47 = ( one_one @ A ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % gcd.bottom_right_bottom
% 0.25/0.47 thf(fact_2544_apfst__conv,axiom,
% 0.25/0.47 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
% 0.25/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
% 0.25/0.47 = ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % apfst_conv
% 0.25/0.47 thf(fact_2545_apfst__eq__conv,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
% 0.25/0.47 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 0.25/0.47 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 0.25/0.47 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 0.25/0.47 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % apfst_eq_conv
% 0.25/0.47 thf(fact_2546_fst__apfst,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 0.25/0.47 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 0.25/0.47 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_apfst
% 0.25/0.47 thf(fact_2547_snd__apfst,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 0.25/0.47 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 0.25/0.47 = ( product_snd @ C @ A @ X ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_apfst
% 0.25/0.47 thf(fact_2548_Collect__case__prod,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 0.25/0.47 ( ( collect @ ( product_prod @ A @ B )
% 0.25/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.47 @ ^ [A7: A,B5: B] :
% 0.25/0.47 ( ( P @ A7 )
% 0.25/0.47 & ( Q @ B5 ) ) ) )
% 0.25/0.47 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.25/0.47 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_case_prod
% 0.25/0.47 thf(fact_2549_is__unit__gcd__iff,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( semiring_gcd @ A )
% 0.25/0.47 => ! [A3: A,B2: A] :
% 0.25/0.47 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 0.25/0.47 = ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.25/0.47 = ( one_one @ A ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % is_unit_gcd_iff
% 0.25/0.47 thf(fact_2550_Sigma__empty1,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 0.25/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_empty1
% 0.25/0.47 thf(fact_2551_Sigma__empty2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ A] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 0.25/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_empty2
% 0.25/0.47 thf(fact_2552_Times__empty,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.47 ( ( ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : B6 )
% 0.25/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.47 = ( ( A4
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 | ( B6
% 0.25/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Times_empty
% 0.25/0.47 thf(fact_2553_Sigma__UNIV__cancel,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ A,X10: set @ B] :
% 0.25/0.47 ( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : X10 )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 0.25/0.47 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_UNIV_cancel
% 0.25/0.47 thf(fact_2554_Compl__Times__UNIV2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ A] :
% 0.25/0.47 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 0.25/0.47 = ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A4 )
% 0.25/0.47 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Compl_Times_UNIV2
% 0.25/0.47 thf(fact_2555_Compl__Times__UNIV1,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ B] :
% 0.25/0.47 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.25/0.47 @ ^ [Uu: A] : A4 ) )
% 0.25/0.47 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.25/0.47 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Compl_Times_UNIV1
% 0.25/0.47 thf(fact_2556_snd__comp__apfst,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.25/0.47 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.25/0.47 = ( product_snd @ A @ B ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_comp_apfst
% 0.25/0.47 thf(fact_2557_UNIV__Times__UNIV,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.25/0.47 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 0.25/0.47 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % UNIV_Times_UNIV
% 0.25/0.47 thf(fact_2558_apfst__id,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType] :
% 0.25/0.47 ( ( product_apfst @ A @ A @ B @ ( id @ A ) )
% 0.25/0.47 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % apfst_id
% 0.25/0.47 thf(fact_2559_pairself__image__cart,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,F: B > A,A4: set @ B,B6: set @ B] :
% 0.25/0.47 ( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F )
% 0.25/0.47 @ ( product_Sigma @ B @ B @ A4
% 0.25/0.47 @ ^ [Uu: B] : B6 ) )
% 0.25/0.47 = ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A4 )
% 0.25/0.47 @ ^ [Uu: A] : ( image2 @ B @ A @ F @ B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % pairself_image_cart
% 0.25/0.47 thf(fact_2560_fst__image__times,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,B6: set @ B,A4: set @ A] :
% 0.25/0.47 ( ( ( B6
% 0.25/0.47 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.47 => ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : B6 ) )
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.47 & ( ( B6
% 0.25/0.47 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.47 => ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : B6 ) )
% 0.25/0.47 = A4 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_image_times
% 0.25/0.47 thf(fact_2561_snd__image__times,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
% 0.25/0.47 ( ( ( A4
% 0.25/0.47 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.47 => ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.25/0.47 @ ( product_Sigma @ B @ A @ A4
% 0.25/0.47 @ ^ [Uu: B] : B6 ) )
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.47 & ( ( A4
% 0.25/0.47 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.47 => ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.25/0.47 @ ( product_Sigma @ B @ A @ A4
% 0.25/0.47 @ ^ [Uu: B] : B6 ) )
% 0.25/0.47 = B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % snd_image_times
% 0.25/0.47 thf(fact_2562_fst__comp__apfst,axiom,
% 0.25/0.47 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 0.25/0.47 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 0.25/0.47 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % fst_comp_apfst
% 0.25/0.47 thf(fact_2563_fun_Omap__transfer,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,D: $tType,G3: $tType,F5: $tType,Rb2: A > F5 > $o,Sd: B > G3 > $o] :
% 0.25/0.47 ( bNF_rel_fun @ ( A > B ) @ ( F5 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F5 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F5 @ B @ G3 @ Rb2 @ Sd )
% 0.25/0.47 @ ( bNF_rel_fun @ ( D > A ) @ ( D > F5 ) @ ( D > B ) @ ( D > G3 )
% 0.25/0.47 @ ( bNF_rel_fun @ D @ D @ A @ F5
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ Rb2 )
% 0.25/0.47 @ ( bNF_rel_fun @ D @ D @ B @ G3
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ Sd ) )
% 0.25/0.47 @ ( comp @ A @ B @ D )
% 0.25/0.47 @ ( comp @ F5 @ G3 @ D ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.map_transfer
% 0.25/0.47 thf(fact_2564_comp__transfer,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,E: $tType,F5: $tType,D: $tType,C: $tType,B6: A > C > $o,C5: B > D > $o,A4: E > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F5 > C ) > F5 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B6 @ C5 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F5 > C ) @ ( E > B ) @ ( F5 > D ) @ ( bNF_rel_fun @ E @ F5 @ A @ C @ A4 @ B6 ) @ ( bNF_rel_fun @ E @ F5 @ B @ D @ A4 @ C5 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F5 ) ) ).
% 0.25/0.47
% 0.25/0.47 % comp_transfer
% 0.25/0.47 thf(fact_2565_power__transfer,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( ( power @ B )
% 0.25/0.47 & ( power @ A ) )
% 0.25/0.47 => ! [R5: A > B > $o] :
% 0.25/0.47 ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.25/0.47 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 0.25/0.47 => ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R5
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ R5 )
% 0.25/0.47 @ ( power_power @ A )
% 0.25/0.47 @ ( power_power @ B ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % power_transfer
% 0.25/0.47 thf(fact_2566_times__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( times_times @ int )
% 0.25/0.47 @ ( times_times @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % times_integer.rsp
% 0.25/0.47 thf(fact_2567_times__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( times_times @ nat )
% 0.25/0.47 @ ( times_times @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % times_natural.rsp
% 0.25/0.47 thf(fact_2568_Times__eq__cancel2,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B6: set @ B] :
% 0.25/0.47 ( ( member @ A @ X @ C5 )
% 0.25/0.47 => ( ( ( product_Sigma @ B @ A @ A4
% 0.25/0.47 @ ^ [Uu: B] : C5 )
% 0.25/0.47 = ( product_Sigma @ B @ A @ B6
% 0.25/0.47 @ ^ [Uu: B] : C5 ) )
% 0.25/0.47 = ( A4 = B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Times_eq_cancel2
% 0.25/0.47 thf(fact_2569_Sigma__cong,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: A > ( set @ B ),D4: A > ( set @ B )] :
% 0.25/0.47 ( ( A4 = B6 )
% 0.25/0.47 => ( ! [X2: A] :
% 0.25/0.47 ( ( member @ A @ X2 @ B6 )
% 0.25/0.47 => ( ( C5 @ X2 )
% 0.25/0.47 = ( D4 @ X2 ) ) )
% 0.25/0.47 => ( ( product_Sigma @ A @ B @ A4 @ C5 )
% 0.25/0.47 = ( product_Sigma @ A @ B @ B6 @ D4 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_cong
% 0.25/0.47 thf(fact_2570_divide__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( divide_divide @ nat )
% 0.25/0.47 @ ( divide_divide @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % divide_natural.rsp
% 0.25/0.47 thf(fact_2571_divide__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( divide_divide @ int )
% 0.25/0.47 @ ( divide_divide @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % divide_integer.rsp
% 0.25/0.47 thf(fact_2572_or__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se1065995026697491101ons_or @ nat )
% 0.25/0.47 @ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % or_natural.rsp
% 0.25/0.47 thf(fact_2573_or__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se1065995026697491101ons_or @ int )
% 0.25/0.47 @ ( bit_se1065995026697491101ons_or @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % or_integer.rsp
% 0.25/0.47 thf(fact_2574_set__bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se5668285175392031749et_bit @ int )
% 0.25/0.47 @ ( bit_se5668285175392031749et_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % set_bit_integer.rsp
% 0.25/0.47 thf(fact_2575_set__bit__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se5668285175392031749et_bit @ nat )
% 0.25/0.47 @ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % set_bit_natural.rsp
% 0.25/0.47 thf(fact_2576_xor__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se5824344971392196577ns_xor @ int )
% 0.25/0.47 @ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % xor_integer.rsp
% 0.25/0.47 thf(fact_2577_xor__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se5824344971392196577ns_xor @ nat )
% 0.25/0.47 @ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % xor_natural.rsp
% 0.25/0.47 thf(fact_2578_and__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se5824344872417868541ns_and @ int )
% 0.25/0.47 @ ( bit_se5824344872417868541ns_and @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % and_integer.rsp
% 0.25/0.47 thf(fact_2579_and__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se5824344872417868541ns_and @ nat )
% 0.25/0.47 @ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % and_natural.rsp
% 0.25/0.47 thf(fact_2580_modulo__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( modulo_modulo @ int )
% 0.25/0.47 @ ( modulo_modulo @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % modulo_integer.rsp
% 0.25/0.47 thf(fact_2581_modulo__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( modulo_modulo @ nat )
% 0.25/0.47 @ ( modulo_modulo @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % modulo_natural.rsp
% 0.25/0.47 thf(fact_2582_less__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( ord_less @ int )
% 0.25/0.47 @ ( ord_less @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % less_integer.rsp
% 0.25/0.47 thf(fact_2583_less__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( ord_less @ nat )
% 0.25/0.47 @ ( ord_less @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % less_natural.rsp
% 0.25/0.47 thf(fact_2584_unset__bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se2638667681897837118et_bit @ int )
% 0.25/0.47 @ ( bit_se2638667681897837118et_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % unset_bit_integer.rsp
% 0.25/0.47 thf(fact_2585_take__bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se2584673776208193580ke_bit @ int )
% 0.25/0.47 @ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % take_bit_integer.rsp
% 0.25/0.47 thf(fact_2586_flip__bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se8732182000553998342ip_bit @ int )
% 0.25/0.47 @ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % flip_bit_integer.rsp
% 0.25/0.47 thf(fact_2587_flip__bit__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se8732182000553998342ip_bit @ nat )
% 0.25/0.47 @ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % flip_bit_natural.rsp
% 0.25/0.47 thf(fact_2588_push__bit__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se4730199178511100633sh_bit @ nat )
% 0.25/0.47 @ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % push_bit_natural.rsp
% 0.25/0.47 thf(fact_2589_push__bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se4730199178511100633sh_bit @ int )
% 0.25/0.47 @ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % push_bit_integer.rsp
% 0.25/0.47 thf(fact_2590_plus__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( plus_plus @ int )
% 0.25/0.47 @ ( plus_plus @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % plus_integer.rsp
% 0.25/0.47 thf(fact_2591_plus__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( plus_plus @ nat )
% 0.25/0.47 @ ( plus_plus @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % plus_natural.rsp
% 0.25/0.47 thf(fact_2592_drop__bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( bit_se4197421643247451524op_bit @ int )
% 0.25/0.47 @ ( bit_se4197421643247451524op_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % drop_bit_integer.rsp
% 0.25/0.47 thf(fact_2593_minus__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( minus_minus @ nat )
% 0.25/0.47 @ ( minus_minus @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % minus_natural.rsp
% 0.25/0.47 thf(fact_2594_minus__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( minus_minus @ int )
% 0.25/0.47 @ ( minus_minus @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % minus_integer.rsp
% 0.25/0.47 thf(fact_2595_less__eq__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( ord_less_eq @ int )
% 0.25/0.47 @ ( ord_less_eq @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % less_eq_integer.rsp
% 0.25/0.47 thf(fact_2596_less__eq__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ( ord_less_eq @ nat )
% 0.25/0.47 @ ( ord_less_eq @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % less_eq_natural.rsp
% 0.25/0.47 thf(fact_2597_SigmaE,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A4: set @ A,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 0.25/0.47 => ~ ! [X2: A] :
% 0.25/0.47 ( ( member @ A @ X2 @ A4 )
% 0.25/0.47 => ! [Y4: B] :
% 0.25/0.47 ( ( member @ B @ Y4 @ ( B6 @ X2 ) )
% 0.25/0.47 => ( C2
% 0.25/0.47 != ( product_Pair @ A @ B @ X2 @ Y4 ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SigmaE
% 0.25/0.47 thf(fact_2598_SigmaD1,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 0.25/0.47 => ( member @ A @ A3 @ A4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % SigmaD1
% 0.25/0.47 thf(fact_2599_SigmaD2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 0.25/0.47 => ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SigmaD2
% 0.25/0.47 thf(fact_2600_SigmaE2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 0.25/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 0.25/0.47 => ~ ( ( member @ A @ A3 @ A4 )
% 0.25/0.47 => ~ ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % SigmaE2
% 0.25/0.47 thf(fact_2601_Suc_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ suc
% 0.25/0.47 @ suc ) ).
% 0.25/0.47
% 0.25/0.47 % Suc.rsp
% 0.25/0.47 thf(fact_2602_sub_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
% 0.25/0.47 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bNF_rel_fun @ num @ num @ int @ int
% 0.25/0.47 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.47 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) )
% 0.25/0.47 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % sub.rsp
% 0.25/0.47 thf(fact_2603_transfer__rule__numeral,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( ( monoid_add @ B )
% 0.25/0.47 & ( semiring_numeral @ B )
% 0.25/0.47 & ( monoid_add @ A )
% 0.25/0.47 & ( semiring_numeral @ A ) )
% 0.25/0.47 => ! [R5: A > B > $o] :
% 0.25/0.47 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.25/0.47 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.25/0.47 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.25/0.47 => ( bNF_rel_fun @ num @ num @ A @ B
% 0.25/0.47 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.47 @ R5
% 0.25/0.47 @ ( numeral_numeral @ A )
% 0.25/0.47 @ ( numeral_numeral @ B ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % transfer_rule_numeral
% 0.25/0.47 thf(fact_2604_uminus__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( uminus_uminus @ int )
% 0.25/0.47 @ ( uminus_uminus @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % uminus_integer.rsp
% 0.25/0.47 thf(fact_2605_integer__of__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ int @ int
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( semiring_1_of_nat @ int )
% 0.25/0.47 @ ( semiring_1_of_nat @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % integer_of_natural.rsp
% 0.25/0.47 thf(fact_2606_Sigma__Diff__distrib1,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I4 @ J4 ) @ C5 )
% 0.25/0.47 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_Diff_distrib1
% 0.25/0.47 thf(fact_2607_natural__of__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ nat @ nat
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ nat2
% 0.25/0.47 @ nat2 ) ).
% 0.25/0.47
% 0.25/0.47 % natural_of_integer.rsp
% 0.25/0.47 thf(fact_2608_abs__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( abs_abs @ int )
% 0.25/0.47 @ ( abs_abs @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % abs_integer.rsp
% 0.25/0.47 thf(fact_2609_bit__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bit_se5641148757651400278ts_bit @ int )
% 0.25/0.47 @ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % bit_integer.rsp
% 0.25/0.47 thf(fact_2610_not__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bit_ri4277139882892585799ns_not @ int )
% 0.25/0.47 @ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % not_integer.rsp
% 0.25/0.47 thf(fact_2611_sgn__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( sgn_sgn @ int )
% 0.25/0.47 @ ( sgn_sgn @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % sgn_integer.rsp
% 0.25/0.47 thf(fact_2612_mask__integer_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ int @ int
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bit_se2239418461657761734s_mask @ int )
% 0.25/0.47 @ ( bit_se2239418461657761734s_mask @ int ) ) ).
% 0.25/0.47
% 0.25/0.47 % mask_integer.rsp
% 0.25/0.47 thf(fact_2613_mask__natural_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.47 @ ( bit_se2239418461657761734s_mask @ nat )
% 0.25/0.47 @ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% 0.25/0.47
% 0.25/0.47 % mask_natural.rsp
% 0.25/0.47 thf(fact_2614_transfer__rule__of__int,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType] :
% 0.25/0.47 ( ( ( ring_1 @ B )
% 0.25/0.47 & ( ring_1 @ A ) )
% 0.25/0.47 => ! [R5: A > B > $o] :
% 0.25/0.47 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.25/0.47 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.25/0.47 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.25/0.47 => ( ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
% 0.25/0.47 => ( bNF_rel_fun @ int @ int @ A @ B
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ R5
% 0.25/0.47 @ ( ring_1_of_int @ A )
% 0.25/0.47 @ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % transfer_rule_of_int
% 0.25/0.47 thf(fact_2615_fun_Orel__map_I1_J,axiom,
% 0.25/0.47 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
% 0.25/0.47 ( ( bNF_rel_fun @ D @ D @ C @ B
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ Sb
% 0.25/0.47 @ ( comp @ A @ C @ D @ I @ X )
% 0.25/0.47 @ Y )
% 0.25/0.47 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
% 0.25/0.47 @ X
% 0.25/0.47 @ Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.rel_map(1)
% 0.25/0.47 thf(fact_2616_fun_Orel__map_I2_J,axiom,
% 0.25/0.47 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G: B > C,Y: D > B] :
% 0.25/0.47 ( ( bNF_rel_fun @ D @ D @ A @ C
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ Sa
% 0.25/0.47 @ X
% 0.25/0.47 @ ( comp @ B @ C @ D @ G @ Y ) )
% 0.25/0.47 = ( bNF_rel_fun @ D @ D @ A @ B
% 0.25/0.47 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
% 0.25/0.47 @ X
% 0.25/0.47 @ Y ) ) ).
% 0.25/0.47
% 0.25/0.47 % fun.rel_map(2)
% 0.25/0.47 thf(fact_2617_times__eq__iff,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
% 0.25/0.47 ( ( ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : B6 )
% 0.25/0.47 = ( product_Sigma @ A @ B @ C5
% 0.25/0.47 @ ^ [Uu: A] : D4 ) )
% 0.25/0.47 = ( ( ( A4 = C5 )
% 0.25/0.47 & ( B6 = D4 ) )
% 0.25/0.47 | ( ( ( A4
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 | ( B6
% 0.25/0.47 = ( bot_bot @ ( set @ B ) ) ) )
% 0.25/0.47 & ( ( C5
% 0.25/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.47 | ( D4
% 0.25/0.47 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % times_eq_iff
% 0.25/0.47 thf(fact_2618_dup_Orsp,axiom,
% 0.25/0.47 ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.47 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
% 0.25/0.47 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ).
% 0.25/0.47
% 0.25/0.47 % dup.rsp
% 0.25/0.47 thf(fact_2619_Sigma__Diff__distrib2,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ I4
% 0.25/0.47 @ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
% 0.25/0.47 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_Diff_distrib2
% 0.25/0.47 thf(fact_2620_Times__Diff__distrib1,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.47 @ ^ [Uu: A] : C5 )
% 0.25/0.47 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.47 @ ^ [Uu: A] : C5 )
% 0.25/0.47 @ ( product_Sigma @ A @ B @ B6
% 0.25/0.47 @ ^ [Uu: A] : C5 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Times_Diff_distrib1
% 0.25/0.47 thf(fact_2621_Sigma__Union,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,X10: set @ ( set @ A ),B6: A > ( set @ B )] :
% 0.25/0.47 ( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X10 ) @ B6 )
% 0.25/0.47 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.47 @ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.47 @ ^ [A9: set @ A] : ( product_Sigma @ A @ B @ A9 @ B6 )
% 0.25/0.47 @ X10 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Sigma_Union
% 0.25/0.47 thf(fact_2622_Collect__case__prod__Sigma,axiom,
% 0.25/0.47 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 0.25/0.47 ( ( collect @ ( product_prod @ A @ B )
% 0.25/0.47 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.47 @ ^ [X3: A,Y3: B] :
% 0.25/0.47 ( ( P @ X3 )
% 0.25/0.47 & ( Q @ X3 @ Y3 ) ) ) )
% 0.25/0.47 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 0.25/0.47 @ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % Collect_case_prod_Sigma
% 0.25/0.47 thf(fact_2623_apfst__compose,axiom,
% 0.25/0.47 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
% 0.25/0.47 ( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
% 0.25/0.47 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% 0.25/0.47
% 0.25/0.47 % apfst_compose
% 0.25/0.47 thf(fact_2624_apsnd__apfst__commute,axiom,
% 0.25/0.47 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P5: product_prod @ D @ C] :
% 0.25/0.47 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P5 ) )
% 0.25/0.47 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P5 ) ) ) ).
% 0.25/0.47
% 0.25/0.47 % apsnd_apfst_commute
% 0.25/0.47 thf(fact_2625_gcd__add__mult,axiom,
% 0.25/0.47 ! [A: $tType] :
% 0.25/0.47 ( ( semiring_gcd @ A )
% 0.25/0.47 => ! [M2: A,K: A,N: A] :
% 0.25/0.47 ( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K @ M2 ) @ N ) )
% 0.25/0.47 = ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_add_mult
% 0.25/0.48 thf(fact_2626_Sigma__mono,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,C5: set @ A,B6: A > ( set @ B ),D4: A > ( set @ B )] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
% 0.25/0.48 => ( ! [X2: A] :
% 0.25/0.48 ( ( member @ A @ X2 @ A4 )
% 0.25/0.48 => ( ord_less_eq @ ( set @ B ) @ ( B6 @ X2 ) @ ( D4 @ X2 ) ) )
% 0.25/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) @ ( product_Sigma @ A @ B @ C5 @ D4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_mono
% 0.25/0.48 thf(fact_2627_bezout__int,axiom,
% 0.25/0.48 ! [X: int,Y: int] :
% 0.25/0.48 ? [U3: int,V4: int] :
% 0.25/0.48 ( ( plus_plus @ int @ ( times_times @ int @ U3 @ X ) @ ( times_times @ int @ V4 @ Y ) )
% 0.25/0.48 = ( gcd_gcd @ int @ X @ Y ) ) ).
% 0.25/0.48
% 0.25/0.48 % bezout_int
% 0.25/0.48 thf(fact_2628_Times__subset__cancel2,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B6: set @ B] :
% 0.25/0.48 ( ( member @ A @ X @ C5 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ( product_Sigma @ B @ A @ A4
% 0.25/0.48 @ ^ [Uu: B] : C5 )
% 0.25/0.48 @ ( product_Sigma @ B @ A @ B6
% 0.25/0.48 @ ^ [Uu: B] : C5 ) )
% 0.25/0.48 = ( ord_less_eq @ ( set @ B ) @ A4 @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Times_subset_cancel2
% 0.25/0.48 thf(fact_2629_UN__Times__distrib,axiom,
% 0.25/0.48 ! [C: $tType,D: $tType,B: $tType,A: $tType,E4: C > ( set @ A ),F6: D > ( set @ B ),A4: set @ C,B6: set @ D] :
% 0.25/0.48 ( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ^ [A7: C,B5: D] :
% 0.25/0.48 ( product_Sigma @ A @ B @ ( E4 @ A7 )
% 0.25/0.48 @ ^ [Uu: A] : ( F6 @ B5 ) ) )
% 0.25/0.48 @ ( product_Sigma @ C @ D @ A4
% 0.25/0.48 @ ^ [Uu: C] : B6 ) ) )
% 0.25/0.48 = ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E4 @ A4 ) )
% 0.25/0.48 @ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F6 @ B6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % UN_Times_distrib
% 0.25/0.48 thf(fact_2630_mem__Times__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ X
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : B6 ) )
% 0.25/0.48 = ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
% 0.25/0.48 & ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % mem_Times_iff
% 0.25/0.48 thf(fact_2631_in__prod__fst__sndI,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
% 0.25/0.48 ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
% 0.25/0.48 => ( ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ B ) @ X
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : B6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % in_prod_fst_sndI
% 0.25/0.48 thf(fact_2632_swap__product,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 0.25/0.48 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [I3: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I3 ) )
% 0.25/0.48 @ ( product_Sigma @ B @ A @ A4
% 0.25/0.48 @ ^ [Uu: B] : B6 ) )
% 0.25/0.48 = ( product_Sigma @ A @ B @ B6
% 0.25/0.48 @ ^ [Uu: A] : A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % swap_product
% 0.25/0.48 thf(fact_2633_Sigma__empty__iff,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,I4: set @ A,X10: A > ( set @ B )] :
% 0.25/0.48 ( ( ( product_Sigma @ A @ B @ I4 @ X10 )
% 0.25/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.48 = ( ! [X3: A] :
% 0.25/0.48 ( ( member @ A @ X3 @ I4 )
% 0.25/0.48 => ( ( X10 @ X3 )
% 0.25/0.48 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_empty_iff
% 0.25/0.48 thf(fact_2634_gcd__mult__unit2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 0.25/0.48 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_mult_unit2
% 0.25/0.48 thf(fact_2635_gcd__mult__unit1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 0.25/0.48 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_mult_unit1
% 0.25/0.48 thf(fact_2636_gcd__div__unit1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 0.25/0.48 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_div_unit1
% 0.25/0.48 thf(fact_2637_gcd__div__unit2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 0.25/0.48 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_div_unit2
% 0.25/0.48 thf(fact_2638_less__int_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 0.25/0.48 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 0.25/0.48 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 0.25/0.48 @ ( ord_less @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % less_int.transfer
% 0.25/0.48 thf(fact_2639_less__eq__int_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 0.25/0.48 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 0.25/0.48 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 0.25/0.48 @ ( ord_less_eq @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % less_eq_int.transfer
% 0.25/0.48 thf(fact_2640_plus__int_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
% 0.25/0.48 @ ( plus_plus @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % plus_int.transfer
% 0.25/0.48 thf(fact_2641_minus__int_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
% 0.25/0.48 @ ( minus_minus @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % minus_int.transfer
% 0.25/0.48 thf(fact_2642_times__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,A4: set @ A,C5: set @ B,B6: set @ A,D4: set @ B] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : C5 )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ B6
% 0.25/0.48 @ ^ [Uu: A] : D4 ) )
% 0.25/0.48 = ( ( A4
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.48 | ( C5
% 0.25/0.48 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.48 | ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 & ( ord_less_eq @ ( set @ B ) @ C5 @ D4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_subset_iff
% 0.25/0.48 thf(fact_2643_image__paired__Times,axiom,
% 0.25/0.48 ! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A4: set @ C,B6: set @ D] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 0.25/0.48 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.25/0.48 @ ( product_Sigma @ C @ D @ A4
% 0.25/0.48 @ ^ [Uu: C] : B6 ) )
% 0.25/0.48 = ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A4 )
% 0.25/0.48 @ ^ [Uu: A] : ( image2 @ D @ B @ G @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_paired_Times
% 0.25/0.48 thf(fact_2644_fst__image__Sigma,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,B6: A > ( set @ B )] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 0.25/0.48 = ( collect @ A
% 0.25/0.48 @ ^ [X3: A] :
% 0.25/0.48 ( ( member @ A @ X3 @ A4 )
% 0.25/0.48 & ( ( B6 @ X3 )
% 0.25/0.48 != ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fst_image_Sigma
% 0.25/0.48 thf(fact_2645_Sigma__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( product_Sigma @ A @ B )
% 0.25/0.48 = ( ^ [A9: set @ A,B9: A > ( set @ B )] :
% 0.25/0.48 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ^ [X3: A] :
% 0.25/0.48 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.48 @ ( B9 @ X3 ) ) )
% 0.25/0.48 @ A9 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_def
% 0.25/0.48 thf(fact_2646_times__int_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
% 0.25/0.48 @ ( times_times @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_int.transfer
% 0.25/0.48 thf(fact_2647_num__of__integer_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ int @ int @ num @ num
% 0.25/0.48 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.48 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 0.25/0.48 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % num_of_integer.rsp
% 0.25/0.48 thf(fact_2648_snd__image__Sigma,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,A4: set @ B,B6: B > ( set @ A )] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A4 @ B6 ) )
% 0.25/0.48 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % snd_image_Sigma
% 0.25/0.48 thf(fact_2649_subset__fst__snd,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4
% 0.25/0.48 @ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
% 0.25/0.48 @ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % subset_fst_snd
% 0.25/0.48 thf(fact_2650_int__transfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ pcr_int
% 0.25/0.48 @ ^ [N5: nat] : ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) )
% 0.25/0.48 @ ( semiring_1_of_nat @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % int_transfer
% 0.25/0.48 thf(fact_2651_uminus__int_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
% 0.25/0.48 @ ( uminus_uminus @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_int.transfer
% 0.25/0.48 thf(fact_2652_eq__fst__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,A3: A,P5: product_prod @ A @ B] :
% 0.25/0.48 ( ( A3
% 0.25/0.48 = ( product_fst @ A @ B @ P5 ) )
% 0.25/0.48 = ( ? [B5: B] :
% 0.25/0.48 ( P5
% 0.25/0.48 = ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % eq_fst_iff
% 0.25/0.48 thf(fact_2653_transfer__rule__of__nat,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( semiring_1 @ B )
% 0.25/0.48 & ( semiring_1 @ A ) )
% 0.25/0.48 => ! [R5: A > B > $o] :
% 0.25/0.48 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.25/0.48 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.25/0.48 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.25/0.48 => ( bNF_rel_fun @ nat @ nat @ A @ B
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ R5
% 0.25/0.48 @ ( semiring_1_of_nat @ A )
% 0.25/0.48 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % transfer_rule_of_nat
% 0.25/0.48 thf(fact_2654_transfer__rule__of__bool,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ( zero_neq_one @ B )
% 0.25/0.48 & ( zero_neq_one @ A ) )
% 0.25/0.48 => ! [R5: A > B > $o] :
% 0.25/0.48 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.25/0.48 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.25/0.48 => ( bNF_rel_fun @ $o @ $o @ A @ B
% 0.25/0.48 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.25/0.48 @ R5
% 0.25/0.48 @ ( zero_neq_one_of_bool @ A )
% 0.25/0.48 @ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % transfer_rule_of_bool
% 0.25/0.48 thf(fact_2655_plus__rat_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 0.25/0.48 @ ( plus_plus @ rat ) ) ).
% 0.25/0.48
% 0.25/0.48 % plus_rat.transfer
% 0.25/0.48 thf(fact_2656_inverse__rat_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] :
% 0.25/0.48 ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( ( product_fst @ int @ int @ X3 )
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
% 0.25/0.48 @ ( inverse_inverse @ rat ) ) ).
% 0.25/0.48
% 0.25/0.48 % inverse_rat.transfer
% 0.25/0.48 thf(fact_2657_Id__on__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( id_on @ A )
% 0.25/0.48 = ( ^ [A9: set @ A] :
% 0.25/0.48 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.48 @ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.48 @ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.25/0.48 @ A9 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_on_def
% 0.25/0.48 thf(fact_2658_times__rat_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 0.25/0.48 @ ( times_times @ rat ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_rat.transfer
% 0.25/0.48 thf(fact_2659_gcd__1__nat,axiom,
% 0.25/0.48 ! [M2: nat] :
% 0.25/0.48 ( ( gcd_gcd @ nat @ M2 @ ( one_one @ nat ) )
% 0.25/0.48 = ( one_one @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_1_nat
% 0.25/0.48 thf(fact_2660_Id__onI,axiom,
% 0.25/0.48 ! [A: $tType,A3: A,A4: set @ A] :
% 0.25/0.48 ( ( member @ A @ A3 @ A4 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_onI
% 0.25/0.48 thf(fact_2661_unset__bit__natural_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( bit_se2638667681897837118et_bit @ nat )
% 0.25/0.48 @ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % unset_bit_natural.rsp
% 0.25/0.48 thf(fact_2662_take__bit__natural_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( bit_se2584673776208193580ke_bit @ nat )
% 0.25/0.48 @ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % take_bit_natural.rsp
% 0.25/0.48 thf(fact_2663_bit__natural_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 0.25/0.48 @ ( bit_se5641148757651400278ts_bit @ nat )
% 0.25/0.48 @ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % bit_natural.rsp
% 0.25/0.48 thf(fact_2664_drop__bit__natural_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( bit_se4197421643247451524op_bit @ nat )
% 0.25/0.48 @ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % drop_bit_natural.rsp
% 0.25/0.48 thf(fact_2665_Id__on__iff,axiom,
% 0.25/0.48 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A4 ) )
% 0.25/0.48 = ( ( X = Y )
% 0.25/0.48 & ( member @ A @ X @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_on_iff
% 0.25/0.48 thf(fact_2666_Id__on__eqI,axiom,
% 0.25/0.48 ! [A: $tType,A3: A,B2: A,A4: set @ A] :
% 0.25/0.48 ( ( A3 = B2 )
% 0.25/0.48 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_on_eqI
% 0.25/0.48 thf(fact_2667_Id__onE,axiom,
% 0.25/0.48 ! [A: $tType,C2: product_prod @ A @ A,A4: set @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A4 ) )
% 0.25/0.48 => ~ ! [X2: A] :
% 0.25/0.48 ( ( member @ A @ X2 @ A4 )
% 0.25/0.48 => ( C2
% 0.25/0.48 != ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_onE
% 0.25/0.48 thf(fact_2668_bezout__nat,axiom,
% 0.25/0.48 ! [A3: nat,B2: nat] :
% 0.25/0.48 ( ( A3
% 0.25/0.48 != ( zero_zero @ nat ) )
% 0.25/0.48 => ? [X2: nat,Y4: nat] :
% 0.25/0.48 ( ( times_times @ nat @ A3 @ X2 )
% 0.25/0.48 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % bezout_nat
% 0.25/0.48 thf(fact_2669_one__rat_Otransfer,axiom,
% 0.25/0.48 pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% 0.25/0.48
% 0.25/0.48 % one_rat.transfer
% 0.25/0.48 thf(fact_2670_zero__rat_Otransfer,axiom,
% 0.25/0.48 pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% 0.25/0.48
% 0.25/0.48 % zero_rat.transfer
% 0.25/0.48 thf(fact_2671_bezw__aux,axiom,
% 0.25/0.48 ! [X: nat,Y: nat] :
% 0.25/0.48 ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y ) )
% 0.25/0.48 = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % bezw_aux
% 0.25/0.48 thf(fact_2672_uminus__rat_Otransfer,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
% 0.25/0.48 @ ( uminus_uminus @ rat ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_rat.transfer
% 0.25/0.48 thf(fact_2673_gcd__nat_Opelims,axiom,
% 0.25/0.48 ! [X: nat,Xa: nat,Y: nat] :
% 0.25/0.48 ( ( ( gcd_gcd @ nat @ X @ Xa )
% 0.25/0.48 = Y )
% 0.25/0.48 => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
% 0.25/0.48 => ~ ( ( ( ( Xa
% 0.25/0.48 = ( zero_zero @ nat ) )
% 0.25/0.48 => ( Y = X ) )
% 0.25/0.48 & ( ( Xa
% 0.25/0.48 != ( zero_zero @ nat ) )
% 0.25/0.48 => ( Y
% 0.25/0.48 = ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
% 0.25/0.48 => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % gcd_nat.pelims
% 0.25/0.48 thf(fact_2674_o__rsp_I1_J,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,E: $tType,F5: $tType,D: $tType,C: $tType,R23: A > C > $o,R32: B > D > $o,R12: E > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F5 > C ) > F5 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R23 @ R32 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F5 > C ) @ ( E > B ) @ ( F5 > D ) @ ( bNF_rel_fun @ E @ F5 @ A @ C @ R12 @ R23 ) @ ( bNF_rel_fun @ E @ F5 @ B @ D @ R12 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % o_rsp(1)
% 0.25/0.48 thf(fact_2675_o__rsp_I2_J,axiom,
% 0.25/0.48 ! [E: $tType,F5: $tType,H8: $tType,G3: $tType,R12: E > F5 > $o] :
% 0.25/0.48 ( bNF_rel_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( E > G3 ) > E > H8 ) @ ( ( F5 > G3 ) > F5 > H8 )
% 0.25/0.48 @ ^ [Y5: G3 > H8,Z4: G3 > H8] : ( Y5 = Z4 )
% 0.25/0.48 @ ( bNF_rel_fun @ ( E > G3 ) @ ( F5 > G3 ) @ ( E > H8 ) @ ( F5 > H8 )
% 0.25/0.48 @ ( bNF_rel_fun @ E @ F5 @ G3 @ G3 @ R12
% 0.25/0.48 @ ^ [Y5: G3,Z4: G3] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( bNF_rel_fun @ E @ F5 @ H8 @ H8 @ R12
% 0.25/0.48 @ ^ [Y5: H8,Z4: H8] : ( Y5 = Z4 ) ) )
% 0.25/0.48 @ ( comp @ G3 @ H8 @ E )
% 0.25/0.48 @ ( comp @ G3 @ H8 @ F5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % o_rsp(2)
% 0.25/0.48 thf(fact_2676_apfst__convE,axiom,
% 0.25/0.48 ! [C: $tType,A: $tType,B: $tType,Q3: product_prod @ A @ B,F: C > A,P5: product_prod @ C @ B] :
% 0.25/0.48 ( ( Q3
% 0.25/0.48 = ( product_apfst @ C @ A @ B @ F @ P5 ) )
% 0.25/0.48 => ~ ! [X2: C,Y4: B] :
% 0.25/0.48 ( ( P5
% 0.25/0.48 = ( product_Pair @ C @ B @ X2 @ Y4 ) )
% 0.25/0.48 => ( Q3
% 0.25/0.48 != ( product_Pair @ A @ B @ ( F @ X2 ) @ Y4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % apfst_convE
% 0.25/0.48 thf(fact_2677_UN__le__eq__Un0,axiom,
% 0.25/0.48 ! [A: $tType,M3: nat > ( set @ A ),N: nat] :
% 0.25/0.48 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_ord_atMost @ nat @ N ) ) )
% 0.25/0.48 = ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M3 @ ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % UN_le_eq_Un0
% 0.25/0.48 thf(fact_2678_times__int_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_int.rsp
% 0.25/0.48 thf(fact_2679_sup__Some,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( sup @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( sup_sup @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 0.25/0.48 = ( some @ A @ ( sup_sup @ A @ X @ Y ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_Some
% 0.25/0.48 thf(fact_2680_intrel__iff,axiom,
% 0.25/0.48 ! [X: nat,Y: nat,U: nat,V: nat] :
% 0.25/0.48 ( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V ) )
% 0.25/0.48 = ( ( plus_plus @ nat @ X @ V )
% 0.25/0.48 = ( plus_plus @ nat @ U @ Y ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % intrel_iff
% 0.25/0.48 thf(fact_2681_insert__Times__insert,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B6: set @ B] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
% 0.25/0.48 @ ^ [Uu: A] : ( insert2 @ B @ B2 @ B6 ) )
% 0.25/0.48 = ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
% 0.25/0.48 @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : ( insert2 @ B @ B2 @ B6 ) )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
% 0.25/0.48 @ ^ [Uu: A] : B6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % insert_Times_insert
% 0.25/0.48 thf(fact_2682_Sigma__Un__distrib1,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I4 @ J4 ) @ C5 )
% 0.25/0.48 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_Un_distrib1
% 0.25/0.48 thf(fact_2683_Sigma__Un__distrib2,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ I4
% 0.25/0.48 @ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
% 0.25/0.48 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_Un_distrib2
% 0.25/0.48 thf(fact_2684_Times__Un__distrib1,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 @ ^ [Uu: A] : C5 )
% 0.25/0.48 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : C5 )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ B6
% 0.25/0.48 @ ^ [Uu: A] : C5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Times_Un_distrib1
% 0.25/0.48 thf(fact_2685_set__diff__diff__left,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,B6: set @ A,C5: set @ A] :
% 0.25/0.48 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.48 = ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B6 @ C5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % set_diff_diff_left
% 0.25/0.48 thf(fact_2686_sup__option__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( sup @ A )
% 0.25/0.48 => ( ( sup_sup @ ( option @ A ) )
% 0.25/0.48 = ( ^ [X3: option @ A,Y3: option @ A] :
% 0.25/0.48 ( case_option @ ( option @ A ) @ A @ Y3
% 0.25/0.48 @ ^ [X9: A] :
% 0.25/0.48 ( case_option @ ( option @ A ) @ A @ X3
% 0.25/0.48 @ ^ [Z5: A] : ( some @ A @ ( sup_sup @ A @ X9 @ Z5 ) )
% 0.25/0.48 @ Y3 )
% 0.25/0.48 @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_option_def
% 0.25/0.48 thf(fact_2687_ivl__disj__un__two__touch_I4_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two_touch(4)
% 0.25/0.48 thf(fact_2688_ivl__disj__un__two_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(3)
% 0.25/0.48 thf(fact_2689_max__ext__additive,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,B6: set @ A,R5: set @ ( product_prod @ A @ A ),C5: set @ A,D4: set @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B6 ) @ ( max_ext @ A @ R5 ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C5 @ D4 ) @ ( max_ext @ A @ R5 ) )
% 0.25/0.48 => ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ A ) @ B6 @ D4 ) ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % max_ext_additive
% 0.25/0.48 thf(fact_2690_zero__int_Orsp,axiom,
% 0.25/0.48 intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % zero_int.rsp
% 0.25/0.48 thf(fact_2691_ivl__disj__un__two_I7_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(7)
% 0.25/0.48 thf(fact_2692_card__Un__le,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,B6: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % card_Un_le
% 0.25/0.48 thf(fact_2693_ivl__disj__un__one_I2_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 0.25/0.48 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_one(2)
% 0.25/0.48 thf(fact_2694_atLeastLessThan__add__Un,axiom,
% 0.25/0.48 ! [I: nat,J: nat,K: nat] :
% 0.25/0.48 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.48 => ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
% 0.25/0.48 = ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastLessThan_add_Un
% 0.25/0.48 thf(fact_2695_uminus__int_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_int.rsp
% 0.25/0.48 thf(fact_2696_one__int_Orsp,axiom,
% 0.25/0.48 intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % one_int.rsp
% 0.25/0.48 thf(fact_2697_ivl__disj__un__two__touch_I2_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two_touch(2)
% 0.25/0.48 thf(fact_2698_ivl__disj__un__two_I1_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(1)
% 0.25/0.48 thf(fact_2699_ivl__disj__un__singleton_I2_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [U: A] :
% 0.25/0.48 ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ U ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_singleton(2)
% 0.25/0.48 thf(fact_2700_ivl__disj__un__one_I4_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_one(4)
% 0.25/0.48 thf(fact_2701_ivl__disj__un__one_I1_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.25/0.48 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_one(1)
% 0.25/0.48 thf(fact_2702_intrel__def,axiom,
% 0.25/0.48 ( intrel
% 0.25/0.48 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] :
% 0.25/0.48 ( ( plus_plus @ nat @ X3 @ V2 )
% 0.25/0.48 = ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % intrel_def
% 0.25/0.48 thf(fact_2703_ivl__disj__un__singleton_I6_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_singleton(6)
% 0.25/0.48 thf(fact_2704_ivl__disj__un__singleton_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_singleton(3)
% 0.25/0.48 thf(fact_2705_ivl__disj__un__two_I4_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(4)
% 0.25/0.48 thf(fact_2706_less__int_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 0.25/0.48 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 0.25/0.48 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % less_int.rsp
% 0.25/0.48 thf(fact_2707_less__eq__int_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 0.25/0.48 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 0.25/0.48 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ $o
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % less_eq_int.rsp
% 0.25/0.48 thf(fact_2708_plus__int_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % plus_int.rsp
% 0.25/0.48 thf(fact_2709_minus__int_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
% 0.25/0.48 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 0.25/0.48 @ ^ [X3: nat,Y3: nat] :
% 0.25/0.48 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 0.25/0.48 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % minus_int.rsp
% 0.25/0.48 thf(fact_2710_max__weak__def,axiom,
% 0.25/0.48 ( fun_max_weak
% 0.25/0.48 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % max_weak_def
% 0.25/0.48 thf(fact_2711_min__weak__def,axiom,
% 0.25/0.48 ( fun_min_weak
% 0.25/0.48 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % min_weak_def
% 0.25/0.48 thf(fact_2712_fold__union__pair,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,B6: set @ A,X: B,A4: set @ ( product_prod @ B @ A )] :
% 0.25/0.48 ( ( finite_finite2 @ A @ B6 )
% 0.25/0.48 => ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.48 @ B6 ) )
% 0.25/0.48 @ A4 )
% 0.25/0.48 = ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 0.25/0.48 @ A4
% 0.25/0.48 @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fold_union_pair
% 0.25/0.48 thf(fact_2713_Field__insert,axiom,
% 0.25/0.48 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
% 0.25/0.48 = ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Field_insert
% 0.25/0.48 thf(fact_2714_set__to__map__insert,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
% 0.25/0.48 => ( ( set_to_map @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ Kv @ S2 ) )
% 0.25/0.48 = ( fun_upd @ A @ ( option @ B ) @ ( set_to_map @ A @ B @ S2 ) @ ( product_fst @ A @ B @ Kv ) @ ( some @ B @ ( product_snd @ A @ B @ Kv ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % set_to_map_insert
% 0.25/0.48 thf(fact_2715_plus__rat_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % plus_rat.rsp
% 0.25/0.48 thf(fact_2716_inverse__rat_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] :
% 0.25/0.48 ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( ( product_fst @ int @ int @ X3 )
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] :
% 0.25/0.48 ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( ( product_fst @ int @ int @ X3 )
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % inverse_rat.rsp
% 0.25/0.48 thf(fact_2717_fun__upd__upd,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z2: B] :
% 0.25/0.48 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z2 )
% 0.25/0.48 = ( fun_upd @ A @ B @ F @ X @ Z2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_upd
% 0.25/0.48 thf(fact_2718_fun__upd__triv,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.25/0.48 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 0.25/0.48 = F ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_triv
% 0.25/0.48 thf(fact_2719_fun__upd__apply,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( fun_upd @ B @ A )
% 0.25/0.48 = ( ^ [F2: B > A,X3: B,Y3: A,Z5: B] : ( if @ A @ ( Z5 = X3 ) @ Y3 @ ( F2 @ Z5 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_apply
% 0.25/0.48 thf(fact_2720_image__update,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X: A,A4: set @ A,F: A > B,N: B] :
% 0.25/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.25/0.48 => ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F @ X @ N ) @ A4 )
% 0.25/0.48 = ( image2 @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_update
% 0.25/0.48 thf(fact_2721_empty__upd__none,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X: A] :
% 0.25/0.48 ( ( fun_upd @ A @ ( option @ B )
% 0.25/0.48 @ ^ [X3: A] : ( none @ B )
% 0.25/0.48 @ X
% 0.25/0.48 @ ( none @ B ) )
% 0.25/0.48 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % empty_upd_none
% 0.25/0.48 thf(fact_2722_sup__None__1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( sup @ A )
% 0.25/0.48 => ! [Y: option @ A] :
% 0.25/0.48 ( ( sup_sup @ ( option @ A ) @ ( none @ A ) @ Y )
% 0.25/0.48 = Y ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_None_1
% 0.25/0.48 thf(fact_2723_sup__None__2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( sup @ A )
% 0.25/0.48 => ! [X: option @ A] :
% 0.25/0.48 ( ( sup_sup @ ( option @ A ) @ X @ ( none @ A ) )
% 0.25/0.48 = X ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_None_2
% 0.25/0.48 thf(fact_2724_finite__Field__eq__finite,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( finite_finite2 @ A @ ( field2 @ A @ R5 ) )
% 0.25/0.48 = ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % finite_Field_eq_finite
% 0.25/0.48 thf(fact_2725_image__map__upd,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X: A,A4: set @ A,M2: A > ( option @ B ),Y: B] :
% 0.25/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.25/0.48 => ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ A4 )
% 0.25/0.48 = ( image2 @ A @ ( option @ B ) @ M2 @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_map_upd
% 0.25/0.48 thf(fact_2726_map__option__o__map__upd,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,F: C > B,M2: A > ( option @ C ),A3: A,B2: C] :
% 0.25/0.48 ( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( fun_upd @ A @ ( option @ C ) @ M2 @ A3 @ ( some @ C @ B2 ) ) )
% 0.25/0.48 = ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ M2 ) @ A3 @ ( some @ B @ ( F @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_option_o_map_upd
% 0.25/0.48 thf(fact_2727_fun__upd__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( fun_upd @ A @ B )
% 0.25/0.48 = ( ^ [F2: A > B,A7: A,B5: B,X3: A] : ( if @ B @ ( X3 = A7 ) @ B5 @ ( F2 @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_def
% 0.25/0.48 thf(fact_2728_fun__upd__eqD,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z2: B] :
% 0.25/0.48 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.25/0.48 = ( fun_upd @ A @ B @ G @ X @ Z2 ) )
% 0.25/0.48 => ( Y = Z2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_eqD
% 0.25/0.48 thf(fact_2729_fun__upd__idem,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 0.25/0.48 ( ( ( F @ X )
% 0.25/0.48 = Y )
% 0.25/0.48 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 0.25/0.48 = F ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_idem
% 0.25/0.48 thf(fact_2730_fun__upd__same,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 0.25/0.48 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 0.25/0.48 = Y ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_same
% 0.25/0.48 thf(fact_2731_fun__upd__other,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,Z2: A,X: A,F: A > B,Y: B] :
% 0.25/0.48 ( ( Z2 != X )
% 0.25/0.48 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z2 )
% 0.25/0.48 = ( F @ Z2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_other
% 0.25/0.48 thf(fact_2732_fun__upd__twist,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,A3: A,C2: A,M2: A > B,B2: B,D3: B] :
% 0.25/0.48 ( ( A3 != C2 )
% 0.25/0.48 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ A3 @ B2 ) @ C2 @ D3 )
% 0.25/0.48 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ C2 @ D3 ) @ A3 @ B2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_twist
% 0.25/0.48 thf(fact_2733_fun__upd__idem__iff,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 0.25/0.48 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 0.25/0.48 = F )
% 0.25/0.48 = ( ( F @ X )
% 0.25/0.48 = Y ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_idem_iff
% 0.25/0.48 thf(fact_2734_map__upd__Some__unfold,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: B > ( option @ A ),A3: B,B2: A,X: B,Y: A] :
% 0.25/0.48 ( ( ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) @ X )
% 0.25/0.48 = ( some @ A @ Y ) )
% 0.25/0.48 = ( ( ( X = A3 )
% 0.25/0.48 & ( B2 = Y ) )
% 0.25/0.48 | ( ( X != A3 )
% 0.25/0.48 & ( ( M2 @ X )
% 0.25/0.48 = ( some @ A @ Y ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_upd_Some_unfold
% 0.25/0.48 thf(fact_2735_map__upd__triv,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X: A] :
% 0.25/0.48 ( ( ( T2 @ K )
% 0.25/0.48 = ( some @ A @ X ) )
% 0.25/0.48 => ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X ) )
% 0.25/0.48 = T2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_upd_triv
% 0.25/0.48 thf(fact_2736_map__upd__eqD1,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,X: B,N: A > ( option @ B ),Y: B] :
% 0.25/0.48 ( ( ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ X ) )
% 0.25/0.48 = ( fun_upd @ A @ ( option @ B ) @ N @ A3 @ ( some @ B @ Y ) ) )
% 0.25/0.48 => ( X = Y ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_upd_eqD1
% 0.25/0.48 thf(fact_2737_FieldI2,axiom,
% 0.25/0.48 ! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
% 0.25/0.48 => ( member @ A @ J @ ( field2 @ A @ R5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % FieldI2
% 0.25/0.48 thf(fact_2738_FieldI1,axiom,
% 0.25/0.48 ! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
% 0.25/0.48 => ( member @ A @ I @ ( field2 @ A @ R5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % FieldI1
% 0.25/0.48 thf(fact_2739_sup__Un__eq2,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ( sup_sup @ ( A > B > $o )
% 0.25/0.48 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
% 0.25/0.48 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
% 0.25/0.48 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_Un_eq2
% 0.25/0.48 thf(fact_2740_fun__upd__comp,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,F: C > B,G: A > C,X: A,Y: C] :
% 0.25/0.48 ( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G @ X @ Y ) )
% 0.25/0.48 = ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G ) @ X @ ( F @ Y ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_comp
% 0.25/0.48 thf(fact_2741_map__upd__nonempty,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X: B] :
% 0.25/0.48 ( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X ) )
% 0.25/0.48 != ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_upd_nonempty
% 0.25/0.48 thf(fact_2742_one__rat_Orsp,axiom,
% 0.25/0.48 ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % one_rat.rsp
% 0.25/0.48 thf(fact_2743_R__subset__Field,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.48 @ ( product_Sigma @ A @ A @ ( field2 @ A @ R5 )
% 0.25/0.48 @ ^ [Uu: A] : ( field2 @ A @ R5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % R_subset_Field
% 0.25/0.48 thf(fact_2744_fst__in__Field,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_fst @ A @ A ) @ R5 ) @ ( field2 @ A @ R5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % fst_in_Field
% 0.25/0.48 thf(fact_2745_snd__in__Field,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_snd @ A @ A ) @ R5 ) @ ( field2 @ A @ R5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % snd_in_Field
% 0.25/0.48 thf(fact_2746_zero__rat_Orsp,axiom,
% 0.25/0.48 ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % zero_rat.rsp
% 0.25/0.48 thf(fact_2747_Fract_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
% 0.25/0.48 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.48 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
% 0.25/0.48 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.48 @ ratrel )
% 0.25/0.48 @ ^ [A7: int,B5: int] :
% 0.25/0.48 ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( B5
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ A7 @ B5 ) )
% 0.25/0.48 @ ^ [A7: int,B5: int] :
% 0.25/0.48 ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( B5
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ A7 @ B5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Fract.rsp
% 0.25/0.48 thf(fact_2748_sum_Oeq__fold,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( comm_monoid_add @ A )
% 0.25/0.48 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 0.25/0.48 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sum.eq_fold
% 0.25/0.48 thf(fact_2749_prod_Oeq__fold,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( comm_monoid_mult @ A )
% 0.25/0.48 => ( ( groups7121269368397514597t_prod @ B @ A )
% 0.25/0.48 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % prod.eq_fold
% 0.25/0.48 thf(fact_2750_finite__range__updI,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F: B > ( option @ A ),A3: B,B2: A] :
% 0.25/0.48 ( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F @ ( top_top @ ( set @ B ) ) ) )
% 0.25/0.48 => ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % finite_range_updI
% 0.25/0.48 thf(fact_2751_fun__upd__image,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X: B,A4: set @ B,F: B > A,Y: A] :
% 0.25/0.48 ( ( ( member @ B @ X @ A4 )
% 0.25/0.48 => ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A4 )
% 0.25/0.48 = ( insert2 @ A @ Y @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 0.25/0.48 & ( ~ ( member @ B @ X @ A4 )
% 0.25/0.48 => ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A4 )
% 0.25/0.48 = ( image2 @ B @ A @ F @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_image
% 0.25/0.48 thf(fact_2752_times__rat_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_rat.rsp
% 0.25/0.48 thf(fact_2753_uminus__rat_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_rat.rsp
% 0.25/0.48 thf(fact_2754_product__fold,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.48 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.48 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.48 => ( ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : B6 )
% 0.25/0.48 = ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ^ [X3: A,Z5: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.25/0.48 @ Z5
% 0.25/0.48 @ B6 )
% 0.25/0.48 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
% 0.25/0.48 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % product_fold
% 0.25/0.48 thf(fact_2755_Id__on__fold,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A] :
% 0.25/0.48 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.48 => ( ( id_on @ A @ A4 )
% 0.25/0.48 = ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.48 @ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) )
% 0.25/0.48 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.48 @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_on_fold
% 0.25/0.48 thf(fact_2756_plus__rat_Oabs__eq,axiom,
% 0.25/0.48 ! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
% 0.25/0.48 ( ( ratrel @ Xa @ Xa )
% 0.25/0.48 => ( ( ratrel @ X @ X )
% 0.25/0.48 => ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.25/0.48 = ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % plus_rat.abs_eq
% 0.25/0.48 thf(fact_2757_inverse__rat_Oabs__eq,axiom,
% 0.25/0.48 ! [X: product_prod @ int @ int] :
% 0.25/0.48 ( ( ratrel @ X @ X )
% 0.25/0.48 => ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
% 0.25/0.48 = ( abs_Rat
% 0.25/0.48 @ ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( ( product_fst @ int @ int @ X )
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % inverse_rat.abs_eq
% 0.25/0.48 thf(fact_2758_times__rat_Oabs__eq,axiom,
% 0.25/0.48 ! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
% 0.25/0.48 ( ( ratrel @ Xa @ Xa )
% 0.25/0.48 => ( ( ratrel @ X @ X )
% 0.25/0.48 => ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 0.25/0.48 = ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_rat.abs_eq
% 0.25/0.48 thf(fact_2759_uminus__rat_Oabs__eq,axiom,
% 0.25/0.48 ! [X: product_prod @ int @ int] :
% 0.25/0.48 ( ( ratrel @ X @ X )
% 0.25/0.48 => ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
% 0.25/0.48 = ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_rat.abs_eq
% 0.25/0.48 thf(fact_2760_graph__map__upd,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: A > ( option @ B ),K: A,V: B] :
% 0.25/0.48 ( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
% 0.25/0.48 = ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % graph_map_upd
% 0.25/0.48 thf(fact_2761_insert__relcomp__union__fold,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X10: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
% 0.25/0.48 => ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S2 ) @ X10 )
% 0.25/0.48 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
% 0.25/0.48 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 0.25/0.48 @ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( if @ ( set @ ( product_prod @ C @ B ) )
% 0.25/0.48 @ ( ( product_snd @ C @ A @ X )
% 0.25/0.48 = W2 )
% 0.25/0.48 @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A10 )
% 0.25/0.48 @ A10 ) )
% 0.25/0.48 @ X10
% 0.25/0.48 @ S2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % insert_relcomp_union_fold
% 0.25/0.48 thf(fact_2762_graph__empty,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( graph @ A @ B
% 0.25/0.48 @ ^ [X3: A] : ( none @ B ) )
% 0.25/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % graph_empty
% 0.25/0.48 thf(fact_2763_relcomp_Ocases,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
% 0.25/0.48 => ~ ! [B3: B] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B3 ) @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ A22 ) @ S ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcomp.cases
% 0.25/0.48 thf(fact_2764_relcomp_Osimps,axiom,
% 0.25/0.48 ! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
% 0.25/0.48 = ( ? [A7: A,B5: B,C3: C] :
% 0.25/0.48 ( ( A1 = A7 )
% 0.25/0.48 & ( A22 = C3 )
% 0.25/0.48 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R )
% 0.25/0.48 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C3 ) @ S ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcomp.simps
% 0.25/0.48 thf(fact_2765_relcomp_OrelcompI,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),C2: C,S: set @ ( product_prod @ B @ C )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S )
% 0.25/0.48 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcomp.relcompI
% 0.25/0.48 thf(fact_2766_relcompE,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S ) )
% 0.25/0.48 => ~ ! [X2: A,Y4: C,Z3: B] :
% 0.25/0.48 ( ( Xz
% 0.25/0.48 = ( product_Pair @ A @ B @ X2 @ Z3 ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y4 ) @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcompE
% 0.25/0.48 thf(fact_2767_relcompEpair,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R @ S ) )
% 0.25/0.48 => ~ ! [B3: C] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B3 ) @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B3 @ C2 ) @ S ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcompEpair
% 0.25/0.48 thf(fact_2768_in__graphI,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: B > ( option @ A ),K: B,V: A] :
% 0.25/0.48 ( ( ( M2 @ K )
% 0.25/0.48 = ( some @ A @ V ) )
% 0.25/0.48 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % in_graphI
% 0.25/0.48 thf(fact_2769_in__graphD,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M2 ) )
% 0.25/0.48 => ( ( M2 @ K )
% 0.25/0.48 = ( some @ B @ V ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % in_graphD
% 0.25/0.48 thf(fact_2770_max__ext__compat,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
% 0.25/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S2 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % max_ext_compat
% 0.25/0.48 thf(fact_2771_min__ext__compat,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
% 0.25/0.48 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S2 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % min_ext_compat
% 0.25/0.48 thf(fact_2772_relcomp__fold,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
% 0.25/0.48 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R5 )
% 0.25/0.48 => ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
% 0.25/0.48 => ( ( relcomp @ A @ B @ C @ R5 @ S2 )
% 0.25/0.48 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
% 0.25/0.48 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
% 0.25/0.48 @ ^ [X3: A,Y3: B,A9: set @ ( product_prod @ A @ C )] :
% 0.25/0.48 ( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
% 0.25/0.48 @ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
% 0.25/0.48 @ ^ [W2: B,Z5: C,A10: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W2 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Z5 ) @ A10 ) @ A10 ) )
% 0.25/0.48 @ A9
% 0.25/0.48 @ S2 ) )
% 0.25/0.48 @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
% 0.25/0.48 @ R5 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcomp_fold
% 0.25/0.48 thf(fact_2773_one__rat__def,axiom,
% 0.25/0.48 ( ( one_one @ rat )
% 0.25/0.48 = ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % one_rat_def
% 0.25/0.48 thf(fact_2774_graph__fun__upd__None,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A] :
% 0.25/0.48 ( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) )
% 0.25/0.48 = ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [E5: product_prod @ A @ B] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ E5 @ ( graph @ A @ B @ M2 ) )
% 0.25/0.48 & ( ( product_fst @ A @ B @ E5 )
% 0.25/0.48 != K ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % graph_fun_upd_None
% 0.25/0.48 thf(fact_2775_insert__relcomp__fold,axiom,
% 0.25/0.48 ! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R5: set @ ( product_prod @ C @ A )] :
% 0.25/0.48 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
% 0.25/0.48 => ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R5 ) @ S2 )
% 0.25/0.48 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
% 0.25/0.48 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 0.25/0.48 @ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( if @ ( set @ ( product_prod @ C @ B ) )
% 0.25/0.48 @ ( ( product_snd @ C @ A @ X )
% 0.25/0.48 = W2 )
% 0.25/0.48 @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A10 )
% 0.25/0.48 @ A10 ) )
% 0.25/0.48 @ ( relcomp @ C @ A @ B @ R5 @ S2 )
% 0.25/0.48 @ S2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % insert_relcomp_fold
% 0.25/0.48 thf(fact_2776_zero__rat__def,axiom,
% 0.25/0.48 ( ( zero_zero @ rat )
% 0.25/0.48 = ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % zero_rat_def
% 0.25/0.48 thf(fact_2777_plus__rat__def,axiom,
% 0.25/0.48 ( ( plus_plus @ rat )
% 0.25/0.48 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % plus_rat_def
% 0.25/0.48 thf(fact_2778_inverse__rat__def,axiom,
% 0.25/0.48 ( ( inverse_inverse @ rat )
% 0.25/0.48 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] :
% 0.25/0.48 ( if @ ( product_prod @ int @ int )
% 0.25/0.48 @ ( ( product_fst @ int @ int @ X3 )
% 0.25/0.48 = ( zero_zero @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.48 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % inverse_rat_def
% 0.25/0.48 thf(fact_2779_times__rat__def,axiom,
% 0.25/0.48 ( ( times_times @ rat )
% 0.25/0.48 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % times_rat_def
% 0.25/0.48 thf(fact_2780_uminus__rat__def,axiom,
% 0.25/0.48 ( ( uminus_uminus @ rat )
% 0.25/0.48 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 0.25/0.48 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_rat_def
% 0.25/0.48 thf(fact_2781_comp__fun__commute__relcomp__fold,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
% 0.25/0.48 => ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
% 0.25/0.48 @ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 0.25/0.48 @ ^ [X3: C,Y3: A,A9: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
% 0.25/0.48 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 0.25/0.48 @ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W2 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Z5 ) @ A10 ) @ A10 ) )
% 0.25/0.48 @ A9
% 0.25/0.48 @ S2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % comp_fun_commute_relcomp_fold
% 0.25/0.48 thf(fact_2782_restrict__upd__same,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
% 0.25/0.48 ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.48 = ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_upd_same
% 0.25/0.48 thf(fact_2783_restrict__map__UNIV,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: A > ( option @ B )] :
% 0.25/0.48 ( ( restrict_map @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.48 = F ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_UNIV
% 0.25/0.48 thf(fact_2784_restrict__out,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X: A,A4: set @ A,M2: A > ( option @ B )] :
% 0.25/0.48 ( ~ ( member @ A @ X @ A4 )
% 0.25/0.48 => ( ( restrict_map @ A @ B @ M2 @ A4 @ X )
% 0.25/0.48 = ( none @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_out
% 0.25/0.48 thf(fact_2785_restrict__map__empty,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,D4: set @ A] :
% 0.25/0.48 ( ( restrict_map @ A @ B
% 0.25/0.48 @ ^ [X3: A] : ( none @ B )
% 0.25/0.48 @ D4 )
% 0.25/0.48 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_empty
% 0.25/0.48 thf(fact_2786_restrict__map__to__empty,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.48 ( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.48 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_to_empty
% 0.25/0.48 thf(fact_2787_fun__upd__None__restrict,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X: A,D4: set @ A,M2: A > ( option @ B )] :
% 0.25/0.48 ( ( ( member @ A @ X @ D4 )
% 0.25/0.48 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
% 0.25/0.48 = ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
% 0.25/0.48 & ( ~ ( member @ A @ X @ D4 )
% 0.25/0.48 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
% 0.25/0.48 = ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fun_upd_None_restrict
% 0.25/0.48 thf(fact_2788_le__map__restrict,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( order @ B )
% 0.25/0.48 => ! [M2: A > ( option @ B ),X10: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M2 @ X10 ) @ M2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % le_map_restrict
% 0.25/0.48 thf(fact_2789_restrict__map__subset__eq,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B ),R5: set @ A,M7: A > ( option @ B ),R7: set @ A] :
% 0.25/0.48 ( ( ( restrict_map @ A @ B @ M2 @ R5 )
% 0.25/0.48 = M7 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ R7 @ R5 )
% 0.25/0.48 => ( ( restrict_map @ A @ B @ M2 @ R7 )
% 0.25/0.48 = ( restrict_map @ A @ B @ M7 @ R7 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_subset_eq
% 0.25/0.48 thf(fact_2790_restrict__map__eq_I2_J,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: set @ B,K: B,V: A] :
% 0.25/0.48 ( ( ( restrict_map @ B @ A @ M2 @ A4 @ K )
% 0.25/0.48 = ( some @ A @ V ) )
% 0.25/0.48 = ( ( ( M2 @ K )
% 0.25/0.48 = ( some @ A @ V ) )
% 0.25/0.48 & ( member @ B @ K @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_eq(2)
% 0.25/0.48 thf(fact_2791_restrict__map__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( restrict_map @ A @ B )
% 0.25/0.48 = ( ^ [M: A > ( option @ B ),A9: set @ A,X3: A] : ( if @ ( option @ B ) @ ( member @ A @ X3 @ A9 ) @ ( M @ X3 ) @ ( none @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_def
% 0.25/0.48 thf(fact_2792_comp__fun__commute_Ointro,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.25/0.48 ( ! [Y4: A,X2: A] :
% 0.25/0.48 ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 0.25/0.48 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.25/0.48 => ( finite6289374366891150609ommute @ A @ B @ F ) ) ).
% 0.25/0.48
% 0.25/0.48 % comp_fun_commute.intro
% 0.25/0.48 thf(fact_2793_comp__fun__commute_Ocomp__fun__commute,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 0.25/0.48 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 0.25/0.48 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 0.25/0.48 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % comp_fun_commute.comp_fun_commute
% 0.25/0.48 thf(fact_2794_comp__fun__commute__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( finite6289374366891150609ommute @ A @ B )
% 0.25/0.48 = ( ^ [F2: A > B > B] :
% 0.25/0.48 ! [Y3: A,X3: A] :
% 0.25/0.48 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 0.25/0.48 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % comp_fun_commute_def
% 0.25/0.48 thf(fact_2795_graph__restrictD_I1_J,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B ),A4: set @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A4 ) ) )
% 0.25/0.48 => ( member @ A @ K @ A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % graph_restrictD(1)
% 0.25/0.48 thf(fact_2796_map__restrict__insert__none__simp,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: B > ( option @ A ),X: B,S: set @ B] :
% 0.25/0.48 ( ( ( M2 @ X )
% 0.25/0.48 = ( none @ A ) )
% 0.25/0.48 => ( ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ ( insert2 @ B @ X @ S ) ) )
% 0.25/0.48 = ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ S ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_restrict_insert_none_simp
% 0.25/0.48 thf(fact_2797_restrict__map__upd,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: A > ( option @ B ),S2: set @ A,K: A,V: B] :
% 0.25/0.48 ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F @ S2 ) @ K @ ( some @ B @ V ) )
% 0.25/0.48 = ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ K @ ( some @ B @ V ) ) @ ( insert2 @ A @ K @ S2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_map_upd
% 0.25/0.48 thf(fact_2798_graph__restrictD_I2_J,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B ),A4: set @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A4 ) ) )
% 0.25/0.48 => ( ( M2 @ K )
% 0.25/0.48 = ( some @ B @ V ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % graph_restrictD(2)
% 0.25/0.48 thf(fact_2799_restrict__complement__singleton__eq,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
% 0.25/0.48 ( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.48 = ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % restrict_complement_singleton_eq
% 0.25/0.48 thf(fact_2800_map__upd__eq__restrict,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A] :
% 0.25/0.48 ( ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( none @ B ) )
% 0.25/0.48 = ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_upd_eq_restrict
% 0.25/0.48 thf(fact_2801_comp__fun__commute__product__fold,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,B6: set @ A] :
% 0.25/0.48 ( ( finite_finite2 @ A @ B6 )
% 0.25/0.48 => ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ^ [X3: B,Z5: set @ ( product_prod @ B @ A )] :
% 0.25/0.48 ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.48 @ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
% 0.25/0.48 @ Z5
% 0.25/0.48 @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % comp_fun_commute_product_fold
% 0.25/0.48 thf(fact_2802_Sup__fin_Oeq__fold_H,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ( ( lattic5882676163264333800up_fin @ A )
% 0.25/0.48 = ( ^ [A9: set @ A] :
% 0.25/0.48 ( the2 @ A
% 0.25/0.48 @ ( finite_fold @ A @ ( option @ A )
% 0.25/0.48 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( sup_sup @ A @ X3 ) @ Y3 ) )
% 0.25/0.48 @ ( none @ A )
% 0.25/0.48 @ A9 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sup_fin.eq_fold'
% 0.25/0.48 thf(fact_2803_ran__map__upd,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
% 0.25/0.48 ( ( ( M2 @ A3 )
% 0.25/0.48 = ( none @ A ) )
% 0.25/0.48 => ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) ) )
% 0.25/0.48 = ( insert2 @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ran_map_upd
% 0.25/0.48 thf(fact_2804_Gcd__fin_Oeq__fold,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ( ( semiring_gcd_Gcd_fin @ A )
% 0.25/0.48 = ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A9 ) @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Gcd_fin.eq_fold
% 0.25/0.48 thf(fact_2805_Max_Oeq__fold_H,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ( ( lattic643756798349783984er_Max @ A )
% 0.25/0.48 = ( ^ [A9: set @ A] :
% 0.25/0.48 ( the2 @ A
% 0.25/0.48 @ ( finite_fold @ A @ ( option @ A )
% 0.25/0.48 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_max @ A @ X3 ) @ Y3 ) )
% 0.25/0.48 @ ( none @ A )
% 0.25/0.48 @ A9 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Max.eq_fold'
% 0.25/0.48 thf(fact_2806_Min_Oeq__fold_H,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ( ( lattic643756798350308766er_Min @ A )
% 0.25/0.48 = ( ^ [A9: set @ A] :
% 0.25/0.48 ( the2 @ A
% 0.25/0.48 @ ( finite_fold @ A @ ( option @ A )
% 0.25/0.48 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_min @ A @ X3 ) @ Y3 ) )
% 0.25/0.48 @ ( none @ A )
% 0.25/0.48 @ A9 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Min.eq_fold'
% 0.25/0.48 thf(fact_2807_pairself__image__eq,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: B > A,P: B > B > $o] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
% 0.25/0.48 = ( collect @ ( product_prod @ A @ A )
% 0.25/0.48 @ ^ [Uu: product_prod @ A @ A] :
% 0.25/0.48 ? [A7: B,B5: B] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( product_Pair @ A @ A @ ( F @ A7 ) @ ( F @ B5 ) ) )
% 0.25/0.48 & ( P @ A7 @ B5 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % pairself_image_eq
% 0.25/0.48 thf(fact_2808_eq__or__mem__image__simp,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,F: B > A,A3: B,B6: set @ B] :
% 0.25/0.48 ( ( collect @ A
% 0.25/0.48 @ ^ [Uu: A] :
% 0.25/0.48 ? [L3: B] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( F @ L3 ) )
% 0.25/0.48 & ( ( L3 = A3 )
% 0.25/0.48 | ( member @ B @ L3 @ B6 ) ) ) )
% 0.25/0.48 = ( insert2 @ A @ ( F @ A3 )
% 0.25/0.48 @ ( collect @ A
% 0.25/0.48 @ ^ [Uu: A] :
% 0.25/0.48 ? [L3: B] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( F @ L3 ) )
% 0.25/0.48 & ( member @ B @ L3 @ B6 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % eq_or_mem_image_simp
% 0.25/0.48 thf(fact_2809_Gcd__fin_Oinfinite,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ! [A4: set @ A] :
% 0.25/0.48 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.48 => ( ( semiring_gcd_Gcd_fin @ A @ A4 )
% 0.25/0.48 = ( one_one @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Gcd_fin.infinite
% 0.25/0.48 thf(fact_2810_is__unit__Gcd__fin__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semiring_gcd @ A )
% 0.25/0.48 => ! [A4: set @ A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( one_one @ A ) )
% 0.25/0.48 = ( ( semiring_gcd_Gcd_fin @ A @ A4 )
% 0.25/0.48 = ( one_one @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % is_unit_Gcd_fin_iff
% 0.25/0.48 thf(fact_2811_Eps__Opt__eq__None,axiom,
% 0.25/0.48 ! [A: $tType,P: A > $o] :
% 0.25/0.48 ( ( ( eps_Opt @ A @ P )
% 0.25/0.48 = ( none @ A ) )
% 0.25/0.48 = ( ~ ? [X8: A] : ( P @ X8 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Eps_Opt_eq_None
% 0.25/0.48 thf(fact_2812_ran__empty,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ran @ B @ A
% 0.25/0.48 @ ^ [X3: B] : ( none @ A ) )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ran_empty
% 0.25/0.48 thf(fact_2813_map__update__eta__repair_I2_J,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
% 0.25/0.48 ( ( ( M2 @ K )
% 0.25/0.48 = ( none @ B ) )
% 0.25/0.48 => ( ( ran @ A @ B
% 0.25/0.48 @ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
% 0.25/0.48 = ( insert2 @ B @ V @ ( ran @ A @ B @ M2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_update_eta_repair(2)
% 0.25/0.48 thf(fact_2814_ran__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( ran @ A @ B )
% 0.25/0.48 = ( ^ [M: A > ( option @ B )] :
% 0.25/0.48 ( collect @ B
% 0.25/0.48 @ ^ [B5: B] :
% 0.25/0.48 ? [A7: A] :
% 0.25/0.48 ( ( M @ A7 )
% 0.25/0.48 = ( some @ B @ B5 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ran_def
% 0.25/0.48 thf(fact_2815_ranI,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
% 0.25/0.48 ( ( ( M2 @ A3 )
% 0.25/0.48 = ( some @ A @ B2 ) )
% 0.25/0.48 => ( member @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ranI
% 0.25/0.48 thf(fact_2816_fs__contract,axiom,
% 0.25/0.48 ! [B: $tType,C: $tType,A: $tType,F: A > B > C,S2: set @ C] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.25/0.48 @ ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [Uu: product_prod @ A @ B] :
% 0.25/0.48 ? [P6: product_prod @ A @ B] :
% 0.25/0.48 ( ( Uu = P6 )
% 0.25/0.48 & ( member @ C @ ( F @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) @ S2 ) ) ) )
% 0.25/0.48 = ( collect @ A
% 0.25/0.48 @ ^ [A7: A] :
% 0.25/0.48 ? [B5: B] : ( member @ C @ ( F @ A7 @ B5 ) @ S2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fs_contract
% 0.25/0.48 thf(fact_2817_ran__restrictD,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,Y: A,M2: B > ( option @ A ),A4: set @ B] :
% 0.25/0.48 ( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M2 @ A4 ) ) )
% 0.25/0.48 => ? [X2: B] :
% 0.25/0.48 ( ( member @ B @ X2 @ A4 )
% 0.25/0.48 & ( ( M2 @ X2 )
% 0.25/0.48 = ( some @ A @ Y ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ran_restrictD
% 0.25/0.48 thf(fact_2818_Un__interval,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [B1: A,B22: A,B32: A,F: A > B] :
% 0.25/0.48 ( ( ord_less_eq @ A @ B1 @ B22 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ B22 @ B32 )
% 0.25/0.48 => ( ( sup_sup @ ( set @ B )
% 0.25/0.48 @ ( collect @ B
% 0.25/0.48 @ ^ [Uu: B] :
% 0.25/0.48 ? [I3: A] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( F @ I3 ) )
% 0.25/0.48 & ( ord_less_eq @ A @ B1 @ I3 )
% 0.25/0.48 & ( ord_less @ A @ I3 @ B22 ) ) )
% 0.25/0.48 @ ( collect @ B
% 0.25/0.48 @ ^ [Uu: B] :
% 0.25/0.48 ? [I3: A] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( F @ I3 ) )
% 0.25/0.48 & ( ord_less_eq @ A @ B22 @ I3 )
% 0.25/0.48 & ( ord_less @ A @ I3 @ B32 ) ) ) )
% 0.25/0.48 = ( collect @ B
% 0.25/0.48 @ ^ [Uu: B] :
% 0.25/0.48 ? [I3: A] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( F @ I3 ) )
% 0.25/0.48 & ( ord_less_eq @ A @ B1 @ I3 )
% 0.25/0.48 & ( ord_less @ A @ I3 @ B32 ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Un_interval
% 0.25/0.48 thf(fact_2819_fold__atLeastAtMost__nat,axiom,
% 0.25/0.48 ! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
% 0.25/0.48 ( ( finite6289374366891150609ommute @ nat @ A @ F )
% 0.25/0.48 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 0.25/0.48 = ( finite_fold @ nat @ A @ F @ Acc2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fold_atLeastAtMost_nat
% 0.25/0.48 thf(fact_2820_Min_Oinfinite,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A4: set @ A] :
% 0.25/0.48 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.48 => ( ( lattic643756798350308766er_Min @ A @ A4 )
% 0.25/0.48 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Min.infinite
% 0.25/0.48 thf(fact_2821_relcomp__unfold,axiom,
% 0.25/0.48 ! [B: $tType,C: $tType,A: $tType] :
% 0.25/0.48 ( ( relcomp @ A @ C @ B )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
% 0.25/0.48 ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.48 @ ^ [X3: A,Z5: B] :
% 0.25/0.48 ? [Y3: C] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R3 )
% 0.25/0.48 & ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z5 ) @ S4 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relcomp_unfold
% 0.25/0.48 thf(fact_2822_Max_Oinfinite,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A4: set @ A] :
% 0.25/0.48 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.48 => ( ( lattic643756798349783984er_Max @ A @ A4 )
% 0.25/0.48 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Max.infinite
% 0.25/0.48 thf(fact_2823_graph__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( graph @ A @ B )
% 0.25/0.48 = ( ^ [M: A > ( option @ B )] :
% 0.25/0.48 ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [Uu: product_prod @ A @ B] :
% 0.25/0.48 ? [A7: A,B5: B] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( product_Pair @ A @ B @ A7 @ B5 ) )
% 0.25/0.48 & ( ( M @ A7 )
% 0.25/0.48 = ( some @ B @ B5 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % graph_def
% 0.25/0.48 thf(fact_2824_Sup__fin_Oinfinite,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( semilattice_sup @ A )
% 0.25/0.48 => ! [A4: set @ A] :
% 0.25/0.48 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.48 => ( ( lattic5882676163264333800up_fin @ A @ A4 )
% 0.25/0.48 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sup_fin.infinite
% 0.25/0.48 thf(fact_2825_Min__add__commute,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( linord4140545234300271783up_add @ A )
% 0.25/0.48 => ! [S2: set @ B,F: B > A,K: A] :
% 0.25/0.48 ( ( finite_finite2 @ B @ S2 )
% 0.25/0.48 => ( ( S2
% 0.25/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.48 => ( ( lattic643756798350308766er_Min @ A
% 0.25/0.48 @ ( image2 @ B @ A
% 0.25/0.48 @ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
% 0.25/0.48 @ S2 ) )
% 0.25/0.48 = ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Min_add_commute
% 0.25/0.48 thf(fact_2826_card__le__Suc__Max,axiom,
% 0.25/0.48 ! [S2: set @ nat] :
% 0.25/0.48 ( ( finite_finite2 @ nat @ S2 )
% 0.25/0.48 => ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % card_le_Suc_Max
% 0.25/0.48 thf(fact_2827_Max__add__commute,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( linord4140545234300271783up_add @ A )
% 0.25/0.48 => ! [S2: set @ B,F: B > A,K: A] :
% 0.25/0.48 ( ( finite_finite2 @ B @ S2 )
% 0.25/0.48 => ( ( S2
% 0.25/0.48 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.48 => ( ( lattic643756798349783984er_Max @ A
% 0.25/0.48 @ ( image2 @ B @ A
% 0.25/0.48 @ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
% 0.25/0.48 @ S2 ) )
% 0.25/0.48 = ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Max_add_commute
% 0.25/0.48 thf(fact_2828_set__to__map__ran,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,S2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( set_to_map @ B @ A @ S2 ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % set_to_map_ran
% 0.25/0.48 thf(fact_2829_map__to__set__upd,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
% 0.25/0.48 ( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
% 0.25/0.48 = ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V )
% 0.25/0.48 @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M2 )
% 0.25/0.48 @ ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [Uu: product_prod @ A @ B] :
% 0.25/0.48 ? [V5: B] :
% 0.25/0.48 ( Uu
% 0.25/0.48 = ( product_Pair @ A @ B @ K @ V5 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_upd
% 0.25/0.48 thf(fact_2830_brk__rel__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( brk_rel @ A @ B )
% 0.25/0.48 = ( ^ [R6: set @ ( product_prod @ A @ B )] :
% 0.25/0.48 ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
% 0.25/0.48 @ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
% 0.25/0.48 @ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
% 0.25/0.48 ? [X3: A,Y3: B] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) )
% 0.25/0.48 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 ) ) )
% 0.25/0.48 @ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
% 0.25/0.48 @ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
% 0.25/0.48 ? [X3: A,Y3: B] :
% 0.25/0.48 ( Uu
% 0.25/0.48 = ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % brk_rel_def
% 0.25/0.48 thf(fact_2831_image2__def,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType] :
% 0.25/0.48 ( ( bNF_Greatest_image2 @ C @ A @ B )
% 0.25/0.48 = ( ^ [A9: set @ C,F2: C > A,G2: C > B] :
% 0.25/0.48 ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ^ [Uu: product_prod @ A @ B] :
% 0.25/0.48 ? [A7: C] :
% 0.25/0.48 ( ( Uu
% 0.25/0.48 = ( product_Pair @ A @ B @ ( F2 @ A7 ) @ ( G2 @ A7 ) ) )
% 0.25/0.48 & ( member @ C @ A7 @ A9 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image2_def
% 0.25/0.48 thf(fact_2832_greaterThan__Suc,axiom,
% 0.25/0.48 ! [K: nat] :
% 0.25/0.48 ( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
% 0.25/0.48 = ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert2 @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_Suc
% 0.25/0.48 thf(fact_2833_refl__on__singleton,axiom,
% 0.25/0.48 ! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_on_singleton
% 0.25/0.48 thf(fact_2834_time__refines,axiom,
% 0.25/0.48 ! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.25/0.48 ( ( refine_Imp_refines @ A @ C2 @ C6 )
% 0.25/0.48 => ( ~ ( time_fails @ A @ C6 @ H3 )
% 0.25/0.48 => ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H3 ) @ ( time_time @ A @ C6 @ H3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % time_refines
% 0.25/0.48 thf(fact_2835_greaterThan__eq__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( ( set_ord_greaterThan @ A @ X )
% 0.25/0.48 = ( set_ord_greaterThan @ A @ Y ) )
% 0.25/0.48 = ( X = Y ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_eq_iff
% 0.25/0.48 thf(fact_2836_greaterThan__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [I: A,K: A] :
% 0.25/0.48 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
% 0.25/0.48 = ( ord_less @ A @ K @ I ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_iff
% 0.25/0.48 thf(fact_2837_Inf__greaterThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ( comple6319245703460814977attice @ A )
% 0.25/0.48 & ( dense_linorder @ A ) )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
% 0.25/0.48 = X ) ) ).
% 0.25/0.48
% 0.25/0.48 % Inf_greaterThan
% 0.25/0.48 thf(fact_2838_greaterThan__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
% 0.25/0.48 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_subset_iff
% 0.25/0.48 thf(fact_2839_Compl__greaterThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [K: A] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ K ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Compl_greaterThan
% 0.25/0.48 thf(fact_2840_Compl__atMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [K: A] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
% 0.25/0.48 = ( set_ord_greaterThan @ A @ K ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Compl_atMost
% 0.25/0.48 thf(fact_2841_map__to__set__empty,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( map_to_set @ A @ B
% 0.25/0.48 @ ^ [X3: A] : ( none @ B ) )
% 0.25/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_empty
% 0.25/0.48 thf(fact_2842_Sup__greaterThanAtLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( comple6319245703460814977attice @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 0.25/0.48 => ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
% 0.25/0.48 = ( top_top @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sup_greaterThanAtLeast
% 0.25/0.48 thf(fact_2843_image__uminus__greaterThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
% 0.25/0.48 = ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_greaterThan
% 0.25/0.48 thf(fact_2844_image__uminus__lessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
% 0.25/0.48 = ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_lessThan
% 0.25/0.48 thf(fact_2845_refl__onD2,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.48 ( ( refl_on @ A @ A4 @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.48 => ( member @ A @ Y @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_onD2
% 0.25/0.48 thf(fact_2846_refl__onD1,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.48 ( ( refl_on @ A @ A4 @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.48 => ( member @ A @ X @ A4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_onD1
% 0.25/0.48 thf(fact_2847_refl__onD,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.48 ( ( refl_on @ A @ A4 @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_onD
% 0.25/0.48 thf(fact_2848_greaterThan__non__empty,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_top @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( set_ord_greaterThan @ A @ X )
% 0.25/0.48 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_non_empty
% 0.25/0.48 thf(fact_2849_infinite__Ioi,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ( linorder @ A )
% 0.25/0.48 & ( no_top @ A ) )
% 0.25/0.48 => ! [A3: A] :
% 0.25/0.48 ~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % infinite_Ioi
% 0.25/0.48 thf(fact_2850_greaterThan__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ( ( set_ord_greaterThan @ A )
% 0.25/0.48 = ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_def
% 0.25/0.48 thf(fact_2851_fails__refines,axiom,
% 0.25/0.48 ! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 0.25/0.48 ( ( refine_Imp_refines @ A @ C2 @ C6 )
% 0.25/0.48 => ( ( time_fails @ A @ C2 @ H3 )
% 0.25/0.48 => ( time_fails @ A @ C6 @ H3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % fails_refines
% 0.25/0.48 thf(fact_2852_map__to__set__inverse,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.48 ( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M2 ) )
% 0.25/0.48 = M2 ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_inverse
% 0.25/0.48 thf(fact_2853_map__to__set__empty__iff_I1_J,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.48 ( ( ( map_to_set @ A @ B @ M2 )
% 0.25/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.48 = ( M2
% 0.25/0.48 = ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_empty_iff(1)
% 0.25/0.48 thf(fact_2854_map__to__set__empty__iff_I2_J,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.48 ( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
% 0.25/0.48 = ( map_to_set @ A @ B @ M2 ) )
% 0.25/0.48 = ( M2
% 0.25/0.48 = ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_empty_iff(2)
% 0.25/0.48 thf(fact_2855_image2__eqI,axiom,
% 0.25/0.48 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A4: set @ B] :
% 0.25/0.48 ( ( B2
% 0.25/0.48 = ( F @ X ) )
% 0.25/0.48 => ( ( C2
% 0.25/0.48 = ( G @ X ) )
% 0.25/0.48 => ( ( member @ B @ X @ A4 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A4 @ F @ G ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image2_eqI
% 0.25/0.48 thf(fact_2856_refl__on__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( refl_on @ A )
% 0.25/0.48 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
% 0.25/0.48 @ ( product_Sigma @ A @ A @ A9
% 0.25/0.48 @ ^ [Uu: A] : A9 ) )
% 0.25/0.48 & ! [X3: A] :
% 0.25/0.48 ( ( member @ A @ X3 @ A9 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_on_def
% 0.25/0.48 thf(fact_2857_refl__onI,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.48 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.48 @ ^ [Uu: A] : A4 ) )
% 0.25/0.48 => ( ! [X2: A] :
% 0.25/0.48 ( ( member @ A @ X2 @ A4 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R ) )
% 0.25/0.48 => ( refl_on @ A @ A4 @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_onI
% 0.25/0.48 thf(fact_2858_map__to__set__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( map_to_set @ A @ B )
% 0.25/0.48 = ( ^ [M: A > ( option @ B )] :
% 0.25/0.48 ( collect @ ( product_prod @ A @ B )
% 0.25/0.48 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.48 @ ^ [K4: A,V2: B] :
% 0.25/0.48 ( ( M @ K4 )
% 0.25/0.48 = ( some @ B @ V2 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_def
% 0.25/0.48 thf(fact_2859_INT__greaterThan__UNIV,axiom,
% 0.25/0.48 ( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 0.25/0.48 = ( bot_bot @ ( set @ nat ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % INT_greaterThan_UNIV
% 0.25/0.48 thf(fact_2860_map__to__set__ran,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType] :
% 0.25/0.48 ( ( ran @ B @ A )
% 0.25/0.48 = ( ^ [M: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % map_to_set_ran
% 0.25/0.48 thf(fact_2861_greaterThan__0,axiom,
% 0.25/0.48 ( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
% 0.25/0.48 = ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_0
% 0.25/0.48 thf(fact_2862_refines__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( refine_Imp_refines @ A )
% 0.25/0.48 = ( ^ [P6: heap_Time_Heap @ A,Q5: heap_Time_Heap @ A] :
% 0.25/0.48 ! [H: heap_ext @ product_unit] :
% 0.25/0.48 ( case_option @ $o @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ $true
% 0.25/0.48 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ $o
% 0.25/0.48 @ ^ [R3: A] :
% 0.25/0.48 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ $o
% 0.25/0.48 @ ^ [H7: heap_ext @ product_unit,T3: nat] :
% 0.25/0.48 ( ( heap_Time_execute @ A @ P6 @ H )
% 0.25/0.48 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T3 ) ) ) ) ) )
% 0.25/0.48 @ ( heap_Time_execute @ A @ Q5 @ H ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refines_def
% 0.25/0.48 thf(fact_2863_refines__empty,axiom,
% 0.25/0.48 ! [A: $tType,M2: heap_Time_Heap @ A] :
% 0.25/0.48 ( refine_Imp_refines @ A @ M2
% 0.25/0.48 @ ( heap_Time_Heap2 @ A
% 0.25/0.48 @ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refines_empty
% 0.25/0.48 thf(fact_2864_refines__bind,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,M2: heap_Time_Heap @ A,M7: heap_Time_Heap @ A,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B )] :
% 0.25/0.48 ( ( refine_Imp_refines @ A @ M2 @ M7 )
% 0.25/0.48 => ( ! [X2: A] : ( refine_Imp_refines @ B @ ( F @ X2 ) @ ( F3 @ X2 ) )
% 0.25/0.48 => ( refine_Imp_refines @ B @ ( heap_Time_bind @ A @ B @ M2 @ F ) @ ( heap_Time_bind @ A @ B @ M7 @ F3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refines_bind
% 0.25/0.48 thf(fact_2865_refl__on__domain,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( refl_on @ A @ A4 @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.48 & ( member @ A @ B2 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refl_on_domain
% 0.25/0.48 thf(fact_2866_refines__option,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A3: option @ A,A6: option @ A,M1: heap_Time_Heap @ B,M12: heap_Time_Heap @ B,M22: A > ( heap_Time_Heap @ B ),M23: A > ( heap_Time_Heap @ B )] :
% 0.25/0.48 ( ( A3 = A6 )
% 0.25/0.48 => ( ( refine_Imp_refines @ B @ M1 @ M12 )
% 0.25/0.48 => ( ! [X2: A] : ( refine_Imp_refines @ B @ ( M22 @ X2 ) @ ( M23 @ X2 ) )
% 0.25/0.48 => ( refine_Imp_refines @ B @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M1 @ M22 @ A3 ) @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M12 @ M23 @ A6 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % refines_option
% 0.25/0.48 thf(fact_2867_linear__order__on__singleton,axiom,
% 0.25/0.48 ! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % linear_order_on_singleton
% 0.25/0.48 thf(fact_2868_Total__subset__Id,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( total_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) )
% 0.25/0.48 => ( ( R
% 0.25/0.48 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.25/0.48 | ? [A5: A] :
% 0.25/0.48 ( R
% 0.25/0.48 = ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Total_subset_Id
% 0.25/0.48 thf(fact_2869_relation__of__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order_relation_of @ A )
% 0.25/0.48 = ( ^ [P4: A > A > $o,A9: set @ A] :
% 0.25/0.48 ( collect @ ( product_prod @ A @ A )
% 0.25/0.48 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.48 @ ^ [A7: A,B5: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 )
% 0.25/0.48 @ ( product_Sigma @ A @ A @ A9
% 0.25/0.48 @ ^ [Uu: A] : A9 ) )
% 0.25/0.48 & ( P4 @ A7 @ B5 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % relation_of_def
% 0.25/0.48 thf(fact_2870_euclidean__size__times__nonunit,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( A3
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ( B2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_times_nonunit
% 0.25/0.48 thf(fact_2871_surj__swap,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.48 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % surj_swap
% 0.25/0.48 thf(fact_2872_euclidean__size__integer_Orep__eq,axiom,
% 0.25/0.48 ( ( euclid6346220572633701492n_size @ code_integer )
% 0.25/0.48 = ( ^ [X3: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_integer.rep_eq
% 0.25/0.48 thf(fact_2873_swap__swap,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
% 0.25/0.48 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P5 ) )
% 0.25/0.48 = P5 ) ).
% 0.25/0.48
% 0.25/0.48 % swap_swap
% 0.25/0.48 thf(fact_2874_pair__in__Id__conv,axiom,
% 0.25/0.48 ! [A: $tType,A3: A,B2: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
% 0.25/0.48 = ( A3 = B2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % pair_in_Id_conv
% 0.25/0.48 thf(fact_2875_IdI,axiom,
% 0.25/0.48 ! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% 0.25/0.48
% 0.25/0.48 % IdI
% 0.25/0.48 thf(fact_2876_swap__simp,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X: B,Y: A] :
% 0.25/0.48 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 0.25/0.48 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 0.25/0.48
% 0.25/0.48 % swap_simp
% 0.25/0.48 thf(fact_2877_case__swap,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P5: product_prod @ C @ B] :
% 0.25/0.48 ( ( product_case_prod @ B @ C @ A
% 0.25/0.48 @ ^ [Y3: B,X3: C] : ( F @ X3 @ Y3 )
% 0.25/0.48 @ ( product_swap @ C @ B @ P5 ) )
% 0.25/0.48 = ( product_case_prod @ C @ B @ A @ F @ P5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % case_swap
% 0.25/0.48 thf(fact_2878_swap__comp__swap,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
% 0.25/0.48 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % swap_comp_swap
% 0.25/0.48 thf(fact_2879_bijective__Id,axiom,
% 0.25/0.48 ! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% 0.25/0.48
% 0.25/0.48 % bijective_Id
% 0.25/0.48 thf(fact_2880_euclidean__size__1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid5411537665997757685th_nat @ A )
% 0.25/0.48 => ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
% 0.25/0.48 = ( one_one @ nat ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_1
% 0.25/0.48 thf(fact_2881_fst__swap,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 0.25/0.48 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 0.25/0.48 = ( product_snd @ B @ A @ X ) ) ).
% 0.25/0.48
% 0.25/0.48 % fst_swap
% 0.25/0.48 thf(fact_2882_snd__swap,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 0.25/0.48 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 0.25/0.48 = ( product_fst @ A @ B @ X ) ) ).
% 0.25/0.48
% 0.25/0.48 % snd_swap
% 0.25/0.48 thf(fact_2883_pair__in__swap__image,axiom,
% 0.25/0.48 ! [A: $tType,B: $tType,Y: A,X: B,A4: set @ ( product_prod @ B @ A )] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A4 ) )
% 0.25/0.48 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % pair_in_swap_image
% 0.25/0.48 thf(fact_2884_euclidean__size__integer_Oabs__eq,axiom,
% 0.25/0.48 ! [X: int] :
% 0.25/0.48 ( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X ) )
% 0.25/0.48 = ( euclid6346220572633701492n_size @ int @ X ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_integer.abs_eq
% 0.25/0.48 thf(fact_2885_BNF__Greatest__Fixpoint_OIdD,axiom,
% 0.25/0.48 ! [A: $tType,A3: A,B2: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
% 0.25/0.48 => ( A3 = B2 ) ) ).
% 0.25/0.48
% 0.25/0.48 % BNF_Greatest_Fixpoint.IdD
% 0.25/0.48 thf(fact_2886_IdE,axiom,
% 0.25/0.48 ! [A: $tType,P5: product_prod @ A @ A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ P5 @ ( id2 @ A ) )
% 0.25/0.48 => ~ ! [X2: A] :
% 0.25/0.48 ( P5
% 0.25/0.48 != ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % IdE
% 0.25/0.48 thf(fact_2887_euclidean__size__natural_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ( euclid6346220572633701492n_size @ nat )
% 0.25/0.48 @ ( euclid6346220572633701492n_size @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_natural.rsp
% 0.25/0.48 thf(fact_2888_Linear__order__in__diff__Id,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.48 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Linear_order_in_diff_Id
% 0.25/0.48 thf(fact_2889_euclidean__size__unit,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.25/0.48 => ! [A3: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ( euclid6346220572633701492n_size @ A @ A3 )
% 0.25/0.48 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_unit
% 0.25/0.48 thf(fact_2890_euclidean__size__integer__def,axiom,
% 0.25/0.48 ( ( euclid6346220572633701492n_size @ code_integer )
% 0.25/0.48 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_integer_def
% 0.25/0.48 thf(fact_2891_euclidean__size__integer_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ int @ int @ nat @ nat
% 0.25/0.48 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ( euclid6346220572633701492n_size @ int )
% 0.25/0.48 @ ( euclid6346220572633701492n_size @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_integer.rsp
% 0.25/0.48 thf(fact_2892_Id__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( id2 @ A )
% 0.25/0.48 = ( collect @ ( product_prod @ A @ A )
% 0.25/0.48 @ ^ [P6: product_prod @ A @ A] :
% 0.25/0.48 ? [X3: A] :
% 0.25/0.48 ( P6
% 0.25/0.48 = ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Id_def
% 0.25/0.48 thf(fact_2893_unit__iff__euclidean__size,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.25/0.48 => ! [A3: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 = ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 0.25/0.48 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
% 0.25/0.48 & ( A3
% 0.25/0.48 != ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % unit_iff_euclidean_size
% 0.25/0.48 thf(fact_2894_euclidean__size__times__unit,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3725896446679973847miring @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.48 => ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.25/0.48 = ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % euclidean_size_times_unit
% 0.25/0.48 thf(fact_2895_product__swap,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
% 0.25/0.48 ( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 0.25/0.48 @ ( product_Sigma @ B @ A @ A4
% 0.25/0.48 @ ^ [Uu: B] : B6 ) )
% 0.25/0.48 = ( product_Sigma @ A @ B @ B6
% 0.25/0.48 @ ^ [Uu: A] : A4 ) ) ).
% 0.25/0.48
% 0.25/0.48 % product_swap
% 0.25/0.48 thf(fact_2896_prod_Oswap__def,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( product_swap @ A @ B )
% 0.25/0.48 = ( ^ [P6: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P6 ) @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % prod.swap_def
% 0.25/0.48 thf(fact_2897_reflcl__set__eq,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( sup_sup @ ( A > A > $o )
% 0.25/0.48 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 0.25/0.48 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.25/0.48 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % reflcl_set_eq
% 0.25/0.48 thf(fact_2898_divmod__cases,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.25/0.48 => ! [B2: A,A3: A] :
% 0.25/0.48 ( ( ( B2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.25/0.48 = ( zero_zero @ A ) )
% 0.25/0.48 => ( A3
% 0.25/0.48 != ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
% 0.25/0.48 => ( ( ( B2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ! [Q2: A,R2: A] :
% 0.25/0.48 ( ( ( euclid7384307370059645450egment @ A @ R2 )
% 0.25/0.48 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.25/0.48 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.25/0.48 => ( ( R2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 0.25/0.48 = Q2 )
% 0.25/0.48 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.25/0.48 = R2 )
% 0.25/0.48 => ( A3
% 0.25/0.48 != ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R2 ) ) ) ) ) ) ) )
% 0.25/0.48 => ( B2
% 0.25/0.48 = ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % divmod_cases
% 0.25/0.48 thf(fact_2899_mod__eqI,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.25/0.48 => ! [B2: A,R: A,Q3: A,A3: A] :
% 0.25/0.48 ( ( B2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 0.25/0.48 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.25/0.48 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.25/0.48 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R )
% 0.25/0.48 = A3 )
% 0.25/0.48 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 0.25/0.48 = R ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % mod_eqI
% 0.25/0.48 thf(fact_2900_div__bounded,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.25/0.48 => ! [B2: A,R: A,Q3: A] :
% 0.25/0.48 ( ( B2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 0.25/0.48 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.25/0.48 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.25/0.48 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R ) @ B2 )
% 0.25/0.48 = Q3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % div_bounded
% 0.25/0.48 thf(fact_2901_div__eqI,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.25/0.48 => ! [B2: A,R: A,Q3: A,A3: A] :
% 0.25/0.48 ( ( B2
% 0.25/0.48 != ( zero_zero @ A ) )
% 0.25/0.48 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 0.25/0.48 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 0.25/0.48 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 0.25/0.48 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R )
% 0.25/0.48 = A3 )
% 0.25/0.48 => ( ( divide_divide @ A @ A3 @ B2 )
% 0.25/0.48 = Q3 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % div_eqI
% 0.25/0.48 thf(fact_2902_division__segment__integer_Orep__eq,axiom,
% 0.25/0.48 ! [X: code_integer] :
% 0.25/0.48 ( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X ) )
% 0.25/0.48 = ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_integer.rep_eq
% 0.25/0.48 thf(fact_2903_division__segment__1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid5411537665997757685th_nat @ A )
% 0.25/0.48 => ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
% 0.25/0.48 = ( one_one @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_1
% 0.25/0.48 thf(fact_2904_division__segment__numeral,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid5411537665997757685th_nat @ A )
% 0.25/0.48 => ! [K: num] :
% 0.25/0.48 ( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
% 0.25/0.48 = ( one_one @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_numeral
% 0.25/0.48 thf(fact_2905_division__segment__of__nat,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid5411537665997757685th_nat @ A )
% 0.25/0.48 => ! [N: nat] :
% 0.25/0.48 ( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
% 0.25/0.48 = ( one_one @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_of_nat
% 0.25/0.48 thf(fact_2906_division__segment__integer_Oabs__eq,axiom,
% 0.25/0.48 ! [X: int] :
% 0.25/0.48 ( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X ) )
% 0.25/0.48 = ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_integer.abs_eq
% 0.25/0.48 thf(fact_2907_division__segment__nat__def,axiom,
% 0.25/0.48 ( ( euclid7384307370059645450egment @ nat )
% 0.25/0.48 = ( ^ [N5: nat] : ( one_one @ nat ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_nat_def
% 0.25/0.48 thf(fact_2908_division__segment__integer_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ int @ int @ int @ int
% 0.25/0.48 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.48 @ ( euclid7384307370059645450egment @ int )
% 0.25/0.48 @ ( euclid7384307370059645450egment @ int ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_integer.rsp
% 0.25/0.48 thf(fact_2909_division__segment__natural_Orsp,axiom,
% 0.25/0.48 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.48 @ ( euclid7384307370059645450egment @ nat )
% 0.25/0.48 @ ( euclid7384307370059645450egment @ nat ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_natural.rsp
% 0.25/0.48 thf(fact_2910_division__segment__integer__def,axiom,
% 0.25/0.48 ( ( euclid7384307370059645450egment @ code_integer )
% 0.25/0.48 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % division_segment_integer_def
% 0.25/0.48 thf(fact_2911_is__unit__division__segment,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( euclid3128863361964157862miring @ A )
% 0.25/0.48 => ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % is_unit_division_segment
% 0.25/0.48 thf(fact_2912_Linear__order__wf__diff__Id,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.48 => ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
% 0.25/0.48 = ( ! [A9: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( A9
% 0.25/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.48 => ? [X3: A] :
% 0.25/0.48 ( ( member @ A @ X3 @ A9 )
% 0.25/0.48 & ! [Y3: A] :
% 0.25/0.48 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Linear_order_wf_diff_Id
% 0.25/0.48 thf(fact_2913_bsqr__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( bNF_Wellorder_bsqr @ A )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
% 0.25/0.48 @ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
% 0.25/0.48 @ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
% 0.25/0.48 @ ^ [A12: A,A23: A] :
% 0.25/0.48 ( product_case_prod @ A @ A @ $o
% 0.25/0.48 @ ^ [B12: A,B23: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B12 @ ( insert2 @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R3 ) )
% 0.25/0.48 & ( ( ( A12 = B12 )
% 0.25/0.48 & ( A23 = B23 ) )
% 0.25/0.48 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
% 0.25/0.48 | ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 )
% 0.25/0.48 = ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) )
% 0.25/0.48 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B12 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
% 0.25/0.48 | ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 )
% 0.25/0.48 = ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) )
% 0.25/0.48 & ( A12 = B12 )
% 0.25/0.48 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % bsqr_def
% 0.25/0.48 thf(fact_2914_wo__rel_Ocases__Total3,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
% 0.25/0.48 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) )
% 0.25/0.48 => ( Phi @ A3 @ B2 ) )
% 0.25/0.48 => ( ( ( A3 = B2 )
% 0.25/0.48 => ( Phi @ A3 @ B2 ) )
% 0.25/0.48 => ( Phi @ A3 @ B2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.cases_Total3
% 0.25/0.48 thf(fact_2915_ivl__disj__un__two_I5_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(5)
% 0.25/0.48 thf(fact_2916_ivl__disj__un__singleton_I4_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_singleton(4)
% 0.25/0.48 thf(fact_2917_finite__greaterThanAtMost,axiom,
% 0.25/0.48 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% 0.25/0.48
% 0.25/0.48 % finite_greaterThanAtMost
% 0.25/0.48 thf(fact_2918_finite__greaterThanAtMost__int,axiom,
% 0.25/0.48 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% 0.25/0.48
% 0.25/0.48 % finite_greaterThanAtMost_int
% 0.25/0.48 thf(fact_2919_finite__greaterThanAtMost__integer,axiom,
% 0.25/0.48 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% 0.25/0.48
% 0.25/0.48 % finite_greaterThanAtMost_integer
% 0.25/0.48 thf(fact_2920_card__greaterThanAtMost,axiom,
% 0.25/0.48 ! [L: nat,U: nat] :
% 0.25/0.48 ( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
% 0.25/0.48 = ( minus_minus @ nat @ U @ L ) ) ).
% 0.25/0.48
% 0.25/0.48 % card_greaterThanAtMost
% 0.25/0.48 thf(fact_2921_greaterThanAtMost__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [I: A,L: A,U: A] :
% 0.25/0.48 ( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.25/0.48 = ( ( ord_less @ A @ L @ I )
% 0.25/0.48 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_iff
% 0.25/0.48 thf(fact_2922_greaterThanAtMost__empty,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [L: A,K: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ K )
% 0.25/0.48 => ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_empty
% 0.25/0.48 thf(fact_2923_greaterThanAtMost__empty__iff2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [K: A,L: A] :
% 0.25/0.48 ( ( ( bot_bot @ ( set @ A ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ K @ L ) )
% 0.25/0.48 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_empty_iff2
% 0.25/0.48 thf(fact_2924_greaterThanAtMost__empty__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [K: A,L: A] :
% 0.25/0.48 ( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.48 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_empty_iff
% 0.25/0.48 thf(fact_2925_infinite__Ioc__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( dense_linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
% 0.25/0.48 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % infinite_Ioc_iff
% 0.25/0.48 thf(fact_2926_image__add__greaterThanAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_semidom @ A )
% 0.25/0.48 => ! [C2: A,A3: A,B2: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_add_greaterThanAtMost
% 0.25/0.48 thf(fact_2927_Sup__greaterThanAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( comple6319245703460814977attice @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( ord_less @ A @ X @ Y )
% 0.25/0.48 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
% 0.25/0.48 = Y ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sup_greaterThanAtMost
% 0.25/0.48 thf(fact_2928_Inf__greaterThanAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ( comple6319245703460814977attice @ A )
% 0.25/0.48 & ( dense_linorder @ A ) )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( ord_less @ A @ X @ Y )
% 0.25/0.48 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
% 0.25/0.48 = X ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Inf_greaterThanAtMost
% 0.25/0.48 thf(fact_2929_brk__rel__wf,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( wf @ A @ R5 )
% 0.25/0.48 => ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % brk_rel_wf
% 0.25/0.48 thf(fact_2930_card__greaterThanAtMost__int,axiom,
% 0.25/0.48 ! [L: int,U: int] :
% 0.25/0.48 ( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
% 0.25/0.48 = ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % card_greaterThanAtMost_int
% 0.25/0.48 thf(fact_2931_image__minus__const__greaterThanAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_idom @ A )
% 0.25/0.48 => ! [C2: A,A3: A,B2: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_minus_const_greaterThanAtMost
% 0.25/0.48 thf(fact_2932_image__diff__atLeastLessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_idom @ A )
% 0.25/0.48 => ! [C2: A,A3: A,B2: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_diff_atLeastLessThan
% 0.25/0.48 thf(fact_2933_image__uminus__greaterThanAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_greaterThanAtMost
% 0.25/0.48 thf(fact_2934_image__uminus__atLeastLessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_atLeastLessThan
% 0.25/0.48 thf(fact_2935_reduction__pairI,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( wf @ A @ R5 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
% 0.25/0.48 => ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % reduction_pairI
% 0.25/0.48 thf(fact_2936_wo__rel_Omax2__equals1,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 0.25/0.48 = A3 )
% 0.25/0.48 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.max2_equals1
% 0.25/0.48 thf(fact_2937_wo__rel_Omax2__equals2,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 0.25/0.48 = B2 )
% 0.25/0.48 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.max2_equals2
% 0.25/0.48 thf(fact_2938_wo__rel_Omax2__greater,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
% 0.25/0.48 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.max2_greater
% 0.25/0.48 thf(fact_2939_wf__induct__rule,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ( ! [X2: A] :
% 0.25/0.48 ( ! [Y6: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 0.25/0.48 => ( P @ Y6 ) )
% 0.25/0.48 => ( P @ X2 ) )
% 0.25/0.48 => ( P @ A3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_induct_rule
% 0.25/0.48 thf(fact_2940_wf__eq__minimal,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( wf @ A )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ! [Q6: set @ A] :
% 0.25/0.48 ( ? [X3: A] : ( member @ A @ X3 @ Q6 )
% 0.25/0.48 => ? [X3: A] :
% 0.25/0.48 ( ( member @ A @ X3 @ Q6 )
% 0.25/0.48 & ! [Y3: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
% 0.25/0.48 => ~ ( member @ A @ Y3 @ Q6 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_eq_minimal
% 0.25/0.48 thf(fact_2941_wf__not__refl,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_not_refl
% 0.25/0.48 thf(fact_2942_wf__not__sym,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_not_sym
% 0.25/0.48 thf(fact_2943_wf__irrefl,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_irrefl
% 0.25/0.48 thf(fact_2944_wf__induct,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ( ! [X2: A] :
% 0.25/0.48 ( ! [Y6: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 0.25/0.48 => ( P @ Y6 ) )
% 0.25/0.48 => ( P @ X2 ) )
% 0.25/0.48 => ( P @ A3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_induct
% 0.25/0.48 thf(fact_2945_wf__asym,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
% 0.25/0.48 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_asym
% 0.25/0.48 thf(fact_2946_wfUNIVI,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ! [P7: A > $o,X2: A] :
% 0.25/0.48 ( ! [Xa2: A] :
% 0.25/0.48 ( ! [Y4: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
% 0.25/0.48 => ( P7 @ Y4 ) )
% 0.25/0.48 => ( P7 @ Xa2 ) )
% 0.25/0.48 => ( P7 @ X2 ) )
% 0.25/0.48 => ( wf @ A @ R ) ) ).
% 0.25/0.48
% 0.25/0.48 % wfUNIVI
% 0.25/0.48 thf(fact_2947_wfI__min,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ! [X2: A,Q7: set @ A] :
% 0.25/0.48 ( ( member @ A @ X2 @ Q7 )
% 0.25/0.48 => ? [Xa2: A] :
% 0.25/0.48 ( ( member @ A @ Xa2 @ Q7 )
% 0.25/0.48 & ! [Y4: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R5 )
% 0.25/0.48 => ~ ( member @ A @ Y4 @ Q7 ) ) ) )
% 0.25/0.48 => ( wf @ A @ R5 ) ) ).
% 0.25/0.48
% 0.25/0.48 % wfI_min
% 0.25/0.48 thf(fact_2948_wfE__min,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
% 0.25/0.48 ( ( wf @ A @ R5 )
% 0.25/0.48 => ( ( member @ A @ X @ Q )
% 0.25/0.48 => ~ ! [Z3: A] :
% 0.25/0.48 ( ( member @ A @ Z3 @ Q )
% 0.25/0.48 => ~ ! [Y6: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R5 )
% 0.25/0.48 => ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wfE_min
% 0.25/0.48 thf(fact_2949_wf__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( wf @ A )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ! [P4: A > $o] :
% 0.25/0.48 ( ! [X3: A] :
% 0.25/0.48 ( ! [Y3: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
% 0.25/0.48 => ( P4 @ Y3 ) )
% 0.25/0.48 => ( P4 @ X3 ) )
% 0.25/0.48 => ! [X8: A] : ( P4 @ X8 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_def
% 0.25/0.48 thf(fact_2950_wo__rel_Omax2__def,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.48 => ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 0.25/0.48 = B2 ) )
% 0.25/0.48 & ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.48 => ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 0.25/0.48 = A3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.max2_def
% 0.25/0.48 thf(fact_2951_wo__rel_Owell__order__induct,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ! [X2: A] :
% 0.25/0.48 ( ! [Y6: A] :
% 0.25/0.48 ( ( ( Y6 != X2 )
% 0.25/0.48 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R ) )
% 0.25/0.48 => ( P @ Y6 ) )
% 0.25/0.48 => ( P @ X2 ) )
% 0.25/0.48 => ( P @ A3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.well_order_induct
% 0.25/0.48 thf(fact_2952_Ioc__inj,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.25/0.48 & ( ord_less_eq @ A @ D3 @ C2 ) )
% 0.25/0.48 | ( ( A3 = C2 )
% 0.25/0.48 & ( B2 = D3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Ioc_inj
% 0.25/0.48 thf(fact_2953_wo__rel_OTOTALS,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ! [X7: A] :
% 0.25/0.48 ( ( member @ A @ X7 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ! [Xa2: A] :
% 0.25/0.48 ( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa2 ) @ R )
% 0.25/0.48 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X7 ) @ R ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.TOTALS
% 0.25/0.48 thf(fact_2954_wo__rel_Omax2__greater__among,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
% 0.25/0.48 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
% 0.25/0.48 & ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.max2_greater_among
% 0.25/0.48 thf(fact_2955_wfE__min_H,axiom,
% 0.25/0.48 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),Q: set @ A] :
% 0.25/0.48 ( ( wf @ A @ R5 )
% 0.25/0.48 => ( ( Q
% 0.25/0.48 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.48 => ~ ! [Z3: A] :
% 0.25/0.48 ( ( member @ A @ Z3 @ Q )
% 0.25/0.48 => ~ ! [Y6: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R5 )
% 0.25/0.48 => ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wfE_min'
% 0.25/0.48 thf(fact_2956_infinite__Ioc,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( dense_linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.48 => ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % infinite_Ioc
% 0.25/0.48 thf(fact_2957_Ioc__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.25/0.48 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.48 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Ioc_subset_iff
% 0.25/0.48 thf(fact_2958_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.25/0.48 ! [L: nat,U: nat] :
% 0.25/0.48 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastSucAtMost_greaterThanAtMost
% 0.25/0.48 thf(fact_2959_ivl__disj__un__two_I6_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(6)
% 0.25/0.48 thf(fact_2960_well__order__induct__imp,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ! [X2: A] :
% 0.25/0.48 ( ! [Y6: A] :
% 0.25/0.48 ( ( ( Y6 != X2 )
% 0.25/0.48 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R ) )
% 0.25/0.48 => ( ( member @ A @ Y6 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( P @ Y6 ) ) )
% 0.25/0.48 => ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( P @ X2 ) ) )
% 0.25/0.48 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( P @ A3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % well_order_induct_imp
% 0.25/0.48 thf(fact_2961_wf__bounded__measure,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
% 0.25/0.48 ( ! [A5: A,B3: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R )
% 0.25/0.48 => ( ( ord_less_eq @ nat @ ( Ub @ B3 ) @ ( Ub @ A5 ) )
% 0.25/0.48 & ( ord_less_eq @ nat @ ( F @ B3 ) @ ( Ub @ A5 ) )
% 0.25/0.48 & ( ord_less @ nat @ ( F @ A5 ) @ ( F @ B3 ) ) ) )
% 0.25/0.48 => ( wf @ A @ R ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_bounded_measure
% 0.25/0.48 thf(fact_2962_wfI,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.48 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.48 @ ^ [Uu: A] : B6 ) )
% 0.25/0.48 => ( ! [X2: A,P7: A > $o] :
% 0.25/0.48 ( ! [Xa2: A] :
% 0.25/0.48 ( ! [Y4: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
% 0.25/0.48 => ( P7 @ Y4 ) )
% 0.25/0.48 => ( P7 @ Xa2 ) )
% 0.25/0.48 => ( ( member @ A @ X2 @ A4 )
% 0.25/0.48 => ( ( member @ A @ X2 @ B6 )
% 0.25/0.48 => ( P7 @ X2 ) ) ) )
% 0.25/0.48 => ( wf @ A @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wfI
% 0.25/0.48 thf(fact_2963_ivl__disj__un__two_I8_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(8)
% 0.25/0.48 thf(fact_2964_ivl__disj__un__one_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_one(3)
% 0.25/0.48 thf(fact_2965_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.25/0.48 ! [L: int,U: int] :
% 0.25/0.48 ( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.25/0.48 thf(fact_2966_ivl__disj__un__one_I5_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 0.25/0.48 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_one(5)
% 0.25/0.48 thf(fact_2967_wf__eq__minimal2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( wf @ A )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ! [A9: set @ A] :
% 0.25/0.48 ( ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R3 ) )
% 0.25/0.48 & ( A9
% 0.25/0.48 != ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.48 => ? [X3: A] :
% 0.25/0.48 ( ( member @ A @ X3 @ A9 )
% 0.25/0.48 & ! [Y3: A] :
% 0.25/0.48 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.48 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_eq_minimal2
% 0.25/0.48 thf(fact_2968_wf__bounded__set,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
% 0.25/0.48 ( ! [A5: A,B3: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R )
% 0.25/0.48 => ( ( finite_finite2 @ B @ ( Ub @ A5 ) )
% 0.25/0.48 & ( ord_less_eq @ ( set @ B ) @ ( Ub @ B3 ) @ ( Ub @ A5 ) )
% 0.25/0.48 & ( ord_less_eq @ ( set @ B ) @ ( F @ B3 ) @ ( Ub @ A5 ) )
% 0.25/0.48 & ( ord_less @ ( set @ B ) @ ( F @ A5 ) @ ( F @ B3 ) ) ) )
% 0.25/0.48 => ( wf @ A @ R ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_bounded_set
% 0.25/0.48 thf(fact_2969_wo__rel_Ocases__Total,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.48 => ( Phi @ A3 @ B2 ) )
% 0.25/0.48 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
% 0.25/0.48 => ( Phi @ A3 @ B2 ) )
% 0.25/0.48 => ( Phi @ A3 @ B2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.cases_Total
% 0.25/0.48 thf(fact_2970_sum_Ohead,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( comm_monoid_add @ A )
% 0.25/0.48 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.48 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.48 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.48 = ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sum.head
% 0.25/0.48 thf(fact_2971_prod_Ohead,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( comm_monoid_mult @ A )
% 0.25/0.48 => ! [M2: nat,N: nat,G: nat > A] :
% 0.25/0.48 ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.48 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.48 = ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % prod.head
% 0.25/0.48 thf(fact_2972_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( dense_linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.48 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_subseteq_atLeastAtMost_iff
% 0.25/0.48 thf(fact_2973_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( dense_linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.48 & ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_subseteq_atLeastLessThan_iff
% 0.25/0.48 thf(fact_2974_ivl__disj__un__two__touch_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two_touch(3)
% 0.25/0.48 thf(fact_2975_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ( ( set_or3652927894154168847AtMost @ A )
% 0.25/0.48 = ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert2 @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanAtMost_eq_atLeastAtMost_diff
% 0.25/0.48 thf(fact_2976_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( dense_linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( ( ord_less @ A @ A3 @ B2 )
% 0.25/0.48 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.25/0.48 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThanLessThan_subseteq_greaterThanAtMost_iff
% 0.25/0.48 thf(fact_2977_ivl__disj__un__two_I2_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two(2)
% 0.25/0.48 thf(fact_2978_ivl__disj__un__two__touch_I1_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( ord_less @ A @ L @ M2 )
% 0.25/0.48 => ( ( ord_less @ A @ M2 @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_two_touch(1)
% 0.25/0.48 thf(fact_2979_wf__bounded__supset,axiom,
% 0.25/0.48 ! [A: $tType,S2: set @ A] :
% 0.25/0.48 ( ( finite_finite2 @ A @ S2 )
% 0.25/0.48 => ( wf @ ( set @ A )
% 0.25/0.48 @ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.25/0.48 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 0.25/0.48 @ ^ [Q8: set @ A,Q6: set @ A] :
% 0.25/0.48 ( ( ord_less @ ( set @ A ) @ Q6 @ Q8 )
% 0.25/0.48 & ( ord_less_eq @ ( set @ A ) @ Q8 @ S2 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_bounded_supset
% 0.25/0.48 thf(fact_2980_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
% 0.25/0.48 ! [L: code_integer,U: code_integer] :
% 0.25/0.48 ( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastPlusOneAtMost_greaterThanAtMost_integer
% 0.25/0.48 thf(fact_2981_ivl__disj__un__singleton_I5_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.48 => ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_un_singleton(5)
% 0.25/0.48 thf(fact_2982_wf__iff__no__infinite__down__chain,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( wf @ A )
% 0.25/0.48 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.48 ~ ? [F2: nat > A] :
% 0.25/0.48 ! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) @ R3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_iff_no_infinite_down_chain
% 0.25/0.48 thf(fact_2983_wf__no__infinite__down__chainE,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
% 0.25/0.48 ( ( wf @ A @ R )
% 0.25/0.48 => ~ ! [K5: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K5 ) ) @ ( F @ K5 ) ) @ R ) ) ).
% 0.25/0.48
% 0.25/0.48 % wf_no_infinite_down_chainE
% 0.25/0.48 thf(fact_2984_wo__rel_Ominim__least,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ B2 @ B6 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) @ B2 ) @ R ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.minim_least
% 0.25/0.48 thf(fact_2985_wo__rel_Oequals__minim,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
% 0.25/0.48 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 0.25/0.48 => ( ( member @ A @ A3 @ B6 )
% 0.25/0.48 => ( ! [B3: A] :
% 0.25/0.48 ( ( member @ A @ B3 @ B6 )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) )
% 0.25/0.48 => ( A3
% 0.25/0.48 = ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % wo_rel.equals_minim
% 0.25/0.48 thf(fact_2986_dependent__wf__choice,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
% 0.25/0.48 ( ( wf @ A @ R5 )
% 0.25/0.48 => ( ! [F4: A > B,G4: A > B,X2: A,R2: B] :
% 0.25/0.48 ( ! [Z6: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X2 ) @ R5 )
% 0.25/0.48 => ( ( F4 @ Z6 )
% 0.25/0.48 = ( G4 @ Z6 ) ) )
% 0.25/0.48 => ( ( P @ F4 @ X2 @ R2 )
% 0.25/0.48 = ( P @ G4 @ X2 @ R2 ) ) )
% 0.25/0.48 => ( ! [X2: A,F4: A > B] :
% 0.25/0.48 ( ! [Y6: A] :
% 0.25/0.48 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R5 )
% 0.25/0.48 => ( P @ F4 @ Y6 @ ( F4 @ Y6 ) ) )
% 0.25/0.48 => ? [X_1: B] : ( P @ F4 @ X2 @ X_1 ) )
% 0.25/0.48 => ? [F4: A > B] :
% 0.25/0.48 ! [X7: A] : ( P @ F4 @ X7 @ ( F4 @ X7 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % dependent_wf_choice
% 0.25/0.48 thf(fact_2987_of__rat__neg__one,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( field_char_0 @ A )
% 0.25/0.48 => ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 0.25/0.48 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % of_rat_neg_one
% 0.25/0.48 thf(fact_2988_Ref__Time_Oupdate__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( heap @ A )
% 0.25/0.48 => ( ( ref_update @ A )
% 0.25/0.48 = ( ^ [R3: ref @ A,V2: A] :
% 0.25/0.48 ( heap_Time_heap @ product_unit
% 0.25/0.48 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ V2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Ref_Time.update_def
% 0.25/0.48 thf(fact_2989_sum__div__partition,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType] :
% 0.25/0.48 ( ( euclid4440199948858584721cancel @ A )
% 0.25/0.48 => ! [A4: set @ B,F: B > A,B2: A] :
% 0.25/0.48 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.48 => ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A4 ) @ B2 )
% 0.25/0.48 = ( plus_plus @ A
% 0.25/0.48 @ ( groups7311177749621191930dd_sum @ B @ A
% 0.25/0.48 @ ^ [A7: B] : ( divide_divide @ A @ ( F @ A7 ) @ B2 )
% 0.25/0.48 @ ( inf_inf @ ( set @ B ) @ A4
% 0.25/0.48 @ ( collect @ B
% 0.25/0.48 @ ^ [A7: B] : ( dvd_dvd @ A @ B2 @ ( F @ A7 ) ) ) ) )
% 0.25/0.48 @ ( divide_divide @ A
% 0.25/0.48 @ ( groups7311177749621191930dd_sum @ B @ A @ F
% 0.25/0.48 @ ( inf_inf @ ( set @ B ) @ A4
% 0.25/0.48 @ ( collect @ B
% 0.25/0.48 @ ^ [A7: B] :
% 0.25/0.48 ~ ( dvd_dvd @ A @ B2 @ ( F @ A7 ) ) ) ) )
% 0.25/0.48 @ B2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % sum_div_partition
% 0.25/0.48 thf(fact_2990_bsqr__max2,axiom,
% 0.25/0.48 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A1: A,A22: A,B1: A,B22: A] :
% 0.25/0.48 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr @ A @ R ) )
% 0.25/0.48 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B22 ) ) @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % bsqr_max2
% 0.25/0.48 thf(fact_2991_UN__atLeast__UNIV,axiom,
% 0.25/0.48 ( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 0.25/0.48 = ( top_top @ ( set @ nat ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % UN_atLeast_UNIV
% 0.25/0.48 thf(fact_2992_atLeast__eq__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( ( set_ord_atLeast @ A @ X )
% 0.25/0.48 = ( set_ord_atLeast @ A @ Y ) )
% 0.25/0.48 = ( X = Y ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_eq_iff
% 0.25/0.48 thf(fact_2993_unit__abs__eta__conv,axiom,
% 0.25/0.48 ! [A: $tType,F: product_unit > A] :
% 0.25/0.48 ( ( ^ [U2: product_unit] : ( F @ product_Unity ) )
% 0.25/0.48 = F ) ).
% 0.25/0.48
% 0.25/0.48 % unit_abs_eta_conv
% 0.25/0.48 thf(fact_2994_inf__Some,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( inf @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( inf_inf @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 0.25/0.48 = ( some @ A @ ( inf_inf @ A @ X @ Y ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % inf_Some
% 0.25/0.48 thf(fact_2995_atLeast__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ! [I: A,K: A] :
% 0.25/0.48 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
% 0.25/0.48 = ( ord_less_eq @ A @ K @ I ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_iff
% 0.25/0.48 thf(fact_2996_Inf__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( comple6319245703460814977attice @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X ) )
% 0.25/0.48 = X ) ) ).
% 0.25/0.48
% 0.25/0.48 % Inf_atLeast
% 0.25/0.48 thf(fact_2997_atLeast__subset__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [X: A,Y: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
% 0.25/0.48 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_subset_iff
% 0.25/0.48 thf(fact_2998_image__add__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_semidom @ A )
% 0.25/0.48 => ! [K: A,I: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
% 0.25/0.48 = ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_add_atLeast
% 0.25/0.48 thf(fact_2999_Sup__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( comple6319245703460814977attice @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
% 0.25/0.48 = ( top_top @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sup_atLeast
% 0.25/0.48 thf(fact_3000_inter__compl__diff__conv,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 0.25/0.48 = ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ).
% 0.25/0.48
% 0.25/0.48 % inter_compl_diff_conv
% 0.25/0.48 thf(fact_3001_Int__atMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atMost
% 0.25/0.48 thf(fact_3002_Compl__lessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [K: A] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
% 0.25/0.48 = ( set_ord_atLeast @ A @ K ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Compl_lessThan
% 0.25/0.48 thf(fact_3003_Compl__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [K: A] :
% 0.25/0.48 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
% 0.25/0.48 = ( set_ord_lessThan @ A @ K ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Compl_atLeast
% 0.25/0.48 thf(fact_3004_one__eq__of__rat__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( field_char_0 @ A )
% 0.25/0.48 => ! [A3: rat] :
% 0.25/0.48 ( ( ( one_one @ A )
% 0.25/0.48 = ( field_char_0_of_rat @ A @ A3 ) )
% 0.25/0.48 = ( ( one_one @ rat )
% 0.25/0.48 = A3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % one_eq_of_rat_iff
% 0.25/0.48 thf(fact_3005_of__rat__eq__1__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( field_char_0 @ A )
% 0.25/0.48 => ! [A3: rat] :
% 0.25/0.48 ( ( ( field_char_0_of_rat @ A @ A3 )
% 0.25/0.48 = ( one_one @ A ) )
% 0.25/0.48 = ( A3
% 0.25/0.48 = ( one_one @ rat ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % of_rat_eq_1_iff
% 0.25/0.48 thf(fact_3006_of__rat__1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( field_char_0 @ A )
% 0.25/0.48 => ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
% 0.25/0.48 = ( one_one @ A ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % of_rat_1
% 0.25/0.48 thf(fact_3007_atLeast__0,axiom,
% 0.25/0.48 ( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
% 0.25/0.48 = ( top_top @ ( set @ nat ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_0
% 0.25/0.48 thf(fact_3008_Icc__subset__Ici__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [L: A,H3: A,L4: A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) @ ( set_ord_atLeast @ A @ L4 ) )
% 0.25/0.48 = ( ~ ( ord_less_eq @ A @ L @ H3 )
% 0.25/0.48 | ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Icc_subset_Ici_iff
% 0.25/0.48 thf(fact_3009_Int__atLeastAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atLeastAtMost
% 0.25/0.48 thf(fact_3010_Int__atLeastLessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atLeastLessThan
% 0.25/0.48 thf(fact_3011_Int__atLeastAtMostR1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atLeastAtMostR1
% 0.25/0.48 thf(fact_3012_Int__atLeastAtMostL1,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atLeastAtMostL1
% 0.25/0.48 thf(fact_3013_image__minus__const__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_idom @ A )
% 0.25/0.48 => ! [C2: A,A3: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A3 ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_minus_const_atLeast
% 0.25/0.48 thf(fact_3014_image__minus__const__AtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_idom @ A )
% 0.25/0.48 => ! [C2: A,B2: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B2 ) )
% 0.25/0.48 = ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_minus_const_AtMost
% 0.25/0.48 thf(fact_3015_image__uminus__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
% 0.25/0.48 = ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_atLeast
% 0.25/0.48 thf(fact_3016_image__uminus__atMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ordered_ab_group_add @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
% 0.25/0.48 = ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % image_uminus_atMost
% 0.25/0.48 thf(fact_3017_Int__atLeastAtMostL2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C2 ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atLeastAtMostL2
% 0.25/0.48 thf(fact_3018_Int__atLeastAtMostR2,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,C2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ D3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_atLeastAtMostR2
% 0.25/0.48 thf(fact_3019_Int__greaterThanAtMost,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_greaterThanAtMost
% 0.25/0.48 thf(fact_3020_Int__greaterThanLessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
% 0.25/0.48 = ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Int_greaterThanLessThan
% 0.25/0.48 thf(fact_3021_one__le__of__rat__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_field @ A )
% 0.25/0.48 => ! [R: rat] :
% 0.25/0.48 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 0.25/0.48 = ( ord_less_eq @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % one_le_of_rat_iff
% 0.25/0.48 thf(fact_3022_of__rat__le__1__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_field @ A )
% 0.25/0.48 => ! [R: rat] :
% 0.25/0.48 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 0.25/0.48 = ( ord_less_eq @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % of_rat_le_1_iff
% 0.25/0.48 thf(fact_3023_one__less__of__rat__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_field @ A )
% 0.25/0.48 => ! [R: rat] :
% 0.25/0.48 ( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 0.25/0.48 = ( ord_less @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % one_less_of_rat_iff
% 0.25/0.48 thf(fact_3024_of__rat__less__1__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linordered_field @ A )
% 0.25/0.48 => ! [R: rat] :
% 0.25/0.48 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 0.25/0.48 = ( ord_less @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % of_rat_less_1_iff
% 0.25/0.48 thf(fact_3025_atLeastLessThan__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ( ( set_or7035219750837199246ssThan @ A )
% 0.25/0.48 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L3 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastLessThan_def
% 0.25/0.48 thf(fact_3026_well__order__on__domain,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.48 ( ( order_well_order_on @ A @ A4 @ R )
% 0.25/0.48 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.48 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.48 & ( member @ A @ B2 @ A4 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % well_order_on_domain
% 0.25/0.48 thf(fact_3027_Inf__unit__def,axiom,
% 0.25/0.48 ( ( complete_Inf_Inf @ product_unit )
% 0.25/0.48 = ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% 0.25/0.48
% 0.25/0.48 % Inf_unit_def
% 0.25/0.48 thf(fact_3028_Sup__unit__def,axiom,
% 0.25/0.48 ( ( complete_Sup_Sup @ product_unit )
% 0.25/0.48 = ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sup_unit_def
% 0.25/0.48 thf(fact_3029_Times__Int__distrib1,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 @ ^ [Uu: A] : C5 )
% 0.25/0.48 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : C5 )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ B6
% 0.25/0.48 @ ^ [Uu: A] : C5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Times_Int_distrib1
% 0.25/0.48 thf(fact_3030_Sigma__Int__distrib2,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ I4
% 0.25/0.48 @ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
% 0.25/0.48 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_Int_distrib2
% 0.25/0.48 thf(fact_3031_Times__Int__Times,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
% 0.25/0.48 ( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.48 @ ^ [Uu: A] : B6 )
% 0.25/0.48 @ ( product_Sigma @ A @ B @ C5
% 0.25/0.48 @ ^ [Uu: A] : D4 ) )
% 0.25/0.48 = ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ C5 )
% 0.25/0.48 @ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B6 @ D4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Times_Int_Times
% 0.25/0.48 thf(fact_3032_Sigma__Int__distrib1,axiom,
% 0.25/0.48 ! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
% 0.25/0.48 ( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I4 @ J4 ) @ C5 )
% 0.25/0.48 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % Sigma_Int_distrib1
% 0.25/0.48 thf(fact_3033_old_Ounit_Oexhaust,axiom,
% 0.25/0.48 ! [Y: product_unit] : ( Y = product_Unity ) ).
% 0.25/0.48
% 0.25/0.48 % old.unit.exhaust
% 0.25/0.48 thf(fact_3034_top__unit__def,axiom,
% 0.25/0.48 ( ( top_top @ product_unit )
% 0.25/0.48 = product_Unity ) ).
% 0.25/0.48
% 0.25/0.48 % top_unit_def
% 0.25/0.48 thf(fact_3035_uminus__unit__def,axiom,
% 0.25/0.48 ( ( uminus_uminus @ product_unit )
% 0.25/0.48 = ( ^ [Uu3: product_unit] : product_Unity ) ) ).
% 0.25/0.48
% 0.25/0.48 % uminus_unit_def
% 0.25/0.48 thf(fact_3036_bot__unit__def,axiom,
% 0.25/0.48 ( ( bot_bot @ product_unit )
% 0.25/0.48 = product_Unity ) ).
% 0.25/0.48
% 0.25/0.48 % bot_unit_def
% 0.25/0.48 thf(fact_3037_ivl__disj__int__one_I8_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_int_one(8)
% 0.25/0.48 thf(fact_3038_sup__unit__def,axiom,
% 0.25/0.48 ( ( sup_sup @ product_unit )
% 0.25/0.48 = ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% 0.25/0.48
% 0.25/0.48 % sup_unit_def
% 0.25/0.48 thf(fact_3039_atLeastAtMost__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ( ( set_or1337092689740270186AtMost @ A )
% 0.25/0.48 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L3 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeastAtMost_def
% 0.25/0.48 thf(fact_3040_ivl__disj__int__one_I6_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [L: A,U: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_int_one(6)
% 0.25/0.48 thf(fact_3041_disjointI,axiom,
% 0.25/0.48 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 0.25/0.48 ( ! [X2: A] :
% 0.25/0.48 ( ( member @ A @ X2 @ A3 )
% 0.25/0.48 => ~ ( member @ A @ X2 @ B2 ) )
% 0.25/0.48 => ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % disjointI
% 0.25/0.48 thf(fact_3042_inter__eq__subsetI,axiom,
% 0.25/0.48 ! [A: $tType,S2: set @ A,S7: set @ A,A4: set @ A,B6: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ S2 @ S7 )
% 0.25/0.48 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ S7 )
% 0.25/0.48 = ( inf_inf @ ( set @ A ) @ B6 @ S7 ) )
% 0.25/0.48 => ( ( inf_inf @ ( set @ A ) @ A4 @ S2 )
% 0.25/0.48 = ( inf_inf @ ( set @ A ) @ B6 @ S2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % inter_eq_subsetI
% 0.25/0.48 thf(fact_3043_not__empty__eq__Ici__eq__empty,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( preorder @ A )
% 0.25/0.48 => ! [L: A] :
% 0.25/0.48 ( ( bot_bot @ ( set @ A ) )
% 0.25/0.48 != ( set_ord_atLeast @ A @ L ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_empty_eq_Ici_eq_empty
% 0.25/0.48 thf(fact_3044_infinite__Ici,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ( linorder @ A )
% 0.25/0.48 & ( no_top @ A ) )
% 0.25/0.48 => ! [A3: A] :
% 0.25/0.48 ~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % infinite_Ici
% 0.25/0.48 thf(fact_3045_not__UNIV__eq__Ici,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_bot @ A )
% 0.25/0.48 => ! [L4: A] :
% 0.25/0.48 ( ( top_top @ ( set @ A ) )
% 0.25/0.48 != ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_UNIV_eq_Ici
% 0.25/0.48 thf(fact_3046_not__Ici__eq__Icc,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_top @ A )
% 0.25/0.48 => ! [L4: A,L: A,H3: A] :
% 0.25/0.48 ( ( set_ord_atLeast @ A @ L4 )
% 0.25/0.48 != ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Ici_eq_Icc
% 0.25/0.48 thf(fact_3047_not__Iic__eq__Ici,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_top @ A )
% 0.25/0.48 => ! [H3: A,L4: A] :
% 0.25/0.48 ( ( set_ord_atMost @ A @ H3 )
% 0.25/0.48 != ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Iic_eq_Ici
% 0.25/0.48 thf(fact_3048_atMost__Int__atLeast,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [N: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
% 0.25/0.48 = ( insert2 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atMost_Int_atLeast
% 0.25/0.48 thf(fact_3049_atLeast__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ord @ A )
% 0.25/0.48 => ( ( set_ord_atLeast @ A )
% 0.25/0.48 = ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_def
% 0.25/0.48 thf(fact_3050_inf__option__def,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( inf @ A )
% 0.25/0.48 => ( ( inf_inf @ ( option @ A ) )
% 0.25/0.48 = ( ^ [X3: option @ A,Y3: option @ A] :
% 0.25/0.48 ( case_option @ ( option @ A ) @ A @ ( none @ A )
% 0.25/0.48 @ ^ [Z5: A] :
% 0.25/0.48 ( case_option @ ( option @ A ) @ A @ ( none @ A )
% 0.25/0.48 @ ^ [Aa2: A] : ( some @ A @ ( inf_inf @ A @ Z5 @ Aa2 ) )
% 0.25/0.48 @ Y3 )
% 0.25/0.48 @ X3 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % inf_option_def
% 0.25/0.48 thf(fact_3051_translation__Int,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( ab_group_add @ A )
% 0.25/0.48 => ! [A3: A,S: set @ A,T2: set @ A] :
% 0.25/0.48 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
% 0.25/0.48 = ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % translation_Int
% 0.25/0.48 thf(fact_3052_disjoint__mono,axiom,
% 0.25/0.48 ! [A: $tType,A3: set @ A,A6: set @ A,B2: set @ A,B4: set @ A] :
% 0.25/0.48 ( ( ord_less_eq @ ( set @ A ) @ A3 @ A6 )
% 0.25/0.48 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ B4 )
% 0.25/0.48 => ( ( ( inf_inf @ ( set @ A ) @ A6 @ B4 )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.48 => ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % disjoint_mono
% 0.25/0.48 thf(fact_3053_disjoint__alt__simp2,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.48 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 != A4 )
% 0.25/0.48 = ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % disjoint_alt_simp2
% 0.25/0.48 thf(fact_3054_disjoint__alt__simp1,axiom,
% 0.25/0.48 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.48 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 = A4 )
% 0.25/0.48 = ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % disjoint_alt_simp1
% 0.25/0.48 thf(fact_3055_ivl__disj__int__two_I3_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_int_two(3)
% 0.25/0.48 thf(fact_3056_ivl__disj__int__two_I6_J,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order @ A )
% 0.25/0.48 => ! [L: A,M2: A,U: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 0.25/0.48 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % ivl_disj_int_two(6)
% 0.25/0.48 thf(fact_3057_greaterThan__Int__greaterThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
% 0.25/0.48 = ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % greaterThan_Int_greaterThan
% 0.25/0.48 thf(fact_3058_atLeast__eq__UNIV__iff,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( order_bot @ A )
% 0.25/0.48 => ! [X: A] :
% 0.25/0.48 ( ( ( set_ord_atLeast @ A @ X )
% 0.25/0.48 = ( top_top @ ( set @ A ) ) )
% 0.25/0.48 = ( X
% 0.25/0.48 = ( bot_bot @ A ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % atLeast_eq_UNIV_iff
% 0.25/0.48 thf(fact_3059_not__UNIV__le__Ici,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_bot @ A )
% 0.25/0.48 => ! [L: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_UNIV_le_Ici
% 0.25/0.48 thf(fact_3060_UNIV__unit,axiom,
% 0.25/0.48 ( ( top_top @ ( set @ product_unit ) )
% 0.25/0.48 = ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % UNIV_unit
% 0.25/0.48 thf(fact_3061_not__Ici__le__Icc,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_top @ A )
% 0.25/0.48 => ! [L: A,L4: A,H4: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Ici_le_Icc
% 0.25/0.48 thf(fact_3062_lessThan__Int__lessThan,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( linorder @ A )
% 0.25/0.48 => ! [A3: A,B2: A] :
% 0.25/0.48 ( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
% 0.25/0.48 = ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % lessThan_Int_lessThan
% 0.25/0.48 thf(fact_3063_not__Iic__le__Ici,axiom,
% 0.25/0.48 ! [A: $tType] :
% 0.25/0.48 ( ( no_bot @ A )
% 0.25/0.48 => ! [H3: A,L4: A] :
% 0.25/0.48 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H3 ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 0.25/0.48
% 0.25/0.48 % not_Iic_le_Ici
% 0.25/0.48 thf(fact_3064_not__Ici__le__Iic,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( no_top @ A )
% 0.25/0.49 => ! [L: A,H4: A] :
% 0.25/0.49 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % not_Ici_le_Iic
% 0.25/0.49 thf(fact_3065_translation__subtract__Int,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( ab_group_add @ A )
% 0.25/0.49 => ! [A3: A,S: set @ A,T2: set @ A] :
% 0.25/0.49 ( ( image2 @ A @ A
% 0.25/0.49 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.49 @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
% 0.25/0.49 = ( inf_inf @ ( set @ A )
% 0.25/0.49 @ ( image2 @ A @ A
% 0.25/0.49 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.49 @ S )
% 0.25/0.49 @ ( image2 @ A @ A
% 0.25/0.49 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 0.25/0.49 @ T2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % translation_subtract_Int
% 0.25/0.49 thf(fact_3066_of__rat__add,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( field_char_0 @ A )
% 0.25/0.49 => ! [A3: rat,B2: rat] :
% 0.25/0.49 ( ( field_char_0_of_rat @ A @ ( plus_plus @ rat @ A3 @ B2 ) )
% 0.25/0.49 = ( plus_plus @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % of_rat_add
% 0.25/0.49 thf(fact_3067_Ioi__le__Ico,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( preorder @ A )
% 0.25/0.49 => ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Ioi_le_Ico
% 0.25/0.49 thf(fact_3068_atLeast__Suc__greaterThan,axiom,
% 0.25/0.49 ! [K: nat] :
% 0.25/0.49 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 0.25/0.49 = ( set_ord_greaterThan @ nat @ K ) ) ).
% 0.25/0.49
% 0.25/0.49 % atLeast_Suc_greaterThan
% 0.25/0.49 thf(fact_3069_ivl__disj__int__two_I7_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,M2: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_two(7)
% 0.25/0.49 thf(fact_3070_disjoint__alt__simp3,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ A4 )
% 0.25/0.49 = ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.49 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % disjoint_alt_simp3
% 0.25/0.49 thf(fact_3071_Ioc__disjoint,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [A3: A,B2: A,C2: A,D3: A] :
% 0.25/0.49 ( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 0.25/0.49 | ( ord_less_eq @ A @ D3 @ C2 )
% 0.25/0.49 | ( ord_less_eq @ A @ B2 @ C2 )
% 0.25/0.49 | ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Ioc_disjoint
% 0.25/0.49 thf(fact_3072_ivl__disj__int__one_I4_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_one(4)
% 0.25/0.49 thf(fact_3073_ivl__disj__int__one_I2_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_one(2)
% 0.25/0.49 thf(fact_3074_wait__def,axiom,
% 0.25/0.49 ( heap_Time_wait
% 0.25/0.49 = ( ^ [N5: nat] :
% 0.25/0.49 ( heap_Time_Heap2 @ product_unit
% 0.25/0.49 @ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N5 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % wait_def
% 0.25/0.49 thf(fact_3075_ivl__disj__int__two_I8_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,M2: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_two(8)
% 0.25/0.49 thf(fact_3076_ivl__disj__int__two_I5_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,M2: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_two(5)
% 0.25/0.49 thf(fact_3077_ivl__disj__int__two_I4_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,M2: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_two(4)
% 0.25/0.49 thf(fact_3078_ivl__disj__int__two_I1_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,M2: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_two(1)
% 0.25/0.49 thf(fact_3079_ivl__disj__int__one_I3_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_one(3)
% 0.25/0.49 thf(fact_3080_ivl__disj__int__one_I7_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_one(7)
% 0.25/0.49 thf(fact_3081_ivl__disj__int__one_I1_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_one(1)
% 0.25/0.49 thf(fact_3082_ivl__disj__int__two_I2_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,M2: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_two(2)
% 0.25/0.49 thf(fact_3083_ivl__disj__int__one_I5_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_int_one(5)
% 0.25/0.49 thf(fact_3084_ivl__disj__un__one_I8_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.49 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 0.25/0.49 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_un_one(8)
% 0.25/0.49 thf(fact_3085_prod_Ointer__restrict,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_mult @ A )
% 0.25/0.49 => ! [A4: set @ B,G: B > A,B6: set @ B] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 = ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.49 @ ^ [X3: B] : ( if @ A @ ( member @ B @ X3 @ B6 ) @ ( G @ X3 ) @ ( one_one @ A ) )
% 0.25/0.49 @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % prod.inter_restrict
% 0.25/0.49 thf(fact_3086_greaterThanAtMost__def,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( ord @ A )
% 0.25/0.49 => ( ( set_or3652927894154168847AtMost @ A )
% 0.25/0.49 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L3 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % greaterThanAtMost_def
% 0.25/0.49 thf(fact_3087_greaterThanLessThan__eq,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( ord @ A )
% 0.25/0.49 => ( ( set_or5935395276787703475ssThan @ A )
% 0.25/0.49 = ( ^ [A7: A,B5: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A7 ) @ ( set_ord_lessThan @ A @ B5 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % greaterThanLessThan_eq
% 0.25/0.49 thf(fact_3088_greaterThanLessThan__def,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( ord @ A )
% 0.25/0.49 => ( ( set_or5935395276787703475ssThan @ A )
% 0.25/0.49 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L3 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % greaterThanLessThan_def
% 0.25/0.49 thf(fact_3089_Ici__subset__Ioi__iff,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [A3: A,B2: A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
% 0.25/0.49 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Ici_subset_Ioi_iff
% 0.25/0.49 thf(fact_3090_execute__assert_H,axiom,
% 0.25/0.49 ! [P: $o,H3: heap_ext @ product_unit] :
% 0.25/0.49 ( ( P
% 0.25/0.49 => ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
% 0.25/0.49 = ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) )
% 0.25/0.49 & ( ~ P
% 0.25/0.49 => ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
% 0.25/0.49 = ( none @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % execute_assert'
% 0.25/0.49 thf(fact_3091_sum_Ounion__inter,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.49 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) )
% 0.25/0.49 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.union_inter
% 0.25/0.49 thf(fact_3092_sum_OInt__Diff,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [A4: set @ B,G: B > A,B6: set @ B] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
% 0.25/0.49 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.Int_Diff
% 0.25/0.49 thf(fact_3093_prod_Omono__neutral__cong,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_mult @ A )
% 0.25/0.49 => ! [T6: set @ B,S2: set @ B,H3: B > A,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ T6 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ S2 )
% 0.25/0.49 => ( ! [I2: B] :
% 0.25/0.49 ( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 0.25/0.49 => ( ( H3 @ I2 )
% 0.25/0.49 = ( one_one @ A ) ) )
% 0.25/0.49 => ( ! [I2: B] :
% 0.25/0.49 ( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S2 @ T6 ) )
% 0.25/0.49 => ( ( G @ I2 )
% 0.25/0.49 = ( one_one @ A ) ) )
% 0.25/0.49 => ( ! [X2: B] :
% 0.25/0.49 ( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S2 @ T6 ) )
% 0.25/0.49 => ( ( G @ X2 )
% 0.25/0.49 = ( H3 @ X2 ) ) )
% 0.25/0.49 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 0.25/0.49 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % prod.mono_neutral_cong
% 0.25/0.49 thf(fact_3094_Iio__Int__singleton,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [X: A,K: A] :
% 0.25/0.49 ( ( ( ord_less @ A @ X @ K )
% 0.25/0.49 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.49 & ( ~ ( ord_less @ A @ X @ K )
% 0.25/0.49 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Iio_Int_singleton
% 0.25/0.49 thf(fact_3095_card__Un__Int,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ A @ B6 )
% 0.25/0.49 => ( ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) )
% 0.25/0.49 = ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % card_Un_Int
% 0.25/0.49 thf(fact_3096_sum_OIf__cases,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [A4: set @ B,P: B > $o,H3: B > A,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ B @ A
% 0.25/0.49 @ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G @ X3 ) )
% 0.25/0.49 @ A4 )
% 0.25/0.49 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.If_cases
% 0.25/0.49 thf(fact_3097_ivl__disj__un__singleton_I1_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [L: A] :
% 0.25/0.49 ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
% 0.25/0.49 = ( set_ord_atLeast @ A @ L ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_un_singleton(1)
% 0.25/0.49 thf(fact_3098_ivl__disj__un__one_I7_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( ord_less_eq @ A @ L @ U )
% 0.25/0.49 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 0.25/0.49 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_un_one(7)
% 0.25/0.49 thf(fact_3099_ivl__disj__un__one_I6_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [L: A,U: A] :
% 0.25/0.49 ( ( ord_less @ A @ L @ U )
% 0.25/0.49 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 0.25/0.49 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ivl_disj_un_one(6)
% 0.25/0.49 thf(fact_3100_Linear__order__Well__order__iff,axiom,
% 0.25/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.49 => ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.49 = ( ! [A9: set @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
% 0.25/0.49 => ( ( A9
% 0.25/0.49 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ? [X3: A] :
% 0.25/0.49 ( ( member @ A @ X3 @ A9 )
% 0.25/0.49 & ! [Y3: A] :
% 0.25/0.49 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Linear_order_Well_order_iff
% 0.25/0.49 thf(fact_3101_sum_Ounion__inter__neutral,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.49 => ( ! [X2: B] :
% 0.25/0.49 ( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 => ( ( G @ X2 )
% 0.25/0.49 = ( zero_zero @ A ) ) )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.union_inter_neutral
% 0.25/0.49 thf(fact_3102_sum__Un,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( ab_group_add @ A )
% 0.25/0.49 => ! [A4: set @ B,B6: set @ B,F: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum_Un
% 0.25/0.49 thf(fact_3103_sum_Ounion__disjoint,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.49 => ( ( ( inf_inf @ ( set @ B ) @ A4 @ B6 )
% 0.25/0.49 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.union_disjoint
% 0.25/0.49 thf(fact_3104_prod_Ounion__inter__neutral,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_mult @ A )
% 0.25/0.49 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.49 => ( ! [X2: B] :
% 0.25/0.49 ( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 => ( ( G @ X2 )
% 0.25/0.49 = ( one_one @ A ) ) )
% 0.25/0.49 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % prod.union_inter_neutral
% 0.25/0.49 thf(fact_3105_execute__update,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( heap @ A )
% 0.25/0.49 => ! [R: ref @ A,V: A,H3: heap_ext @ product_unit] :
% 0.25/0.49 ( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R @ V ) @ H3 )
% 0.25/0.49 = ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R @ V @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % execute_update
% 0.25/0.49 thf(fact_3106_sum_Ounion__diff2,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 0.25/0.49 = ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.union_diff2
% 0.25/0.49 thf(fact_3107_sum__Un2,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ B )
% 0.25/0.49 => ! [A4: set @ A,B6: set @ A,F: A > B] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum_Un2
% 0.25/0.49 thf(fact_3108_card__Un__disjoint,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ A @ B6 )
% 0.25/0.49 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % card_Un_disjoint
% 0.25/0.49 thf(fact_3109_sum__Un__nat,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,B6: set @ A,F: A > nat] :
% 0.25/0.49 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.49 => ( ( finite_finite2 @ A @ B6 )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum_Un_nat
% 0.25/0.49 thf(fact_3110_atLeast__Suc,axiom,
% 0.25/0.49 ! [K: nat] :
% 0.25/0.49 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 0.25/0.49 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert2 @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % atLeast_Suc
% 0.25/0.49 thf(fact_3111_old_Ounit_Orec,axiom,
% 0.25/0.49 ! [T: $tType,F1: T] :
% 0.25/0.49 ( ( product_rec_unit @ T @ F1 @ product_Unity )
% 0.25/0.49 = F1 ) ).
% 0.25/0.49
% 0.25/0.49 % old.unit.rec
% 0.25/0.49 thf(fact_3112_default__unit__def,axiom,
% 0.25/0.49 ( ( default_default @ product_unit )
% 0.25/0.49 = product_Unity ) ).
% 0.25/0.49
% 0.25/0.49 % default_unit_def
% 0.25/0.49 thf(fact_3113_CODE__ABORT__def,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( cODE_ABORT @ A )
% 0.25/0.49 = ( ^ [F2: product_unit > A] : ( F2 @ product_Unity ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % CODE_ABORT_def
% 0.25/0.49 thf(fact_3114_Inf__fin_Oeq__fold_H,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( semilattice_inf @ A )
% 0.25/0.49 => ( ( lattic7752659483105999362nf_fin @ A )
% 0.25/0.49 = ( ^ [A9: set @ A] :
% 0.25/0.49 ( the2 @ A
% 0.25/0.49 @ ( finite_fold @ A @ ( option @ A )
% 0.25/0.49 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( inf_inf @ A @ X3 ) @ Y3 ) )
% 0.25/0.49 @ ( none @ A )
% 0.25/0.49 @ A9 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Inf_fin.eq_fold'
% 0.25/0.49 thf(fact_3115_image__split__eq__Sigma,axiom,
% 0.25/0.49 ! [C: $tType,B: $tType,A: $tType,F: C > A,G: C > B,A4: set @ C] :
% 0.25/0.49 ( ( image2 @ C @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.25/0.49 @ A4 )
% 0.25/0.49 = ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A4 )
% 0.25/0.49 @ ^ [X3: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % image_split_eq_Sigma
% 0.25/0.49 thf(fact_3116_atLeast__empty__triv,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % atLeast_empty_triv
% 0.25/0.49 thf(fact_3117_inf__None__2,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( inf @ A )
% 0.25/0.49 => ! [X: option @ A] :
% 0.25/0.49 ( ( inf_inf @ ( option @ A ) @ X @ ( none @ A ) )
% 0.25/0.49 = ( none @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inf_None_2
% 0.25/0.49 thf(fact_3118_inf__None__1,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( inf @ A )
% 0.25/0.49 => ! [Y: option @ A] :
% 0.25/0.49 ( ( inf_inf @ ( option @ A ) @ ( none @ A ) @ Y )
% 0.25/0.49 = ( none @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inf_None_1
% 0.25/0.49 thf(fact_3119_vimage__id,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( vimage @ A @ A @ ( id @ A ) )
% 0.25/0.49 = ( id @ ( set @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_id
% 0.25/0.49 thf(fact_3120_set_Ocomp,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
% 0.25/0.49 ( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
% 0.25/0.49 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set.comp
% 0.25/0.49 thf(fact_3121_set_Ocompositionality,axiom,
% 0.25/0.49 ! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
% 0.25/0.49 ( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
% 0.25/0.49 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% 0.25/0.49
% 0.25/0.49 % set.compositionality
% 0.25/0.49 thf(fact_3122_vimage__comp,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X: set @ C] :
% 0.25/0.49 ( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X ) )
% 0.25/0.49 = ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_comp
% 0.25/0.49 thf(fact_3123_inf__unit__def,axiom,
% 0.25/0.49 ( ( inf_inf @ product_unit )
% 0.25/0.49 = ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% 0.25/0.49
% 0.25/0.49 % inf_unit_def
% 0.25/0.49 thf(fact_3124_inf__Int__eq2,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( inf_inf @ ( A > B > $o )
% 0.25/0.49 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
% 0.25/0.49 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
% 0.25/0.49 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inf_Int_eq2
% 0.25/0.49 thf(fact_3125_surj__image__vimage__eq,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
% 0.25/0.49 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 = ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
% 0.25/0.49 = A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % surj_image_vimage_eq
% 0.25/0.49 thf(fact_3126_set_Oidentity,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( vimage @ A @ A
% 0.25/0.49 @ ^ [X3: A] : X3 )
% 0.25/0.49 = ( id @ ( set @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set.identity
% 0.25/0.49 thf(fact_3127_vimage__Times,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A4: set @ B,B6: set @ C] :
% 0.25/0.49 ( ( vimage @ A @ ( product_prod @ B @ C ) @ F
% 0.25/0.49 @ ( product_Sigma @ B @ C @ A4
% 0.25/0.49 @ ^ [Uu: B] : B6 ) )
% 0.25/0.49 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A4 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_Times
% 0.25/0.49 thf(fact_3128_Pair__vimage__Sigma,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,X: B,A4: set @ B,F: B > ( set @ A )] :
% 0.25/0.49 ( ( ( member @ B @ X @ A4 )
% 0.25/0.49 => ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F ) )
% 0.25/0.49 = ( F @ X ) ) )
% 0.25/0.49 & ( ~ ( member @ B @ X @ A4 )
% 0.25/0.49 => ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Pair_vimage_Sigma
% 0.25/0.49 thf(fact_3129_surj__vimage__empty,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
% 0.25/0.49 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 = ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ( vimage @ B @ A @ F @ A4 )
% 0.25/0.49 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.49 = ( A4
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % surj_vimage_empty
% 0.25/0.49 thf(fact_3130_vimage__subsetD,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > A,B6: set @ A,A4: set @ B] :
% 0.25/0.49 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 = ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ B6 ) @ A4 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_subsetD
% 0.25/0.49 thf(fact_3131_vimage__fst,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ A] :
% 0.25/0.49 ( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
% 0.25/0.49 = ( product_Sigma @ A @ B @ A4
% 0.25/0.49 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_fst
% 0.25/0.49 thf(fact_3132_vimage__snd,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ B] :
% 0.25/0.49 ( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 )
% 0.25/0.49 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_snd
% 0.25/0.49 thf(fact_3133_Inf__fin_Oinfinite,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( semilattice_inf @ A )
% 0.25/0.49 => ! [A4: set @ A] :
% 0.25/0.49 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.49 => ( ( lattic7752659483105999362nf_fin @ A @ A4 )
% 0.25/0.49 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Inf_fin.infinite
% 0.25/0.49 thf(fact_3134_rtrancl__last__visit__node,axiom,
% 0.25/0.49 ! [A: $tType,S: A,S3: A,R5: set @ ( product_prod @ A @ A ),Sh: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ S3 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( ( ( S != Sh )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ S3 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
% 0.25/0.49 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ Sh ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S3 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_last_visit_node
% 0.25/0.49 thf(fact_3135_floor__rat__def,axiom,
% 0.25/0.49 ( ( archim6421214686448440834_floor @ rat )
% 0.25/0.49 = ( ^ [X3: rat] :
% 0.25/0.49 ( the @ int
% 0.25/0.49 @ ^ [Z5: int] :
% 0.25/0.49 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z5 ) @ X3 )
% 0.25/0.49 & ( ord_less @ rat @ X3 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % floor_rat_def
% 0.25/0.49 thf(fact_3136_rat__floor__lemma,axiom,
% 0.25/0.49 ! [A3: int,B2: int] :
% 0.25/0.49 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
% 0.25/0.49 & ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rat_floor_lemma
% 0.25/0.49 thf(fact_3137_num__of__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.49 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 0.25/0.49 @ code_num_of_integer ) ).
% 0.25/0.49
% 0.25/0.49 % num_of_integer.transfer
% 0.25/0.49 thf(fact_3138_prod__decode__triangle__add,axiom,
% 0.25/0.49 ! [K: nat,M2: nat] :
% 0.25/0.49 ( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) )
% 0.25/0.49 = ( nat_prod_decode_aux @ K @ M2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % prod_decode_triangle_add
% 0.25/0.49 thf(fact_3139_wf__insert,axiom,
% 0.25/0.49 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
% 0.25/0.49 = ( ( wf @ A @ R )
% 0.25/0.49 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % wf_insert
% 0.25/0.49 thf(fact_3140_add__rat,axiom,
% 0.25/0.49 ! [B2: int,D3: int,A3: int,C2: int] :
% 0.25/0.49 ( ( B2
% 0.25/0.49 != ( zero_zero @ int ) )
% 0.25/0.49 => ( ( D3
% 0.25/0.49 != ( zero_zero @ int ) )
% 0.25/0.49 => ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D3 ) )
% 0.25/0.49 = ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % add_rat
% 0.25/0.49 thf(fact_3141_r__le__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,S2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % r_le_rtrancl
% 0.25/0.49 thf(fact_3142_rtrancl__mono__mp,axiom,
% 0.25/0.49 ! [A: $tType,U4: set @ ( product_prod @ A @ A ),V6: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U4 @ V6 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ U4 ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ V6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_mono_mp
% 0.25/0.49 thf(fact_3143_rtrancl__mono__rightI,axiom,
% 0.25/0.49 ! [A: $tType,S2: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ S7 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S7 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_mono_rightI
% 0.25/0.49 thf(fact_3144_converse__rtranclE_H,axiom,
% 0.25/0.49 ! [A: $tType,U: A,V: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( ( U != V )
% 0.25/0.49 => ~ ! [Vh: A] :
% 0.25/0.49 ( ( U != Vh )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R5 )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_rtranclE'
% 0.25/0.49 thf(fact_3145_rtrancl_Ocases,axiom,
% 0.25/0.49 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( A22 != A1 )
% 0.25/0.49 => ~ ! [B3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl.cases
% 0.25/0.49 thf(fact_3146_rtrancl_Osimps,axiom,
% 0.25/0.49 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 = ( ? [A7: A] :
% 0.25/0.49 ( ( A1 = A7 )
% 0.25/0.49 & ( A22 = A7 ) )
% 0.25/0.49 | ? [A7: A,B5: A,C3: A] :
% 0.25/0.49 ( ( A1 = A7 )
% 0.25/0.49 & ( A22 = C3 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl.simps
% 0.25/0.49 thf(fact_3147_rtrancl_Ortrancl__refl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl.rtrancl_refl
% 0.25/0.49 thf(fact_3148_rtrancl_Ortrancl__into__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl.rtrancl_into_rtrancl
% 0.25/0.49 thf(fact_3149_rtranclE,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( A3 != B2 )
% 0.25/0.49 => ~ ! [Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtranclE
% 0.25/0.49 thf(fact_3150_rtrancl__trans,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_trans
% 0.25/0.49 thf(fact_3151_rtrancl__induct,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( P @ A3 )
% 0.25/0.49 => ( ! [Y4: A,Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.49 => ( ( P @ Y4 )
% 0.25/0.49 => ( P @ Z3 ) ) ) )
% 0.25/0.49 => ( P @ B2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_induct
% 0.25/0.49 thf(fact_3152_converse__rtranclE,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( X != Z2 )
% 0.25/0.49 => ~ ! [Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_rtranclE
% 0.25/0.49 thf(fact_3153_converse__rtrancl__induct,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( P @ B2 )
% 0.25/0.49 => ( ! [Y4: A,Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( P @ Z3 )
% 0.25/0.49 => ( P @ Y4 ) ) ) )
% 0.25/0.49 => ( P @ A3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_rtrancl_induct
% 0.25/0.49 thf(fact_3154_converse__rtrancl__into__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_rtrancl_into_rtrancl
% 0.25/0.49 thf(fact_3155_in__rtrancl__insert,axiom,
% 0.25/0.49 ! [A: $tType,X: product_prod @ A @ A,R5: set @ ( product_prod @ A @ A ),R: product_prod @ A @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ R @ R5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % in_rtrancl_insert
% 0.25/0.49 thf(fact_3156_integer_Orep__transfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.49 @ ^ [X3: int] : X3
% 0.25/0.49 @ code_int_of_integer ) ).
% 0.25/0.49
% 0.25/0.49 % integer.rep_transfer
% 0.25/0.49 thf(fact_3157_integer_Oid__abs__transfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ int @ int @ code_integer
% 0.25/0.49 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.49 @ code_pcr_integer
% 0.25/0.49 @ ^ [X3: int] : X3
% 0.25/0.49 @ code_integer_of_int ) ).
% 0.25/0.49
% 0.25/0.49 % integer.id_abs_transfer
% 0.25/0.49 thf(fact_3158_rtrancl__Un__separator__converseE,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
% 0.25/0.49 => ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ Q )
% 0.25/0.49 => ( Y4 = X2 ) ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_Un_separator_converseE
% 0.25/0.49 thf(fact_3159_rtrancl__Un__separatorE,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
% 0.25/0.49 => ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ P ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ Q )
% 0.25/0.49 => ( X2 = Y4 ) ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_Un_separatorE
% 0.25/0.49 thf(fact_3160_zero__integer_Otransfer,axiom,
% 0.25/0.49 code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % zero_integer.transfer
% 0.25/0.49 thf(fact_3161_converse__rtrancl__induct2,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ( P @ Bx @ By )
% 0.25/0.49 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
% 0.25/0.49 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ( P @ Aa3 @ Ba )
% 0.25/0.49 => ( P @ A5 @ B3 ) ) ) )
% 0.25/0.49 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_rtrancl_induct2
% 0.25/0.49 thf(fact_3162_converse__rtranclE2,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ( ( product_Pair @ A @ B @ Xa @ Xb )
% 0.25/0.49 != ( product_Pair @ A @ B @ Za2 @ Zb ) )
% 0.25/0.49 => ~ ! [A5: A,B3: B] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ R )
% 0.25/0.49 => ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_rtranclE2
% 0.25/0.49 thf(fact_3163_rtrancl__induct2,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ( P @ Ax @ Ay )
% 0.25/0.49 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
% 0.25/0.49 => ( ( P @ A5 @ B3 )
% 0.25/0.49 => ( P @ Aa3 @ Ba ) ) ) )
% 0.25/0.49 => ( P @ Bx @ By ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_induct2
% 0.25/0.49 thf(fact_3164_rtrancl__sub__insert__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R5 ) @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_sub_insert_rtrancl
% 0.25/0.49 thf(fact_3165_plus__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( plus_plus @ int ) @ ( plus_plus @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % plus_integer.transfer
% 0.25/0.49 thf(fact_3166_quotient__of__eq,axiom,
% 0.25/0.49 ! [A3: int,B2: int,P5: int,Q3: int] :
% 0.25/0.49 ( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
% 0.25/0.49 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.25/0.49 => ( ( fract @ P5 @ Q3 )
% 0.25/0.49 = ( fract @ A3 @ B2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % quotient_of_eq
% 0.25/0.49 thf(fact_3167_less__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.49 @ ( ord_less @ int )
% 0.25/0.49 @ ( ord_less @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % less_integer.transfer
% 0.25/0.49 thf(fact_3168_one__integer_Otransfer,axiom,
% 0.25/0.49 code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % one_integer.transfer
% 0.25/0.49 thf(fact_3169_less__eq__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.49 @ ( ord_less_eq @ int )
% 0.25/0.49 @ ( ord_less_eq @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % less_eq_integer.transfer
% 0.25/0.49 thf(fact_3170_uminus__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % uminus_integer.transfer
% 0.25/0.49 thf(fact_3171_times__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( times_times @ int ) @ ( times_times @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % times_integer.transfer
% 0.25/0.49 thf(fact_3172_minus__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( minus_minus @ int ) @ ( minus_minus @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % minus_integer.transfer
% 0.25/0.49 thf(fact_3173_abs__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( abs_abs @ int ) @ ( abs_abs @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % abs_integer.transfer
% 0.25/0.49 thf(fact_3174_nat__of__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ nat2
% 0.25/0.49 @ code_nat_of_integer ) ).
% 0.25/0.49
% 0.25/0.49 % nat_of_integer.transfer
% 0.25/0.49 thf(fact_3175_normalize__eq,axiom,
% 0.25/0.49 ! [A3: int,B2: int,P5: int,Q3: int] :
% 0.25/0.49 ( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
% 0.25/0.49 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.25/0.49 => ( ( fract @ P5 @ Q3 )
% 0.25/0.49 = ( fract @ A3 @ B2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % normalize_eq
% 0.25/0.49 thf(fact_3176_integer__of__nat_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ code_pcr_integer
% 0.25/0.49 @ ( semiring_1_of_nat @ int )
% 0.25/0.49 @ code_integer_of_nat ) ).
% 0.25/0.49
% 0.25/0.49 % integer_of_nat.transfer
% 0.25/0.49 thf(fact_3177_divide__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( divide_divide @ int ) @ ( divide_divide @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % divide_integer.transfer
% 0.25/0.49 thf(fact_3178_modulo__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( modulo_modulo @ int ) @ ( modulo_modulo @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % modulo_integer.transfer
% 0.25/0.49 thf(fact_3179_sgn__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( sgn_sgn @ int ) @ ( sgn_sgn @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % sgn_integer.transfer
% 0.25/0.49 thf(fact_3180_dup_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
% 0.25/0.49 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
% 0.25/0.49 @ code_dup ) ).
% 0.25/0.49
% 0.25/0.49 % dup.transfer
% 0.25/0.49 thf(fact_3181_take__bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.25/0.49 @ ( bit_se2584673776208193580ke_bit @ int )
% 0.25/0.49 @ ( bit_se2584673776208193580ke_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % take_bit_integer.transfer
% 0.25/0.49 thf(fact_3182_not__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( bit_ri4277139882892585799ns_not @ int ) @ ( bit_ri4277139882892585799ns_not @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % not_integer.transfer
% 0.25/0.49 thf(fact_3183_and__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344872417868541ns_and @ int ) @ ( bit_se5824344872417868541ns_and @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % and_integer.transfer
% 0.25/0.49 thf(fact_3184_bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bit_se5641148757651400278ts_bit @ int )
% 0.25/0.49 @ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % bit_integer.transfer
% 0.25/0.49 thf(fact_3185_or__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se1065995026697491101ons_or @ int ) @ ( bit_se1065995026697491101ons_or @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % or_integer.transfer
% 0.25/0.49 thf(fact_3186_xor__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344971392196577ns_xor @ int ) @ ( bit_se5824344971392196577ns_xor @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % xor_integer.transfer
% 0.25/0.49 thf(fact_3187_drop__bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.25/0.49 @ ( bit_se4197421643247451524op_bit @ int )
% 0.25/0.49 @ ( bit_se4197421643247451524op_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % drop_bit_integer.transfer
% 0.25/0.49 thf(fact_3188_push__bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.25/0.49 @ ( bit_se4730199178511100633sh_bit @ int )
% 0.25/0.49 @ ( bit_se4730199178511100633sh_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % push_bit_integer.transfer
% 0.25/0.49 thf(fact_3189_mask__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ code_pcr_integer
% 0.25/0.49 @ ( bit_se2239418461657761734s_mask @ int )
% 0.25/0.49 @ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % mask_integer.transfer
% 0.25/0.49 thf(fact_3190_unset__bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.25/0.49 @ ( bit_se2638667681897837118et_bit @ int )
% 0.25/0.49 @ ( bit_se2638667681897837118et_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % unset_bit_integer.transfer
% 0.25/0.49 thf(fact_3191_set__bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.25/0.49 @ ( bit_se5668285175392031749et_bit @ int )
% 0.25/0.49 @ ( bit_se5668285175392031749et_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_bit_integer.transfer
% 0.25/0.49 thf(fact_3192_division__segment__integer_Otransfer,axiom,
% 0.25/0.49 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% 0.25/0.49
% 0.25/0.49 % division_segment_integer.transfer
% 0.25/0.49 thf(fact_3193_flip__bit__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 0.25/0.49 @ ( bit_se8732182000553998342ip_bit @ int )
% 0.25/0.49 @ ( bit_se8732182000553998342ip_bit @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % flip_bit_integer.transfer
% 0.25/0.49 thf(fact_3194_euclidean__size__integer_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 0.25/0.49 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.49 @ ( euclid6346220572633701492n_size @ int )
% 0.25/0.49 @ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% 0.25/0.49
% 0.25/0.49 % euclidean_size_integer.transfer
% 0.25/0.49 thf(fact_3195_trancl__subset__Sigma__aux,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.49 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.49 @ ^ [Uu: A] : A4 ) )
% 0.25/0.49 => ( ( A3 = B2 )
% 0.25/0.49 | ( member @ A @ A3 @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_subset_Sigma_aux
% 0.25/0.49 thf(fact_3196_Restr__rtrancl__mono,axiom,
% 0.25/0.49 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 0.25/0.49 @ ( product_Sigma @ A @ A @ U4
% 0.25/0.49 @ ^ [Uu: A] : U4 ) ) ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Restr_rtrancl_mono
% 0.25/0.49 thf(fact_3197_quotient__of__Fract,axiom,
% 0.25/0.49 ! [A3: int,B2: int] :
% 0.25/0.49 ( ( quotient_of @ ( fract @ A3 @ B2 ) )
% 0.25/0.49 = ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % quotient_of_Fract
% 0.25/0.49 thf(fact_3198_sub_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
% 0.25/0.49 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ num @ num @ int @ code_integer
% 0.25/0.49 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 0.25/0.49 @ code_pcr_integer )
% 0.25/0.49 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) )
% 0.25/0.49 @ code_sub ) ).
% 0.25/0.49
% 0.25/0.49 % sub.transfer
% 0.25/0.49 thf(fact_3199_rtrancl__mapI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A3: A,B2: A,E4: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ E4 ) )
% 0.25/0.49 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F ) @ E4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_mapI
% 0.25/0.49 thf(fact_3200_Fract_Oabs__eq,axiom,
% 0.25/0.49 ( fract
% 0.25/0.49 = ( ^ [Xa4: int,X3: int] :
% 0.25/0.49 ( abs_Rat
% 0.25/0.49 @ ( if @ ( product_prod @ int @ int )
% 0.25/0.49 @ ( X3
% 0.25/0.49 = ( zero_zero @ int ) )
% 0.25/0.49 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.49 @ ( product_Pair @ int @ int @ Xa4 @ X3 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Fract.abs_eq
% 0.25/0.49 thf(fact_3201_rtrancl__last__touch,axiom,
% 0.25/0.49 ! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( ( member @ A @ Q3 @ S2 )
% 0.25/0.49 => ~ ! [Qt: A] :
% 0.25/0.49 ( ( member @ A @ Qt @ S2 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_last_touch
% 0.25/0.49 thf(fact_3202_rtrancl__last__visit_H,axiom,
% 0.25/0.49 ! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : S2 ) ) ) )
% 0.25/0.49 => ~ ! [Qt: A] :
% 0.25/0.49 ( ( member @ A @ Qt @ S2 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_last_visit'
% 0.25/0.49 thf(fact_3203_rtrancl__insert,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
% 0.25/0.49 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R )
% 0.25/0.49 @ ( collect @ ( product_prod @ A @ A )
% 0.25/0.49 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.49 @ ^ [X3: A,Y3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_insert
% 0.25/0.49 thf(fact_3204_Fract_Otransfer,axiom,
% 0.25/0.49 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
% 0.25/0.49 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.49 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
% 0.25/0.49 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.49 @ pcr_rat )
% 0.25/0.49 @ ^ [A7: int,B5: int] :
% 0.25/0.49 ( if @ ( product_prod @ int @ int )
% 0.25/0.49 @ ( B5
% 0.25/0.49 = ( zero_zero @ int ) )
% 0.25/0.49 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.49 @ ( product_Pair @ int @ int @ A7 @ B5 ) )
% 0.25/0.49 @ fract ) ).
% 0.25/0.49
% 0.25/0.49 % Fract.transfer
% 0.25/0.49 thf(fact_3205_Fract__add__one,axiom,
% 0.25/0.49 ! [N: int,M2: int] :
% 0.25/0.49 ( ( N
% 0.25/0.49 != ( zero_zero @ int ) )
% 0.25/0.49 => ( ( fract @ ( plus_plus @ int @ M2 @ N ) @ N )
% 0.25/0.49 = ( plus_plus @ rat @ ( fract @ M2 @ N ) @ ( one_one @ rat ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Fract_add_one
% 0.25/0.49 thf(fact_3206_Fract__def,axiom,
% 0.25/0.49 ( fract
% 0.25/0.49 = ( map_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat ) @ ( id @ int ) @ ( map_fun @ int @ int @ ( product_prod @ int @ int ) @ rat @ ( id @ int ) @ abs_Rat )
% 0.25/0.49 @ ^ [A7: int,B5: int] :
% 0.25/0.49 ( if @ ( product_prod @ int @ int )
% 0.25/0.49 @ ( B5
% 0.25/0.49 = ( zero_zero @ int ) )
% 0.25/0.49 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 0.25/0.49 @ ( product_Pair @ int @ int @ A7 @ B5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Fract_def
% 0.25/0.49 thf(fact_3207_old_Orec__unit__def,axiom,
% 0.25/0.49 ! [T: $tType] :
% 0.25/0.49 ( ( product_rec_unit @ T )
% 0.25/0.49 = ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % old.rec_unit_def
% 0.25/0.49 thf(fact_3208_old_Orec__prod__def,axiom,
% 0.25/0.49 ! [T: $tType,B: $tType,A: $tType] :
% 0.25/0.49 ( ( product_rec_prod @ A @ B @ T )
% 0.25/0.49 = ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % old.rec_prod_def
% 0.25/0.49 thf(fact_3209_list__decode_Opinduct,axiom,
% 0.25/0.49 ! [A0: nat,P: nat > $o] :
% 0.25/0.49 ( ( accp @ nat @ nat_list_decode_rel @ A0 )
% 0.25/0.49 => ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 0.25/0.49 => ( P @ ( zero_zero @ nat ) ) )
% 0.25/0.49 => ( ! [N3: nat] :
% 0.25/0.49 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) )
% 0.25/0.49 => ( ! [X7: nat,Y6: nat] :
% 0.25/0.49 ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
% 0.25/0.49 = ( nat_prod_decode @ N3 ) )
% 0.25/0.49 => ( P @ Y6 ) )
% 0.25/0.49 => ( P @ ( suc @ N3 ) ) ) )
% 0.25/0.49 => ( P @ A0 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % list_decode.pinduct
% 0.25/0.49 thf(fact_3210_rtrancl__restrictI,axiom,
% 0.25/0.49 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : R5 ) ) ) )
% 0.25/0.49 => ( ~ ( member @ A @ U @ R5 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_restrictI
% 0.25/0.49 thf(fact_3211_inj__on__vimage__singleton,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,A3: B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A4 )
% 0.25/0.49 @ ( insert2 @ A
% 0.25/0.49 @ ( the @ A
% 0.25/0.49 @ ^ [X3: A] :
% 0.25/0.49 ( ( member @ A @ X3 @ A4 )
% 0.25/0.49 & ( ( F @ X3 )
% 0.25/0.49 = A3 ) ) )
% 0.25/0.49 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_vimage_singleton
% 0.25/0.49 thf(fact_3212_inj__on__empty,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_empty
% 0.25/0.49 thf(fact_3213_inj__uminus,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( ordered_ab_group_add @ A )
% 0.25/0.49 => ! [A4: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_uminus
% 0.25/0.49 thf(fact_3214_rel__restrict__empty,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( rel_restrict @ A @ R5 @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 = R5 ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_empty
% 0.25/0.49 thf(fact_3215_The__split__eq,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.25/0.49 ( ( the @ ( product_prod @ A @ B )
% 0.25/0.49 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.49 @ ^ [X9: A,Y8: B] :
% 0.25/0.49 ( ( X = X9 )
% 0.25/0.49 & ( Y = Y8 ) ) ) )
% 0.25/0.49 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 0.25/0.49
% 0.25/0.49 % The_split_eq
% 0.25/0.49 thf(fact_3216_inj__apfst,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 0.25/0.49 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.49 = ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_apfst
% 0.25/0.49 thf(fact_3217_inj__apsnd,axiom,
% 0.25/0.49 ! [A: $tType,C: $tType,B: $tType,F: B > C] :
% 0.25/0.49 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.49 = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_apsnd
% 0.25/0.49 thf(fact_3218_inj__on__apfst,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > C,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F )
% 0.25/0.49 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.49 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 0.25/0.49 = ( inj_on @ A @ C @ F @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_apfst
% 0.25/0.49 thf(fact_3219_inj__on__apsnd,axiom,
% 0.25/0.49 ! [A: $tType,C: $tType,B: $tType,F: B > C,A4: set @ B] :
% 0.25/0.49 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F )
% 0.25/0.49 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : A4 ) )
% 0.25/0.49 = ( inj_on @ B @ C @ F @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_apsnd
% 0.25/0.49 thf(fact_3220_inj__on__insert,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A3: A,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( insert2 @ A @ A3 @ A4 ) )
% 0.25/0.49 = ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 & ~ ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_insert
% 0.25/0.49 thf(fact_3221_inj__on__Int,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 | ( inj_on @ A @ B @ F @ B6 ) )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_Int
% 0.25/0.49 thf(fact_3222_fun_Oinj__map,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,D: $tType,F: A > B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % fun.inj_map
% 0.25/0.49 thf(fact_3223_inj__on__imageI2,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F3: C > B,F: A > C,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F ) @ A4 )
% 0.25/0.49 => ( inj_on @ A @ C @ F @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_imageI2
% 0.25/0.49 thf(fact_3224_inj__on__of__nat,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( semiring_char_0 @ A )
% 0.25/0.49 => ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_of_nat
% 0.25/0.49 thf(fact_3225_inj__on__add,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( cancel_semigroup_add @ A )
% 0.25/0.49 => ! [A3: A,A4: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_add
% 0.25/0.49 thf(fact_3226_inj__on__image__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ A,G: A > B,F: A > A] :
% 0.25/0.49 ( ! [X2: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ A4 )
% 0.25/0.49 => ! [Xa3: A] :
% 0.25/0.49 ( ( member @ A @ Xa3 @ A4 )
% 0.25/0.49 => ( ( ( G @ ( F @ X2 ) )
% 0.25/0.49 = ( G @ ( F @ Xa3 ) ) )
% 0.25/0.49 = ( ( G @ X2 )
% 0.25/0.49 = ( G @ Xa3 ) ) ) ) )
% 0.25/0.49 => ( ( inj_on @ A @ A @ F @ A4 )
% 0.25/0.49 => ( ( inj_on @ A @ B @ G @ ( image2 @ A @ A @ F @ A4 ) )
% 0.25/0.49 = ( inj_on @ A @ B @ G @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_image_iff
% 0.25/0.49 thf(fact_3227_inj__on__id2,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A] :
% 0.25/0.49 ( inj_on @ A @ A
% 0.25/0.49 @ ^ [X3: A] : X3
% 0.25/0.49 @ A4 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_id2
% 0.25/0.49 thf(fact_3228_inj__on__inverseI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ A,G: B > A,F: A > B] :
% 0.25/0.49 ( ! [X2: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ A4 )
% 0.25/0.49 => ( ( G @ ( F @ X2 ) )
% 0.25/0.49 = X2 ) )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_inverseI
% 0.25/0.49 thf(fact_3229_inj__on__contraD,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( X != Y )
% 0.25/0.49 => ( ( member @ A @ X @ A4 )
% 0.25/0.49 => ( ( member @ A @ Y @ A4 )
% 0.25/0.49 => ( ( F @ X )
% 0.25/0.49 != ( F @ Y ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_contraD
% 0.25/0.49 thf(fact_3230_inj__on__eq__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( member @ A @ X @ A4 )
% 0.25/0.49 => ( ( member @ A @ Y @ A4 )
% 0.25/0.49 => ( ( ( F @ X )
% 0.25/0.49 = ( F @ Y ) )
% 0.25/0.49 = ( X = Y ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_eq_iff
% 0.25/0.49 thf(fact_3231_inj__on__cong,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ A,F: A > B,G: A > B] :
% 0.25/0.49 ( ! [A5: A] :
% 0.25/0.49 ( ( member @ A @ A5 @ A4 )
% 0.25/0.49 => ( ( F @ A5 )
% 0.25/0.49 = ( G @ A5 ) ) )
% 0.25/0.49 => ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 = ( inj_on @ A @ B @ G @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_cong
% 0.25/0.49 thf(fact_3232_inj__on__def,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( inj_on @ A @ B )
% 0.25/0.49 = ( ^ [F2: A > B,A9: set @ A] :
% 0.25/0.49 ! [X3: A] :
% 0.25/0.49 ( ( member @ A @ X3 @ A9 )
% 0.25/0.49 => ! [Y3: A] :
% 0.25/0.49 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.49 => ( ( ( F2 @ X3 )
% 0.25/0.49 = ( F2 @ Y3 ) )
% 0.25/0.49 => ( X3 = Y3 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_def
% 0.25/0.49 thf(fact_3233_inj__onI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 0.25/0.49 ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ A4 )
% 0.25/0.49 => ( ( member @ A @ Y4 @ A4 )
% 0.25/0.49 => ( ( ( F @ X2 )
% 0.25/0.49 = ( F @ Y4 ) )
% 0.25/0.49 => ( X2 = Y4 ) ) ) )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_onI
% 0.25/0.49 thf(fact_3234_inj__onD,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( ( F @ X )
% 0.25/0.49 = ( F @ Y ) )
% 0.25/0.49 => ( ( member @ A @ X @ A4 )
% 0.25/0.49 => ( ( member @ A @ Y @ A4 )
% 0.25/0.49 => ( X = Y ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_onD
% 0.25/0.49 thf(fact_3235_inj__on__diff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_diff
% 0.25/0.49 thf(fact_3236_inj__on__id,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_id
% 0.25/0.49 thf(fact_3237_inj__Some,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_Some
% 0.25/0.49 thf(fact_3238_inj__on__subset,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_subset
% 0.25/0.49 thf(fact_3239_subset__inj__on,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ B6 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % subset_inj_on
% 0.25/0.49 thf(fact_3240_inj__fun,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( inj_on @ A @ ( C > B )
% 0.25/0.49 @ ^ [X3: A,Y3: C] : ( F @ X3 )
% 0.25/0.49 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_fun
% 0.25/0.49 thf(fact_3241_inj__def,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 = ( ! [X3: A,Y3: A] :
% 0.25/0.49 ( ( ( F @ X3 )
% 0.25/0.49 = ( F @ Y3 ) )
% 0.25/0.49 => ( X3 = Y3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_def
% 0.25/0.49 thf(fact_3242_inj__eq,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ( F @ X )
% 0.25/0.49 = ( F @ Y ) )
% 0.25/0.49 = ( X = Y ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_eq
% 0.25/0.49 thf(fact_3243_injI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.49 ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( ( F @ X2 )
% 0.25/0.49 = ( F @ Y4 ) )
% 0.25/0.49 => ( X2 = Y4 ) )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % injI
% 0.25/0.49 thf(fact_3244_injD,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ( F @ X )
% 0.25/0.49 = ( F @ Y ) )
% 0.25/0.49 => ( X = Y ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % injD
% 0.25/0.49 thf(fact_3245_inj__swap,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A4 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_swap
% 0.25/0.49 thf(fact_3246_inj__on__convol__ident,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,X10: set @ A] :
% 0.25/0.49 ( inj_on @ A @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( F @ X3 ) )
% 0.25/0.49 @ X10 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_convol_ident
% 0.25/0.49 thf(fact_3247_inj__Pair_I1_J,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
% 0.25/0.49 ( inj_on @ A @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( C2 @ X3 ) )
% 0.25/0.49 @ S2 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_Pair(1)
% 0.25/0.49 thf(fact_3248_inj__Pair_I2_J,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
% 0.25/0.49 ( inj_on @ A @ ( product_prod @ B @ A )
% 0.25/0.49 @ ^ [X3: A] : ( product_Pair @ B @ A @ ( C2 @ X3 ) @ X3 )
% 0.25/0.49 @ S2 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_Pair(2)
% 0.25/0.49 thf(fact_3249_inj__on__add_H,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( cancel_semigroup_add @ A )
% 0.25/0.49 => ! [A3: A,A4: set @ A] :
% 0.25/0.49 ( inj_on @ A @ A
% 0.25/0.49 @ ^ [B5: A] : ( plus_plus @ A @ B5 @ A3 )
% 0.25/0.49 @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_add'
% 0.25/0.49 thf(fact_3250_rel__restrict__lift,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R5 ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_lift
% 0.25/0.49 thf(fact_3251_rel__restrictI,axiom,
% 0.25/0.49 ! [A: $tType,X: A,R5: set @ A,Y: A,E4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ~ ( member @ A @ X @ R5 )
% 0.25/0.49 => ( ~ ( member @ A @ Y @ R5 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrictI
% 0.25/0.49 thf(fact_3252_rel__restrict__notR_I1_J,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R5 ) )
% 0.25/0.49 => ~ ( member @ A @ X @ R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_notR(1)
% 0.25/0.49 thf(fact_3253_rel__restrict__notR_I2_J,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R5 ) )
% 0.25/0.49 => ~ ( member @ A @ Y @ R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_notR(2)
% 0.25/0.49 thf(fact_3254_rel__restrict__union,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( rel_restrict @ A @ R5 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( rel_restrict @ A @ ( rel_restrict @ A @ R5 @ A4 ) @ B6 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_union
% 0.25/0.49 thf(fact_3255_rel__restrict__sub,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 ) @ R5 ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_sub
% 0.25/0.49 thf(fact_3256_rel__restrict__mono,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A4 @ R5 ) @ ( rel_restrict @ A @ B6 @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_mono
% 0.25/0.49 thf(fact_3257_finite__rel__restrict,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 0.25/0.49 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
% 0.25/0.49 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R5 @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_rel_restrict
% 0.25/0.49 thf(fact_3258_option_Oinj__map,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.inj_map
% 0.25/0.49 thf(fact_3259_linorder__inj__onI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( order @ A )
% 0.25/0.49 => ! [A4: set @ A,F: A > B] :
% 0.25/0.49 ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( ord_less @ A @ X2 @ Y4 )
% 0.25/0.49 => ( ( member @ A @ X2 @ A4 )
% 0.25/0.49 => ( ( member @ A @ Y4 @ A4 )
% 0.25/0.49 => ( ( F @ X2 )
% 0.25/0.49 != ( F @ Y4 ) ) ) ) )
% 0.25/0.49 => ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ A4 )
% 0.25/0.49 => ( ( member @ A @ Y4 @ A4 )
% 0.25/0.49 => ( ( ord_less_eq @ A @ X2 @ Y4 )
% 0.25/0.49 | ( ord_less_eq @ A @ Y4 @ X2 ) ) ) )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % linorder_inj_onI
% 0.25/0.49 thf(fact_3260_linorder__injI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( linorder @ A )
% 0.25/0.49 => ! [F: A > B] :
% 0.25/0.49 ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( ord_less @ A @ X2 @ Y4 )
% 0.25/0.49 => ( ( F @ X2 )
% 0.25/0.49 != ( F @ Y4 ) ) )
% 0.25/0.49 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % linorder_injI
% 0.25/0.49 thf(fact_3261_inj__add__left,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( cancel_semigroup_add @ A )
% 0.25/0.49 => ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_add_left
% 0.25/0.49 thf(fact_3262_inj__on__image__eq__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ C5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
% 0.25/0.49 => ( ( ( image2 @ A @ B @ F @ A4 )
% 0.25/0.49 = ( image2 @ A @ B @ F @ B6 ) )
% 0.25/0.49 = ( A4 = B6 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_image_eq_iff
% 0.25/0.49 thf(fact_3263_inj__on__image__mem__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: A,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ B6 )
% 0.25/0.49 => ( ( member @ A @ A3 @ B6 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.49 => ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 = ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_image_mem_iff
% 0.25/0.49 thf(fact_3264_inj__img__insertE,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: B,B6: set @ B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ~ ( member @ B @ X @ B6 )
% 0.25/0.49 => ( ( ( insert2 @ B @ X @ B6 )
% 0.25/0.49 = ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 => ~ ! [X5: A,A11: set @ A] :
% 0.25/0.49 ( ~ ( member @ A @ X5 @ A11 )
% 0.25/0.49 => ( ( A4
% 0.25/0.49 = ( insert2 @ A @ X5 @ A11 ) )
% 0.25/0.49 => ( ( X
% 0.25/0.49 = ( F @ X5 ) )
% 0.25/0.49 => ( B6
% 0.25/0.49 != ( image2 @ A @ B @ F @ A11 ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_img_insertE
% 0.25/0.49 thf(fact_3265_range__ex1__eq,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,B2: B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( member @ B @ B2 @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 0.25/0.49 = ( ? [X3: A] :
% 0.25/0.49 ( ( B2
% 0.25/0.49 = ( F @ X3 ) )
% 0.25/0.49 & ! [Y3: A] :
% 0.25/0.49 ( ( B2
% 0.25/0.49 = ( F @ Y3 ) )
% 0.25/0.49 => ( Y3 = X3 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % range_ex1_eq
% 0.25/0.49 thf(fact_3266_inj__image__eq__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ( image2 @ A @ B @ F @ A4 )
% 0.25/0.49 = ( image2 @ A @ B @ F @ B6 ) )
% 0.25/0.49 = ( A4 = B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_image_eq_iff
% 0.25/0.49 thf(fact_3267_inj__image__mem__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A3: A,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 = ( member @ A @ A3 @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_image_mem_iff
% 0.25/0.49 thf(fact_3268_comp__inj__on,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: B > C] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( inj_on @ B @ C @ G @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % comp_inj_on
% 0.25/0.49 thf(fact_3269_inj__on__imageI,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A4 )
% 0.25/0.49 => ( inj_on @ C @ B @ G @ ( image2 @ A @ C @ F @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_imageI
% 0.25/0.49 thf(fact_3270_comp__inj__on__iff,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,F3: B > C] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( inj_on @ B @ C @ F3 @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % comp_inj_on_iff
% 0.25/0.49 thf(fact_3271_inj__on__Un__image__eq__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 => ( ( ( image2 @ A @ B @ F @ A4 )
% 0.25/0.49 = ( image2 @ A @ B @ F @ B6 ) )
% 0.25/0.49 = ( A4 = B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_Un_image_eq_iff
% 0.25/0.49 thf(fact_3272_inj__compose,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( inj_on @ C @ A @ G @ ( top_top @ ( set @ C ) ) )
% 0.25/0.49 => ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_compose
% 0.25/0.49 thf(fact_3273_inj__on__strict__subset,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ B6 )
% 0.25/0.49 => ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.49 => ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_strict_subset
% 0.25/0.49 thf(fact_3274_inj__on__fun__updI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,Y: B,X: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ~ ( member @ B @ Y @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 => ( inj_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_fun_updI
% 0.25/0.49 thf(fact_3275_inj__on__fst__map__to__set,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_fst_map_to_set
% 0.25/0.49 thf(fact_3276_inj__diff__right,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( ab_group_add @ A )
% 0.25/0.49 => ! [A3: A] :
% 0.25/0.49 ( inj_on @ A @ A
% 0.25/0.49 @ ^ [B5: A] : ( minus_minus @ A @ B5 @ A3 )
% 0.25/0.49 @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_diff_right
% 0.25/0.49 thf(fact_3277_inj__singleton,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A] :
% 0.25/0.49 ( inj_on @ A @ ( set @ A )
% 0.25/0.49 @ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 @ A4 ) ).
% 0.25/0.49
% 0.25/0.49 % inj_singleton
% 0.25/0.49 thf(fact_3278_finite__Collect,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,S2: set @ A,F: B > A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ S2 )
% 0.25/0.49 => ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( finite_finite2 @ B
% 0.25/0.49 @ ( collect @ B
% 0.25/0.49 @ ^ [A7: B] : ( member @ A @ ( F @ A7 ) @ S2 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_Collect
% 0.25/0.49 thf(fact_3279_swap__inj__on,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
% 0.25/0.49 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 0.25/0.49 @ ^ [I3: A,J2: B] : ( product_Pair @ B @ A @ J2 @ I3 ) )
% 0.25/0.49 @ A4 ) ).
% 0.25/0.49
% 0.25/0.49 % swap_inj_on
% 0.25/0.49 thf(fact_3280_rel__restrict__mono2,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ A,S2: set @ A,A4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ R5 @ S2 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A4 @ S2 ) @ ( rel_restrict @ A @ A4 @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_mono2
% 0.25/0.49 thf(fact_3281_inj__image__subset__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) )
% 0.25/0.49 = ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_image_subset_iff
% 0.25/0.49 thf(fact_3282_inj__on__image__Int,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ C5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
% 0.25/0.49 => ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_image_Int
% 0.25/0.49 thf(fact_3283_image__Int,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % image_Int
% 0.25/0.49 thf(fact_3284_inj__on__image__set__diff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ C5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
% 0.25/0.49 => ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_image_set_diff
% 0.25/0.49 thf(fact_3285_sum_Oreindex,axiom,
% 0.25/0.49 ! [C: $tType,A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_add @ A )
% 0.25/0.49 => ! [H3: B > C,A4: set @ B,G: C > A] :
% 0.25/0.49 ( ( inj_on @ B @ C @ H3 @ A4 )
% 0.25/0.49 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 0.25/0.49 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum.reindex
% 0.25/0.49 thf(fact_3286_image__set__diff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % image_set_diff
% 0.25/0.49 thf(fact_3287_prod_Oreindex,axiom,
% 0.25/0.49 ! [C: $tType,A: $tType,B: $tType] :
% 0.25/0.49 ( ( comm_monoid_mult @ A )
% 0.25/0.49 => ! [H3: B > C,A4: set @ B,G: C > A] :
% 0.25/0.49 ( ( inj_on @ B @ C @ H3 @ A4 )
% 0.25/0.49 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 0.25/0.49 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % prod.reindex
% 0.25/0.49 thf(fact_3288_inj__vimage__image__eq,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 = A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_vimage_image_eq
% 0.25/0.49 thf(fact_3289_rel__restrict__def,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( rel_restrict @ A )
% 0.25/0.49 = ( ^ [R6: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 0.25/0.49 ( collect @ ( product_prod @ A @ A )
% 0.25/0.49 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.49 @ ^ [V2: A,W2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ R6 )
% 0.25/0.49 & ~ ( member @ A @ V2 @ A9 )
% 0.25/0.49 & ~ ( member @ A @ W2 @ A9 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_def
% 0.25/0.49 thf(fact_3290_finite__imp__inj__to__nat__seg_H,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.49 => ~ ! [F4: A > nat] :
% 0.25/0.49 ( ? [N3: nat] :
% 0.25/0.49 ( ( image2 @ A @ nat @ F4 @ A4 )
% 0.25/0.49 = ( collect @ nat
% 0.25/0.49 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
% 0.25/0.49 => ~ ( inj_on @ A @ nat @ F4 @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_imp_inj_to_nat_seg'
% 0.25/0.49 thf(fact_3291_set__to__map__inverse,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
% 0.25/0.49 => ( ( map_to_set @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
% 0.25/0.49 = S2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_to_map_inverse
% 0.25/0.49 thf(fact_3292_rel__restrict__Int__empty,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( ( inf_inf @ ( set @ A ) @ A4 @ ( field2 @ A @ R5 ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ( rel_restrict @ A @ R5 @ A4 )
% 0.25/0.49 = R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_Int_empty
% 0.25/0.49 thf(fact_3293_Field__rel__restrict,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R5 ) @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % Field_rel_restrict
% 0.25/0.49 thf(fact_3294_vimage__subsetI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,B6: set @ B,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % vimage_subsetI
% 0.25/0.49 thf(fact_3295_inj__image__Compl__subset,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_image_Compl_subset
% 0.25/0.49 thf(fact_3296_The__case__prod,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.25/0.49 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
% 0.25/0.49 = ( the @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % The_case_prod
% 0.25/0.49 thf(fact_3297_inj__on__disjoint__Un,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,G: A > B,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( inj_on @ A @ B @ G @ B6 )
% 0.25/0.49 => ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ G @ B6 ) )
% 0.25/0.49 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.49 => ( inj_on @ A @ B
% 0.25/0.49 @ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.25/0.49 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_disjoint_Un
% 0.25/0.49 thf(fact_3298_set__to__map__simp,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),K: A,V: B] :
% 0.25/0.49 ( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
% 0.25/0.49 => ( ( ( set_to_map @ A @ B @ S2 @ K )
% 0.25/0.49 = ( some @ B @ V ) )
% 0.25/0.49 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ S2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_to_map_simp
% 0.25/0.49 thf(fact_3299_rel__restrict__compl,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 0.25/0.49 ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 ) @ ( rel_restrict @ A @ R5 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) )
% 0.25/0.49 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_compl
% 0.25/0.49 thf(fact_3300_homo__rel__restrict__mono,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),B6: set @ A,A4: set @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ B6
% 0.25/0.49 @ ^ [Uu: A] : B6 ) )
% 0.25/0.49 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 )
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A4 )
% 0.25/0.49 @ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % homo_rel_restrict_mono
% 0.25/0.49 thf(fact_3301_rel__restrict__alt__def,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( rel_restrict @ A )
% 0.25/0.49 = ( ^ [R6: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 0.25/0.49 ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R6
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A9 )
% 0.25/0.49 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ A9 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_alt_def
% 0.25/0.49 thf(fact_3302_inj__on__Un,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.49 = ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 & ( inj_on @ A @ B @ F @ B6 )
% 0.25/0.49 & ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) )
% 0.25/0.49 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_Un
% 0.25/0.49 thf(fact_3303_inj__vimage__singleton,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A3: B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
% 0.25/0.49 @ ( insert2 @ A
% 0.25/0.49 @ ( the @ A
% 0.25/0.49 @ ^ [X3: A] :
% 0.25/0.49 ( ( F @ X3 )
% 0.25/0.49 = A3 ) )
% 0.25/0.49 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_vimage_singleton
% 0.25/0.49 thf(fact_3304_If__the__inv__into__f__f,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,I: A,C5: set @ A,G: A > B,X: A] :
% 0.25/0.49 ( ( member @ A @ I @ C5 )
% 0.25/0.49 => ( ( inj_on @ A @ B @ G @ C5 )
% 0.25/0.49 => ( ( comp @ B @ A @ A
% 0.25/0.49 @ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X )
% 0.25/0.49 @ G
% 0.25/0.49 @ I )
% 0.25/0.49 = ( id @ A @ I ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % If_the_inv_into_f_f
% 0.25/0.49 thf(fact_3305_ran__map__upd__Some,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z2: A] :
% 0.25/0.49 ( ( ( M2 @ X )
% 0.25/0.49 = ( some @ A @ Y ) )
% 0.25/0.49 => ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
% 0.25/0.49 => ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
% 0.25/0.49 => ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z2 ) ) )
% 0.25/0.49 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ran_map_upd_Some
% 0.25/0.49 thf(fact_3306_trancl__multi__insert,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),X10: set @ A,M2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
% 0.25/0.49 @ ( transitive_trancl @ A
% 0.25/0.49 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.49 @ ( product_Sigma @ A @ A @ X10
% 0.25/0.49 @ ^ [Uu: A] : ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ~ ! [X2: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ X10 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_multi_insert
% 0.25/0.49 thf(fact_3307_trancl__multi__insert2,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),M2: A,X10: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
% 0.25/0.49 @ ( transitive_trancl @ A
% 0.25/0.49 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 @ ^ [Uu: A] : X10 ) ) ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ~ ! [X2: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ X10 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ M2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_multi_insert2
% 0.25/0.49 thf(fact_3308_the__inv__f__o__f__id,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,Z2: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F @ Z2 )
% 0.25/0.49 = ( id @ A @ Z2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_f_o_f_id
% 0.25/0.49 thf(fact_3309_restrict__map__self,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( restrict_map @ A @ B @ M2 @ ( dom @ A @ B @ M2 ) )
% 0.25/0.49 = M2 ) ).
% 0.25/0.49
% 0.25/0.49 % restrict_map_self
% 0.25/0.49 thf(fact_3310_same__fst__trancl,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,P: A > $o,R5: A > ( set @ ( product_prod @ B @ B ) )] :
% 0.25/0.49 ( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R5 ) )
% 0.25/0.49 = ( same_fst @ A @ B @ P
% 0.25/0.49 @ ^ [X3: A] : ( transitive_trancl @ B @ ( R5 @ X3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % same_fst_trancl
% 0.25/0.49 thf(fact_3311_dom__eq__empty__conv,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > ( option @ B )] :
% 0.25/0.49 ( ( ( dom @ A @ B @ F )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 = ( F
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_eq_empty_conv
% 0.25/0.49 thf(fact_3312_fun__upd__None__if__notin__dom,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,K: A,M2: A > ( option @ B )] :
% 0.25/0.49 ( ~ ( member @ A @ K @ ( dom @ A @ B @ M2 ) )
% 0.25/0.49 => ( ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) )
% 0.25/0.49 = M2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % fun_upd_None_if_notin_dom
% 0.25/0.49 thf(fact_3313_the__inv__into__onto,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( image2 @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 = A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_into_onto
% 0.25/0.49 thf(fact_3314_dom__empty,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( dom @ A @ B
% 0.25/0.49 @ ^ [X3: A] : ( none @ B ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_empty
% 0.25/0.49 thf(fact_3315_map__update__eta__repair_I1_J,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( dom @ A @ B
% 0.25/0.49 @ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
% 0.25/0.49 = ( insert2 @ A @ K @ ( dom @ A @ B @ M2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_update_eta_repair(1)
% 0.25/0.49 thf(fact_3316_dom__const_H,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.49 ( ( dom @ A @ B
% 0.25/0.49 @ ^ [X3: A] : ( some @ B @ ( F @ X3 ) ) )
% 0.25/0.49 = ( top_top @ ( set @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_const'
% 0.25/0.49 thf(fact_3317_dom__mmupd,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K2: set @ A,V: B] :
% 0.25/0.49 ( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M2 @ K2 @ V ) )
% 0.25/0.49 = ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ K2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_mmupd
% 0.25/0.49 thf(fact_3318_trancl__single,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A] :
% 0.25/0.49 ( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
% 0.25/0.49 = ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_single
% 0.25/0.49 thf(fact_3319_restrict__map__inv,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > ( option @ B )] :
% 0.25/0.49 ( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F ) ) )
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % restrict_map_inv
% 0.25/0.49 thf(fact_3320_dom__fun__upd,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Y: option @ B,F: A > ( option @ B ),X: A] :
% 0.25/0.49 ( ( ( Y
% 0.25/0.49 = ( none @ B ) )
% 0.25/0.49 => ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
% 0.25/0.49 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.25/0.49 & ( ( Y
% 0.25/0.49 != ( none @ B ) )
% 0.25/0.49 => ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
% 0.25/0.49 = ( insert2 @ A @ X @ ( dom @ A @ B @ F ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_fun_upd
% 0.25/0.49 thf(fact_3321_domI,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
% 0.25/0.49 ( ( ( M2 @ A3 )
% 0.25/0.49 = ( some @ A @ B2 ) )
% 0.25/0.49 => ( member @ B @ A3 @ ( dom @ B @ A @ M2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % domI
% 0.25/0.49 thf(fact_3322_domD,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
% 0.25/0.49 => ? [B3: B] :
% 0.25/0.49 ( ( M2 @ A3 )
% 0.25/0.49 = ( some @ B @ B3 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % domD
% 0.25/0.49 thf(fact_3323_domIff,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
% 0.25/0.49 = ( ( M2 @ A3 )
% 0.25/0.49 != ( none @ B ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % domIff
% 0.25/0.49 thf(fact_3324_nempty__dom,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,E2: A > ( option @ B )] :
% 0.25/0.49 ( ( E2
% 0.25/0.49 != ( ^ [X3: A] : ( none @ B ) ) )
% 0.25/0.49 => ~ ! [M5: A] :
% 0.25/0.49 ~ ( member @ A @ M5 @ ( dom @ A @ B @ E2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % nempty_dom
% 0.25/0.49 thf(fact_3325_trancl__sub,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( transitive_trancl @ A @ R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_sub
% 0.25/0.49 thf(fact_3326_trancl__mono__mp,axiom,
% 0.25/0.49 ! [A: $tType,U4: set @ ( product_prod @ A @ A ),V6: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U4 @ V6 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U4 ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_mono_mp
% 0.25/0.49 thf(fact_3327_trancl_Ocases,axiom,
% 0.25/0.49 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R )
% 0.25/0.49 => ~ ! [B3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl.cases
% 0.25/0.49 thf(fact_3328_trancl_Osimps,axiom,
% 0.25/0.49 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 = ( ? [A7: A,B5: A] :
% 0.25/0.49 ( ( A1 = A7 )
% 0.25/0.49 & ( A22 = B5 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R ) )
% 0.25/0.49 | ? [A7: A,B5: A,C3: A] :
% 0.25/0.49 ( ( A1 = A7 )
% 0.25/0.49 & ( A22 = C3 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl.simps
% 0.25/0.49 thf(fact_3329_trancl_Or__into__trancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl.r_into_trancl
% 0.25/0.49 thf(fact_3330_tranclE,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.49 => ~ ! [C4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C4 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C4 @ B2 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % tranclE
% 0.25/0.49 thf(fact_3331_trancl__trans,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_trans
% 0.25/0.49 thf(fact_3332_trancl__induct,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ! [Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ R )
% 0.25/0.49 => ( P @ Y4 ) )
% 0.25/0.49 => ( ! [Y4: A,Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.49 => ( ( P @ Y4 )
% 0.25/0.49 => ( P @ Z3 ) ) ) )
% 0.25/0.49 => ( P @ B2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_induct
% 0.25/0.49 thf(fact_3333_r__r__into__trancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R5 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R5 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % r_r_into_trancl
% 0.25/0.49 thf(fact_3334_converse__tranclE,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R )
% 0.25/0.49 => ~ ! [Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_tranclE
% 0.25/0.49 thf(fact_3335_irrefl__trancl__rD,axiom,
% 0.25/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.49 ( ! [X2: A] :
% 0.25/0.49 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.49 => ( X != Y ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % irrefl_trancl_rD
% 0.25/0.49 thf(fact_3336_Transitive__Closure_Otrancl__into__trancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Transitive_Closure.trancl_into_trancl
% 0.25/0.49 thf(fact_3337_trancl__into__trancl2,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_into_trancl2
% 0.25/0.49 thf(fact_3338_trancl__trans__induct,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 0.25/0.49 => ( P @ X2 @ Y4 ) )
% 0.25/0.49 => ( ! [X2: A,Y4: A,Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( P @ X2 @ Y4 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( P @ Y4 @ Z3 )
% 0.25/0.49 => ( P @ X2 @ Z3 ) ) ) ) )
% 0.25/0.49 => ( P @ X @ Y ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_trans_induct
% 0.25/0.49 thf(fact_3339_converse__trancl__induct,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ! [Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R )
% 0.25/0.49 => ( P @ Y4 ) )
% 0.25/0.49 => ( ! [Y4: A,Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( P @ Z3 )
% 0.25/0.49 => ( P @ Y4 ) ) ) )
% 0.25/0.49 => ( P @ A3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % converse_trancl_induct
% 0.25/0.49 thf(fact_3340_dom__def,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( dom @ A @ B )
% 0.25/0.49 = ( ^ [M: A > ( option @ B )] :
% 0.25/0.49 ( collect @ A
% 0.25/0.49 @ ^ [A7: A] :
% 0.25/0.49 ( ( M @ A7 )
% 0.25/0.49 != ( none @ B ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_def
% 0.25/0.49 thf(fact_3341_the__inv__into__f__f,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( member @ A @ X @ A4 )
% 0.25/0.49 => ( ( the_inv_into @ A @ B @ A4 @ F @ ( F @ X ) )
% 0.25/0.49 = X ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_into_f_f
% 0.25/0.49 thf(fact_3342_the__inv__into__f__eq,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( ( F @ X )
% 0.25/0.49 = Y )
% 0.25/0.49 => ( ( member @ A @ X @ A4 )
% 0.25/0.49 => ( ( the_inv_into @ A @ B @ A4 @ F @ Y )
% 0.25/0.49 = X ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_into_f_eq
% 0.25/0.49 thf(fact_3343_insert__dom,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Y: A] :
% 0.25/0.49 ( ( ( F @ X )
% 0.25/0.49 = ( some @ A @ Y ) )
% 0.25/0.49 => ( ( insert2 @ B @ X @ ( dom @ B @ A @ F ) )
% 0.25/0.49 = ( dom @ B @ A @ F ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % insert_dom
% 0.25/0.49 thf(fact_3344_trancl__rtrancl__trancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_rtrancl_trancl
% 0.25/0.49 thf(fact_3345_rtrancl__trancl__trancl,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_trancl_trancl
% 0.25/0.49 thf(fact_3346_rtrancl__into__trancl2,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_into_trancl2
% 0.25/0.49 thf(fact_3347_rtrancl__into__trancl1,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_into_trancl1
% 0.25/0.49 thf(fact_3348_rtrancl__eq__or__trancl,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 = ( ( X = Y )
% 0.25/0.49 | ( ( X != Y )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_eq_or_trancl
% 0.25/0.49 thf(fact_3349_trancl__into__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_into_rtrancl
% 0.25/0.49 thf(fact_3350_tranclD2,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) )
% 0.25/0.49 => ? [Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % tranclD2
% 0.25/0.49 thf(fact_3351_rtranclD,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( ( A3 = B2 )
% 0.25/0.49 | ( ( A3 != B2 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtranclD
% 0.25/0.49 thf(fact_3352_tranclD,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) )
% 0.25/0.49 => ? [Z3: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R5 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % tranclD
% 0.25/0.49 thf(fact_3353_map__dom__ran__finite,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M3: A > ( option @ B )] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
% 0.25/0.49 => ( finite_finite2 @ B @ ( ran @ A @ B @ M3 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_dom_ran_finite
% 0.25/0.49 thf(fact_3354_rel__restrict__trancl__notR_I2_J,axiom,
% 0.25/0.49 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) )
% 0.25/0.49 => ~ ( member @ A @ W @ R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_trancl_notR(2)
% 0.25/0.49 thf(fact_3355_rel__restrict__trancl__notR_I1_J,axiom,
% 0.25/0.49 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) )
% 0.25/0.49 => ~ ( member @ A @ V @ R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_trancl_notR(1)
% 0.25/0.49 thf(fact_3356_rel__restrict__trancl__mem,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R5 ) ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_trancl_mem
% 0.25/0.49 thf(fact_3357_trancl__induct2,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ! [A5: A,B3: B] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ R )
% 0.25/0.49 => ( P @ A5 @ B3 ) )
% 0.25/0.49 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 0.25/0.49 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
% 0.25/0.49 => ( ( P @ A5 @ B3 )
% 0.25/0.49 => ( P @ Aa3 @ Ba ) ) ) )
% 0.25/0.49 => ( P @ Bx @ By ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_induct2
% 0.25/0.49 thf(fact_3358_card__map__to__set,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( finite_card @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
% 0.25/0.49 = ( finite_card @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % card_map_to_set
% 0.25/0.49 thf(fact_3359_trancl__sub__insert__trancl,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R5 ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_sub_insert_trancl
% 0.25/0.49 thf(fact_3360_rel__restrict__trancl__sub,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),R5: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R5 ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R5 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_trancl_sub
% 0.25/0.49 thf(fact_3361_le__map__dom__mono,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( preorder @ B )
% 0.25/0.49 => ! [M2: A > ( option @ B ),M7: A > ( option @ B )] :
% 0.25/0.49 ( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ ( dom @ A @ B @ M7 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % le_map_dom_mono
% 0.25/0.49 thf(fact_3362_the__inv__into__def,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( the_inv_into @ A @ B )
% 0.25/0.49 = ( ^ [A9: set @ A,F2: A > B,X3: B] :
% 0.25/0.49 ( the @ A
% 0.25/0.49 @ ^ [Y3: A] :
% 0.25/0.49 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.49 & ( ( F2 @ Y3 )
% 0.25/0.49 = X3 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_into_def
% 0.25/0.49 thf(fact_3363_finite__map__freshness,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: A > ( option @ B )] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( dom @ A @ B @ F ) )
% 0.25/0.49 => ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ? [X2: A] :
% 0.25/0.49 ( ( F @ X2 )
% 0.25/0.49 = ( none @ B ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_map_freshness
% 0.25/0.49 thf(fact_3364_dom__minus,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,A4: set @ B] :
% 0.25/0.49 ( ( ( F @ X )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 => ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ ( insert2 @ B @ X @ A4 ) )
% 0.25/0.49 = ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_minus
% 0.25/0.49 thf(fact_3365_restrict__map__eq_I1_J,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: set @ B,K: B] :
% 0.25/0.49 ( ( ( restrict_map @ B @ A @ M2 @ A4 @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 = ( ~ ( member @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M2 ) @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % restrict_map_eq(1)
% 0.25/0.49 thf(fact_3366_map__card__eq__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M3: A > ( option @ B ),X: A,Y: A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
% 0.25/0.49 => ( ( ( finite_card @ A @ ( dom @ A @ B @ M3 ) )
% 0.25/0.49 = ( finite_card @ B @ ( ran @ A @ B @ M3 ) ) )
% 0.25/0.49 => ( ( member @ A @ X @ ( dom @ A @ B @ M3 ) )
% 0.25/0.49 => ( ( ( M3 @ X )
% 0.25/0.49 = ( M3 @ Y ) )
% 0.25/0.49 = ( X = Y ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_card_eq_iff
% 0.25/0.49 thf(fact_3367_wf__linord__ex__has__least,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M2: B > A] :
% 0.25/0.49 ( ( wf @ A @ R )
% 0.25/0.49 => ( ! [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
% 0.25/0.49 => ( ( P @ K )
% 0.25/0.49 => ? [X2: B] :
% 0.25/0.49 ( ( P @ X2 )
% 0.25/0.49 & ! [Y6: B] :
% 0.25/0.49 ( ( P @ Y6 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X2 ) @ ( M2 @ Y6 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % wf_linord_ex_has_least
% 0.25/0.49 thf(fact_3368_trancl__union__outside,axiom,
% 0.25/0.49 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E4 ) )
% 0.25/0.49 => ? [X2: A,Y4: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ U4 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ W ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_union_outside
% 0.25/0.49 thf(fact_3369_trancl__over__edgeE,axiom,
% 0.25/0.49 ! [A: $tType,U: A,W: A,V1: A,V22: A,E4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E4 ) ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ E4 ) )
% 0.25/0.49 => ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E4 ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_over_edgeE
% 0.25/0.49 thf(fact_3370_finite__map__to__set,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
% 0.25/0.49 = ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_map_to_set
% 0.25/0.49 thf(fact_3371_map__to__set__dom,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( dom @ A @ B )
% 0.25/0.49 = ( ^ [M: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_to_set_dom
% 0.25/0.49 thf(fact_3372_inj__on__the__inv__into,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_the_inv_into
% 0.25/0.49 thf(fact_3373_f__the__inv__into__f,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,Y: B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( member @ B @ Y @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 => ( ( F @ ( the_inv_into @ A @ B @ A4 @ F @ Y ) )
% 0.25/0.49 = Y ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % f_the_inv_into_f
% 0.25/0.49 thf(fact_3374_the__inv__f__f,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 0.25/0.49 = X ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_f_f
% 0.25/0.49 thf(fact_3375_set__to__map__dom,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( dom @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
% 0.25/0.49 = ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_to_map_dom
% 0.25/0.49 thf(fact_3376_Restr__trancl__mono,axiom,
% 0.25/0.49 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
% 0.25/0.49 @ ( transitive_trancl @ A
% 0.25/0.49 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 0.25/0.49 @ ( product_Sigma @ A @ A @ U4
% 0.25/0.49 @ ^ [Uu: A] : U4 ) ) ) )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Restr_trancl_mono
% 0.25/0.49 thf(fact_3377_inj__on__map__the,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,D4: set @ A,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ D4 @ ( dom @ A @ B @ M2 ) )
% 0.25/0.49 => ( ( inj_on @ A @ ( option @ B ) @ M2 @ D4 )
% 0.25/0.49 => ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M2 ) @ D4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_map_the
% 0.25/0.49 thf(fact_3378_finite__Map__induct,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
% 0.25/0.49 => ( ( P
% 0.25/0.49 @ ^ [X3: A] : ( none @ B ) )
% 0.25/0.49 => ( ! [K5: A,V4: B,M5: A > ( option @ B )] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M5 ) )
% 0.25/0.49 => ( ~ ( member @ A @ K5 @ ( dom @ A @ B @ M5 ) )
% 0.25/0.49 => ( ( P @ M5 )
% 0.25/0.49 => ( P @ ( fun_upd @ A @ ( option @ B ) @ M5 @ K5 @ ( some @ B @ V4 ) ) ) ) ) )
% 0.25/0.49 => ( P @ M2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_Map_induct
% 0.25/0.49 thf(fact_3379_ran__is__image,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( ran @ B @ A )
% 0.25/0.49 = ( ^ [M8: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M8 ) @ ( dom @ B @ A @ M8 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ran_is_image
% 0.25/0.49 thf(fact_3380_graph__eq__to__snd__dom,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( graph @ A @ B )
% 0.25/0.49 = ( ^ [M: A > ( option @ B )] :
% 0.25/0.49 ( image2 @ A @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( the2 @ B @ ( M @ X3 ) ) )
% 0.25/0.49 @ ( dom @ A @ B @ M ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % graph_eq_to_snd_dom
% 0.25/0.49 thf(fact_3381_trancl__insert2,axiom,
% 0.25/0.49 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
% 0.25/0.49 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
% 0.25/0.49 @ ( collect @ ( product_prod @ A @ A )
% 0.25/0.49 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.49 @ ^ [X3: A,Y3: A] :
% 0.25/0.49 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 | ( X3 = A3 ) )
% 0.25/0.49 & ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.49 | ( Y3 = B2 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_insert2
% 0.25/0.49 thf(fact_3382_rel__restrict__Sigma__sub,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,R5: set @ A] :
% 0.25/0.49 ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.49 @ ( rel_restrict @ A
% 0.25/0.49 @ ( transitive_trancl @ A
% 0.25/0.49 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.49 @ ^ [Uu: A] : A4 ) )
% 0.25/0.49 @ R5 )
% 0.25/0.49 @ ( transitive_trancl @ A
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A4 @ R5 )
% 0.25/0.49 @ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ A4 @ R5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_Sigma_sub
% 0.25/0.49 thf(fact_3383_the__inv__into__into,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: B,B6: set @ A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( member @ B @ X @ ( image2 @ A @ B @ F @ A4 ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.49 => ( member @ A @ ( the_inv_into @ A @ B @ A4 @ F @ X ) @ B6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_into_into
% 0.25/0.49 thf(fact_3384_the__inv__into__comp,axiom,
% 0.25/0.49 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A4: set @ C,X: B] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) )
% 0.25/0.49 => ( ( inj_on @ C @ A @ G @ A4 )
% 0.25/0.49 => ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) ) )
% 0.25/0.49 => ( ( the_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F @ G ) @ X )
% 0.25/0.49 = ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A4 @ G ) @ ( the_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A4 ) @ F ) @ X ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % the_inv_into_comp
% 0.25/0.49 thf(fact_3385_dom__eq__singleton__conv,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
% 0.25/0.49 ( ( ( dom @ A @ B @ F )
% 0.25/0.49 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( ? [V2: B] :
% 0.25/0.49 ( F
% 0.25/0.49 = ( fun_upd @ A @ ( option @ B )
% 0.25/0.49 @ ^ [X3: A] : ( none @ B )
% 0.25/0.49 @ X
% 0.25/0.49 @ ( some @ B @ V2 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % dom_eq_singleton_conv
% 0.25/0.49 thf(fact_3386_rtrancl__last__visit,axiom,
% 0.25/0.49 ! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : S2 ) ) ) )
% 0.25/0.49 => ~ ! [Qt: A] :
% 0.25/0.49 ( ( member @ A @ Qt @ S2 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_trancl @ A @ R5 ) )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
% 0.25/0.49 @ ( transitive_rtrancl @ A
% 0.25/0.49 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 0.25/0.49 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.49 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_last_visit
% 0.25/0.49 thf(fact_3387_trancl__insert,axiom,
% 0.25/0.49 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
% 0.25/0.49 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
% 0.25/0.49 @ ( collect @ ( product_prod @ A @ A )
% 0.25/0.49 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.49 @ ^ [A7: A,B5: A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_insert
% 0.25/0.49 thf(fact_3388_finite__trancl__ntranl,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
% 0.25/0.49 => ( ( transitive_trancl @ A @ R5 )
% 0.25/0.49 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) @ ( one_one @ nat ) ) @ R5 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_trancl_ntranl
% 0.25/0.49 thf(fact_3389_ran__add,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.25/0.49 ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( dom @ A @ B @ G ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ran @ A @ B @ ( map_add @ A @ B @ F @ G ) )
% 0.25/0.49 = ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F ) @ ( ran @ A @ B @ G ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ran_add
% 0.25/0.49 thf(fact_3390_eq__f__restr__ss__eq,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,S: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ S @ ( dom @ A @ B @ ( F @ A4 ) ) )
% 0.25/0.49 => ( ( A4
% 0.25/0.49 = ( restrict_map @ A @ B @ ( F @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S ) ) )
% 0.25/0.49 = ( ( map_le @ A @ B @ A4 @ ( F @ A4 ) )
% 0.25/0.49 & ( S
% 0.25/0.49 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % eq_f_restr_ss_eq
% 0.25/0.49 thf(fact_3391_eq__f__restr__conv,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,S: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
% 0.25/0.49 ( ( ( ord_less_eq @ ( set @ A ) @ S @ ( dom @ A @ B @ ( F @ A4 ) ) )
% 0.25/0.49 & ( A4
% 0.25/0.49 = ( restrict_map @ A @ B @ ( F @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S ) ) ) )
% 0.25/0.49 = ( ( map_le @ A @ B @ A4 @ ( F @ A4 ) )
% 0.25/0.49 & ( S
% 0.25/0.49 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % eq_f_restr_conv
% 0.25/0.49 thf(fact_3392_type__copy__set__map0,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F5: $tType,Rep: A > B,Abs: B > A,S2: B > ( set @ D ),M3: C > B,F: E > D,S7: C > ( set @ E ),G: F5 > C] :
% 0.25/0.49 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( ( ( comp @ B @ ( set @ D ) @ C @ S2 @ M3 )
% 0.25/0.49 = ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F ) @ S7 ) )
% 0.25/0.49 => ( ( comp @ A @ ( set @ D ) @ F5 @ ( comp @ B @ ( set @ D ) @ A @ S2 @ Rep ) @ ( comp @ C @ A @ F5 @ ( comp @ B @ A @ C @ Abs @ M3 ) @ G ) )
% 0.25/0.49 = ( comp @ ( set @ E ) @ ( set @ D ) @ F5 @ ( image2 @ E @ D @ F ) @ ( comp @ C @ ( set @ E ) @ F5 @ S7 @ G ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_copy_set_map0
% 0.25/0.49 thf(fact_3393_map__add__find__right,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,N: B > ( option @ A ),K: B,Xx: A,M2: B > ( option @ A )] :
% 0.25/0.49 ( ( ( N @ K )
% 0.25/0.49 = ( some @ A @ Xx ) )
% 0.25/0.49 => ( ( map_add @ B @ A @ M2 @ N @ K )
% 0.25/0.49 = ( some @ A @ Xx ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_find_right
% 0.25/0.49 thf(fact_3394_map__add__eq__empty__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.25/0.49 ( ( ( map_add @ A @ B @ F @ G )
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) )
% 0.25/0.49 = ( ( F
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) )
% 0.25/0.49 & ( G
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_eq_empty_iff
% 0.25/0.49 thf(fact_3395_map__add__None,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B] :
% 0.25/0.49 ( ( ( map_add @ B @ A @ M2 @ N @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 = ( ( ( N @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 & ( ( M2 @ K )
% 0.25/0.49 = ( none @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_None
% 0.25/0.49 thf(fact_3396_empty__map__add,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( map_add @ A @ B
% 0.25/0.49 @ ^ [X3: A] : ( none @ B )
% 0.25/0.49 @ M2 )
% 0.25/0.49 = M2 ) ).
% 0.25/0.49
% 0.25/0.49 % empty_map_add
% 0.25/0.49 thf(fact_3397_map__add__empty,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( map_add @ A @ B @ M2
% 0.25/0.49 @ ^ [X3: A] : ( none @ B ) )
% 0.25/0.49 = M2 ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_empty
% 0.25/0.49 thf(fact_3398_empty__eq__map__add__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.25/0.49 ( ( ( ^ [X3: A] : ( none @ B ) )
% 0.25/0.49 = ( map_add @ A @ B @ F @ G ) )
% 0.25/0.49 = ( ( F
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) )
% 0.25/0.49 & ( G
% 0.25/0.49 = ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % empty_eq_map_add_iff
% 0.25/0.49 thf(fact_3399_map__add__upd,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: A > ( option @ B ),G: A > ( option @ B ),X: A,Y: B] :
% 0.25/0.49 ( ( map_add @ A @ B @ F @ ( fun_upd @ A @ ( option @ B ) @ G @ X @ ( some @ B @ Y ) ) )
% 0.25/0.49 = ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F @ G ) @ X @ ( some @ B @ Y ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_upd
% 0.25/0.49 thf(fact_3400_map__add__upd__left,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,M2: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
% 0.25/0.49 ( ~ ( member @ A @ M2 @ ( dom @ A @ B @ E22 ) )
% 0.25/0.49 => ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M2 @ ( some @ B @ U1 ) ) @ E22 )
% 0.25/0.49 = ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M2 @ ( some @ B @ U1 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_upd_left
% 0.25/0.49 thf(fact_3401_le__map__mmupd__not__dom,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,M2: A > ( option @ B ),K2: set @ A,V: B] : ( map_le @ A @ B @ M2 @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ).
% 0.25/0.49
% 0.25/0.49 % le_map_mmupd_not_dom
% 0.25/0.49 thf(fact_3402_map__add__find__left,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,G: B > ( option @ A ),K: B,F: B > ( option @ A )] :
% 0.25/0.49 ( ( ( G @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 => ( ( map_add @ B @ A @ F @ G @ K )
% 0.25/0.49 = ( F @ K ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_find_left
% 0.25/0.49 thf(fact_3403_map__add__left__None,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > ( option @ A ),K: B,G: B > ( option @ A )] :
% 0.25/0.49 ( ( ( F @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 => ( ( map_add @ B @ A @ F @ G @ K )
% 0.25/0.49 = ( G @ K ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_left_None
% 0.25/0.49 thf(fact_3404_map__leI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
% 0.25/0.49 ( ! [X2: A,V4: B] :
% 0.25/0.49 ( ( ( M1 @ X2 )
% 0.25/0.49 = ( some @ B @ V4 ) )
% 0.25/0.49 => ( ( M22 @ X2 )
% 0.25/0.49 = ( some @ B @ V4 ) ) )
% 0.25/0.49 => ( map_le @ A @ B @ M1 @ M22 ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_leI
% 0.25/0.49 thf(fact_3405_map__leD,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),K: A,V: B] :
% 0.25/0.49 ( ( map_le @ A @ B @ M1 @ M22 )
% 0.25/0.49 => ( ( ( M1 @ K )
% 0.25/0.49 = ( some @ B @ V ) )
% 0.25/0.49 => ( ( M22 @ K )
% 0.25/0.49 = ( some @ B @ V ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_leD
% 0.25/0.49 thf(fact_3406_map__add__first__le,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( order @ B )
% 0.25/0.49 => ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B )] :
% 0.25/0.49 ( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
% 0.25/0.49 => ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_first_le
% 0.25/0.49 thf(fact_3407_map__le__empty,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,G: A > ( option @ B )] :
% 0.25/0.49 ( map_le @ A @ B
% 0.25/0.49 @ ^ [X3: A] : ( none @ B )
% 0.25/0.49 @ G ) ).
% 0.25/0.49
% 0.25/0.49 % map_le_empty
% 0.25/0.49 thf(fact_3408_map__add__Some__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
% 0.25/0.49 ( ( ( map_add @ B @ A @ M2 @ N @ K )
% 0.25/0.49 = ( some @ A @ X ) )
% 0.25/0.49 = ( ( ( N @ K )
% 0.25/0.49 = ( some @ A @ X ) )
% 0.25/0.49 | ( ( ( N @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 & ( ( M2 @ K )
% 0.25/0.49 = ( some @ A @ X ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_Some_iff
% 0.25/0.49 thf(fact_3409_map__add__SomeD,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
% 0.25/0.49 ( ( ( map_add @ B @ A @ M2 @ N @ K )
% 0.25/0.49 = ( some @ A @ X ) )
% 0.25/0.49 => ( ( ( N @ K )
% 0.25/0.49 = ( some @ A @ X ) )
% 0.25/0.49 | ( ( ( N @ K )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 & ( ( M2 @ K )
% 0.25/0.49 = ( some @ A @ X ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_SomeD
% 0.25/0.49 thf(fact_3410_type__copy__map__comp0__undo,axiom,
% 0.25/0.49 ! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F5: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F5,Abs3: F5 > E,M3: F5 > D,M13: B > D,M24: F5 > B] :
% 0.25/0.49 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
% 0.25/0.49 => ( ( type_definition @ E @ F5 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F5 ) ) )
% 0.25/0.49 => ( ( ( comp @ F5 @ C @ E @ ( comp @ D @ C @ F5 @ Abs2 @ M3 ) @ Rep3 )
% 0.25/0.49 = ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M13 ) @ Rep ) @ ( comp @ F5 @ A @ E @ ( comp @ B @ A @ F5 @ Abs @ M24 ) @ Rep3 ) ) )
% 0.25/0.49 => ( ( comp @ B @ D @ F5 @ M13 @ M24 )
% 0.25/0.49 = M3 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_copy_map_comp0_undo
% 0.25/0.49 thf(fact_3411_type__copy__map__comp0,axiom,
% 0.25/0.49 ! [F5: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M3: C > D,M13: B > D,M24: C > B,F: D > F5,G: E > C] :
% 0.25/0.49 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( ( M3
% 0.25/0.49 = ( comp @ B @ D @ C @ M13 @ M24 ) )
% 0.25/0.49 => ( ( comp @ C @ F5 @ E @ ( comp @ D @ F5 @ C @ F @ M3 ) @ G )
% 0.25/0.49 = ( comp @ A @ F5 @ E @ ( comp @ B @ F5 @ A @ ( comp @ D @ F5 @ B @ F @ M13 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M24 ) @ G ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_copy_map_comp0
% 0.25/0.49 thf(fact_3412_upd__None__map__le,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > ( option @ B ),X: A] : ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) @ F ) ).
% 0.25/0.49
% 0.25/0.49 % upd_None_map_le
% 0.25/0.49 thf(fact_3413_map__add__def,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( map_add @ A @ B )
% 0.25/0.49 = ( ^ [M14: A > ( option @ B ),M25: A > ( option @ B ),X3: A] : ( case_option @ ( option @ B ) @ B @ ( M14 @ X3 ) @ ( some @ B ) @ ( M25 @ X3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_def
% 0.25/0.49 thf(fact_3414_map__add__left__comm,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A4: A > ( option @ B ),B6: A > ( option @ B ),C5: A > ( option @ B )] :
% 0.25/0.49 ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A4 ) @ ( dom @ A @ B @ B6 ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ( map_add @ A @ B @ A4 @ ( map_add @ A @ B @ B6 @ C5 ) )
% 0.25/0.49 = ( map_add @ A @ B @ B6 @ ( map_add @ A @ B @ A4 @ C5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_left_comm
% 0.25/0.49 thf(fact_3415_type__copy__map__id0,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M3: B > B] :
% 0.25/0.49 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( ( M3
% 0.25/0.49 = ( id @ B ) )
% 0.25/0.49 => ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M3 ) @ Rep )
% 0.25/0.49 = ( id @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_copy_map_id0
% 0.25/0.49 thf(fact_3416_type__copy__Abs__o__Rep,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
% 0.25/0.49 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( ( comp @ B @ A @ A @ Abs @ Rep )
% 0.25/0.49 = ( id @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_copy_Abs_o_Rep
% 0.25/0.49 thf(fact_3417_type__copy__Rep__o__Abs,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
% 0.25/0.49 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.49 => ( ( comp @ A @ B @ B @ Rep @ Abs )
% 0.25/0.49 = ( id @ B ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_copy_Rep_o_Abs
% 0.25/0.49 thf(fact_3418_type__definition__integer,axiom,
% 0.25/0.49 type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% 0.25/0.49
% 0.25/0.49 % type_definition_integer
% 0.25/0.49 thf(fact_3419_map__le__imp__upd__le,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),X: A,Y: B] :
% 0.25/0.49 ( ( map_le @ A @ B @ M1 @ M22 )
% 0.25/0.49 => ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M1 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M22 @ X @ ( some @ B @ Y ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_le_imp_upd_le
% 0.25/0.49 thf(fact_3420_map__add__distinct__le,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( preorder @ B )
% 0.25/0.49 => ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B ),N2: A > ( option @ B )] :
% 0.25/0.49 ( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
% 0.25/0.49 => ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N2 )
% 0.25/0.49 => ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M7 ) @ ( dom @ A @ B @ N2 ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N2 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_distinct_le
% 0.25/0.49 thf(fact_3421_map__mmupd__update__less,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,K2: set @ A,K9: set @ A,M2: A > ( option @ B ),V: B] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ K2 @ K9 )
% 0.25/0.49 => ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M2 ) ) @ V ) @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K9 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_mmupd_update_less
% 0.25/0.49 thf(fact_3422_trancl__set__ntrancl,axiom,
% 0.25/0.49 ! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
% 0.25/0.49 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_set_ntrancl
% 0.25/0.49 thf(fact_3423_pred__nat__trancl__eq__le,axiom,
% 0.25/0.49 ! [M2: nat,N: nat] :
% 0.25/0.49 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
% 0.25/0.49 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 0.25/0.49
% 0.25/0.49 % pred_nat_trancl_eq_le
% 0.25/0.49 thf(fact_3424_less__eq,axiom,
% 0.25/0.49 ! [M2: nat,N: nat] :
% 0.25/0.49 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
% 0.25/0.49 = ( ord_less @ nat @ M2 @ N ) ) ).
% 0.25/0.49
% 0.25/0.49 % less_eq
% 0.25/0.49 thf(fact_3425_Gcd__fin__def,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( semiring_gcd @ A )
% 0.25/0.49 => ( ( semiring_gcd_Gcd_fin @ A )
% 0.25/0.49 = ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Gcd_fin_def
% 0.25/0.49 thf(fact_3426_relImage__def,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType] :
% 0.25/0.49 ( ( bNF_Gr4221423524335903396lImage @ B @ A )
% 0.25/0.49 = ( ^ [R6: set @ ( product_prod @ B @ B ),F2: B > A] :
% 0.25/0.49 ( collect @ ( product_prod @ A @ A )
% 0.25/0.49 @ ^ [Uu: product_prod @ A @ A] :
% 0.25/0.49 ? [A12: B,A23: B] :
% 0.25/0.49 ( ( Uu
% 0.25/0.49 = ( product_Pair @ A @ A @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) )
% 0.25/0.49 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R6 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % relImage_def
% 0.25/0.49 thf(fact_3427_set__map__filter,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,G: B > ( option @ A ),Xs: list @ B] :
% 0.25/0.49 ( ( set2 @ A @ ( map_filter @ B @ A @ G @ Xs ) )
% 0.25/0.49 = ( collect @ A
% 0.25/0.49 @ ^ [Y3: A] :
% 0.25/0.49 ? [X3: B] :
% 0.25/0.49 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
% 0.25/0.49 & ( ( G @ X3 )
% 0.25/0.49 = ( some @ A @ Y3 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_map_filter
% 0.25/0.49 thf(fact_3428_ran__map__of,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( map_of @ B @ A @ Xs ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ran_map_of
% 0.25/0.49 thf(fact_3429_trancl__restrict__reachable,axiom,
% 0.25/0.49 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E4 ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ S2 ) @ S2 )
% 0.25/0.49 => ( ( member @ A @ U @ S2 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
% 0.25/0.49 @ ( transitive_trancl @ A
% 0.25/0.49 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 0.25/0.49 @ ( product_Sigma @ A @ A @ S2
% 0.25/0.49 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_restrict_reachable
% 0.25/0.49 thf(fact_3430_pair__vimage__is__Image,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,U: B,E4: set @ ( product_prod @ B @ A )] :
% 0.25/0.49 ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E4 )
% 0.25/0.49 = ( image @ B @ A @ E4 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % pair_vimage_is_Image
% 0.25/0.49 thf(fact_3431_set__bind__option,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,X: option @ B,F: B > ( option @ A )] :
% 0.25/0.49 ( ( set_option @ A @ ( bind @ B @ A @ X @ F ) )
% 0.25/0.49 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F ) @ ( set_option @ B @ X ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_bind_option
% 0.25/0.49 thf(fact_3432_ImageI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),A4: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 0.25/0.49 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.49 => ( member @ B @ B2 @ ( image @ A @ B @ R @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ImageI
% 0.25/0.49 thf(fact_3433_elem__set,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Xo: option @ A] :
% 0.25/0.49 ( ( member @ A @ X @ ( set_option @ A @ Xo ) )
% 0.25/0.49 = ( Xo
% 0.25/0.49 = ( some @ A @ X ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % elem_set
% 0.25/0.49 thf(fact_3434_set__empty__eq,axiom,
% 0.25/0.49 ! [A: $tType,Xo: option @ A] :
% 0.25/0.49 ( ( ( set_option @ A @ Xo )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 = ( Xo
% 0.25/0.49 = ( none @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_empty_eq
% 0.25/0.49 thf(fact_3435_Image__singleton__iff,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A3: B] :
% 0.25/0.49 ( ( member @ A @ B2 @ ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 0.25/0.49 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) ) ).
% 0.25/0.49
% 0.25/0.49 % Image_singleton_iff
% 0.25/0.49 thf(fact_3436_rev__ImageI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( member @ A @ A3 @ A4 )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 0.25/0.49 => ( member @ B @ B2 @ ( image @ A @ B @ R @ A4 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rev_ImageI
% 0.25/0.49 thf(fact_3437_Image__iff,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A4: set @ B] :
% 0.25/0.49 ( ( member @ A @ B2 @ ( image @ B @ A @ R @ A4 ) )
% 0.25/0.49 = ( ? [X3: B] :
% 0.25/0.49 ( ( member @ B @ X3 @ A4 )
% 0.25/0.49 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Image_iff
% 0.25/0.49 thf(fact_3438_ImageE,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A4: set @ B] :
% 0.25/0.49 ( ( member @ A @ B2 @ ( image @ B @ A @ R @ A4 ) )
% 0.25/0.49 => ~ ! [X2: B] :
% 0.25/0.49 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R )
% 0.25/0.49 => ~ ( member @ B @ X2 @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ImageE
% 0.25/0.49 thf(fact_3439_ospec,axiom,
% 0.25/0.49 ! [A: $tType,A4: option @ A,P: A > $o,X: A] :
% 0.25/0.49 ( ! [X2: A] :
% 0.25/0.49 ( ( member @ A @ X2 @ ( set_option @ A @ A4 ) )
% 0.25/0.49 => ( P @ X2 ) )
% 0.25/0.49 => ( ( A4
% 0.25/0.49 = ( some @ A @ X ) )
% 0.25/0.49 => ( P @ X ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % ospec
% 0.25/0.49 thf(fact_3440_option_Oset__intros,axiom,
% 0.25/0.49 ! [A: $tType,X22: A] : ( member @ A @ X22 @ ( set_option @ A @ ( some @ A @ X22 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.set_intros
% 0.25/0.49 thf(fact_3441_option_Oset__cases,axiom,
% 0.25/0.49 ! [A: $tType,E2: A,A3: option @ A] :
% 0.25/0.49 ( ( member @ A @ E2 @ ( set_option @ A @ A3 ) )
% 0.25/0.49 => ( A3
% 0.25/0.49 = ( some @ A @ E2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.set_cases
% 0.25/0.49 thf(fact_3442_map__option__idI,axiom,
% 0.25/0.49 ! [A: $tType,X: option @ A,F: A > A] :
% 0.25/0.49 ( ! [Y4: A] :
% 0.25/0.49 ( ( member @ A @ Y4 @ ( set_option @ A @ X ) )
% 0.25/0.49 => ( ( F @ Y4 )
% 0.25/0.49 = Y4 ) )
% 0.25/0.49 => ( ( map_option @ A @ A @ F @ X )
% 0.25/0.49 = X ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_option_idI
% 0.25/0.49 thf(fact_3443_option_Oinj__map__strong,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,X: option @ A,Xa: option @ A,F: A > B,Fa: A > B] :
% 0.25/0.49 ( ! [Z3: A,Za: A] :
% 0.25/0.49 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 0.25/0.49 => ( ( member @ A @ Za @ ( set_option @ A @ Xa ) )
% 0.25/0.49 => ( ( ( F @ Z3 )
% 0.25/0.49 = ( Fa @ Za ) )
% 0.25/0.49 => ( Z3 = Za ) ) ) )
% 0.25/0.49 => ( ( ( map_option @ A @ B @ F @ X )
% 0.25/0.49 = ( map_option @ A @ B @ Fa @ Xa ) )
% 0.25/0.49 => ( X = Xa ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.inj_map_strong
% 0.25/0.49 thf(fact_3444_option_Omap__cong0,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,X: option @ A,F: A > B,G: A > B] :
% 0.25/0.49 ( ! [Z3: A] :
% 0.25/0.49 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 0.25/0.49 => ( ( F @ Z3 )
% 0.25/0.49 = ( G @ Z3 ) ) )
% 0.25/0.49 => ( ( map_option @ A @ B @ F @ X )
% 0.25/0.49 = ( map_option @ A @ B @ G @ X ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.map_cong0
% 0.25/0.49 thf(fact_3445_option_Omap__cong,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
% 0.25/0.49 ( ( X = Ya )
% 0.25/0.49 => ( ! [Z3: A] :
% 0.25/0.49 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 0.25/0.49 => ( ( F @ Z3 )
% 0.25/0.49 = ( G @ Z3 ) ) )
% 0.25/0.49 => ( ( map_option @ A @ B @ F @ X )
% 0.25/0.49 = ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.map_cong
% 0.25/0.49 thf(fact_3446_bind__option__cong,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 0.25/0.49 ( ( X = Y )
% 0.25/0.49 => ( ! [Z3: A] :
% 0.25/0.49 ( ( member @ A @ Z3 @ ( set_option @ A @ Y ) )
% 0.25/0.49 => ( ( F @ Z3 )
% 0.25/0.49 = ( G @ Z3 ) ) )
% 0.25/0.49 => ( ( bind @ A @ B @ X @ F )
% 0.25/0.49 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % bind_option_cong
% 0.25/0.49 thf(fact_3447_rtrancl__image__advance__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,Q3: A,R5: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
% 0.25/0.49 ( ( member @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.49 => ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_image_advance_rtrancl
% 0.25/0.49 thf(fact_3448_rtrancl__image__advance,axiom,
% 0.25/0.49 ! [A: $tType,Q3: A,R5: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
% 0.25/0.49 ( ( member @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ R5 )
% 0.25/0.49 => ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_image_advance
% 0.25/0.49 thf(fact_3449_finite__Image__subset,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ B,C5: set @ ( product_prod @ B @ A )] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( image @ B @ A @ A4 @ B6 ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C5 @ A4 )
% 0.25/0.49 => ( finite_finite2 @ A @ ( image @ B @ A @ C5 @ B6 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_Image_subset
% 0.25/0.49 thf(fact_3450_rtrancl__image__unfold__right,axiom,
% 0.25/0.49 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V6 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V6 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_image_unfold_right
% 0.25/0.49 thf(fact_3451_rtrancl__reachable__induct,axiom,
% 0.25/0.49 ! [A: $tType,I4: set @ A,INV: set @ A,E4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ I4 @ INV )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ INV ) @ INV )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I4 ) @ INV ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_reachable_induct
% 0.25/0.49 thf(fact_3452_trancl__Image__unfold__right,axiom,
% 0.25/0.49 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 0.25/0.49 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S2 )
% 0.25/0.49 = ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ S2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_Image_unfold_right
% 0.25/0.49 thf(fact_3453_trancl__Image__unfold__left,axiom,
% 0.25/0.49 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 0.25/0.49 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S2 )
% 0.25/0.49 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( image @ A @ A @ E4 @ S2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_Image_unfold_left
% 0.25/0.49 thf(fact_3454_option_Osimps_I14_J,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( set_option @ A @ ( none @ A ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.simps(14)
% 0.25/0.49 thf(fact_3455_option_Oset__map,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,V: option @ A] :
% 0.25/0.49 ( ( set_option @ B @ ( map_option @ A @ B @ F @ V ) )
% 0.25/0.49 = ( image2 @ A @ B @ F @ ( set_option @ A @ V ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.set_map
% 0.25/0.49 thf(fact_3456_option_Oset__sel,axiom,
% 0.25/0.49 ! [A: $tType,A3: option @ A] :
% 0.25/0.49 ( ( A3
% 0.25/0.49 != ( none @ A ) )
% 0.25/0.49 => ( member @ A @ ( the2 @ A @ A3 ) @ ( set_option @ A @ A3 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.set_sel
% 0.25/0.49 thf(fact_3457_Image__empty__rtrancl__Image__id,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V: A] :
% 0.25/0.49 ( ( ( image @ A @ A @ R5 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Image_empty_rtrancl_Image_id
% 0.25/0.49 thf(fact_3458_Image__empty__trancl__Image__empty,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V: A] :
% 0.25/0.49 ( ( ( image @ A @ A @ R5 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.49 => ( ( image @ A @ A @ ( transitive_trancl @ A @ R5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.49 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Image_empty_trancl_Image_empty
% 0.25/0.49 thf(fact_3459_reachable__mono,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A ),X10: set @ A,X11: set @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R7 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ X10 @ X11 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ X10 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R7 ) @ X11 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % reachable_mono
% 0.25/0.49 thf(fact_3460_Image__subset__snd__image,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A4 @ B6 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % Image_subset_snd_image
% 0.25/0.49 thf(fact_3461_trancl__image__by__rtrancl,axiom,
% 0.25/0.49 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),Vi: set @ A] :
% 0.25/0.49 ( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ Vi ) @ Vi )
% 0.25/0.49 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ Vi ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_image_by_rtrancl
% 0.25/0.49 thf(fact_3462_weak__map__of__SomeI,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
% 0.25/0.49 => ? [X2: B] :
% 0.25/0.49 ( ( map_of @ A @ B @ L @ K )
% 0.25/0.49 = ( some @ B @ X2 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % weak_map_of_SomeI
% 0.25/0.49 thf(fact_3463_map__of__SomeD,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
% 0.25/0.49 ( ( ( map_of @ B @ A @ Xs @ K )
% 0.25/0.49 = ( some @ A @ Y ) )
% 0.25/0.49 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_SomeD
% 0.25/0.49 thf(fact_3464_Image__singleton,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
% 0.25/0.49 ( ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
% 0.25/0.49 = ( collect @ A
% 0.25/0.49 @ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B5 ) @ R ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Image_singleton
% 0.25/0.49 thf(fact_3465_option_Osimps_I15_J,axiom,
% 0.25/0.49 ! [A: $tType,X22: A] :
% 0.25/0.49 ( ( set_option @ A @ ( some @ A @ X22 ) )
% 0.25/0.49 = ( insert2 @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % option.simps(15)
% 0.25/0.49 thf(fact_3466_rtrancl__apply__insert,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,S2: set @ A] :
% 0.25/0.49 ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( insert2 @ A @ X @ S2 ) )
% 0.25/0.49 = ( insert2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( sup_sup @ ( set @ A ) @ S2 @ ( image @ A @ A @ R5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_apply_insert
% 0.25/0.49 thf(fact_3467_rtrancl__Image__in__Field,axiom,
% 0.25/0.49 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ V6 ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R5 ) @ V6 ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_Image_in_Field
% 0.25/0.49 thf(fact_3468_E__closed__restr__reach__cases,axiom,
% 0.25/0.49 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R5 ) @ R5 )
% 0.25/0.49 => ( ~ ( member @ A @ V @ R5 )
% 0.25/0.49 => ~ ( ~ ( member @ A @ U @ R5 )
% 0.25/0.49 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % E_closed_restr_reach_cases
% 0.25/0.49 thf(fact_3469_rel__restrict__tranclI,axiom,
% 0.25/0.49 ! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E4 ) )
% 0.25/0.49 => ( ~ ( member @ A @ X @ R5 )
% 0.25/0.49 => ( ~ ( member @ A @ Y @ R5 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R5 ) @ R5 )
% 0.25/0.49 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rel_restrict_tranclI
% 0.25/0.49 thf(fact_3470_refl__on__reflcl__Image,axiom,
% 0.25/0.49 ! [A: $tType,B6: set @ A,A4: set @ ( product_prod @ A @ A ),C5: set @ A] :
% 0.25/0.49 ( ( refl_on @ A @ B6 @ A4 )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ B6 )
% 0.25/0.49 => ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) @ C5 )
% 0.25/0.49 = ( image @ A @ A @ A4 @ C5 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % refl_on_reflcl_Image
% 0.25/0.49 thf(fact_3471_map__of__eq__None__iff,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X: B] :
% 0.25/0.49 ( ( ( map_of @ B @ A @ Xys @ X )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 = ( ~ ( member @ B @ X @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_eq_None_iff
% 0.25/0.49 thf(fact_3472_finite__reachable__advance,axiom,
% 0.25/0.49 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),V0: A,V: A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
% 0.25/0.49 => ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_reachable_advance
% 0.25/0.49 thf(fact_3473_rtrancl__Image__advance__ss,axiom,
% 0.25/0.49 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % rtrancl_Image_advance_ss
% 0.25/0.49 thf(fact_3474_trancl__Image__advance__ss,axiom,
% 0.25/0.49 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
% 0.25/0.49 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
% 0.25/0.49 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % trancl_Image_advance_ss
% 0.25/0.49 thf(fact_3475_subset__Image1__Image1__iff,axiom,
% 0.25/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.49 ( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.49 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.49 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.49 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % subset_Image1_Image1_iff
% 0.25/0.49 thf(fact_3476_map__of__distinct__upd,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
% 0.25/0.49 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 0.25/0.49 => ( ( map_add @ A @ B
% 0.25/0.49 @ ( fun_upd @ A @ ( option @ B )
% 0.25/0.49 @ ^ [X3: A] : ( none @ B )
% 0.25/0.49 @ X
% 0.25/0.49 @ ( some @ B @ Y ) )
% 0.25/0.49 @ ( map_of @ A @ B @ Xs ) )
% 0.25/0.49 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_distinct_upd
% 0.25/0.49 thf(fact_3477_subset__Image__Image__iff,axiom,
% 0.25/0.49 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 0.25/0.49 ( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ A4 ) @ ( image @ A @ A @ R @ B6 ) )
% 0.25/0.49 = ( ! [X3: A] :
% 0.25/0.49 ( ( member @ A @ X3 @ A4 )
% 0.25/0.49 => ? [Y3: A] :
% 0.25/0.49 ( ( member @ A @ Y3 @ B6 )
% 0.25/0.49 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R ) ) ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % subset_Image_Image_iff
% 0.25/0.49 thf(fact_3478_map__of__map__restrict,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,Ks: list @ A] :
% 0.25/0.49 ( ( map_of @ A @ B
% 0.25/0.49 @ ( map @ A @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( F @ K4 ) )
% 0.25/0.49 @ Ks ) )
% 0.25/0.49 = ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F ) @ ( set2 @ A @ Ks ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_map_restrict
% 0.25/0.49 thf(fact_3479_map__of__mapk__SomeI,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,C: $tType,F: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X: C] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.49 => ( ( ( map_of @ A @ C @ T2 @ K )
% 0.25/0.49 = ( some @ C @ X ) )
% 0.25/0.49 => ( ( map_of @ B @ C
% 0.25/0.49 @ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
% 0.25/0.49 @ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
% 0.25/0.49 @ ^ [K4: A] : ( product_Pair @ B @ C @ ( F @ K4 ) ) )
% 0.25/0.49 @ T2 )
% 0.25/0.49 @ ( F @ K ) )
% 0.25/0.49 = ( some @ C @ X ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_mapk_SomeI
% 0.25/0.49 thf(fact_3480_map__comp__map,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
% 0.25/0.49 ( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
% 0.25/0.49 = ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_comp_map
% 0.25/0.49 thf(fact_3481_map__map,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs: list @ C] :
% 0.25/0.49 ( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs ) )
% 0.25/0.49 = ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_map
% 0.25/0.49 thf(fact_3482_List_Omap_Ocompositionality,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
% 0.25/0.49 ( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
% 0.25/0.49 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% 0.25/0.49
% 0.25/0.49 % List.map.compositionality
% 0.25/0.49 thf(fact_3483_list_Omap__comp,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
% 0.25/0.49 ( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
% 0.25/0.49 = ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.25/0.49
% 0.25/0.49 % list.map_comp
% 0.25/0.49 thf(fact_3484_List_Omap_Ocomp,axiom,
% 0.25/0.49 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 0.25/0.49 ( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
% 0.25/0.49 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % List.map.comp
% 0.25/0.49 thf(fact_3485_map__fst__mk__snd,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,K: B,L: list @ A] :
% 0.25/0.49 ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 0.25/0.49 @ ( map @ A @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
% 0.25/0.49 @ L ) )
% 0.25/0.49 = L ) ).
% 0.25/0.49
% 0.25/0.49 % map_fst_mk_snd
% 0.25/0.49 thf(fact_3486_map__snd__mk__fst,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,K: B,L: list @ A] :
% 0.25/0.49 ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
% 0.25/0.49 = L ) ).
% 0.25/0.49
% 0.25/0.49 % map_snd_mk_fst
% 0.25/0.49 thf(fact_3487_set__oo__map__alt,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.49 ( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F ) )
% 0.25/0.49 = ( ^ [L3: list @ A] : ( image2 @ A @ B @ F @ ( set2 @ A @ L3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_oo_map_alt
% 0.25/0.49 thf(fact_3488_Id__on__set,axiom,
% 0.25/0.49 ! [A: $tType,Xs: list @ A] :
% 0.25/0.49 ( ( id_on @ A @ ( set2 @ A @ Xs ) )
% 0.25/0.49 = ( set2 @ ( product_prod @ A @ A )
% 0.25/0.49 @ ( map @ A @ ( product_prod @ A @ A )
% 0.25/0.49 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 0.25/0.49 @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Id_on_set
% 0.25/0.49 thf(fact_3489_map__of__map__keys,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
% 0.25/0.49 ( ( ( set2 @ A @ Xs )
% 0.25/0.49 = ( dom @ A @ B @ M2 ) )
% 0.25/0.49 => ( ( map_of @ A @ B
% 0.25/0.49 @ ( map @ A @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( the2 @ B @ ( M2 @ K4 ) ) )
% 0.25/0.49 @ Xs ) )
% 0.25/0.49 = M2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_map_keys
% 0.25/0.49 thf(fact_3490_map__of__map,axiom,
% 0.25/0.49 ! [B: $tType,C: $tType,A: $tType,F: C > B,Xs: list @ ( product_prod @ A @ C )] :
% 0.25/0.49 ( ( map_of @ A @ B
% 0.25/0.49 @ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
% 0.25/0.49 @ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
% 0.25/0.49 @ ^ [K4: A,V2: C] : ( product_Pair @ A @ B @ K4 @ ( F @ V2 ) ) )
% 0.25/0.49 @ Xs ) )
% 0.25/0.49 = ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_map
% 0.25/0.49 thf(fact_3491_inj__on__map__inv__f,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,L: list @ A,A4: set @ A,F: A > B] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A4 )
% 0.25/0.49 => ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( map @ B @ A @ ( inv_on @ A @ B @ F @ A4 ) @ ( map @ A @ B @ F @ L ) )
% 0.25/0.49 = L ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inj_on_map_inv_f
% 0.25/0.49 thf(fact_3492_Misc_Oran__distinct,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
% 0.25/0.49 => ( ( ran @ A @ B @ ( map_of @ A @ B @ Al ) )
% 0.25/0.49 = ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Misc.ran_distinct
% 0.25/0.49 thf(fact_3493_map__of__eq__Some__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
% 0.25/0.49 => ( ( ( map_of @ A @ B @ Xys @ X )
% 0.25/0.49 = ( some @ B @ Y ) )
% 0.25/0.49 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_eq_Some_iff
% 0.25/0.49 thf(fact_3494_Some__eq__map__of__iff,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
% 0.25/0.49 => ( ( ( some @ B @ Y )
% 0.25/0.49 = ( map_of @ A @ B @ Xys @ X ) )
% 0.25/0.49 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % Some_eq_map_of_iff
% 0.25/0.49 thf(fact_3495_map__of__is__SomeI,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
% 0.25/0.49 => ( ( map_of @ A @ B @ Xys @ X )
% 0.25/0.49 = ( some @ B @ Y ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_is_SomeI
% 0.25/0.49 thf(fact_3496_inv__on__f__f,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A] :
% 0.25/0.49 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.49 => ( ( member @ A @ X @ A4 )
% 0.25/0.49 => ( ( inv_on @ A @ B @ F @ A4 @ ( F @ X ) )
% 0.25/0.49 = X ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % inv_on_f_f
% 0.25/0.49 thf(fact_3497_distinct__mapI,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > A,L: list @ B] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
% 0.25/0.49 => ( distinct @ B @ L ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_mapI
% 0.25/0.49 thf(fact_3498_distinct__finite__set,axiom,
% 0.25/0.49 ! [A: $tType,X: set @ A] :
% 0.25/0.49 ( finite_finite2 @ ( list @ A )
% 0.25/0.49 @ ( collect @ ( list @ A )
% 0.25/0.49 @ ^ [Ys: list @ A] :
% 0.25/0.49 ( ( ( set2 @ A @ Ys )
% 0.25/0.49 = X )
% 0.25/0.49 & ( distinct @ A @ Ys ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_finite_set
% 0.25/0.49 thf(fact_3499_distinct__map__eq,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > A,L: list @ B,X: B,Y: B] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
% 0.25/0.49 => ( ( ( F @ X )
% 0.25/0.49 = ( F @ Y ) )
% 0.25/0.49 => ( ( member @ B @ X @ ( set2 @ B @ L ) )
% 0.25/0.49 => ( ( member @ B @ Y @ ( set2 @ B @ L ) )
% 0.25/0.49 => ( X = Y ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_map_eq
% 0.25/0.49 thf(fact_3500_finite__set__image,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ ( list @ A )] :
% 0.25/0.49 ( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A4 ) )
% 0.25/0.49 => ( ! [Xs2: list @ A] :
% 0.25/0.49 ( ( member @ ( list @ A ) @ Xs2 @ A4 )
% 0.25/0.49 => ( distinct @ A @ Xs2 ) )
% 0.25/0.49 => ( finite_finite2 @ ( list @ A ) @ A4 ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % finite_set_image
% 0.25/0.49 thf(fact_3501_inv__on__f__range,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Y: A,F: B > A,A4: set @ B] :
% 0.25/0.49 ( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A4 ) )
% 0.25/0.49 => ( member @ B @ ( inv_on @ B @ A @ F @ A4 @ Y ) @ A4 ) ) ).
% 0.25/0.49
% 0.25/0.49 % inv_on_f_range
% 0.25/0.49 thf(fact_3502_f__inv__on__f,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Y: A,F: B > A,A4: set @ B] :
% 0.25/0.49 ( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A4 ) )
% 0.25/0.49 => ( ( F @ ( inv_on @ B @ A @ F @ A4 @ Y ) )
% 0.25/0.49 = Y ) ) ).
% 0.25/0.49
% 0.25/0.49 % f_inv_on_f
% 0.25/0.49 thf(fact_3503_distinct__finite__subset,axiom,
% 0.25/0.49 ! [A: $tType,X: set @ A] :
% 0.25/0.49 ( ( finite_finite2 @ A @ X )
% 0.25/0.49 => ( finite_finite2 @ ( list @ A )
% 0.25/0.49 @ ( collect @ ( list @ A )
% 0.25/0.49 @ ^ [Ys: list @ A] :
% 0.25/0.49 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys ) @ X )
% 0.25/0.49 & ( distinct @ A @ Ys ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_finite_subset
% 0.25/0.49 thf(fact_3504_distinct__map__fstD,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( Y = Z2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_map_fstD
% 0.25/0.49 thf(fact_3505_eq__key__imp__eq__value,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( V1 = V22 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % eq_key_imp_eq_value
% 0.25/0.49 thf(fact_3506_map__to__set__map__of,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
% 0.25/0.49 => ( ( map_to_set @ A @ B @ ( map_of @ A @ B @ L ) )
% 0.25/0.49 = ( set2 @ ( product_prod @ A @ B ) @ L ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_to_set_map_of
% 0.25/0.49 thf(fact_3507_map__of__map__to__set,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M2: A > ( option @ B )] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
% 0.25/0.49 => ( ( ( map_of @ A @ B @ L )
% 0.25/0.49 = M2 )
% 0.25/0.49 = ( ( set2 @ ( product_prod @ A @ B ) @ L )
% 0.25/0.49 = ( map_to_set @ A @ B @ M2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_map_to_set
% 0.25/0.49 thf(fact_3508_set__map__of__compr,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
% 0.25/0.49 = ( collect @ ( product_prod @ A @ B )
% 0.25/0.49 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.49 @ ^ [K4: A,V2: B] :
% 0.25/0.49 ( ( map_of @ A @ B @ Xs @ K4 )
% 0.25/0.49 = ( some @ B @ V2 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % set_map_of_compr
% 0.25/0.49 thf(fact_3509_card__lists__distinct__length__eq,axiom,
% 0.25/0.49 ! [A: $tType,A4: set @ A,K: nat] :
% 0.25/0.49 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.49 => ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
% 0.25/0.49 => ( ( finite_card @ ( list @ A )
% 0.25/0.49 @ ( collect @ ( list @ A )
% 0.25/0.49 @ ^ [Xs3: list @ A] :
% 0.25/0.49 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.25/0.49 = K )
% 0.25/0.49 & ( distinct @ A @ Xs3 )
% 0.25/0.49 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
% 0.25/0.49 = ( groups7121269368397514597t_prod @ nat @ nat
% 0.25/0.49 @ ^ [X3: nat] : X3
% 0.25/0.49 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % card_lists_distinct_length_eq
% 0.25/0.49 thf(fact_3510_card__lists__distinct__length__eq_H,axiom,
% 0.25/0.49 ! [A: $tType,K: nat,A4: set @ A] :
% 0.25/0.49 ( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
% 0.25/0.49 => ( ( finite_card @ ( list @ A )
% 0.25/0.49 @ ( collect @ ( list @ A )
% 0.25/0.49 @ ^ [Xs3: list @ A] :
% 0.25/0.49 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.25/0.49 = K )
% 0.25/0.49 & ( distinct @ A @ Xs3 )
% 0.25/0.49 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
% 0.25/0.49 = ( groups7121269368397514597t_prod @ nat @ nat
% 0.25/0.49 @ ^ [X3: nat] : X3
% 0.25/0.49 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % card_lists_distinct_length_eq'
% 0.25/0.49 thf(fact_3511_distinct__foldl__invar,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,S2: list @ A,I4: ( set @ A ) > B > $o,Sigma_0: B,F: B > A > B] :
% 0.25/0.49 ( ( distinct @ A @ S2 )
% 0.25/0.49 => ( ( I4 @ ( set2 @ A @ S2 ) @ Sigma_0 )
% 0.25/0.49 => ( ! [X2: A,It: set @ A,Sigma: B] :
% 0.25/0.49 ( ( member @ A @ X2 @ It )
% 0.25/0.49 => ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S2 ) )
% 0.25/0.49 => ( ( I4 @ It @ Sigma )
% 0.25/0.49 => ( I4 @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F @ Sigma @ X2 ) ) ) ) )
% 0.25/0.49 => ( I4 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F @ Sigma_0 @ S2 ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_foldl_invar
% 0.25/0.49 thf(fact_3512_map__of__Some__filter__not__in,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A,P: ( product_prod @ B @ A ) > $o] :
% 0.25/0.49 ( ( ( map_of @ B @ A @ Xs @ K )
% 0.25/0.49 = ( some @ A @ V ) )
% 0.25/0.49 => ( ~ ( P @ ( product_Pair @ B @ A @ K @ V ) )
% 0.25/0.49 => ( ( distinct @ B @ ( map @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ Xs ) )
% 0.25/0.49 => ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ P @ Xs ) @ K )
% 0.25/0.49 = ( none @ A ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_Some_filter_not_in
% 0.25/0.49 thf(fact_3513_map__add__map__of__foldr,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,M2: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
% 0.25/0.49 ( ( map_add @ A @ B @ M2 @ ( map_of @ A @ B @ Ps ) )
% 0.25/0.49 = ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
% 0.25/0.49 @ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
% 0.25/0.49 @ ^ [K4: A,V2: B,M: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M @ K4 @ ( some @ B @ V2 ) ) )
% 0.25/0.49 @ Ps
% 0.25/0.49 @ M2 ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_add_map_of_foldr
% 0.25/0.49 thf(fact_3514_foldl__length,axiom,
% 0.25/0.49 ! [A: $tType,L: list @ A] :
% 0.25/0.49 ( ( foldl @ nat @ A
% 0.25/0.49 @ ^ [I3: nat,X3: A] : ( suc @ I3 )
% 0.25/0.49 @ ( zero_zero @ nat )
% 0.25/0.49 @ L )
% 0.25/0.49 = ( size_size @ ( list @ A ) @ L ) ) ).
% 0.25/0.49
% 0.25/0.49 % foldl_length
% 0.25/0.49 thf(fact_3515_foldl__length__aux,axiom,
% 0.25/0.49 ! [A: $tType,A3: nat,L: list @ A] :
% 0.25/0.49 ( ( foldl @ nat @ A
% 0.25/0.49 @ ^ [I3: nat,X3: A] : ( suc @ I3 )
% 0.25/0.49 @ A3
% 0.25/0.49 @ L )
% 0.25/0.49 = ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % foldl_length_aux
% 0.25/0.49 thf(fact_3516_foldl__A1__eq,axiom,
% 0.25/0.49 ! [A: $tType,F: A > A > A,N: A,I: A,Ww: list @ A] :
% 0.25/0.49 ( ! [E3: A] :
% 0.25/0.49 ( ( F @ N @ E3 )
% 0.25/0.49 = E3 )
% 0.25/0.49 => ( ! [E3: A] :
% 0.25/0.49 ( ( F @ E3 @ N )
% 0.25/0.49 = E3 )
% 0.25/0.49 => ( ! [A5: A,B3: A,C4: A] :
% 0.25/0.49 ( ( F @ A5 @ ( F @ B3 @ C4 ) )
% 0.25/0.49 = ( F @ ( F @ A5 @ B3 ) @ C4 ) )
% 0.25/0.49 => ( ( foldl @ A @ A @ F @ I @ Ww )
% 0.25/0.49 = ( F @ I @ ( foldl @ A @ A @ F @ N @ Ww ) ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % foldl_A1_eq
% 0.25/0.49 thf(fact_3517_sum__length__filter__compl,axiom,
% 0.25/0.49 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 0.25/0.49 ( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) )
% 0.25/0.49 @ ( size_size @ ( list @ A )
% 0.25/0.49 @ ( filter @ A
% 0.25/0.49 @ ^ [X3: A] :
% 0.25/0.49 ~ ( P @ X3 )
% 0.25/0.49 @ Xs ) ) )
% 0.25/0.49 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 0.25/0.49
% 0.25/0.49 % sum_length_filter_compl
% 0.25/0.49 thf(fact_3518_comp__fun__commute_Ofoldr__conv__foldl,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B > B,Xs: list @ A,A3: B] :
% 0.25/0.49 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 0.25/0.49 => ( ( foldr @ A @ B @ F @ Xs @ A3 )
% 0.25/0.49 = ( foldl @ B @ A
% 0.25/0.49 @ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
% 0.25/0.49 @ A3
% 0.25/0.49 @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % comp_fun_commute.foldr_conv_foldl
% 0.25/0.49 thf(fact_3519_comp__fun__commute_Ofoldl__f__commute,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,F: A > B > B,A3: A,B2: B,Xs: list @ A] :
% 0.25/0.49 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 0.25/0.49 => ( ( F @ A3
% 0.25/0.49 @ ( foldl @ B @ A
% 0.25/0.49 @ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
% 0.25/0.49 @ B2
% 0.25/0.49 @ Xs ) )
% 0.25/0.49 = ( foldl @ B @ A
% 0.25/0.49 @ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
% 0.25/0.49 @ ( F @ A3 @ B2 )
% 0.25/0.49 @ Xs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % comp_fun_commute.foldl_f_commute
% 0.25/0.49 thf(fact_3520_map__of__None__filterD,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X: B,P: ( product_prod @ B @ A ) > $o] :
% 0.25/0.49 ( ( ( map_of @ B @ A @ Xs @ X )
% 0.25/0.49 = ( none @ A ) )
% 0.25/0.49 => ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ P @ Xs ) @ X )
% 0.25/0.49 = ( none @ A ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_None_filterD
% 0.25/0.49 thf(fact_3521_fst__foldl,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,C: $tType,F: A > C > A,G: A > B > C > B,A3: A,B2: B,Xs: list @ C] :
% 0.25/0.49 ( ( product_fst @ A @ B
% 0.25/0.49 @ ( foldl @ ( product_prod @ A @ B ) @ C
% 0.25/0.49 @ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
% 0.25/0.49 @ ^ [A7: A,B5: B,X3: C] : ( product_Pair @ A @ B @ ( F @ A7 @ X3 ) @ ( G @ A7 @ B5 @ X3 ) ) )
% 0.25/0.49 @ ( product_Pair @ A @ B @ A3 @ B2 )
% 0.25/0.49 @ Xs ) )
% 0.25/0.49 = ( foldl @ A @ C @ F @ A3 @ Xs ) ) ).
% 0.25/0.49
% 0.25/0.49 % fst_foldl
% 0.25/0.49 thf(fact_3522_foldl__absorb1,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( monoid_mult @ A )
% 0.25/0.49 => ! [X: A,Zs: list @ A] :
% 0.25/0.49 ( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
% 0.25/0.49 = ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % foldl_absorb1
% 0.25/0.49 thf(fact_3523_foldl__un__empty__eq,axiom,
% 0.25/0.49 ! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
% 0.25/0.49 ( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
% 0.25/0.49 = ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % foldl_un_empty_eq
% 0.25/0.49 thf(fact_3524_distinct__map__fst__filterI,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),P: ( product_prod @ A @ B ) > $o] :
% 0.25/0.49 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 0.25/0.49 => ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( filter @ ( product_prod @ A @ B ) @ P @ Xs ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_map_fst_filterI
% 0.25/0.49 thf(fact_3525_map__of__filter__in,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z2: A,P: B > A > $o] :
% 0.25/0.49 ( ( ( map_of @ B @ A @ Xs @ K )
% 0.25/0.49 = ( some @ A @ Z2 ) )
% 0.25/0.49 => ( ( P @ K @ Z2 )
% 0.25/0.49 => ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs ) @ K )
% 0.25/0.49 = ( some @ A @ Z2 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_of_filter_in
% 0.25/0.49 thf(fact_3526_distinct__length__le,axiom,
% 0.25/0.49 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 0.25/0.49 ( ( distinct @ A @ Ys2 )
% 0.25/0.49 => ( ( ( set2 @ A @ Ys2 )
% 0.25/0.49 = ( set2 @ A @ Xs ) )
% 0.25/0.49 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % distinct_length_le
% 0.25/0.49 thf(fact_3527_foldl__set,axiom,
% 0.25/0.49 ! [A: $tType,L: list @ ( set @ A )] :
% 0.25/0.49 ( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
% 0.25/0.49 = ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.49 @ ( collect @ ( set @ A )
% 0.25/0.49 @ ^ [X3: set @ A] : ( member @ ( set @ A ) @ X3 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % foldl_set
% 0.25/0.49 thf(fact_3528_map__filter__map__filter,axiom,
% 0.25/0.49 ! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs: list @ B] :
% 0.25/0.49 ( ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) )
% 0.25/0.49 = ( map_filter @ B @ A
% 0.25/0.49 @ ^ [X3: B] : ( if @ ( option @ A ) @ ( P @ X3 ) @ ( some @ A @ ( F @ X3 ) ) @ ( none @ A ) )
% 0.25/0.49 @ Xs ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_filter_map_filter
% 0.25/0.49 thf(fact_3529_map__filter__def,axiom,
% 0.25/0.49 ! [B: $tType,A: $tType] :
% 0.25/0.49 ( ( map_filter @ A @ B )
% 0.25/0.49 = ( ^ [F2: A > ( option @ B ),Xs3: list @ A] :
% 0.25/0.49 ( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F2 )
% 0.25/0.49 @ ( filter @ A
% 0.25/0.49 @ ^ [X3: A] :
% 0.25/0.49 ( ( F2 @ X3 )
% 0.25/0.49 != ( none @ B ) )
% 0.25/0.49 @ Xs3 ) ) ) ) ).
% 0.25/0.49
% 0.25/0.49 % map_filter_def
% 0.25/0.49 thf(fact_3530_execute__of__list,axiom,
% 0.25/0.49 ! [A: $tType] :
% 0.25/0.49 ( ( heap @ A )
% 0.25/0.49 => ! [Xs: list @ A,H3: heap_ext @ product_unit] :
% 0.25/0.49 ( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 )
% 0.25/0.49 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
% 0.25/0.50 @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % execute_of_list
% 0.25/0.50 thf(fact_3531_mergesort__by__rel__split__length,axiom,
% 0.25/0.50 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
% 0.25/0.50 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs1 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( modulo_modulo @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 0.25/0.50 & ( ( size_size @ ( list @ A ) @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
% 0.25/0.50 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs22 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split_length
% 0.25/0.50 thf(fact_3532_card__disjoint__shuffles,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
% 0.25/0.50 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.50 => ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.25/0.50 = ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % card_disjoint_shuffles
% 0.25/0.50 thf(fact_3533_of__list__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ( ( array_of_list @ A )
% 0.25/0.50 = ( ^ [Xs3: list @ A] :
% 0.25/0.50 ( heap_Time_heap @ ( array @ A )
% 0.25/0.50 @ ^ [H: heap_ext @ product_unit] :
% 0.25/0.50 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
% 0.25/0.50 @ ( array_alloc @ A @ Xs3 @ H ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % of_list_def
% 0.25/0.50 thf(fact_3534_slice__len,axiom,
% 0.25/0.50 ! [A: $tType,From: nat,To: nat,Xs: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ From @ To )
% 0.25/0.50 => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs ) )
% 0.25/0.50 = ( minus_minus @ nat @ To @ From ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % slice_len
% 0.25/0.50 thf(fact_3535_slice__complete,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A] :
% 0.25/0.50 ( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
% 0.25/0.50 = Xs ) ).
% 0.25/0.50
% 0.25/0.50 % slice_complete
% 0.25/0.50 thf(fact_3536_foldr__length,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( foldr @ A @ nat
% 0.25/0.50 @ ^ [X3: A] : suc
% 0.25/0.50 @ L
% 0.25/0.50 @ ( zero_zero @ nat ) )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ L ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldr_length
% 0.25/0.50 thf(fact_3537_length__shuffles,axiom,
% 0.25/0.50 ! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.25/0.50 => ( ( size_size @ ( list @ A ) @ Zs )
% 0.25/0.50 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_shuffles
% 0.25/0.50 thf(fact_3538_foldr__length__aux,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A,A3: nat] :
% 0.25/0.50 ( ( foldr @ A @ nat
% 0.25/0.50 @ ^ [X3: A] : suc
% 0.25/0.50 @ L
% 0.25/0.50 @ A3 )
% 0.25/0.50 = ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldr_length_aux
% 0.25/0.50 thf(fact_3539_effect__of__listI,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ! [A3: array @ A,H4: heap_ext @ product_unit,Xs: list @ A,H3: heap_ext @ product_unit] :
% 0.25/0.50 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
% 0.25/0.50 = ( array_alloc @ A @ Xs @ H3 ) )
% 0.25/0.50 => ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % effect_of_listI
% 0.25/0.50 thf(fact_3540_execute__make,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ! [N: nat,F: nat > A,H3: heap_ext @ product_unit] :
% 0.25/0.50 ( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 )
% 0.25/0.50 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 0.25/0.50 @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % execute_make
% 0.25/0.50 thf(fact_3541_execute__new,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ! [N: nat,X: A,H3: heap_ext @ product_unit] :
% 0.25/0.50 ( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 )
% 0.25/0.50 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 0.25/0.50 @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % execute_new
% 0.25/0.50 thf(fact_3542_Array__Time_Omake__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ( ( array_make @ A )
% 0.25/0.50 = ( ^ [N5: nat,F2: nat > A] :
% 0.25/0.50 ( heap_Time_heap @ ( array @ A )
% 0.25/0.50 @ ^ [H: heap_ext @ product_unit] :
% 0.25/0.50 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) )
% 0.25/0.50 @ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Array_Time.make_def
% 0.25/0.50 thf(fact_3543_new__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ( ( array_new @ A )
% 0.25/0.50 = ( ^ [N5: nat,X3: A] :
% 0.25/0.50 ( heap_Time_heap @ ( array @ A )
% 0.25/0.50 @ ^ [H: heap_ext @ product_unit] :
% 0.25/0.50 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.50 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) )
% 0.25/0.50 @ ( array_alloc @ A @ ( replicate @ A @ N5 @ X3 ) @ H ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % new_def
% 0.25/0.50 thf(fact_3544_upt__0__eq__Nil__conv,axiom,
% 0.25/0.50 ! [J: nat] :
% 0.25/0.50 ( ( ( upt @ ( zero_zero @ nat ) @ J )
% 0.25/0.50 = ( nil @ nat ) )
% 0.25/0.50 = ( J
% 0.25/0.50 = ( zero_zero @ nat ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_0_eq_Nil_conv
% 0.25/0.50 thf(fact_3545_map__of__eq__empty__iff,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( ( map_of @ A @ B @ Xys )
% 0.25/0.50 = ( ^ [X3: A] : ( none @ B ) ) )
% 0.25/0.50 = ( Xys
% 0.25/0.50 = ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_eq_empty_iff
% 0.25/0.50 thf(fact_3546_slice__Nil,axiom,
% 0.25/0.50 ! [A: $tType,Begin: nat,End: nat] :
% 0.25/0.50 ( ( slice @ A @ Begin @ End @ ( nil @ A ) )
% 0.25/0.50 = ( nil @ A ) ) ).
% 0.25/0.50
% 0.25/0.50 % slice_Nil
% 0.25/0.50 thf(fact_3547_slice__eq__bounds__empty,axiom,
% 0.25/0.50 ! [A: $tType,I: nat,Xs: list @ A] :
% 0.25/0.50 ( ( slice @ A @ I @ I @ Xs )
% 0.25/0.50 = ( nil @ A ) ) ).
% 0.25/0.50
% 0.25/0.50 % slice_eq_bounds_empty
% 0.25/0.50 thf(fact_3548_empty__eq__map__of__iff,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( ( ^ [X3: A] : ( none @ B ) )
% 0.25/0.50 = ( map_of @ A @ B @ Xys ) )
% 0.25/0.50 = ( Xys
% 0.25/0.50 = ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % empty_eq_map_of_iff
% 0.25/0.50 thf(fact_3549_length__ge__1__conv,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 = ( L
% 0.25/0.50 != ( nil @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_ge_1_conv
% 0.25/0.50 thf(fact_3550_map__fst__mk__fst,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,K: A,L: list @ B] :
% 0.25/0.50 ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
% 0.25/0.50 = ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_fst_mk_fst
% 0.25/0.50 thf(fact_3551_map__snd__mk__snd,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,K: A,L: list @ B] :
% 0.25/0.50 ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 0.25/0.50 @ ( map @ B @ ( product_prod @ B @ A )
% 0.25/0.50 @ ^ [X3: B] : ( product_Pair @ B @ A @ X3 @ K )
% 0.25/0.50 @ L ) )
% 0.25/0.50 = ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_snd_mk_snd
% 0.25/0.50 thf(fact_3552_map__add__upt_H,axiom,
% 0.25/0.50 ! [Ofs: nat,A3: nat,B2: nat] :
% 0.25/0.50 ( ( map @ nat @ nat
% 0.25/0.50 @ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
% 0.25/0.50 @ ( upt @ A3 @ B2 ) )
% 0.25/0.50 = ( upt @ ( plus_plus @ nat @ A3 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_add_upt'
% 0.25/0.50 thf(fact_3553_len__greater__imp__nonempty,axiom,
% 0.25/0.50 ! [A: $tType,X: nat,L: list @ A] :
% 0.25/0.50 ( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( L
% 0.25/0.50 != ( nil @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % len_greater_imp_nonempty
% 0.25/0.50 thf(fact_3554_map__add__upt,axiom,
% 0.25/0.50 ! [N: nat,M2: nat] :
% 0.25/0.50 ( ( map @ nat @ nat
% 0.25/0.50 @ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N )
% 0.25/0.50 @ ( upt @ ( zero_zero @ nat ) @ M2 ) )
% 0.25/0.50 = ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_add_upt
% 0.25/0.50 thf(fact_3555_map__of__Cons__code_I1_J,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,K: B] :
% 0.25/0.50 ( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K )
% 0.25/0.50 = ( none @ A ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_Cons_code(1)
% 0.25/0.50 thf(fact_3556_map__of_Osimps_I1_J,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
% 0.25/0.50 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of.simps(1)
% 0.25/0.50 thf(fact_3557_mergesort__by__rel__split_Osimps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,Xs1: list @ A,Xs22: list @ A] :
% 0.25/0.50 ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) )
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split.simps(1)
% 0.25/0.50 thf(fact_3558_upt__filter__extend,axiom,
% 0.25/0.50 ! [U: nat,U5: nat,P: nat > $o] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ U @ U5 )
% 0.25/0.50 => ( ! [I2: nat] :
% 0.25/0.50 ( ( ( ord_less_eq @ nat @ U @ I2 )
% 0.25/0.50 & ( ord_less @ nat @ I2 @ U5 ) )
% 0.25/0.50 => ~ ( P @ I2 ) )
% 0.25/0.50 => ( ( filter @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
% 0.25/0.50 = ( filter @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_filter_extend
% 0.25/0.50 thf(fact_3559_effect__newI,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ! [A3: array @ A,H4: heap_ext @ product_unit,N: nat,X: A,H3: heap_ext @ product_unit] :
% 0.25/0.50 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
% 0.25/0.50 = ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) )
% 0.25/0.50 => ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % effect_newI
% 0.25/0.50 thf(fact_3560_effect__makeI,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ! [A3: array @ A,H4: heap_ext @ product_unit,F: nat > A,N: nat,H3: heap_ext @ product_unit] :
% 0.25/0.50 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
% 0.25/0.50 = ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) )
% 0.25/0.50 => ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % effect_makeI
% 0.25/0.50 thf(fact_3561_mergesort__by__rel_Opinduct,axiom,
% 0.25/0.50 ! [A: $tType,A0: A > A > $o,A1: list @ A,P: ( A > A > $o ) > ( list @ A ) > $o] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A1 ) )
% 0.25/0.50 => ( ! [R8: A > A > $o,Xs2: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs2 ) )
% 0.25/0.50 => ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( P @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
% 0.25/0.50 => ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( P @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
% 0.25/0.50 => ( P @ R8 @ Xs2 ) ) ) )
% 0.25/0.50 => ( P @ A0 @ A1 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel.pinduct
% 0.25/0.50 thf(fact_3562_mergesort__by__rel_Oelims,axiom,
% 0.25/0.50 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
% 0.25/0.50 ( ( ( mergesort_by_rel @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( Y = Xa ) )
% 0.25/0.50 & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel.elims
% 0.25/0.50 thf(fact_3563_mergesort__by__rel_Osimps,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( mergesort_by_rel @ A )
% 0.25/0.50 = ( ^ [R6: A > A > $o,Xs3: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs3 @ ( merges9089515139780605204_merge @ A @ R6 @ ( mergesort_by_rel @ A @ R6 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel.simps
% 0.25/0.50 thf(fact_3564_enumerate__replicate__eq,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,M2: nat,A3: A] :
% 0.25/0.50 ( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A3 ) )
% 0.25/0.50 = ( map @ nat @ ( product_prod @ nat @ A )
% 0.25/0.50 @ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
% 0.25/0.50 @ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % enumerate_replicate_eq
% 0.25/0.50 thf(fact_3565_nth__step__trancl,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,R5: set @ ( product_prod @ A @ A ),N: nat,M2: nat] :
% 0.25/0.50 ( ! [N3: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ N3 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.25/0.50 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N3 ) ) @ ( nth @ A @ Xs @ N3 ) ) @ R5 ) )
% 0.25/0.50 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( ord_less @ nat @ M2 @ N )
% 0.25/0.50 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M2 ) ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_step_trancl
% 0.25/0.50 thf(fact_3566_mergesort__by__rel__simps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o] :
% 0.25/0.50 ( ( mergesort_by_rel @ A @ R5 @ ( nil @ A ) )
% 0.25/0.50 = ( nil @ A ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_simps(1)
% 0.25/0.50 thf(fact_3567_set__mergesort__by__rel,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 0.25/0.50 ( ( set2 @ A @ ( mergesort_by_rel @ A @ R5 @ Xs ) )
% 0.25/0.50 = ( set2 @ A @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_mergesort_by_rel
% 0.25/0.50 thf(fact_3568_mergesort__by__rel__merge__simps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,Ys2: list @ A] :
% 0.25/0.50 ( ( merges9089515139780605204_merge @ A @ R5 @ ( nil @ A ) @ Ys2 )
% 0.25/0.50 = Ys2 ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge_simps(3)
% 0.25/0.50 thf(fact_3569_nth__upt,axiom,
% 0.25/0.50 ! [I: nat,K: nat,J: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
% 0.25/0.50 => ( ( nth @ nat @ ( upt @ I @ J ) @ K )
% 0.25/0.50 = ( plus_plus @ nat @ I @ K ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_upt
% 0.25/0.50 thf(fact_3570_set__mergesort__by__rel__merge,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
% 0.25/0.50 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_mergesort_by_rel_merge
% 0.25/0.50 thf(fact_3571_map__fst__enumerate,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.25/0.50 ( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
% 0.25/0.50 = ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_fst_enumerate
% 0.25/0.50 thf(fact_3572_nth__image__indices,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( image2 @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
% 0.25/0.50 = ( set2 @ A @ L ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_image_indices
% 0.25/0.50 thf(fact_3573_ran__nth__set__encoding__conv,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( ran @ nat @ A
% 0.25/0.50 @ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I3 ) ) @ ( none @ A ) ) )
% 0.25/0.50 = ( set2 @ A @ L ) ) ).
% 0.25/0.50
% 0.25/0.50 % ran_nth_set_encoding_conv
% 0.25/0.50 thf(fact_3574_mergesort__by__rel_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
% 0.25/0.50 ~ ! [R8: A > A > $o,Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs2 ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel.cases
% 0.25/0.50 thf(fact_3575_map__eq__nth__eq,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: B > A,L: list @ B,L4: list @ B,I: nat] :
% 0.25/0.50 ( ( ( map @ B @ A @ F @ L )
% 0.25/0.50 = ( map @ B @ A @ F @ L4 ) )
% 0.25/0.50 => ( ( F @ ( nth @ B @ L @ I ) )
% 0.25/0.50 = ( F @ ( nth @ B @ L4 @ I ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_eq_nth_eq
% 0.25/0.50 thf(fact_3576_nth__enumerate__eq,axiom,
% 0.25/0.50 ! [A: $tType,M2: nat,Xs: list @ A,N: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M2 )
% 0.25/0.50 = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_enumerate_eq
% 0.25/0.50 thf(fact_3577_mergesort__by__rel__merge__simps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 0.25/0.50 ( ( merges9089515139780605204_merge @ A @ R5 @ Xs @ ( nil @ A ) )
% 0.25/0.50 = Xs ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge_simps(2)
% 0.25/0.50 thf(fact_3578_obtain__list__from__elements,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,P: A > nat > $o] :
% 0.25/0.50 ( ! [I2: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I2 @ N )
% 0.25/0.50 => ? [Li: A] : ( P @ Li @ I2 ) )
% 0.25/0.50 => ~ ! [L5: list @ A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ L5 )
% 0.25/0.50 = N )
% 0.25/0.50 => ~ ! [I6: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I6 @ N )
% 0.25/0.50 => ( P @ ( nth @ A @ L5 @ I6 ) @ I6 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % obtain_list_from_elements
% 0.25/0.50 thf(fact_3579_in__set__enumerate__eq,axiom,
% 0.25/0.50 ! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
% 0.25/0.50 = ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
% 0.25/0.50 & ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
% 0.25/0.50 & ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
% 0.25/0.50 = ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_enumerate_eq
% 0.25/0.50 thf(fact_3580_all__set__conv__nth,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A,P: A > $o] :
% 0.25/0.50 ( ( ! [X3: A] :
% 0.25/0.50 ( ( member @ A @ X3 @ ( set2 @ A @ L ) )
% 0.25/0.50 => ( P @ X3 ) ) )
% 0.25/0.50 = ( ! [I3: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( P @ ( nth @ A @ L @ I3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % all_set_conv_nth
% 0.25/0.50 thf(fact_3581_set__image__eq__pointwiseI,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F: A > B] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ L )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ L4 ) )
% 0.25/0.50 => ( ! [I2: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( F @ ( nth @ A @ L @ I2 ) )
% 0.25/0.50 = ( F @ ( nth @ A @ L4 @ I2 ) ) ) )
% 0.25/0.50 => ( ( image2 @ A @ B @ F @ ( set2 @ A @ L ) )
% 0.25/0.50 = ( image2 @ A @ B @ F @ ( set2 @ A @ L4 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_image_eq_pointwiseI
% 0.25/0.50 thf(fact_3582_in__set__image__conv__nth,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: B > A,X: B,L: list @ B] :
% 0.25/0.50 ( ( member @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ L ) ) )
% 0.25/0.50 = ( ? [I3: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L ) )
% 0.25/0.50 & ( ( F @ ( nth @ B @ L @ I3 ) )
% 0.25/0.50 = ( F @ X ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_image_conv_nth
% 0.25/0.50 thf(fact_3583_distinct__idx,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: B > A,L: list @ B,I: nat,J: nat] :
% 0.25/0.50 ( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
% 0.25/0.50 => ( ( ( F @ ( nth @ B @ L @ I ) )
% 0.25/0.50 = ( F @ ( nth @ B @ L @ J ) ) )
% 0.25/0.50 => ( I = J ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % distinct_idx
% 0.25/0.50 thf(fact_3584_array__of__list__make,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( heap @ A )
% 0.25/0.50 => ( ( array_of_list @ A )
% 0.25/0.50 = ( ^ [Xs3: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( nth @ A @ Xs3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % array_of_list_make
% 0.25/0.50 thf(fact_3585_nth__map__upt,axiom,
% 0.25/0.50 ! [A: $tType,I: nat,N: nat,M2: nat,F: nat > A] :
% 0.25/0.50 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
% 0.25/0.50 => ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M2 @ N ) ) @ I )
% 0.25/0.50 = ( F @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_map_upt
% 0.25/0.50 thf(fact_3586_mergesort__by__rel_Opsimps,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs ) )
% 0.25/0.50 => ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( ( mergesort_by_rel @ A @ R5 @ Xs )
% 0.25/0.50 = Xs ) )
% 0.25/0.50 & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( ( mergesort_by_rel @ A @ R5 @ Xs )
% 0.25/0.50 = ( merges9089515139780605204_merge @ A @ R5 @ ( mergesort_by_rel @ A @ R5 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R5 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel.psimps
% 0.25/0.50 thf(fact_3587_mergesort__by__rel_Opelims,axiom,
% 0.25/0.50 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
% 0.25/0.50 ( ( ( mergesort_by_rel @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.50 => ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( Y = Xa ) )
% 0.25/0.50 & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel.pelims
% 0.25/0.50 thf(fact_3588_enumerate__map__upt,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,F: nat > A,M2: nat] :
% 0.25/0.50 ( ( enumerate @ A @ N @ ( map @ nat @ A @ F @ ( upt @ N @ M2 ) ) )
% 0.25/0.50 = ( map @ nat @ ( product_prod @ nat @ A )
% 0.25/0.50 @ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F @ K4 ) )
% 0.25/0.50 @ ( upt @ N @ M2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % enumerate_map_upt
% 0.25/0.50 thf(fact_3589_map__upt__eqI,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,N: nat,M2: nat,F: nat > A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( minus_minus @ nat @ N @ M2 ) )
% 0.25/0.50 => ( ! [I2: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( nth @ A @ Xs @ I2 )
% 0.25/0.50 = ( F @ ( plus_plus @ nat @ M2 @ I2 ) ) ) )
% 0.25/0.50 => ( ( map @ nat @ A @ F @ ( upt @ M2 @ N ) )
% 0.25/0.50 = Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_upt_eqI
% 0.25/0.50 thf(fact_3590_slice__nth,axiom,
% 0.25/0.50 ! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ From @ To )
% 0.25/0.50 => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
% 0.25/0.50 => ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
% 0.25/0.50 = ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % slice_nth
% 0.25/0.50 thf(fact_3591_product__nth,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
% 0.25/0.50 => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
% 0.25/0.50 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) @ ( nth @ B @ Ys2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % product_nth
% 0.25/0.50 thf(fact_3592_mergesort__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( ord @ A )
% 0.25/0.50 => ( ( mergesort @ A )
% 0.25/0.50 = ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_def
% 0.25/0.50 thf(fact_3593_sorted__in__between,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [I: nat,J: nat,L: list @ A,X: A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ J )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.25/0.50 => ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X )
% 0.25/0.50 => ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
% 0.25/0.50 => ~ ! [K5: nat] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ I @ K5 )
% 0.25/0.50 => ( ( ord_less @ nat @ K5 @ J )
% 0.25/0.50 => ( ( ord_less_eq @ A @ ( nth @ A @ L @ K5 ) @ X )
% 0.25/0.50 => ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_in_between
% 0.25/0.50 thf(fact_3594_Un__set__drop__extend,axiom,
% 0.25/0.50 ! [A: $tType,J: nat,L: list @ ( set @ A )] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
% 0.25/0.50 => ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J @ L ) ) ) )
% 0.25/0.50 = ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Un_set_drop_extend
% 0.25/0.50 thf(fact_3595_mergesort__by__rel__simps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,X1: A,X22: A,Xs: list @ A] :
% 0.25/0.50 ( ( mergesort_by_rel @ A @ R5 @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
% 0.25/0.50 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 0.25/0.50 @ ^ [Xs12: list @ A,Xs23: list @ A] : ( merges9089515139780605204_merge @ A @ R5 @ ( mergesort_by_rel @ A @ R5 @ Xs12 ) @ ( mergesort_by_rel @ A @ R5 @ Xs23 ) )
% 0.25/0.50 @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X22 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_simps(3)
% 0.25/0.50 thf(fact_3596_drop__drop,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
% 0.25/0.50 ( ( drop @ A @ N @ ( drop @ A @ M2 @ Xs ) )
% 0.25/0.50 = ( drop @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % drop_drop
% 0.25/0.50 thf(fact_3597_sorted__wrt__mergesort__by__rel__merge,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ! [X2: A,Y4: A] :
% 0.25/0.50 ( ( R5 @ X2 @ Y4 )
% 0.25/0.50 | ( R5 @ Y4 @ X2 ) )
% 0.25/0.50 => ( ! [X2: A,Y4: A,Z3: A] :
% 0.25/0.50 ( ( R5 @ X2 @ Y4 )
% 0.25/0.50 => ( ( R5 @ Y4 @ Z3 )
% 0.25/0.50 => ( R5 @ X2 @ Z3 ) ) )
% 0.25/0.50 => ( ( sorted_wrt @ A @ R5 @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
% 0.25/0.50 = ( ( sorted_wrt @ A @ R5 @ Xs )
% 0.25/0.50 & ( sorted_wrt @ A @ R5 @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt_mergesort_by_rel_merge
% 0.25/0.50 thf(fact_3598_mergesort__by__rel__simps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,X: A] :
% 0.25/0.50 ( ( mergesort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.25/0.50 = ( cons @ A @ X @ ( nil @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_simps(2)
% 0.25/0.50 thf(fact_3599_enumerate__simps_I2_J,axiom,
% 0.25/0.50 ! [B: $tType,N: nat,X: B,Xs: list @ B] :
% 0.25/0.50 ( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
% 0.25/0.50 = ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % enumerate_simps(2)
% 0.25/0.50 thf(fact_3600_sorted__wrt__map__linord,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( sorted_wrt @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
% 0.25/0.50 @ L )
% 0.25/0.50 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt_map_linord
% 0.25/0.50 thf(fact_3601_nth__Cons__numeral,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,V: num] :
% 0.25/0.50 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
% 0.25/0.50 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_Cons_numeral
% 0.25/0.50 thf(fact_3602_drop__Cons__numeral,axiom,
% 0.25/0.50 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % drop_Cons_numeral
% 0.25/0.50 thf(fact_3603_nth__drop,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,Xs: list @ A,I: nat] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
% 0.25/0.50 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_drop
% 0.25/0.50 thf(fact_3604_nth__Cons__pos,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.25/0.50 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.25/0.50 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_Cons_pos
% 0.25/0.50 thf(fact_3605_Misc_Ofoldr__Cons,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A] :
% 0.25/0.50 ( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
% 0.25/0.50 = Xs ) ).
% 0.25/0.50
% 0.25/0.50 % Misc.foldr_Cons
% 0.25/0.50 thf(fact_3606_map__consI_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,W: list @ A,F: B > A,Ww: list @ B,A3: B] :
% 0.25/0.50 ( ( W
% 0.25/0.50 = ( map @ B @ A @ F @ Ww ) )
% 0.25/0.50 => ( ( cons @ A @ ( F @ A3 ) @ W )
% 0.25/0.50 = ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_consI(1)
% 0.25/0.50 thf(fact_3607_map__eq__consE,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
% 0.25/0.50 ( ( ( map @ B @ A @ F @ Ls )
% 0.25/0.50 = ( cons @ A @ Fa @ Fl ) )
% 0.25/0.50 => ~ ! [A5: B,L5: list @ B] :
% 0.25/0.50 ( ( Ls
% 0.25/0.50 = ( cons @ B @ A5 @ L5 ) )
% 0.25/0.50 => ( ( ( F @ A5 )
% 0.25/0.50 = Fa )
% 0.25/0.50 => ( ( map @ B @ A @ F @ L5 )
% 0.25/0.50 != Fl ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_eq_consE
% 0.25/0.50 thf(fact_3608_neq__NilE,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( L
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( L
% 0.25/0.50 != ( cons @ A @ X2 @ Xs2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % neq_NilE
% 0.25/0.50 thf(fact_3609_list__2pre__induct,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
% 0.25/0.50 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.25/0.50 => ( ! [E3: A,W12: list @ A,W23: list @ B] :
% 0.25/0.50 ( ( P @ W12 @ W23 )
% 0.25/0.50 => ( P @ ( cons @ A @ E3 @ W12 ) @ W23 ) )
% 0.25/0.50 => ( ! [E3: B,W13: list @ A,W24: list @ B] :
% 0.25/0.50 ( ( P @ W13 @ W24 )
% 0.25/0.50 => ( P @ W13 @ ( cons @ B @ E3 @ W24 ) ) )
% 0.25/0.50 => ( P @ W1 @ W22 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_2pre_induct
% 0.25/0.50 thf(fact_3610_list__induct__first2,axiom,
% 0.25/0.50 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 0.25/0.50 ( ( P @ ( nil @ A ) )
% 0.25/0.50 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.50 => ( ! [X13: A,X23: A,Xs2: list @ A] :
% 0.25/0.50 ( ( P @ Xs2 )
% 0.25/0.50 => ( P @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
% 0.25/0.50 => ( P @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_induct_first2
% 0.25/0.50 thf(fact_3611_mergesort__by__rel__merge__induct,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R5: A > B > $o,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
% 0.25/0.50 => ( ! [X_12: list @ B] : ( P @ ( nil @ A ) @ X_12 )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 0.25/0.50 ( ( R5 @ X2 @ Y4 )
% 0.25/0.50 => ( ( P @ Xs2 @ ( cons @ B @ Y4 @ Ys3 ) )
% 0.25/0.50 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 0.25/0.50 ( ~ ( R5 @ X2 @ Y4 )
% 0.25/0.50 => ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
% 0.25/0.50 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
% 0.25/0.50 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge_induct
% 0.25/0.50 thf(fact_3612_sorted__wrt__mergesort__by__rel,axiom,
% 0.25/0.50 ! [X14: $tType,R5: X14 > X14 > $o,Xs: list @ X14] :
% 0.25/0.50 ( ! [X2: X14,Y4: X14] :
% 0.25/0.50 ( ( R5 @ X2 @ Y4 )
% 0.25/0.50 | ( R5 @ Y4 @ X2 ) )
% 0.25/0.50 => ( ! [X2: X14,Y4: X14,Z3: X14] :
% 0.25/0.50 ( ( R5 @ X2 @ Y4 )
% 0.25/0.50 => ( ( R5 @ Y4 @ Z3 )
% 0.25/0.50 => ( R5 @ X2 @ Z3 ) ) )
% 0.25/0.50 => ( sorted_wrt @ X14 @ R5 @ ( mergesort_by_rel @ X14 @ R5 @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt_mergesort_by_rel
% 0.25/0.50 thf(fact_3613_list__tail__coinc,axiom,
% 0.25/0.50 ! [A: $tType,N12: A,R1: list @ A,N23: A,R22: list @ A] :
% 0.25/0.50 ( ( ( cons @ A @ N12 @ R1 )
% 0.25/0.50 = ( cons @ A @ N23 @ R22 ) )
% 0.25/0.50 => ( ( N12 = N23 )
% 0.25/0.50 & ( R1 = R22 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_tail_coinc
% 0.25/0.50 thf(fact_3614_drop__eq__ConsD,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,Xs: list @ A,X: A,Xs4: list @ A] :
% 0.25/0.50 ( ( ( drop @ A @ N @ Xs )
% 0.25/0.50 = ( cons @ A @ X @ Xs4 ) )
% 0.25/0.50 => ( ( drop @ A @ ( suc @ N ) @ Xs )
% 0.25/0.50 = Xs4 ) ) ).
% 0.25/0.50
% 0.25/0.50 % drop_eq_ConsD
% 0.25/0.50 thf(fact_3615_drop__Cons_H,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ( N
% 0.25/0.50 = ( zero_zero @ nat ) )
% 0.25/0.50 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( cons @ A @ X @ Xs ) ) )
% 0.25/0.50 & ( ( N
% 0.25/0.50 != ( zero_zero @ nat ) )
% 0.25/0.50 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % drop_Cons'
% 0.25/0.50 thf(fact_3616_mergesort__by__rel__merge__simps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( ( R5 @ X @ Y )
% 0.25/0.50 => ( ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.25/0.50 = ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) ) )
% 0.25/0.50 & ( ~ ( R5 @ X @ Y )
% 0.25/0.50 => ( ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.25/0.50 = ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge_simps(1)
% 0.25/0.50 thf(fact_3617_sorted__filter_H,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A,P: A > $o] :
% 0.25/0.50 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.25/0.50 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( filter @ A @ P @ L ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_filter'
% 0.25/0.50 thf(fact_3618_sorted__mergesort__by__rel,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_mergesort_by_rel
% 0.25/0.50 thf(fact_3619_map__filter__simps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Xs: list @ B] :
% 0.25/0.50 ( ( map_filter @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
% 0.25/0.50 = ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs )
% 0.25/0.50 @ ^ [Y3: A] : ( cons @ A @ Y3 @ ( map_filter @ B @ A @ F @ Xs ) )
% 0.25/0.50 @ ( F @ X ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_filter_simps(1)
% 0.25/0.50 thf(fact_3620_zipf_Ocases,axiom,
% 0.25/0.50 ! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
% 0.25/0.50 ( ! [F4: A > B > C] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
% 0.25/0.50 => ( ! [F4: A > B > C,A5: A,As: list @ A,B3: B,Bs: list @ B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 0.25/0.50 => ( ! [A5: A > B > C,V4: A,Va: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
% 0.25/0.50 => ~ ! [A5: A > B > C,V4: B,Va: list @ B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zipf.cases
% 0.25/0.50 thf(fact_3621_merge_Ocases,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.50 ( ! [L22: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
% 0.25/0.50 => ( ! [V4: A,Va: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) )
% 0.25/0.50 => ~ ! [X13: A,L12: list @ A,X23: A,L22: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge.cases
% 0.25/0.50 thf(fact_3622_list__all__zip_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
% 0.25/0.50 ( ! [P7: A > B > $o] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
% 0.25/0.50 => ( ! [P7: A > B > $o,A5: A,As: list @ A,B3: B,Bs: list @ B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 0.25/0.50 => ( ! [P7: A > B > $o,V4: A,Va: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
% 0.25/0.50 => ~ ! [P7: A > B > $o,V4: B,Va: list @ B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_all_zip.cases
% 0.25/0.50 thf(fact_3623_partition__rev_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
% 0.25/0.50 ( ! [P7: A > $o,Yes: list @ A,No: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
% 0.25/0.50 => ~ ! [P7: A > $o,Yes: list @ A,No: list @ A,X2: A,Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % partition_rev.cases
% 0.25/0.50 thf(fact_3624_quicksort__by__rel_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
% 0.25/0.50 ( ! [R8: A > A > $o,Sl: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
% 0.25/0.50 => ~ ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % quicksort_by_rel.cases
% 0.25/0.50 thf(fact_3625_mergesort__by__rel__merge_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
% 0.25/0.50 ( ! [R8: A > A > $o,X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) )
% 0.25/0.50 => ( ! [R8: A > A > $o,Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
% 0.25/0.50 => ~ ! [R8: A > A > $o,V4: A,Va: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge.cases
% 0.25/0.50 thf(fact_3626_mergesort__by__rel__split_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
% 0.25/0.50 ( ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) )
% 0.25/0.50 => ( ! [Xs13: list @ A,Xs24: list @ A,X2: A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
% 0.25/0.50 => ~ ! [Xs13: list @ A,Xs24: list @ A,X13: A,X23: A,Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split.cases
% 0.25/0.50 thf(fact_3627_shuffles_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.50 ( ! [Ys3: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 0.25/0.50 => ( ! [Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % shuffles.cases
% 0.25/0.50 thf(fact_3628_map__tailrec__rev_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
% 0.25/0.50 ( ! [F4: A > B,Bs: list @ B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
% 0.25/0.50 => ~ ! [F4: A > B,A5: A,As: list @ A,Bs: list @ B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Bs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_tailrec_rev.cases
% 0.25/0.50 thf(fact_3629_foldr_Ofoldr__Cons,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( foldr @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( comp @ B @ B @ B @ ( F @ X ) @ ( foldr @ A @ B @ F @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldr.foldr_Cons
% 0.25/0.50 thf(fact_3630_arg__min__list_Ocases,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( linorder @ B )
% 0.25/0.50 => ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
% 0.25/0.50 ( ! [F4: A > B,X2: A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
% 0.25/0.50 => ( ! [F4: A > B,X2: A,Y4: A,Zs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
% 0.25/0.50 => ~ ! [A5: A > B] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ A5 @ ( nil @ A ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % arg_min_list.cases
% 0.25/0.50 thf(fact_3631_successively_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
% 0.25/0.50 ( ! [P7: A > A > $o] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
% 0.25/0.50 => ( ! [P7: A > A > $o,X2: A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
% 0.25/0.50 => ~ ! [P7: A > A > $o,X2: A,Y4: A,Xs2: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % successively.cases
% 0.25/0.50 thf(fact_3632_sorted__wrt_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
% 0.25/0.50 ( ! [P7: A > A > $o] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
% 0.25/0.50 => ~ ! [P7: A > A > $o,X2: A,Ys3: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt.cases
% 0.25/0.50 thf(fact_3633_mergesort__by__rel__merge_Oelims,axiom,
% 0.25/0.50 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 0.25/0.50 ( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ! [Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( ( Xb
% 0.25/0.50 = ( cons @ A @ Y4 @ Ys3 ) )
% 0.25/0.50 => ~ ( ( ( X @ X2 @ Y4 )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
% 0.25/0.50 & ( ~ ( X @ X2 @ Y4 )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ) ) ) )
% 0.25/0.50 => ( ( ( Xb
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( Y != Xa ) )
% 0.25/0.50 => ~ ( ( Xa
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ! [V4: A,Va: list @ A] :
% 0.25/0.50 ( ( Xb
% 0.25/0.50 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge.elims
% 0.25/0.50 thf(fact_3634_mergesort__by__rel__merge_Osimps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,R5: A > A > $o,V: A,Va2: list @ A] :
% 0.25/0.50 ( ( merges9089515139780605204_merge @ A @ R5 @ ( nil @ A ) @ ( cons @ A @ V @ Va2 ) )
% 0.25/0.50 = ( cons @ A @ V @ Va2 ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge.simps(3)
% 0.25/0.50 thf(fact_3635_mergesort__by__rel__split_Osimps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,X1: A,X22: A,Xs: list @ A] :
% 0.25/0.50 ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
% 0.25/0.50 = ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs1 ) @ ( cons @ A @ X22 @ Xs22 ) ) @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split.simps(3)
% 0.25/0.50 thf(fact_3636_filter__conv__foldr,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( filter @ A )
% 0.25/0.50 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 0.25/0.50 ( foldr @ A @ ( list @ A )
% 0.25/0.50 @ ^ [X3: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P4 @ X3 ) @ ( cons @ A @ X3 @ Xt ) @ Xt )
% 0.25/0.50 @ Xs3
% 0.25/0.50 @ ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % filter_conv_foldr
% 0.25/0.50 thf(fact_3637_sorted__wrt01,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,P: A > A > $o] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.25/0.50 => ( sorted_wrt @ A @ P @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt01
% 0.25/0.50 thf(fact_3638_length__compl__induct,axiom,
% 0.25/0.50 ! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
% 0.25/0.50 ( ( P @ ( nil @ A ) )
% 0.25/0.50 => ( ! [E3: A,L5: list @ A] :
% 0.25/0.50 ( ! [Ll: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L5 ) )
% 0.25/0.50 => ( P @ Ll ) )
% 0.25/0.50 => ( P @ ( cons @ A @ E3 @ L5 ) ) )
% 0.25/0.50 => ( P @ L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_compl_induct
% 0.25/0.50 thf(fact_3639_list__decomp__1,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ L )
% 0.25/0.50 = ( one_one @ nat ) )
% 0.25/0.50 => ? [A5: A] :
% 0.25/0.50 ( L
% 0.25/0.50 = ( cons @ A @ A5 @ ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_decomp_1
% 0.25/0.50 thf(fact_3640_map__of__Cons__code_I2_J,axiom,
% 0.25/0.50 ! [C: $tType,B: $tType,L: B,K: B,V: C,Ps: list @ ( product_prod @ B @ C )] :
% 0.25/0.50 ( ( ( L = K )
% 0.25/0.50 => ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
% 0.25/0.50 = ( some @ C @ V ) ) )
% 0.25/0.50 & ( ( L != K )
% 0.25/0.50 => ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
% 0.25/0.50 = ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_Cons_code(2)
% 0.25/0.50 thf(fact_3641_mergesort__by__rel__split_Osimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,X: A] :
% 0.25/0.50 ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs22 ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split.simps(2)
% 0.25/0.50 thf(fact_3642_mergesort__by__rel__split_Oelims,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.50 ( ( ( merges295452479951948502_split @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ( ( Xa
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) ) ) )
% 0.25/0.50 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ! [X2: A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) ) ) )
% 0.25/0.50 => ~ ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ! [X13: A,X23: A,Xs2: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X23 @ Xs24 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split.elims
% 0.25/0.50 thf(fact_3643_sorted01,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [Xs: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.25/0.50 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted01
% 0.25/0.50 thf(fact_3644_list_Osize_I4_J,axiom,
% 0.25/0.50 ! [A: $tType,X21: A,X222: list @ A] :
% 0.25/0.50 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
% 0.25/0.50 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list.size(4)
% 0.25/0.50 thf(fact_3645_nth__Cons_H,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ( N
% 0.25/0.50 = ( zero_zero @ nat ) )
% 0.25/0.50 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.25/0.50 = X ) )
% 0.25/0.50 & ( ( N
% 0.25/0.50 != ( zero_zero @ nat ) )
% 0.25/0.50 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.25/0.50 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_Cons'
% 0.25/0.50 thf(fact_3646_upt__eq__Cons__conv,axiom,
% 0.25/0.50 ! [I: nat,J: nat,X: nat,Xs: list @ nat] :
% 0.25/0.50 ( ( ( upt @ I @ J )
% 0.25/0.50 = ( cons @ nat @ X @ Xs ) )
% 0.25/0.50 = ( ( ord_less @ nat @ I @ J )
% 0.25/0.50 & ( I = X )
% 0.25/0.50 & ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
% 0.25/0.50 = Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_eq_Cons_conv
% 0.25/0.50 thf(fact_3647_in__set__drop__conv__nth,axiom,
% 0.25/0.50 ! [A: $tType,X: A,N: nat,L: list @ A] :
% 0.25/0.50 ( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
% 0.25/0.50 = ( ? [I3: nat] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ N @ I3 )
% 0.25/0.50 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 & ( X
% 0.25/0.50 = ( nth @ A @ L @ I3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_drop_conv_nth
% 0.25/0.50 thf(fact_3648_set__drop__conv,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,L: list @ A] :
% 0.25/0.50 ( ( set2 @ A @ ( drop @ A @ N @ L ) )
% 0.25/0.50 = ( collect @ A
% 0.25/0.50 @ ^ [Uu: A] :
% 0.25/0.50 ? [I3: nat] :
% 0.25/0.50 ( ( Uu
% 0.25/0.50 = ( nth @ A @ L @ I3 ) )
% 0.25/0.50 & ( ord_less_eq @ nat @ N @ I3 )
% 0.25/0.50 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_drop_conv
% 0.25/0.50 thf(fact_3649_list__decomp__2,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ L )
% 0.25/0.50 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.50 => ? [A5: A,B3: A] :
% 0.25/0.50 ( L
% 0.25/0.50 = ( cons @ A @ A5 @ ( cons @ A @ B3 @ ( nil @ A ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_decomp_2
% 0.25/0.50 thf(fact_3650_nth__non__equal__first__eq,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
% 0.25/0.50 ( ( X != Y )
% 0.25/0.50 => ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 0.25/0.50 = Y )
% 0.25/0.50 = ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.25/0.50 = Y )
% 0.25/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_non_equal_first_eq
% 0.25/0.50 thf(fact_3651_Cons__replicate__eq,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
% 0.25/0.50 ( ( ( cons @ A @ X @ Xs )
% 0.25/0.50 = ( replicate @ A @ N @ Y ) )
% 0.25/0.50 = ( ( X = Y )
% 0.25/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.25/0.50 & ( Xs
% 0.25/0.50 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Cons_replicate_eq
% 0.25/0.50 thf(fact_3652_distinct__sorted__mono,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A,I: nat,J: nat] :
% 0.25/0.50 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.25/0.50 => ( ( distinct @ A @ L )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ J )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % distinct_sorted_mono
% 0.25/0.50 thf(fact_3653_distinct__sorted__strict__mono__iff,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A,I: nat,J: nat] :
% 0.25/0.50 ( ( distinct @ A @ L )
% 0.25/0.50 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
% 0.25/0.50 = ( ord_less @ nat @ I @ J ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % distinct_sorted_strict_mono_iff
% 0.25/0.50 thf(fact_3654_distinct__sorted__mono__iff,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A,I: nat,J: nat] :
% 0.25/0.50 ( ( distinct @ A @ L )
% 0.25/0.50 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
% 0.25/0.50 = ( ord_less_eq @ nat @ I @ J ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % distinct_sorted_mono_iff
% 0.25/0.50 thf(fact_3655_map__of_Osimps_I2_J,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P5 @ Ps ) )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P5 ) @ ( some @ B @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of.simps(2)
% 0.25/0.50 thf(fact_3656_slice__Cons,axiom,
% 0.25/0.50 ! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ( ( Begin
% 0.25/0.50 = ( zero_zero @ nat ) )
% 0.25/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
% 0.25/0.50 => ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
% 0.25/0.50 & ( ~ ( ( Begin
% 0.25/0.50 = ( zero_zero @ nat ) )
% 0.25/0.50 & ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
% 0.25/0.50 => ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % slice_Cons
% 0.25/0.50 thf(fact_3657_mergesort__by__rel__split_Opelims,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.50 ( ( ( merges295452479951948502_split @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.50 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ( ( Xa
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) ) ) ) )
% 0.25/0.50 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ! [X2: A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) )
% 0.25/0.50 => ~ ! [Xs13: list @ A,Xs24: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 0.25/0.50 => ! [X13: A,X23: A,Xs2: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X23 @ Xs24 ) ) @ Xs2 ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_split.pelims
% 0.25/0.50 thf(fact_3658_list__decode_Opelims,axiom,
% 0.25/0.50 ! [X: nat,Y: list @ nat] :
% 0.25/0.50 ( ( ( nat_list_decode @ X )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ nat @ nat_list_decode_rel @ X )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( zero_zero @ nat ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( nil @ nat ) )
% 0.25/0.50 => ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
% 0.25/0.50 => ~ ! [N3: nat] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( suc @ N3 ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 0.25/0.50 @ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
% 0.25/0.50 @ ( nat_prod_decode @ N3 ) ) )
% 0.25/0.50 => ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_decode.pelims
% 0.25/0.50 thf(fact_3659_mergesort__by__rel__merge_Opelims,axiom,
% 0.25/0.50 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 0.25/0.50 ( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ! [Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( ( Xb
% 0.25/0.50 = ( cons @ A @ Y4 @ Ys3 ) )
% 0.25/0.50 => ( ( ( ( X @ X2 @ Y4 )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
% 0.25/0.50 & ( ~ ( X @ X2 @ Y4 )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ) )
% 0.25/0.50 => ( ( ( Xb
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y = Xa )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
% 0.25/0.50 => ~ ( ( Xa
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ! [V4: A,Va: list @ A] :
% 0.25/0.50 ( ( Xb
% 0.25/0.50 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_by_rel_merge.pelims
% 0.25/0.50 thf(fact_3660_subset__eq__mset__impl_Ocases,axiom,
% 0.25/0.50 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.50 ( ! [Ys3: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % subset_eq_mset_impl.cases
% 0.25/0.50 thf(fact_3661_drop__upt,axiom,
% 0.25/0.50 ! [M2: nat,I: nat,J: nat] :
% 0.25/0.50 ( ( drop @ nat @ M2 @ ( upt @ I @ J ) )
% 0.25/0.50 = ( upt @ ( plus_plus @ nat @ I @ M2 ) @ J ) ) ).
% 0.25/0.50
% 0.25/0.50 % drop_upt
% 0.25/0.50 thf(fact_3662_natural__zero__minus__one,axiom,
% 0.25/0.50 ( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
% 0.25/0.50 = ( zero_zero @ code_natural ) ) ).
% 0.25/0.50
% 0.25/0.50 % natural_zero_minus_one
% 0.25/0.50 thf(fact_3663_list__decode_Opsimps_I1_J,axiom,
% 0.25/0.50 ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 0.25/0.50 => ( ( nat_list_decode @ ( zero_zero @ nat ) )
% 0.25/0.50 = ( nil @ nat ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_decode.psimps(1)
% 0.25/0.50 thf(fact_3664_list__decode_Opsimps_I2_J,axiom,
% 0.25/0.50 ! [N: nat] :
% 0.25/0.50 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
% 0.25/0.50 => ( ( nat_list_decode @ ( suc @ N ) )
% 0.25/0.50 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 0.25/0.50 @ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
% 0.25/0.50 @ ( nat_prod_decode @ N ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_decode.psimps(2)
% 0.25/0.50 thf(fact_3665_list__encode_Oelims,axiom,
% 0.25/0.50 ! [X: list @ nat,Y: nat] :
% 0.25/0.50 ( ( ( nat_list_encode @ X )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ nat ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( zero_zero @ nat ) ) )
% 0.25/0.50 => ~ ! [X2: nat,Xs2: list @ nat] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ nat @ X2 @ Xs2 ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_encode.elims
% 0.25/0.50 thf(fact_3666_mergesort__remdups__correct,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A] :
% 0.25/0.50 ( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
% 0.25/0.50 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_remdups @ A @ L ) )
% 0.25/0.50 & ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
% 0.25/0.50 = ( set2 @ A @ L ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_remdups_correct
% 0.25/0.50 thf(fact_3667_filter__upt__last,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,L: list @ A,Js: list @ nat,J: nat,I: nat] :
% 0.25/0.50 ( ( ( filter @ nat
% 0.25/0.50 @ ^ [K4: nat] : ( P @ ( nth @ A @ L @ K4 ) )
% 0.25/0.50 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
% 0.25/0.50 = ( append @ nat @ Js @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
% 0.25/0.50 => ( ( ord_less @ nat @ J @ I )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % filter_upt_last
% 0.25/0.50 thf(fact_3668_empty__append__eq__id,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( append @ A @ ( nil @ A ) )
% 0.25/0.50 = ( ^ [X3: list @ A] : X3 ) ) ).
% 0.25/0.50
% 0.25/0.50 % empty_append_eq_id
% 0.25/0.50 thf(fact_3669_op__conc__empty__img__id,axiom,
% 0.25/0.50 ! [A: $tType,L6: set @ ( list @ A )] :
% 0.25/0.50 ( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L6 )
% 0.25/0.50 = L6 ) ).
% 0.25/0.50
% 0.25/0.50 % op_conc_empty_img_id
% 0.25/0.50 thf(fact_3670_list__e__eq__lel_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,L1: list @ A,E6: A,L2: list @ A,E2: A] :
% 0.25/0.50 ( ( ( append @ A @ L1 @ ( cons @ A @ E6 @ L2 ) )
% 0.25/0.50 = ( cons @ A @ E2 @ ( nil @ A ) ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 & ( E6 = E2 )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_e_eq_lel(2)
% 0.25/0.50 thf(fact_3671_list__e__eq__lel_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,E2: A,L1: list @ A,E6: A,L2: list @ A] :
% 0.25/0.50 ( ( ( cons @ A @ E2 @ ( nil @ A ) )
% 0.25/0.50 = ( append @ A @ L1 @ ( cons @ A @ E6 @ L2 ) ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 & ( E6 = E2 )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_e_eq_lel(1)
% 0.25/0.50 thf(fact_3672_list__se__match_I4_J,axiom,
% 0.25/0.50 ! [A: $tType,L2: list @ A,A3: A,L1: list @ A] :
% 0.25/0.50 ( ( L2
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( ( cons @ A @ A3 @ ( nil @ A ) )
% 0.25/0.50 = ( append @ A @ L1 @ L2 ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_se_match(4)
% 0.25/0.50 thf(fact_3673_list__se__match_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,L1: list @ A,A3: A,L2: list @ A] :
% 0.25/0.50 ( ( L1
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( ( cons @ A @ A3 @ ( nil @ A ) )
% 0.25/0.50 = ( append @ A @ L1 @ L2 ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( nil @ A ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_se_match(3)
% 0.25/0.50 thf(fact_3674_list__se__match_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,L2: list @ A,L1: list @ A,A3: A] :
% 0.25/0.50 ( ( L2
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( ( append @ A @ L1 @ L2 )
% 0.25/0.50 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_se_match(2)
% 0.25/0.50 thf(fact_3675_list__se__match_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,L1: list @ A,L2: list @ A,A3: A] :
% 0.25/0.50 ( ( L1
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( ( append @ A @ L1 @ L2 )
% 0.25/0.50 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( nil @ A ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_se_match(1)
% 0.25/0.50 thf(fact_3676_list__ee__eq__leel_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,L1: list @ A,E12: A,E23: A,L2: list @ A,E1: A,E22: A] :
% 0.25/0.50 ( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) )
% 0.25/0.50 = ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 & ( E1 = E12 )
% 0.25/0.50 & ( E22 = E23 )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_ee_eq_leel(2)
% 0.25/0.50 thf(fact_3677_list__ee__eq__leel_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,E1: A,E22: A,L1: list @ A,E12: A,E23: A,L2: list @ A] :
% 0.25/0.50 ( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
% 0.25/0.50 = ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) ) )
% 0.25/0.50 = ( ( L1
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 & ( E1 = E12 )
% 0.25/0.50 & ( E22 = E23 )
% 0.25/0.50 & ( L2
% 0.25/0.50 = ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_ee_eq_leel(1)
% 0.25/0.50 thf(fact_3678_length__append,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) )
% 0.25/0.50 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_append
% 0.25/0.50 thf(fact_3679_upt__merge,axiom,
% 0.25/0.50 ! [I: nat,J: nat,K: nat] :
% 0.25/0.50 ( ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.50 & ( ord_less_eq @ nat @ J @ K ) )
% 0.25/0.50 => ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
% 0.25/0.50 = ( upt @ I @ K ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_merge
% 0.25/0.50 thf(fact_3680_nth__append__first,axiom,
% 0.25/0.50 ! [A: $tType,I: nat,L: list @ A,L4: list @ A] :
% 0.25/0.50 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.50 => ( ( nth @ A @ ( append @ A @ L @ L4 ) @ I )
% 0.25/0.50 = ( nth @ A @ L @ I ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_append_first
% 0.25/0.50 thf(fact_3681_nth__append__length__plus,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,N: nat] :
% 0.25/0.50 ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
% 0.25/0.50 = ( nth @ A @ Ys2 @ N ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_append_length_plus
% 0.25/0.50 thf(fact_3682_list__match__lel__lel,axiom,
% 0.25/0.50 ! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
% 0.25/0.50 ( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
% 0.25/0.50 = ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
% 0.25/0.50 => ( ! [C21: list @ A] :
% 0.25/0.50 ( ( C1
% 0.25/0.50 = ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
% 0.25/0.50 => ( C23
% 0.25/0.50 != ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
% 0.25/0.50 => ( ( ( C12 = C1 )
% 0.25/0.50 => ( ( Qs2 = Qs )
% 0.25/0.50 => ( C23 != C22 ) ) )
% 0.25/0.50 => ~ ! [C212: list @ A] :
% 0.25/0.50 ( ( C12
% 0.25/0.50 = ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
% 0.25/0.50 => ( C22
% 0.25/0.50 != ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_match_lel_lel
% 0.25/0.50 thf(fact_3683_foldl__conc__empty__eq,axiom,
% 0.25/0.50 ! [A: $tType,I: list @ A,Ww: list @ ( list @ A )] :
% 0.25/0.50 ( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I @ Ww )
% 0.25/0.50 = ( append @ A @ I @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_conc_empty_eq
% 0.25/0.50 thf(fact_3684_Misc_Omap__eq__append__conv,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
% 0.25/0.50 ( ( ( map @ B @ A @ F @ Ls )
% 0.25/0.50 = ( append @ A @ Fl @ Fl2 ) )
% 0.25/0.50 = ( ? [L3: list @ B,L7: list @ B] :
% 0.25/0.50 ( ( Ls
% 0.25/0.50 = ( append @ B @ L3 @ L7 ) )
% 0.25/0.50 & ( ( map @ B @ A @ F @ L3 )
% 0.25/0.50 = Fl )
% 0.25/0.50 & ( ( map @ B @ A @ F @ L7 )
% 0.25/0.50 = Fl2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Misc.map_eq_append_conv
% 0.25/0.50 thf(fact_3685_Misc_Oappend__eq__map__conv,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
% 0.25/0.50 ( ( ( append @ A @ Fl @ Fl2 )
% 0.25/0.50 = ( map @ B @ A @ F @ Ls ) )
% 0.25/0.50 = ( ? [L3: list @ B,L7: list @ B] :
% 0.25/0.50 ( ( Ls
% 0.25/0.50 = ( append @ B @ L3 @ L7 ) )
% 0.25/0.50 & ( ( map @ B @ A @ F @ L3 )
% 0.25/0.50 = Fl )
% 0.25/0.50 & ( ( map @ B @ A @ F @ L7 )
% 0.25/0.50 = Fl2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Misc.append_eq_map_conv
% 0.25/0.50 thf(fact_3686_map__eq__appendE,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
% 0.25/0.50 ( ( ( map @ B @ A @ F @ Ls )
% 0.25/0.50 = ( append @ A @ Fl @ Fl2 ) )
% 0.25/0.50 => ~ ! [L5: list @ B,L8: list @ B] :
% 0.25/0.50 ( ( Ls
% 0.25/0.50 = ( append @ B @ L5 @ L8 ) )
% 0.25/0.50 => ( ( ( map @ B @ A @ F @ L5 )
% 0.25/0.50 = Fl )
% 0.25/0.50 => ( ( map @ B @ A @ F @ L8 )
% 0.25/0.50 != Fl2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_eq_appendE
% 0.25/0.50 thf(fact_3687_append__eq__mapE,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
% 0.25/0.50 ( ( ( append @ A @ Fl @ Fl2 )
% 0.25/0.50 = ( map @ B @ A @ F @ Ls ) )
% 0.25/0.50 => ~ ! [L5: list @ B,L8: list @ B] :
% 0.25/0.50 ( ( Ls
% 0.25/0.50 = ( append @ B @ L5 @ L8 ) )
% 0.25/0.50 => ( ( ( map @ B @ A @ F @ L5 )
% 0.25/0.50 = Fl )
% 0.25/0.50 => ( ( map @ B @ A @ F @ L8 )
% 0.25/0.50 != Fl2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % append_eq_mapE
% 0.25/0.50 thf(fact_3688_enumerate__append__eq,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys2 ) )
% 0.25/0.50 = ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % enumerate_append_eq
% 0.25/0.50 thf(fact_3689_neq__Nil__revE,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( L
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ~ ! [Ll2: list @ A,E3: A] :
% 0.25/0.50 ( L
% 0.25/0.50 != ( append @ A @ Ll2 @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % neq_Nil_revE
% 0.25/0.50 thf(fact_3690_rev__induct2_H,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ B ) )
% 0.25/0.50 => ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 0.25/0.50 ( ( P @ Xs2 @ Ys3 )
% 0.25/0.50 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
% 0.25/0.50 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % rev_induct2'
% 0.25/0.50 thf(fact_3691_neq__Nil__rev__conv,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( L
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 = ( ? [Xs3: list @ A,X3: A] :
% 0.25/0.50 ( L
% 0.25/0.50 = ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % neq_Nil_rev_conv
% 0.25/0.50 thf(fact_3692_rev__nonempty__induct2_H,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( Ys2
% 0.25/0.50 != ( nil @ B ) )
% 0.25/0.50 => ( ! [X2: A,Y4: B] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Y4: B] :
% 0.25/0.50 ( ( Xs2
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
% 0.25/0.50 => ( ! [X2: A,Y4: B,Ys3: list @ B] :
% 0.25/0.50 ( ( Ys3
% 0.25/0.50 != ( nil @ B ) )
% 0.25/0.50 => ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 0.25/0.50 ( ( P @ Xs2 @ Ys3 )
% 0.25/0.50 => ( ( Xs2
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( Ys3
% 0.25/0.50 != ( nil @ B ) )
% 0.25/0.50 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
% 0.25/0.50 => ( P @ Xs @ Ys2 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % rev_nonempty_induct2'
% 0.25/0.50 thf(fact_3693_list__Cons__eq__append__cases,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.25/0.50 ( ( ( cons @ A @ X @ Xs )
% 0.25/0.50 = ( append @ A @ Ys2 @ Zs ) )
% 0.25/0.50 => ( ( ( Ys2
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( Zs
% 0.25/0.50 != ( cons @ A @ X @ Xs ) ) )
% 0.25/0.50 => ~ ! [Ys4: list @ A] :
% 0.25/0.50 ( ( Ys2
% 0.25/0.50 = ( cons @ A @ X @ Ys4 ) )
% 0.25/0.50 => ( ( append @ A @ Ys4 @ Zs )
% 0.25/0.50 != Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_Cons_eq_append_cases
% 0.25/0.50 thf(fact_3694_list__append__eq__Cons__cases,axiom,
% 0.25/0.50 ! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ( append @ A @ Ys2 @ Zs )
% 0.25/0.50 = ( cons @ A @ X @ Xs ) )
% 0.25/0.50 => ( ( ( Ys2
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( Zs
% 0.25/0.50 != ( cons @ A @ X @ Xs ) ) )
% 0.25/0.50 => ~ ! [Ys4: list @ A] :
% 0.25/0.50 ( ( Ys2
% 0.25/0.50 = ( cons @ A @ X @ Ys4 ) )
% 0.25/0.50 => ( ( append @ A @ Ys4 @ Zs )
% 0.25/0.50 != Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_append_eq_Cons_cases
% 0.25/0.50 thf(fact_3695_in__set__list__format,axiom,
% 0.25/0.50 ! [A: $tType,E2: A,L: list @ A] :
% 0.25/0.50 ( ( member @ A @ E2 @ ( set2 @ A @ L ) )
% 0.25/0.50 => ~ ! [L12: list @ A,L22: list @ A] :
% 0.25/0.50 ( L
% 0.25/0.50 != ( append @ A @ L12 @ ( cons @ A @ E2 @ L22 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_list_format
% 0.25/0.50 thf(fact_3696_xy__in__set__cases,axiom,
% 0.25/0.50 ! [A: $tType,X: A,L: list @ A,Y: A] :
% 0.25/0.50 ( ( member @ A @ X @ ( set2 @ A @ L ) )
% 0.25/0.50 => ( ( member @ A @ Y @ ( set2 @ A @ L ) )
% 0.25/0.50 => ( ( ( X = Y )
% 0.25/0.50 => ! [L12: list @ A,L22: list @ A] :
% 0.25/0.50 ( L
% 0.25/0.50 != ( append @ A @ L12 @ ( cons @ A @ Y @ L22 ) ) ) )
% 0.25/0.50 => ( ( ( X != Y )
% 0.25/0.50 => ! [L12: list @ A,L22: list @ A,L32: list @ A] :
% 0.25/0.50 ( L
% 0.25/0.50 != ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
% 0.25/0.50 => ~ ( ( X != Y )
% 0.25/0.50 => ! [L12: list @ A,L22: list @ A,L32: list @ A] :
% 0.25/0.50 ( L
% 0.25/0.50 != ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % xy_in_set_cases
% 0.25/0.50 thf(fact_3697_map__consI_I2_J,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,W: list @ A,L: list @ A,F: B > A,Ww: list @ B,A3: B] :
% 0.25/0.50 ( ( ( append @ A @ W @ L )
% 0.25/0.50 = ( append @ A @ ( map @ B @ A @ F @ Ww ) @ L ) )
% 0.25/0.50 => ( ( cons @ A @ ( F @ A3 ) @ ( append @ A @ W @ L ) )
% 0.25/0.50 = ( append @ A @ ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) @ L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_consI(2)
% 0.25/0.50 thf(fact_3698_list__rest__coinc,axiom,
% 0.25/0.50 ! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
% 0.25/0.50 => ( ( ( append @ A @ S1 @ R1 )
% 0.25/0.50 = ( append @ A @ S22 @ R22 ) )
% 0.25/0.50 => ? [R1p: list @ A] :
% 0.25/0.50 ( R22
% 0.25/0.50 = ( append @ A @ R1p @ R1 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_rest_coinc
% 0.25/0.50 thf(fact_3699_set__union__code,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
% 0.25/0.50 = ( set2 @ A @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_union_code
% 0.25/0.50 thf(fact_3700_distinct__match,axiom,
% 0.25/0.50 ! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
% 0.25/0.50 ( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) ) )
% 0.25/0.50 => ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
% 0.25/0.50 = ( append @ A @ Al2 @ ( cons @ A @ E2 @ Bl2 ) ) )
% 0.25/0.50 = ( ( Al = Al2 )
% 0.25/0.50 & ( Bl = Bl2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % distinct_match
% 0.25/0.50 thf(fact_3701_upt__eq__append__conv,axiom,
% 0.25/0.50 ! [I: nat,J: nat,Xs: list @ nat,Ys2: list @ nat] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.50 => ( ( ( upt @ I @ J )
% 0.25/0.50 = ( append @ nat @ Xs @ Ys2 ) )
% 0.25/0.50 = ( ? [K4: nat] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ I @ K4 )
% 0.25/0.50 & ( ord_less_eq @ nat @ K4 @ J )
% 0.25/0.50 & ( ( upt @ I @ K4 )
% 0.25/0.50 = Xs )
% 0.25/0.50 & ( ( upt @ K4 @ J )
% 0.25/0.50 = Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_eq_append_conv
% 0.25/0.50 thf(fact_3702_replicate__add,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,M2: nat,X: A] :
% 0.25/0.50 ( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X )
% 0.25/0.50 = ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M2 @ X ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % replicate_add
% 0.25/0.50 thf(fact_3703_upt__append,axiom,
% 0.25/0.50 ! [I: nat,J: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I @ J )
% 0.25/0.50 => ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
% 0.25/0.50 = ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_append
% 0.25/0.50 thf(fact_3704_upt__add__eq__append,axiom,
% 0.25/0.50 ! [I: nat,J: nat,K: nat] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.50 => ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
% 0.25/0.50 = ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_add_eq_append
% 0.25/0.50 thf(fact_3705_foldl__rule,axiom,
% 0.25/0.50 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
% 0.25/0.50 ( ( I4 @ Sigma_0 @ ( nil @ A ) @ L0 )
% 0.25/0.50 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 0.25/0.50 ( ( L0
% 0.25/0.50 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 0.25/0.50 => ( ( I4 @ Sigma @ L12 @ ( cons @ A @ X2 @ L22 ) )
% 0.25/0.50 => ( I4 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L12 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
% 0.25/0.50 => ( I4 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_rule
% 0.25/0.50 thf(fact_3706_foldl__rule__P,axiom,
% 0.25/0.50 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
% 0.25/0.50 ( ( I4 @ Sigma_0 @ ( nil @ A ) @ L0 )
% 0.25/0.50 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 0.25/0.50 ( ( L0
% 0.25/0.50 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 0.25/0.50 => ( ( I4 @ Sigma @ L12 @ ( cons @ A @ X2 @ L22 ) )
% 0.25/0.50 => ( I4 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L12 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
% 0.25/0.50 => ( ! [Sigma: Sigma2] :
% 0.25/0.50 ( ( I4 @ Sigma @ L0 @ ( nil @ A ) )
% 0.25/0.50 => ( P @ Sigma ) )
% 0.25/0.50 => ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_rule_P
% 0.25/0.50 thf(fact_3707_foldl__rule__aux,axiom,
% 0.25/0.50 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
% 0.25/0.50 ( ( I4 @ Sigma_0 @ L0 )
% 0.25/0.50 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 0.25/0.50 ( ( L0
% 0.25/0.50 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 0.25/0.50 => ( ( I4 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
% 0.25/0.50 => ( I4 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
% 0.25/0.50 => ( I4 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_rule_aux
% 0.25/0.50 thf(fact_3708_foldl__rule__aux__P,axiom,
% 0.25/0.50 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
% 0.25/0.50 ( ( I4 @ Sigma_0 @ L0 )
% 0.25/0.50 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 0.25/0.50 ( ( L0
% 0.25/0.50 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 0.25/0.50 => ( ( I4 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
% 0.25/0.50 => ( I4 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
% 0.25/0.50 => ( ! [Sigma: Sigma2] :
% 0.25/0.50 ( ( I4 @ Sigma @ ( nil @ A ) )
% 0.25/0.50 => ( P @ Sigma ) )
% 0.25/0.50 => ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_rule_aux_P
% 0.25/0.50 thf(fact_3709_map__by__foldl,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: A > B,L: list @ A] :
% 0.25/0.50 ( ( foldl @ ( list @ B ) @ A
% 0.25/0.50 @ ^ [L3: list @ B,X3: A] : ( append @ B @ L3 @ ( cons @ B @ ( F @ X3 ) @ ( nil @ B ) ) )
% 0.25/0.50 @ ( nil @ B )
% 0.25/0.50 @ L )
% 0.25/0.50 = ( map @ A @ B @ F @ L ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_by_foldl
% 0.25/0.50 thf(fact_3710_length__Suc__rev__conv,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( suc @ N ) )
% 0.25/0.50 = ( ? [Ys: list @ A,Y3: A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 = ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
% 0.25/0.50 & ( ( size_size @ ( list @ A ) @ Ys )
% 0.25/0.50 = N ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_Suc_rev_conv
% 0.25/0.50 thf(fact_3711_length__compl__rev__induct,axiom,
% 0.25/0.50 ! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
% 0.25/0.50 ( ( P @ ( nil @ A ) )
% 0.25/0.50 => ( ! [L5: list @ A,E3: A] :
% 0.25/0.50 ( ! [Ll: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L5 ) )
% 0.25/0.50 => ( P @ Ll ) )
% 0.25/0.50 => ( P @ ( append @ A @ L5 @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) )
% 0.25/0.50 => ( P @ L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_compl_rev_induct
% 0.25/0.50 thf(fact_3712_not__distinct__split__distinct,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A] :
% 0.25/0.50 ( ~ ( distinct @ A @ Xs )
% 0.25/0.50 => ~ ! [Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( ( distinct @ A @ Ys3 )
% 0.25/0.50 => ( ( member @ A @ Y4 @ ( set2 @ A @ Ys3 ) )
% 0.25/0.50 => ! [Zs2: list @ A] :
% 0.25/0.50 ( Xs
% 0.25/0.50 != ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % not_distinct_split_distinct
% 0.25/0.50 thf(fact_3713_filter__eq__snocD,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,L: list @ A,L4: list @ A,X: A] :
% 0.25/0.50 ( ( ( filter @ A @ P @ L )
% 0.25/0.50 = ( append @ A @ L4 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.25/0.50 => ( ( member @ A @ X @ ( set2 @ A @ L ) )
% 0.25/0.50 & ( P @ X ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % filter_eq_snocD
% 0.25/0.50 thf(fact_3714_replicate__Suc__conv__snoc,axiom,
% 0.25/0.50 ! [A: $tType,N: nat,X: A] :
% 0.25/0.50 ( ( replicate @ A @ ( suc @ N ) @ X )
% 0.25/0.50 = ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % replicate_Suc_conv_snoc
% 0.25/0.50 thf(fact_3715_slice__prepend,axiom,
% 0.25/0.50 ! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ I @ K )
% 0.25/0.50 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( slice @ A @ I @ K @ Xs )
% 0.25/0.50 = ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( append @ A @ Ys2 @ Xs ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % slice_prepend
% 0.25/0.50 thf(fact_3716_sorted__append__bigger,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [Xs: list @ A,Y: A] :
% 0.25/0.50 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.25/0.50 => ( ! [X2: A] :
% 0.25/0.50 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 => ( ord_less_eq @ A @ X2 @ Y ) )
% 0.25/0.50 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_append_bigger
% 0.25/0.50 thf(fact_3717_upt__eq__lel__conv,axiom,
% 0.25/0.50 ! [L: nat,H3: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
% 0.25/0.50 ( ( ( upt @ L @ H3 )
% 0.25/0.50 = ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
% 0.25/0.50 = ( ( Is1
% 0.25/0.50 = ( upt @ L @ I ) )
% 0.25/0.50 & ( Is2
% 0.25/0.50 = ( upt @ ( suc @ I ) @ H3 ) )
% 0.25/0.50 & ( ord_less_eq @ nat @ L @ I )
% 0.25/0.50 & ( ord_less @ nat @ I @ H3 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upt_eq_lel_conv
% 0.25/0.50 thf(fact_3718_list__encode_Osimps_I2_J,axiom,
% 0.25/0.50 ! [X: nat,Xs: list @ nat] :
% 0.25/0.50 ( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
% 0.25/0.50 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_encode.simps(2)
% 0.25/0.50 thf(fact_3719_list__encode_Opelims,axiom,
% 0.25/0.50 ! [X: list @ nat,Y: nat] :
% 0.25/0.50 ( ( ( nat_list_encode @ X )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ nat ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( zero_zero @ nat ) )
% 0.25/0.50 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
% 0.25/0.50 => ~ ! [X2: nat,Xs2: list @ nat] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ nat @ X2 @ Xs2 ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_encode.pelims
% 0.25/0.50 thf(fact_3720_map__upds__append1,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M2: A > ( option @ B ),X: A] :
% 0.25/0.50 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_upds_append1
% 0.25/0.50 thf(fact_3721_lexn__conv,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( lexn @ A )
% 0.25/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A ),N5: nat] :
% 0.25/0.50 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 0.25/0.50 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.25/0.50 = N5 )
% 0.25/0.50 & ( ( size_size @ ( list @ A ) @ Ys )
% 0.25/0.50 = N5 )
% 0.25/0.50 & ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
% 0.25/0.50 ( ( Xs3
% 0.25/0.50 = ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
% 0.25/0.50 & ( Ys
% 0.25/0.50 = ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
% 0.25/0.50 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexn_conv
% 0.25/0.50 thf(fact_3722_drop__last__conv,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( L
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
% 0.25/0.50 = ( cons @ A @ ( last @ A @ L ) @ ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % drop_last_conv
% 0.25/0.50 thf(fact_3723_Misc_Olast__in__set,axiom,
% 0.25/0.50 ! [A: $tType,L: list @ A] :
% 0.25/0.50 ( ( L
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( member @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Misc.last_in_set
% 0.25/0.50 thf(fact_3724_last__upt,axiom,
% 0.25/0.50 ! [I: nat,J: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I @ J )
% 0.25/0.50 => ( ( last @ nat @ ( upt @ I @ J ) )
% 0.25/0.50 = ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % last_upt
% 0.25/0.50 thf(fact_3725_map__upds__Cons,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
% 0.25/0.50 ( ( map_upds @ A @ B @ M2 @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
% 0.25/0.50 = ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs2 ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_upds_Cons
% 0.25/0.50 thf(fact_3726_map__upds__twist,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,A3: A,As2: list @ A,M2: A > ( option @ B ),B2: B,Bs2: list @ B] :
% 0.25/0.50 ( ~ ( member @ A @ A3 @ ( set2 @ A @ As2 ) )
% 0.25/0.50 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs2 )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As2 @ Bs2 ) @ A3 @ ( some @ B @ B2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_upds_twist
% 0.25/0.50 thf(fact_3727_last__filter,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( P @ ( last @ A @ Xs ) )
% 0.25/0.50 => ( ( last @ A @ ( filter @ A @ P @ Xs ) )
% 0.25/0.50 = ( last @ A @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % last_filter
% 0.25/0.50 thf(fact_3728_lexn__length,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),N: nat] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexn @ A @ R @ N ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = N )
% 0.25/0.50 & ( ( size_size @ ( list @ A ) @ Ys2 )
% 0.25/0.50 = N ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexn_length
% 0.25/0.50 thf(fact_3729_product_Osimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 0.25/0.50 = ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys2 ) @ ( product @ A @ B @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % product.simps(2)
% 0.25/0.50 thf(fact_3730_map__of__Some__split,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A] :
% 0.25/0.50 ( ( ( map_of @ B @ A @ Xs @ K )
% 0.25/0.50 = ( some @ A @ V ) )
% 0.25/0.50 => ? [Ys3: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 = ( append @ ( product_prod @ B @ A ) @ Ys3 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs2 ) ) )
% 0.25/0.50 & ( ( map_of @ B @ A @ Ys3 @ K )
% 0.25/0.50 = ( none @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_Some_split
% 0.25/0.50 thf(fact_3731_last__conv__nth,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( last @ A @ Xs )
% 0.25/0.50 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % last_conv_nth
% 0.25/0.50 thf(fact_3732_map__of__distinct__lookup,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
% 0.25/0.50 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 0.25/0.50 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) @ X )
% 0.25/0.50 = ( some @ B @ Y ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_distinct_lookup
% 0.25/0.50 thf(fact_3733_map__of__distinct__upd3,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B,Y7: B] :
% 0.25/0.50 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 0.25/0.50 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ Ys2 ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_distinct_upd3
% 0.25/0.50 thf(fact_3734_map__of__distinct__upd2,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
% 0.25/0.50 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 0.25/0.50 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_distinct_upd2
% 0.25/0.50 thf(fact_3735_map__of__distinct__upd4,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
% 0.25/0.50 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 0.25/0.50 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys2 ) )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) ) @ X @ ( none @ B ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_distinct_upd4
% 0.25/0.50 thf(fact_3736_lex__conv,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( lex @ A )
% 0.25/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 0.25/0.50 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.25/0.50 & ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
% 0.25/0.50 ( ( Xs3
% 0.25/0.50 = ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
% 0.25/0.50 & ( Ys
% 0.25/0.50 = ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
% 0.25/0.50 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lex_conv
% 0.25/0.50 thf(fact_3737_shuffles_Opelims,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
% 0.25/0.50 ( ( ( shuffles @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 0.25/0.50 => ( ( ( Xa
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ! [Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ Y4 @ Ys3 ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y4 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % shuffles.pelims
% 0.25/0.50 thf(fact_3738_set__relcomp,axiom,
% 0.25/0.50 ! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
% 0.25/0.50 ( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
% 0.25/0.50 = ( set2 @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( concat @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ^ [Xy: product_prod @ A @ C] :
% 0.25/0.50 ( concat @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ^ [Yz: product_prod @ C @ B] :
% 0.25/0.50 ( if @ ( list @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ( ( product_snd @ A @ C @ Xy )
% 0.25/0.50 = ( product_fst @ C @ B @ Yz ) )
% 0.25/0.50 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
% 0.25/0.50 @ ( nil @ ( product_prod @ A @ B ) ) )
% 0.25/0.50 @ Yzs ) )
% 0.25/0.50 @ Xys ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_relcomp
% 0.25/0.50 thf(fact_3739_natural_Osimps_I4_J,axiom,
% 0.25/0.50 ! [T: $tType,F1: T,F22: code_natural > T] :
% 0.25/0.50 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.25/0.50 = F1 ) ).
% 0.25/0.50
% 0.25/0.50 % natural.simps(4)
% 0.25/0.50 thf(fact_3740_Cons__in__lex,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( lex @ A @ R ) )
% 0.25/0.50 = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.50 & ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ys2 ) ) )
% 0.25/0.50 | ( ( X = Y )
% 0.25/0.50 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Cons_in_lex
% 0.25/0.50 thf(fact_3741_concat__filter__neq__Nil,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.25/0.50 ( ( concat @ A
% 0.25/0.50 @ ( filter @ ( list @ A )
% 0.25/0.50 @ ^ [Ys: list @ A] :
% 0.25/0.50 ( Ys
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 @ Xs ) )
% 0.25/0.50 = ( concat @ A @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % concat_filter_neq_Nil
% 0.25/0.50 thf(fact_3742_Nil__notin__lex,axiom,
% 0.25/0.50 ! [A: $tType,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) @ ( lex @ A @ R ) ) ).
% 0.25/0.50
% 0.25/0.50 % Nil_notin_lex
% 0.25/0.50 thf(fact_3743_Nil2__notin__lex,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% 0.25/0.50
% 0.25/0.50 % Nil2_notin_lex
% 0.25/0.50 thf(fact_3744_lex__append__leftI,axiom,
% 0.25/0.50 ! [A: $tType,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lex_append_leftI
% 0.25/0.50 thf(fact_3745_foldl__foldl__conv__concat,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: A > B > A,A3: A,Xs: list @ ( list @ B )] :
% 0.25/0.50 ( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F ) @ A3 @ Xs )
% 0.25/0.50 = ( foldl @ A @ B @ F @ A3 @ ( concat @ B @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_foldl_conv_concat
% 0.25/0.50 thf(fact_3746_product__concat__map,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( product @ A @ B )
% 0.25/0.50 = ( ^ [Xs3: list @ A,Ys: list @ B] :
% 0.25/0.50 ( concat @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys )
% 0.25/0.50 @ Xs3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % product_concat_map
% 0.25/0.50 thf(fact_3747_lex__append__left__iff,axiom,
% 0.25/0.50 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.25/0.50 ( ! [X2: A] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 0.25/0.50 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
% 0.25/0.50 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lex_append_left_iff
% 0.25/0.50 thf(fact_3748_lex__append__leftD,axiom,
% 0.25/0.50 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.25/0.50 ( ! [X2: A] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 0.25/0.50 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lex_append_leftD
% 0.25/0.50 thf(fact_3749_lex__append__rightI,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ Vs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Us ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lex_append_rightI
% 0.25/0.50 thf(fact_3750_map__of__concat,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
% 0.25/0.50 ( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
% 0.25/0.50 = ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
% 0.25/0.50 @ ^ [Xs3: list @ ( product_prod @ A @ B ),F2: A > ( option @ B )] : ( map_add @ A @ B @ F2 @ ( map_of @ A @ B @ Xs3 ) )
% 0.25/0.50 @ Xss
% 0.25/0.50 @ ^ [X3: A] : ( none @ B ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_concat
% 0.25/0.50 thf(fact_3751_comm__append__is__replicate,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( Ys2
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( ( append @ A @ Xs @ Ys2 )
% 0.25/0.50 = ( append @ A @ Ys2 @ Xs ) )
% 0.25/0.50 => ? [N3: nat,Zs2: list @ A] :
% 0.25/0.50 ( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
% 0.25/0.50 & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
% 0.25/0.50 = ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % comm_append_is_replicate
% 0.25/0.50 thf(fact_3752_shuffles_Opinduct,axiom,
% 0.25/0.50 ! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
% 0.25/0.50 => ( ! [Ys3: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 0.25/0.50 => ( P @ ( nil @ A ) @ Ys3 ) )
% 0.25/0.50 => ( ! [Xs2: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
% 0.25/0.50 => ( P @ Xs2 @ ( nil @ A ) ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
% 0.25/0.50 => ( ( P @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) )
% 0.25/0.50 => ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
% 0.25/0.50 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
% 0.25/0.50 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % shuffles.pinduct
% 0.25/0.50 thf(fact_3753_shuffles_Opsimps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,Ys2: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
% 0.25/0.50 => ( ( shuffles @ A @ ( nil @ A ) @ Ys2 )
% 0.25/0.50 = ( insert2 @ ( list @ A ) @ Ys2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % shuffles.psimps(1)
% 0.25/0.50 thf(fact_3754_shuffles_Opsimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
% 0.25/0.50 => ( ( shuffles @ A @ Xs @ ( nil @ A ) )
% 0.25/0.50 = ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % shuffles.psimps(2)
% 0.25/0.50 thf(fact_3755_distinct__concat_H,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ ( list @ A )] :
% 0.25/0.50 ( ( distinct @ ( list @ A )
% 0.25/0.50 @ ( filter @ ( list @ A )
% 0.25/0.50 @ ^ [Ys: list @ A] :
% 0.25/0.50 ( Ys
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 @ Xs ) )
% 0.25/0.50 => ( ! [Ys3: list @ A] :
% 0.25/0.50 ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( distinct @ A @ Ys3 ) )
% 0.25/0.50 => ( ! [Ys3: list @ A,Zs2: list @ A] :
% 0.25/0.50 ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( Ys3 != Zs2 )
% 0.25/0.50 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs2 ) )
% 0.25/0.50 = ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.25/0.50 => ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % distinct_concat'
% 0.25/0.50 thf(fact_3756_shuffles_Opsimps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) )
% 0.25/0.50 => ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 0.25/0.50 = ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % shuffles.psimps(3)
% 0.25/0.50 thf(fact_3757_product__code,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys2 ) )
% 0.25/0.50 = ( set2 @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( concat @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys2 )
% 0.25/0.50 @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % product_code
% 0.25/0.50 thf(fact_3758_lenlex__conv,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( lenlex @ A )
% 0.25/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 0.25/0.50 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 0.25/0.50 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys ) )
% 0.25/0.50 | ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ys ) )
% 0.25/0.50 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lex @ A @ R3 ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lenlex_conv
% 0.25/0.50 thf(fact_3759_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.25/0.50 ! [N: nat,J: nat,I: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
% 0.25/0.50 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
% 0.25/0.50 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_sorted_list_of_set_greaterThanLessThan
% 0.25/0.50 thf(fact_3760_splice_Opinduct,axiom,
% 0.25/0.50 ! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
% 0.25/0.50 => ( ! [Ys3: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 0.25/0.50 => ( P @ ( nil @ A ) @ Ys3 ) )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) )
% 0.25/0.50 => ( ( P @ Ys3 @ Xs2 )
% 0.25/0.50 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) )
% 0.25/0.50 => ( P @ A0 @ A1 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % splice.pinduct
% 0.25/0.50 thf(fact_3761_Nil__lenlex__iff1,axiom,
% 0.25/0.50 ! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
% 0.25/0.50 = ( Ns
% 0.25/0.50 != ( nil @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Nil_lenlex_iff1
% 0.25/0.50 thf(fact_3762_lenlex__irreflexive,axiom,
% 0.25/0.50 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 0.25/0.50 ( ! [X2: A] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 0.25/0.50 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lenlex_irreflexive
% 0.25/0.50 thf(fact_3763_Nil__lenlex__iff2,axiom,
% 0.25/0.50 ! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% 0.25/0.50
% 0.25/0.50 % Nil_lenlex_iff2
% 0.25/0.50 thf(fact_3764_lenlex__length,axiom,
% 0.25/0.50 ! [A: $tType,Ms: list @ A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) )
% 0.25/0.50 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lenlex_length
% 0.25/0.50 thf(fact_3765_lenlex__append1,axiom,
% 0.25/0.50 ! [A: $tType,Us: list @ A,Xs: list @ A,R5: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R5 ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ Vs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys2 ) ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lenlex_append1
% 0.25/0.50 thf(fact_3766_member__product,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A4 @ B6 ) )
% 0.25/0.50 = ( member @ ( product_prod @ A @ B ) @ X
% 0.25/0.50 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.50 @ ^ [Uu: A] : B6 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % member_product
% 0.25/0.50 thf(fact_3767_Product__Type_Oproduct__def,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( product_product @ A @ B )
% 0.25/0.50 = ( ^ [A9: set @ A,B9: set @ B] :
% 0.25/0.50 ( product_Sigma @ A @ B @ A9
% 0.25/0.50 @ ^ [Uu: A] : B9 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Product_Type.product_def
% 0.25/0.50 thf(fact_3768_lenlex__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( lenlex @ A )
% 0.25/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R3 ) )
% 0.25/0.50 @ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lenlex_def
% 0.25/0.50 thf(fact_3769_Cons__lenlex__iff,axiom,
% 0.25/0.50 ! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
% 0.25/0.50 = ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
% 0.25/0.50 | ( ( ( size_size @ ( list @ A ) @ Ms )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ns ) )
% 0.25/0.50 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N ) @ R ) )
% 0.25/0.50 | ( ( M2 = N )
% 0.25/0.50 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Cons_lenlex_iff
% 0.25/0.50 thf(fact_3770_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.25/0.50 ! [N: nat,J: nat,I: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
% 0.25/0.50 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
% 0.25/0.50 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_sorted_list_of_set_greaterThanAtMost
% 0.25/0.50 thf(fact_3771_splice_Opelims,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
% 0.25/0.50 ( ( ( splice @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y = Xa )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( cons @ A @ X2 @ ( splice @ A @ Xa @ Xs2 ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % splice.pelims
% 0.25/0.50 thf(fact_3772_sorted__find__Min,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [Xs: list @ A,P: A > $o] :
% 0.25/0.50 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.25/0.50 => ( ? [X7: A] :
% 0.25/0.50 ( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 & ( P @ X7 ) )
% 0.25/0.50 => ( ( find @ A @ P @ Xs )
% 0.25/0.50 = ( some @ A
% 0.25/0.50 @ ( lattic643756798350308766er_Min @ A
% 0.25/0.50 @ ( collect @ A
% 0.25/0.50 @ ^ [X3: A] :
% 0.25/0.50 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 & ( P @ X3 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_find_Min
% 0.25/0.50 thf(fact_3773_part__def,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ( ( linorder_part @ B @ A )
% 0.25/0.50 = ( ^ [F2: B > A,Pivot: A,Xs3: list @ B] :
% 0.25/0.50 ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
% 0.25/0.50 @ ( filter @ B
% 0.25/0.50 @ ^ [X3: B] : ( ord_less @ A @ ( F2 @ X3 ) @ Pivot )
% 0.25/0.50 @ Xs3 )
% 0.25/0.50 @ ( product_Pair @ ( list @ B ) @ ( list @ B )
% 0.25/0.50 @ ( filter @ B
% 0.25/0.50 @ ^ [X3: B] :
% 0.25/0.50 ( ( F2 @ X3 )
% 0.25/0.50 = Pivot )
% 0.25/0.50 @ Xs3 )
% 0.25/0.50 @ ( filter @ B
% 0.25/0.50 @ ^ [X3: B] : ( ord_less @ A @ Pivot @ ( F2 @ X3 ) )
% 0.25/0.50 @ Xs3 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % part_def
% 0.25/0.50 thf(fact_3774_merge__list__correct,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [Ls: list @ ( list @ A ),As2: list @ ( list @ A )] :
% 0.25/0.50 ( ! [L5: list @ A] :
% 0.25/0.50 ( ( member @ ( list @ A ) @ L5 @ ( set2 @ ( list @ A ) @ Ls ) )
% 0.25/0.50 => ( ( distinct @ A @ L5 )
% 0.25/0.50 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L5 ) ) )
% 0.25/0.50 => ( ! [L5: list @ A] :
% 0.25/0.50 ( ( member @ ( list @ A ) @ L5 @ ( set2 @ ( list @ A ) @ As2 ) )
% 0.25/0.50 => ( ( distinct @ A @ L5 )
% 0.25/0.50 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L5 ) ) )
% 0.25/0.50 => ( ( distinct @ A @ ( merge_list @ A @ As2 @ Ls ) )
% 0.25/0.50 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As2 @ Ls ) )
% 0.25/0.50 & ( ( set2 @ A @ ( merge_list @ A @ As2 @ Ls ) )
% 0.25/0.50 = ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As2 @ Ls ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list_correct
% 0.25/0.50 thf(fact_3775_length__splice,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) )
% 0.25/0.50 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_splice
% 0.25/0.50 thf(fact_3776_splice__replicate,axiom,
% 0.25/0.50 ! [A: $tType,M2: nat,X: A,N: nat] :
% 0.25/0.50 ( ( splice @ A @ ( replicate @ A @ M2 @ X ) @ ( replicate @ A @ N @ X ) )
% 0.25/0.50 = ( replicate @ A @ ( plus_plus @ nat @ M2 @ N ) @ X ) ) ).
% 0.25/0.50
% 0.25/0.50 % splice_replicate
% 0.25/0.50 thf(fact_3777_merge__list_Osimps_I4_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [La: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
% 0.25/0.50 ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.simps(4)
% 0.25/0.50 thf(fact_3778_merge__list_Osimps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [La: list @ A,Acc22: list @ ( list @ A )] :
% 0.25/0.50 ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) )
% 0.25/0.50 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.simps(3)
% 0.25/0.50 thf(fact_3779_merge__list_Osimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A] :
% 0.25/0.50 ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 = L ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.simps(2)
% 0.25/0.50 thf(fact_3780_merge__list_Osimps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
% 0.25/0.50 = ( nil @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.simps(1)
% 0.25/0.50 thf(fact_3781_find__SomeD_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 0.25/0.50 ( ( ( find @ A @ P @ Xs )
% 0.25/0.50 = ( some @ A @ X ) )
% 0.25/0.50 => ( P @ X ) ) ).
% 0.25/0.50
% 0.25/0.50 % find_SomeD(1)
% 0.25/0.50 thf(fact_3782_find_Osimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ( P @ X )
% 0.25/0.50 => ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( some @ A @ X ) ) )
% 0.25/0.50 & ( ~ ( P @ X )
% 0.25/0.50 => ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( find @ A @ P @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % find.simps(2)
% 0.25/0.50 thf(fact_3783_find_Osimps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,Uu2: A > $o] :
% 0.25/0.50 ( ( find @ A @ Uu2 @ ( nil @ A ) )
% 0.25/0.50 = ( none @ A ) ) ).
% 0.25/0.50
% 0.25/0.50 % find.simps(1)
% 0.25/0.50 thf(fact_3784_find__SomeD_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 0.25/0.50 ( ( ( find @ A @ P @ Xs )
% 0.25/0.50 = ( some @ A @ X ) )
% 0.25/0.50 => ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % find_SomeD(2)
% 0.25/0.50 thf(fact_3785_find__None__iff2,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 0.25/0.50 ( ( ( none @ A )
% 0.25/0.50 = ( find @ A @ P @ Xs ) )
% 0.25/0.50 = ( ~ ? [X3: A] :
% 0.25/0.50 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 & ( P @ X3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % find_None_iff2
% 0.25/0.50 thf(fact_3786_find__None__iff,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 0.25/0.50 ( ( ( find @ A @ P @ Xs )
% 0.25/0.50 = ( none @ A ) )
% 0.25/0.50 = ( ~ ? [X3: A] :
% 0.25/0.50 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 & ( P @ X3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % find_None_iff
% 0.25/0.50 thf(fact_3787_part__code_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [F: B > A,Pivot2: A] :
% 0.25/0.50 ( ( linorder_part @ B @ A @ F @ Pivot2 @ ( nil @ B ) )
% 0.25/0.50 = ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % part_code(1)
% 0.25/0.50 thf(fact_3788_find__Some__iff,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 0.25/0.50 ( ( ( find @ A @ P @ Xs )
% 0.25/0.50 = ( some @ A @ X ) )
% 0.25/0.50 = ( ? [I3: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 & ( P @ ( nth @ A @ Xs @ I3 ) )
% 0.25/0.50 & ( X
% 0.25/0.50 = ( nth @ A @ Xs @ I3 ) )
% 0.25/0.50 & ! [J2: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ J2 @ I3 )
% 0.25/0.50 => ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % find_Some_iff
% 0.25/0.50 thf(fact_3789_find__Some__iff2,axiom,
% 0.25/0.50 ! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
% 0.25/0.50 ( ( ( some @ A @ X )
% 0.25/0.50 = ( find @ A @ P @ Xs ) )
% 0.25/0.50 = ( ? [I3: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 & ( P @ ( nth @ A @ Xs @ I3 ) )
% 0.25/0.50 & ( X
% 0.25/0.50 = ( nth @ A @ Xs @ I3 ) )
% 0.25/0.50 & ! [J2: nat] :
% 0.25/0.50 ( ( ord_less @ nat @ J2 @ I3 )
% 0.25/0.50 => ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % find_Some_iff2
% 0.25/0.50 thf(fact_3790_mergesort__remdups__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ( ( mergesort_remdups @ A )
% 0.25/0.50 = ( ^ [Xs3: list @ A] :
% 0.25/0.50 ( merge_list @ A @ ( nil @ ( list @ A ) )
% 0.25/0.50 @ ( map @ A @ ( list @ A )
% 0.25/0.50 @ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
% 0.25/0.50 @ Xs3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % mergesort_remdups_def
% 0.25/0.50 thf(fact_3791_splice_Opsimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys2 ) )
% 0.25/0.50 => ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 0.25/0.50 = ( cons @ A @ X @ ( splice @ A @ Ys2 @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % splice.psimps(2)
% 0.25/0.50 thf(fact_3792_splice_Opsimps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType,Ys2: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
% 0.25/0.50 => ( ( splice @ A @ ( nil @ A ) @ Ys2 )
% 0.25/0.50 = Ys2 ) ) ).
% 0.25/0.50
% 0.25/0.50 % splice.psimps(1)
% 0.25/0.50 thf(fact_3793_part__code_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [F: B > A,Pivot2: A,X: B,Xs: list @ B] :
% 0.25/0.50 ( ( linorder_part @ B @ A @ F @ Pivot2 @ ( cons @ B @ X @ Xs ) )
% 0.25/0.50 = ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
% 0.25/0.50 @ ^ [Lts: list @ B] :
% 0.25/0.50 ( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
% 0.25/0.50 @ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F @ X ) @ Pivot2 ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot2 @ ( F @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
% 0.25/0.50 @ ( linorder_part @ B @ A @ F @ Pivot2 @ Xs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % part_code(2)
% 0.25/0.50 thf(fact_3794_extract__SomeE,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A,Ys2: list @ A,Y: A,Zs: list @ A] :
% 0.25/0.50 ( ( ( extract @ A @ P @ Xs )
% 0.25/0.50 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
% 0.25/0.50 => ( ( Xs
% 0.25/0.50 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ Zs ) ) )
% 0.25/0.50 & ( P @ Y )
% 0.25/0.50 & ~ ? [X7: A] :
% 0.25/0.50 ( ( member @ A @ X7 @ ( set2 @ A @ Ys2 ) )
% 0.25/0.50 & ( P @ X7 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % extract_SomeE
% 0.25/0.50 thf(fact_3795_extract__Some__iff,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,Xs: list @ A,Ys2: list @ A,Y: A,Zs: list @ A] :
% 0.25/0.50 ( ( ( extract @ A @ P @ Xs )
% 0.25/0.50 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
% 0.25/0.50 = ( ( Xs
% 0.25/0.50 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ Zs ) ) )
% 0.25/0.50 & ( P @ Y )
% 0.25/0.50 & ~ ? [X3: A] :
% 0.25/0.50 ( ( member @ A @ X3 @ ( set2 @ A @ Ys2 ) )
% 0.25/0.50 & ( P @ X3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % extract_Some_iff
% 0.25/0.50 thf(fact_3796_extract__Cons__code,axiom,
% 0.25/0.50 ! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
% 0.25/0.50 ( ( ( P @ X )
% 0.25/0.50 => ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
% 0.25/0.50 & ( ~ ( P @ X )
% 0.25/0.50 => ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.50 = ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 0.25/0.50 @ ^ [Ys: list @ A] :
% 0.25/0.50 ( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 0.25/0.50 @ ^ [Y3: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs3 ) ) ) ) )
% 0.25/0.50 @ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % extract_Cons_code
% 0.25/0.50 thf(fact_3797_merge__list_Ocases,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [X: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
% 0.25/0.50 ( ( X
% 0.25/0.50 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 => ( ! [L5: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.50 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 => ( ! [La2: list @ A,Acc23: list @ ( list @ A ),L5: list @ A] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.50 => ~ ! [Acc23: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 0.25/0.50 ( X
% 0.25/0.50 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.cases
% 0.25/0.50 thf(fact_3798_subset__eq__mset__impl_Oelims,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
% 0.25/0.50 ( ( ( subset_eq_mset_impl @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( some @ $o
% 0.25/0.50 @ ( Xa
% 0.25/0.50 != ( nil @ A ) ) ) ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ( Y
% 0.25/0.50 != ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
% 0.25/0.50 @ ^ [Ys1: list @ A] :
% 0.25/0.50 ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
% 0.25/0.50 @ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
% 0.25/0.50 @ ( extract @ A
% 0.25/0.50 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.25/0.50 @ X2 )
% 0.25/0.50 @ Xa ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % subset_eq_mset_impl.elims
% 0.25/0.50 thf(fact_3799_lexord__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( lexord @ A )
% 0.25/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 0.25/0.50 @ ^ [X3: list @ A,Y3: list @ A] :
% 0.25/0.50 ? [A7: A,V2: list @ A] :
% 0.25/0.50 ( ( Y3
% 0.25/0.50 = ( append @ A @ X3 @ ( cons @ A @ A7 @ V2 ) ) )
% 0.25/0.50 | ? [U2: list @ A,B5: A,C3: A,W2: list @ A,Z5: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R3 )
% 0.25/0.50 & ( X3
% 0.25/0.50 = ( append @ A @ U2 @ ( cons @ A @ B5 @ W2 ) ) )
% 0.25/0.50 & ( Y3
% 0.25/0.50 = ( append @ A @ U2 @ ( cons @ A @ C3 @ Z5 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_def
% 0.25/0.50 thf(fact_3800_list__collect__set__alt,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType] :
% 0.25/0.50 ( ( list_collect_set @ B @ A )
% 0.25/0.50 = ( ^ [F2: B > ( set @ A ),L3: list @ B] :
% 0.25/0.50 ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.50 @ ( collect @ ( set @ A )
% 0.25/0.50 @ ^ [Uu: set @ A] :
% 0.25/0.50 ? [I3: nat] :
% 0.25/0.50 ( ( Uu
% 0.25/0.50 = ( F2 @ ( nth @ B @ L3 @ I3 ) ) )
% 0.25/0.50 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L3 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_alt
% 0.25/0.50 thf(fact_3801_subset__eq__mset__impl_Opelims,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
% 0.25/0.50 ( ( ( subset_eq_mset_impl @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( some @ $o
% 0.25/0.50 @ ( Xa
% 0.25/0.50 != ( nil @ A ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 0.25/0.50 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
% 0.25/0.50 @ ^ [Ys1: list @ A] :
% 0.25/0.50 ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
% 0.25/0.50 @ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
% 0.25/0.50 @ ( extract @ A
% 0.25/0.50 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.25/0.50 @ X2 )
% 0.25/0.50 @ Xa ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % subset_eq_mset_impl.pelims
% 0.25/0.50 thf(fact_3802_list__collect__set__simps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ ( nil @ B ) ) )
% 0.25/0.50 = ( F @ A3 ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_simps(2)
% 0.25/0.50 thf(fact_3803_list__collect__set__simps_I1_J,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( nil @ B ) )
% 0.25/0.50 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_simps(1)
% 0.25/0.50 thf(fact_3804_list__collect__set__simps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B,L: list @ B] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ L ) )
% 0.25/0.50 = ( sup_sup @ ( set @ A ) @ ( F @ A3 ) @ ( list_collect_set @ B @ A @ F @ L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_simps(3)
% 0.25/0.50 thf(fact_3805_list__collect__set__simps_I4_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,F: B > ( set @ A ),L: list @ B,L4: list @ B] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( append @ B @ L @ L4 ) )
% 0.25/0.50 = ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ L ) @ ( list_collect_set @ B @ A @ F @ L4 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_simps(4)
% 0.25/0.50 thf(fact_3806_lexord__cons__cons,axiom,
% 0.25/0.50 ! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R ) )
% 0.25/0.50 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.50 | ( ( A3 = B2 )
% 0.25/0.50 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_cons_cons
% 0.25/0.50 thf(fact_3807_lexord__Nil__left,axiom,
% 0.25/0.50 ! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
% 0.25/0.50 = ( ? [A7: A,X3: list @ A] :
% 0.25/0.50 ( Y
% 0.25/0.50 = ( cons @ A @ A7 @ X3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_Nil_left
% 0.25/0.50 thf(fact_3808_list__collect__set__map__simps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ ( nil @ C ) ) ) )
% 0.25/0.50 = ( F @ ( X @ A3 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_map_simps(2)
% 0.25/0.50 thf(fact_3809_list__collect__set__map__simps_I1_J,axiom,
% 0.25/0.50 ! [C: $tType,B: $tType,A: $tType,F: B > ( set @ A ),X: C > B] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( nil @ C ) ) )
% 0.25/0.50 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_map_simps(1)
% 0.25/0.50 thf(fact_3810_list__collect__set__map__simps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C,L: list @ C] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ L ) ) )
% 0.25/0.50 = ( sup_sup @ ( set @ A ) @ ( F @ ( X @ A3 ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_map_simps(3)
% 0.25/0.50 thf(fact_3811_list__collect__set__map__simps_I4_J,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,L: list @ C,L4: list @ C] :
% 0.25/0.50 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( append @ C @ L @ L4 ) ) )
% 0.25/0.50 = ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L4 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_map_simps(4)
% 0.25/0.50 thf(fact_3812_lexord__irreflexive,axiom,
% 0.25/0.50 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 0.25/0.50 ( ! [X2: A] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 0.25/0.50 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_irreflexive
% 0.25/0.50 thf(fact_3813_lexord__linear,axiom,
% 0.25/0.50 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
% 0.25/0.50 ( ! [A5: A,B3: A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R )
% 0.25/0.50 | ( A5 = B3 )
% 0.25/0.50 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R ) )
% 0.25/0.50 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 0.25/0.50 | ( X = Y )
% 0.25/0.50 | ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_linear
% 0.25/0.50 thf(fact_3814_lexord__Nil__right,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_Nil_right
% 0.25/0.50 thf(fact_3815_lexord__append__leftI,axiom,
% 0.25/0.50 ! [A: $tType,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_append_leftI
% 0.25/0.50 thf(fact_3816_lexord__partial__trans,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Ys2: list @ A,Zs: list @ A] :
% 0.25/0.50 ( ! [X2: A,Y4: A,Z3: A] :
% 0.25/0.50 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 0.25/0.50 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.50 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) ) )
% 0.25/0.50 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R ) )
% 0.25/0.50 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_partial_trans
% 0.25/0.50 thf(fact_3817_lexord__append__leftD,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) )
% 0.25/0.50 => ( ! [A5: A] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_append_leftD
% 0.25/0.50 thf(fact_3818_lexord__append__rightI,axiom,
% 0.25/0.50 ! [A: $tType,Y: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ? [B8: A,Z6: list @ A] :
% 0.25/0.50 ( Y
% 0.25/0.50 = ( cons @ A @ B8 @ Z6 ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_append_rightI
% 0.25/0.50 thf(fact_3819_lexord__sufE,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Zs: list @ A,Ys2: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R ) )
% 0.25/0.50 => ( ( Xs != Ys2 )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ Zs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Qs ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_sufE
% 0.25/0.50 thf(fact_3820_list__collect__set__as__map,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType] :
% 0.25/0.50 ( ( list_collect_set @ B @ A )
% 0.25/0.50 = ( ^ [F2: B > ( set @ A ),L3: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F2 @ L3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_as_map
% 0.25/0.50 thf(fact_3821_lexord__lex,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R ) )
% 0.25/0.50 = ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 0.25/0.50 & ( ( size_size @ ( list @ A ) @ X )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_lex
% 0.25/0.50 thf(fact_3822_lexord__append__left__rightI,axiom,
% 0.25/0.50 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_append_left_rightI
% 0.25/0.50 thf(fact_3823_lexord__same__pref__iff,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
% 0.25/0.50 = ( ? [X3: A] :
% 0.25/0.50 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R ) )
% 0.25/0.50 | ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_same_pref_iff
% 0.25/0.50 thf(fact_3824_lexord__sufI,axiom,
% 0.25/0.50 ! [A: $tType,U: list @ A,W: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z2: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R ) )
% 0.25/0.50 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W @ Z2 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % lexord_sufI
% 0.25/0.50 thf(fact_3825_list__collect__set__def,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType] :
% 0.25/0.50 ( ( list_collect_set @ B @ A )
% 0.25/0.50 = ( ^ [F2: B > ( set @ A ),L3: list @ B] :
% 0.25/0.50 ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.50 @ ( collect @ ( set @ A )
% 0.25/0.50 @ ^ [Uu: set @ A] :
% 0.25/0.50 ? [A7: B] :
% 0.25/0.50 ( ( Uu
% 0.25/0.50 = ( F2 @ A7 ) )
% 0.25/0.50 & ( member @ B @ A7 @ ( set2 @ B @ L3 ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % list_collect_set_def
% 0.25/0.50 thf(fact_3826_subset__eq__mset__impl_Osimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( subset_eq_mset_impl @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 0.25/0.50 = ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
% 0.25/0.50 @ ^ [Ys1: list @ A] :
% 0.25/0.50 ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
% 0.25/0.50 @ ^ [X3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
% 0.25/0.50 @ ( extract @ A
% 0.25/0.50 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.25/0.50 @ X )
% 0.25/0.50 @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % subset_eq_mset_impl.simps(2)
% 0.25/0.50 thf(fact_3827_List_Olexordp__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( lexordp @ A )
% 0.25/0.50 = ( ^ [R3: A > A > $o,Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % List.lexordp_def
% 0.25/0.50 thf(fact_3828_upto_Opelims,axiom,
% 0.25/0.50 ! [X: int,Xa: int,Y: list @ int] :
% 0.25/0.50 ( ( ( upto @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
% 0.25/0.50 => ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ X @ Xa )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( nil @ int ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto.pelims
% 0.25/0.50 thf(fact_3829_upto_Opsimps,axiom,
% 0.25/0.50 ! [I: int,J: int] :
% 0.25/0.50 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
% 0.25/0.50 => ( ( ( ord_less_eq @ int @ I @ J )
% 0.25/0.50 => ( ( upto @ I @ J )
% 0.25/0.50 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ I @ J )
% 0.25/0.50 => ( ( upto @ I @ J )
% 0.25/0.50 = ( nil @ int ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto.psimps
% 0.25/0.50 thf(fact_3830_listrel1__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( listrel1 @ A )
% 0.25/0.50 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.25/0.50 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 0.25/0.50 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 0.25/0.50 ? [Us2: list @ A,Z5: A,Z8: A,Vs2: list @ A] :
% 0.25/0.50 ( ( Xs3
% 0.25/0.50 = ( append @ A @ Us2 @ ( cons @ A @ Z5 @ Vs2 ) ) )
% 0.25/0.50 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ Z8 ) @ R3 )
% 0.25/0.50 & ( Ys
% 0.25/0.50 = ( append @ A @ Us2 @ ( cons @ A @ Z8 @ Vs2 ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1_def
% 0.25/0.50 thf(fact_3831_Cons__listrel1__Cons,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.50 & ( Xs = Ys2 ) )
% 0.25/0.50 | ( ( X = Y )
% 0.25/0.50 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Cons_listrel1_Cons
% 0.25/0.50 thf(fact_3832_nth__upto,axiom,
% 0.25/0.50 ! [I: int,K: nat,J: int] :
% 0.25/0.50 ( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
% 0.25/0.50 => ( ( nth @ int @ ( upto @ I @ J ) @ K )
% 0.25/0.50 = ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_upto
% 0.25/0.50 thf(fact_3833_length__upto,axiom,
% 0.25/0.50 ! [I: int,J: int] :
% 0.25/0.50 ( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
% 0.25/0.50 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % length_upto
% 0.25/0.50 thf(fact_3834_upto__rec__numeral_I1_J,axiom,
% 0.25/0.50 ! [M2: num,N: num] :
% 0.25/0.50 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 = ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 = ( nil @ int ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_rec_numeral(1)
% 0.25/0.50 thf(fact_3835_upto__rec__numeral_I2_J,axiom,
% 0.25/0.50 ! [M2: num,N: num] :
% 0.25/0.50 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 = ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 = ( nil @ int ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_rec_numeral(2)
% 0.25/0.50 thf(fact_3836_upto__rec__numeral_I3_J,axiom,
% 0.25/0.50 ! [M2: num,N: num] :
% 0.25/0.50 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 0.25/0.50 = ( nil @ int ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_rec_numeral(3)
% 0.25/0.50 thf(fact_3837_upto__rec__numeral_I4_J,axiom,
% 0.25/0.50 ! [M2: num,N: num] :
% 0.25/0.50 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.50 = ( nil @ int ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_rec_numeral(4)
% 0.25/0.50 thf(fact_3838_listrel1I2,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1I2
% 0.25/0.50 thf(fact_3839_rtrancl__listrel1__ConsI1,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % rtrancl_listrel1_ConsI1
% 0.25/0.50 thf(fact_3840_not__listrel1__Nil,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% 0.25/0.50
% 0.25/0.50 % not_listrel1_Nil
% 0.25/0.50 thf(fact_3841_not__Nil__listrel1,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% 0.25/0.50
% 0.25/0.50 % not_Nil_listrel1
% 0.25/0.50 thf(fact_3842_rtrancl__listrel1__eq__len,axiom,
% 0.25/0.50 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
% 0.25/0.50 => ( ( size_size @ ( list @ A ) @ X )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Y ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % rtrancl_listrel1_eq_len
% 0.25/0.50 thf(fact_3843_listrel1__eq__len,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 => ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1_eq_len
% 0.25/0.50 thf(fact_3844_append__listrel1I,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
% 0.25/0.50 ( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 & ( Us = Vs ) )
% 0.25/0.50 | ( ( Xs = Ys2 )
% 0.25/0.50 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % append_listrel1I
% 0.25/0.50 thf(fact_3845_upto__split2,axiom,
% 0.25/0.50 ! [I: int,J: int,K: int] :
% 0.25/0.50 ( ( ord_less_eq @ int @ I @ J )
% 0.25/0.50 => ( ( ord_less_eq @ int @ J @ K )
% 0.25/0.50 => ( ( upto @ I @ K )
% 0.25/0.50 = ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_split2
% 0.25/0.50 thf(fact_3846_upto__rec1,axiom,
% 0.25/0.50 ! [I: int,J: int] :
% 0.25/0.50 ( ( ord_less_eq @ int @ I @ J )
% 0.25/0.50 => ( ( upto @ I @ J )
% 0.25/0.50 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_rec1
% 0.25/0.50 thf(fact_3847_upto_Osimps,axiom,
% 0.25/0.50 ( upto
% 0.25/0.50 = ( ^ [I3: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J2 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto.simps
% 0.25/0.50 thf(fact_3848_upto_Oelims,axiom,
% 0.25/0.50 ! [X: int,Xa: int,Y: list @ int] :
% 0.25/0.50 ( ( ( upto @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( ( ord_less_eq @ int @ X @ Xa )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
% 0.25/0.50 & ( ~ ( ord_less_eq @ int @ X @ Xa )
% 0.25/0.50 => ( Y
% 0.25/0.50 = ( nil @ int ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto.elims
% 0.25/0.50 thf(fact_3849_Cons__listrel1E2,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 => ( ! [X2: A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 = ( cons @ A @ X2 @ Ys2 ) )
% 0.25/0.50 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
% 0.25/0.50 => ~ ! [Zs2: list @ A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 = ( cons @ A @ Y @ Zs2 ) )
% 0.25/0.50 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Cons_listrel1E2
% 0.25/0.50 thf(fact_3850_Cons__listrel1E1,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 => ( ! [Y4: A] :
% 0.25/0.50 ( ( Ys2
% 0.25/0.50 = ( cons @ A @ Y4 @ Xs ) )
% 0.25/0.50 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R ) )
% 0.25/0.50 => ~ ! [Zs2: list @ A] :
% 0.25/0.50 ( ( Ys2
% 0.25/0.50 = ( cons @ A @ X @ Zs2 ) )
% 0.25/0.50 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % Cons_listrel1E1
% 0.25/0.50 thf(fact_3851_listrel1I1,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1I1
% 0.25/0.50 thf(fact_3852_greaterThanAtMost__upto,axiom,
% 0.25/0.50 ( ( set_or3652927894154168847AtMost @ int )
% 0.25/0.50 = ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % greaterThanAtMost_upto
% 0.25/0.50 thf(fact_3853_listrel1I,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.50 => ( ( Xs
% 0.25/0.50 = ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
% 0.25/0.50 => ( ( Ys2
% 0.25/0.50 = ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1I
% 0.25/0.50 thf(fact_3854_listrel1E,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 => ~ ! [X2: A,Y4: A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 0.25/0.50 => ! [Us3: list @ A,Vs3: list @ A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 = ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
% 0.25/0.50 => ( Ys2
% 0.25/0.50 != ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1E
% 0.25/0.50 thf(fact_3855_upto__split3,axiom,
% 0.25/0.50 ! [I: int,J: int,K: int] :
% 0.25/0.50 ( ( ord_less_eq @ int @ I @ J )
% 0.25/0.50 => ( ( ord_less_eq @ int @ J @ K )
% 0.25/0.50 => ( ( upto @ I @ K )
% 0.25/0.50 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % upto_split3
% 0.25/0.50 thf(fact_3856_rtrancl__listrel1__ConsI2,axiom,
% 0.25/0.50 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.50 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
% 0.25/0.50 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % rtrancl_listrel1_ConsI2
% 0.25/0.50 thf(fact_3857_greaterThanLessThan__upto,axiom,
% 0.25/0.50 ( ( set_or5935395276787703475ssThan @ int )
% 0.25/0.50 = ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % greaterThanLessThan_upto
% 0.25/0.50 thf(fact_3858_snoc__listrel1__snoc__iff,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.50 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 = ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.50 & ( X = Y ) )
% 0.25/0.50 | ( ( Xs = Ys2 )
% 0.25/0.50 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % snoc_listrel1_snoc_iff
% 0.25/0.50 thf(fact_3859_listrel1p__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( listrel1p @ A )
% 0.25/0.50 = ( ^ [R3: A > A > $o,Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % listrel1p_def
% 0.25/0.50 thf(fact_3860_sorted__wrt__map__rev__linord,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( sorted_wrt @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y3 ) @ ( product_fst @ A @ B @ X3 ) )
% 0.25/0.50 @ L )
% 0.25/0.50 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt_map_rev_linord
% 0.25/0.50 thf(fact_3861_set__zip,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 0.25/0.50 = ( collect @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [Uu: product_prod @ A @ B] :
% 0.25/0.50 ? [I3: nat] :
% 0.25/0.50 ( ( Uu
% 0.25/0.50 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys2 @ I3 ) ) )
% 0.25/0.50 & ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_zip
% 0.25/0.50 thf(fact_3862_arg__min__list_Opelims,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( linorder @ B )
% 0.25/0.50 => ! [X: A > B,Xa: list @ A,Y: A] :
% 0.25/0.50 ( ( ( arg_min_list @ A @ B @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.50 => ( ! [X2: A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.50 => ( ( Y = X2 )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 0.25/0.50 => ( ! [X2: A,Y4: A,Zs2: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) ) )
% 0.25/0.50 => ~ ( ( Xa
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( undefined @ A ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % arg_min_list.pelims
% 0.25/0.50 thf(fact_3863_zip__eq__zip__same__len,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A6: list @ A,B4: list @ B] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ A3 )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ B2 ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ A6 )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ B4 ) )
% 0.25/0.50 => ( ( ( zip @ A @ B @ A3 @ B2 )
% 0.25/0.50 = ( zip @ A @ B @ A6 @ B4 ) )
% 0.25/0.50 = ( ( A3 = A6 )
% 0.25/0.50 & ( B2 = B4 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_eq_zip_same_len
% 0.25/0.50 thf(fact_3864_sorted__wrt__rev__linord,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A] :
% 0.25/0.50 ( ( sorted_wrt @ A
% 0.25/0.50 @ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
% 0.25/0.50 @ L )
% 0.25/0.50 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % sorted_wrt_rev_linord
% 0.25/0.50 thf(fact_3865_zip__Cons__Cons,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys2: list @ B] :
% 0.25/0.50 ( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) )
% 0.25/0.50 = ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_Cons_Cons
% 0.25/0.50 thf(fact_3866_map__of__rev__distinct,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,M2: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M2 ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M2 ) )
% 0.25/0.50 = ( map_of @ A @ B @ M2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_rev_distinct
% 0.25/0.50 thf(fact_3867_zip__replicate,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
% 0.25/0.50 ( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
% 0.25/0.50 = ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_replicate
% 0.25/0.50 thf(fact_3868_map__of__zip__is__None,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
% 0.25/0.50 = ( none @ B ) )
% 0.25/0.50 = ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_zip_is_None
% 0.25/0.50 thf(fact_3869_nth__zip,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
% 0.25/0.50 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nth_zip
% 0.25/0.50 thf(fact_3870_zip__assoc,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C] :
% 0.25/0.50 ( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys2 @ Zs ) )
% 0.25/0.50 = ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 0.25/0.50 @ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 0.25/0.50 @ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
% 0.25/0.50 @ ^ [X3: A,Y3: B,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
% 0.25/0.50 @ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys2 ) @ Zs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_assoc
% 0.25/0.50 thf(fact_3871_pair__list__split,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,L: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ~ ! [L12: list @ A,L22: list @ B] :
% 0.25/0.50 ( ( L
% 0.25/0.50 = ( zip @ A @ B @ L12 @ L22 ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ L12 )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ L22 ) )
% 0.25/0.50 => ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ L )
% 0.25/0.50 != ( size_size @ ( list @ B ) @ L22 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % pair_list_split
% 0.25/0.50 thf(fact_3872_zip__inj,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A6: list @ A,B4: list @ B] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ A3 )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ B2 ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ A6 )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ B4 ) )
% 0.25/0.50 => ( ( ( zip @ A @ B @ A3 @ B2 )
% 0.25/0.50 = ( zip @ A @ B @ A6 @ B4 ) )
% 0.25/0.50 => ( ( A3 = A6 )
% 0.25/0.50 & ( B2 = B4 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_inj
% 0.25/0.50 thf(fact_3873_zip__left__commute,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C] :
% 0.25/0.50 ( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys2 @ Zs ) )
% 0.25/0.50 = ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 0.25/0.50 @ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 0.25/0.50 @ ^ [Y3: B] :
% 0.25/0.50 ( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 0.25/0.50 @ ^ [X3: A,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
% 0.25/0.50 @ ( zip @ B @ ( product_prod @ A @ C ) @ Ys2 @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_left_commute
% 0.25/0.50 thf(fact_3874_zip__same__conv__map,axiom,
% 0.25/0.50 ! [A: $tType,Xs: list @ A] :
% 0.25/0.50 ( ( zip @ A @ A @ Xs @ Xs )
% 0.25/0.50 = ( map @ A @ ( product_prod @ A @ A )
% 0.25/0.50 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 0.25/0.50 @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_same_conv_map
% 0.25/0.50 thf(fact_3875_set__zip__rightD,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 0.25/0.50 => ( member @ B @ Y @ ( set2 @ B @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_zip_rightD
% 0.25/0.50 thf(fact_3876_set__zip__leftD,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 0.25/0.50 => ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_zip_leftD
% 0.25/0.50 thf(fact_3877_in__set__zipE,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 0.25/0.50 => ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.50 => ~ ( member @ B @ Y @ ( set2 @ B @ Ys2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_zipE
% 0.25/0.50 thf(fact_3878_zip__same,axiom,
% 0.25/0.50 ! [A: $tType,A3: A,B2: A,Xs: list @ A] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
% 0.25/0.50 = ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
% 0.25/0.50 & ( A3 = B2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_same
% 0.25/0.50 thf(fact_3879_zip__eq__ConsE,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
% 0.25/0.50 ( ( ( zip @ A @ B @ Xs @ Ys2 )
% 0.25/0.50 = ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
% 0.25/0.50 => ~ ! [X2: A,Xs6: list @ A] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 = ( cons @ A @ X2 @ Xs6 ) )
% 0.25/0.50 => ! [Y4: B,Ys4: list @ B] :
% 0.25/0.50 ( ( Ys2
% 0.25/0.50 = ( cons @ B @ Y4 @ Ys4 ) )
% 0.25/0.50 => ( ( Xy2
% 0.25/0.50 = ( product_Pair @ A @ B @ X2 @ Y4 ) )
% 0.25/0.50 => ( Xys
% 0.25/0.50 != ( zip @ A @ B @ Xs6 @ Ys4 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_eq_ConsE
% 0.25/0.50 thf(fact_3880_set__zip__cart,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L4: list @ B] :
% 0.25/0.50 ( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L4 ) ) )
% 0.25/0.50 => ( member @ ( product_prod @ A @ B ) @ X
% 0.25/0.50 @ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
% 0.25/0.50 @ ^ [Uu: A] : ( set2 @ B @ L4 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % set_zip_cart
% 0.25/0.50 thf(fact_3881_zip__commute,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( zip @ A @ B )
% 0.25/0.50 = ( ^ [Xs3: list @ A,Ys: list @ B] :
% 0.25/0.50 ( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) )
% 0.25/0.50 @ ( zip @ B @ A @ Ys @ Xs3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_commute
% 0.25/0.50 thf(fact_3882_last__zip,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( Xs
% 0.25/0.50 != ( nil @ A ) )
% 0.25/0.50 => ( ( Ys2
% 0.25/0.50 != ( nil @ B ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 0.25/0.50 = ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys2 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % last_zip
% 0.25/0.50 thf(fact_3883_in__set__impl__in__set__zip1,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.50 => ~ ! [Y4: B] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_impl_in_set_zip1
% 0.25/0.50 thf(fact_3884_in__set__impl__in__set__zip2,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Y: B] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( member @ B @ Y @ ( set2 @ B @ Ys2 ) )
% 0.25/0.50 => ~ ! [X2: A] :
% 0.25/0.50 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % in_set_impl_in_set_zip2
% 0.25/0.50 thf(fact_3885_map__of__zip__is__Some,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.50 = ( ? [Y3: B] :
% 0.25/0.50 ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
% 0.25/0.50 = ( some @ B @ Y3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_zip_is_Some
% 0.25/0.50 thf(fact_3886_map__upds__fold__map__upd,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType] :
% 0.25/0.50 ( ( map_upds @ A @ B )
% 0.25/0.50 = ( ^ [M: A > ( option @ B ),Ks2: list @ A,Vs2: list @ B] :
% 0.25/0.50 ( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [N5: A > ( option @ B )] :
% 0.25/0.50 ( product_case_prod @ A @ B @ ( A > ( option @ B ) )
% 0.25/0.50 @ ^ [K4: A,V2: B] : ( fun_upd @ A @ ( option @ B ) @ N5 @ K4 @ ( some @ B @ V2 ) ) )
% 0.25/0.50 @ M
% 0.25/0.50 @ ( zip @ A @ B @ Ks2 @ Vs2 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_upds_fold_map_upd
% 0.25/0.50 thf(fact_3887_enumerate__eq__zip,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( enumerate @ A )
% 0.25/0.50 = ( ^ [N5: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N5 @ ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % enumerate_eq_zip
% 0.25/0.50 thf(fact_3888_map__zip__map,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys2: list @ C] :
% 0.25/0.50 ( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys2 ) )
% 0.25/0.50 = ( map @ ( product_prod @ D @ C ) @ A
% 0.25/0.50 @ ( product_case_prod @ D @ C @ A
% 0.25/0.50 @ ^ [X3: D,Y3: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X3 ) @ Y3 ) ) )
% 0.25/0.50 @ ( zip @ D @ C @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_zip_map
% 0.25/0.50 thf(fact_3889_map__zip__map2,axiom,
% 0.25/0.50 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys2: list @ D] :
% 0.25/0.50 ( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys2 ) ) )
% 0.25/0.50 = ( map @ ( product_prod @ B @ D ) @ A
% 0.25/0.50 @ ( product_case_prod @ B @ D @ A
% 0.25/0.50 @ ^ [X3: B,Y3: D] : ( F @ ( product_Pair @ B @ C @ X3 @ ( G @ Y3 ) ) ) )
% 0.25/0.50 @ ( zip @ B @ D @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_zip_map2
% 0.25/0.50 thf(fact_3890_foldl__snd__zip,axiom,
% 0.25/0.50 ! [B: $tType,C: $tType,A: $tType,Ys2: list @ A,Xs: list @ B,F: C > A > C,B2: C] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ B ) @ Xs ) )
% 0.25/0.50 => ( ( foldl @ C @ ( product_prod @ B @ A )
% 0.25/0.50 @ ^ [B5: C] :
% 0.25/0.50 ( product_case_prod @ B @ A @ C
% 0.25/0.50 @ ^ [X3: B] : ( F @ B5 ) )
% 0.25/0.50 @ B2
% 0.25/0.50 @ ( zip @ B @ A @ Xs @ Ys2 ) )
% 0.25/0.50 = ( foldl @ C @ A @ F @ B2 @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldl_snd_zip
% 0.25/0.50 thf(fact_3891_map__zip2,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,K: A,L: list @ B] :
% 0.25/0.50 ( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L )
% 0.25/0.50 = ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) @ L ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_zip2
% 0.25/0.50 thf(fact_3892_map__zip1,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,K: B,L: list @ A] :
% 0.25/0.50 ( ( map @ A @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
% 0.25/0.50 @ L )
% 0.25/0.50 = ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_zip1
% 0.25/0.50 thf(fact_3893_map__prod__fun__zip,axiom,
% 0.25/0.50 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,Xs: list @ C,Ys2: list @ D] :
% 0.25/0.50 ( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 0.25/0.50 @ ( zip @ C @ D @ Xs @ Ys2 ) )
% 0.25/0.50 = ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ ( map @ D @ B @ G @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_prod_fun_zip
% 0.25/0.50 thf(fact_3894_zip__map2,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F: C > B,Ys2: list @ C] :
% 0.25/0.50 ( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F @ Ys2 ) )
% 0.25/0.50 = ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: A,Y3: C] : ( product_Pair @ A @ B @ X3 @ ( F @ Y3 ) ) )
% 0.25/0.50 @ ( zip @ A @ C @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_map2
% 0.25/0.50 thf(fact_3895_zip__map1,axiom,
% 0.25/0.50 ! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: list @ C,Ys2: list @ B] :
% 0.25/0.50 ( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ Ys2 )
% 0.25/0.50 = ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
% 0.25/0.50 @ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
% 0.25/0.50 @ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) ) )
% 0.25/0.50 @ ( zip @ C @ B @ Xs @ Ys2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_map1
% 0.25/0.50 thf(fact_3896_map__of__zip__map,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
% 0.25/0.50 ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F @ Xs ) ) )
% 0.25/0.50 = ( ^ [X3: A] : ( if @ ( option @ B ) @ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F @ X3 ) ) @ ( none @ B ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_zip_map
% 0.25/0.50 thf(fact_3897_map__of__zip__upd,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Ys2: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y: B,Z2: B] :
% 0.25/0.50 ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ( ( size_size @ ( list @ B ) @ Zs )
% 0.25/0.50 = ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.50 => ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.50 => ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y ) )
% 0.25/0.50 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z2 ) ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 0.25/0.50 = ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_zip_upd
% 0.25/0.50 thf(fact_3898_map__of__zip__nth,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
% 0.25/0.50 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.50 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( distinct @ A @ Xs )
% 0.25/0.50 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.50 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
% 0.25/0.50 = ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % map_of_zip_nth
% 0.25/0.50 thf(fact_3899_foldr__snd__zip,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,C: $tType,Ys2: list @ A,Xs: list @ B,F: A > C > C,B2: C] :
% 0.25/0.50 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ B ) @ Xs ) )
% 0.25/0.50 => ( ( foldr @ ( product_prod @ B @ A ) @ C
% 0.25/0.50 @ ( product_case_prod @ B @ A @ ( C > C )
% 0.25/0.50 @ ^ [X3: B] : F )
% 0.25/0.50 @ ( zip @ B @ A @ Xs @ Ys2 )
% 0.25/0.50 @ B2 )
% 0.25/0.50 = ( foldr @ A @ C @ F @ Ys2 @ B2 ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % foldr_snd_zip
% 0.25/0.50 thf(fact_3900_nths__shift__lemma,axiom,
% 0.25/0.50 ! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
% 0.25/0.50 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.25/0.50 @ ( filter @ ( product_prod @ A @ nat )
% 0.25/0.50 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A4 )
% 0.25/0.50 @ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
% 0.25/0.50 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 0.25/0.50 @ ( filter @ ( product_prod @ A @ nat )
% 0.25/0.50 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A4 )
% 0.25/0.50 @ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % nths_shift_lemma
% 0.25/0.50 thf(fact_3901_remove__rev__alt__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( remove_rev @ A )
% 0.25/0.50 = ( ^ [X3: A,Xs3: list @ A] :
% 0.25/0.50 ( filter @ A
% 0.25/0.50 @ ^ [Y3: A] : ( Y3 != X3 )
% 0.25/0.50 @ ( rev @ A @ Xs3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % remove_rev_alt_def
% 0.25/0.50 thf(fact_3902_zip__Cons1,axiom,
% 0.25/0.50 ! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.50 ( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 0.25/0.50 = ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ^ [Y3: B,Ys: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( zip @ A @ B @ Xs @ Ys ) )
% 0.25/0.50 @ Ys2 ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_Cons1
% 0.25/0.50 thf(fact_3903_zip__Cons,axiom,
% 0.25/0.50 ! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys2: list @ B] :
% 0.25/0.50 ( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys2 ) )
% 0.25/0.50 = ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
% 0.25/0.50 @ ^ [Z5: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z5 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys2 ) )
% 0.25/0.50 @ Xs ) ) ).
% 0.25/0.50
% 0.25/0.50 % zip_Cons
% 0.25/0.50 thf(fact_3904_filter__rev__alt,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( filter_rev @ A )
% 0.25/0.50 = ( ^ [P4: A > $o,L3: list @ A] : ( filter @ A @ P4 @ ( rev @ A @ L3 ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % filter_rev_alt
% 0.25/0.50 thf(fact_3905_remove__rev__def,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( remove_rev @ A )
% 0.25/0.50 = ( ^ [X3: A] :
% 0.25/0.50 ( filter_rev @ A
% 0.25/0.50 @ ( comp @ $o @ $o @ A @ (~)
% 0.25/0.50 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.25/0.50 @ X3 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % remove_rev_def
% 0.25/0.50 thf(fact_3906_min__list_Opelims,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( ord @ A )
% 0.25/0.50 => ! [X: list @ A,Y: A] :
% 0.25/0.50 ( ( ( min_list @ A @ X )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
% 0.25/0.50 => ( ! [X2: A,Xs2: list @ A] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( case_list @ A @ A @ X2
% 0.25/0.50 @ ^ [A7: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
% 0.25/0.50 @ Xs2 ) )
% 0.25/0.50 => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.25/0.50 => ~ ( ( X
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( undefined @ A ) )
% 0.25/0.50 => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % min_list.pelims
% 0.25/0.50 thf(fact_3907_merge__list_Opsimps_I2_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [L: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.50 => ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 = L ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.psimps(2)
% 0.25/0.50 thf(fact_3908_merge__list_Opsimps_I3_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [La: list @ A,Acc22: list @ ( list @ A )] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 => ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) )
% 0.25/0.50 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.psimps(3)
% 0.25/0.50 thf(fact_3909_merge__list_Opsimps_I4_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [La: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.50 => ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.psimps(4)
% 0.25/0.50 thf(fact_3910_merge__list_Opsimps_I1_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 => ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
% 0.25/0.50 = ( nil @ A ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.psimps(1)
% 0.25/0.50 thf(fact_3911_merge__list_Opelims,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
% 0.25/0.50 ( ( ( merge_list @ A @ X @ Xa )
% 0.25/0.50 = Y )
% 0.25/0.50 => ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ Xa ) )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ ( list @ A ) ) )
% 0.25/0.50 => ( ( Xa
% 0.25/0.50 = ( nil @ ( list @ A ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( nil @ A ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
% 0.25/0.50 => ( ( ( X
% 0.25/0.50 = ( nil @ ( list @ A ) ) )
% 0.25/0.50 => ! [L5: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 => ( ( Y = L5 )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
% 0.25/0.50 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 0.25/0.50 => ( ( Xa
% 0.25/0.50 = ( nil @ ( list @ A ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
% 0.25/0.50 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 0.25/0.50 ( ( X
% 0.25/0.50 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 0.25/0.50 => ! [L5: list @ A] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
% 0.25/0.50 => ~ ! [L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 0.25/0.50 ( ( Xa
% 0.25/0.50 = ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
% 0.25/0.50 => ( ( Y
% 0.25/0.50 = ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls2 ) )
% 0.25/0.50 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.pelims
% 0.25/0.50 thf(fact_3912_merge__list_Opsimps_I5_J,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( linorder @ A )
% 0.25/0.50 => ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls: list @ ( list @ A )] :
% 0.25/0.50 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) ) )
% 0.25/0.50 => ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) )
% 0.25/0.50 = ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls ) ) ) ) ).
% 0.25/0.50
% 0.25/0.50 % merge_list.psimps(5)
% 0.25/0.50 thf(fact_3913_partition__filter__conv,axiom,
% 0.25/0.50 ! [A: $tType] :
% 0.25/0.50 ( ( partition @ A )
% 0.25/0.50 = ( ^ [F2: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter @ A @ F2 @ Xs3 ) @ ( filter @ A @ ( comp @ $o @ $o @ A @ (~) @ F2 ) @ Xs3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_filter_conv
% 0.25/0.51 thf(fact_3914_partition__rev__filter__conv,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,Xs: list @ A] :
% 0.25/0.51 ( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ Xs )
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( rev @ A @ ( filter @ A @ P @ Xs ) ) @ Yes2 ) @ ( append @ A @ ( rev @ A @ ( filter @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) @ No2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_rev_filter_conv
% 0.25/0.51 thf(fact_3915_merge_Osimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [L2: list @ A] :
% 0.25/0.51 ( ( merge @ A @ ( nil @ A ) @ L2 )
% 0.25/0.51 = L2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge.simps(1)
% 0.25/0.51 thf(fact_3916_merge_Osimps_I3_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [X1: A,X22: A,L1: list @ A,L2: list @ A] :
% 0.25/0.51 ( ( ( ord_less @ A @ X1 @ X22 )
% 0.25/0.51 => ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 0.25/0.51 = ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X22 @ L2 ) ) ) ) )
% 0.25/0.51 & ( ~ ( ord_less @ A @ X1 @ X22 )
% 0.25/0.51 => ( ( ( X1 = X22 )
% 0.25/0.51 => ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 0.25/0.51 = ( cons @ A @ X1 @ ( merge @ A @ L1 @ L2 ) ) ) )
% 0.25/0.51 & ( ( X1 != X22 )
% 0.25/0.51 => ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 0.25/0.51 = ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge.simps(3)
% 0.25/0.51 thf(fact_3917_merge_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [V: A,Va2: list @ A] :
% 0.25/0.51 ( ( merge @ A @ ( cons @ A @ V @ Va2 ) @ ( nil @ A ) )
% 0.25/0.51 = ( cons @ A @ V @ Va2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge.simps(2)
% 0.25/0.51 thf(fact_3918_partition__rev_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) ) @ Xs ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_rev.simps(2)
% 0.25/0.51 thf(fact_3919_partition__rev_Osimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A] :
% 0.25/0.51 ( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( nil @ A ) )
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_rev.simps(1)
% 0.25/0.51 thf(fact_3920_merge__list_Osimps_I5_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls: list @ ( list @ A )] :
% 0.25/0.51 ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) )
% 0.25/0.51 = ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge_list.simps(5)
% 0.25/0.51 thf(fact_3921_partition_Osimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o] :
% 0.25/0.51 ( ( partition @ A @ P @ ( nil @ A ) )
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition.simps(1)
% 0.25/0.51 thf(fact_3922_partition__P,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
% 0.25/0.51 ( ( ( partition @ A @ P @ Xs )
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
% 0.25/0.51 => ( ! [X7: A] :
% 0.25/0.51 ( ( member @ A @ X7 @ ( set2 @ A @ Yes2 ) )
% 0.25/0.51 => ( P @ X7 ) )
% 0.25/0.51 & ! [X7: A] :
% 0.25/0.51 ( ( member @ A @ X7 @ ( set2 @ A @ No2 ) )
% 0.25/0.51 => ~ ( P @ X7 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_P
% 0.25/0.51 thf(fact_3923_merge_Oelims,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [X: list @ A,Xa: list @ A,Y: list @ A] :
% 0.25/0.51 ( ( ( merge @ A @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( Y != Xa ) )
% 0.25/0.51 => ( ! [V4: A,Va: list @ A] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.51 => ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( cons @ A @ V4 @ Va ) ) ) )
% 0.25/0.51 => ~ ! [X13: A,L12: list @ A] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ A @ X13 @ L12 ) )
% 0.25/0.51 => ! [X23: A,L22: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ X23 @ L22 ) )
% 0.25/0.51 => ~ ( ( ( ord_less @ A @ X13 @ X23 )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
% 0.25/0.51 & ( ~ ( ord_less @ A @ X13 @ X23 )
% 0.25/0.51 => ( ( ( X13 = X23 )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ L22 ) ) ) )
% 0.25/0.51 & ( ( X13 != X23 )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X13 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge.elims
% 0.25/0.51 thf(fact_3924_partition_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 0.25/0.51 @ ^ [Yes3: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X @ No3 ) ) )
% 0.25/0.51 @ ( partition @ A @ P @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition.simps(2)
% 0.25/0.51 thf(fact_3925_partition__rev_Oelims,axiom,
% 0.25/0.51 ! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.51 ( ( ( partition_rev @ A @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ! [Yes: list @ A,No: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 0.25/0.51 => ( ( Xb
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) ) ) )
% 0.25/0.51 => ~ ! [Yes: list @ A,No: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 0.25/0.51 => ! [X2: A,Xs2: list @ A] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_rev.elims
% 0.25/0.51 thf(fact_3926_partition__set,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
% 0.25/0.51 ( ( ( partition @ A @ P @ Xs )
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
% 0.25/0.51 => ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes2 ) @ ( set2 @ A @ No2 ) )
% 0.25/0.51 = ( set2 @ A @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_set
% 0.25/0.51 thf(fact_3927_inv__image__partition,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,P: A > $o,Ys2: list @ A] :
% 0.25/0.51 ( ! [X2: A] :
% 0.25/0.51 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.51 => ( P @ X2 ) )
% 0.25/0.51 => ( ! [Y4: A] :
% 0.25/0.51 ( ( member @ A @ Y4 @ ( set2 @ A @ Ys2 ) )
% 0.25/0.51 => ~ ( P @ Y4 ) )
% 0.25/0.51 => ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
% 0.25/0.51 = ( shuffles @ A @ Xs @ Ys2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % inv_image_partition
% 0.25/0.51 thf(fact_3928_merge__list_Oelims,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
% 0.25/0.51 ( ( ( merge_list @ A @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ( ( Xa
% 0.25/0.51 = ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( nil @ A ) ) ) )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ! [L5: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 => ( Y != L5 ) ) )
% 0.25/0.51 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 0.25/0.51 => ( ( Xa
% 0.25/0.51 = ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) )
% 0.25/0.51 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 0.25/0.51 => ! [L5: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) )
% 0.25/0.51 => ~ ! [L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge_list.elims
% 0.25/0.51 thf(fact_3929_merge__correct,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [L1: list @ A,L2: list @ A] :
% 0.25/0.51 ( ( ( distinct @ A @ L1 )
% 0.25/0.51 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L1 ) )
% 0.25/0.51 => ( ( ( distinct @ A @ L2 )
% 0.25/0.51 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) )
% 0.25/0.51 => ( ( distinct @ A @ ( merge @ A @ L1 @ L2 ) )
% 0.25/0.51 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge @ A @ L1 @ L2 ) )
% 0.25/0.51 & ( ( set2 @ A @ ( merge @ A @ L1 @ L2 ) )
% 0.25/0.51 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge_correct
% 0.25/0.51 thf(fact_3930_merge__list_Opinduct,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [A0: list @ ( list @ A ),A1: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ A0 @ A1 ) )
% 0.25/0.51 => ( ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 => ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 => ( ! [L5: list @ A] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.51 => ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.51 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 => ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 0.25/0.51 => ( P @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.51 => ( ! [La2: list @ A,Acc23: list @ ( list @ A ),L5: list @ A] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.51 => ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) )
% 0.25/0.51 => ( P @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) )
% 0.25/0.51 => ( ! [Acc23: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) )
% 0.25/0.51 => ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ Acc23 ) @ Ls2 )
% 0.25/0.51 => ( P @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) )
% 0.25/0.51 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge_list.pinduct
% 0.25/0.51 thf(fact_3931_partition__rev_Opelims,axiom,
% 0.25/0.51 ! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 0.25/0.51 ( ( ( partition_rev @ A @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ Xa @ Xb ) ) )
% 0.25/0.51 => ( ! [Yes: list @ A,No: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 0.25/0.51 => ( ( Xb
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) ) ) ) )
% 0.25/0.51 => ~ ! [Yes: list @ A,No: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 0.25/0.51 => ! [X2: A,Xs2: list @ A] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % partition_rev.pelims
% 0.25/0.51 thf(fact_3932_merge_Opelims,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [X: list @ A,Xa: list @ A,Y: list @ A] :
% 0.25/0.51 ( ( ( merge @ A @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Y = Xa )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 0.25/0.51 => ( ! [V4: A,Va: list @ A] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.51 => ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) ) ) ) )
% 0.25/0.51 => ~ ! [X13: A,L12: list @ A] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ A @ X13 @ L12 ) )
% 0.25/0.51 => ! [X23: A,L22: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ X23 @ L22 ) )
% 0.25/0.51 => ( ( ( ( ord_less @ A @ X13 @ X23 )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
% 0.25/0.51 & ( ~ ( ord_less @ A @ X13 @ X23 )
% 0.25/0.51 => ( ( ( X13 = X23 )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ L22 ) ) ) )
% 0.25/0.51 & ( ( X13 != X23 )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X13 @ L12 ) @ L22 ) ) ) ) ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % merge.pelims
% 0.25/0.51 thf(fact_3933_transpose_Opinduct,axiom,
% 0.25/0.51 ! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
% 0.25/0.51 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
% 0.25/0.51 => ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ( P @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 => ( ! [Xss2: list @ ( list @ A )] :
% 0.25/0.51 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
% 0.25/0.51 => ( ( P @ Xss2 )
% 0.25/0.51 => ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
% 0.25/0.51 => ( ! [X2: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
% 0.25/0.51 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) )
% 0.25/0.51 => ( ( P
% 0.25/0.51 @ ( cons @ ( list @ A ) @ Xs2
% 0.25/0.51 @ ( concat @ ( list @ A )
% 0.25/0.51 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 0.25/0.51 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
% 0.25/0.51 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 @ Xss2 ) ) ) )
% 0.25/0.51 => ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) ) ) )
% 0.25/0.51 => ( P @ A0 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % transpose.pinduct
% 0.25/0.51 thf(fact_3934_quicksort__by__rel_Oelims,axiom,
% 0.25/0.51 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 0.25/0.51 ( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( Xb
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( Y != Xa ) )
% 0.25/0.51 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 0.25/0.51 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
% 0.25/0.51 @ ( partition_rev @ A
% 0.25/0.51 @ ^ [Y3: A] : ( X @ Y3 @ X2 )
% 0.25/0.51 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 0.25/0.51 @ Xs2 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.elims
% 0.25/0.51 thf(fact_3935_set__quicksort__by__rel,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,Xs: list @ A] :
% 0.25/0.51 ( ( set2 @ A @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs ) )
% 0.25/0.51 = ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % set_quicksort_by_rel
% 0.25/0.51 thf(fact_3936_quicksort__by__rel_Osimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Sl2: list @ A] :
% 0.25/0.51 ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( nil @ A ) )
% 0.25/0.51 = Sl2 ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.simps(1)
% 0.25/0.51 thf(fact_3937_quicksort__by__rel__remove__acc,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( quicksort_by_rel @ A )
% 0.25/0.51 = ( ^ [R6: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R6 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel_remove_acc
% 0.25/0.51 thf(fact_3938_quicksort__by__rel__remove__acc__guared,axiom,
% 0.25/0.51 ! [A: $tType,Sl2: list @ A,R5: A > A > $o,Xs: list @ A] :
% 0.25/0.51 ( ( Sl2
% 0.25/0.51 != ( nil @ A ) )
% 0.25/0.51 => ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs )
% 0.25/0.51 = ( append @ A @ ( quicksort_by_rel @ A @ R5 @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel_remove_acc_guared
% 0.25/0.51 thf(fact_3939_sorted__wrt__quicksort__by__rel,axiom,
% 0.25/0.51 ! [X14: $tType,R5: X14 > X14 > $o,Xs: list @ X14] :
% 0.25/0.51 ( ! [X2: X14,Y4: X14] :
% 0.25/0.51 ( ( R5 @ X2 @ Y4 )
% 0.25/0.51 | ( R5 @ Y4 @ X2 ) )
% 0.25/0.51 => ( ! [X2: X14,Y4: X14,Z3: X14] :
% 0.25/0.51 ( ( R5 @ X2 @ Y4 )
% 0.25/0.51 => ( ( R5 @ Y4 @ Z3 )
% 0.25/0.51 => ( R5 @ X2 @ Z3 ) ) )
% 0.25/0.51 => ( sorted_wrt @ X14 @ R5 @ ( quicksort_by_rel @ X14 @ R5 @ ( nil @ X14 ) @ Xs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % sorted_wrt_quicksort_by_rel
% 0.25/0.51 thf(fact_3940_sorted__quicksort__by__rel,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( linorder @ A )
% 0.25/0.51 => ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % sorted_quicksort_by_rel
% 0.25/0.51 thf(fact_3941_quicksort__by__rel_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 0.25/0.51 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs_b ) ) @ Xs_s )
% 0.25/0.51 @ ( partition_rev @ A
% 0.25/0.51 @ ^ [Y3: A] : ( R5 @ Y3 @ X )
% 0.25/0.51 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 0.25/0.51 @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.simps(2)
% 0.25/0.51 thf(fact_3942_transpose_Opsimps_I3_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
% 0.25/0.51 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
% 0.25/0.51 => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
% 0.25/0.51 = ( cons @ ( list @ A )
% 0.25/0.51 @ ( cons @ A @ X
% 0.25/0.51 @ ( concat @ A
% 0.25/0.51 @ ( map @ ( list @ A ) @ ( list @ A )
% 0.25/0.51 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
% 0.25/0.51 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
% 0.25/0.51 @ Xss ) ) )
% 0.25/0.51 @ ( transpose @ A
% 0.25/0.51 @ ( cons @ ( list @ A ) @ Xs
% 0.25/0.51 @ ( concat @ ( list @ A )
% 0.25/0.51 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 0.25/0.51 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
% 0.25/0.51 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 @ Xss ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % transpose.psimps(3)
% 0.25/0.51 thf(fact_3943_transpose_Opelims,axiom,
% 0.25/0.51 ! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
% 0.25/0.51 ( ( ( transpose @ A @ X )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
% 0.25/0.51 => ( ! [Xss2: list @ ( list @ A )] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( transpose @ A @ Xss2 ) )
% 0.25/0.51 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
% 0.25/0.51 => ~ ! [X2: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( cons @ ( list @ A )
% 0.25/0.51 @ ( cons @ A @ X2
% 0.25/0.51 @ ( concat @ A
% 0.25/0.51 @ ( map @ ( list @ A ) @ ( list @ A )
% 0.25/0.51 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
% 0.25/0.51 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
% 0.25/0.51 @ Xss2 ) ) )
% 0.25/0.51 @ ( transpose @ A
% 0.25/0.51 @ ( cons @ ( list @ A ) @ Xs2
% 0.25/0.51 @ ( concat @ ( list @ A )
% 0.25/0.51 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 0.25/0.51 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
% 0.25/0.51 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
% 0.25/0.51 @ Xss2 ) ) ) ) ) )
% 0.25/0.51 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % transpose.pelims
% 0.25/0.51 thf(fact_3944_quicksort__by__rel_Opsimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
% 0.25/0.51 => ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 0.25/0.51 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs_b ) ) @ Xs_s )
% 0.25/0.51 @ ( partition_rev @ A
% 0.25/0.51 @ ^ [Y3: A] : ( R5 @ Y3 @ X )
% 0.25/0.51 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 0.25/0.51 @ Xs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.psimps(2)
% 0.25/0.51 thf(fact_3945_quicksort__by__rel_Opelims,axiom,
% 0.25/0.51 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 0.25/0.51 ( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
% 0.25/0.51 => ( ( ( Xb
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Y = Xa )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
% 0.25/0.51 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 0.25/0.51 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
% 0.25/0.51 @ ( partition_rev @ A
% 0.25/0.51 @ ^ [Y3: A] : ( X @ Y3 @ X2 )
% 0.25/0.51 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 0.25/0.51 @ Xs2 ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.pelims
% 0.25/0.51 thf(fact_3946_transpose_Opsimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
% 0.25/0.51 => ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
% 0.25/0.51 = ( nil @ ( list @ A ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % transpose.psimps(1)
% 0.25/0.51 thf(fact_3947_transpose_Opsimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,Xss: list @ ( list @ A )] :
% 0.25/0.51 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.25/0.51 => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 0.25/0.51 = ( transpose @ A @ Xss ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % transpose.psimps(2)
% 0.25/0.51 thf(fact_3948_quicksort__by__rel_Opsimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Sl2: list @ A] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
% 0.25/0.51 => ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( nil @ A ) )
% 0.25/0.51 = Sl2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.psimps(1)
% 0.25/0.51 thf(fact_3949_quicksort__by__rel_Opinduct,axiom,
% 0.25/0.51 ! [A: $tType,A0: A > A > $o,A1: list @ A,A22: list @ A,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ A0 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A1 @ A22 ) ) )
% 0.25/0.51 => ( ! [R8: A > A > $o,Sl: list @ A] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
% 0.25/0.51 => ( P @ R8 @ Sl @ ( nil @ A ) ) )
% 0.25/0.51 => ( ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.25/0.51 => ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
% 0.25/0.51 ( ( Xa2
% 0.25/0.51 = ( partition_rev @ A
% 0.25/0.51 @ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
% 0.25/0.51 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 0.25/0.51 @ Xs2 ) )
% 0.25/0.51 => ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
% 0.25/0.51 = Xa2 )
% 0.25/0.51 => ( P @ R8 @ Sl @ Y6 ) ) )
% 0.25/0.51 => ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
% 0.25/0.51 ( ( Xa2
% 0.25/0.51 = ( partition_rev @ A
% 0.25/0.51 @ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
% 0.25/0.51 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 0.25/0.51 @ Xs2 ) )
% 0.25/0.51 => ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
% 0.25/0.51 = Xa2 )
% 0.25/0.51 => ( P @ R8 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R8 @ Sl @ Y6 ) ) @ Xb2 ) ) )
% 0.25/0.51 => ( P @ R8 @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) )
% 0.25/0.51 => ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel.pinduct
% 0.25/0.51 thf(fact_3950_filter__rev__aux__alt,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( filter_rev_aux @ A )
% 0.25/0.51 = ( ^ [A7: list @ A,P4: A > $o,L3: list @ A] : ( append @ A @ ( filter @ A @ P4 @ ( rev @ A @ L3 ) ) @ A7 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % filter_rev_aux_alt
% 0.25/0.51 thf(fact_3951_lexord__take__index__conv,axiom,
% 0.25/0.51 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 0.25/0.51 = ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
% 0.25/0.51 & ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
% 0.25/0.51 = X ) )
% 0.25/0.51 | ? [I3: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
% 0.25/0.51 & ( ( take @ A @ I3 @ X )
% 0.25/0.51 = ( take @ A @ I3 @ Y ) )
% 0.25/0.51 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % lexord_take_index_conv
% 0.25/0.51 thf(fact_3952_last__take__nth__conv,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,L: list @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( N
% 0.25/0.51 != ( zero_zero @ nat ) )
% 0.25/0.51 => ( ( last @ A @ ( take @ A @ N @ L ) )
% 0.25/0.51 = ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % last_take_nth_conv
% 0.25/0.51 thf(fact_3953_revg__fun,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( revg @ A )
% 0.25/0.51 = ( ^ [A7: list @ A] : ( append @ A @ ( rev @ A @ A7 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % revg_fun
% 0.25/0.51 thf(fact_3954_take__upt,axiom,
% 0.25/0.51 ! [I: nat,M2: nat,N: nat] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M2 ) @ N )
% 0.25/0.51 => ( ( take @ nat @ M2 @ ( upt @ I @ N ) )
% 0.25/0.51 = ( upt @ I @ ( plus_plus @ nat @ I @ M2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_upt
% 0.25/0.51 thf(fact_3955_take__Cons__numeral,axiom,
% 0.25/0.51 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_Cons_numeral
% 0.25/0.51 thf(fact_3956_take__drop,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
% 0.25/0.51 ( ( take @ A @ N @ ( drop @ A @ M2 @ Xs ) )
% 0.25/0.51 = ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_drop
% 0.25/0.51 thf(fact_3957_revg_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,As2: list @ A,B2: list @ A] :
% 0.25/0.51 ( ( revg @ A @ ( cons @ A @ A3 @ As2 ) @ B2 )
% 0.25/0.51 = ( revg @ A @ As2 @ ( cons @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % revg.simps(2)
% 0.25/0.51 thf(fact_3958_revg_Osimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,B2: list @ A] :
% 0.25/0.51 ( ( revg @ A @ ( nil @ A ) @ B2 )
% 0.25/0.51 = B2 ) ).
% 0.25/0.51
% 0.25/0.51 % revg.simps(1)
% 0.25/0.51 thf(fact_3959_filter__rev__aux_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,X: A,A3: list @ A,Xs: list @ A] :
% 0.25/0.51 ( ( ( P @ X )
% 0.25/0.51 => ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( filter_rev_aux @ A @ ( cons @ A @ X @ A3 ) @ P @ Xs ) ) )
% 0.25/0.51 & ( ~ ( P @ X )
% 0.25/0.51 => ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( filter_rev_aux @ A @ A3 @ P @ Xs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % filter_rev_aux.simps(2)
% 0.25/0.51 thf(fact_3960_filter__rev__aux_Osimps_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,A3: list @ A,P: A > $o] :
% 0.25/0.51 ( ( filter_rev_aux @ A @ A3 @ P @ ( nil @ A ) )
% 0.25/0.51 = A3 ) ).
% 0.25/0.51
% 0.25/0.51 % filter_rev_aux.simps(1)
% 0.25/0.51 thf(fact_3961_drop__take__drop__unsplit,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,J: nat,L: list @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.51 => ( ( append @ A @ ( drop @ A @ I @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
% 0.25/0.51 = ( drop @ A @ I @ L ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % drop_take_drop_unsplit
% 0.25/0.51 thf(fact_3962_take__add,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,J: nat,Xs: list @ A] :
% 0.25/0.51 ( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
% 0.25/0.51 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_add
% 0.25/0.51 thf(fact_3963_slice__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( slice @ A )
% 0.25/0.51 = ( ^ [From2: nat,To2: nat,List2: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % slice_def
% 0.25/0.51 thf(fact_3964_filter__upt__take__conv,axiom,
% 0.25/0.51 ! [A: $tType,P: A > $o,M2: nat,L: list @ A,N: nat] :
% 0.25/0.51 ( ( filter @ nat
% 0.25/0.51 @ ^ [I3: nat] : ( P @ ( nth @ A @ ( take @ A @ M2 @ L ) @ I3 ) )
% 0.25/0.51 @ ( upt @ N @ M2 ) )
% 0.25/0.51 = ( filter @ nat
% 0.25/0.51 @ ^ [I3: nat] : ( P @ ( nth @ A @ L @ I3 ) )
% 0.25/0.51 @ ( upt @ N @ M2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % filter_upt_take_conv
% 0.25/0.51 thf(fact_3965_zip__replicate1,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,N: nat,X: A,Ys2: list @ B] :
% 0.25/0.51 ( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys2 )
% 0.25/0.51 = ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zip_replicate1
% 0.25/0.51 thf(fact_3966_revg_Oelims,axiom,
% 0.25/0.51 ! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
% 0.25/0.51 ( ( ( revg @ A @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( Y != Xa ) )
% 0.25/0.51 => ~ ! [A5: A,As: list @ A] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ A @ A5 @ As ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( revg @ A @ As @ ( cons @ A @ A5 @ Xa ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % revg.elims
% 0.25/0.51 thf(fact_3967_zip__replicate2,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
% 0.25/0.51 ( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
% 0.25/0.51 = ( map @ A @ ( product_prod @ A @ B )
% 0.25/0.51 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ Y )
% 0.25/0.51 @ ( take @ A @ N @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zip_replicate2
% 0.25/0.51 thf(fact_3968_take__Cons_H,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( ( N
% 0.25/0.51 = ( zero_zero @ nat ) )
% 0.25/0.51 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( nil @ A ) ) )
% 0.25/0.51 & ( ( N
% 0.25/0.51 != ( zero_zero @ nat ) )
% 0.25/0.51 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_Cons'
% 0.25/0.51 thf(fact_3969_filter__nth__ex__nth,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,P: A > $o,Xs: list @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) )
% 0.25/0.51 => ? [M5: nat] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ N @ M5 )
% 0.25/0.51 & ( ord_less @ nat @ M5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 & ( ( nth @ A @ ( filter @ A @ P @ Xs ) @ N )
% 0.25/0.51 = ( nth @ A @ Xs @ M5 ) )
% 0.25/0.51 & ( ( filter @ A @ P @ ( take @ A @ M5 @ Xs ) )
% 0.25/0.51 = ( take @ A @ N @ ( filter @ A @ P @ Xs ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % filter_nth_ex_nth
% 0.25/0.51 thf(fact_3970_map__upd__upds__conv__if,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,X: A,Ys2: list @ B,Xs: list @ A,F: A > ( option @ B ),Y: B] :
% 0.25/0.51 ( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
% 0.25/0.51 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y ) ) @ Xs @ Ys2 )
% 0.25/0.51 = ( map_upds @ A @ B @ F @ Xs @ Ys2 ) ) )
% 0.25/0.51 & ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
% 0.25/0.51 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y ) ) @ Xs @ Ys2 )
% 0.25/0.51 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F @ Xs @ Ys2 ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % map_upd_upds_conv_if
% 0.25/0.51 thf(fact_3971_Union__take__drop__id,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,L: list @ ( set @ A )] :
% 0.25/0.51 ( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
% 0.25/0.51 = ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Union_take_drop_id
% 0.25/0.51 thf(fact_3972_lex__take__index,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
% 0.25/0.51 => ~ ! [I2: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 0.25/0.51 => ( ( ( take @ A @ I2 @ Xs )
% 0.25/0.51 = ( take @ A @ I2 @ Ys2 ) )
% 0.25/0.51 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys2 @ I2 ) ) @ R ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % lex_take_index
% 0.25/0.51 thf(fact_3973_map__nth__upt__drop__take__conv,axiom,
% 0.25/0.51 ! [A: $tType,N6: nat,L: list @ A,M3: nat] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ N6 @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M3 @ N6 ) )
% 0.25/0.51 = ( drop @ A @ M3 @ ( take @ A @ N6 @ L ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % map_nth_upt_drop_take_conv
% 0.25/0.51 thf(fact_3974_filter__rev__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( filter_rev @ A )
% 0.25/0.51 = ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % filter_rev_def
% 0.25/0.51 thf(fact_3975_revg_Opelims,axiom,
% 0.25/0.51 ! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
% 0.25/0.51 ( ( ( revg @ A @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Y = Xa )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 0.25/0.51 => ~ ! [A5: A,As: list @ A] :
% 0.25/0.51 ( ( X
% 0.25/0.51 = ( cons @ A @ A5 @ As ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( revg @ A @ As @ ( cons @ A @ A5 @ Xa ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A5 @ As ) @ Xa ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % revg.pelims
% 0.25/0.51 thf(fact_3976_take__butlast__conv,axiom,
% 0.25/0.51 ! [A: $tType,L: list @ A] :
% 0.25/0.51 ( ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
% 0.25/0.51 = ( butlast @ A @ L ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_butlast_conv
% 0.25/0.51 thf(fact_3977_take__minus__one__conv__butlast,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,L: list @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( take @ A @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
% 0.25/0.51 = ( butlast @ A @ ( take @ A @ N @ L ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_minus_one_conv_butlast
% 0.25/0.51 thf(fact_3978_take__update__last,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,List: list @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
% 0.25/0.51 => ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X )
% 0.25/0.51 = ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_update_last
% 0.25/0.51 thf(fact_3979_take__update,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,L: list @ A,I: nat,X: A] :
% 0.25/0.51 ( ( take @ A @ N @ ( list_update @ A @ L @ I @ X ) )
% 0.25/0.51 = ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X ) ) ).
% 0.25/0.51
% 0.25/0.51 % take_update
% 0.25/0.51 thf(fact_3980_drop__upd__irrelevant,axiom,
% 0.25/0.51 ! [A: $tType,M2: nat,N: nat,L: list @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ M2 @ N )
% 0.25/0.51 => ( ( drop @ A @ N @ ( list_update @ A @ L @ M2 @ X ) )
% 0.25/0.51 = ( drop @ A @ N @ L ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % drop_upd_irrelevant
% 0.25/0.51 thf(fact_3981_nth__update__invalid,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,L: list @ A,J: nat,X: A] :
% 0.25/0.51 ( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I )
% 0.25/0.51 = ( nth @ A @ L @ I ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % nth_update_invalid
% 0.25/0.51 thf(fact_3982_length__butlast,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A] :
% 0.25/0.51 ( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
% 0.25/0.51 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % length_butlast
% 0.25/0.51 thf(fact_3983_butlast__upd__last__eq,axiom,
% 0.25/0.51 ! [A: $tType,L: list @ A,X: A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( list_update @ A @ ( butlast @ A @ L ) @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ X )
% 0.25/0.51 = ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_upd_last_eq
% 0.25/0.51 thf(fact_3984_butlast__update_H,axiom,
% 0.25/0.51 ! [A: $tType,L: list @ A,I: nat,X: A] :
% 0.25/0.51 ( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X )
% 0.25/0.51 = ( butlast @ A @ ( list_update @ A @ L @ I @ X ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_update'
% 0.25/0.51 thf(fact_3985_distinct__butlast__swap,axiom,
% 0.25/0.51 ! [A: $tType,Pq: list @ A,I: nat] :
% 0.25/0.51 ( ( distinct @ A @ Pq )
% 0.25/0.51 => ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I @ ( last @ A @ Pq ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % distinct_butlast_swap
% 0.25/0.51 thf(fact_3986_zip__update,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys2: list @ B,Y: B] :
% 0.25/0.51 ( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys2 @ I @ Y ) )
% 0.25/0.51 = ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zip_update
% 0.25/0.51 thf(fact_3987_butlast__list__update,axiom,
% 0.25/0.51 ! [A: $tType,K: nat,Xs: list @ A,X: A] :
% 0.25/0.51 ( ( ( K
% 0.25/0.51 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.25/0.51 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.25/0.51 = ( butlast @ A @ Xs ) ) )
% 0.25/0.51 & ( ( K
% 0.25/0.51 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.25/0.51 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.25/0.51 = ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_list_update
% 0.25/0.51 thf(fact_3988_in__set__upd__eq__aux,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
% 0.25/0.51 = ( ( X = Y )
% 0.25/0.51 | ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % in_set_upd_eq_aux
% 0.25/0.51 thf(fact_3989_in__set__upd__cases,axiom,
% 0.25/0.51 ! [A: $tType,X: A,L: list @ A,I: nat,Y: A] :
% 0.25/0.51 ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
% 0.25/0.51 => ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( X != Y ) )
% 0.25/0.51 => ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % in_set_upd_cases
% 0.25/0.51 thf(fact_3990_in__set__upd__eq,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
% 0.25/0.51 = ( ( X = Y )
% 0.25/0.51 | ( ( member @ A @ X @ ( set2 @ A @ L ) )
% 0.25/0.51 & ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % in_set_upd_eq
% 0.25/0.51 thf(fact_3991_nth__list__update_H,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,J: nat,L: list @ A,X: A] :
% 0.25/0.51 ( ( ( ( I = J )
% 0.25/0.51 & ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
% 0.25/0.51 => ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
% 0.25/0.51 = X ) )
% 0.25/0.51 & ( ~ ( ( I = J )
% 0.25/0.51 & ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
% 0.25/0.51 => ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
% 0.25/0.51 = ( nth @ A @ L @ J ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % nth_list_update'
% 0.25/0.51 thf(fact_3992_butlast__subset,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,A4: set @ A] :
% 0.25/0.51 ( ( Xs
% 0.25/0.51 != ( nil @ A ) )
% 0.25/0.51 => ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
% 0.25/0.51 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A4 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_subset
% 0.25/0.51 thf(fact_3993_butlast__eq__cons__conv,axiom,
% 0.25/0.51 ! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( ( butlast @ A @ L )
% 0.25/0.51 = ( cons @ A @ X @ Xs ) )
% 0.25/0.51 = ( ? [Xl: A] :
% 0.25/0.51 ( L
% 0.25/0.51 = ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_eq_cons_conv
% 0.25/0.51 thf(fact_3994_butlast__eq__consE,axiom,
% 0.25/0.51 ! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
% 0.25/0.51 ( ( ( butlast @ A @ L )
% 0.25/0.51 = ( cons @ A @ X @ Xs ) )
% 0.25/0.51 => ~ ! [Xl2: A] :
% 0.25/0.51 ( L
% 0.25/0.51 != ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_eq_consE
% 0.25/0.51 thf(fact_3995_butlast__upt,axiom,
% 0.25/0.51 ! [M2: nat,N: nat] :
% 0.25/0.51 ( ( butlast @ nat @ ( upt @ M2 @ N ) )
% 0.25/0.51 = ( upt @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_upt
% 0.25/0.51 thf(fact_3996_map__upd__eq,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: A] :
% 0.25/0.51 ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( F @ ( nth @ A @ L @ I ) )
% 0.25/0.51 = ( F @ X ) ) )
% 0.25/0.51 => ( ( map @ A @ B @ F @ ( list_update @ A @ L @ I @ X ) )
% 0.25/0.51 = ( map @ A @ B @ F @ L ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % map_upd_eq
% 0.25/0.51 thf(fact_3997_snoc__eq__iff__butlast_H,axiom,
% 0.25/0.51 ! [A: $tType,Ys2: list @ A,Xs: list @ A,X: A] :
% 0.25/0.51 ( ( Ys2
% 0.25/0.51 = ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 0.25/0.51 = ( ( Ys2
% 0.25/0.51 != ( nil @ A ) )
% 0.25/0.51 & ( ( butlast @ A @ Ys2 )
% 0.25/0.51 = Xs )
% 0.25/0.51 & ( ( last @ A @ Ys2 )
% 0.25/0.51 = X ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % snoc_eq_iff_butlast'
% 0.25/0.51 thf(fact_3998_butlast__conv__take,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( butlast @ A )
% 0.25/0.51 = ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_conv_take
% 0.25/0.51 thf(fact_3999_insert__swap__set__eq,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,L: list @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( insert2 @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X ) ) )
% 0.25/0.51 = ( insert2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % insert_swap_set_eq
% 0.25/0.51 thf(fact_4000_rev__update,axiom,
% 0.25/0.51 ! [A: $tType,K: nat,Xs: list @ A,Y: A] :
% 0.25/0.51 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
% 0.25/0.51 = ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % rev_update
% 0.25/0.51 thf(fact_4001_last__list__update,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,K: nat,X: A] :
% 0.25/0.51 ( ( Xs
% 0.25/0.51 != ( nil @ A ) )
% 0.25/0.51 => ( ( ( K
% 0.25/0.51 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.25/0.51 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.25/0.51 = X ) )
% 0.25/0.51 & ( ( K
% 0.25/0.51 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 0.25/0.51 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.25/0.51 = ( last @ A @ Xs ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % last_list_update
% 0.25/0.51 thf(fact_4002_butlast__take,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
% 0.25/0.51 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % butlast_take
% 0.25/0.51 thf(fact_4003_listrel1__iff__update,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.51 = ( ? [Y3: A,N5: nat] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N5 ) @ Y3 ) @ R )
% 0.25/0.51 & ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 & ( Ys2
% 0.25/0.51 = ( list_update @ A @ Xs @ N5 @ Y3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel1_iff_update
% 0.25/0.51 thf(fact_4004_map__distinct__upd__conv,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: B] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ( ( distinct @ A @ L )
% 0.25/0.51 => ( ( list_update @ B @ ( map @ A @ B @ F @ L ) @ I @ X )
% 0.25/0.51 = ( map @ A @ B @ ( fun_upd @ A @ B @ F @ ( nth @ A @ L @ I ) @ X ) @ L ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % map_distinct_upd_conv
% 0.25/0.51 thf(fact_4005_foldl__list__update,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: B > A > B,A3: B,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( ( foldl @ B @ A @ F @ A3 @ ( list_update @ A @ Xs @ N @ X ) )
% 0.25/0.51 = ( foldl @ B @ A @ F @ ( F @ ( foldl @ B @ A @ F @ A3 @ ( take @ A @ N @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % foldl_list_update
% 0.25/0.51 thf(fact_4006_case__natural__def,axiom,
% 0.25/0.51 ! [T: $tType] :
% 0.25/0.51 ( ( code_case_natural @ T )
% 0.25/0.51 = ( ^ [F12: T,F23: code_natural > T] :
% 0.25/0.51 ( code_rec_natural @ T @ F12
% 0.25/0.51 @ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % case_natural_def
% 0.25/0.51 thf(fact_4007_nth__rotate,axiom,
% 0.25/0.51 ! [A: $tType,N: nat,Xs: list @ A,M2: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( ( nth @ A @ ( rotate @ A @ M2 @ Xs ) @ N )
% 0.25/0.51 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % nth_rotate
% 0.25/0.51 thf(fact_4008_list_Osize__gen_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
% 0.25/0.51 ( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
% 0.25/0.51 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % list.size_gen(2)
% 0.25/0.51 thf(fact_4009_rotate1__length01,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.25/0.51 => ( ( rotate1 @ A @ Xs )
% 0.25/0.51 = Xs ) ) ).
% 0.25/0.51
% 0.25/0.51 % rotate1_length01
% 0.25/0.51 thf(fact_4010_natural_Osimps_I6_J,axiom,
% 0.25/0.51 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 0.25/0.51 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 0.25/0.51 = F1 ) ).
% 0.25/0.51
% 0.25/0.51 % natural.simps(6)
% 0.25/0.51 thf(fact_4011_size__list__append,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.51 ( ( size_list @ A @ F @ ( append @ A @ Xs @ Ys2 ) )
% 0.25/0.51 = ( plus_plus @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ F @ Ys2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_list_append
% 0.25/0.51 thf(fact_4012_rotate__length01,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 0.25/0.51 => ( ( rotate @ A @ N @ Xs )
% 0.25/0.51 = Xs ) ) ).
% 0.25/0.51
% 0.25/0.51 % rotate_length01
% 0.25/0.51 thf(fact_4013_rotate__rotate,axiom,
% 0.25/0.51 ! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
% 0.25/0.51 ( ( rotate @ A @ M2 @ ( rotate @ A @ N @ Xs ) )
% 0.25/0.51 = ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) @ Xs ) ) ).
% 0.25/0.51
% 0.25/0.51 % rotate_rotate
% 0.25/0.51 thf(fact_4014_rotate__add,axiom,
% 0.25/0.51 ! [A: $tType,M2: nat,N: nat] :
% 0.25/0.51 ( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) )
% 0.25/0.51 = ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % rotate_add
% 0.25/0.51 thf(fact_4015_rec__natural__def,axiom,
% 0.25/0.51 ! [T: $tType] :
% 0.25/0.51 ( ( code_rec_natural @ T )
% 0.25/0.51 = ( ^ [F12: T,F23: code_natural > T > T,X3: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % rec_natural_def
% 0.25/0.51 thf(fact_4016_lenlex__append2,axiom,
% 0.25/0.51 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.51 ( ( irrefl @ A @ R5 )
% 0.25/0.51 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys2 ) ) @ ( lenlex @ A @ R5 ) )
% 0.25/0.51 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % lenlex_append2
% 0.25/0.51 thf(fact_4017_zipf__zip,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,L1: list @ A,L2: list @ B] :
% 0.25/0.51 ( ( ( size_size @ ( list @ A ) @ L1 )
% 0.25/0.51 = ( size_size @ ( list @ B ) @ L2 ) )
% 0.25/0.51 => ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L1 @ L2 )
% 0.25/0.51 = ( zip @ A @ B @ L1 @ L2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zipf_zip
% 0.25/0.51 thf(fact_4018_map__tailrec__rev_Opelims,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
% 0.25/0.51 ( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 0.25/0.51 => ( ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Y = Xb )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
% 0.25/0.51 => ~ ! [A5: A,As: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ A5 @ As ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A5 ) @ Xb ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % map_tailrec_rev.pelims
% 0.25/0.51 thf(fact_4019_lexord__same__pref__if__irrefl,axiom,
% 0.25/0.51 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 0.25/0.51 ( ( irrefl @ A @ R )
% 0.25/0.51 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
% 0.25/0.51 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % lexord_same_pref_if_irrefl
% 0.25/0.51 thf(fact_4020_irreflI,axiom,
% 0.25/0.51 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ! [A5: A] :
% 0.25/0.51 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R5 )
% 0.25/0.51 => ( irrefl @ A @ R5 ) ) ).
% 0.25/0.51
% 0.25/0.51 % irreflI
% 0.25/0.51 thf(fact_4021_irrefl__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( irrefl @ A )
% 0.25/0.51 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ! [A7: A] :
% 0.25/0.51 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % irrefl_def
% 0.25/0.51 thf(fact_4022_zipf_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
% 0.25/0.51 ( ( zipf @ A @ B @ C @ F @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
% 0.25/0.51 = ( cons @ C @ ( F @ A3 @ B2 ) @ ( zipf @ A @ B @ C @ F @ As2 @ Bs2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zipf.simps(2)
% 0.25/0.51 thf(fact_4023_zipf_Osimps_I1_J,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,C: $tType,F: A > B > C] :
% 0.25/0.51 ( ( zipf @ A @ B @ C @ F @ ( nil @ A ) @ ( nil @ B ) )
% 0.25/0.51 = ( nil @ C ) ) ).
% 0.25/0.51
% 0.25/0.51 % zipf.simps(1)
% 0.25/0.51 thf(fact_4024_lexl__not__refl,axiom,
% 0.25/0.51 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A] :
% 0.25/0.51 ( ( irrefl @ A @ R )
% 0.25/0.51 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % lexl_not_refl
% 0.25/0.51 thf(fact_4025_zipf_Oelims,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,C: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
% 0.25/0.51 ( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Xb
% 0.25/0.51 = ( nil @ B ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( nil @ C ) ) ) )
% 0.25/0.51 => ( ! [A5: A,As: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ A5 @ As ) )
% 0.25/0.51 => ! [B3: B,Bs: list @ B] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( cons @ C @ ( X @ A5 @ B3 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs ) ) ) ) )
% 0.25/0.51 => ( ( ? [V4: A,Va: list @ A] :
% 0.25/0.51 ( Xa
% 0.25/0.51 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.51 => ( ( Xb
% 0.25/0.51 = ( nil @ B ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( undefined @ ( list @ C ) ) ) ) )
% 0.25/0.51 => ~ ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ? [V4: B,Va: list @ B] :
% 0.25/0.51 ( Xb
% 0.25/0.51 = ( cons @ B @ V4 @ Va ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 != ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zipf.elims
% 0.25/0.51 thf(fact_4026_zipf_Opelims,axiom,
% 0.25/0.51 ! [C: $tType,A: $tType,B: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
% 0.25/0.51 ( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 0.25/0.51 => ( ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( ( Xb
% 0.25/0.51 = ( nil @ B ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( nil @ C ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
% 0.25/0.51 => ( ! [A5: A,As: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ A5 @ As ) )
% 0.25/0.51 => ! [B3: B,Bs: list @ B] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( cons @ C @ ( X @ A5 @ B3 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) ) ) ) )
% 0.25/0.51 => ( ! [V4: A,Va: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.51 => ( ( Xb
% 0.25/0.51 = ( nil @ B ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( undefined @ ( list @ C ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
% 0.25/0.51 => ~ ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ! [V4: B,Va: list @ B] :
% 0.25/0.51 ( ( Xb
% 0.25/0.51 = ( cons @ B @ V4 @ Va ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( undefined @ ( list @ C ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % zipf.pelims
% 0.25/0.51 thf(fact_4027_count__list_Osimps_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A,Y: A,Xs: list @ A] :
% 0.25/0.51 ( ( ( X = Y )
% 0.25/0.51 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
% 0.25/0.51 = ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
% 0.25/0.51 & ( ( X != Y )
% 0.25/0.51 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
% 0.25/0.51 = ( count_list @ A @ Xs @ Y ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % count_list.simps(2)
% 0.25/0.51 thf(fact_4028_effect__makeE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [N: nat,F: nat > A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 @ H4 @ R @ N2 )
% 0.25/0.51 => ~ ( ( R
% 0.25/0.51 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) )
% 0.25/0.51 => ( ( ( array_get @ A @ H4 @ R )
% 0.25/0.51 = ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
% 0.25/0.51 => ( ( array_present @ A @ H4 @ R )
% 0.25/0.51 => ( ~ ( array_present @ A @ H3 @ R )
% 0.25/0.51 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 0.25/0.51 != N2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_makeE
% 0.25/0.51 thf(fact_4029_listrel__iff__nth,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 = ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.51 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.51 & ! [N5: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N5 ) @ ( nth @ B @ Ys2 @ N5 ) ) @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_iff_nth
% 0.25/0.51 thf(fact_4030_listrel__rtrancl__refl,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_rtrancl_refl
% 0.25/0.51 thf(fact_4031_Array__Time_Oget__alloc,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [Xs: list @ A,H3: heap_ext @ product_unit,Ys2: list @ A] :
% 0.25/0.51 ( ( array_get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H3 ) ) )
% 0.25/0.51 = Xs ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.get_alloc
% 0.25/0.51 thf(fact_4032_Array__Time_Opresent__alloc,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [Xs: list @ A,H3: heap_ext @ product_unit] : ( array_present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.present_alloc
% 0.25/0.51 thf(fact_4033_Array__Time_Onot__present__alloc,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H3: heap_ext @ product_unit,Xs: list @ A] :
% 0.25/0.51 ~ ( array_present @ A @ H3 @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.not_present_alloc
% 0.25/0.51 thf(fact_4034_listrel_ONil,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel.Nil
% 0.25/0.51 thf(fact_4035_listrel__Nil1,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ( Xs
% 0.25/0.51 = ( nil @ B ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_Nil1
% 0.25/0.51 thf(fact_4036_listrel__Nil2,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ( Xs
% 0.25/0.51 = ( nil @ A ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_Nil2
% 0.25/0.51 thf(fact_4037_listrel__eq__len,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.51 = ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_eq_len
% 0.25/0.51 thf(fact_4038_listrel__rtrancl__trans,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Zs: list @ A] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
% 0.25/0.51 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
% 0.25/0.51 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_rtrancl_trans
% 0.25/0.51 thf(fact_4039_listrel_OCons,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,X: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys2: list @ B] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
% 0.25/0.51 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel.Cons
% 0.25/0.51 thf(fact_4040_listrel__Cons1,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,Y: A,Ys2: list @ A,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys2 ) @ Xs ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ~ ! [Y4: B,Ys3: list @ B] :
% 0.25/0.51 ( ( Xs
% 0.25/0.51 = ( cons @ B @ Y4 @ Ys3 ) )
% 0.25/0.51 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y4 ) @ R )
% 0.25/0.51 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys2 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_Cons1
% 0.25/0.51 thf(fact_4041_listrel__Cons2,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.51 ( ( Xs
% 0.25/0.51 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.51 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
% 0.25/0.51 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_Cons2
% 0.25/0.51 thf(fact_4042_listrel__reflcl__if__listrel1,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 0.25/0.51 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_reflcl_if_listrel1
% 0.25/0.51 thf(fact_4043_rtrancl__listrel1__if__listrel,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ R ) )
% 0.25/0.51 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % rtrancl_listrel1_if_listrel
% 0.25/0.51 thf(fact_4044_effect__of__listE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [Xs: list @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 @ H4 @ R @ N2 )
% 0.25/0.51 => ~ ( ( R
% 0.25/0.51 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) )
% 0.25/0.51 => ( ( ( array_get @ A @ H4 @ R )
% 0.25/0.51 = Xs )
% 0.25/0.51 => ( ( array_present @ A @ H4 @ R )
% 0.25/0.51 => ( ~ ( array_present @ A @ H3 @ R )
% 0.25/0.51 => ( N2
% 0.25/0.51 != ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_of_listE
% 0.25/0.51 thf(fact_4045_effect__newE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [N: nat,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 @ H4 @ R @ N2 )
% 0.25/0.51 => ~ ( ( R
% 0.25/0.51 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) )
% 0.25/0.51 => ( ( ( array_get @ A @ H4 @ R )
% 0.25/0.51 = ( replicate @ A @ N @ X ) )
% 0.25/0.51 => ( ( array_present @ A @ H4 @ R )
% 0.25/0.51 => ( ~ ( array_present @ A @ H3 @ R )
% 0.25/0.51 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 0.25/0.51 != N2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_newE
% 0.25/0.51 thf(fact_4046_listrel_Ocases,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 => ( ( ( A1
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( A22
% 0.25/0.51 != ( nil @ B ) ) )
% 0.25/0.51 => ~ ! [X2: A,Y4: B,Xs2: list @ A] :
% 0.25/0.51 ( ( A1
% 0.25/0.51 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.51 => ! [Ys3: list @ B] :
% 0.25/0.51 ( ( A22
% 0.25/0.51 = ( cons @ B @ Y4 @ Ys3 ) )
% 0.25/0.51 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 0.25/0.51 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel.cases
% 0.25/0.51 thf(fact_4047_listrel_Osimps,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 = ( ( ( A1
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 & ( A22
% 0.25/0.51 = ( nil @ B ) ) )
% 0.25/0.51 | ? [X3: A,Y3: B,Xs3: list @ A,Ys: list @ B] :
% 0.25/0.51 ( ( A1
% 0.25/0.51 = ( cons @ A @ X3 @ Xs3 ) )
% 0.25/0.51 & ( A22
% 0.25/0.51 = ( cons @ B @ Y3 @ Ys ) )
% 0.25/0.51 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 0.25/0.51 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel.simps
% 0.25/0.51 thf(fact_4048_listrel__iff__zip,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 0.25/0.51 = ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.51 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.51 & ! [X3: product_prod @ A @ B] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 0.25/0.51 => ( product_case_prod @ A @ B @ $o
% 0.25/0.51 @ ^ [Y3: A,Z5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R )
% 0.25/0.51 @ X3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_iff_zip
% 0.25/0.51 thf(fact_4049_present__alloc__noteq,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType] :
% 0.25/0.51 ( ( ( heap @ B )
% 0.25/0.51 & ( heap @ A ) )
% 0.25/0.51 => ! [H3: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
% 0.25/0.51 ( ( array_present @ A @ H3 @ A3 )
% 0.25/0.51 => ( array_noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ B @ Xs @ H3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % present_alloc_noteq
% 0.25/0.51 thf(fact_4050_listrel__def,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( listrel @ A @ B )
% 0.25/0.51 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
% 0.25/0.51 @ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
% 0.25/0.51 @ ( listrelp @ A @ B
% 0.25/0.51 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrel_def
% 0.25/0.51 thf(fact_4051_set,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [Ls: list @ A,H3: heap_ext @ product_unit,New_ls: list @ A] :
% 0.25/0.51 ( ( array_set @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H3 ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H3 ) ) )
% 0.25/0.51 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ New_ls @ H3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % set
% 0.25/0.51 thf(fact_4052_Array__Time_Oset__same,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [R: array @ A,X: list @ A,Y: list @ A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_set @ A @ R @ X @ ( array_set @ A @ R @ Y @ H3 ) )
% 0.25/0.51 = ( array_set @ A @ R @ X @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.set_same
% 0.25/0.51 thf(fact_4053_Array__Time_Ounequal,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,A6: array @ A] :
% 0.25/0.51 ( ( A3 != A6 )
% 0.25/0.51 = ( array_noteq @ A @ A @ A3 @ A6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.unequal
% 0.25/0.51 thf(fact_4054_Array__Time_Oget__set__eq,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [R: array @ A,X: list @ A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_get @ A @ ( array_set @ A @ R @ X @ H3 ) @ R )
% 0.25/0.51 = X ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.get_set_eq
% 0.25/0.51 thf(fact_4055_Array__Time_Oget__set__neq,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [R: array @ A,S: array @ B,X: list @ B,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_noteq @ A @ B @ R @ S )
% 0.25/0.51 => ( ( array_get @ A @ ( array_set @ B @ S @ X @ H3 ) @ R )
% 0.25/0.51 = ( array_get @ A @ H3 @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.get_set_neq
% 0.25/0.51 thf(fact_4056_Array__Time_Oset__set__swap,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [R: array @ A,R4: array @ B,X: list @ A,X6: list @ B,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_noteq @ A @ B @ R @ R4 )
% 0.25/0.51 => ( ( array_set @ A @ R @ X @ ( array_set @ B @ R4 @ X6 @ H3 ) )
% 0.25/0.51 = ( array_set @ B @ R4 @ X6 @ ( array_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.set_set_swap
% 0.25/0.51 thf(fact_4057_Array__Time_Onoteq__irrefl,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [R: array @ A] :
% 0.25/0.51 ~ ( array_noteq @ A @ A @ R @ R ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.noteq_irrefl
% 0.25/0.51 thf(fact_4058_Array__Time_Onoteq__sym,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [A3: array @ A,B2: array @ B] :
% 0.25/0.51 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 0.25/0.51 => ( array_noteq @ B @ A @ B2 @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.noteq_sym
% 0.25/0.51 thf(fact_4059_Field__not__elem,axiom,
% 0.25/0.51 ! [A: $tType,V: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ~ ( member @ A @ V @ ( field2 @ A @ R5 ) )
% 0.25/0.51 => ! [X7: product_prod @ A @ A] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ A ) @ X7 @ R5 )
% 0.25/0.51 => ( product_case_prod @ A @ A @ $o
% 0.25/0.51 @ ^ [Y3: A,Z5: A] :
% 0.25/0.51 ( ( Y3 != V )
% 0.25/0.51 & ( Z5 != V ) )
% 0.25/0.51 @ X7 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Field_not_elem
% 0.25/0.51 thf(fact_4060_refl__on__def_H,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( refl_on @ A )
% 0.25/0.51 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.51 ( ! [X3: product_prod @ A @ A] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ A ) @ X3 @ R3 )
% 0.25/0.51 => ( product_case_prod @ A @ A @ $o
% 0.25/0.51 @ ^ [Y3: A,Z5: A] :
% 0.25/0.51 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.51 & ( member @ A @ Z5 @ A9 ) )
% 0.25/0.51 @ X3 ) )
% 0.25/0.51 & ! [X3: A] :
% 0.25/0.51 ( ( member @ A @ X3 @ A9 )
% 0.25/0.51 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % refl_on_def'
% 0.25/0.51 thf(fact_4061_listrelp__listrel__eq,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( listrelp @ A @ B
% 0.25/0.51 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.25/0.51 = ( ^ [X3: list @ A,Y3: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X3 @ Y3 ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % listrelp_listrel_eq
% 0.25/0.51 thf(fact_4062_Under__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( order_Under @ A )
% 0.25/0.51 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 0.25/0.51 ( collect @ A
% 0.25/0.51 @ ^ [B5: A] :
% 0.25/0.51 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 0.25/0.51 & ! [X3: A] :
% 0.25/0.51 ( ( member @ A @ X3 @ A9 )
% 0.25/0.51 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X3 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Under_def
% 0.25/0.51 thf(fact_4063_UnderS__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( order_UnderS @ A )
% 0.25/0.51 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 0.25/0.51 ( collect @ A
% 0.25/0.51 @ ^ [B5: A] :
% 0.25/0.51 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 0.25/0.51 & ! [X3: A] :
% 0.25/0.51 ( ( member @ A @ X3 @ A9 )
% 0.25/0.51 => ( ( B5 != X3 )
% 0.25/0.51 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % UnderS_def
% 0.25/0.51 thf(fact_4064_sorted__wrt_Opelims_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 0.25/0.51 ( ( sorted_wrt @ A @ X @ Xa )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.51 => ( ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
% 0.25/0.51 => ~ ! [X2: A,Ys3: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ X2 @ Ys3 ) )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
% 0.25/0.51 => ~ ( ! [Xa2: A] :
% 0.25/0.51 ( ( member @ A @ Xa2 @ ( set2 @ A @ Ys3 ) )
% 0.25/0.51 => ( X @ X2 @ Xa2 ) )
% 0.25/0.51 & ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % sorted_wrt.pelims(2)
% 0.25/0.51 thf(fact_4065_sorted__wrt_Opelims_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
% 0.25/0.51 ( ( ( sorted_wrt @ A @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.51 => ( ( ( Xa
% 0.25/0.51 = ( nil @ A ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
% 0.25/0.51 => ~ ! [X2: A,Ys3: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ X2 @ Ys3 ) )
% 0.25/0.51 => ( ( Y
% 0.25/0.51 = ( ! [Y3: A] :
% 0.25/0.51 ( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
% 0.25/0.51 => ( X @ X2 @ Y3 ) )
% 0.25/0.51 & ( sorted_wrt @ A @ X @ Ys3 ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % sorted_wrt.pelims(1)
% 0.25/0.51 thf(fact_4066_sorted__wrt_Opelims_I3_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 0.25/0.51 ( ~ ( sorted_wrt @ A @ X @ Xa )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.51 => ~ ! [X2: A,Ys3: list @ A] :
% 0.25/0.51 ( ( Xa
% 0.25/0.51 = ( cons @ A @ X2 @ Ys3 ) )
% 0.25/0.51 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
% 0.25/0.51 => ( ! [Xa3: A] :
% 0.25/0.51 ( ( member @ A @ Xa3 @ ( set2 @ A @ Ys3 ) )
% 0.25/0.51 => ( X @ X2 @ Xa3 ) )
% 0.25/0.51 & ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % sorted_wrt.pelims(3)
% 0.25/0.51 thf(fact_4067_Above__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( order_Above @ A )
% 0.25/0.51 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 0.25/0.51 ( collect @ A
% 0.25/0.51 @ ^ [B5: A] :
% 0.25/0.51 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 0.25/0.51 & ! [X3: A] :
% 0.25/0.51 ( ( member @ A @ X3 @ A9 )
% 0.25/0.51 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B5 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Above_def
% 0.25/0.51 thf(fact_4068_execute__nth_I1_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( heap_Time_execute @ A @ ( array_nth @ A @ A3 @ I ) @ H3 )
% 0.25/0.51 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_nth(1)
% 0.25/0.51 thf(fact_4069_Array__Time_Olength__alloc,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [Xs: list @ A,H3: heap_ext @ product_unit,Ys2: list @ A] :
% 0.25/0.51 ( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H3 ) ) )
% 0.25/0.51 = ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.length_alloc
% 0.25/0.51 thf(fact_4070_Array__Time_Onth__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_nth @ A )
% 0.25/0.51 = ( ^ [A7: array @ A,I3: nat] :
% 0.25/0.51 ( heap_Time_guard @ A
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.nth_def
% 0.25/0.51 thf(fact_4071_success__nthI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( heap_Time_success @ A @ ( array_nth @ A @ A3 @ I ) @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % success_nthI
% 0.25/0.51 thf(fact_4072_fails__array__nth,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [P5: array @ A,I: nat,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( time_fails @ A @ ( array_nth @ A @ P5 @ I ) @ H3 )
% 0.25/0.51 = ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ P5 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % fails_array_nth
% 0.25/0.51 thf(fact_4073_execute__nth_I2_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 0.25/0.51 => ( ( heap_Time_execute @ A @ ( array_nth @ A @ A3 @ I ) @ H3 )
% 0.25/0.51 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_nth(2)
% 0.25/0.51 thf(fact_4074_length__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_length @ A )
% 0.25/0.51 = ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H @ A7 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % length_def
% 0.25/0.51 thf(fact_4075_effect__nthE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,I: nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ A @ ( array_nth @ A @ A3 @ I ) @ H3 @ H4 @ R @ N )
% 0.25/0.51 => ~ ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( R
% 0.25/0.51 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 0.25/0.51 => ( ( H4 = H3 )
% 0.25/0.51 => ( N
% 0.25/0.51 != ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_nthE
% 0.25/0.51 thf(fact_4076_effect__nthI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( H4 = H3 )
% 0.25/0.51 => ( ( R
% 0.25/0.51 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 0.25/0.51 => ( ( N
% 0.25/0.51 = ( one_one @ nat ) )
% 0.25/0.51 => ( heap_Time_effect @ A @ ( array_nth @ A @ A3 @ I ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_nthI
% 0.25/0.51 thf(fact_4077_nth_H__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_nth2 @ A )
% 0.25/0.51 = ( ^ [A7: array @ A] : ( comp @ nat @ ( heap_Time_Heap @ A ) @ code_integer @ ( array_nth @ A @ A7 ) @ code_nat_of_integer ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % nth'_def
% 0.25/0.51 thf(fact_4078_freeze__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_freeze @ A )
% 0.25/0.51 = ( ^ [A7: array @ A] :
% 0.25/0.51 ( heap_Time_heap @ ( list @ A )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H @ A7 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H @ A7 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % freeze_def
% 0.25/0.51 thf(fact_4079_execute__map__entry_I2_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,F: A > A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 0.25/0.51 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 )
% 0.25/0.51 = ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_map_entry(2)
% 0.25/0.51 thf(fact_4080_execute__freeze,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( heap_Time_execute @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 )
% 0.25/0.51 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H3 @ A3 ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_freeze
% 0.25/0.51 thf(fact_4081_success__freezeI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 ) ) ).
% 0.25/0.51
% 0.25/0.51 % success_freezeI
% 0.25/0.51 thf(fact_4082_success__map__entryI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,F: A > A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( heap_Time_success @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % success_map_entryI
% 0.25/0.51 thf(fact_4083_effect__freezeE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: list @ A,N: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 @ H4 @ R @ N )
% 0.25/0.51 => ~ ( ( H4 = H3 )
% 0.25/0.51 => ( ( R
% 0.25/0.51 = ( array_get @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( N
% 0.25/0.51 != ( plus_plus @ nat @ ( array_length @ A @ H3 @ A3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_freezeE
% 0.25/0.51 thf(fact_4084_effect__freezeI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: list @ A,A3: array @ A,N: nat] :
% 0.25/0.51 ( ( H4 = H3 )
% 0.25/0.51 => ( ( R
% 0.25/0.51 = ( array_get @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( N
% 0.25/0.51 = ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 @ H4 @ R @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_freezeI
% 0.25/0.51 thf(fact_4085_execute__map__entry_I1_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,F: A > A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 )
% 0.25/0.51 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_map_entry(1)
% 0.25/0.51 thf(fact_4086_map__entry__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_map_entry @ A )
% 0.25/0.51 = ( ^ [I3: nat,F2: A > A,A7: array @ A] :
% 0.25/0.51 ( heap_Time_guard @ ( array @ A )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) ) @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % map_entry_def
% 0.25/0.51 thf(fact_4087_effect__map__entryI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,F: A > A,R: array @ A,N: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) )
% 0.25/0.51 => ( ( R = A3 )
% 0.25/0.51 => ( ( N
% 0.25/0.51 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.51 => ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_map_entryI
% 0.25/0.51 thf(fact_4088_effect__map__entryE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,F: A > A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 @ H4 @ R @ N )
% 0.25/0.51 => ~ ( ( R = A3 )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) )
% 0.25/0.51 => ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( N
% 0.25/0.51 != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_map_entryE
% 0.25/0.51 thf(fact_4089_length__update,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [B2: array @ B,I: nat,V: B,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_length @ A @ ( array_update @ B @ B2 @ I @ V @ H3 ) )
% 0.25/0.51 = ( array_length @ A @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % length_update
% 0.25/0.51 thf(fact_4090_Array__Time_Opresent__update,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [B2: array @ B,I: nat,V: B,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_present @ A @ ( array_update @ B @ B2 @ I @ V @ H3 ) )
% 0.25/0.51 = ( array_present @ A @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.present_update
% 0.25/0.51 thf(fact_4091_get__update__elem__neqIndex,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,J: nat,A3: array @ A,V: A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( I != J )
% 0.25/0.51 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A3 @ J @ V @ H3 ) @ A3 ) @ I )
% 0.25/0.51 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % get_update_elem_neqIndex
% 0.25/0.51 thf(fact_4092_get__update__eq,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,I: nat,V: A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_get @ A @ ( array_update @ A @ A3 @ I @ V @ H3 ) @ A3 )
% 0.25/0.51 = ( list_update @ A @ ( array_get @ A @ H3 @ A3 ) @ I @ V ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % get_update_eq
% 0.25/0.51 thf(fact_4093_nth__update__neq,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [A3: array @ A,B2: array @ B,J: nat,V: B,H3: heap_ext @ product_unit,I: nat] :
% 0.25/0.51 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 0.25/0.51 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B2 @ J @ V @ H3 ) @ A3 ) @ I )
% 0.25/0.51 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % nth_update_neq
% 0.25/0.51 thf(fact_4094_update__swap__neqIndex,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,I7: nat,A3: array @ A,V: A,V7: A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( I != I7 )
% 0.25/0.51 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ A @ A3 @ I7 @ V7 @ H3 ) )
% 0.25/0.51 = ( array_update @ A @ A3 @ I7 @ V7 @ ( array_update @ A @ A3 @ I @ V @ H3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % update_swap_neqIndex
% 0.25/0.51 thf(fact_4095_update__swap__neq,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( ( heap @ A )
% 0.25/0.51 & ( heap @ B ) )
% 0.25/0.51 => ! [A3: array @ A,A6: array @ B,I: nat,V: A,I7: nat,V7: B,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( array_noteq @ A @ B @ A3 @ A6 )
% 0.25/0.51 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ B @ A6 @ I7 @ V7 @ H3 ) )
% 0.25/0.51 = ( array_update @ B @ A6 @ I7 @ V7 @ ( array_update @ A @ A3 @ I @ V @ H3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % update_swap_neq
% 0.25/0.51 thf(fact_4096_Array__Time_Oupdate__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_update @ A )
% 0.25/0.51 = ( ^ [A7: array @ A,I3: nat,X3: A,H: heap_ext @ product_unit] : ( array_set @ A @ A7 @ ( list_update @ A @ ( array_get @ A @ H @ A7 ) @ I3 @ X3 ) @ H ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Array_Time.update_def
% 0.25/0.51 thf(fact_4097_execute__swap_I1_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 )
% 0.25/0.51 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ X @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_swap(1)
% 0.25/0.51 thf(fact_4098_swap__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_swap @ A )
% 0.25/0.51 = ( ^ [I3: nat,X3: A,A7: array @ A] :
% 0.25/0.51 ( heap_Time_guard @ A
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ X3 @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % swap_def
% 0.25/0.51 thf(fact_4099_effect__swapI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,X: A,R: A,N: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( array_update @ A @ A3 @ I @ X @ H3 ) )
% 0.25/0.51 => ( ( R
% 0.25/0.51 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 0.25/0.51 => ( ( N
% 0.25/0.51 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.51 => ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_swapI
% 0.25/0.51 thf(fact_4100_effect__swapE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,X: A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 @ H4 @ R @ N )
% 0.25/0.51 => ~ ( ( R
% 0.25/0.51 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( array_update @ A @ A3 @ I @ X @ H3 ) )
% 0.25/0.51 => ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( N
% 0.25/0.51 != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_swapE
% 0.25/0.51 thf(fact_4101_success__swapI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( heap_Time_success @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % success_swapI
% 0.25/0.51 thf(fact_4102_execute__swap_I2_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,X: A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 0.25/0.51 => ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 )
% 0.25/0.51 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_swap(2)
% 0.25/0.51 thf(fact_4103_execute__upd_I1_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 )
% 0.25/0.51 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ X @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_upd(1)
% 0.25/0.51 thf(fact_4104_upd__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_upd @ A )
% 0.25/0.51 = ( ^ [I3: nat,X3: A,A7: array @ A] :
% 0.25/0.51 ( heap_Time_guard @ ( array @ A )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ X3 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % upd_def
% 0.25/0.51 thf(fact_4105_execute__upd_I2_J,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,X: A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 0.25/0.51 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 )
% 0.25/0.51 = ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_upd(2)
% 0.25/0.51 thf(fact_4106_effect__updI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,V: A,N: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( array_update @ A @ A3 @ I @ V @ H3 ) )
% 0.25/0.51 => ( ( N
% 0.25/0.51 = ( one_one @ nat ) )
% 0.25/0.51 => ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V @ A3 ) @ H3 @ H4 @ A3 @ N ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_updI
% 0.25/0.51 thf(fact_4107_upd__ureturn,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,X: A,A3: array @ A] :
% 0.25/0.51 ( ( heap_Time_bind @ ( array @ A ) @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 )
% 0.25/0.51 @ ^ [Uu: array @ A] : ( heap_Time_ureturn @ ( array @ A ) @ A3 ) )
% 0.25/0.51 = ( array_upd @ A @ I @ X @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % upd_ureturn
% 0.25/0.51 thf(fact_4108_success__updI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( heap_Time_success @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % success_updI
% 0.25/0.51 thf(fact_4109_fails__array__upd,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,X: A,P5: array @ A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P5 ) @ H3 )
% 0.25/0.51 = ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ P5 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % fails_array_upd
% 0.25/0.51 thf(fact_4110_effect__updE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [I: nat,V: A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V @ A3 ) @ H3 @ H4 @ R @ N )
% 0.25/0.51 => ~ ( ( R = A3 )
% 0.25/0.51 => ( ( H4
% 0.25/0.51 = ( array_update @ A @ A3 @ I @ V @ H3 ) )
% 0.25/0.51 => ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( N
% 0.25/0.51 != ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_updE
% 0.25/0.51 thf(fact_4111_upd_H__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_upd2 @ A )
% 0.25/0.51 = ( ^ [A7: array @ A,I3: code_integer,X3: A] :
% 0.25/0.51 ( heap_Time_bind @ ( array @ A ) @ product_unit @ ( array_upd @ A @ ( code_nat_of_integer @ I3 ) @ X3 @ A7 )
% 0.25/0.51 @ ^ [Uu: array @ A] : ( heap_Time_ureturn @ product_unit @ product_Unity ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % upd'_def
% 0.25/0.51 thf(fact_4112_execute__len,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ( ( heap_Time_execute @ nat @ ( array_len @ A @ A3 ) @ H3 )
% 0.25/0.51 = ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % execute_len
% 0.25/0.51 thf(fact_4113_in__measures_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A,Y: A,F: A > nat,Fs: list @ ( A > nat )] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
% 0.25/0.51 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 0.25/0.51 | ( ( ( F @ X )
% 0.25/0.51 = ( F @ Y ) )
% 0.25/0.51 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % in_measures(2)
% 0.25/0.51 thf(fact_4114_mset__mergesort__by__rel__split,axiom,
% 0.25/0.51 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs1 ) ) @ ( mset @ A @ Xs22 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_mergesort_by_rel_split
% 0.25/0.51 thf(fact_4115_union__eq__empty,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N6 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ( M3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( N6
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_eq_empty
% 0.25/0.51 thf(fact_4116_empty__eq__union,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( ( zero_zero @ ( multiset @ A ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.51 = ( ( M3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( N6
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % empty_eq_union
% 0.25/0.51 thf(fact_4117_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.25/0.51 ( ( ( zero_zero @ ( multiset @ A ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
% 0.25/0.51 = ( ( X
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( Y
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.zero_eq_add_iff_both_eq_0
% 0.25/0.51 thf(fact_4118_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ( X
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( Y
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_eq_0_iff_both_eq_0
% 0.25/0.51 thf(fact_4119_diff__diff__add__mset,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,P: multiset @ A] :
% 0.25/0.51 ( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) @ P )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ P ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % diff_diff_add_mset
% 0.25/0.51 thf(fact_4120_mergesort__by__rel__permutes,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 0.25/0.51 ( ( mset @ A @ ( mergesort_by_rel @ A @ R5 @ Xs ) )
% 0.25/0.51 = ( mset @ A @ Xs ) ) ).
% 0.25/0.51
% 0.25/0.51 % mergesort_by_rel_permutes
% 0.25/0.51 thf(fact_4121_in__measures_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,X: A,Y: A] :
% 0.25/0.51 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % in_measures(1)
% 0.25/0.51 thf(fact_4122_mset__append,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.51 ( ( mset @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_append
% 0.25/0.51 thf(fact_4123_quicksort__by__rel__permutes,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,Xs: list @ A] :
% 0.25/0.51 ( ( mset @ A @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs ) )
% 0.25/0.51 = ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % quicksort_by_rel_permutes
% 0.25/0.51 thf(fact_4124_mset__mergesort__by__rel__merge,axiom,
% 0.25/0.51 ! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.51 ( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_mergesort_by_rel_merge
% 0.25/0.51 thf(fact_4125_success__lenI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ nat @ ( array_len @ A @ A3 ) @ H3 ) ) ).
% 0.25/0.51
% 0.25/0.51 % success_lenI
% 0.25/0.51 thf(fact_4126_fails__array__len,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [P5: array @ A,H3: heap_ext @ product_unit] :
% 0.25/0.51 ~ ( time_fails @ nat @ ( array_len @ A @ P5 ) @ H3 ) ) ).
% 0.25/0.51
% 0.25/0.51 % fails_array_len
% 0.25/0.51 thf(fact_4127_mset__distrib,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ A4 @ B6 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.51 => ~ ! [Am: multiset @ A,An: multiset @ A] :
% 0.25/0.51 ( ( A4
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
% 0.25/0.51 => ! [Bm: multiset @ A,Bn: multiset @ A] :
% 0.25/0.51 ( ( B6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
% 0.25/0.51 => ( ( M3
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
% 0.25/0.51 => ( N6
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_distrib
% 0.25/0.51 thf(fact_4128_mset__shuffles,axiom,
% 0.25/0.51 ! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.51 ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
% 0.25/0.51 => ( ( mset @ A @ Zs )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_shuffles
% 0.25/0.51 thf(fact_4129_union__less__mono,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( preorder @ A )
% 0.25/0.51 => ! [A4: multiset @ A,C5: multiset @ A,B6: multiset @ A,D4: multiset @ A] :
% 0.25/0.51 ( ( ord_less @ ( multiset @ A ) @ A4 @ C5 )
% 0.25/0.51 => ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
% 0.25/0.51 => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_less_mono
% 0.25/0.51 thf(fact_4130_union__le__mono2,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( preorder @ A )
% 0.25/0.51 => ! [B6: multiset @ A,D4: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
% 0.25/0.51 => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_le_mono2
% 0.25/0.51 thf(fact_4131_union__le__mono1,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( preorder @ A )
% 0.25/0.51 => ! [B6: multiset @ A,D4: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
% 0.25/0.51 => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ D4 @ C5 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_le_mono1
% 0.25/0.51 thf(fact_4132_multi__union__self__other__eq,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,X10: multiset @ A,Y9: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ A4 @ X10 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ A4 @ Y9 ) )
% 0.25/0.51 => ( X10 = Y9 ) ) ).
% 0.25/0.51
% 0.25/0.51 % multi_union_self_other_eq
% 0.25/0.51 thf(fact_4133_union__right__cancel,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,K2: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ M3 @ K2 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) )
% 0.25/0.51 = ( M3 = N6 ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_right_cancel
% 0.25/0.51 thf(fact_4134_union__left__cancel,axiom,
% 0.25/0.51 ! [A: $tType,K2: multiset @ A,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M3 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ K2 @ N6 ) )
% 0.25/0.51 = ( M3 = N6 ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_left_cancel
% 0.25/0.51 thf(fact_4135_union__commute,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) )
% 0.25/0.51 = ( ^ [M8: multiset @ A,N9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N9 @ M8 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_commute
% 0.25/0.51 thf(fact_4136_union__lcomm,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,K2: multiset @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ N6 @ ( plus_plus @ ( multiset @ A ) @ M3 @ K2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_lcomm
% 0.25/0.51 thf(fact_4137_union__assoc,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,K2: multiset @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ K2 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_assoc
% 0.25/0.51 thf(fact_4138_empty__neutral_I1_J,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 0.25/0.51 = X ) ).
% 0.25/0.51
% 0.25/0.51 % empty_neutral(1)
% 0.25/0.51 thf(fact_4139_empty__neutral_I2_J,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = X ) ).
% 0.25/0.51
% 0.25/0.51 % empty_neutral(2)
% 0.25/0.51 thf(fact_4140_diff__union__cancelR,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ N6 )
% 0.25/0.51 = M3 ) ).
% 0.25/0.51
% 0.25/0.51 % diff_union_cancelR
% 0.25/0.51 thf(fact_4141_diff__union__cancelL,axiom,
% 0.25/0.51 ! [A: $tType,N6: multiset @ A,M3: multiset @ A] :
% 0.25/0.51 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N6 @ M3 ) @ N6 )
% 0.25/0.51 = M3 ) ).
% 0.25/0.51
% 0.25/0.51 % diff_union_cancelL
% 0.25/0.51 thf(fact_4142_Multiset_Odiff__add,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,Q: multiset @ A] :
% 0.25/0.51 ( ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ Q ) )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) @ Q ) ) ).
% 0.25/0.51
% 0.25/0.51 % Multiset.diff_add
% 0.25/0.51 thf(fact_4143_union__diff__assoc,axiom,
% 0.25/0.51 ! [A: $tType,C5: multiset @ A,B6: multiset @ A,A4: multiset @ A] :
% 0.25/0.51 ( ( ( minus_minus @ ( multiset @ A ) @ C5 @ B6 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ A4 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_diff_assoc
% 0.25/0.51 thf(fact_4144_comp__fun__commute__plus__mset,axiom,
% 0.25/0.51 ! [A: $tType] : ( finite6289374366891150609ommute @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % comp_fun_commute_plus_mset
% 0.25/0.51 thf(fact_4145_len__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ( ( array_len @ A )
% 0.25/0.51 = ( ^ [A7: array @ A] :
% 0.25/0.51 ( heap_Time_tap @ nat
% 0.25/0.51 @ ^ [H: heap_ext @ product_unit] : ( array_length @ A @ H @ A7 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % len_def
% 0.25/0.51 thf(fact_4146_measures__less,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
% 0.25/0.51 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 0.25/0.51 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % measures_less
% 0.25/0.51 thf(fact_4147_measures__lesseq,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
% 0.25/0.51 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
% 0.25/0.51 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % measures_lesseq
% 0.25/0.51 thf(fact_4148_effect__lengthE,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: nat,N: nat] :
% 0.25/0.51 ( ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H3 @ H4 @ R @ N )
% 0.25/0.51 => ~ ( ( R
% 0.25/0.51 = ( array_length @ A @ H4 @ A3 ) )
% 0.25/0.51 => ( ( H4 = H3 )
% 0.25/0.51 => ( N
% 0.25/0.51 != ( one_one @ nat ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_lengthE
% 0.25/0.51 thf(fact_4149_effect__lengthI,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( heap @ A )
% 0.25/0.51 => ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: nat,A3: array @ A,N: nat] :
% 0.25/0.51 ( ( H4 = H3 )
% 0.25/0.51 => ( ( R
% 0.25/0.51 = ( array_length @ A @ H3 @ A3 ) )
% 0.25/0.51 => ( ( N
% 0.25/0.51 = ( one_one @ nat ) )
% 0.25/0.51 => ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % effect_lengthI
% 0.25/0.51 thf(fact_4150_len_H__def,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType] :
% 0.25/0.51 ( ( ( heap @ B )
% 0.25/0.51 & ( semiring_1 @ A ) )
% 0.25/0.51 => ( ( array_len2 @ B @ A )
% 0.25/0.51 = ( ^ [A7: array @ B] :
% 0.25/0.51 ( heap_Time_bind @ nat @ A @ ( array_len @ B @ A7 )
% 0.25/0.51 @ ^ [N5: nat] : ( heap_Time_ureturn @ A @ ( semiring_1_of_nat @ A @ N5 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % len'_def
% 0.25/0.51 thf(fact_4151_mset__zip__take__Cons__drop__twice,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,J: nat,X: A,Y: B] :
% 0.25/0.51 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.51 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.51 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.51 => ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys2 ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys2 ) ) ) ) )
% 0.25/0.51 = ( add_mset @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_zip_take_Cons_drop_twice
% 0.25/0.51 thf(fact_4152_image__mset__map__of,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
% 0.25/0.51 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 0.25/0.51 => ( ( image_mset @ A @ B
% 0.25/0.51 @ ^ [I3: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I3 ) )
% 0.25/0.51 @ ( mset @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 0.25/0.51 = ( mset @ B @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % image_mset_map_of
% 0.25/0.51 thf(fact_4153_mset__set__Union,axiom,
% 0.25/0.51 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.51 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.51 => ( ( finite_finite2 @ A @ B6 )
% 0.25/0.51 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.51 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.51 => ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A4 ) @ ( mset_set @ A @ B6 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_set_Union
% 0.25/0.51 thf(fact_4154_union__mset__add__mset__right,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,A3: A,B6: multiset @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ A3 @ B6 ) )
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_mset_add_mset_right
% 0.25/0.51 thf(fact_4155_union__mset__add__mset__left,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ A4 ) @ B6 )
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_mset_add_mset_left
% 0.25/0.51 thf(fact_4156_image__mset__union,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,F: B > A,M3: multiset @ B,N6: multiset @ B] :
% 0.25/0.51 ( ( image_mset @ B @ A @ F @ ( plus_plus @ ( multiset @ B ) @ M3 @ N6 ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M3 ) @ ( image_mset @ B @ A @ F @ N6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % image_mset_union
% 0.25/0.51 thf(fact_4157_mset__map__split__orig,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M13: multiset @ A,M24: multiset @ A] :
% 0.25/0.51 ( ( ( image_mset @ B @ A @ F @ P )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ M13 @ M24 ) )
% 0.25/0.51 => ~ ! [P1: multiset @ B,P22: multiset @ B] :
% 0.25/0.51 ( ( P
% 0.25/0.51 = ( plus_plus @ ( multiset @ B ) @ P1 @ P22 ) )
% 0.25/0.51 => ( ( ( image_mset @ B @ A @ F @ P1 )
% 0.25/0.51 = M13 )
% 0.25/0.51 => ( ( image_mset @ B @ A @ F @ P22 )
% 0.25/0.51 != M24 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_map_split_orig
% 0.25/0.51 thf(fact_4158_add__mset__add__single,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( add_mset @ A )
% 0.25/0.51 = ( ^ [A7: A,A9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A9 @ ( add_mset @ A @ A7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % add_mset_add_single
% 0.25/0.51 thf(fact_4159_union__is__single,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,A3: A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N6 )
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = ( ( ( M3
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 & ( N6
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 | ( ( M3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( N6
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_is_single
% 0.25/0.51 thf(fact_4160_single__is__union,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.51 = ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = M3 )
% 0.25/0.51 & ( N6
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 | ( ( M3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = N6 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % single_is_union
% 0.25/0.51 thf(fact_4161_add__mset__replicate__mset__safe,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,M3: multiset @ B,A3: B] :
% 0.25/0.51 ( ( nO_MATCH @ ( multiset @ A ) @ ( multiset @ B ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
% 0.25/0.51 => ( ( add_mset @ B @ A3 @ M3 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ B ) @ ( add_mset @ B @ A3 @ ( zero_zero @ ( multiset @ B ) ) ) @ M3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % add_mset_replicate_mset_safe
% 0.25/0.51 thf(fact_4162_mset__single__cases,axiom,
% 0.25/0.51 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ S @ C2 )
% 0.25/0.51 = ( add_mset @ A @ R4 @ C6 ) )
% 0.25/0.51 => ( ( ( S = R4 )
% 0.25/0.51 => ( C2 != C6 ) )
% 0.25/0.51 => ~ ( ( C6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( ( C2
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 != ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_single_cases
% 0.25/0.51 thf(fact_4163_mset__single__cases_H,axiom,
% 0.25/0.51 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ S @ C2 )
% 0.25/0.51 = ( add_mset @ A @ R4 @ C6 ) )
% 0.25/0.51 => ( ( ( S = R4 )
% 0.25/0.51 => ( C2 != C6 ) )
% 0.25/0.51 => ~ ! [Cc: multiset @ A] :
% 0.25/0.51 ( ( C6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
% 0.25/0.51 => ( ( C2
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
% 0.25/0.51 => ( ( ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = Cc )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 != Cc ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_single_cases'
% 0.25/0.51 thf(fact_4164_mset__single__cases2,axiom,
% 0.25/0.51 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ S @ C2 )
% 0.25/0.51 = ( add_mset @ A @ R4 @ C6 ) )
% 0.25/0.51 => ( ( ( S = R4 )
% 0.25/0.51 => ( C2 != C6 ) )
% 0.25/0.51 => ~ ( ( C6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 => ( ( C2
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 != ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_single_cases2
% 0.25/0.51 thf(fact_4165_mset__single__cases2_H,axiom,
% 0.25/0.51 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ S @ C2 )
% 0.25/0.51 = ( add_mset @ A @ R4 @ C6 ) )
% 0.25/0.51 => ( ( ( S = R4 )
% 0.25/0.51 => ( C2 != C6 ) )
% 0.25/0.51 => ~ ! [Cc: multiset @ A] :
% 0.25/0.51 ( ( C6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 => ( ( C2
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 => ( ( ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = Cc )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 != Cc ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_single_cases2'
% 0.25/0.51 thf(fact_4166_mset__unplusm__dist__cases,axiom,
% 0.25/0.51 ! [A: $tType,S: A,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 0.25/0.51 => ( ( ( B6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( A4
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
% 0.25/0.51 => ~ ( ( C5
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( A4
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_unplusm_dist_cases
% 0.25/0.51 thf(fact_4167_mset__unplusm__dist__cases2,axiom,
% 0.25/0.51 ! [A: $tType,B6: multiset @ A,C5: multiset @ A,S: A,A4: multiset @ A] :
% 0.25/0.51 ( ( ( plus_plus @ ( multiset @ A ) @ B6 @ C5 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 ) )
% 0.25/0.51 => ( ( ( B6
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( A4
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
% 0.25/0.51 => ~ ( ( C5
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( A4
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_unplusm_dist_cases2
% 0.25/0.51 thf(fact_4168_image__mset_Ocompositionality,axiom,
% 0.25/0.51 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Multiset: multiset @ A] :
% 0.25/0.51 ( ( image_mset @ B @ C @ F @ ( image_mset @ A @ B @ G @ Multiset ) )
% 0.25/0.51 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Multiset ) ) ).
% 0.25/0.51
% 0.25/0.51 % image_mset.compositionality
% 0.25/0.51 thf(fact_4169_multiset_Omap__comp,axiom,
% 0.25/0.51 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: multiset @ A] :
% 0.25/0.51 ( ( image_mset @ B @ C @ G @ ( image_mset @ A @ B @ F @ V ) )
% 0.25/0.51 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.25/0.51
% 0.25/0.51 % multiset.map_comp
% 0.25/0.51 thf(fact_4170_image__mset_Ocomp,axiom,
% 0.25/0.51 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 0.25/0.51 ( ( comp @ ( multiset @ B ) @ ( multiset @ C ) @ ( multiset @ A ) @ ( image_mset @ B @ C @ F ) @ ( image_mset @ A @ B @ G ) )
% 0.25/0.51 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % image_mset.comp
% 0.25/0.51 thf(fact_4171_mset__map__id,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,F: B > A,G: A > B,X10: multiset @ A] :
% 0.25/0.51 ( ! [X2: A] :
% 0.25/0.51 ( ( F @ ( G @ X2 ) )
% 0.25/0.51 = X2 )
% 0.25/0.51 => ( ( image_mset @ B @ A @ F @ ( image_mset @ A @ B @ G @ X10 ) )
% 0.25/0.51 = X10 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_map_id
% 0.25/0.51 thf(fact_4172_mset__union__2__elem,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,B2: A,C2: A,M3: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ A3 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = ( add_mset @ A @ C2 @ M3 ) )
% 0.25/0.51 => ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = M3 )
% 0.25/0.51 & ( B2 = C2 ) )
% 0.25/0.51 | ( ( A3 = C2 )
% 0.25/0.51 & ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = M3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_union_2_elem
% 0.25/0.51 thf(fact_4173_wcount__add__mset,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,X: A,M3: multiset @ A,A3: A] :
% 0.25/0.51 ( ( wcount @ A @ F @ ( add_mset @ A @ X @ M3 ) @ A3 )
% 0.25/0.51 = ( plus_plus @ nat @ ( if @ nat @ ( X = A3 ) @ ( suc @ ( F @ A3 ) ) @ ( zero_zero @ nat ) ) @ ( wcount @ A @ F @ M3 @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % wcount_add_mset
% 0.25/0.51 thf(fact_4174_subset__mset_Osum__list__update,axiom,
% 0.25/0.51 ! [A: $tType,K: nat,Xs: list @ ( multiset @ A ),X: multiset @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Xs ) )
% 0.25/0.51 => ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( list_update @ ( multiset @ A ) @ Xs @ K @ X ) )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X ) @ ( nth @ ( multiset @ A ) @ Xs @ K ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_list_update
% 0.25/0.51 thf(fact_4175_size__multiset__add__mset,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,A3: A,M3: multiset @ A] :
% 0.25/0.51 ( ( size_multiset @ A @ F @ ( add_mset @ A @ A3 @ M3 ) )
% 0.25/0.51 = ( plus_plus @ nat @ ( suc @ ( F @ A3 ) ) @ ( size_multiset @ A @ F @ M3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_multiset_add_mset
% 0.25/0.51 thf(fact_4176_subset__mset_Osum__eq__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,F6: set @ B,F: B > ( multiset @ A )] :
% 0.25/0.51 ( ( finite_finite2 @ B @ F6 )
% 0.25/0.51 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ F6 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ! [X3: B] :
% 0.25/0.51 ( ( member @ B @ X3 @ F6 )
% 0.25/0.51 => ( ( F @ X3 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_eq_0_iff
% 0.25/0.51 thf(fact_4177_size__multiset__union,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( size_multiset @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.51 = ( plus_plus @ nat @ ( size_multiset @ A @ F @ M3 ) @ ( size_multiset @ A @ F @ N6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_multiset_union
% 0.25/0.51 thf(fact_4178_subset__mset_Osum__list__eq__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,Ns: list @ ( multiset @ A )] :
% 0.25/0.51 ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ! [X3: multiset @ A] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Ns ) )
% 0.25/0.51 => ( X3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_list_eq_0_iff
% 0.25/0.51 thf(fact_4179_wcount__union,axiom,
% 0.25/0.51 ! [A: $tType,F: A > nat,M3: multiset @ A,N6: multiset @ A,A3: A] :
% 0.25/0.51 ( ( wcount @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ A3 )
% 0.25/0.51 = ( plus_plus @ nat @ ( wcount @ A @ F @ M3 @ A3 ) @ ( wcount @ A @ F @ N6 @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % wcount_union
% 0.25/0.51 thf(fact_4180_subset__mset_Oelem__le__sum__list,axiom,
% 0.25/0.51 ! [A: $tType,K: nat,Ns: list @ ( multiset @ A )] :
% 0.25/0.51 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Ns ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( nth @ ( multiset @ A ) @ Ns @ K ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.elem_le_sum_list
% 0.25/0.51 thf(fact_4181_subset__mset_Osum__mono2,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,B6: set @ B,A4: set @ B,F: B > ( multiset @ A )] :
% 0.25/0.51 ( ( finite_finite2 @ B @ B6 )
% 0.25/0.51 => ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
% 0.25/0.51 => ( ! [B3: B] :
% 0.25/0.51 ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ B3 ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ B6 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_mono2
% 0.25/0.51 thf(fact_4182_next_Osimps,axiom,
% 0.25/0.51 ! [V: code_natural,W: code_natural] :
% 0.25/0.51 ( ( next @ ( product_Pair @ code_natural @ code_natural @ V @ W ) )
% 0.25/0.51 = ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % next.simps
% 0.25/0.51 thf(fact_4183_find__dropWhile,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( find @ A )
% 0.25/0.51 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 0.25/0.51 ( case_list @ ( option @ A ) @ A @ ( none @ A )
% 0.25/0.51 @ ^ [X3: A,Xa4: list @ A] : ( some @ A @ X3 )
% 0.25/0.51 @ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % find_dropWhile
% 0.25/0.51 thf(fact_4184_mset__subset__eq__mono__add__right__cancel,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,C5: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ A4 @ B6 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_mono_add_right_cancel
% 0.25/0.51 thf(fact_4185_mset__subset__eq__mono__add__left__cancel,axiom,
% 0.25/0.51 ! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ A4 @ B6 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_mono_add_left_cancel
% 0.25/0.51 thf(fact_4186_subset__mset_Oadd__le__cancel__right,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_le_cancel_right
% 0.25/0.51 thf(fact_4187_subset__mset_Oadd__le__cancel__left,axiom,
% 0.25/0.51 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_le_cancel_left
% 0.25/0.51 thf(fact_4188_subset__mset_Ole__add__same__cancel2,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.le_add_same_cancel2
% 0.25/0.51 thf(fact_4189_subset__mset_Ole__add__same__cancel1,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.le_add_same_cancel1
% 0.25/0.51 thf(fact_4190_subset__mset_Oadd__le__same__cancel2,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ B2 )
% 0.25/0.51 = ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_le_same_cancel2
% 0.25/0.51 thf(fact_4191_subset__mset_Oadd__le__same__cancel1,axiom,
% 0.25/0.51 ! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) @ B2 )
% 0.25/0.51 = ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_le_same_cancel1
% 0.25/0.51 thf(fact_4192_mset__subset__eq__multiset__union__diff__commute,axiom,
% 0.25/0.51 ! [A: $tType,B6: multiset @ A,A4: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ B6 @ A4 )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_multiset_union_diff_commute
% 0.25/0.51 thf(fact_4193_subset__mset_Oadd__diff__assoc2,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ C2 )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_diff_assoc2
% 0.25/0.51 thf(fact_4194_subset__mset_Oadd__diff__assoc,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_diff_assoc
% 0.25/0.51 thf(fact_4195_subset__mset_Odiff__add,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ A3 )
% 0.25/0.51 = B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.diff_add
% 0.25/0.51 thf(fact_4196_mset__le__add__mset__decr__left1,axiom,
% 0.25/0.51 ! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_add_mset_decr_left1
% 0.25/0.51 thf(fact_4197_mset__le__incr__right2,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_incr_right2
% 0.25/0.51 thf(fact_4198_mset__le__incr__right1,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_incr_right1
% 0.25/0.51 thf(fact_4199_mset__le__decr__left2,axiom,
% 0.25/0.51 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_decr_left2
% 0.25/0.51 thf(fact_4200_mset__le__decr__left1,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_decr_left1
% 0.25/0.51 thf(fact_4201_mset__union__subset,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ A4 @ C5 )
% 0.25/0.51 & ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_union_subset
% 0.25/0.51 thf(fact_4202_mset__le__distrib,axiom,
% 0.25/0.51 ! [A: $tType,X10: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ X10 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 0.25/0.51 => ~ ! [Xa5: multiset @ A,Xb4: multiset @ A] :
% 0.25/0.51 ( ( X10
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ Xa5 @ Xb4 ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ Xa5 @ A4 )
% 0.25/0.51 => ~ ( subseteq_mset @ A @ Xb4 @ B6 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_distrib
% 0.25/0.51 thf(fact_4203_mset__le__addE,axiom,
% 0.25/0.51 ! [A: $tType,Xs: multiset @ A,Ys2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ Xs @ Ys2 )
% 0.25/0.51 => ~ ! [Zs2: multiset @ A] :
% 0.25/0.51 ( Ys2
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_addE
% 0.25/0.51 thf(fact_4204_mset__subset__eq__exists__conv,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( subseteq_mset @ A )
% 0.25/0.51 = ( ^ [A9: multiset @ A,B9: multiset @ A] :
% 0.25/0.51 ? [C7: multiset @ A] :
% 0.25/0.51 ( B9
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ A9 @ C7 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_exists_conv
% 0.25/0.51 thf(fact_4205_mset__subset__eq__add__right,axiom,
% 0.25/0.51 ! [A: $tType,B6: multiset @ A,A4: multiset @ A] : ( subseteq_mset @ A @ B6 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_add_right
% 0.25/0.51 thf(fact_4206_mset__subset__eq__mono__add,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A,D4: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A4 @ B6 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ C5 @ D4 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ D4 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_mono_add
% 0.25/0.51 thf(fact_4207_mset__subset__eq__add__left,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] : ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_subset_eq_add_left
% 0.25/0.51 thf(fact_4208_subset__mset_Oadd__le__imp__le__right,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_le_imp_le_right
% 0.25/0.51 thf(fact_4209_subset__mset_Oadd__le__imp__le__left,axiom,
% 0.25/0.51 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 0.25/0.51 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_le_imp_le_left
% 0.25/0.51 thf(fact_4210_subset__mset_Oadd__right__mono,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_right_mono
% 0.25/0.51 thf(fact_4211_subset__mset_Oadd__left__mono,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_left_mono
% 0.25/0.51 thf(fact_4212_subset__mset_Ole__iff__add,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( subseteq_mset @ A )
% 0.25/0.51 = ( ^ [A7: multiset @ A,B5: multiset @ A] :
% 0.25/0.51 ? [C3: multiset @ A] :
% 0.25/0.51 ( B5
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ A7 @ C3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.le_iff_add
% 0.25/0.51 thf(fact_4213_subset__mset_Oless__eqE,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ~ ! [C4: multiset @ A] :
% 0.25/0.51 ( B2
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.less_eqE
% 0.25/0.51 thf(fact_4214_subset__mset_Oadd__mono,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ C2 @ D3 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_mono
% 0.25/0.51 thf(fact_4215_full__exhaustive__natural_H_Ocases,axiom,
% 0.25/0.51 ! [X: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 0.25/0.51 ~ ! [F4: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % full_exhaustive_natural'.cases
% 0.25/0.51 thf(fact_4216_full__exhaustive__fun_H_Ocases,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType] :
% 0.25/0.51 ( ( ( quickc3360725361186068524ustive @ B )
% 0.25/0.51 & ( cl_HOL_Oequal @ A )
% 0.25/0.51 & ( quickc3360725361186068524ustive @ A ) )
% 0.25/0.51 => ! [X: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 0.25/0.51 ~ ! [F4: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % full_exhaustive_fun'.cases
% 0.25/0.51 thf(fact_4217_exhaustive__natural_H_Ocases,axiom,
% 0.25/0.51 ! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 0.25/0.51 ~ ! [F4: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % exhaustive_natural'.cases
% 0.25/0.51 thf(fact_4218_exhaustive__fun_H_Ocases,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType] :
% 0.25/0.51 ( ( ( quickc658316121487927005ustive @ B )
% 0.25/0.51 & ( cl_HOL_Oequal @ A )
% 0.25/0.51 & ( quickc658316121487927005ustive @ A ) )
% 0.25/0.51 => ! [X: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 0.25/0.51 ~ ! [F4: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % exhaustive_fun'.cases
% 0.25/0.51 thf(fact_4219_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
% 0.25/0.51 ! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
% 0.25/0.51 ~ ! [F4: code_natural > A,N3: code_natural,M5: code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M5 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Lazy_Sequence.iterate_upto.cases
% 0.25/0.51 thf(fact_4220_iter_Ocases,axiom,
% 0.25/0.51 ! [A: $tType,X: product_prod @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )] :
% 0.25/0.51 ~ ! [Random: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ),Nrandom: code_natural,Seed: product_prod @ code_natural @ code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Random @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Nrandom @ Seed ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % iter.cases
% 0.25/0.51 thf(fact_4221_log_Ocases,axiom,
% 0.25/0.51 ! [X: product_prod @ code_natural @ code_natural] :
% 0.25/0.51 ~ ! [B3: code_natural,I2: code_natural] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ code_natural @ code_natural @ B3 @ I2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % log.cases
% 0.25/0.51 thf(fact_4222_mset__le__subtract,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A4 @ B6 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_subtract
% 0.25/0.51 thf(fact_4223_mset__le__add__mset__decr__left2,axiom,
% 0.25/0.51 ! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_add_mset_decr_left2
% 0.25/0.51 thf(fact_4224_mset__le__single__cases,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,A3: A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ M3 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 => ( ( M3
% 0.25/0.51 != ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( M3
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_single_cases
% 0.25/0.51 thf(fact_4225_mset__le__add__mset,axiom,
% 0.25/0.51 ! [A: $tType,X: A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ C5 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C5 )
% 0.25/0.51 & ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_add_mset
% 0.25/0.51 thf(fact_4226_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ( X
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( Y
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_nonpos_eq_0_iff
% 0.25/0.51 thf(fact_4227_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 0.25/0.51 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ Y )
% 0.25/0.51 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ( X
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 & ( Y
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_nonneg_eq_0_iff
% 0.25/0.51 thf(fact_4228_subset__mset_Oadd__nonpos__nonpos,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_nonpos_nonpos
% 0.25/0.51 thf(fact_4229_subset__mset_Oadd__nonneg__nonneg,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_nonneg_nonneg
% 0.25/0.51 thf(fact_4230_subset__mset_Oadd__increasing2,axiom,
% 0.25/0.51 ! [A: $tType,C2: multiset @ A,B2: multiset @ A,A3: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ C2 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ B2 @ A3 )
% 0.25/0.51 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_increasing2
% 0.25/0.51 thf(fact_4231_subset__mset_Oadd__decreasing2,axiom,
% 0.25/0.51 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_decreasing2
% 0.25/0.51 thf(fact_4232_subset__mset_Oadd__increasing,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 0.25/0.51 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_increasing
% 0.25/0.51 thf(fact_4233_subset__mset_Oadd__decreasing,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ C2 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_decreasing
% 0.25/0.51 thf(fact_4234_mset__le__subtract__left,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,X10: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ X10 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ B6 @ ( minus_minus @ ( multiset @ A ) @ X10 @ A4 ) )
% 0.25/0.51 & ( subseteq_mset @ A @ A4 @ X10 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_subtract_left
% 0.25/0.51 thf(fact_4235_mset__le__subtract__right,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,X10: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ X10 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ A4 @ ( minus_minus @ ( multiset @ A ) @ X10 @ B6 ) )
% 0.25/0.51 & ( subseteq_mset @ A @ B6 @ X10 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_subtract_right
% 0.25/0.51 thf(fact_4236_multiset__diff__union__assoc,axiom,
% 0.25/0.51 ! [A: $tType,C5: multiset @ A,B6: multiset @ A,A4: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ C5 @ B6 )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ A4 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % multiset_diff_union_assoc
% 0.25/0.51 thf(fact_4237_subset__eq__diff__conv,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,C5: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 )
% 0.25/0.51 = ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_eq_diff_conv
% 0.25/0.51 thf(fact_4238_subset__mset_Ole__imp__diff__is__add,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( ( minus_minus @ ( multiset @ A ) @ B2 @ A3 )
% 0.25/0.51 = C2 )
% 0.25/0.51 = ( B2
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.le_imp_diff_is_add
% 0.25/0.51 thf(fact_4239_subset__mset_Oadd__diff__inverse,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 0.25/0.51 = B2 ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.add_diff_inverse
% 0.25/0.51 thf(fact_4240_subset__mset_Odiff__diff__right,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.diff_diff_right
% 0.25/0.51 thf(fact_4241_subset__mset_Odiff__add__assoc2,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ C2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.diff_add_assoc2
% 0.25/0.51 thf(fact_4242_subset__mset_Odiff__add__assoc,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A3 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.diff_add_assoc
% 0.25/0.51 thf(fact_4243_subset__mset_Ole__diff__conv2,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 0.25/0.51 = ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.le_diff_conv2
% 0.25/0.51 thf(fact_4244_subset__mset_Ole__add__diff,axiom,
% 0.25/0.51 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.51 => ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.le_add_diff
% 0.25/0.51 thf(fact_4245_mset__map__split__orig__le,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M13: multiset @ A,M24: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P ) @ ( plus_plus @ ( multiset @ A ) @ M13 @ M24 ) )
% 0.25/0.51 => ~ ! [P1: multiset @ B,P22: multiset @ B] :
% 0.25/0.51 ( ( P
% 0.25/0.51 = ( plus_plus @ ( multiset @ B ) @ P1 @ P22 ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P1 ) @ M13 )
% 0.25/0.51 => ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P22 ) @ M24 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_map_split_orig_le
% 0.25/0.51 thf(fact_4246_mset__le__subtract__add__mset__right,axiom,
% 0.25/0.51 ! [A: $tType,X: A,B6: multiset @ A,X10: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ X10 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ B6 ) )
% 0.25/0.51 & ( subseteq_mset @ A @ B6 @ X10 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_subtract_add_mset_right
% 0.25/0.51 thf(fact_4247_mset__le__subtract__add__mset__left,axiom,
% 0.25/0.51 ! [A: $tType,X: A,B6: multiset @ A,X10: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ X10 )
% 0.25/0.51 => ( ( subseteq_mset @ A @ B6 @ ( minus_minus @ ( multiset @ A ) @ X10 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 & ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X10 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_subtract_add_mset_left
% 0.25/0.51 thf(fact_4248_subset__mset_Osum__nonneg,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A )] :
% 0.25/0.51 ( ! [X2: B] :
% 0.25/0.51 ( ( member @ B @ X2 @ A4 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X2 ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_nonneg
% 0.25/0.51 thf(fact_4249_subset__mset_Osum__nonpos,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,A4: set @ B,F: B > ( multiset @ A )] :
% 0.25/0.51 ( ! [X2: B] :
% 0.25/0.51 ( ( member @ B @ X2 @ A4 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( F @ X2 ) @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_nonpos
% 0.25/0.51 thf(fact_4250_subset__mset_Osum__list__nonneg,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ ( multiset @ A )] :
% 0.25/0.51 ( ! [X2: multiset @ A] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_list_nonneg
% 0.25/0.51 thf(fact_4251_subset__mset_Osum__list__nonpos,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ ( multiset @ A )] :
% 0.25/0.51 ( ! [X2: multiset @ A] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 0.25/0.51 => ( subseteq_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_list_nonpos
% 0.25/0.51 thf(fact_4252_subset__mset_Omember__le__sum__list,axiom,
% 0.25/0.51 ! [A: $tType,X: multiset @ A,Xs: list @ ( multiset @ A )] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 0.25/0.51 => ( subseteq_mset @ A @ X @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.member_le_sum_list
% 0.25/0.51 thf(fact_4253_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,Xs: list @ ( multiset @ A )] :
% 0.25/0.51 ( ! [X2: multiset @ A] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 ) )
% 0.25/0.51 => ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ! [X3: multiset @ A] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 0.25/0.51 => ( X3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_list_nonneg_eq_0_iff
% 0.25/0.51 thf(fact_4254_subset__mset_Osum__mono,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,K2: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
% 0.25/0.51 ( ! [I2: B] :
% 0.25/0.51 ( ( member @ B @ I2 @ K2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ K2 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ K2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_mono
% 0.25/0.51 thf(fact_4255_subset__mset_Osum__le__included,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,C: $tType,S: set @ B,T2: set @ C,G: C > ( multiset @ A ),I: C > B,F: B > ( multiset @ A )] :
% 0.25/0.51 ( ( finite_finite2 @ B @ S )
% 0.25/0.51 => ( ( finite_finite2 @ C @ T2 )
% 0.25/0.51 => ( ! [X2: C] :
% 0.25/0.51 ( ( member @ C @ X2 @ T2 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( G @ X2 ) ) )
% 0.25/0.51 => ( ! [X2: B] :
% 0.25/0.51 ( ( member @ B @ X2 @ S )
% 0.25/0.51 => ? [Xa2: C] :
% 0.25/0.51 ( ( member @ C @ Xa2 @ T2 )
% 0.25/0.51 & ( ( I @ Xa2 )
% 0.25/0.51 = X2 )
% 0.25/0.51 & ( subseteq_mset @ A @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ C @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ T2 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_le_included
% 0.25/0.51 thf(fact_4256_subset__mset_Osum__nonneg__eq__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A )] :
% 0.25/0.51 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.51 => ( ! [X2: B] :
% 0.25/0.51 ( ( member @ B @ X2 @ A4 )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X2 ) ) )
% 0.25/0.51 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ! [X3: B] :
% 0.25/0.51 ( ( member @ B @ X3 @ A4 )
% 0.25/0.51 => ( ( F @ X3 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_nonneg_eq_0_iff
% 0.25/0.51 thf(fact_4257_subset__mset_Osum__nonneg__0,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,S: set @ B,F: B > ( multiset @ A ),I: B] :
% 0.25/0.51 ( ( finite_finite2 @ B @ S )
% 0.25/0.51 => ( ! [I2: B] :
% 0.25/0.51 ( ( member @ B @ I2 @ S )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 0.25/0.51 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ( ( member @ B @ I @ S )
% 0.25/0.51 => ( ( F @ I )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_nonneg_0
% 0.25/0.51 thf(fact_4258_subset__mset_Osum__nonneg__leq__bound,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType,S: set @ B,F: B > ( multiset @ A ),B6: multiset @ A,I: B] :
% 0.25/0.51 ( ( finite_finite2 @ B @ S )
% 0.25/0.51 => ( ! [I2: B] :
% 0.25/0.51 ( ( member @ B @ I2 @ S )
% 0.25/0.51 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 0.25/0.51 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S )
% 0.25/0.51 = B6 )
% 0.25/0.51 => ( ( member @ B @ I @ S )
% 0.25/0.51 => ( subseteq_mset @ A @ ( F @ I ) @ B6 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_nonneg_leq_bound
% 0.25/0.51 thf(fact_4259_minus__shift__def,axiom,
% 0.25/0.51 ( minus_shift
% 0.25/0.51 = ( ^ [R3: code_natural,K4: code_natural,L3: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K4 @ L3 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R3 @ K4 ) @ L3 ) @ ( minus_minus @ code_natural @ K4 @ L3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % minus_shift_def
% 0.25/0.51 thf(fact_4260_split__seed__def,axiom,
% 0.25/0.51 ( split_seed
% 0.25/0.51 = ( ^ [S4: product_prod @ code_natural @ code_natural] :
% 0.25/0.51 ( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.51 @ ^ [V2: code_natural,W2: code_natural] :
% 0.25/0.51 ( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.51 @ ^ [V5: code_natural,W3: code_natural] : ( product_Pair @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_Pair @ code_natural @ code_natural @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W3 ) @ ( product_Pair @ code_natural @ code_natural @ V5 @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
% 0.25/0.51 @ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S4 ) ) )
% 0.25/0.51 @ S4 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % split_seed_def
% 0.25/0.51 thf(fact_4261_log_Osimps,axiom,
% 0.25/0.51 ( log
% 0.25/0.51 = ( ^ [B5: code_natural,I3: code_natural] :
% 0.25/0.51 ( if @ code_natural
% 0.25/0.51 @ ( ( ord_less_eq @ code_natural @ B5 @ ( one_one @ code_natural ) )
% 0.25/0.51 | ( ord_less @ code_natural @ I3 @ B5 ) )
% 0.25/0.51 @ ( one_one @ code_natural )
% 0.25/0.51 @ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B5 @ ( divide_divide @ code_natural @ I3 @ B5 ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % log.simps
% 0.25/0.51 thf(fact_4262_log_Oelims,axiom,
% 0.25/0.51 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 0.25/0.51 ( ( ( log @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.25/0.51 | ( ord_less @ code_natural @ Xa @ X ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( one_one @ code_natural ) ) )
% 0.25/0.51 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.25/0.51 | ( ord_less @ code_natural @ Xa @ X ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % log.elims
% 0.25/0.51 thf(fact_4263_extract__def,axiom,
% 0.25/0.51 ! [A: $tType] :
% 0.25/0.51 ( ( extract @ A )
% 0.25/0.51 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 0.25/0.51 ( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 0.25/0.51 @ ^ [Y3: A,Ys: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Ys ) ) )
% 0.25/0.51 @ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % extract_def
% 0.25/0.51 thf(fact_4264_drop__takeWhile,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,P: A > $o,L: list @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
% 0.25/0.51 => ( ( drop @ A @ I @ ( takeWhile @ A @ P @ L ) )
% 0.25/0.51 = ( takeWhile @ A @ P @ ( drop @ A @ I @ L ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % drop_takeWhile
% 0.25/0.51 thf(fact_4265_eq__len__takeWhile__conv,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,P: A > $o,L: list @ A] :
% 0.25/0.51 ( ( I
% 0.25/0.51 = ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
% 0.25/0.51 = ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 & ! [J2: nat] :
% 0.25/0.51 ( ( ord_less @ nat @ J2 @ I )
% 0.25/0.51 => ( P @ ( nth @ A @ L @ J2 ) ) )
% 0.25/0.51 & ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 => ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % eq_len_takeWhile_conv
% 0.25/0.51 thf(fact_4266_less__length__takeWhile__conv,axiom,
% 0.25/0.51 ! [A: $tType,I: nat,P: A > $o,L: list @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
% 0.25/0.51 = ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 0.25/0.51 & ! [J2: nat] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ J2 @ I )
% 0.25/0.51 => ( P @ ( nth @ A @ L @ J2 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % less_length_takeWhile_conv
% 0.25/0.51 thf(fact_4267_length__dropWhile__takeWhile,axiom,
% 0.25/0.51 ! [A: $tType,X: nat,P: A > $o,Xs: list @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
% 0.25/0.51 => ( ord_less @ nat @ ( plus_plus @ nat @ X @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % length_dropWhile_takeWhile
% 0.25/0.51 thf(fact_4268_dropWhile__nth,axiom,
% 0.25/0.51 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 0.25/0.51 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
% 0.25/0.51 => ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
% 0.25/0.51 = ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % dropWhile_nth
% 0.25/0.51 thf(fact_4269_inc__shift__def,axiom,
% 0.25/0.51 ( inc_shift
% 0.25/0.51 = ( ^ [V2: code_natural,K4: code_natural] : ( if @ code_natural @ ( V2 = K4 ) @ ( one_one @ code_natural ) @ ( plus_plus @ code_natural @ K4 @ ( one_one @ code_natural ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % inc_shift_def
% 0.25/0.51 thf(fact_4270_log_Opelims,axiom,
% 0.25/0.51 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 0.25/0.51 ( ( ( log @ X @ Xa )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
% 0.25/0.51 => ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.25/0.51 | ( ord_less @ code_natural @ Xa @ X ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( one_one @ code_natural ) ) )
% 0.25/0.51 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 0.25/0.51 | ( ord_less @ code_natural @ Xa @ X ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % log.pelims
% 0.25/0.51 thf(fact_4271_Random_Orange__def,axiom,
% 0.25/0.51 ( range
% 0.25/0.51 = ( ^ [K4: code_natural] :
% 0.25/0.51 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.51 @ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
% 0.25/0.51 @ ^ [L3: code_natural] :
% 0.25/0.51 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
% 0.25/0.51 @ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V2 @ ( times_times @ code_natural @ L3 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.25/0.51 @ ( one_one @ code_natural ) )
% 0.25/0.51 @ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V2 @ K4 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Random.range_def
% 0.25/0.51 thf(fact_4272_Predicate_Oiterate__upto_Opinduct,axiom,
% 0.25/0.51 ! [A: $tType,A0: code_natural > A,A1: code_natural,A22: code_natural,P: ( code_natural > A ) > code_natural > code_natural > $o] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ A0 @ ( product_Pair @ code_natural @ code_natural @ A1 @ A22 ) ) )
% 0.25/0.51 => ( ! [F4: code_natural > A,N3: code_natural,M5: code_natural] :
% 0.25/0.51 ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M5 ) ) )
% 0.25/0.51 => ( ! [X7: product_unit] :
% 0.25/0.51 ( ~ ( ord_less @ code_natural @ M5 @ N3 )
% 0.25/0.51 => ( P @ F4 @ ( plus_plus @ code_natural @ N3 @ ( one_one @ code_natural ) ) @ M5 ) )
% 0.25/0.51 => ( P @ F4 @ N3 @ M5 ) ) )
% 0.25/0.51 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % Predicate.iterate_upto.pinduct
% 0.25/0.51 thf(fact_4273_mset__size2elem,axiom,
% 0.25/0.51 ! [A: $tType,P: multiset @ A,Q3: A,Q4: A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.51 => ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
% 0.25/0.51 => ( P
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_size2elem
% 0.25/0.51 thf(fact_4274_scomp__apply,axiom,
% 0.25/0.51 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 0.25/0.51 ( ( product_scomp @ B @ C @ D @ A )
% 0.25/0.51 = ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X3: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % scomp_apply
% 0.25/0.51 thf(fact_4275_size__union,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.51 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ N6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_union
% 0.25/0.51 thf(fact_4276_scomp__scomp,axiom,
% 0.25/0.51 ! [A: $tType,C: $tType,D: $tType,B: $tType,F5: $tType,E: $tType,F: A > ( product_prod @ E @ F5 ),G: E > F5 > ( product_prod @ C @ D ),H3: C > D > B] :
% 0.25/0.51 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F5 @ ( product_prod @ C @ D ) @ F @ G ) @ H3 )
% 0.25/0.51 = ( product_scomp @ A @ E @ F5 @ B @ F
% 0.25/0.51 @ ^ [X3: E] : ( product_scomp @ F5 @ C @ D @ B @ ( G @ X3 ) @ H3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % scomp_scomp
% 0.25/0.51 thf(fact_4277_scomp__Pair,axiom,
% 0.25/0.51 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 0.25/0.51 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 0.25/0.51 = X ) ).
% 0.25/0.51
% 0.25/0.51 % scomp_Pair
% 0.25/0.51 thf(fact_4278_Pair__scomp,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
% 0.25/0.51 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
% 0.25/0.51 = ( F @ X ) ) ).
% 0.25/0.51
% 0.25/0.51 % Pair_scomp
% 0.25/0.51 thf(fact_4279_scomp__def,axiom,
% 0.25/0.51 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.25/0.51 ( ( product_scomp @ A @ B @ C @ D )
% 0.25/0.51 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X3: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X3 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % scomp_def
% 0.25/0.51 thf(fact_4280_size__1__singleton__mset,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A] :
% 0.25/0.51 ( ( ( size_size @ ( multiset @ A ) @ M3 )
% 0.25/0.51 = ( one_one @ nat ) )
% 0.25/0.51 => ? [A5: A] :
% 0.25/0.51 ( M3
% 0.25/0.51 = ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_1_singleton_mset
% 0.25/0.51 thf(fact_4281_size__single,axiom,
% 0.25/0.51 ! [A: $tType,B2: A] :
% 0.25/0.51 ( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = ( one_one @ nat ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_single
% 0.25/0.51 thf(fact_4282_iterate_Oelims,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
% 0.25/0.51 ( ( ( iterate @ B @ A @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( ( X
% 0.25/0.51 = ( zero_zero @ code_natural ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( product_Pair @ B @ A @ Xb ) ) )
% 0.25/0.51 & ( ( X
% 0.25/0.51 != ( zero_zero @ code_natural ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % iterate.elims
% 0.25/0.51 thf(fact_4283_iterate_Osimps,axiom,
% 0.25/0.51 ! [B: $tType,A: $tType] :
% 0.25/0.51 ( ( iterate @ B @ A )
% 0.25/0.51 = ( ^ [K4: code_natural,F2: B > A > ( product_prod @ B @ A ),X3: B] :
% 0.25/0.51 ( if @ ( A > ( product_prod @ B @ A ) )
% 0.25/0.51 @ ( K4
% 0.25/0.51 = ( zero_zero @ code_natural ) )
% 0.25/0.51 @ ( product_Pair @ B @ A @ X3 )
% 0.25/0.51 @ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F2 @ X3 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K4 @ ( one_one @ code_natural ) ) @ F2 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % iterate.simps
% 0.25/0.51 thf(fact_4284_size__mset__SucE,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A,N: nat] :
% 0.25/0.51 ( ( ( size_size @ ( multiset @ A ) @ A4 )
% 0.25/0.51 = ( suc @ N ) )
% 0.25/0.51 => ~ ! [A5: A,B10: multiset @ A] :
% 0.25/0.51 ( ( A4
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) @ B10 ) )
% 0.25/0.51 => ( ( size_size @ ( multiset @ A ) @ B10 )
% 0.25/0.51 != N ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_mset_SucE
% 0.25/0.51 thf(fact_4285_scomp__unfold,axiom,
% 0.25/0.51 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 0.25/0.51 ( ( product_scomp @ A @ B @ C @ D )
% 0.25/0.51 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X3: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X3 ) ) @ ( product_snd @ B @ C @ ( F2 @ X3 ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % scomp_unfold
% 0.25/0.51 thf(fact_4286_mset__size__le1__cases,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( suc @ ( zero_zero @ nat ) ) )
% 0.25/0.51 => ( ( M3
% 0.25/0.51 != ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 => ~ ! [M5: A] :
% 0.25/0.51 ( M3
% 0.25/0.51 != ( add_mset @ A @ M5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_size_le1_cases
% 0.25/0.51 thf(fact_4287_iterate_Opelims,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
% 0.25/0.51 ( ( ( iterate @ B @ A @ X @ Xa @ Xb )
% 0.25/0.51 = Y )
% 0.25/0.51 => ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
% 0.25/0.51 => ~ ( ( ( ( X
% 0.25/0.51 = ( zero_zero @ code_natural ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( product_Pair @ B @ A @ Xb ) ) )
% 0.25/0.51 & ( ( X
% 0.25/0.51 != ( zero_zero @ code_natural ) )
% 0.25/0.51 => ( Y
% 0.25/0.51 = ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
% 0.25/0.51 => ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % iterate.pelims
% 0.25/0.51 thf(fact_4288_size__Diff__singleton__if,axiom,
% 0.25/0.51 ! [A: $tType,X: A,A4: multiset @ A] :
% 0.25/0.51 ( ( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
% 0.25/0.51 => ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 = ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( one_one @ nat ) ) ) )
% 0.25/0.51 & ( ~ ( member @ A @ X @ ( set_mset @ A @ A4 ) )
% 0.25/0.51 => ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 = ( size_size @ ( multiset @ A ) @ A4 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_Diff_singleton_if
% 0.25/0.51 thf(fact_4289_size__Diff__singleton,axiom,
% 0.25/0.51 ! [A: $tType,X: A,M3: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
% 0.25/0.51 => ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 = ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_Diff_singleton
% 0.25/0.51 thf(fact_4290_size__diff__se,axiom,
% 0.25/0.51 ! [A: $tType,T2: A,S2: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
% 0.25/0.51 => ( ( size_size @ ( multiset @ A ) @ S2 )
% 0.25/0.51 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % size_diff_se
% 0.25/0.51 thf(fact_4291_set__mset__union,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.51 ( ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.51 = ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ M3 ) @ ( set_mset @ A @ N6 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % set_mset_union
% 0.25/0.51 thf(fact_4292_mset__diff__cancel1elem,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,B6: multiset @ A] :
% 0.25/0.51 ( ~ ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B6 )
% 0.25/0.51 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_diff_cancel1elem
% 0.25/0.51 thf(fact_4293_ex__Melem__conv,axiom,
% 0.25/0.51 ! [A: $tType,A4: multiset @ A] :
% 0.25/0.51 ( ( ? [X3: A] : ( member @ A @ X3 @ ( set_mset @ A @ A4 ) ) )
% 0.25/0.51 = ( A4
% 0.25/0.51 != ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % ex_Melem_conv
% 0.25/0.51 thf(fact_4294_union__iff,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 0.25/0.51 = ( ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 0.25/0.51 | ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % union_iff
% 0.25/0.51 thf(fact_4295_mset__un__cases,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 0.25/0.51 => ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 0.25/0.51 => ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_un_cases
% 0.25/0.51 thf(fact_4296_mset__left__cancel__union,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 0.25/0.51 => ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 0.25/0.51 => ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_left_cancel_union
% 0.25/0.51 thf(fact_4297_mset__right__cancel__union,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 0.25/0.51 => ( ~ ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 0.25/0.51 => ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_right_cancel_union
% 0.25/0.51 thf(fact_4298_iterate_Ocases,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
% 0.25/0.51 ~ ! [K5: code_natural,F4: B > A > ( product_prod @ B @ A ),X2: B] :
% 0.25/0.51 ( X
% 0.25/0.51 != ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K5 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F4 @ X2 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % iterate.cases
% 0.25/0.51 thf(fact_4299_image__mset__cong__pair,axiom,
% 0.25/0.51 ! [C: $tType,B: $tType,A: $tType,M3: multiset @ ( product_prod @ A @ B ),F: A > B > C,G: A > B > C] :
% 0.25/0.51 ( ! [X2: A,Y4: B] :
% 0.25/0.51 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M3 ) )
% 0.25/0.51 => ( ( F @ X2 @ Y4 )
% 0.25/0.51 = ( G @ X2 @ Y4 ) ) )
% 0.25/0.51 => ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ M3 )
% 0.25/0.51 = ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G ) @ M3 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % image_mset_cong_pair
% 0.25/0.51 thf(fact_4300_multi__member__skip,axiom,
% 0.25/0.51 ! [A: $tType,X: A,XS: multiset @ A,Y: A] :
% 0.25/0.51 ( ( member @ A @ X @ ( set_mset @ A @ XS ) )
% 0.25/0.51 => ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % multi_member_skip
% 0.25/0.51 thf(fact_4301_multi__member__this,axiom,
% 0.25/0.51 ! [A: $tType,X: A,XS: multiset @ A] : ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % multi_member_this
% 0.25/0.51 thf(fact_4302_mset__right__cancel__elem,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B2: A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 0.25/0.51 => ( ( A3 != B2 )
% 0.25/0.51 => ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_right_cancel_elem
% 0.25/0.51 thf(fact_4303_mset__left__cancel__elem,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,B2: A,A4: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 ) ) )
% 0.25/0.51 => ( ( A3 != B2 )
% 0.25/0.51 => ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_left_cancel_elem
% 0.25/0.51 thf(fact_4304_mset__le__mono__add__single,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,Ys2: multiset @ A,B2: A,Ws: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ Ys2 ) )
% 0.25/0.51 => ( ( member @ A @ B2 @ ( set_mset @ A @ Ws ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys2 @ Ws ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_le_mono_add_single
% 0.25/0.51 thf(fact_4305_mset__union__subset__s,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B6 ) @ C5 )
% 0.25/0.51 => ( ( member @ A @ A3 @ ( set_mset @ A @ C5 ) )
% 0.25/0.51 & ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_union_subset_s
% 0.25/0.51 thf(fact_4306_mset__2dist2__cases,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,B2: A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.51 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 0.25/0.51 => ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A4 )
% 0.25/0.51 => ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B6 )
% 0.25/0.51 => ( ( ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 0.25/0.51 => ~ ( member @ A @ B2 @ ( set_mset @ A @ B6 ) ) )
% 0.25/0.51 => ~ ( ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 0.25/0.51 => ~ ( member @ A @ B2 @ ( set_mset @ A @ A4 ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_2dist2_cases
% 0.25/0.51 thf(fact_4307_insert__DiffM2,axiom,
% 0.25/0.51 ! [A: $tType,X: A,M3: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = M3 ) ) ).
% 0.25/0.51
% 0.25/0.51 % insert_DiffM2
% 0.25/0.51 thf(fact_4308_diff__union__single__conv,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,J4: multiset @ A,I4: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ J4 ) )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ I4 @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % diff_union_single_conv
% 0.25/0.51 thf(fact_4309_mset__un__single__un__cases,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.51 ( ( ( add_mset @ A @ A3 @ A4 )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 0.25/0.51 => ( ( ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 0.25/0.51 => ( A4
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
% 0.25/0.51 => ~ ( ( member @ A @ A3 @ ( set_mset @ A @ C5 ) )
% 0.25/0.51 => ( A4
% 0.25/0.51 != ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_un_single_un_cases
% 0.25/0.51 thf(fact_4310_diff__union__single__conv2,axiom,
% 0.25/0.51 ! [A: $tType,A3: A,J4: multiset @ A,I4: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ A3 @ ( set_mset @ A @ J4 ) )
% 0.25/0.51 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J4 @ I4 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 0.25/0.51 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I4 ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % diff_union_single_conv2
% 0.25/0.51 thf(fact_4311_mset__union__diff__comm,axiom,
% 0.25/0.51 ! [A: $tType,T2: A,S2: multiset @ A,T6: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
% 0.25/0.51 => ( ( plus_plus @ ( multiset @ A ) @ T6 @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T6 @ S2 ) @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_union_diff_comm
% 0.25/0.51 thf(fact_4312_mset__contains__eq,axiom,
% 0.25/0.51 ! [A: $tType,M2: A,M3: multiset @ A] :
% 0.25/0.51 ( ( member @ A @ M2 @ ( set_mset @ A @ M3 ) )
% 0.25/0.51 = ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 0.25/0.51 = M3 ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_contains_eq
% 0.25/0.51 thf(fact_4313_mset__size1elem,axiom,
% 0.25/0.51 ! [A: $tType,P: multiset @ A,Q3: A] :
% 0.25/0.51 ( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
% 0.25/0.51 => ( ( member @ A @ Q3 @ ( set_mset @ A @ P ) )
% 0.25/0.51 => ( P
% 0.25/0.51 = ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % mset_size1elem
% 0.25/0.51 thf(fact_4314_subset__mset_Osum__mset__mono,axiom,
% 0.25/0.51 ! [A: $tType,B: $tType,K2: multiset @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
% 0.25/0.51 ( ! [I2: B] :
% 0.25/0.51 ( ( member @ B @ I2 @ ( set_mset @ B @ K2 ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 0.25/0.51 => ( subseteq_mset @ A @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ F @ K2 ) ) @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ G @ K2 ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_mset_mono
% 0.25/0.51 thf(fact_4315_subset__mset_Osum__mset__0__iff,axiom,
% 0.25/0.51 ! [A: $tType,M3: multiset @ ( multiset @ A )] :
% 0.25/0.51 ( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.51 = ( ! [X3: multiset @ A] :
% 0.25/0.51 ( ( member @ ( multiset @ A ) @ X3 @ ( set_mset @ ( multiset @ A ) @ M3 ) )
% 0.25/0.51 => ( X3
% 0.25/0.51 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % subset_mset.sum_mset_0_iff
% 0.25/0.51 thf(fact_4316_wmsI,axiom,
% 0.25/0.51 ! [A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.51 ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 0.25/0.51 | ( ( A4
% 0.25/0.51 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 0.25/0.51 & ( B6
% 0.25/0.51 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) )
% 0.25/0.51 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ B6 ) ) @ ms_weak ) ) ).
% 0.25/0.51
% 0.25/0.51 % wmsI
% 0.25/0.51 thf(fact_4317_smsI,axiom,
% 0.25/0.51 ! [A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.51 ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 0.25/0.51 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ B6 ) ) @ ms_strict ) ) ).
% 0.25/0.51
% 0.25/0.51 % smsI
% 0.25/0.51 thf(fact_4318_ms__reduction__pair,axiom,
% 0.25/0.51 fun_reduction_pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ms_strict @ ms_weak ) ).
% 0.25/0.51
% 0.25/0.51 % ms_reduction_pair
% 0.25/0.51 thf(fact_4319_ms__strictI,axiom,
% 0.25/0.51 ! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.51 ( ( pw_leq @ Z9 @ Z10 )
% 0.25/0.51 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 0.25/0.51 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B6 ) ) @ ms_strict ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % ms_strictI
% 0.25/0.51 thf(fact_4320_ms__weakI1,axiom,
% 0.25/0.51 ! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.51 ( ( pw_leq @ Z9 @ Z10 )
% 0.25/0.51 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 0.25/0.51 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B6 ) ) @ ms_weak ) ) ) ).
% 0.25/0.51
% 0.25/0.51 % ms_weakI1
% 0.25/0.51 thf(fact_4321_pw__leq__split,axiom,
% 0.25/0.51 ! [X10: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.51 ( ( pw_leq @ X10 @ Y9 )
% 0.25/0.51 => ? [A15: multiset @ ( product_prod @ nat @ nat ),B10: multiset @ ( product_prod @ nat @ nat ),Z11: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.51 ( ( X10
% 0.25/0.51 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A15 @ Z11 ) )
% 0.25/0.51 & ( Y9
% 0.25/0.51 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B10 @ Z11 ) )
% 0.25/0.51 & ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A15 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B10 ) ) @ fun_max_strict )
% 0.25/0.51 | ( ( B10
% 0.25/0.51 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 0.25/0.51 & ( A15
% 0.25/0.51 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pw_leq_split
% 0.25/0.52 thf(fact_4322_ms__weakI2,axiom,
% 0.25/0.52 ! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( pw_leq @ Z9 @ Z10 )
% 0.25/0.52 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% 0.25/0.52
% 0.25/0.52 % ms_weakI2
% 0.25/0.52 thf(fact_4323_pw__leq__lstep,axiom,
% 0.25/0.52 ! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 0.25/0.52 => ( pw_leq @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pw_leq_lstep
% 0.25/0.52 thf(fact_4324_pw__leq__step,axiom,
% 0.25/0.52 ! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X10: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 0.25/0.52 => ( ( pw_leq @ X10 @ Y9 )
% 0.25/0.52 => ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X10 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y9 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pw_leq_step
% 0.25/0.52 thf(fact_4325_pw__leq_Osimps,axiom,
% 0.25/0.52 ( pw_leq
% 0.25/0.52 = ( ^ [A12: multiset @ ( product_prod @ nat @ nat ),A23: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( ( A12
% 0.25/0.52 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 0.25/0.52 & ( A23
% 0.25/0.52 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
% 0.25/0.52 | ? [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat,X8: multiset @ ( product_prod @ nat @ nat ),Y10: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( A12
% 0.25/0.52 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X8 ) )
% 0.25/0.52 & ( A23
% 0.25/0.52 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y10 ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
% 0.25/0.52 & ( pw_leq @ X8 @ Y10 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pw_leq.simps
% 0.25/0.52 thf(fact_4326_pw__leq_Ocases,axiom,
% 0.25/0.52 ! [A1: multiset @ ( product_prod @ nat @ nat ),A22: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( pw_leq @ A1 @ A22 )
% 0.25/0.52 => ( ( ( A1
% 0.25/0.52 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 0.25/0.52 => ( A22
% 0.25/0.52 != ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
% 0.25/0.52 => ~ ! [X2: product_prod @ nat @ nat,Y4: product_prod @ nat @ nat,X15: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( A1
% 0.25/0.52 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X15 ) )
% 0.25/0.52 => ! [Y11: multiset @ ( product_prod @ nat @ nat )] :
% 0.25/0.52 ( ( A22
% 0.25/0.52 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y4 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y11 ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y4 ) @ fun_pair_leq )
% 0.25/0.52 => ~ ( pw_leq @ X15 @ Y11 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pw_leq.cases
% 0.25/0.52 thf(fact_4327_mult1E,axiom,
% 0.25/0.52 ! [A: $tType,N6: multiset @ A,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ R ) )
% 0.25/0.52 => ~ ! [A5: A,M0: multiset @ A] :
% 0.25/0.52 ( ( M3
% 0.25/0.52 = ( add_mset @ A @ A5 @ M0 ) )
% 0.25/0.52 => ! [K10: multiset @ A] :
% 0.25/0.52 ( ( N6
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ M0 @ K10 ) )
% 0.25/0.52 => ~ ! [B8: A] :
% 0.25/0.52 ( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A5 ) @ R ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult1E
% 0.25/0.52 thf(fact_4328_mult1I,axiom,
% 0.25/0.52 ! [A: $tType,M3: multiset @ A,A3: A,M02: multiset @ A,N6: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( M3
% 0.25/0.52 = ( add_mset @ A @ A3 @ M02 ) )
% 0.25/0.52 => ( ( N6
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ M02 @ K2 ) )
% 0.25/0.52 => ( ! [B3: A] :
% 0.25/0.52 ( ( member @ A @ B3 @ ( set_mset @ A @ K2 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ R ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult1I
% 0.25/0.52 thf(fact_4329_less__add,axiom,
% 0.25/0.52 ! [A: $tType,N6: multiset @ A,A3: A,M02: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ ( add_mset @ A @ A3 @ M02 ) ) @ ( mult1 @ A @ R ) )
% 0.25/0.52 => ( ? [M9: multiset @ A] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M9 @ M02 ) @ ( mult1 @ A @ R ) )
% 0.25/0.52 & ( N6
% 0.25/0.52 = ( add_mset @ A @ A3 @ M9 ) ) )
% 0.25/0.52 | ? [K10: multiset @ A] :
% 0.25/0.52 ( ! [B8: A] :
% 0.25/0.52 ( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A3 ) @ R ) )
% 0.25/0.52 & ( N6
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ M02 @ K10 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_add
% 0.25/0.52 thf(fact_4330_one__step__implies__mult,axiom,
% 0.25/0.52 ! [A: $tType,J4: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
% 0.25/0.52 ( ( J4
% 0.25/0.52 != ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ! [X2: A] :
% 0.25/0.52 ( ( member @ A @ X2 @ ( set_mset @ A @ K2 ) )
% 0.25/0.52 => ? [Xa2: A] :
% 0.25/0.52 ( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R ) ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) ) @ ( mult @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % one_step_implies_mult
% 0.25/0.52 thf(fact_4331_not__less__empty,axiom,
% 0.25/0.52 ! [A: $tType,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_less_empty
% 0.25/0.52 thf(fact_4332_mult1__union,axiom,
% 0.25/0.52 ! [A: $tType,B6: multiset @ A,D4: multiset @ A,R: set @ ( product_prod @ A @ A ),C5: multiset @ A] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B6 @ D4 ) @ ( mult1 @ A @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) @ ( mult1 @ A @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult1_union
% 0.25/0.52 thf(fact_4333_mult1__lessE,axiom,
% 0.25/0.52 ! [A: $tType,N6: multiset @ A,M3: multiset @ A,R: A > A > $o] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) )
% 0.25/0.52 => ( ( asymp @ A @ R )
% 0.25/0.52 => ~ ! [A5: A,M0: multiset @ A] :
% 0.25/0.52 ( ( M3
% 0.25/0.52 = ( add_mset @ A @ A5 @ M0 ) )
% 0.25/0.52 => ! [K10: multiset @ A] :
% 0.25/0.52 ( ( N6
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ M0 @ K10 ) )
% 0.25/0.52 => ( ~ ( member @ A @ A5 @ ( set_mset @ A @ K10 ) )
% 0.25/0.52 => ~ ! [B8: A] :
% 0.25/0.52 ( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
% 0.25/0.52 => ( R @ B8 @ A5 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult1_lessE
% 0.25/0.52 thf(fact_4334_in__mset__fold__plus__iff,axiom,
% 0.25/0.52 ! [A: $tType,X: A,M3: multiset @ A,NN: multiset @ ( multiset @ A )] :
% 0.25/0.52 ( ( member @ A @ X @ ( set_mset @ A @ ( fold_mset @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ M3 @ NN ) ) )
% 0.25/0.52 = ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
% 0.25/0.52 | ? [N9: multiset @ A] :
% 0.25/0.52 ( ( member @ ( multiset @ A ) @ N9 @ ( set_mset @ ( multiset @ A ) @ NN ) )
% 0.25/0.52 & ( member @ A @ X @ ( set_mset @ A @ N9 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % in_mset_fold_plus_iff
% 0.25/0.52 thf(fact_4335_mult__implies__one__step,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.52 ( ( trans @ A @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ N6 ) @ ( mult @ A @ R ) )
% 0.25/0.52 => ? [I8: multiset @ A,J5: multiset @ A] :
% 0.25/0.52 ( ( N6
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
% 0.25/0.52 & ? [K10: multiset @ A] :
% 0.25/0.52 ( ( M3
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ I8 @ K10 ) )
% 0.25/0.52 & ( J5
% 0.25/0.52 != ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 & ! [X7: A] :
% 0.25/0.52 ( ( member @ A @ X7 @ ( set_mset @ A @ K10 ) )
% 0.25/0.52 => ? [Xa3: A] :
% 0.25/0.52 ( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult_implies_one_step
% 0.25/0.52 thf(fact_4336_mult1__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( mult1 @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
% 0.25/0.52 @ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
% 0.25/0.52 @ ^ [N9: multiset @ A,M8: multiset @ A] :
% 0.25/0.52 ? [A7: A,M03: multiset @ A,K3: multiset @ A] :
% 0.25/0.52 ( ( M8
% 0.25/0.52 = ( add_mset @ A @ A7 @ M03 ) )
% 0.25/0.52 & ( N9
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ M03 @ K3 ) )
% 0.25/0.52 & ! [B5: A] :
% 0.25/0.52 ( ( member @ A @ B5 @ ( set_mset @ A @ K3 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult1_def
% 0.25/0.52 thf(fact_4337_trans__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( trans @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ! [X3: A,Y3: A,Z5: A] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z5 ) @ R3 )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z5 ) @ R3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trans_def
% 0.25/0.52 thf(fact_4338_transI,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ! [X2: A,Y4: A,Z3: A] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) )
% 0.25/0.52 => ( trans @ A @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % transI
% 0.25/0.52 thf(fact_4339_transE,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 0.25/0.52 ( ( trans @ A @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % transE
% 0.25/0.52 thf(fact_4340_transD,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 0.25/0.52 ( ( trans @ A @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % transD
% 0.25/0.52 thf(fact_4341_lexord__trans,axiom,
% 0.25/0.52 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A ),Z2: list @ A] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R ) )
% 0.25/0.52 => ( ( trans @ A @ R )
% 0.25/0.52 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % lexord_trans
% 0.25/0.52 thf(fact_4342_lenlex__trans,axiom,
% 0.25/0.52 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A ),Z2: list @ A] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R ) )
% 0.25/0.52 => ( ( trans @ A @ R )
% 0.25/0.52 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % lenlex_trans
% 0.25/0.52 thf(fact_4343_union__fold__mset__add__mset,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( plus_plus @ ( multiset @ A ) )
% 0.25/0.52 = ( fold_mset @ A @ ( multiset @ A ) @ ( add_mset @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % union_fold_mset_add_mset
% 0.25/0.52 thf(fact_4344_trans__singleton,axiom,
% 0.25/0.52 ! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trans_singleton
% 0.25/0.52 thf(fact_4345_trans__rtrancl__eq__reflcl,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( trans @ A @ A4 )
% 0.25/0.52 => ( ( transitive_rtrancl @ A @ A4 )
% 0.25/0.52 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trans_rtrancl_eq_reflcl
% 0.25/0.52 thf(fact_4346_comp__fun__commute_Ofold__mset__union,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B > B,S: B,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.52 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 0.25/0.52 => ( ( fold_mset @ A @ B @ F @ S @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.52 = ( fold_mset @ A @ B @ F @ ( fold_mset @ A @ B @ F @ S @ M3 ) @ N6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % comp_fun_commute.fold_mset_union
% 0.25/0.52 thf(fact_4347_trans__join,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( trans @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ! [X3: product_prod @ A @ A] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ X3 @ R3 )
% 0.25/0.52 => ( product_case_prod @ A @ A @ $o
% 0.25/0.52 @ ^ [Y3: A,Y13: A] :
% 0.25/0.52 ! [Z5: product_prod @ A @ A] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ Z5 @ R3 )
% 0.25/0.52 => ( product_case_prod @ A @ A @ $o
% 0.25/0.52 @ ^ [Y23: A,Aa2: A] :
% 0.25/0.52 ( ( Y13 = Y23 )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa2 ) @ R3 ) )
% 0.25/0.52 @ Z5 ) )
% 0.25/0.52 @ X3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trans_join
% 0.25/0.52 thf(fact_4348_Image__absorb__rtrancl,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ A,C5: set @ A] :
% 0.25/0.52 ( ( trans @ A @ A4 )
% 0.25/0.52 => ( ( refl_on @ A @ B6 @ A4 )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ B6 )
% 0.25/0.52 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A4 ) @ C5 )
% 0.25/0.52 = ( image @ A @ A @ A4 @ C5 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Image_absorb_rtrancl
% 0.25/0.52 thf(fact_4349_wf__finite__segments,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( irrefl @ A @ R )
% 0.25/0.52 => ( ( trans @ A @ R )
% 0.25/0.52 => ( ! [X2: A] :
% 0.25/0.52 ( finite_finite2 @ A
% 0.25/0.52 @ ( collect @ A
% 0.25/0.52 @ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) ) )
% 0.25/0.52 => ( wf @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % wf_finite_segments
% 0.25/0.52 thf(fact_4350_mult__cancel__add__mset,axiom,
% 0.25/0.52 ! [A: $tType,S: set @ ( product_prod @ A @ A ),Uu2: A,X10: multiset @ A,Y9: multiset @ A] :
% 0.25/0.52 ( ( trans @ A @ S )
% 0.25/0.52 => ( ( irrefl @ A @ S )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu2 @ X10 ) @ ( add_mset @ A @ Uu2 @ Y9 ) ) @ ( mult @ A @ S ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult_cancel_add_mset
% 0.25/0.52 thf(fact_4351_mult__cancel,axiom,
% 0.25/0.52 ! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Z9: multiset @ A,Y9: multiset @ A] :
% 0.25/0.52 ( ( trans @ A @ S )
% 0.25/0.52 => ( ( irrefl @ A @ S )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X10 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y9 @ Z9 ) ) @ ( mult @ A @ S ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult_cancel
% 0.25/0.52 thf(fact_4352_mult__cancel__max,axiom,
% 0.25/0.52 ! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Y9: multiset @ A] :
% 0.25/0.52 ( ( trans @ A @ S )
% 0.25/0.52 => ( ( irrefl @ A @ S )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ Y9 ) @ ( minus_minus @ ( multiset @ A ) @ Y9 @ X10 ) ) @ ( mult @ A @ S ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult_cancel_max
% 0.25/0.52 thf(fact_4353_multeqp__code__iff__reflcl__mult,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),P: A > A > $o,N6: multiset @ A,M3: multiset @ A] :
% 0.25/0.52 ( ( irrefl @ A @ R5 )
% 0.25/0.52 => ( ( trans @ A @ R5 )
% 0.25/0.52 => ( ! [X2: A,Y4: A] :
% 0.25/0.52 ( ( P @ X2 @ Y4 )
% 0.25/0.52 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R5 ) )
% 0.25/0.52 => ( ( multeqp_code @ A @ P @ N6 @ M3 )
% 0.25/0.52 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R5 ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % multeqp_code_iff_reflcl_mult
% 0.25/0.52 thf(fact_4354_multp__code__iff__mult,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),P: A > A > $o,N6: multiset @ A,M3: multiset @ A] :
% 0.25/0.52 ( ( irrefl @ A @ R5 )
% 0.25/0.52 => ( ( trans @ A @ R5 )
% 0.25/0.52 => ( ! [X2: A,Y4: A] :
% 0.25/0.52 ( ( P @ X2 @ Y4 )
% 0.25/0.52 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R5 ) )
% 0.25/0.52 => ( ( multp_code @ A @ P @ N6 @ M3 )
% 0.25/0.52 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult @ A @ R5 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % multp_code_iff_mult
% 0.25/0.52 thf(fact_4355_max__ext__eq,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( max_ext @ A )
% 0.25/0.52 = ( ^ [R6: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.25/0.52 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 0.25/0.52 @ ^ [X8: set @ A,Y10: set @ A] :
% 0.25/0.52 ( ( finite_finite2 @ A @ X8 )
% 0.25/0.52 & ( finite_finite2 @ A @ Y10 )
% 0.25/0.52 & ( Y10
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 & ! [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ X8 )
% 0.25/0.52 => ? [Y3: A] :
% 0.25/0.52 ( ( member @ A @ Y3 @ Y10 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % max_ext_eq
% 0.25/0.52 thf(fact_4356_init__seg__of__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( init_seg_of @ A )
% 0.25/0.52 = ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.52 @ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
% 0.25/0.52 @ ^ [R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 )
% 0.25/0.52 & ! [A7: A,B5: A,C3: A] :
% 0.25/0.52 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ S4 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R3 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % init_seg_of_def
% 0.25/0.52 thf(fact_4357_refl__on__init__seg__of,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( init_seg_of @ A ) ) ).
% 0.25/0.52
% 0.25/0.52 % refl_on_init_seg_of
% 0.25/0.52 thf(fact_4358_antisym__init__seg__of,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ R ) @ ( init_seg_of @ A ) )
% 0.25/0.52 => ( R = S ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % antisym_init_seg_of
% 0.25/0.52 thf(fact_4359_trans__init__seg__of,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ T2 ) @ ( init_seg_of @ A ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trans_init_seg_of
% 0.25/0.52 thf(fact_4360_Image__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( image @ A @ B )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ B ),S4: set @ A] :
% 0.25/0.52 ( collect @ B
% 0.25/0.52 @ ^ [Y3: B] :
% 0.25/0.52 ? [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ S4 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Image_def
% 0.25/0.52 thf(fact_4361_initial__segment__of__Diff,axiom,
% 0.25/0.52 ! [A: $tType,P5: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ Q3 ) @ ( init_seg_of @ A ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ S ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q3 @ S ) ) @ ( init_seg_of @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % initial_segment_of_Diff
% 0.25/0.52 thf(fact_4362_min__ext__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( min_ext @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 0.25/0.52 @ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
% 0.25/0.52 ? [X8: set @ A,Y10: set @ A] :
% 0.25/0.52 ( ( Uu
% 0.25/0.52 = ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y10 ) )
% 0.25/0.52 & ( X8
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 & ! [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ Y10 )
% 0.25/0.52 => ? [Y3: A] :
% 0.25/0.52 ( ( member @ A @ Y3 @ X8 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % min_ext_def
% 0.25/0.52 thf(fact_4363_map__project__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( map_project @ A @ B )
% 0.25/0.52 = ( ^ [F2: A > ( option @ B ),A9: set @ A] :
% 0.25/0.52 ( collect @ B
% 0.25/0.52 @ ^ [B5: B] :
% 0.25/0.52 ? [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ A9 )
% 0.25/0.52 & ( ( F2 @ X3 )
% 0.25/0.52 = ( some @ B @ B5 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % map_project_def
% 0.25/0.52 thf(fact_4364_Chains__init__seg__of__Union,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R5 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
% 0.25/0.52 => ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R @ R5 )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R5 ) ) @ ( init_seg_of @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Chains_init_seg_of_Union
% 0.25/0.52 thf(fact_4365_mult__cancel__max0,axiom,
% 0.25/0.52 ! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Y9: multiset @ A] :
% 0.25/0.52 ( ( trans @ A @ S )
% 0.25/0.52 => ( ( irrefl @ A @ S )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ ( inter_mset @ A @ X10 @ Y9 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y9 @ ( inter_mset @ A @ X10 @ Y9 ) ) ) @ ( mult @ A @ S ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mult_cancel_max0
% 0.25/0.52 thf(fact_4366_sum__mset_Oremove,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [X: A,A4: multiset @ A] :
% 0.25/0.52 ( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
% 0.25/0.52 => ( ( comm_m7189776963980413722m_mset @ A @ A4 )
% 0.25/0.52 = ( plus_plus @ A @ X @ ( comm_m7189776963980413722m_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.remove
% 0.25/0.52 thf(fact_4367_sum__mset_Oadd__mset,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [X: A,N6: multiset @ A] :
% 0.25/0.52 ( ( comm_m7189776963980413722m_mset @ A @ ( add_mset @ A @ X @ N6 ) )
% 0.25/0.52 = ( plus_plus @ A @ X @ ( comm_m7189776963980413722m_mset @ A @ N6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.add_mset
% 0.25/0.52 thf(fact_4368_sum__mset_Ounion,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.52 ( ( comm_m7189776963980413722m_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.52 = ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ M3 ) @ ( comm_m7189776963980413722m_mset @ A @ N6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.union
% 0.25/0.52 thf(fact_4369_sum__mset_Oinsert,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [G: B > A,X: B,A4: multiset @ B] :
% 0.25/0.52 ( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X @ A4 ) ) )
% 0.25/0.52 = ( plus_plus @ A @ ( G @ X ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.insert
% 0.25/0.52 thf(fact_4370_inter__union__distrib__right,axiom,
% 0.25/0.52 ! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( plus_plus @ ( multiset @ A ) @ C5 @ ( inter_mset @ A @ A4 @ B6 ) )
% 0.25/0.52 = ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % inter_union_distrib_right
% 0.25/0.52 thf(fact_4371_inter__union__distrib__left,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.52 ( ( plus_plus @ ( multiset @ A ) @ ( inter_mset @ A @ A4 @ B6 ) @ C5 )
% 0.25/0.52 = ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % inter_union_distrib_left
% 0.25/0.52 thf(fact_4372_inter__subset__eq__union,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] : ( subseteq_mset @ A @ ( inter_mset @ A @ A4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % inter_subset_eq_union
% 0.25/0.52 thf(fact_4373_inter__mset__empty__distrib__left,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.52 ( ( ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 = ( ( ( inter_mset @ A @ A4 @ C5 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 & ( ( inter_mset @ A @ B6 @ C5 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % inter_mset_empty_distrib_left
% 0.25/0.52 thf(fact_4374_inter__mset__empty__distrib__right,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.52 ( ( ( inter_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 = ( ( ( inter_mset @ A @ A4 @ B6 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 & ( ( inter_mset @ A @ A4 @ C5 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % inter_mset_empty_distrib_right
% 0.25/0.52 thf(fact_4375_sum__mset_Odistrib,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [G: B > A,H3: B > A,A4: multiset @ B] :
% 0.25/0.52 ( ( comm_m7189776963980413722m_mset @ A
% 0.25/0.52 @ ( image_mset @ B @ A
% 0.25/0.52 @ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H3 @ X3 ) )
% 0.25/0.52 @ A4 ) )
% 0.25/0.52 = ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ H3 @ A4 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.distrib
% 0.25/0.52 thf(fact_4376_size__eq__sum__mset,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( size_size @ ( multiset @ A ) )
% 0.25/0.52 = ( ^ [M8: multiset @ A] :
% 0.25/0.52 ( comm_m7189776963980413722m_mset @ nat
% 0.25/0.52 @ ( image_mset @ A @ nat
% 0.25/0.52 @ ^ [A7: A] : ( one_one @ nat )
% 0.25/0.52 @ M8 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % size_eq_sum_mset
% 0.25/0.52 thf(fact_4377_multiset__union__diff__commute,axiom,
% 0.25/0.52 ! [A: $tType,B6: multiset @ A,C5: multiset @ A,A4: multiset @ A] :
% 0.25/0.52 ( ( ( inter_mset @ A @ B6 @ C5 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % multiset_union_diff_commute
% 0.25/0.52 thf(fact_4378_sum__mset_Oeq__fold,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ( ( comm_m7189776963980413722m_mset @ A )
% 0.25/0.52 = ( fold_mset @ A @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.eq_fold
% 0.25/0.52 thf(fact_4379_Chains__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( chains @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( collect @ ( set @ A )
% 0.25/0.52 @ ^ [C7: set @ A] :
% 0.25/0.52 ! [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ C7 )
% 0.25/0.52 => ! [Y3: A] :
% 0.25/0.52 ( ( member @ A @ Y3 @ C7 )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 0.25/0.52 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Chains_def
% 0.25/0.52 thf(fact_4380_Chains__subset_H,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 0.25/0.52 => ( ord_less_eq @ ( set @ ( set @ A ) )
% 0.25/0.52 @ ( collect @ ( set @ A )
% 0.25/0.52 @ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.52 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) )
% 0.25/0.52 @ ( chains @ A @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Chains_subset'
% 0.25/0.52 thf(fact_4381_Chains__subset,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R )
% 0.25/0.52 @ ( collect @ ( set @ A )
% 0.25/0.52 @ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.52 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Chains_subset
% 0.25/0.52 thf(fact_4382_Chains__alt__def,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 0.25/0.52 => ( ( chains @ A @ R )
% 0.25/0.52 = ( collect @ ( set @ A )
% 0.25/0.52 @ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
% 0.25/0.52 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Chains_alt_def
% 0.25/0.52 thf(fact_4383_iter_H_Ocases,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( quickcheck_random @ A )
% 0.25/0.52 => ! [X: product_prod @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) )] :
% 0.25/0.52 ~ ! [T4: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
% 0.25/0.52 ( X
% 0.25/0.52 != ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T4 @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Nrandom @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Sz @ Seed ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % iter'.cases
% 0.25/0.52 thf(fact_4384_multp__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( multp @ A )
% 0.25/0.52 = ( ^ [R3: A > A > $o,M8: multiset @ A,N9: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M8 @ N9 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % multp_def
% 0.25/0.52 thf(fact_4385_AboveS__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( order_AboveS @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 0.25/0.52 ( collect @ A
% 0.25/0.52 @ ^ [B5: A] :
% 0.25/0.52 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 0.25/0.52 & ! [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ A9 )
% 0.25/0.52 => ( ( B5 != X3 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B5 ) @ R3 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % AboveS_def
% 0.25/0.52 thf(fact_4386_lexordp__conv__lexord,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( linorder @ A )
% 0.25/0.52 => ( ( ord_lexordp @ A )
% 0.25/0.52 = ( ^ [Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % lexordp_conv_lexord
% 0.25/0.52 thf(fact_4387_select__weight__cons__zero,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Xs: list @ ( product_prod @ code_natural @ A )] :
% 0.25/0.52 ( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X ) @ Xs ) )
% 0.25/0.52 = ( select_weight @ A @ Xs ) ) ).
% 0.25/0.52
% 0.25/0.52 % select_weight_cons_zero
% 0.25/0.52 thf(fact_4388_one__step__implies__multp,axiom,
% 0.25/0.52 ! [A: $tType,J4: multiset @ A,K2: multiset @ A,R: A > A > $o,I4: multiset @ A] :
% 0.25/0.52 ( ( J4
% 0.25/0.52 != ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ! [X2: A] :
% 0.25/0.52 ( ( member @ A @ X2 @ ( set_mset @ A @ K2 ) )
% 0.25/0.52 => ? [Xa2: A] :
% 0.25/0.52 ( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
% 0.25/0.52 & ( R @ X2 @ Xa2 ) ) )
% 0.25/0.52 => ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % one_step_implies_multp
% 0.25/0.52 thf(fact_4389_select__weight__select,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) )
% 0.25/0.52 = ( select @ A @ Xs ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % select_weight_select
% 0.25/0.52 thf(fact_4390_wo__rel_Osuc__greater,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
% 0.25/0.52 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( ( ( order_AboveS @ A @ R @ B6 )
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 => ( ( member @ A @ B2 @ B6 )
% 0.25/0.52 => ( ( ( bNF_Wellorder_wo_suc @ A @ R @ B6 )
% 0.25/0.52 != B2 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) ) @ R ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % wo_rel.suc_greater
% 0.25/0.52 thf(fact_4391_wo__rel_Oequals__suc__AboveS,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
% 0.25/0.52 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B6 ) )
% 0.25/0.52 => ( ! [A16: A] :
% 0.25/0.52 ( ( member @ A @ A16 @ ( order_AboveS @ A @ R @ B6 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A16 ) @ R ) )
% 0.25/0.52 => ( A3
% 0.25/0.52 = ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % wo_rel.equals_suc_AboveS
% 0.25/0.52 thf(fact_4392_wo__rel_Osuc__least__AboveS,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B6: set @ A] :
% 0.25/0.52 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.52 => ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B6 ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) @ A3 ) @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % wo_rel.suc_least_AboveS
% 0.25/0.52 thf(fact_4393_wo__rel_Osuc__ofilter__in,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
% 0.25/0.52 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.52 => ( ( order_ofilter @ A @ R @ A4 )
% 0.25/0.52 => ( ( ( order_AboveS @ A @ R @ A4 )
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ A4 ) ) @ R )
% 0.25/0.52 => ( ( B2
% 0.25/0.52 != ( bNF_Wellorder_wo_suc @ A @ R @ A4 ) )
% 0.25/0.52 => ( member @ A @ B2 @ A4 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % wo_rel.suc_ofilter_in
% 0.25/0.52 thf(fact_4394_slice__head,axiom,
% 0.25/0.52 ! [A: $tType,From: nat,To: nat,Xs: list @ A] :
% 0.25/0.52 ( ( ord_less @ nat @ From @ To )
% 0.25/0.52 => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.52 => ( ( hd @ A @ ( slice @ A @ From @ To @ Xs ) )
% 0.25/0.52 = ( nth @ A @ Xs @ From ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % slice_head
% 0.25/0.52 thf(fact_4395_gen__length__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( gen_length @ A )
% 0.25/0.52 = ( ^ [N5: nat,Xs3: list @ A] : ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % gen_length_def
% 0.25/0.52 thf(fact_4396_subset__mset_Osum__pos2,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,I4: set @ B,I: B,F: B > ( multiset @ A )] :
% 0.25/0.52 ( ( finite_finite2 @ B @ I4 )
% 0.25/0.52 => ( ( member @ B @ I @ I4 )
% 0.25/0.52 => ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I ) )
% 0.25/0.52 => ( ! [I2: B] :
% 0.25/0.52 ( ( member @ B @ I2 @ I4 )
% 0.25/0.52 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 0.25/0.52 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I4 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.sum_pos2
% 0.25/0.52 thf(fact_4397_subset__mset_Oadd__less__cancel__right,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 0.25/0.52 = ( subset_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_cancel_right
% 0.25/0.52 thf(fact_4398_subset__mset_Oadd__less__cancel__left,axiom,
% 0.25/0.52 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 0.25/0.52 = ( subset_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_cancel_left
% 0.25/0.52 thf(fact_4399_subset__mset_Oadd__less__same__cancel1,axiom,
% 0.25/0.52 ! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) @ B2 )
% 0.25/0.52 = ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_same_cancel1
% 0.25/0.52 thf(fact_4400_subset__mset_Oadd__less__same__cancel2,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ B2 )
% 0.25/0.52 = ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_same_cancel2
% 0.25/0.52 thf(fact_4401_subset__mset_Oless__add__same__cancel1,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
% 0.25/0.52 = ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.less_add_same_cancel1
% 0.25/0.52 thf(fact_4402_subset__mset_Oless__add__same__cancel2,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) )
% 0.25/0.52 = ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.less_add_same_cancel2
% 0.25/0.52 thf(fact_4403_subset__mset_Oadd__less__imp__less__right,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 0.25/0.52 => ( subset_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_imp_less_right
% 0.25/0.52 thf(fact_4404_subset__mset_Oadd__less__imp__less__left,axiom,
% 0.25/0.52 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 0.25/0.52 => ( subset_mset @ A @ A3 @ B2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_imp_less_left
% 0.25/0.52 thf(fact_4405_subset__mset_Oadd__strict__right__mono,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ B2 )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_strict_right_mono
% 0.25/0.52 thf(fact_4406_subset__mset_Oadd__strict__left__mono,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ B2 )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_strict_left_mono
% 0.25/0.52 thf(fact_4407_subset__mset_Oadd__strict__mono,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ B2 )
% 0.25/0.52 => ( ( subset_mset @ A @ C2 @ D3 )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_strict_mono
% 0.25/0.52 thf(fact_4408_hd__zip,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( Ys2
% 0.25/0.52 != ( nil @ B ) )
% 0.25/0.52 => ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 0.25/0.52 = ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys2 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % hd_zip
% 0.25/0.52 thf(fact_4409_subset__mset_Oadd__le__less__mono,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 0.25/0.52 ( ( subseteq_mset @ A @ A3 @ B2 )
% 0.25/0.52 => ( ( subset_mset @ A @ C2 @ D3 )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_le_less_mono
% 0.25/0.52 thf(fact_4410_subset__mset_Oadd__less__le__mono,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ B2 )
% 0.25/0.52 => ( ( subseteq_mset @ A @ C2 @ D3 )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_less_le_mono
% 0.25/0.52 thf(fact_4411_subset__mset_OlessE,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ B2 )
% 0.25/0.52 => ~ ! [C4: multiset @ A] :
% 0.25/0.52 ( ( B2
% 0.25/0.52 = ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) )
% 0.25/0.52 => ( C4
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.lessE
% 0.25/0.52 thf(fact_4412_subset__mset_Oadd__neg__neg,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_neg_neg
% 0.25/0.52 thf(fact_4413_subset__mset_Oadd__pos__pos,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.52 => ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 0.25/0.52 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_pos_pos
% 0.25/0.52 thf(fact_4414_subset__mset_Opos__add__strict,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.52 => ( ( subset_mset @ A @ B2 @ C2 )
% 0.25/0.52 => ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.pos_add_strict
% 0.25/0.52 thf(fact_4415_subset__implies__mult,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( subset_mset @ A @ A4 @ B6 )
% 0.25/0.52 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A4 @ B6 ) @ ( mult @ A @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_implies_mult
% 0.25/0.52 thf(fact_4416_subset__mset_Oadd__strict__increasing2,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.52 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.52 => ( ( subset_mset @ A @ B2 @ C2 )
% 0.25/0.52 => ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_strict_increasing2
% 0.25/0.52 thf(fact_4417_subset__mset_Oadd__strict__increasing,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.52 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 0.25/0.52 => ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_strict_increasing
% 0.25/0.52 thf(fact_4418_subset__mset_Oadd__pos__nonneg,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.52 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 0.25/0.52 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_pos_nonneg
% 0.25/0.52 thf(fact_4419_subset__mset_Oadd__nonpos__neg,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_nonpos_neg
% 0.25/0.52 thf(fact_4420_subset__mset_Oadd__nonneg__pos,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 0.25/0.52 => ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 0.25/0.52 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_nonneg_pos
% 0.25/0.52 thf(fact_4421_subset__mset_Oadd__neg__nonpos,axiom,
% 0.25/0.52 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 0.25/0.52 ( ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.add_neg_nonpos
% 0.25/0.52 thf(fact_4422_sorted__hd__min,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( linorder @ A )
% 0.25/0.52 => ! [Xs: list @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 0.25/0.52 => ! [X7: A] :
% 0.25/0.52 ( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
% 0.25/0.52 => ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X7 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sorted_hd_min
% 0.25/0.52 thf(fact_4423_sorted__hd__last,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( linorder @ A )
% 0.25/0.52 => ! [L: list @ A] :
% 0.25/0.52 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 0.25/0.52 => ( ( L
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ord_less_eq @ A @ ( hd @ A @ L ) @ ( last @ A @ L ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sorted_hd_last
% 0.25/0.52 thf(fact_4424_hd__last__singletonI,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( ( hd @ A @ Xs )
% 0.25/0.52 = ( last @ A @ Xs ) )
% 0.25/0.52 => ( ( distinct @ A @ Xs )
% 0.25/0.52 => ( Xs
% 0.25/0.52 = ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % hd_last_singletonI
% 0.25/0.52 thf(fact_4425_hd__butlast,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( ord_less @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.52 => ( ( hd @ A @ ( butlast @ A @ Xs ) )
% 0.25/0.52 = ( hd @ A @ Xs ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % hd_butlast
% 0.25/0.52 thf(fact_4426_subset__mset_Osum__pos,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,I4: set @ B,F: B > ( multiset @ A )] :
% 0.25/0.52 ( ( finite_finite2 @ B @ I4 )
% 0.25/0.52 => ( ( I4
% 0.25/0.52 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.52 => ( ! [I2: B] :
% 0.25/0.52 ( ( member @ B @ I2 @ I4 )
% 0.25/0.52 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 0.25/0.52 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I4 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.sum_pos
% 0.25/0.52 thf(fact_4427_subset__mset_Osum__strict__mono,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
% 0.25/0.52 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.52 => ( ( A4
% 0.25/0.52 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.52 => ( ! [X2: B] :
% 0.25/0.52 ( ( member @ B @ X2 @ A4 )
% 0.25/0.52 => ( subset_mset @ A @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.25/0.52 => ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ A4 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.sum_strict_mono
% 0.25/0.52 thf(fact_4428_ofilter__ordLess,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_ofilter @ A @ R @ A4 )
% 0.25/0.52 => ( ( ord_less @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.52 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.52 @ ^ [Uu: A] : A4 ) )
% 0.25/0.52 @ R )
% 0.25/0.52 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ofilter_ordLess
% 0.25/0.52 thf(fact_4429_ofilter__subset__ordLess,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_ofilter @ A @ R @ A4 )
% 0.25/0.52 => ( ( order_ofilter @ A @ R @ B6 )
% 0.25/0.52 => ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.52 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.52 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.52 @ ^ [Uu: A] : A4 ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ B6
% 0.25/0.52 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.52 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ofilter_subset_ordLess
% 0.25/0.52 thf(fact_4430_ofilter__subset__ordLeq,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_ofilter @ A @ R @ A4 )
% 0.25/0.52 => ( ( order_ofilter @ A @ R @ B6 )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.52 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.52 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.52 @ ^ [Uu: A] : A4 ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ B6
% 0.25/0.52 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.52 @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ofilter_subset_ordLeq
% 0.25/0.52 thf(fact_4431_Nitpick_Osize__list__simp_I1_J,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( size_list @ A )
% 0.25/0.52 = ( ^ [F2: A > nat,Xs3: list @ A] :
% 0.25/0.52 ( if @ nat
% 0.25/0.52 @ ( Xs3
% 0.25/0.52 = ( nil @ A ) )
% 0.25/0.52 @ ( zero_zero @ nat )
% 0.25/0.52 @ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Nitpick.size_list_simp(1)
% 0.25/0.52 thf(fact_4432_length__tl,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
% 0.25/0.52 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % length_tl
% 0.25/0.52 thf(fact_4433_in__hd__or__tl__conv,axiom,
% 0.25/0.52 ! [A: $tType,L: list @ A,X: A] :
% 0.25/0.52 ( ( L
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( ( X
% 0.25/0.52 = ( hd @ A @ L ) )
% 0.25/0.52 | ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
% 0.25/0.52 = ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % in_hd_or_tl_conv
% 0.25/0.52 thf(fact_4434_tl__replicate,axiom,
% 0.25/0.52 ! [A: $tType,N: nat,X: A] :
% 0.25/0.52 ( ( tl @ A @ ( replicate @ A @ N @ X ) )
% 0.25/0.52 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % tl_replicate
% 0.25/0.52 thf(fact_4435_rev__split__conv,axiom,
% 0.25/0.52 ! [A: $tType,L: list @ A] :
% 0.25/0.52 ( ( L
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( append @ A @ ( rev @ A @ ( tl @ A @ L ) ) @ ( cons @ A @ ( hd @ A @ L ) @ ( nil @ A ) ) )
% 0.25/0.52 = ( rev @ A @ L ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % rev_split_conv
% 0.25/0.52 thf(fact_4436_ordLess__irreflexive,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_irreflexive
% 0.25/0.52 thf(fact_4437_ordLess__transitive,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_transitive
% 0.25/0.52 thf(fact_4438_ordLeq__transitive,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_transitive
% 0.25/0.52 thf(fact_4439_not__ordLess__ordLeq,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 => ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_ordLess_ordLeq
% 0.25/0.52 thf(fact_4440_ordLess__imp__ordLeq,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_imp_ordLeq
% 0.25/0.52 thf(fact_4441_ordLeq__ordLess__trans,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_ordLess_trans
% 0.25/0.52 thf(fact_4442_ordLess__ordLeq__trans,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_ordLeq_trans
% 0.25/0.52 thf(fact_4443_not__ordLess__iff__ordLeq,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_ordLess_iff_ordLeq
% 0.25/0.52 thf(fact_4444_not__ordLeq__iff__ordLess,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_ordLeq_iff_ordLess
% 0.25/0.52 thf(fact_4445_ordLess__or__ordLeq,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 | ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_or_ordLeq
% 0.25/0.52 thf(fact_4446_in__set__tlD,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Xs: list @ A] :
% 0.25/0.52 ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
% 0.25/0.52 => ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % in_set_tlD
% 0.25/0.52 thf(fact_4447_tl__obtain__elem,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( ( tl @ A @ Xs )
% 0.25/0.52 = ( nil @ A ) )
% 0.25/0.52 => ~ ! [E3: A] :
% 0.25/0.52 ( Xs
% 0.25/0.52 != ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % tl_obtain_elem
% 0.25/0.52 thf(fact_4448_not__hd__in__tl,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Xs: list @ A] :
% 0.25/0.52 ( ( X
% 0.25/0.52 != ( hd @ A @ Xs ) )
% 0.25/0.52 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.52 => ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_hd_in_tl
% 0.25/0.52 thf(fact_4449_tl__last,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( ( tl @ A @ Xs )
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( last @ A @ Xs )
% 0.25/0.52 = ( last @ A @ ( tl @ A @ Xs ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % tl_last
% 0.25/0.52 thf(fact_4450_rev__butlast__is__tl__rev,axiom,
% 0.25/0.52 ! [A: $tType,L: list @ A] :
% 0.25/0.52 ( ( rev @ A @ ( butlast @ A @ L ) )
% 0.25/0.52 = ( tl @ A @ ( rev @ A @ L ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % rev_butlast_is_tl_rev
% 0.25/0.52 thf(fact_4451_tl__subset,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A,A4: set @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
% 0.25/0.52 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A4 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % tl_subset
% 0.25/0.52 thf(fact_4452_Misc_Onth__tl,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A,N: nat] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
% 0.25/0.52 = ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Misc.nth_tl
% 0.25/0.52 thf(fact_4453_list__take__induct__tl2,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,P: B > A > $o] :
% 0.25/0.52 ( ( ( size_size @ ( list @ A ) @ Xs )
% 0.25/0.52 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 0.25/0.52 => ( ! [N3: nat] :
% 0.25/0.52 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.52 => ( P @ ( nth @ B @ Ys2 @ N3 ) @ ( nth @ A @ Xs @ N3 ) ) )
% 0.25/0.52 => ! [N10: nat] :
% 0.25/0.52 ( ( ord_less @ nat @ N10 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
% 0.25/0.52 => ( P @ ( nth @ B @ ( tl @ B @ Ys2 ) @ N10 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N10 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % list_take_induct_tl2
% 0.25/0.52 thf(fact_4454_exists__minim__Well__order,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
% 0.25/0.52 ( ( R5
% 0.25/0.52 != ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
% 0.25/0.52 => ( ! [X2: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 0.25/0.52 => ( order_well_order_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
% 0.25/0.52 => ? [X2: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 0.25/0.52 & ! [Xa2: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % exists_minim_Well_order
% 0.25/0.52 thf(fact_4455_ordLeq__total,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 | ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_total
% 0.25/0.52 thf(fact_4456_ordLeq__reflexive,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_reflexive
% 0.25/0.52 thf(fact_4457_ordLeq__Well__order__simp,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 => ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 & ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_Well_order_simp
% 0.25/0.52 thf(fact_4458_tl__take,axiom,
% 0.25/0.52 ! [A: $tType,N: nat,Xs: list @ A] :
% 0.25/0.52 ( ( tl @ A @ ( take @ A @ N @ Xs ) )
% 0.25/0.52 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % tl_take
% 0.25/0.52 thf(fact_4459_distinct__hd__tl,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A,X: A] :
% 0.25/0.52 ( ( distinct @ A @ Xs )
% 0.25/0.52 => ( ( X
% 0.25/0.52 = ( hd @ A @ Xs ) )
% 0.25/0.52 => ~ ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % distinct_hd_tl
% 0.25/0.52 thf(fact_4460_butlast__rev__tl,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( butlast @ A @ ( rev @ A @ Xs ) )
% 0.25/0.52 = ( rev @ A @ ( tl @ A @ Xs ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % butlast_rev_tl
% 0.25/0.52 thf(fact_4461_finite__ordLess__infinite,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R4 ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % finite_ordLess_infinite
% 0.25/0.52 thf(fact_4462_ordLeq__iff__ordLess__Restr,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 = ( ! [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
% 0.25/0.52 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
% 0.25/0.52 @ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) )
% 0.25/0.52 @ R4 )
% 0.25/0.52 @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_iff_ordLess_Restr
% 0.25/0.52 thf(fact_4463_underS__Restr__ordLess,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( ( field2 @ A @ R )
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.52 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ A3 )
% 0.25/0.52 @ ^ [Uu: A] : ( order_underS @ A @ R @ A3 ) ) )
% 0.25/0.52 @ R )
% 0.25/0.52 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % underS_Restr_ordLess
% 0.25/0.52 thf(fact_4464_horner__sum__append,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_semiring_1 @ A )
% 0.25/0.52 => ! [F: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
% 0.25/0.52 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys2 ) )
% 0.25/0.52 = ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Ys2 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % horner_sum_append
% 0.25/0.52 thf(fact_4465_Code__Numeral_Osub__code_I4_J,axiom,
% 0.25/0.52 ! [N: num] :
% 0.25/0.52 ( ( code_sub @ one2 @ ( bit0 @ N ) )
% 0.25/0.52 = ( code_Neg @ ( bitM @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.sub_code(4)
% 0.25/0.52 thf(fact_4466_horner__sum__simps_I2_J,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_semiring_0 @ A )
% 0.25/0.52 => ! [F: B > A,A3: A,X: B,Xs: list @ B] :
% 0.25/0.52 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
% 0.25/0.52 = ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % horner_sum_simps(2)
% 0.25/0.52 thf(fact_4467_underS__E,axiom,
% 0.25/0.52 ! [A: $tType,I: A,R5: set @ ( product_prod @ A @ A ),J: A] :
% 0.25/0.52 ( ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) )
% 0.25/0.52 => ( ( I != J )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % underS_E
% 0.25/0.52 thf(fact_4468_underS__I,axiom,
% 0.25/0.52 ! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( I != J )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
% 0.25/0.52 => ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % underS_I
% 0.25/0.52 thf(fact_4469_underS__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( order_underS @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 0.25/0.52 ( collect @ A
% 0.25/0.52 @ ^ [B5: A] :
% 0.25/0.52 ( ( B5 != A7 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % underS_def
% 0.25/0.52 thf(fact_4470_Code__Numeral_Odup__code_I3_J,axiom,
% 0.25/0.52 ! [N: num] :
% 0.25/0.52 ( ( code_dup @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.dup_code(3)
% 0.25/0.52 thf(fact_4471_less__integer__code_I3_J,axiom,
% 0.25/0.52 ! [L: num] :
% 0.25/0.52 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(3)
% 0.25/0.52 thf(fact_4472_less__integer__code_I7_J,axiom,
% 0.25/0.52 ! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(7)
% 0.25/0.52 thf(fact_4473_plus__integer__code_I6_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( plus_plus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_Neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_integer_code(6)
% 0.25/0.52 thf(fact_4474_less__eq__integer__code_I7_J,axiom,
% 0.25/0.52 ! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(7)
% 0.25/0.52 thf(fact_4475_less__eq__integer__code_I3_J,axiom,
% 0.25/0.52 ! [L: num] :
% 0.25/0.52 ~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(3)
% 0.25/0.52 thf(fact_4476_less__integer__code_I9_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 0.25/0.52 = ( ord_less @ num @ L @ K ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(9)
% 0.25/0.52 thf(fact_4477_less__eq__integer__code_I9_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 0.25/0.52 = ( ord_less_eq @ num @ L @ K ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(9)
% 0.25/0.52 thf(fact_4478_minus__integer__code_I6_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( minus_minus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_sub @ N @ M2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_integer_code(6)
% 0.25/0.52 thf(fact_4479_divmod__abs__code_I2_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 0.25/0.52 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % divmod_abs_code(2)
% 0.25/0.52 thf(fact_4480_horner__sum__foldr,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_semiring_0 @ A )
% 0.25/0.52 => ( ( groups4207007520872428315er_sum @ B @ A )
% 0.25/0.52 = ( ^ [F2: B > A,A7: A,Xs3: list @ B] :
% 0.25/0.52 ( foldr @ B @ A
% 0.25/0.52 @ ^ [X3: B,B5: A] : ( plus_plus @ A @ ( F2 @ X3 ) @ ( times_times @ A @ A7 @ B5 ) )
% 0.25/0.52 @ Xs3
% 0.25/0.52 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % horner_sum_foldr
% 0.25/0.52 thf(fact_4481_underS__incl__iff,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.52 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) )
% 0.25/0.52 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % underS_incl_iff
% 0.25/0.52 thf(fact_4482_Code__Numeral_Osub__code_I5_J,axiom,
% 0.25/0.52 ! [N: num] :
% 0.25/0.52 ( ( code_sub @ one2 @ ( bit1 @ N ) )
% 0.25/0.52 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.sub_code(5)
% 0.25/0.52 thf(fact_4483_ordLess__iff__ordIso__Restr,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 0.25/0.52 = ( ? [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ ( field2 @ A @ R ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.52 @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4
% 0.25/0.52 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.52 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
% 0.25/0.52 @ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) ) )
% 0.25/0.52 @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_iff_ordIso_Restr
% 0.25/0.52 thf(fact_4484_sum__list__update,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( ordere1170586879665033532d_diff @ A )
% 0.25/0.52 => ! [K: nat,Xs: list @ A,X: A] :
% 0.25/0.52 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.52 => ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 0.25/0.52 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_update
% 0.25/0.52 thf(fact_4485_length__product__lists,axiom,
% 0.25/0.52 ! [B: $tType,Xss: list @ ( list @ B )] :
% 0.25/0.52 ( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
% 0.25/0.52 = ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % length_product_lists
% 0.25/0.52 thf(fact_4486_horner__sum__transfer,axiom,
% 0.25/0.52 ! [C: $tType,A: $tType,B: $tType,D: $tType] :
% 0.25/0.52 ( ( ( comm_semiring_0 @ B )
% 0.25/0.52 & ( comm_semiring_0 @ A ) )
% 0.25/0.52 => ! [A4: A > B > $o,B6: C > D > $o] :
% 0.25/0.52 ( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.25/0.52 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.25/0.52 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 0.25/0.52 => ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B6 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B6 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % horner_sum_transfer
% 0.25/0.52 thf(fact_4487_sum__list_OCons,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_add @ A )
% 0.25/0.52 => ! [X: A,Xs: list @ A] :
% 0.25/0.52 ( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
% 0.25/0.52 = ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list.Cons
% 0.25/0.52 thf(fact_4488_sum__list__append,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_add @ A )
% 0.25/0.52 => ! [Xs: list @ A,Ys2: list @ A] :
% 0.25/0.52 ( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys2 ) )
% 0.25/0.52 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_append
% 0.25/0.52 thf(fact_4489_ordIso__transitive,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_transitive
% 0.25/0.52 thf(fact_4490_ordIso__symmetric,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_symmetric
% 0.25/0.52 thf(fact_4491_sum__list__transfer,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( ( monoid_add @ B )
% 0.25/0.52 & ( monoid_add @ A ) )
% 0.25/0.52 => ! [A4: A > B > $o] :
% 0.25/0.52 ( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 0.25/0.52 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 0.25/0.52 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_transfer
% 0.25/0.52 thf(fact_4492_Misc_Olist__all2__induct,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L4: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
% 0.25/0.52 ( ( list_all2 @ A @ B @ P @ L @ L4 )
% 0.25/0.52 => ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
% 0.25/0.52 => ( ! [X2: A,X5: B,Ls2: list @ A,Ls3: list @ B] :
% 0.25/0.52 ( ( P @ X2 @ X5 )
% 0.25/0.52 => ( ( list_all2 @ A @ B @ P @ Ls2 @ Ls3 )
% 0.25/0.52 => ( ( Q @ Ls2 @ Ls3 )
% 0.25/0.52 => ( Q @ ( cons @ A @ X2 @ Ls2 ) @ ( cons @ B @ X5 @ Ls3 ) ) ) ) )
% 0.25/0.52 => ( Q @ L @ L4 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Misc.list_all2_induct
% 0.25/0.52 thf(fact_4493_ordIso__iff__ordLeq,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 = ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_iff_ordLeq
% 0.25/0.52 thf(fact_4494_ordIso__imp__ordLeq,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_imp_ordLeq
% 0.25/0.52 thf(fact_4495_ordIso__ordLeq__trans,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_ordLeq_trans
% 0.25/0.52 thf(fact_4496_ordLeq__ordIso__trans,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_ordIso_trans
% 0.25/0.52 thf(fact_4497_not__ordLess__ordIso,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 => ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_ordLess_ordIso
% 0.25/0.52 thf(fact_4498_ordIso__ordLess__trans,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_ordLess_trans
% 0.25/0.52 thf(fact_4499_ordLess__ordIso__trans,axiom,
% 0.25/0.52 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLess_ordIso_trans
% 0.25/0.52 thf(fact_4500_sum__list__addf,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [F: B > A,G: B > A,Xs: list @ B] :
% 0.25/0.52 ( ( groups8242544230860333062m_list @ A
% 0.25/0.52 @ ( map @ B @ A
% 0.25/0.52 @ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.25/0.52 @ Xs ) )
% 0.25/0.52 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_addf
% 0.25/0.52 thf(fact_4501_finite__well__order__on__ordIso,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.52 => ( ( order_well_order_on @ A @ A4 @ R )
% 0.25/0.52 => ( ( order_well_order_on @ A @ A4 @ R4 )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % finite_well_order_on_ordIso
% 0.25/0.52 thf(fact_4502_ordIso__reflexive,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordIso_reflexive
% 0.25/0.52 thf(fact_4503_uminus__sum__list__map,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( ab_group_add @ A )
% 0.25/0.52 => ! [F: B > A,Xs: list @ B] :
% 0.25/0.52 ( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
% 0.25/0.52 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % uminus_sum_list_map
% 0.25/0.52 thf(fact_4504_ordLeq__iff__ordLess__or__ordIso,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 = ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 | ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % ordLeq_iff_ordLess_or_ordIso
% 0.25/0.52 thf(fact_4505_sum__list_Oeq__foldr,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_add @ A )
% 0.25/0.52 => ( ( groups8242544230860333062m_list @ A )
% 0.25/0.52 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list.eq_foldr
% 0.25/0.52 thf(fact_4506_internalize__ordLeq,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.52 = ( ? [P6: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % internalize_ordLeq
% 0.25/0.52 thf(fact_4507_internalize__ordLess,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.52 = ( ? [P6: set @ ( product_prod @ B @ B )] :
% 0.25/0.52 ( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.52 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % internalize_ordLess
% 0.25/0.52 thf(fact_4508_card__length__sum__list__rec,axiom,
% 0.25/0.52 ! [M2: nat,N6: nat] :
% 0.25/0.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
% 0.25/0.52 => ( ( finite_card @ ( list @ nat )
% 0.25/0.52 @ ( collect @ ( list @ nat )
% 0.25/0.52 @ ^ [L3: list @ nat] :
% 0.25/0.52 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.25/0.52 = M2 )
% 0.25/0.52 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 0.25/0.52 = N6 ) ) ) )
% 0.25/0.52 = ( plus_plus @ nat
% 0.25/0.52 @ ( finite_card @ ( list @ nat )
% 0.25/0.52 @ ( collect @ ( list @ nat )
% 0.25/0.52 @ ^ [L3: list @ nat] :
% 0.25/0.52 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.25/0.52 = ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) )
% 0.25/0.52 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 0.25/0.52 = N6 ) ) ) )
% 0.25/0.52 @ ( finite_card @ ( list @ nat )
% 0.25/0.52 @ ( collect @ ( list @ nat )
% 0.25/0.52 @ ^ [L3: list @ nat] :
% 0.25/0.52 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.25/0.52 = M2 )
% 0.25/0.52 & ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L3 ) @ ( one_one @ nat ) )
% 0.25/0.52 = N6 ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % card_length_sum_list_rec
% 0.25/0.52 thf(fact_4509_size__list__conv__sum__list,axiom,
% 0.25/0.52 ! [B: $tType] :
% 0.25/0.52 ( ( size_list @ B )
% 0.25/0.52 = ( ^ [F2: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F2 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % size_list_conv_sum_list
% 0.25/0.52 thf(fact_4510_card__length__sum__list,axiom,
% 0.25/0.52 ! [M2: nat,N6: nat] :
% 0.25/0.52 ( ( finite_card @ ( list @ nat )
% 0.25/0.52 @ ( collect @ ( list @ nat )
% 0.25/0.52 @ ^ [L3: list @ nat] :
% 0.25/0.52 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 0.25/0.52 = M2 )
% 0.25/0.52 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 0.25/0.52 = N6 ) ) ) )
% 0.25/0.52 = ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M2 ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % card_length_sum_list
% 0.25/0.52 thf(fact_4511_sum__list__Suc,axiom,
% 0.25/0.52 ! [A: $tType,F: A > nat,Xs: list @ A] :
% 0.25/0.52 ( ( groups8242544230860333062m_list @ nat
% 0.25/0.52 @ ( map @ A @ nat
% 0.25/0.52 @ ^ [X3: A] : ( suc @ ( F @ X3 ) )
% 0.25/0.52 @ Xs ) )
% 0.25/0.52 = ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_Suc
% 0.25/0.52 thf(fact_4512_select__weight__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( select_weight @ A )
% 0.25/0.52 = ( ^ [Xs3: list @ ( product_prod @ code_natural @ A )] :
% 0.25/0.52 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs3 ) ) )
% 0.25/0.52 @ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K4 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % select_weight_def
% 0.25/0.52 thf(fact_4513_prod__list__transfer,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( ( monoid_mult @ B )
% 0.25/0.52 & ( monoid_mult @ A ) )
% 0.25/0.52 => ! [A4: A > B > $o] :
% 0.25/0.52 ( ( A4 @ ( one_one @ A ) @ ( one_one @ B ) )
% 0.25/0.52 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 0.25/0.52 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_list_transfer
% 0.25/0.52 thf(fact_4514_dir__image__ordIso,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.25/0.52 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.52 => ( ( inj_on @ A @ B @ F @ ( field2 @ A @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % dir_image_ordIso
% 0.25/0.52 thf(fact_4515_underS__incr,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.52 ( ( trans @ A @ R )
% 0.25/0.52 => ( ( antisym @ A @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.52 => ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % underS_incr
% 0.25/0.52 thf(fact_4516_prod__list_ONil,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_mult @ A )
% 0.25/0.52 => ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
% 0.25/0.52 = ( one_one @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_list.Nil
% 0.25/0.52 thf(fact_4517_antisymD,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.52 ( ( antisym @ A @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
% 0.25/0.52 => ( A3 = B2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % antisymD
% 0.25/0.52 thf(fact_4518_antisymI,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ! [X2: A,Y4: A] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R )
% 0.25/0.52 => ( X2 = Y4 ) ) )
% 0.25/0.52 => ( antisym @ A @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % antisymI
% 0.25/0.52 thf(fact_4519_antisym__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( antisym @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ! [X3: A,Y3: A] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
% 0.25/0.52 => ( X3 = Y3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % antisym_def
% 0.25/0.52 thf(fact_4520_dir__image__def,axiom,
% 0.25/0.52 ! [A2: $tType,A: $tType] :
% 0.25/0.52 ( ( bNF_We2720479622203943262_image @ A @ A2 )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A ),F2: A > A2] :
% 0.25/0.52 ( collect @ ( product_prod @ A2 @ A2 )
% 0.25/0.52 @ ^ [Uu: product_prod @ A2 @ A2] :
% 0.25/0.52 ? [A7: A,B5: A] :
% 0.25/0.52 ( ( Uu
% 0.25/0.52 = ( product_Pair @ A2 @ A2 @ ( F2 @ A7 ) @ ( F2 @ B5 ) ) )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % dir_image_def
% 0.25/0.52 thf(fact_4521_prod__list_Oeq__foldr,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_mult @ A )
% 0.25/0.52 => ( ( groups5270119922927024881d_list @ A )
% 0.25/0.52 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_list.eq_foldr
% 0.25/0.52 thf(fact_4522_pick__same,axiom,
% 0.25/0.52 ! [A: $tType,L: nat,Xs: list @ A] :
% 0.25/0.52 ( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.52 => ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
% 0.25/0.52 = ( nth @ A @ Xs @ L ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pick_same
% 0.25/0.52 thf(fact_4523_flip__pred,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,A4: set @ ( product_prod @ A @ B ),R5: B > A > $o] :
% 0.25/0.52 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R5 ) ) ) )
% 0.25/0.52 => ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 0.25/0.52 @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
% 0.25/0.52 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 0.25/0.52 @ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) )
% 0.25/0.52 @ A4 )
% 0.25/0.52 @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R5 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % flip_pred
% 0.25/0.52 thf(fact_4524_prod__set__simps_I1_J,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,X: A,Y: B] :
% 0.25/0.52 ( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.25/0.52 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_set_simps(1)
% 0.25/0.52 thf(fact_4525_prod__set__simps_I2_J,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.25/0.52 ( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 0.25/0.52 = ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_set_simps(2)
% 0.25/0.52 thf(fact_4526_natural__of__nat__of__nat,axiom,
% 0.25/0.52 ( code_natural_of_nat
% 0.25/0.52 = ( semiring_1_of_nat @ code_natural ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_nat_of_nat
% 0.25/0.52 thf(fact_4527_division__segment__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [X: nat] :
% 0.25/0.52 ( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % division_segment_natural.abs_eq
% 0.25/0.52 thf(fact_4528_set__bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se5668285175392031749et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % set_bit_natural.abs_eq
% 0.25/0.52 thf(fact_4529_flip__bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % flip_bit_natural.abs_eq
% 0.25/0.52 thf(fact_4530_unset__bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se2638667681897837118et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % unset_bit_natural.abs_eq
% 0.25/0.52 thf(fact_4531_euclidean__size__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [X: nat] :
% 0.25/0.52 ( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( euclid6346220572633701492n_size @ nat @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % euclidean_size_natural.abs_eq
% 0.25/0.52 thf(fact_4532_bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [X: nat] :
% 0.25/0.52 ( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( bit_se5641148757651400278ts_bit @ nat @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % bit_natural.abs_eq
% 0.25/0.52 thf(fact_4533_natural__of__nat__cases,axiom,
% 0.25/0.52 ! [X: code_natural] :
% 0.25/0.52 ~ ! [Y4: nat] :
% 0.25/0.52 ( ( X
% 0.25/0.52 = ( code_natural_of_nat @ Y4 ) )
% 0.25/0.52 => ~ ( member @ nat @ Y4 @ ( top_top @ ( set @ nat ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_nat_cases
% 0.25/0.52 thf(fact_4534_natural__of__nat__induct,axiom,
% 0.25/0.52 ! [P: code_natural > $o,X: code_natural] :
% 0.25/0.52 ( ! [Y4: nat] :
% 0.25/0.52 ( ( member @ nat @ Y4 @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.52 => ( P @ ( code_natural_of_nat @ Y4 ) ) )
% 0.25/0.52 => ( P @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_nat_induct
% 0.25/0.52 thf(fact_4535_natural__of__nat__inject,axiom,
% 0.25/0.52 ! [X: nat,Y: nat] :
% 0.25/0.52 ( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.52 => ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.52 => ( ( ( code_natural_of_nat @ X )
% 0.25/0.52 = ( code_natural_of_nat @ Y ) )
% 0.25/0.52 = ( X = Y ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_nat_inject
% 0.25/0.52 thf(fact_4536_take__bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % take_bit_natural.abs_eq
% 0.25/0.52 thf(fact_4537_and__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % and_natural.abs_eq
% 0.25/0.52 thf(fact_4538_xor__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % xor_natural.abs_eq
% 0.25/0.52 thf(fact_4539_or__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % or_natural.abs_eq
% 0.25/0.52 thf(fact_4540_mask__natural_Oabs__eq,axiom,
% 0.25/0.52 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 0.25/0.52 = ( ^ [X3: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mask_natural.abs_eq
% 0.25/0.52 thf(fact_4541_push__bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % push_bit_natural.abs_eq
% 0.25/0.52 thf(fact_4542_drop__bit__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( bit_se4197421643247451524op_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % drop_bit_natural.abs_eq
% 0.25/0.52 thf(fact_4543_zero__natural__def,axiom,
% 0.25/0.52 ( ( zero_zero @ code_natural )
% 0.25/0.52 = ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % zero_natural_def
% 0.25/0.52 thf(fact_4544_one__natural__def,axiom,
% 0.25/0.52 ( ( one_one @ code_natural )
% 0.25/0.52 = ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % one_natural_def
% 0.25/0.52 thf(fact_4545_less__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( ord_less @ nat @ Xa @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_natural.abs_eq
% 0.25/0.52 thf(fact_4546_plus__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( plus_plus @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_natural.abs_eq
% 0.25/0.52 thf(fact_4547_minus__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( minus_minus @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_natural.abs_eq
% 0.25/0.52 thf(fact_4548_less__eq__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( ord_less_eq @ nat @ Xa @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_natural.abs_eq
% 0.25/0.52 thf(fact_4549_times__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( times_times @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_natural.abs_eq
% 0.25/0.52 thf(fact_4550_divide__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( divide_divide @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % divide_natural.abs_eq
% 0.25/0.52 thf(fact_4551_mask__natural__def,axiom,
% 0.25/0.52 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mask_natural_def
% 0.25/0.52 thf(fact_4552_modulo__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [Xa: nat,X: nat] :
% 0.25/0.52 ( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % modulo_natural.abs_eq
% 0.25/0.52 thf(fact_4553_select__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( select @ A )
% 0.25/0.52 = ( ^ [Xs3: list @ A] :
% 0.25/0.52 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( code_natural_of_nat @ ( size_size @ ( list @ A ) @ Xs3 ) ) )
% 0.25/0.52 @ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K4 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % select_def
% 0.25/0.52 thf(fact_4554_Nat_Oabs__eq,axiom,
% 0.25/0.52 ! [X: int] :
% 0.25/0.52 ( ( code_Nat @ ( code_integer_of_int @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Nat.abs_eq
% 0.25/0.52 thf(fact_4555_Nat__def,axiom,
% 0.25/0.52 ( code_Nat
% 0.25/0.52 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % Nat_def
% 0.25/0.52 thf(fact_4556_natural__of__integer_Oabs__eq,axiom,
% 0.25/0.52 ! [X: int] :
% 0.25/0.52 ( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_integer.abs_eq
% 0.25/0.52 thf(fact_4557_natural__of__nat__of__natural__inverse,axiom,
% 0.25/0.52 ! [N: code_natural] :
% 0.25/0.52 ( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
% 0.25/0.52 = N ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_nat_of_natural_inverse
% 0.25/0.52 thf(fact_4558_nat__of__natural__of__nat__inverse,axiom,
% 0.25/0.52 ! [N: nat] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
% 0.25/0.52 = N ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_of_nat_inverse
% 0.25/0.52 thf(fact_4559_nat__of__natural__inverse,axiom,
% 0.25/0.52 ! [X: code_natural] :
% 0.25/0.52 ( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
% 0.25/0.52 = X ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_inverse
% 0.25/0.52 thf(fact_4560_nat__of__natural__min,axiom,
% 0.25/0.52 ! [K: code_natural,L: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
% 0.25/0.52 = ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_min
% 0.25/0.52 thf(fact_4561_nat__of__natural__max,axiom,
% 0.25/0.52 ! [K: code_natural,L: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
% 0.25/0.52 = ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_max
% 0.25/0.52 thf(fact_4562_of__nat__of__natural,axiom,
% 0.25/0.52 ! [N: code_natural] :
% 0.25/0.52 ( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
% 0.25/0.52 = N ) ).
% 0.25/0.52
% 0.25/0.52 % of_nat_of_natural
% 0.25/0.52 thf(fact_4563_nat__of__natural__of__nat,axiom,
% 0.25/0.52 ! [N: nat] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 0.25/0.52 = N ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_of_nat
% 0.25/0.52 thf(fact_4564_zero__natural_Orep__eq,axiom,
% 0.25/0.52 ( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
% 0.25/0.52 = ( zero_zero @ nat ) ) ).
% 0.25/0.52
% 0.25/0.52 % zero_natural.rep_eq
% 0.25/0.52 thf(fact_4565_one__natural_Orep__eq,axiom,
% 0.25/0.52 ( ( code_nat_of_natural @ ( one_one @ code_natural ) )
% 0.25/0.52 = ( one_one @ nat ) ) ).
% 0.25/0.52
% 0.25/0.52 % one_natural.rep_eq
% 0.25/0.52 thf(fact_4566_plus__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_natural.rep_eq
% 0.25/0.52 thf(fact_4567_minus__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_natural.rep_eq
% 0.25/0.52 thf(fact_4568_nat__of__natural__numeral,axiom,
% 0.25/0.52 ! [K: num] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
% 0.25/0.52 = ( numeral_numeral @ nat @ K ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_numeral
% 0.25/0.52 thf(fact_4569_times__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_natural.rep_eq
% 0.25/0.52 thf(fact_4570_divide__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % divide_natural.rep_eq
% 0.25/0.52 thf(fact_4571_modulo__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( modulo_modulo @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % modulo_natural.rep_eq
% 0.25/0.52 thf(fact_4572_division__segment__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X ) )
% 0.25/0.52 = ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % division_segment_natural.rep_eq
% 0.25/0.52 thf(fact_4573_euclidean__size__natural_Orep__eq,axiom,
% 0.25/0.52 ( ( euclid6346220572633701492n_size @ code_natural )
% 0.25/0.52 = ( ^ [X3: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % euclidean_size_natural.rep_eq
% 0.25/0.52 thf(fact_4574_unset__bit__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se2638667681897837118et_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % unset_bit_natural.rep_eq
% 0.25/0.52 thf(fact_4575_flip__bit__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se8732182000553998342ip_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % flip_bit_natural.rep_eq
% 0.25/0.52 thf(fact_4576_set__bit__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se5668285175392031749et_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % set_bit_natural.rep_eq
% 0.25/0.52 thf(fact_4577_Nat_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_integer] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( code_Nat @ X ) )
% 0.25/0.52 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Nat.rep_eq
% 0.25/0.52 thf(fact_4578_push__bit__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se4730199178511100633sh_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % push_bit_natural.rep_eq
% 0.25/0.52 thf(fact_4579_mask__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X ) )
% 0.25/0.52 = ( bit_se2239418461657761734s_mask @ nat @ X ) ) ).
% 0.25/0.52
% 0.25/0.52 % mask_natural.rep_eq
% 0.25/0.52 thf(fact_4580_and__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % and_natural.rep_eq
% 0.25/0.52 thf(fact_4581_bit__natural_Orep__eq,axiom,
% 0.25/0.52 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 0.25/0.52 = ( ^ [X3: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bit_natural.rep_eq
% 0.25/0.52 thf(fact_4582_nat__of__natural__inject,axiom,
% 0.25/0.52 ! [X: code_natural,Y: code_natural] :
% 0.25/0.52 ( ( ( code_nat_of_natural @ X )
% 0.25/0.52 = ( code_nat_of_natural @ Y ) )
% 0.25/0.52 = ( X = Y ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_inject
% 0.25/0.52 thf(fact_4583_natural__eqI,axiom,
% 0.25/0.52 ! [M2: code_natural,N: code_natural] :
% 0.25/0.52 ( ( ( code_nat_of_natural @ M2 )
% 0.25/0.52 = ( code_nat_of_natural @ N ) )
% 0.25/0.52 => ( M2 = N ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_eqI
% 0.25/0.52 thf(fact_4584_natural__eq__iff,axiom,
% 0.25/0.52 ( ( ^ [Y5: code_natural,Z4: code_natural] : ( Y5 = Z4 ) )
% 0.25/0.52 = ( ^ [M: code_natural,N5: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ M )
% 0.25/0.52 = ( code_nat_of_natural @ N5 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_eq_iff
% 0.25/0.52 thf(fact_4585_natural__cases,axiom,
% 0.25/0.52 ! [M2: code_natural] :
% 0.25/0.52 ~ ! [N3: nat] :
% 0.25/0.52 ( M2
% 0.25/0.52 != ( semiring_1_of_nat @ code_natural @ N3 ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_cases
% 0.25/0.52 thf(fact_4586_nat__of__natural,axiom,
% 0.25/0.52 ! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural
% 0.25/0.52 thf(fact_4587_nat__of__natural__cases,axiom,
% 0.25/0.52 ! [Y: nat] :
% 0.25/0.52 ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.52 => ~ ! [X2: code_natural] :
% 0.25/0.52 ( Y
% 0.25/0.52 != ( code_nat_of_natural @ X2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_cases
% 0.25/0.52 thf(fact_4588_nat__of__natural__induct,axiom,
% 0.25/0.52 ! [Y: nat,P: nat > $o] :
% 0.25/0.52 ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.52 => ( ! [X2: code_natural] : ( P @ ( code_nat_of_natural @ X2 ) )
% 0.25/0.52 => ( P @ Y ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % nat_of_natural_induct
% 0.25/0.52 thf(fact_4589_take__bit__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se2584673776208193580ke_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % take_bit_natural.rep_eq
% 0.25/0.52 thf(fact_4590_or__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % or_natural.rep_eq
% 0.25/0.52 thf(fact_4591_xor__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % xor_natural.rep_eq
% 0.25/0.52 thf(fact_4592_drop__bit__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: nat,Xa: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X @ Xa ) )
% 0.25/0.52 = ( bit_se4197421643247451524op_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % drop_bit_natural.rep_eq
% 0.25/0.52 thf(fact_4593_natural__of__integer_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_integer] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
% 0.25/0.52 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_integer.rep_eq
% 0.25/0.52 thf(fact_4594_less__natural_Orep__eq,axiom,
% 0.25/0.52 ( ( ord_less @ code_natural )
% 0.25/0.52 = ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_natural.rep_eq
% 0.25/0.52 thf(fact_4595_natural__of__nat__inverse,axiom,
% 0.25/0.52 ! [Y: nat] :
% 0.25/0.52 ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 0.25/0.52 => ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
% 0.25/0.52 = Y ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_nat_inverse
% 0.25/0.52 thf(fact_4596_type__definition__natural,axiom,
% 0.25/0.52 type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% 0.25/0.52
% 0.25/0.52 % type_definition_natural
% 0.25/0.52 thf(fact_4597_or__natural__def,axiom,
% 0.25/0.52 ( ( bit_se1065995026697491101ons_or @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se1065995026697491101ons_or @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % or_natural_def
% 0.25/0.52 thf(fact_4598_xor__natural__def,axiom,
% 0.25/0.52 ( ( bit_se5824344971392196577ns_xor @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344971392196577ns_xor @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % xor_natural_def
% 0.25/0.52 thf(fact_4599_and__natural__def,axiom,
% 0.25/0.52 ( ( bit_se5824344872417868541ns_and @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344872417868541ns_and @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % and_natural_def
% 0.25/0.52 thf(fact_4600_less__eq__natural_Orep__eq,axiom,
% 0.25/0.52 ( ( ord_less_eq @ code_natural )
% 0.25/0.52 = ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_natural.rep_eq
% 0.25/0.52 thf(fact_4601_division__segment__natural__def,axiom,
% 0.25/0.52 ( ( euclid7384307370059645450egment @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % division_segment_natural_def
% 0.25/0.52 thf(fact_4602_euclidean__size__natural__def,axiom,
% 0.25/0.52 ( ( euclid6346220572633701492n_size @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % euclidean_size_natural_def
% 0.25/0.52 thf(fact_4603_plus__natural__def,axiom,
% 0.25/0.52 ( ( plus_plus @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( plus_plus @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_natural_def
% 0.25/0.52 thf(fact_4604_minus__natural__def,axiom,
% 0.25/0.52 ( ( minus_minus @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( minus_minus @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_natural_def
% 0.25/0.52 thf(fact_4605_bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bit_natural_def
% 0.25/0.52 thf(fact_4606_less__natural__def,axiom,
% 0.25/0.52 ( ( ord_less @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_natural_def
% 0.25/0.52 thf(fact_4607_times__natural__def,axiom,
% 0.25/0.52 ( ( times_times @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( times_times @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_natural_def
% 0.25/0.52 thf(fact_4608_divide__natural__def,axiom,
% 0.25/0.52 ( ( divide_divide @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( divide_divide @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % divide_natural_def
% 0.25/0.52 thf(fact_4609_take__bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se2584673776208193580ke_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2584673776208193580ke_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % take_bit_natural_def
% 0.25/0.52 thf(fact_4610_push__bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se4730199178511100633sh_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4730199178511100633sh_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % push_bit_natural_def
% 0.25/0.52 thf(fact_4611_less__eq__natural__def,axiom,
% 0.25/0.52 ( ( ord_less_eq @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less_eq @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_natural_def
% 0.25/0.52 thf(fact_4612_modulo__natural__def,axiom,
% 0.25/0.52 ( ( modulo_modulo @ code_natural )
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( modulo_modulo @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % modulo_natural_def
% 0.25/0.52 thf(fact_4613_drop__bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se4197421643247451524op_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4197421643247451524op_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % drop_bit_natural_def
% 0.25/0.52 thf(fact_4614_unset__bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se2638667681897837118et_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2638667681897837118et_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % unset_bit_natural_def
% 0.25/0.52 thf(fact_4615_flip__bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se8732182000553998342ip_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se8732182000553998342ip_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % flip_bit_natural_def
% 0.25/0.52 thf(fact_4616_set__bit__natural__def,axiom,
% 0.25/0.52 ( ( bit_se5668285175392031749et_bit @ code_natural )
% 0.25/0.52 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5668285175392031749et_bit @ nat ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % set_bit_natural_def
% 0.25/0.52 thf(fact_4617_natural__decr,axiom,
% 0.25/0.52 ! [N: code_natural] :
% 0.25/0.52 ( ( N
% 0.25/0.52 != ( zero_zero @ code_natural ) )
% 0.25/0.52 => ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_decr
% 0.25/0.52 thf(fact_4618_natural__of__integer__def,axiom,
% 0.25/0.52 ( code_n4118661773612635043nteger
% 0.25/0.52 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_integer_def
% 0.25/0.52 thf(fact_4619_integer__of__natural__of__integer,axiom,
% 0.25/0.52 ! [K: code_integer] :
% 0.25/0.52 ( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
% 0.25/0.52 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 0.25/0.52
% 0.25/0.52 % integer_of_natural_of_integer
% 0.25/0.52 thf(fact_4620_integer__of__natural_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural] :
% 0.25/0.52 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
% 0.25/0.52 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % integer_of_natural.rep_eq
% 0.25/0.52 thf(fact_4621_int__of__integer__of__natural,axiom,
% 0.25/0.52 ! [N: code_natural] :
% 0.25/0.52 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
% 0.25/0.52 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % int_of_integer_of_natural
% 0.25/0.52 thf(fact_4622_integer__of__natural__of__nat,axiom,
% 0.25/0.52 ! [N: nat] :
% 0.25/0.52 ( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 0.25/0.52 = ( semiring_1_of_nat @ code_integer @ N ) ) ).
% 0.25/0.52
% 0.25/0.52 % integer_of_natural_of_nat
% 0.25/0.52 thf(fact_4623_natural__of__integer__of__natural,axiom,
% 0.25/0.52 ! [N: code_natural] :
% 0.25/0.52 ( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
% 0.25/0.52 = N ) ).
% 0.25/0.52
% 0.25/0.52 % natural_of_integer_of_natural
% 0.25/0.52 thf(fact_4624_integer__of__natural__def,axiom,
% 0.25/0.52 ( code_i5400310926305786745atural
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % integer_of_natural_def
% 0.25/0.52 thf(fact_4625_integer__of__natural_Oabs__eq,axiom,
% 0.25/0.52 ! [X: nat] :
% 0.25/0.52 ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % integer_of_natural.abs_eq
% 0.25/0.52 thf(fact_4626_Code__Numeral_Osize__nat__def,axiom,
% 0.25/0.52 code_size_nat = code_nat_of_natural ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.size_nat_def
% 0.25/0.52 thf(fact_4627_Code__Numeral_OSuc__def,axiom,
% 0.25/0.52 ( code_Suc
% 0.25/0.52 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.Suc_def
% 0.25/0.52 thf(fact_4628_sum__mset_Ounion__disjoint,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [A4: multiset @ B,B6: multiset @ B,G: B > A] :
% 0.25/0.52 ( ( ( inter_mset @ B @ A4 @ B6 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ B ) ) )
% 0.25/0.52 => ( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A4 @ B6 ) ) )
% 0.25/0.52 = ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_mset.union_disjoint
% 0.25/0.52 thf(fact_4629_natural_Oinject,axiom,
% 0.25/0.52 ! [Natural: code_natural,Natural2: code_natural] :
% 0.25/0.52 ( ( ( code_Suc @ Natural )
% 0.25/0.52 = ( code_Suc @ Natural2 ) )
% 0.25/0.52 = ( Natural = Natural2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural.inject
% 0.25/0.52 thf(fact_4630_natural_Osimps_I7_J,axiom,
% 0.25/0.52 ! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
% 0.25/0.52 ( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 0.25/0.52 = ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural.simps(7)
% 0.25/0.52 thf(fact_4631_natural_Osimps_I5_J,axiom,
% 0.25/0.52 ! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
% 0.25/0.52 ( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 0.25/0.52 = ( F22 @ Natural ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural.simps(5)
% 0.25/0.52 thf(fact_4632_Suc_Orep__eq,axiom,
% 0.25/0.52 ! [X: code_natural] :
% 0.25/0.52 ( ( code_nat_of_natural @ ( code_Suc @ X ) )
% 0.25/0.52 = ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Suc.rep_eq
% 0.25/0.52 thf(fact_4633_union__sup__distrib__right,axiom,
% 0.25/0.52 ! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( plus_plus @ ( multiset @ A ) @ C5 @ ( union_mset @ A @ A4 @ B6 ) )
% 0.25/0.52 = ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % union_sup_distrib_right
% 0.25/0.52 thf(fact_4634_sup__union__distrib__left,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 0.25/0.52 ( ( plus_plus @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) @ C5 )
% 0.25/0.52 = ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sup_union_distrib_left
% 0.25/0.52 thf(fact_4635_natural_Oexhaust,axiom,
% 0.25/0.52 ! [Y: code_natural] :
% 0.25/0.52 ( ( Y
% 0.25/0.52 != ( zero_zero @ code_natural ) )
% 0.25/0.52 => ~ ! [Natural3: code_natural] :
% 0.25/0.52 ( Y
% 0.25/0.52 != ( code_Suc @ Natural3 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural.exhaust
% 0.25/0.52 thf(fact_4636_natural_Odistinct_I1_J,axiom,
% 0.25/0.52 ! [Natural2: code_natural] :
% 0.25/0.52 ( ( zero_zero @ code_natural )
% 0.25/0.52 != ( code_Suc @ Natural2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural.distinct(1)
% 0.25/0.52 thf(fact_4637_natural_Odistinct_I2_J,axiom,
% 0.25/0.52 ! [Natural4: code_natural] :
% 0.25/0.52 ( ( code_Suc @ Natural4 )
% 0.25/0.52 != ( zero_zero @ code_natural ) ) ).
% 0.25/0.52
% 0.25/0.52 % natural.distinct(2)
% 0.25/0.52 thf(fact_4638_union__mset__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( union_mset @ A )
% 0.25/0.52 = ( ^ [A9: multiset @ A,B9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A9 @ ( minus_minus @ ( multiset @ A ) @ B9 @ A9 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % union_mset_def
% 0.25/0.52 thf(fact_4639_union__diff__inter__eq__sup,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( inter_mset @ A @ A4 @ B6 ) )
% 0.25/0.52 = ( union_mset @ A @ A4 @ B6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % union_diff_inter_eq_sup
% 0.25/0.52 thf(fact_4640_union__diff__sup__eq__inter,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( union_mset @ A @ A4 @ B6 ) )
% 0.25/0.52 = ( inter_mset @ A @ A4 @ B6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % union_diff_sup_eq_inter
% 0.25/0.52 thf(fact_4641_size__Un__Int,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( size_size @ ( multiset @ A ) @ B6 ) )
% 0.25/0.52 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) ) @ ( size_size @ ( multiset @ A ) @ ( inter_mset @ A @ A4 @ B6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % size_Un_Int
% 0.25/0.52 thf(fact_4642_Suc_Oabs__eq,axiom,
% 0.25/0.52 ! [X: nat] :
% 0.25/0.52 ( ( code_Suc @ ( code_natural_of_nat @ X ) )
% 0.25/0.52 = ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Suc.abs_eq
% 0.25/0.52 thf(fact_4643_Suc__natural__minus__one,axiom,
% 0.25/0.52 ! [N: code_natural] :
% 0.25/0.52 ( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
% 0.25/0.52 = N ) ).
% 0.25/0.52
% 0.25/0.52 % Suc_natural_minus_one
% 0.25/0.52 thf(fact_4644_size__Un__disjoint,axiom,
% 0.25/0.52 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( ( inter_mset @ A @ A4 @ B6 )
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 => ( ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) )
% 0.25/0.52 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( size_size @ ( multiset @ A ) @ B6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % size_Un_disjoint
% 0.25/0.52 thf(fact_4645_random__aux__set_Ocases,axiom,
% 0.25/0.52 ! [X: product_prod @ code_natural @ code_natural] :
% 0.25/0.52 ( ! [J3: code_natural] :
% 0.25/0.52 ( X
% 0.25/0.52 != ( product_Pair @ code_natural @ code_natural @ ( zero_zero @ code_natural ) @ J3 ) )
% 0.25/0.52 => ~ ! [I2: code_natural,J3: code_natural] :
% 0.25/0.52 ( X
% 0.25/0.52 != ( product_Pair @ code_natural @ code_natural @ ( code_Suc @ I2 ) @ J3 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % random_aux_set.cases
% 0.25/0.52 thf(fact_4646_Eps__Opt__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( eps_Opt @ A )
% 0.25/0.52 = ( ^ [P4: A > $o] :
% 0.25/0.52 ( if @ ( option @ A )
% 0.25/0.52 @ ? [X8: A] : ( P4 @ X8 )
% 0.25/0.52 @ ( some @ A @ ( fChoice @ A @ P4 ) )
% 0.25/0.52 @ ( none @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Eps_Opt_def
% 0.25/0.52 thf(fact_4647_acyclic__insert,axiom,
% 0.25/0.52 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
% 0.25/0.52 = ( ( transitive_acyclic @ A @ R )
% 0.25/0.52 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclic_insert
% 0.25/0.52 thf(fact_4648_acyclic__empty,axiom,
% 0.25/0.52 ! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclic_empty
% 0.25/0.52 thf(fact_4649_Eps__case__prod__eq,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.25/0.52 ( ( fChoice @ ( product_prod @ A @ B )
% 0.25/0.52 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.52 @ ^ [X9: A,Y8: B] :
% 0.25/0.52 ( ( X = X9 )
% 0.25/0.52 & ( Y = Y8 ) ) ) )
% 0.25/0.52 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 0.25/0.52
% 0.25/0.52 % Eps_case_prod_eq
% 0.25/0.52 thf(fact_4650_some__insert__self,axiom,
% 0.25/0.52 ! [A: $tType,S2: set @ A] :
% 0.25/0.52 ( ( S2
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 => ( ( insert2 @ A
% 0.25/0.52 @ ( fChoice @ A
% 0.25/0.52 @ ^ [X3: A] : ( member @ A @ X3 @ S2 ) )
% 0.25/0.52 @ S2 )
% 0.25/0.52 = S2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % some_insert_self
% 0.25/0.52 thf(fact_4651_some__elem,axiom,
% 0.25/0.52 ! [A: $tType,S2: set @ A] :
% 0.25/0.52 ( ( S2
% 0.25/0.52 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.52 => ( member @ A
% 0.25/0.52 @ ( fChoice @ A
% 0.25/0.52 @ ^ [X3: A] : ( member @ A @ X3 @ S2 ) )
% 0.25/0.52 @ S2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % some_elem
% 0.25/0.52 thf(fact_4652_cyclic__subset,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ~ ( transitive_acyclic @ A @ R5 )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S2 )
% 0.25/0.52 => ~ ( transitive_acyclic @ A @ S2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % cyclic_subset
% 0.25/0.52 thf(fact_4653_acyclic__union_I1_J,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) )
% 0.25/0.52 => ( transitive_acyclic @ A @ A4 ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclic_union(1)
% 0.25/0.52 thf(fact_4654_acyclic__union_I2_J,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) )
% 0.25/0.52 => ( transitive_acyclic @ A @ B6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclic_union(2)
% 0.25/0.52 thf(fact_4655_inv__on__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( inv_on @ A @ B )
% 0.25/0.52 = ( ^ [F2: A > B,A9: set @ A,X3: B] :
% 0.25/0.52 ( fChoice @ A
% 0.25/0.52 @ ^ [Y3: A] :
% 0.25/0.52 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.52 & ( ( F2 @ Y3 )
% 0.25/0.52 = X3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % inv_on_def
% 0.25/0.52 thf(fact_4656_split__paired__Eps,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( fChoice @ ( product_prod @ A @ B ) )
% 0.25/0.52 = ( ^ [P4: ( product_prod @ A @ B ) > $o] :
% 0.25/0.52 ( fChoice @ ( product_prod @ A @ B )
% 0.25/0.52 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.52 @ ^ [A7: A,B5: B] : ( P4 @ ( product_Pair @ A @ B @ A7 @ B5 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % split_paired_Eps
% 0.25/0.52 thf(fact_4657_acyclicI__order,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( preorder @ A )
% 0.25/0.52 => ! [R: set @ ( product_prod @ B @ B ),F: B > A] :
% 0.25/0.52 ( ! [A5: B,B3: B] :
% 0.25/0.52 ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B3 ) @ R )
% 0.25/0.52 => ( ord_less @ A @ ( F @ B3 ) @ ( F @ A5 ) ) )
% 0.25/0.52 => ( transitive_acyclic @ B @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclicI_order
% 0.25/0.52 thf(fact_4658_subset__mset_OacyclicI__order,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B ),F: B > ( multiset @ A )] :
% 0.25/0.52 ( ! [A5: B,B3: B] :
% 0.25/0.52 ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B3 ) @ R )
% 0.25/0.52 => ( subset_mset @ A @ ( F @ B3 ) @ ( F @ A5 ) ) )
% 0.25/0.52 => ( transitive_acyclic @ B @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % subset_mset.acyclicI_order
% 0.25/0.52 thf(fact_4659_acyclicI,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ! [X2: A] :
% 0.25/0.52 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) )
% 0.25/0.52 => ( transitive_acyclic @ A @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclicI
% 0.25/0.52 thf(fact_4660_acyclic__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( transitive_acyclic @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ! [X3: A] :
% 0.25/0.52 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclic_def
% 0.25/0.52 thf(fact_4661_cyclicE,axiom,
% 0.25/0.52 ! [A: $tType,G: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ~ ( transitive_acyclic @ A @ G )
% 0.25/0.52 => ~ ! [X2: A] :
% 0.25/0.52 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ G ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % cyclicE
% 0.25/0.52 thf(fact_4662_some__theI,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,P: A > B > $o] :
% 0.25/0.52 ( ? [A17: A,X_1: B] : ( P @ A17 @ X_1 )
% 0.25/0.52 => ( ! [B13: B,B24: B] :
% 0.25/0.52 ( ? [A5: A] : ( P @ A5 @ B13 )
% 0.25/0.52 => ( ? [A5: A] : ( P @ A5 @ B24 )
% 0.25/0.52 => ( B13 = B24 ) ) )
% 0.25/0.52 => ( P
% 0.25/0.52 @ ( fChoice @ A
% 0.25/0.52 @ ^ [A7: A] :
% 0.25/0.52 ? [X8: B] : ( P @ A7 @ X8 ) )
% 0.25/0.52 @ ( the @ B
% 0.25/0.52 @ ^ [B5: B] :
% 0.25/0.52 ? [A7: A] : ( P @ A7 @ B5 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % some_theI
% 0.25/0.52 thf(fact_4663_acyclic__insert__cyclic,axiom,
% 0.25/0.52 ! [A: $tType,G: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.52 ( ( transitive_acyclic @ A @ G )
% 0.25/0.52 => ( ~ ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ G ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ G ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % acyclic_insert_cyclic
% 0.25/0.52 thf(fact_4664_fun__of__rel__def,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( fun_of_rel @ B @ A )
% 0.25/0.52 = ( ^ [R6: set @ ( product_prod @ B @ A ),X3: B] :
% 0.25/0.52 ( fChoice @ A
% 0.25/0.52 @ ^ [Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % fun_of_rel_def
% 0.25/0.52 thf(fact_4665_irrefl__tranclI,axiom,
% 0.25/0.52 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A] :
% 0.25/0.52 ( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.52 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.25/0.52 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % irrefl_tranclI
% 0.25/0.52 thf(fact_4666_quicksort_Opelims,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( linorder @ A )
% 0.25/0.52 => ! [X: list @ A,Y: list @ A] :
% 0.25/0.52 ( ( ( linorder_quicksort @ A @ X )
% 0.25/0.52 = Y )
% 0.25/0.52 => ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
% 0.25/0.52 => ( ( ( X
% 0.25/0.52 = ( nil @ A ) )
% 0.25/0.52 => ( ( Y
% 0.25/0.52 = ( nil @ A ) )
% 0.25/0.52 => ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
% 0.25/0.52 => ~ ! [X2: A,Xs2: list @ A] :
% 0.25/0.52 ( ( X
% 0.25/0.52 = ( cons @ A @ X2 @ Xs2 ) )
% 0.25/0.52 => ( ( Y
% 0.25/0.52 = ( append @ A
% 0.25/0.52 @ ( linorder_quicksort @ A
% 0.25/0.52 @ ( filter @ A
% 0.25/0.52 @ ^ [Y3: A] :
% 0.25/0.52 ~ ( ord_less_eq @ A @ X2 @ Y3 )
% 0.25/0.52 @ Xs2 ) )
% 0.25/0.52 @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter @ A @ ( ord_less_eq @ A @ X2 ) @ Xs2 ) ) ) ) )
% 0.25/0.52 => ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % quicksort.pelims
% 0.25/0.52 thf(fact_4667_converse__iff,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
% 0.25/0.52 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % converse_iff
% 0.25/0.52 thf(fact_4668_pair__set__inverse,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,P: B > A > $o] :
% 0.25/0.52 ( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
% 0.25/0.52 = ( collect @ ( product_prod @ A @ B )
% 0.25/0.52 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.52 @ ^ [B5: A,A7: B] : ( P @ A7 @ B5 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % pair_set_inverse
% 0.25/0.52 thf(fact_4669_below__Id__inv,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R5 ) @ ( id2 @ A ) )
% 0.25/0.52 = ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % below_Id_inv
% 0.25/0.52 thf(fact_4670_converseI,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 0.25/0.52 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % converseI
% 0.25/0.52 thf(fact_4671_converseE,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R: set @ ( product_prod @ B @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R ) )
% 0.25/0.52 => ~ ! [X2: B,Y4: A] :
% 0.25/0.52 ( ( Yx
% 0.25/0.52 = ( product_Pair @ A @ B @ Y4 @ X2 ) )
% 0.25/0.52 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y4 ) @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % converseE
% 0.25/0.52 thf(fact_4672_converseD,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % converseD
% 0.25/0.52 thf(fact_4673_converse_Osimps,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 0.25/0.52 = ( ? [A7: A,B5: B] :
% 0.25/0.52 ( ( A1 = B5 )
% 0.25/0.52 & ( A22 = A7 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % converse.simps
% 0.25/0.52 thf(fact_4674_converse_Ocases,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.52 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R ) ) ).
% 0.25/0.52
% 0.25/0.52 % converse.cases
% 0.25/0.52 thf(fact_4675_in__listrel1__converse,axiom,
% 0.25/0.52 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R ) ) )
% 0.25/0.52 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % in_listrel1_converse
% 0.25/0.52 thf(fact_4676_rtrancl__converseI,axiom,
% 0.25/0.52 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % rtrancl_converseI
% 0.25/0.52 thf(fact_4677_rtrancl__converseD,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % rtrancl_converseD
% 0.25/0.52 thf(fact_4678_trancl__converseI,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trancl_converseI
% 0.25/0.52 thf(fact_4679_trancl__converseD,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) )
% 0.25/0.52 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % trancl_converseD
% 0.25/0.52 thf(fact_4680_finite__wf__eq__wf__converse,axiom,
% 0.25/0.52 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.52 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
% 0.25/0.52 => ( ( wf @ A @ ( converse @ A @ A @ R5 ) )
% 0.25/0.52 = ( wf @ A @ R5 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % finite_wf_eq_wf_converse
% 0.25/0.52 thf(fact_4681_converse__unfold,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( converse @ B @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ B @ A )] :
% 0.25/0.52 ( collect @ ( product_prod @ A @ B )
% 0.25/0.52 @ ( product_case_prod @ A @ B @ $o
% 0.25/0.52 @ ^ [Y3: A,X3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % converse_unfold
% 0.25/0.52 thf(fact_4682_conversep__converse__eq,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.52 ( ( conversep @ A @ B
% 0.25/0.52 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.25/0.52 = ( ^ [X3: B,Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ ( converse @ A @ B @ R ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % conversep_converse_eq
% 0.25/0.52 thf(fact_4683_converse__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( converse @ A @ B )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.25/0.52 ( collect @ ( product_prod @ B @ A )
% 0.25/0.52 @ ( product_case_prod @ B @ A @ $o
% 0.25/0.52 @ ( conversep @ A @ B
% 0.25/0.52 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % converse_def
% 0.25/0.52 thf(fact_4684_is__singleton__altdef,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( is_singleton @ A )
% 0.25/0.52 = ( ^ [A9: set @ A] :
% 0.25/0.52 ( ( finite_card @ A @ A9 )
% 0.25/0.52 = ( one_one @ nat ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % is_singleton_altdef
% 0.25/0.52 thf(fact_4685_Code__Numeral_Osub__code_I2_J,axiom,
% 0.25/0.52 ! [M2: num] :
% 0.25/0.52 ( ( code_sub @ ( bit0 @ M2 ) @ one2 )
% 0.25/0.52 = ( code_Pos @ ( bitM @ M2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.sub_code(2)
% 0.25/0.52 thf(fact_4686_su__rel__fun__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B )
% 0.25/0.52 = ( ^ [F7: set @ ( product_prod @ A @ B ),F2: A > B] :
% 0.25/0.52 ( ! [A9: A,B9: B,B11: B] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B11 ) @ F7 )
% 0.25/0.52 => ( B9 = B11 ) ) )
% 0.25/0.52 & ! [A9: A,P4: $o] :
% 0.25/0.52 ( ! [B9: B] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 )
% 0.25/0.52 => P4 )
% 0.25/0.52 => P4 )
% 0.25/0.52 & ! [A9: A] :
% 0.25/0.52 ( ( F2 @ A9 )
% 0.25/0.52 = ( the @ B
% 0.25/0.52 @ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun_def
% 0.25/0.52 thf(fact_4687_Code__Numeral_OPos__def,axiom,
% 0.25/0.52 ( code_Pos
% 0.25/0.52 = ( numeral_numeral @ code_integer ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.Pos_def
% 0.25/0.52 thf(fact_4688_su__rel__fun_Osurjective,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 0.25/0.52 => ~ ! [B10: B] :
% 0.25/0.52 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B10 ) @ F6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.surjective
% 0.25/0.52 thf(fact_4689_su__rel__fun_Ounique,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B,B14: B] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B14 ) @ F6 )
% 0.25/0.52 => ( B6 = B14 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.unique
% 0.25/0.52 thf(fact_4690_su__rel__fun_Orepr2,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 )
% 0.25/0.52 => ( B6
% 0.25/0.52 = ( F @ A4 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.repr2
% 0.25/0.52 thf(fact_4691_su__rel__fun_Orepr1,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 0.25/0.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ ( F @ A4 ) ) @ F6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.repr1
% 0.25/0.52 thf(fact_4692_su__rel__fun_Orepr,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 0.25/0.52 => ( ( ( F @ A4 )
% 0.25/0.52 = B6 )
% 0.25/0.52 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.repr
% 0.25/0.52 thf(fact_4693_Pos__fold_I2_J,axiom,
% 0.25/0.52 ! [K: num] :
% 0.25/0.52 ( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
% 0.25/0.52 = ( code_Pos @ ( bit0 @ K ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Pos_fold(2)
% 0.25/0.52 thf(fact_4694_Pos__fold_I1_J,axiom,
% 0.25/0.52 ( ( numeral_numeral @ code_integer @ one2 )
% 0.25/0.52 = ( code_Pos @ one2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % Pos_fold(1)
% 0.25/0.52 thf(fact_4695_Pos__fold_I3_J,axiom,
% 0.25/0.52 ! [K: num] :
% 0.25/0.52 ( ( numeral_numeral @ code_integer @ ( bit1 @ K ) )
% 0.25/0.52 = ( code_Pos @ ( bit1 @ K ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Pos_fold(3)
% 0.25/0.52 thf(fact_4696_one__integer__code,axiom,
% 0.25/0.52 ( ( one_one @ code_integer )
% 0.25/0.52 = ( code_Pos @ one2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % one_integer_code
% 0.25/0.52 thf(fact_4697_Code__Numeral_Odup__code_I2_J,axiom,
% 0.25/0.52 ! [N: num] :
% 0.25/0.52 ( ( code_dup @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_Pos @ ( bit0 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.dup_code(2)
% 0.25/0.52 thf(fact_4698_less__integer__code_I4_J,axiom,
% 0.25/0.52 ! [K: num] :
% 0.25/0.52 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(4)
% 0.25/0.52 thf(fact_4699_less__integer__code_I2_J,axiom,
% 0.25/0.52 ! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(2)
% 0.25/0.52 thf(fact_4700_plus__integer__code_I3_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( plus_plus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_Pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_integer_code(3)
% 0.25/0.52 thf(fact_4701_less__eq__integer__code_I2_J,axiom,
% 0.25/0.52 ! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(2)
% 0.25/0.52 thf(fact_4702_less__eq__integer__code_I4_J,axiom,
% 0.25/0.52 ! [K: num] :
% 0.25/0.52 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(4)
% 0.25/0.52 thf(fact_4703_less__integer__code_I8_J,axiom,
% 0.25/0.52 ! [K: num,L: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(8)
% 0.25/0.52 thf(fact_4704_less__integer__code_I6_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(6)
% 0.25/0.52 thf(fact_4705_uminus__integer__code_I3_J,axiom,
% 0.25/0.52 ! [M2: num] :
% 0.25/0.52 ( ( uminus_uminus @ code_integer @ ( code_Neg @ M2 ) )
% 0.25/0.52 = ( code_Pos @ M2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % uminus_integer_code(3)
% 0.25/0.52 thf(fact_4706_uminus__integer__code_I2_J,axiom,
% 0.25/0.52 ! [M2: num] :
% 0.25/0.52 ( ( uminus_uminus @ code_integer @ ( code_Pos @ M2 ) )
% 0.25/0.52 = ( code_Neg @ M2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % uminus_integer_code(2)
% 0.25/0.52 thf(fact_4707_Code__Numeral_ONeg__def,axiom,
% 0.25/0.52 ( code_Neg
% 0.25/0.52 = ( ^ [N5: num] : ( uminus_uminus @ code_integer @ ( code_Pos @ N5 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.Neg_def
% 0.25/0.52 thf(fact_4708_less__eq__integer__code_I8_J,axiom,
% 0.25/0.52 ! [K: num,L: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(8)
% 0.25/0.52 thf(fact_4709_less__eq__integer__code_I6_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(6)
% 0.25/0.52 thf(fact_4710_times__integer__code_I3_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( times_times @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_Pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_integer_code(3)
% 0.25/0.52 thf(fact_4711_less__integer__code_I5_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 0.25/0.52 = ( ord_less @ num @ K @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_integer_code(5)
% 0.25/0.52 thf(fact_4712_less__eq__integer__code_I5_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 0.25/0.52 = ( ord_less_eq @ num @ K @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % less_eq_integer_code(5)
% 0.25/0.52 thf(fact_4713_minus__integer__code_I3_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( minus_minus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_sub @ M2 @ N ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_integer_code(3)
% 0.25/0.52 thf(fact_4714_divmod__abs__code_I1_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 0.25/0.52 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % divmod_abs_code(1)
% 0.25/0.52 thf(fact_4715_minus__integer__code_I4_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( minus_minus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_Pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_integer_code(4)
% 0.25/0.52 thf(fact_4716_minus__integer__code_I5_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( minus_minus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_Neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % minus_integer_code(5)
% 0.25/0.52 thf(fact_4717_times__integer__code_I4_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( times_times @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_Neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_integer_code(4)
% 0.25/0.52 thf(fact_4718_times__integer__code_I5_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( times_times @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_Neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_integer_code(5)
% 0.25/0.52 thf(fact_4719_times__integer__code_I6_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( times_times @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_Pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % times_integer_code(6)
% 0.25/0.52 thf(fact_4720_plus__integer__code_I4_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( plus_plus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
% 0.25/0.52 = ( code_sub @ M2 @ N ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_integer_code(4)
% 0.25/0.52 thf(fact_4721_plus__integer__code_I5_J,axiom,
% 0.25/0.52 ! [M2: num,N: num] :
% 0.25/0.52 ( ( plus_plus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
% 0.25/0.52 = ( code_sub @ N @ M2 ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_integer_code(5)
% 0.25/0.52 thf(fact_4722_divmod__abs__code_I3_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
% 0.25/0.52 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % divmod_abs_code(3)
% 0.25/0.52 thf(fact_4723_divmod__abs__code_I4_J,axiom,
% 0.25/0.52 ! [K: num,L: num] :
% 0.25/0.52 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
% 0.25/0.52 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 0.25/0.52
% 0.25/0.52 % divmod_abs_code(4)
% 0.25/0.52 thf(fact_4724_Code__Numeral_Osub__code_I3_J,axiom,
% 0.25/0.52 ! [M2: num] :
% 0.25/0.52 ( ( code_sub @ ( bit1 @ M2 ) @ one2 )
% 0.25/0.52 = ( code_Pos @ ( bit0 @ M2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % Code_Numeral.sub_code(3)
% 0.25/0.52 thf(fact_4725_su__rel__fun_Of__def,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
% 0.25/0.52 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 0.25/0.52 => ( ( F @ A4 )
% 0.25/0.52 = ( the @ B
% 0.25/0.52 @ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B9 ) @ F6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.f_def
% 0.25/0.52 thf(fact_4726_su__rel__fun_Ointro,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B] :
% 0.25/0.52 ( ! [A15: A,B10: B,B15: B] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B10 ) @ F6 )
% 0.25/0.52 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B15 ) @ F6 )
% 0.25/0.52 => ( B10 = B15 ) ) )
% 0.25/0.52 => ( ! [A15: A,P7: $o] :
% 0.25/0.52 ( ! [B16: B] :
% 0.25/0.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B16 ) @ F6 )
% 0.25/0.52 => P7 )
% 0.25/0.52 => P7 )
% 0.25/0.52 => ( ! [A15: A] :
% 0.25/0.52 ( ( F @ A15 )
% 0.25/0.52 = ( the @ B
% 0.25/0.52 @ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B9 ) @ F6 ) ) )
% 0.25/0.52 => ( su_rel_fun @ A @ B @ F6 @ F ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % su_rel_fun.intro
% 0.25/0.52 thf(fact_4727_aboveS__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( order_aboveS @ A )
% 0.25/0.52 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 0.25/0.52 ( collect @ A
% 0.25/0.52 @ ^ [B5: A] :
% 0.25/0.52 ( ( B5 != A7 )
% 0.25/0.52 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % aboveS_def
% 0.25/0.52 thf(fact_4728_relInvImage__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( bNF_Gr7122648621184425601vImage @ A @ B )
% 0.25/0.52 = ( ^ [A9: set @ A,R6: set @ ( product_prod @ B @ B ),F2: A > B] :
% 0.25/0.52 ( collect @ ( product_prod @ A @ A )
% 0.25/0.52 @ ^ [Uu: product_prod @ A @ A] :
% 0.25/0.52 ? [A12: A,A23: A] :
% 0.25/0.52 ( ( Uu
% 0.25/0.52 = ( product_Pair @ A @ A @ A12 @ A23 ) )
% 0.25/0.52 & ( member @ A @ A12 @ A9 )
% 0.25/0.52 & ( member @ A @ A23 @ A9 )
% 0.25/0.52 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R6 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % relInvImage_def
% 0.25/0.52 thf(fact_4729_prod__list__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_mult @ A )
% 0.25/0.52 => ( ( groups5270119922927024881d_list @ A )
% 0.25/0.52 = ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_list_def
% 0.25/0.52 thf(fact_4730_sum__list__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( monoid_add @ A )
% 0.25/0.52 => ( ( groups8242544230860333062m_list @ A )
% 0.25/0.52 = ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_def
% 0.25/0.52 thf(fact_4731_sum__list__map__remove1,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_add @ A )
% 0.25/0.52 => ! [X: B,Xs: list @ B,F: B > A] :
% 0.25/0.52 ( ( member @ B @ X @ ( set2 @ B @ Xs ) )
% 0.25/0.52 => ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) )
% 0.25/0.52 = ( plus_plus @ A @ ( F @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % sum_list_map_remove1
% 0.25/0.52 thf(fact_4732_normalize__stable,axiom,
% 0.25/0.52 ! [Q3: int,P5: int] :
% 0.25/0.52 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q3 )
% 0.25/0.52 => ( ( algebr8660921524188924756oprime @ int @ P5 @ Q3 )
% 0.25/0.52 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.25/0.52 = ( product_Pair @ int @ int @ P5 @ Q3 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % normalize_stable
% 0.25/0.52 thf(fact_4733_distinct__count__atmost__1,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( distinct @ A )
% 0.25/0.52 = ( ^ [X3: list @ A] :
% 0.25/0.52 ! [A7: A] :
% 0.25/0.52 ( ( ( member @ A @ A7 @ ( set2 @ A @ X3 ) )
% 0.25/0.52 => ( ( count @ A @ ( mset @ A @ X3 ) @ A7 )
% 0.25/0.52 = ( one_one @ nat ) ) )
% 0.25/0.52 & ( ~ ( member @ A @ A7 @ ( set2 @ A @ X3 ) )
% 0.25/0.52 => ( ( count @ A @ ( mset @ A @ X3 ) @ A7 )
% 0.25/0.52 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % distinct_count_atmost_1
% 0.25/0.52 thf(fact_4734_coprime__self,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
% 0.25/0.52 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_self
% 0.25/0.52 thf(fact_4735_mset__empty__count,axiom,
% 0.25/0.52 ! [A: $tType,M3: multiset @ A] :
% 0.25/0.52 ( ( ! [P6: A] :
% 0.25/0.52 ( ( count @ A @ M3 @ P6 )
% 0.25/0.52 = ( zero_zero @ nat ) ) )
% 0.25/0.52 = ( M3
% 0.25/0.52 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % mset_empty_count
% 0.25/0.52 thf(fact_4736_coprime__imp__gcd__eq__1,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_gcd @ A )
% 0.25/0.52 => ! [A3: A,B2: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.25/0.52 => ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.25/0.52 = ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_imp_gcd_eq_1
% 0.25/0.52 thf(fact_4737_count__union,axiom,
% 0.25/0.52 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,A3: A] :
% 0.25/0.52 ( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ A3 )
% 0.25/0.52 = ( plus_plus @ nat @ ( count @ A @ M3 @ A3 ) @ ( count @ A @ N6 @ A3 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_union
% 0.25/0.52 thf(fact_4738_coprime__0__right__iff,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
% 0.25/0.52 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_0_right_iff
% 0.25/0.52 thf(fact_4739_coprime__0__left__iff,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
% 0.25/0.52 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_0_left_iff
% 0.25/0.52 thf(fact_4740_coprime__mult__self__right__iff,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A,C2: A,B2: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 0.25/0.52 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.25/0.52 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_mult_self_right_iff
% 0.25/0.52 thf(fact_4741_coprime__mult__self__left__iff,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [C2: A,A3: A,B2: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 0.25/0.52 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.25/0.52 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_mult_self_left_iff
% 0.25/0.52 thf(fact_4742_is__unit__gcd,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_gcd @ A )
% 0.25/0.52 => ! [A3: A,B2: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 0.25/0.52 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % is_unit_gcd
% 0.25/0.52 thf(fact_4743_count__greater__eq__one__iff,axiom,
% 0.25/0.52 ! [A: $tType,M3: multiset @ A,X: A] :
% 0.25/0.52 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( count @ A @ M3 @ X ) )
% 0.25/0.52 = ( member @ A @ X @ ( set_mset @ A @ M3 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_greater_eq_one_iff
% 0.25/0.52 thf(fact_4744_count__mset__set_I1_J,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ A,X: A] :
% 0.25/0.52 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.52 => ( ( member @ A @ X @ A4 )
% 0.25/0.52 => ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
% 0.25/0.52 = ( one_one @ nat ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_mset_set(1)
% 0.25/0.52 thf(fact_4745_coprime__1__left,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_1_left
% 0.25/0.52 thf(fact_4746_coprime__1__right,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_1_right
% 0.25/0.52 thf(fact_4747_coprime__add__one__left,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_gcd @ A )
% 0.25/0.52 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_add_one_left
% 0.25/0.52 thf(fact_4748_coprime__add__one__right,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_gcd @ A )
% 0.25/0.52 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_add_one_right
% 0.25/0.52 thf(fact_4749_coprime__doff__one__right,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( ring_gcd @ A )
% 0.25/0.52 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_doff_one_right
% 0.25/0.52 thf(fact_4750_coprime__diff__one__left,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( ring_gcd @ A )
% 0.25/0.52 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_diff_one_left
% 0.25/0.52 thf(fact_4751_is__unit__right__imp__coprime,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [B2: A,A3: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % is_unit_right_imp_coprime
% 0.25/0.52 thf(fact_4752_is__unit__left__imp__coprime,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A,B2: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % is_unit_left_imp_coprime
% 0.25/0.52 thf(fact_4753_coprime__common__divisor,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.25/0.52 => ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_common_divisor
% 0.25/0.52 thf(fact_4754_coprime__absorb__right,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [Y: A,X: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ Y @ X )
% 0.25/0.52 => ( ( algebr8660921524188924756oprime @ A @ X @ Y )
% 0.25/0.52 = ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_absorb_right
% 0.25/0.52 thf(fact_4755_coprime__imp__coprime,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [C2: A,D3: A,A3: A,B2: A] :
% 0.25/0.52 ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
% 0.25/0.52 => ( ! [E3: A] :
% 0.25/0.52 ( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
% 0.25/0.52 => ( ( dvd_dvd @ A @ E3 @ A3 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ E3 @ B2 )
% 0.25/0.52 => ( dvd_dvd @ A @ E3 @ C2 ) ) ) )
% 0.25/0.52 => ( ! [E3: A] :
% 0.25/0.52 ( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
% 0.25/0.52 => ( ( dvd_dvd @ A @ E3 @ A3 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ E3 @ B2 )
% 0.25/0.52 => ( dvd_dvd @ A @ E3 @ D3 ) ) ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_imp_coprime
% 0.25/0.52 thf(fact_4756_coprime__absorb__left,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [X: A,Y: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ X @ Y )
% 0.25/0.52 => ( ( algebr8660921524188924756oprime @ A @ X @ Y )
% 0.25/0.52 = ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_absorb_left
% 0.25/0.52 thf(fact_4757_not__coprimeI,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [C2: A,A3: A,B2: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ C2 @ A3 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 0.25/0.52 => ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 0.25/0.52 => ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_coprimeI
% 0.25/0.52 thf(fact_4758_not__coprimeE,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A,B2: A] :
% 0.25/0.52 ( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 0.25/0.52 => ~ ! [C4: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ C4 @ A3 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ C4 @ B2 )
% 0.25/0.52 => ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % not_coprimeE
% 0.25/0.52 thf(fact_4759_coprime__def,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ( ( algebr8660921524188924756oprime @ A )
% 0.25/0.52 = ( ^ [A7: A,B5: A] :
% 0.25/0.52 ! [C3: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ C3 @ A7 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ C3 @ B5 )
% 0.25/0.52 => ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_def
% 0.25/0.52 thf(fact_4760_coprimeI,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( algebraic_semidom @ A )
% 0.25/0.52 => ! [A3: A,B2: A] :
% 0.25/0.52 ( ! [C4: A] :
% 0.25/0.52 ( ( dvd_dvd @ A @ C4 @ A3 )
% 0.25/0.52 => ( ( dvd_dvd @ A @ C4 @ B2 )
% 0.25/0.52 => ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprimeI
% 0.25/0.52 thf(fact_4761_plus__multiset_Orep__eq,axiom,
% 0.25/0.52 ! [A: $tType,X: multiset @ A,Xa: multiset @ A] :
% 0.25/0.52 ( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ X @ Xa ) )
% 0.25/0.52 = ( ^ [A7: A] : ( plus_plus @ nat @ ( count @ A @ X @ A7 ) @ ( count @ A @ Xa @ A7 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % plus_multiset.rep_eq
% 0.25/0.52 thf(fact_4762_coprime__iff__gcd__eq__1,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_gcd @ A )
% 0.25/0.52 => ( ( algebr8660921524188924756oprime @ A )
% 0.25/0.52 = ( ^ [A7: A,B5: A] :
% 0.25/0.52 ( ( gcd_gcd @ A @ A7 @ B5 )
% 0.25/0.52 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % coprime_iff_gcd_eq_1
% 0.25/0.52 thf(fact_4763_gcd__eq__1__imp__coprime,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_gcd @ A )
% 0.25/0.52 => ! [A3: A,B2: A] :
% 0.25/0.52 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 0.25/0.52 = ( one_one @ A ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % gcd_eq_1_imp_coprime
% 0.25/0.52 thf(fact_4764_quotient__of__coprime,axiom,
% 0.25/0.52 ! [R: rat,P5: int,Q3: int] :
% 0.25/0.52 ( ( ( quotient_of @ R )
% 0.25/0.52 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ int @ P5 @ Q3 ) ) ).
% 0.25/0.52
% 0.25/0.52 % quotient_of_coprime
% 0.25/0.52 thf(fact_4765_normalize__coprime,axiom,
% 0.25/0.52 ! [R: product_prod @ int @ int,P5: int,Q3: int] :
% 0.25/0.52 ( ( ( normalize @ R )
% 0.25/0.52 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ int @ P5 @ Q3 ) ) ).
% 0.25/0.52
% 0.25/0.52 % normalize_coprime
% 0.25/0.52 thf(fact_4766_invertible__coprime,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( euclid8851590272496341667cancel @ A )
% 0.25/0.52 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.52 ( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 0.25/0.52 = ( one_one @ A ) )
% 0.25/0.52 => ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % invertible_coprime
% 0.25/0.52 thf(fact_4767_count__ne__remove,axiom,
% 0.25/0.52 ! [A: $tType,X: A,T2: A,S2: multiset @ A] :
% 0.25/0.52 ( ( X != T2 )
% 0.25/0.52 => ( ( count @ A @ S2 @ X )
% 0.25/0.52 = ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_ne_remove
% 0.25/0.52 thf(fact_4768_count__in__diffI,axiom,
% 0.25/0.52 ! [A: $tType,N6: multiset @ A,X: A,M3: multiset @ A] :
% 0.25/0.52 ( ! [N3: nat] :
% 0.25/0.52 ( ( count @ A @ N6 @ X )
% 0.25/0.52 != ( plus_plus @ nat @ N3 @ ( count @ A @ M3 @ X ) ) )
% 0.25/0.52 => ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_in_diffI
% 0.25/0.52 thf(fact_4769_count__single,axiom,
% 0.25/0.52 ! [A: $tType,B2: A,A3: A] :
% 0.25/0.52 ( ( ( B2 = A3 )
% 0.25/0.52 => ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
% 0.25/0.52 = ( one_one @ nat ) ) )
% 0.25/0.52 & ( ( B2 != A3 )
% 0.25/0.52 => ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
% 0.25/0.52 = ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_single
% 0.25/0.52 thf(fact_4770_in__diff__countE,axiom,
% 0.25/0.52 ! [A: $tType,X: A,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.52 ( ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) ) )
% 0.25/0.52 => ~ ! [N3: nat] :
% 0.25/0.52 ( ( count @ A @ M3 @ X )
% 0.25/0.52 != ( plus_plus @ nat @ ( suc @ N3 ) @ ( count @ A @ N6 @ X ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % in_diff_countE
% 0.25/0.52 thf(fact_4771_remove1__tl,axiom,
% 0.25/0.52 ! [A: $tType,Xs: list @ A] :
% 0.25/0.52 ( ( Xs
% 0.25/0.52 != ( nil @ A ) )
% 0.25/0.52 => ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
% 0.25/0.52 = ( tl @ A @ Xs ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % remove1_tl
% 0.25/0.52 thf(fact_4772_count__mset__set__finite__iff,axiom,
% 0.25/0.52 ! [A: $tType,S2: set @ A,A3: A] :
% 0.25/0.52 ( ( finite_finite2 @ A @ S2 )
% 0.25/0.52 => ( ( ( member @ A @ A3 @ S2 )
% 0.25/0.52 => ( ( count @ A @ ( mset_set @ A @ S2 ) @ A3 )
% 0.25/0.52 = ( one_one @ nat ) ) )
% 0.25/0.52 & ( ~ ( member @ A @ A3 @ S2 )
% 0.25/0.52 => ( ( count @ A @ ( mset_set @ A @ S2 ) @ A3 )
% 0.25/0.52 = ( zero_zero @ nat ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_mset_set_finite_iff
% 0.25/0.52 thf(fact_4773_count__mset__set_H,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ A,X: A] :
% 0.25/0.52 ( ( ( ( finite_finite2 @ A @ A4 )
% 0.25/0.52 & ( member @ A @ X @ A4 ) )
% 0.25/0.52 => ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
% 0.25/0.52 = ( one_one @ nat ) ) )
% 0.25/0.52 & ( ~ ( ( finite_finite2 @ A @ A4 )
% 0.25/0.52 & ( member @ A @ X @ A4 ) )
% 0.25/0.52 => ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
% 0.25/0.52 = ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % count_mset_set'
% 0.25/0.52 thf(fact_4774_length__remove1,axiom,
% 0.25/0.52 ! [A: $tType,X: A,Xs: list @ A] :
% 0.25/0.52 ( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.52 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 0.25/0.52 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
% 0.25/0.52 & ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.52 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 0.25/0.52 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % length_remove1
% 0.25/0.52 thf(fact_4775_prod__mset__delta_H,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_mult @ A )
% 0.25/0.52 => ! [Y: B,C2: A,A4: multiset @ B] :
% 0.25/0.52 ( ( comm_m9189036328036947845d_mset @ A
% 0.25/0.52 @ ( image_mset @ B @ A
% 0.25/0.52 @ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C2 @ ( one_one @ A ) )
% 0.25/0.52 @ A4 ) )
% 0.25/0.52 = ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_mset_delta'
% 0.25/0.52 thf(fact_4776_prod__mset__delta,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType] :
% 0.25/0.52 ( ( comm_monoid_mult @ A )
% 0.25/0.52 => ! [Y: B,C2: A,A4: multiset @ B] :
% 0.25/0.52 ( ( comm_m9189036328036947845d_mset @ A
% 0.25/0.52 @ ( image_mset @ B @ A
% 0.25/0.52 @ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C2 @ ( one_one @ A ) )
% 0.25/0.52 @ A4 ) )
% 0.25/0.52 = ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_mset_delta
% 0.25/0.52 thf(fact_4777_prod_OatLeastLessThan__reindex,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( ( comm_monoid_mult @ A )
% 0.25/0.52 & ( ord @ B ) )
% 0.25/0.52 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 0.25/0.52 ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.52 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.52 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod.atLeastLessThan_reindex
% 0.25/0.52 thf(fact_4778_bij__betw__id,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ A4 ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_id
% 0.25/0.52 thf(fact_4779_bij__betw__add,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( cancel_semigroup_add @ A )
% 0.25/0.52 => ! [A3: A,A4: set @ A,B6: set @ A] :
% 0.25/0.52 ( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 @ B6 )
% 0.25/0.52 = ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 )
% 0.25/0.52 = B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_add
% 0.25/0.52 thf(fact_4780_prod__mset__empty,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_mult @ A )
% 0.25/0.52 => ( ( comm_m9189036328036947845d_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.52 = ( one_one @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_mset_empty
% 0.25/0.52 thf(fact_4781_prod__mset_Ounion,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_mult @ A )
% 0.25/0.52 => ! [M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.52 ( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.52 = ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M3 ) @ ( comm_m9189036328036947845d_mset @ A @ N6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_mset.union
% 0.25/0.52 thf(fact_4782_prod__mset__Un,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( comm_monoid_mult @ A )
% 0.25/0.52 => ! [A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.52 ( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 0.25/0.52 = ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( comm_m9189036328036947845d_mset @ A @ B6 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_mset_Un
% 0.25/0.52 thf(fact_4783_bij__betw__of__nat,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( semiring_char_0 @ A )
% 0.25/0.52 => ! [N6: set @ nat,A4: set @ A] :
% 0.25/0.52 ( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A4 )
% 0.25/0.52 = ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
% 0.25/0.52 = A4 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_of_nat
% 0.25/0.52 thf(fact_4784_prod__mset_Oneutral__const,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( comm_monoid_mult @ A )
% 0.25/0.52 => ! [A4: multiset @ B] :
% 0.25/0.52 ( ( comm_m9189036328036947845d_mset @ A
% 0.25/0.52 @ ( image_mset @ B @ A
% 0.25/0.52 @ ^ [Uu: B] : ( one_one @ A )
% 0.25/0.52 @ A4 ) )
% 0.25/0.52 = ( one_one @ A ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % prod_mset.neutral_const
% 0.25/0.52 thf(fact_4785_bij__betw__imp__surj__on,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( ( image2 @ A @ B @ F @ A4 )
% 0.25/0.52 = B6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_imp_surj_on
% 0.25/0.52 thf(fact_4786_f__the__inv__into__f__bij__betw,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,X: B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( member @ B @ X @ B6 ) )
% 0.25/0.52 => ( ( F @ ( the_inv_into @ A @ B @ A4 @ F @ X ) )
% 0.25/0.52 = X ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % f_the_inv_into_f_bij_betw
% 0.25/0.52 thf(fact_4787_bij__betw__the__inv__into,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ B6 @ A4 ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_the_inv_into
% 0.25/0.52 thf(fact_4788_bij__betw__id__iff,axiom,
% 0.25/0.52 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.52 ( ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ B6 )
% 0.25/0.52 = ( A4 = B6 ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_id_iff
% 0.25/0.52 thf(fact_4789_bij__betwE,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ! [X7: A] :
% 0.25/0.52 ( ( member @ A @ X7 @ A4 )
% 0.25/0.52 => ( member @ B @ ( F @ X7 ) @ B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betwE
% 0.25/0.52 thf(fact_4790_bij__betw__inv,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B6 @ A4 ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_inv
% 0.25/0.52 thf(fact_4791_bij__betw__cong,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,A4: set @ A,F: A > B,G: A > B,A18: set @ B] :
% 0.25/0.52 ( ! [A5: A] :
% 0.25/0.52 ( ( member @ A @ A5 @ A4 )
% 0.25/0.52 => ( ( F @ A5 )
% 0.25/0.52 = ( G @ A5 ) ) )
% 0.25/0.52 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 0.25/0.52 = ( bij_betw @ A @ B @ G @ A4 @ A18 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_cong
% 0.25/0.52 thf(fact_4792_bij__betw__apply,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,A3: A] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.52 => ( member @ B @ ( F @ A3 ) @ B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_apply
% 0.25/0.52 thf(fact_4793_bij__betw__iff__bijections,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.52 ( ( bij_betw @ A @ B )
% 0.25/0.52 = ( ^ [F2: A > B,A9: set @ A,B9: set @ B] :
% 0.25/0.52 ? [G2: B > A] :
% 0.25/0.52 ( ! [X3: A] :
% 0.25/0.52 ( ( member @ A @ X3 @ A9 )
% 0.25/0.52 => ( ( member @ B @ ( F2 @ X3 ) @ B9 )
% 0.25/0.52 & ( ( G2 @ ( F2 @ X3 ) )
% 0.25/0.52 = X3 ) ) )
% 0.25/0.52 & ! [X3: B] :
% 0.25/0.52 ( ( member @ B @ X3 @ B9 )
% 0.25/0.52 => ( ( member @ A @ ( G2 @ X3 ) @ A9 )
% 0.25/0.52 & ( ( F2 @ ( G2 @ X3 ) )
% 0.25/0.52 = X3 ) ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_iff_bijections
% 0.25/0.52 thf(fact_4794_involuntory__imp__bij,axiom,
% 0.25/0.52 ! [A: $tType,F: A > A] :
% 0.25/0.52 ( ! [X2: A] :
% 0.25/0.52 ( ( F @ ( F @ X2 ) )
% 0.25/0.52 = X2 )
% 0.25/0.52 => ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % involuntory_imp_bij
% 0.25/0.52 thf(fact_4795_bij__pointE,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B,Y: B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.52 => ~ ! [X2: A] :
% 0.25/0.52 ( ( Y
% 0.25/0.52 = ( F @ X2 ) )
% 0.25/0.52 => ~ ! [X16: A] :
% 0.25/0.52 ( ( Y
% 0.25/0.52 = ( F @ X16 ) )
% 0.25/0.52 => ( X16 = X2 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_pointE
% 0.25/0.52 thf(fact_4796_bij__iff,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.52 = ( ! [X3: B] :
% 0.25/0.52 ? [Y3: A] :
% 0.25/0.52 ( ( ( F @ Y3 )
% 0.25/0.52 = X3 )
% 0.25/0.52 & ! [Z5: A] :
% 0.25/0.52 ( ( ( F @ Z5 )
% 0.25/0.52 = X3 )
% 0.25/0.52 => ( Z5 = Y3 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_iff
% 0.25/0.52 thf(fact_4797_bij__betw__imp__inj__on,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( inj_on @ A @ B @ F @ A4 ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_imp_inj_on
% 0.25/0.52 thf(fact_4798_bij__betw__trans,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,C: $tType,F: A > B,A4: set @ A,B6: set @ B,G: B > C,C5: set @ C] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.52 => ( ( bij_betw @ B @ C @ G @ B6 @ C5 )
% 0.25/0.52 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 @ C5 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_trans
% 0.25/0.52 thf(fact_4799_bij__betw__comp__iff,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,C: $tType,F: A > B,A4: set @ A,A18: set @ B,F3: B > C,A19: set @ C] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 0.25/0.52 => ( ( bij_betw @ B @ C @ F3 @ A18 @ A19 )
% 0.25/0.52 = ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A4 @ A19 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_comp_iff
% 0.25/0.52 thf(fact_4800_bij__betw__empty2,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ ( bot_bot @ ( set @ B ) ) )
% 0.25/0.52 => ( A4
% 0.25/0.52 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_empty2
% 0.25/0.52 thf(fact_4801_bij__betw__empty1,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) @ A4 )
% 0.25/0.52 => ( A4
% 0.25/0.52 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_empty1
% 0.25/0.52 thf(fact_4802_bij__betw__byWitness,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,A4: set @ A,F3: B > A,F: A > B,A18: set @ B] :
% 0.25/0.52 ( ! [X2: A] :
% 0.25/0.52 ( ( member @ A @ X2 @ A4 )
% 0.25/0.52 => ( ( F3 @ ( F @ X2 ) )
% 0.25/0.52 = X2 ) )
% 0.25/0.52 => ( ! [X2: B] :
% 0.25/0.52 ( ( member @ B @ X2 @ A18 )
% 0.25/0.52 => ( ( F @ ( F3 @ X2 ) )
% 0.25/0.52 = X2 ) )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ A18 )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A18 ) @ A4 )
% 0.25/0.52 => ( bij_betw @ A @ B @ F @ A4 @ A18 ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_byWitness
% 0.25/0.52 thf(fact_4803_bij__betw__subset,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,A18: set @ B,B6: set @ A,B14: set @ B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 0.25/0.52 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 0.25/0.52 => ( ( ( image2 @ A @ B @ F @ B6 )
% 0.25/0.52 = B14 )
% 0.25/0.52 => ( bij_betw @ A @ B @ F @ B6 @ B14 ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_subset
% 0.25/0.52 thf(fact_4804_bij__betw__imp__surj,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ A4 @ ( top_top @ ( set @ B ) ) )
% 0.25/0.52 => ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.52 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_imp_surj
% 0.25/0.52 thf(fact_4805_bij__is__surj,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.52 => ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.52 = ( top_top @ ( set @ B ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_is_surj
% 0.25/0.52 thf(fact_4806_bij__uminus,axiom,
% 0.25/0.52 ! [A: $tType] :
% 0.25/0.52 ( ( ab_group_add @ A )
% 0.25/0.52 => ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_uminus
% 0.25/0.52 thf(fact_4807_bij__comp,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C] :
% 0.25/0.52 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.52 => ( ( bij_betw @ B @ C @ G @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
% 0.25/0.52 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_comp
% 0.25/0.52 thf(fact_4808_inj__on__imp__bij__betw,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.52 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.52 => ( bij_betw @ A @ B @ F @ A4 @ ( image2 @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % inj_on_imp_bij_betw
% 0.25/0.52 thf(fact_4809_bij__betw__imageI,axiom,
% 0.25/0.52 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.52 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.52 => ( ( ( image2 @ A @ B @ F @ A4 )
% 0.25/0.52 = B6 )
% 0.25/0.52 => ( bij_betw @ A @ B @ F @ A4 @ B6 ) ) ) ).
% 0.25/0.52
% 0.25/0.52 % bij_betw_imageI
% 0.25/0.52 thf(fact_4810_bij__betw__def,axiom,
% 0.25/0.52 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( bij_betw @ A @ B )
% 0.25/0.53 = ( ^ [F2: A > B,A9: set @ A,B9: set @ B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F2 @ A9 )
% 0.25/0.53 & ( ( image2 @ A @ B @ F2 @ A9 )
% 0.25/0.53 = B9 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_betw_def
% 0.25/0.53 thf(fact_4811_bij__betw__iff__card,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.53 => ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B6 ) )
% 0.25/0.53 = ( ( finite_card @ A @ A4 )
% 0.25/0.53 = ( finite_card @ B @ B6 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_betw_iff_card
% 0.25/0.53 thf(fact_4812_finite__same__card__bij,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.53 => ( ( ( finite_card @ A @ A4 )
% 0.25/0.53 = ( finite_card @ B @ B6 ) )
% 0.25/0.53 => ? [H2: A > B] : ( bij_betw @ A @ B @ H2 @ A4 @ B6 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % finite_same_card_bij
% 0.25/0.53 thf(fact_4813_bij__is__inj,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_is_inj
% 0.25/0.53 thf(fact_4814_bij__id,axiom,
% 0.25/0.53 ! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_id
% 0.25/0.53 thf(fact_4815_coprime__common__divisor__nat,axiom,
% 0.25/0.53 ! [A3: nat,B2: nat,X: nat] :
% 0.25/0.53 ( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
% 0.25/0.53 => ( ( dvd_dvd @ nat @ X @ A3 )
% 0.25/0.53 => ( ( dvd_dvd @ nat @ X @ B2 )
% 0.25/0.53 => ( X
% 0.25/0.53 = ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % coprime_common_divisor_nat
% 0.25/0.53 thf(fact_4816_prod__mset_Oneutral,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( comm_monoid_mult @ A )
% 0.25/0.53 => ! [A4: multiset @ A] :
% 0.25/0.53 ( ! [X2: A] :
% 0.25/0.53 ( ( member @ A @ X2 @ ( set_mset @ A @ A4 ) )
% 0.25/0.53 => ( X2
% 0.25/0.53 = ( one_one @ A ) ) )
% 0.25/0.53 => ( ( comm_m9189036328036947845d_mset @ A @ A4 )
% 0.25/0.53 = ( one_one @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % prod_mset.neutral
% 0.25/0.53 thf(fact_4817_bij__betw__comp__iff2,axiom,
% 0.25/0.53 ! [C: $tType,A: $tType,B: $tType,F3: A > B,A18: set @ A,A19: set @ B,F: C > A,A4: set @ C] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F3 @ A18 @ A19 )
% 0.25/0.53 => ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F @ A4 ) @ A18 )
% 0.25/0.53 => ( ( bij_betw @ C @ A @ F @ A4 @ A18 )
% 0.25/0.53 = ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F3 @ F ) @ A4 @ A19 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_betw_comp_iff2
% 0.25/0.53 thf(fact_4818_notIn__Un__bij__betw,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,B2: A,A4: set @ A,F: A > B,A18: set @ B] :
% 0.25/0.53 ( ~ ( member @ A @ B2 @ A4 )
% 0.25/0.53 => ( ~ ( member @ B @ ( F @ B2 ) @ A18 )
% 0.25/0.53 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 0.25/0.53 => ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A18 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % notIn_Un_bij_betw
% 0.25/0.53 thf(fact_4819_notIn__Un__bij__betw3,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,B2: A,A4: set @ A,F: A > B,A18: set @ B] :
% 0.25/0.53 ( ~ ( member @ A @ B2 @ A4 )
% 0.25/0.53 => ( ~ ( member @ B @ ( F @ B2 ) @ A18 )
% 0.25/0.53 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 0.25/0.53 = ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A18 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % notIn_Un_bij_betw3
% 0.25/0.53 thf(fact_4820_bijI,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,F: A > B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 = ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bijI
% 0.25/0.53 thf(fact_4821_bij__def,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,F: A > B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 = ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 & ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 = ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_def
% 0.25/0.53 thf(fact_4822_bij__betw__partition,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,C5: set @ A,B6: set @ B,D4: set @ B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D4 ) )
% 0.25/0.53 => ( ( bij_betw @ A @ B @ F @ C5 @ D4 )
% 0.25/0.53 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ C5 )
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( ( ( inf_inf @ ( set @ B ) @ B6 @ D4 )
% 0.25/0.53 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( bij_betw @ A @ B @ F @ A4 @ B6 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_betw_partition
% 0.25/0.53 thf(fact_4823_bij__betw__combine,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.53 => ( ( bij_betw @ A @ B @ F @ C5 @ D4 )
% 0.25/0.53 => ( ( ( inf_inf @ ( set @ B ) @ B6 @ D4 )
% 0.25/0.53 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D4 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_betw_combine
% 0.25/0.53 thf(fact_4824_o__bij,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,G: B > A,F: A > B] :
% 0.25/0.53 ( ( ( comp @ B @ A @ A @ G @ F )
% 0.25/0.53 = ( id @ A ) )
% 0.25/0.53 => ( ( ( comp @ A @ B @ B @ F @ G )
% 0.25/0.53 = ( id @ B ) )
% 0.25/0.53 => ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % o_bij
% 0.25/0.53 thf(fact_4825_bij__image__Compl__eq,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 0.25/0.53 = ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_image_Compl_eq
% 0.25/0.53 thf(fact_4826_Well__order__iso__copy,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,A18: set @ B] :
% 0.25/0.53 ( ( order_well_order_on @ A @ A4 @ R )
% 0.25/0.53 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 0.25/0.53 => ? [R10: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( order_well_order_on @ B @ A18 @ R10 )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R10 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Well_order_iso_copy
% 0.25/0.53 thf(fact_4827_bij__betw__disjoint__Un,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,C5: set @ B,G: A > B,B6: set @ A,D4: set @ B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ A4 @ C5 )
% 0.25/0.53 => ( ( bij_betw @ A @ B @ G @ B6 @ D4 )
% 0.25/0.53 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( ( ( inf_inf @ ( set @ B ) @ C5 @ D4 )
% 0.25/0.53 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( bij_betw @ A @ B
% 0.25/0.53 @ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.25/0.53 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.53 @ ( sup_sup @ ( set @ B ) @ C5 @ D4 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_betw_disjoint_Un
% 0.25/0.53 thf(fact_4828_is__unit__prod__mset__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( algebraic_semidom @ A )
% 0.25/0.53 => ! [A4: multiset @ A] :
% 0.25/0.53 ( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( one_one @ A ) )
% 0.25/0.53 = ( ! [X3: A] :
% 0.25/0.53 ( ( member @ A @ X3 @ ( set_mset @ A @ A4 ) )
% 0.25/0.53 => ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % is_unit_prod_mset_iff
% 0.25/0.53 thf(fact_4829_prod__mset_Oeq__fold,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( comm_monoid_mult @ A )
% 0.25/0.53 => ( ( comm_m9189036328036947845d_mset @ A )
% 0.25/0.53 = ( fold_mset @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % prod_mset.eq_fold
% 0.25/0.53 thf(fact_4830_vimage__subset__eq,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,B6: set @ B,A4: set @ A] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A4 )
% 0.25/0.53 = ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % vimage_subset_eq
% 0.25/0.53 thf(fact_4831_ex__bij__betw__nat__finite,axiom,
% 0.25/0.53 ! [A: $tType,M3: set @ A] :
% 0.25/0.53 ( ( finite_finite2 @ A @ M3 )
% 0.25/0.53 => ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% 0.25/0.53
% 0.25/0.53 % ex_bij_betw_nat_finite
% 0.25/0.53 thf(fact_4832_ex__bij__betw__nat__finite__1,axiom,
% 0.25/0.53 ! [A: $tType,M3: set @ A] :
% 0.25/0.53 ( ( finite_finite2 @ A @ M3 )
% 0.25/0.53 => ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% 0.25/0.53
% 0.25/0.53 % ex_bij_betw_nat_finite_1
% 0.25/0.53 thf(fact_4833_coprime__diff__one__right__nat,axiom,
% 0.25/0.53 ! [N: nat] :
% 0.25/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.25/0.53 => ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % coprime_diff_one_right_nat
% 0.25/0.53 thf(fact_4834_coprime__diff__one__left__nat,axiom,
% 0.25/0.53 ! [N: nat] :
% 0.25/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 0.25/0.53 => ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% 0.25/0.53
% 0.25/0.53 % coprime_diff_one_left_nat
% 0.25/0.53 thf(fact_4835_prod_Oreindex__bij__betw__not__neutral,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType] :
% 0.25/0.53 ( ( comm_monoid_mult @ A )
% 0.25/0.53 => ! [S7: set @ B,T5: set @ C,H3: B > C,S2: set @ B,T6: set @ C,G: C > A] :
% 0.25/0.53 ( ( finite_finite2 @ B @ S7 )
% 0.25/0.53 => ( ( finite_finite2 @ C @ T5 )
% 0.25/0.53 => ( ( bij_betw @ B @ C @ H3 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 0.25/0.53 => ( ! [A5: B] :
% 0.25/0.53 ( ( member @ B @ A5 @ S7 )
% 0.25/0.53 => ( ( G @ ( H3 @ A5 ) )
% 0.25/0.53 = ( one_one @ A ) ) )
% 0.25/0.53 => ( ! [B3: C] :
% 0.25/0.53 ( ( member @ C @ B3 @ T5 )
% 0.25/0.53 => ( ( G @ B3 )
% 0.25/0.53 = ( one_one @ A ) ) )
% 0.25/0.53 => ( ( groups7121269368397514597t_prod @ B @ A
% 0.25/0.53 @ ^ [X3: B] : ( G @ ( H3 @ X3 ) )
% 0.25/0.53 @ S2 )
% 0.25/0.53 = ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % prod.reindex_bij_betw_not_neutral
% 0.25/0.53 thf(fact_4836_sum_OatLeastAtMost__reindex,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( ( comm_monoid_add @ A )
% 0.25/0.53 & ( ord @ B ) )
% 0.25/0.53 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 0.25/0.53 ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % sum.atLeastAtMost_reindex
% 0.25/0.53 thf(fact_4837_sum_OatLeastLessThan__reindex,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( ( comm_monoid_add @ A )
% 0.25/0.53 & ( ord @ B ) )
% 0.25/0.53 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 0.25/0.53 ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.53 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.53 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % sum.atLeastLessThan_reindex
% 0.25/0.53 thf(fact_4838_prod_OatLeastAtMost__reindex,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( ( comm_monoid_mult @ A )
% 0.25/0.53 & ( ord @ B ) )
% 0.25/0.53 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 0.25/0.53 ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.53 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.53 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % prod.atLeastAtMost_reindex
% 0.25/0.53 thf(fact_4839_antisymp__antisym__eq,axiom,
% 0.25/0.53 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( antisymp @ A
% 0.25/0.53 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.53 = ( antisym @ A @ R ) ) ).
% 0.25/0.53
% 0.25/0.53 % antisymp_antisym_eq
% 0.25/0.53 thf(fact_4840_convol__expand__snd_H,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H3: A > C] :
% 0.25/0.53 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 0.25/0.53 = G )
% 0.25/0.53 => ( ( H3
% 0.25/0.53 = ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 0.25/0.53 = ( ( bNF_convol @ A @ B @ C @ G @ H3 )
% 0.25/0.53 = F ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % convol_expand_snd'
% 0.25/0.53 thf(fact_4841_convol__expand__snd,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
% 0.25/0.53 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 0.25/0.53 = G )
% 0.25/0.53 => ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 0.25/0.53 = F ) ) ).
% 0.25/0.53
% 0.25/0.53 % convol_expand_snd
% 0.25/0.53 thf(fact_4842_convol__def,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.53 ( ( bNF_convol @ A @ B @ C )
% 0.25/0.53 = ( ^ [F2: A > B,G2: A > C,A7: A] : ( product_Pair @ B @ C @ ( F2 @ A7 ) @ ( G2 @ A7 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % convol_def
% 0.25/0.53 thf(fact_4843_bij__swap,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bij_swap
% 0.25/0.53 thf(fact_4844_fst__convol,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
% 0.25/0.53 ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
% 0.25/0.53 = F ) ).
% 0.25/0.53
% 0.25/0.53 % fst_convol
% 0.25/0.53 thf(fact_4845_snd__convol,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
% 0.25/0.53 ( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
% 0.25/0.53 = G ) ).
% 0.25/0.53
% 0.25/0.53 % snd_convol
% 0.25/0.53 thf(fact_4846_convol__o,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H3: A > D] :
% 0.25/0.53 ( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H3 )
% 0.25/0.53 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H3 ) @ ( comp @ D @ C @ A @ G @ H3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % convol_o
% 0.25/0.53 thf(fact_4847_ex__bij__betw__finite__nat,axiom,
% 0.25/0.53 ! [A: $tType,M3: set @ A] :
% 0.25/0.53 ( ( finite_finite2 @ A @ M3 )
% 0.25/0.53 => ? [H2: A > nat] : ( bij_betw @ A @ nat @ H2 @ M3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ex_bij_betw_finite_nat
% 0.25/0.53 thf(fact_4848_map__prod__o__convol__id,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
% 0.25/0.53 ( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X )
% 0.25/0.53 = ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_o_convol_id
% 0.25/0.53 thf(fact_4849_Heap__lub__empty,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
% 0.25/0.53 = ( heap_Time_Heap2 @ A
% 0.25/0.53 @ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Heap_lub_empty
% 0.25/0.53 thf(fact_4850_and_Osemilattice__neutr__axioms,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.53 => ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % and.semilattice_neutr_axioms
% 0.25/0.53 thf(fact_4851_map__prod__ident,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( product_map_prod @ A @ A @ B @ B
% 0.25/0.53 @ ^ [X3: A] : X3
% 0.25/0.53 @ ^ [Y3: B] : Y3 )
% 0.25/0.53 = ( ^ [Z5: product_prod @ A @ B] : Z5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_ident
% 0.25/0.53 thf(fact_4852_map__prod__simp,axiom,
% 0.25/0.53 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A3: C,B2: D] :
% 0.25/0.53 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A3 @ B2 ) )
% 0.25/0.53 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G @ B2 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_simp
% 0.25/0.53 thf(fact_4853_fst__map__prod,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 0.25/0.53 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 0.25/0.53 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % fst_map_prod
% 0.25/0.53 thf(fact_4854_snd__map__prod,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 0.25/0.53 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 0.25/0.53 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % snd_map_prod
% 0.25/0.53 thf(fact_4855_map__prod__imageI,axiom,
% 0.25/0.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R5: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R5 )
% 0.25/0.53 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_imageI
% 0.25/0.53 thf(fact_4856_fst__comp__map__prod,axiom,
% 0.25/0.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
% 0.25/0.53 ( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
% 0.25/0.53 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % fst_comp_map_prod
% 0.25/0.53 thf(fact_4857_snd__comp__map__prod,axiom,
% 0.25/0.53 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
% 0.25/0.53 ( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
% 0.25/0.53 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % snd_comp_map_prod
% 0.25/0.53 thf(fact_4858_prod_Omap__comp,axiom,
% 0.25/0.53 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F5,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
% 0.25/0.53 ( ( product_map_prod @ C @ E @ D @ F5 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 0.25/0.53 = ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F5 @ B @ G22 @ F22 ) @ V ) ) ).
% 0.25/0.53
% 0.25/0.53 % prod.map_comp
% 0.25/0.53 thf(fact_4859_map__prod_Ocompositionality,axiom,
% 0.25/0.53 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D,Prod: product_prod @ A @ B] :
% 0.25/0.53 ( ( product_map_prod @ C @ E @ D @ F5 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H3 @ I @ Prod ) )
% 0.25/0.53 = ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Prod ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod.compositionality
% 0.25/0.53 thf(fact_4860_map__prod_Oidentity,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( product_map_prod @ A @ A @ B @ B
% 0.25/0.53 @ ^ [X3: A] : X3
% 0.25/0.53 @ ^ [X3: B] : X3 )
% 0.25/0.53 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod.identity
% 0.25/0.53 thf(fact_4861_prod__fun__imageE,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G: D > B,R5: set @ ( product_prod @ C @ D )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R5 ) )
% 0.25/0.53 => ~ ! [X2: C,Y4: D] :
% 0.25/0.53 ( ( C2
% 0.25/0.53 = ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ Y4 ) ) )
% 0.25/0.53 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y4 ) @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % prod_fun_imageE
% 0.25/0.53 thf(fact_4862_apfst__def,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType] :
% 0.25/0.53 ( ( product_apfst @ A @ C @ B )
% 0.25/0.53 = ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % apfst_def
% 0.25/0.53 thf(fact_4863_apsnd__def,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType] :
% 0.25/0.53 ( ( product_apsnd @ B @ C @ A )
% 0.25/0.53 = ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % apsnd_def
% 0.25/0.53 thf(fact_4864_map__prod__def,axiom,
% 0.25/0.53 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 0.25/0.53 ( ( product_map_prod @ A @ C @ B @ D )
% 0.25/0.53 = ( ^ [F2: A > C,G2: B > D] :
% 0.25/0.53 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 0.25/0.53 @ ^ [X3: A,Y3: B] : ( product_Pair @ C @ D @ ( F2 @ X3 ) @ ( G2 @ Y3 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_def
% 0.25/0.53 thf(fact_4865_map__prod__compose,axiom,
% 0.25/0.53 ! [D: $tType,C: $tType,A: $tType,E: $tType,F5: $tType,B: $tType,F1: E > C,F22: A > E,G1: F5 > D,G22: B > F5] :
% 0.25/0.53 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F5 @ D @ B @ G1 @ G22 ) )
% 0.25/0.53 = ( comp @ ( product_prod @ E @ F5 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F5 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F5 @ F22 @ G22 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_compose
% 0.25/0.53 thf(fact_4866_map__prod_Ocomp,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,E: $tType,F5: $tType,D: $tType,B: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D] :
% 0.25/0.53 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F5 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F5 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H3 @ I ) )
% 0.25/0.53 = ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod.comp
% 0.25/0.53 thf(fact_4867_map__prod__surj__on,axiom,
% 0.25/0.53 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A4: set @ B,A18: set @ A,G: D > C,B6: set @ D,B14: set @ C] :
% 0.25/0.53 ( ( ( image2 @ B @ A @ F @ A4 )
% 0.25/0.53 = A18 )
% 0.25/0.53 => ( ( ( image2 @ D @ C @ G @ B6 )
% 0.25/0.53 = B14 )
% 0.25/0.53 => ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
% 0.25/0.53 @ ( product_Sigma @ B @ D @ A4
% 0.25/0.53 @ ^ [Uu: B] : B6 ) )
% 0.25/0.53 = ( product_Sigma @ A @ C @ A18
% 0.25/0.53 @ ^ [Uu: A] : B14 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_surj_on
% 0.25/0.53 thf(fact_4868_map__prod__o__convol,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H22: E > C,F: A > D,G: A > E] :
% 0.25/0.53 ( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H22 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
% 0.25/0.53 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H22 @ G ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_o_convol
% 0.25/0.53 thf(fact_4869_map__prod__inj__on,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: C > D,B6: set @ C] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.53 => ( ( inj_on @ C @ D @ G @ B6 )
% 0.25/0.53 => ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G )
% 0.25/0.53 @ ( product_Sigma @ A @ C @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_inj_on
% 0.25/0.53 thf(fact_4870_map__prod__surj,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 0.25/0.53 ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 = ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( ( ( image2 @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 0.25/0.53 = ( top_top @ ( set @ D ) ) )
% 0.25/0.53 => ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 0.25/0.53 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % map_prod_surj
% 0.25/0.53 thf(fact_4871_Zorns__po__lemma,axiom,
% 0.25/0.53 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 => ( ! [C8: set @ A] :
% 0.25/0.53 ( ( member @ ( set @ A ) @ C8 @ ( chains @ A @ R ) )
% 0.25/0.53 => ? [X7: A] :
% 0.25/0.53 ( ( member @ A @ X7 @ ( field2 @ A @ R ) )
% 0.25/0.53 & ! [Xa3: A] :
% 0.25/0.53 ( ( member @ A @ Xa3 @ C8 )
% 0.25/0.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X7 ) @ R ) ) ) )
% 0.25/0.53 => ? [X2: A] :
% 0.25/0.53 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.53 & ! [Xa2: A] :
% 0.25/0.53 ( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R )
% 0.25/0.53 => ( Xa2 = X2 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Zorns_po_lemma
% 0.25/0.53 thf(fact_4872_assert__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( heap_Time_assert @ A )
% 0.25/0.53 = ( ^ [P4: A > $o,X3: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ X3 ) @ ( heap_Time_return @ A @ X3 ) @ ( heap_Time_raise @ A @ ( cons @ char @ ( char2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ $false ) @ ( nil @ char ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % assert_def
% 0.25/0.53 thf(fact_4873_power__int__add__1,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [X: A,M2: int] :
% 0.25/0.53 ( ( ( X
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 | ( M2
% 0.25/0.53 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.25/0.53 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
% 0.25/0.53 = ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_add_1
% 0.25/0.53 thf(fact_4874_power__int__1__left,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [N: int] :
% 0.25/0.53 ( ( power_int @ A @ ( one_one @ A ) @ N )
% 0.25/0.53 = ( one_one @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_1_left
% 0.25/0.53 thf(fact_4875_power__int__0__right,axiom,
% 0.25/0.53 ! [B: $tType] :
% 0.25/0.53 ( ( ( inverse @ B )
% 0.25/0.53 & ( power @ B ) )
% 0.25/0.53 => ! [X: B] :
% 0.25/0.53 ( ( power_int @ B @ X @ ( zero_zero @ int ) )
% 0.25/0.53 = ( one_one @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_0_right
% 0.25/0.53 thf(fact_4876_power__int__minus__one__mult__self,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [M2: int] :
% 0.25/0.53 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_minus_one_mult_self
% 0.25/0.53 thf(fact_4877_power__int__minus__one__mult__self_H,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [M2: int,B2: A] :
% 0.25/0.53 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B2 ) )
% 0.25/0.53 = B2 ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_minus_one_mult_self'
% 0.25/0.53 thf(fact_4878_power__int__add__numeral,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [X: A,M2: num,N: num] :
% 0.25/0.53 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
% 0.25/0.53 = ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_add_numeral
% 0.25/0.53 thf(fact_4879_power__int__add__numeral2,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [X: A,M2: num,N: num,B2: A] :
% 0.25/0.53 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
% 0.25/0.53 = ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_add_numeral2
% 0.25/0.53 thf(fact_4880_power__int__one__over,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [X: A,N: int] :
% 0.25/0.53 ( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
% 0.25/0.53 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_one_over
% 0.25/0.53 thf(fact_4881_power__int__0__left__If,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [M2: int] :
% 0.25/0.53 ( ( ( M2
% 0.25/0.53 = ( zero_zero @ int ) )
% 0.25/0.53 => ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.53 = ( one_one @ A ) ) )
% 0.25/0.53 & ( ( M2
% 0.25/0.53 != ( zero_zero @ int ) )
% 0.25/0.53 => ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
% 0.25/0.53 = ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_0_left_If
% 0.25/0.53 thf(fact_4882_power__int__increasing,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [N: int,N6: int,A3: A] :
% 0.25/0.53 ( ( ord_less_eq @ int @ N @ N6 )
% 0.25/0.53 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.53 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_increasing
% 0.25/0.53 thf(fact_4883_power__int__strict__increasing,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [N: int,N6: int,A3: A] :
% 0.25/0.53 ( ( ord_less @ int @ N @ N6 )
% 0.25/0.53 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.53 => ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_strict_increasing
% 0.25/0.53 thf(fact_4884_power__int__minus__one__minus,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [N: int] :
% 0.25/0.53 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
% 0.25/0.53 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_minus_one_minus
% 0.25/0.53 thf(fact_4885_power__int__minus__one__diff__commute,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [A3: int,B2: int] :
% 0.25/0.53 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
% 0.25/0.53 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_minus_one_diff_commute
% 0.25/0.53 thf(fact_4886_power__int__strict__decreasing,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [N: int,N6: int,A3: A] :
% 0.25/0.53 ( ( ord_less @ int @ N @ N6 )
% 0.25/0.53 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 0.25/0.53 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.53 => ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_strict_decreasing
% 0.25/0.53 thf(fact_4887_one__le__power__int,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [X: A,N: int] :
% 0.25/0.53 ( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
% 0.25/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.25/0.53 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % one_le_power_int
% 0.25/0.53 thf(fact_4888_one__less__power__int,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [A3: A,N: int] :
% 0.25/0.53 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.53 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 0.25/0.53 => ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % one_less_power_int
% 0.25/0.53 thf(fact_4889_power__int__add,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [X: A,M2: int,N: int] :
% 0.25/0.53 ( ( ( X
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 | ( ( plus_plus @ int @ M2 @ N )
% 0.25/0.53 != ( zero_zero @ int ) ) )
% 0.25/0.53 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
% 0.25/0.53 = ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_add
% 0.25/0.53 thf(fact_4890_power__int__minus__left__distrib,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.53 ( ( ( division_ring @ A )
% 0.25/0.53 & ( one @ B )
% 0.25/0.53 & ( uminus @ B ) )
% 0.25/0.53 => ! [X: C,A3: A,N: int] :
% 0.25/0.53 ( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
% 0.25/0.53 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.25/0.53 = ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_minus_left_distrib
% 0.25/0.53 thf(fact_4891_power__int__decreasing,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [N: int,N6: int,A3: A] :
% 0.25/0.53 ( ( ord_less_eq @ int @ N @ N6 )
% 0.25/0.53 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 0.25/0.53 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.53 => ( ( ( A3
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 | ( N6
% 0.25/0.53 != ( zero_zero @ int ) )
% 0.25/0.53 | ( N
% 0.25/0.53 = ( zero_zero @ int ) ) )
% 0.25/0.53 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_decreasing
% 0.25/0.53 thf(fact_4892_power__int__le__one,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [X: A,N: int] :
% 0.25/0.53 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 0.25/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.25/0.53 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 0.25/0.53 => ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_le_one
% 0.25/0.53 thf(fact_4893_power__int__le__imp__le__exp,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [X: A,M2: int,N: int] :
% 0.25/0.53 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 0.25/0.53 => ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
% 0.25/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.25/0.53 => ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_le_imp_le_exp
% 0.25/0.53 thf(fact_4894_power__int__le__imp__less__exp,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( linordered_field @ A )
% 0.25/0.53 => ! [X: A,M2: int,N: int] :
% 0.25/0.53 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 0.25/0.53 => ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
% 0.25/0.53 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 0.25/0.53 => ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_le_imp_less_exp
% 0.25/0.53 thf(fact_4895_power__int__add__1_H,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( division_ring @ A )
% 0.25/0.53 => ! [X: A,M2: int] :
% 0.25/0.53 ( ( ( X
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 | ( M2
% 0.25/0.53 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 0.25/0.53 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
% 0.25/0.53 = ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % power_int_add_1'
% 0.25/0.53 thf(fact_4896_integer__of__char__code,axiom,
% 0.25/0.53 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 0.25/0.53 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 0.25/0.53 = ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % integer_of_char_code
% 0.25/0.53 thf(fact_4897_char__of__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_un5681908812861735899ations @ A )
% 0.25/0.53 => ( ( unique5772411509450598832har_of @ A )
% 0.25/0.53 = ( ^ [N5: A] :
% 0.25/0.53 ( char2
% 0.25/0.53 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( one_one @ nat ) )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % char_of_def
% 0.25/0.53 thf(fact_4898_numeral__unfold__funpow,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_1 @ A )
% 0.25/0.53 => ( ( numeral_numeral @ A )
% 0.25/0.53 = ( ^ [K4: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K4 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % numeral_unfold_funpow
% 0.25/0.53 thf(fact_4899_relpowp__1,axiom,
% 0.25/0.53 ! [A: $tType,P: A > A > $o] :
% 0.25/0.53 ( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
% 0.25/0.53 = P ) ).
% 0.25/0.53
% 0.25/0.53 % relpowp_1
% 0.25/0.53 thf(fact_4900_Suc__funpow,axiom,
% 0.25/0.53 ! [N: nat] :
% 0.25/0.53 ( ( compow @ ( nat > nat ) @ N @ suc )
% 0.25/0.53 = ( plus_plus @ nat @ N ) ) ).
% 0.25/0.53
% 0.25/0.53 % Suc_funpow
% 0.25/0.53 thf(fact_4901_comp__funpow,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,N: nat,F: A > A] :
% 0.25/0.53 ( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
% 0.25/0.53 = ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % comp_funpow
% 0.25/0.53 thf(fact_4902_funpow_Osimps_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,F: A > A] :
% 0.25/0.53 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 0.25/0.53 = ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % funpow.simps(2)
% 0.25/0.53 thf(fact_4903_funpow__Suc__right,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,F: A > A] :
% 0.25/0.53 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 0.25/0.53 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ F ) ) ).
% 0.25/0.53
% 0.25/0.53 % funpow_Suc_right
% 0.25/0.53 thf(fact_4904_funpow__add,axiom,
% 0.25/0.53 ! [A: $tType,M2: nat,N: nat,F: A > A] :
% 0.25/0.53 ( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N ) @ F )
% 0.25/0.53 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F ) @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % funpow_add
% 0.25/0.53 thf(fact_4905_of__nat__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_1 @ A )
% 0.25/0.53 => ( ( semiring_1_of_nat @ A )
% 0.25/0.53 = ( ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % of_nat_def
% 0.25/0.53 thf(fact_4906_numeral__add__unfold__funpow,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_numeral @ A )
% 0.25/0.53 => ! [K: num,A3: A] :
% 0.25/0.53 ( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
% 0.25/0.53 = ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % numeral_add_unfold_funpow
% 0.25/0.53 thf(fact_4907_inj__on__funpow__least,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,F: A > A,S: A] :
% 0.25/0.53 ( ( ( compow @ ( A > A ) @ N @ F @ S )
% 0.25/0.53 = S )
% 0.25/0.53 => ( ! [M5: nat] :
% 0.25/0.53 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M5 )
% 0.25/0.53 => ( ( ord_less @ nat @ M5 @ N )
% 0.25/0.53 => ( ( compow @ ( A > A ) @ M5 @ F @ S )
% 0.25/0.53 != S ) ) )
% 0.25/0.53 => ( inj_on @ nat @ A
% 0.25/0.53 @ ^ [K4: nat] : ( compow @ ( A > A ) @ K4 @ F @ S )
% 0.25/0.53 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inj_on_funpow_least
% 0.25/0.53 thf(fact_4908_card__Un__disjnt,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.53 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( finite_finite2 @ A @ B6 )
% 0.25/0.53 => ( ( disjnt @ A @ A4 @ B6 )
% 0.25/0.53 => ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.53 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_Un_disjnt
% 0.25/0.53 thf(fact_4909_plus__multiset__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( plus_plus @ ( multiset @ A ) )
% 0.25/0.53 = ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
% 0.25/0.53 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % plus_multiset_def
% 0.25/0.53 thf(fact_4910_mset__subset__add__iff2,axiom,
% 0.25/0.53 ! [A: $tType,I: nat,J: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 0.25/0.53 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.53 => ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 0.25/0.53 = ( subset_mset @ A @ M2 @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mset_subset_add_iff2
% 0.25/0.53 thf(fact_4911_repeat__mset__distrib2,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,A4: multiset @ A,B6: multiset @ A] :
% 0.25/0.53 ( ( repeat_mset @ A @ N @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ N @ A4 ) @ ( repeat_mset @ A @ N @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % repeat_mset_distrib2
% 0.25/0.53 thf(fact_4912_repeat__mset__Suc,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,M3: multiset @ A] :
% 0.25/0.53 ( ( repeat_mset @ A @ ( suc @ N ) @ M3 )
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ M3 @ ( repeat_mset @ A @ N @ M3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % repeat_mset_Suc
% 0.25/0.53 thf(fact_4913_disjnt__Times1__iff,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,C5: set @ A,A4: set @ B,B6: set @ B] :
% 0.25/0.53 ( ( disjnt @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ C5
% 0.25/0.53 @ ^ [Uu: A] : A4 )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ C5
% 0.25/0.53 @ ^ [Uu: A] : B6 ) )
% 0.25/0.53 = ( ( C5
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 | ( disjnt @ B @ A4 @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % disjnt_Times1_iff
% 0.25/0.53 thf(fact_4914_disjnt__Times2__iff,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,C5: set @ B,B6: set @ A] :
% 0.25/0.53 ( ( disjnt @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : C5 )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ B6
% 0.25/0.53 @ ^ [Uu: A] : C5 ) )
% 0.25/0.53 = ( ( C5
% 0.25/0.53 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 | ( disjnt @ A @ A4 @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % disjnt_Times2_iff
% 0.25/0.53 thf(fact_4915_disjnt__Sigma__iff,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,C5: A > ( set @ B ),B6: set @ A] :
% 0.25/0.53 ( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ C5 ) @ ( product_Sigma @ A @ B @ B6 @ C5 ) )
% 0.25/0.53 = ( ! [X3: A] :
% 0.25/0.53 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
% 0.25/0.53 => ( ( C5 @ X3 )
% 0.25/0.53 = ( bot_bot @ ( set @ B ) ) ) )
% 0.25/0.53 | ( disjnt @ A @ A4 @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % disjnt_Sigma_iff
% 0.25/0.53 thf(fact_4916_left__add__mult__distrib__mset,axiom,
% 0.25/0.53 ! [A: $tType,I: nat,U: multiset @ A,J: nat,K: multiset @ A] :
% 0.25/0.53 ( ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ K ) )
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% 0.25/0.53
% 0.25/0.53 % left_add_mult_distrib_mset
% 0.25/0.53 thf(fact_4917_repeat__mset__distrib,axiom,
% 0.25/0.53 ! [A: $tType,M2: nat,N: nat,A4: multiset @ A] :
% 0.25/0.53 ( ( repeat_mset @ A @ ( plus_plus @ nat @ M2 @ N ) @ A4 )
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ M2 @ A4 ) @ ( repeat_mset @ A @ N @ A4 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % repeat_mset_distrib
% 0.25/0.53 thf(fact_4918_mset__subseteq__add__iff1,axiom,
% 0.25/0.53 ! [A: $tType,J: nat,I: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 0.25/0.53 ( ( ord_less_eq @ nat @ J @ I )
% 0.25/0.53 => ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 0.25/0.53 = ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mset_subseteq_add_iff1
% 0.25/0.53 thf(fact_4919_mset__subseteq__add__iff2,axiom,
% 0.25/0.53 ! [A: $tType,I: nat,J: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 0.25/0.53 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.53 => ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 0.25/0.53 = ( subseteq_mset @ A @ M2 @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mset_subseteq_add_iff2
% 0.25/0.53 thf(fact_4920_mset__subset__add__iff1,axiom,
% 0.25/0.53 ! [A: $tType,J: nat,I: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 0.25/0.53 ( ( ord_less_eq @ nat @ J @ I )
% 0.25/0.53 => ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 0.25/0.53 = ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mset_subset_add_iff1
% 0.25/0.53 thf(fact_4921_plus__multiset_Oabs__eq,axiom,
% 0.25/0.53 ! [A: $tType,Xa: A > nat,X: A > nat] :
% 0.25/0.53 ( ( bNF_eq_onp @ ( A > nat )
% 0.25/0.53 @ ^ [F2: A > nat] :
% 0.25/0.53 ( finite_finite2 @ A
% 0.25/0.53 @ ( collect @ A
% 0.25/0.53 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
% 0.25/0.53 @ Xa
% 0.25/0.53 @ Xa )
% 0.25/0.53 => ( ( bNF_eq_onp @ ( A > nat )
% 0.25/0.53 @ ^ [F2: A > nat] :
% 0.25/0.53 ( finite_finite2 @ A
% 0.25/0.53 @ ( collect @ A
% 0.25/0.53 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
% 0.25/0.53 @ X
% 0.25/0.53 @ X )
% 0.25/0.53 => ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
% 0.25/0.53 = ( abs_multiset @ A
% 0.25/0.53 @ ^ [A7: A] : ( plus_plus @ nat @ ( Xa @ A7 ) @ ( X @ A7 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % plus_multiset.abs_eq
% 0.25/0.53 thf(fact_4922_repeat__mset__distrib__add__mset,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,A3: A,A4: multiset @ A] :
% 0.25/0.53 ( ( repeat_mset @ A @ N @ ( add_mset @ A @ A3 @ A4 ) )
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ ( replicate_mset @ A @ N @ A3 ) @ ( repeat_mset @ A @ N @ A4 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % repeat_mset_distrib_add_mset
% 0.25/0.53 thf(fact_4923_o__prs_I2_J,axiom,
% 0.25/0.53 ! [F5: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H8: $tType,G3: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F5,Rep32: F5 > E] :
% 0.25/0.53 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 0.25/0.53 => ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
% 0.25/0.53 => ( ( quotient3 @ E @ F5 @ R32 @ Abs32 @ Rep32 )
% 0.25/0.53 => ( ( map_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( A > G3 ) > A > H8 ) @ ( ( B > G3 ) > B > H8 ) @ ( id @ ( G3 > H8 ) ) @ ( map_fun @ ( B > G3 ) @ ( A > G3 ) @ ( A > H8 ) @ ( B > H8 ) @ ( map_fun @ A @ B @ G3 @ G3 @ Abs1 @ ( id @ G3 ) ) @ ( map_fun @ B @ A @ H8 @ H8 @ Rep1 @ ( id @ H8 ) ) ) @ ( comp @ G3 @ H8 @ A ) )
% 0.25/0.53 = ( comp @ G3 @ H8 @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % o_prs(2)
% 0.25/0.53 thf(fact_4924_abs__o__rep,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A] :
% 0.25/0.53 ( ( quotient3 @ A @ B @ R5 @ Abs @ Rep )
% 0.25/0.53 => ( ( comp @ A @ B @ B @ Abs @ Rep )
% 0.25/0.53 = ( id @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % abs_o_rep
% 0.25/0.53 thf(fact_4925_o__prs_I1_J,axiom,
% 0.25/0.53 ! [C: $tType,E: $tType,A: $tType,B: $tType,F5: $tType,D: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F5,Rep32: F5 > E] :
% 0.25/0.53 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 0.25/0.53 => ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
% 0.25/0.53 => ( ( quotient3 @ E @ F5 @ R32 @ Abs32 @ Rep32 )
% 0.25/0.53 => ( ( map_fun @ ( D > F5 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F5 ) @ ( map_fun @ C @ D @ F5 @ E @ Abs22 @ Rep32 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F5 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) @ ( map_fun @ B @ A @ E @ F5 @ Rep1 @ Abs32 ) ) @ ( comp @ C @ E @ A ) )
% 0.25/0.53 = ( comp @ D @ F5 @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % o_prs(1)
% 0.25/0.53 thf(fact_4926_plus__multiset_Orsp,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
% 0.25/0.53 @ ( bNF_eq_onp @ ( A > nat )
% 0.25/0.53 @ ^ [F2: A > nat] :
% 0.25/0.53 ( finite_finite2 @ A
% 0.25/0.53 @ ( collect @ A
% 0.25/0.53 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) )
% 0.25/0.53 @ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
% 0.25/0.53 @ ( bNF_eq_onp @ ( A > nat )
% 0.25/0.53 @ ^ [F2: A > nat] :
% 0.25/0.53 ( finite_finite2 @ A
% 0.25/0.53 @ ( collect @ A
% 0.25/0.53 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) )
% 0.25/0.53 @ ( bNF_eq_onp @ ( A > nat )
% 0.25/0.53 @ ^ [F2: A > nat] :
% 0.25/0.53 ( finite_finite2 @ A
% 0.25/0.53 @ ( collect @ A
% 0.25/0.53 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ) )
% 0.25/0.53 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) )
% 0.25/0.53 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % plus_multiset.rsp
% 0.25/0.53 thf(fact_4927_or__not__num__neg_Opelims,axiom,
% 0.25/0.53 ! [X: num,Xa: num,Y: num] :
% 0.25/0.53 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 0.25/0.53 = Y )
% 0.25/0.53 => ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 0.25/0.53 => ( ( ( X = one2 )
% 0.25/0.53 => ( ( Xa = one2 )
% 0.25/0.53 => ( ( Y = one2 )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 0.25/0.53 => ( ( ( X = one2 )
% 0.25/0.53 => ! [M5: num] :
% 0.25/0.53 ( ( Xa
% 0.25/0.53 = ( bit0 @ M5 ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bit1 @ M5 ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M5 ) ) ) ) ) )
% 0.25/0.53 => ( ( ( X = one2 )
% 0.25/0.53 => ! [M5: num] :
% 0.25/0.53 ( ( Xa
% 0.25/0.53 = ( bit1 @ M5 ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bit1 @ M5 ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M5 ) ) ) ) ) )
% 0.25/0.53 => ( ! [N3: num] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( bit0 @ N3 ) )
% 0.25/0.53 => ( ( Xa = one2 )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bit0 @ one2 ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ one2 ) ) ) ) )
% 0.25/0.53 => ( ! [N3: num] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( bit0 @ N3 ) )
% 0.25/0.53 => ! [M5: num] :
% 0.25/0.53 ( ( Xa
% 0.25/0.53 = ( bit0 @ M5 ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 0.25/0.53 => ( ! [N3: num] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( bit0 @ N3 ) )
% 0.25/0.53 => ! [M5: num] :
% 0.25/0.53 ( ( Xa
% 0.25/0.53 = ( bit1 @ M5 ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) )
% 0.25/0.53 => ( ! [N3: num] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( bit1 @ N3 ) )
% 0.25/0.53 => ( ( Xa = one2 )
% 0.25/0.53 => ( ( Y = one2 )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ one2 ) ) ) ) )
% 0.25/0.53 => ( ! [N3: num] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( bit1 @ N3 ) )
% 0.25/0.53 => ! [M5: num] :
% 0.25/0.53 ( ( Xa
% 0.25/0.53 = ( bit0 @ M5 ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 0.25/0.53 => ~ ! [N3: num] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( bit1 @ N3 ) )
% 0.25/0.53 => ! [M5: num] :
% 0.25/0.53 ( ( Xa
% 0.25/0.53 = ( bit1 @ M5 ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 0.25/0.53 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % or_not_num_neg.pelims
% 0.25/0.53 thf(fact_4928_cr__natural__def,axiom,
% 0.25/0.53 ( code_cr_natural
% 0.25/0.53 = ( ^ [X3: nat,Y3: code_natural] :
% 0.25/0.53 ( X3
% 0.25/0.53 = ( code_nat_of_natural @ Y3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % cr_natural_def
% 0.25/0.53 thf(fact_4929_inv__o__cancel,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 0.25/0.53 = ( id @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inv_o_cancel
% 0.25/0.53 thf(fact_4930_o__inv__o__cancel,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,F: A > B,G: A > C] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) @ F )
% 0.25/0.53 = G ) ) ).
% 0.25/0.53
% 0.25/0.53 % o_inv_o_cancel
% 0.25/0.53 thf(fact_4931_inj__map__inv__f,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,L: list @ A] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( map @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( map @ A @ B @ F @ L ) )
% 0.25/0.53 = L ) ) ).
% 0.25/0.53
% 0.25/0.53 % inj_map_inv_f
% 0.25/0.53 thf(fact_4932_inv__into__comp,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A4: set @ C,X: B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) )
% 0.25/0.53 => ( ( inj_on @ C @ A @ G @ A4 )
% 0.25/0.53 => ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) ) )
% 0.25/0.53 => ( ( hilbert_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F @ G ) @ X )
% 0.25/0.53 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A4 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A4 ) @ F ) @ X ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inv_into_comp
% 0.25/0.53 thf(fact_4933_inv__unique__comp,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: B > A,G: A > B] :
% 0.25/0.53 ( ( ( comp @ B @ A @ A @ F @ G )
% 0.25/0.53 = ( id @ A ) )
% 0.25/0.53 => ( ( ( comp @ A @ B @ B @ G @ F )
% 0.25/0.53 = ( id @ B ) )
% 0.25/0.53 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 0.25/0.53 = G ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inv_unique_comp
% 0.25/0.53 thf(fact_4934_o__inv__distrib,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > A] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( ( bij_betw @ C @ A @ G @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F @ G ) )
% 0.25/0.53 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % o_inv_distrib
% 0.25/0.53 thf(fact_4935_surj__iff,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: B > A] :
% 0.25/0.53 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 = ( top_top @ ( set @ A ) ) )
% 0.25/0.53 = ( ( comp @ B @ A @ A @ F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) )
% 0.25/0.53 = ( id @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % surj_iff
% 0.25/0.53 thf(fact_4936_inj__iff,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 = ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 0.25/0.53 = ( id @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inj_iff
% 0.25/0.53 thf(fact_4937_fn__o__inv__fn__is__id,axiom,
% 0.25/0.53 ! [A: $tType,F: A > A,N: nat] :
% 0.25/0.53 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
% 0.25/0.53 = ( ^ [X3: A] : X3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % fn_o_inv_fn_is_id
% 0.25/0.53 thf(fact_4938_inv__fn__o__fn__is__id,axiom,
% 0.25/0.53 ! [A: $tType,F: A > A,N: nat] :
% 0.25/0.53 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N @ F ) )
% 0.25/0.53 = ( ^ [X3: A] : X3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inv_fn_o_fn_is_id
% 0.25/0.53 thf(fact_4939_bijection_Oinv__comp__right,axiom,
% 0.25/0.53 ! [A: $tType,F: A > A] :
% 0.25/0.53 ( ( hilbert_bijection @ A @ F )
% 0.25/0.53 => ( ( comp @ A @ A @ A @ F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
% 0.25/0.53 = ( id @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bijection.inv_comp_right
% 0.25/0.53 thf(fact_4940_bijection_Oinv__comp__left,axiom,
% 0.25/0.53 ! [A: $tType,F: A > A] :
% 0.25/0.53 ( ( hilbert_bijection @ A @ F )
% 0.25/0.53 => ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 0.25/0.53 = ( id @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bijection.inv_comp_left
% 0.25/0.53 thf(fact_4941_finite__reachable__restrictedI,axiom,
% 0.25/0.53 ! [A: $tType,Q: set @ A,I4: set @ A,E4: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( finite_finite2 @ A @ Q )
% 0.25/0.53 => ( ( ord_less_eq @ ( set @ A ) @ I4 @ Q )
% 0.25/0.53 => ( ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ E4 ) @ Q )
% 0.25/0.53 => ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I4 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % finite_reachable_restrictedI
% 0.25/0.53 thf(fact_4942_Range__insert,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A3: B,B2: A,R: set @ ( product_prod @ B @ A )] :
% 0.25/0.53 ( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) )
% 0.25/0.53 = ( insert2 @ A @ B2 @ ( range2 @ B @ A @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range_insert
% 0.25/0.53 thf(fact_4943_Range__iff,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ B @ A )] :
% 0.25/0.53 ( ( member @ A @ A3 @ ( range2 @ B @ A @ R ) )
% 0.25/0.53 = ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A3 ) @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range_iff
% 0.25/0.53 thf(fact_4944_RangeE,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
% 0.25/0.53 ( ( member @ A @ B2 @ ( range2 @ B @ A @ R ) )
% 0.25/0.53 => ~ ! [A5: B] :
% 0.25/0.53 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B2 ) @ R ) ) ).
% 0.25/0.53
% 0.25/0.53 % RangeE
% 0.25/0.53 thf(fact_4945_Range_Ointros,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 0.25/0.53 => ( member @ B @ B2 @ ( range2 @ A @ B @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range.intros
% 0.25/0.53 thf(fact_4946_Range_Osimps,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
% 0.25/0.53 = ( ? [A7: A,B5: B] :
% 0.25/0.53 ( ( A3 = B5 )
% 0.25/0.53 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range.simps
% 0.25/0.53 thf(fact_4947_Range_Ocases,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
% 0.25/0.53 => ~ ! [A5: A] :
% 0.25/0.53 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range.cases
% 0.25/0.53 thf(fact_4948_trancl__Image__in__Range,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ R5 ) @ V6 ) @ ( range2 @ A @ A @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % trancl_Image_in_Range
% 0.25/0.53 thf(fact_4949_Range__rel__restrict,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R5 ) @ A4 ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range_rel_restrict
% 0.25/0.53 thf(fact_4950_dom__ran__disj__comp,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) )
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( ( relcomp @ A @ A @ A @ R5 @ R5 )
% 0.25/0.53 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % dom_ran_disj_comp
% 0.25/0.53 thf(fact_4951_wf__max,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( wf @ A @ ( converse @ A @ A @ R5 ) )
% 0.25/0.53 => ( ( R5
% 0.25/0.53 != ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.25/0.53 => ~ ! [M5: A] :
% 0.25/0.53 ~ ( member @ A @ M5 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R5 ) @ ( domain @ A @ A @ R5 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % wf_max
% 0.25/0.53 thf(fact_4952_Rangep__Range__eq,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( rangep @ A @ B
% 0.25/0.53 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.25/0.53 = ( ^ [X3: B] : ( member @ B @ X3 @ ( range2 @ A @ B @ R ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Rangep_Range_eq
% 0.25/0.53 thf(fact_4953_Domain__insert,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R ) )
% 0.25/0.53 = ( insert2 @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain_insert
% 0.25/0.53 thf(fact_4954_Domain_Ocases,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 0.25/0.53 => ~ ! [B3: B] :
% 0.25/0.53 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain.cases
% 0.25/0.53 thf(fact_4955_Domain_Osimps,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 0.25/0.53 = ( ? [A7: A,B5: B] :
% 0.25/0.53 ( ( A3 = A7 )
% 0.25/0.53 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain.simps
% 0.25/0.53 thf(fact_4956_Domain_ODomainI,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 0.25/0.53 => ( member @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain.DomainI
% 0.25/0.53 thf(fact_4957_DomainE,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 0.25/0.53 => ~ ! [B3: B] :
% 0.25/0.53 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% 0.25/0.53
% 0.25/0.53 % DomainE
% 0.25/0.53 thf(fact_4958_Domain__iff,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 0.25/0.53 = ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain_iff
% 0.25/0.53 thf(fact_4959_Not__Domain__rtrancl,axiom,
% 0.25/0.53 ! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A ),Y: A] :
% 0.25/0.53 ( ~ ( member @ A @ X @ ( domain @ A @ A @ R5 ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R5 ) )
% 0.25/0.53 = ( X = Y ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Not_Domain_rtrancl
% 0.25/0.53 thf(fact_4960_Domain__unfold,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( domain @ A @ B )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( collect @ A
% 0.25/0.53 @ ^ [X3: A] :
% 0.25/0.53 ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain_unfold
% 0.25/0.53 thf(fact_4961_for__in__RI,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,X: A,R5: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ A @ X @ ( domain @ A @ B @ R5 ) )
% 0.25/0.53 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R5 @ X ) ) @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % for_in_RI
% 0.25/0.53 thf(fact_4962_Domain__rel__restrict,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ A4 ) ) ).
% 0.25/0.53
% 0.25/0.53 % Domain_rel_restrict
% 0.25/0.53 thf(fact_4963_wf__no__path,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) )
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( wf @ A @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % wf_no_path
% 0.25/0.53 thf(fact_4964_wf__min,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( wf @ A @ R5 )
% 0.25/0.53 => ( ( R5
% 0.25/0.53 != ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 0.25/0.53 => ~ ! [M5: A] :
% 0.25/0.53 ~ ( member @ A @ M5 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % wf_min
% 0.25/0.53 thf(fact_4965_Range__def,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( range2 @ A @ B )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( collect @ B
% 0.25/0.53 @ ( rangep @ A @ B
% 0.25/0.53 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Range_def
% 0.25/0.53 thf(fact_4966_and_Omonoid__axioms,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.53 => ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % and.monoid_axioms
% 0.25/0.53 thf(fact_4967_cr__integer__def,axiom,
% 0.25/0.53 ( code_cr_integer
% 0.25/0.53 = ( ^ [X3: int,Y3: code_integer] :
% 0.25/0.53 ( X3
% 0.25/0.53 = ( code_int_of_integer @ Y3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % cr_integer_def
% 0.25/0.53 thf(fact_4968_wo__rel_OisMinim__def,axiom,
% 0.25/0.53 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
% 0.25/0.53 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 0.25/0.53 => ( ( bNF_We4791949203932849705sMinim @ A @ R @ A4 @ B2 )
% 0.25/0.53 = ( ( member @ A @ B2 @ A4 )
% 0.25/0.53 & ! [X3: A] :
% 0.25/0.53 ( ( member @ A @ X3 @ A4 )
% 0.25/0.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X3 ) @ R ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % wo_rel.isMinim_def
% 0.25/0.53 thf(fact_4969_monoid_Oleft__neutral,axiom,
% 0.25/0.53 ! [A: $tType,F: A > A > A,Z2: A,A3: A] :
% 0.25/0.53 ( ( monoid @ A @ F @ Z2 )
% 0.25/0.53 => ( ( F @ Z2 @ A3 )
% 0.25/0.53 = A3 ) ) ).
% 0.25/0.53
% 0.25/0.53 % monoid.left_neutral
% 0.25/0.53 thf(fact_4970_monoid_Oright__neutral,axiom,
% 0.25/0.53 ! [A: $tType,F: A > A > A,Z2: A,A3: A] :
% 0.25/0.53 ( ( monoid @ A @ F @ Z2 )
% 0.25/0.53 => ( ( F @ A3 @ Z2 )
% 0.25/0.53 = A3 ) ) ).
% 0.25/0.53
% 0.25/0.53 % monoid.right_neutral
% 0.25/0.53 thf(fact_4971_add_Omonoid__axioms,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( monoid_add @ A )
% 0.25/0.53 => ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % add.monoid_axioms
% 0.25/0.53 thf(fact_4972_mult_Omonoid__axioms,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( monoid_mult @ A )
% 0.25/0.53 => ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mult.monoid_axioms
% 0.25/0.53 thf(fact_4973_integer_Opcr__cr__eq,axiom,
% 0.25/0.53 code_pcr_integer = code_cr_integer ).
% 0.25/0.53
% 0.25/0.53 % integer.pcr_cr_eq
% 0.25/0.53 thf(fact_4974_multp__implies__one__step,axiom,
% 0.25/0.53 ! [A: $tType,R: A > A > $o,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.53 ( ( transp @ A @ R )
% 0.25/0.53 => ( ( multp @ A @ R @ M3 @ N6 )
% 0.25/0.53 => ? [I8: multiset @ A,J5: multiset @ A] :
% 0.25/0.53 ( ( N6
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
% 0.25/0.53 & ? [K10: multiset @ A] :
% 0.25/0.53 ( ( M3
% 0.25/0.53 = ( plus_plus @ ( multiset @ A ) @ I8 @ K10 ) )
% 0.25/0.53 & ( J5
% 0.25/0.53 != ( zero_zero @ ( multiset @ A ) ) )
% 0.25/0.53 & ! [X7: A] :
% 0.25/0.53 ( ( member @ A @ X7 @ ( set_mset @ A @ K10 ) )
% 0.25/0.53 => ? [Xa3: A] :
% 0.25/0.53 ( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
% 0.25/0.53 & ( R @ X7 @ Xa3 ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % multp_implies_one_step
% 0.25/0.53 thf(fact_4975_relpow__fun__conv,axiom,
% 0.25/0.53 ! [A: $tType,A3: A,B2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 = ( ? [F2: nat > A] :
% 0.25/0.53 ( ( ( F2 @ ( zero_zero @ nat ) )
% 0.25/0.53 = A3 )
% 0.25/0.53 & ( ( F2 @ N )
% 0.25/0.53 = B2 )
% 0.25/0.53 & ! [I3: nat] :
% 0.25/0.53 ( ( ord_less @ nat @ I3 @ N )
% 0.25/0.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R5 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_fun_conv
% 0.25/0.53 thf(fact_4976_irreflp__irrefl__eq,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( irreflp @ A
% 0.25/0.53 @ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) )
% 0.25/0.53 = ( irrefl @ A @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % irreflp_irrefl_eq
% 0.25/0.53 thf(fact_4977_relpow__1,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R5 )
% 0.25/0.53 = R5 ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_1
% 0.25/0.53 thf(fact_4978_multp__cancel,axiom,
% 0.25/0.53 ! [A: $tType,R: A > A > $o,X10: multiset @ A,Z9: multiset @ A,Y9: multiset @ A] :
% 0.25/0.53 ( ( transp @ A @ R )
% 0.25/0.53 => ( ( irreflp @ A @ R )
% 0.25/0.53 => ( ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ X10 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y9 @ Z9 ) )
% 0.25/0.53 = ( multp @ A @ R @ X10 @ Y9 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % multp_cancel
% 0.25/0.53 thf(fact_4979_relpow__Suc__D2_H,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A ),X7: A,Y6: A,Z6: A] :
% 0.25/0.53 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z6 ) @ R5 ) )
% 0.25/0.53 => ? [W4: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ W4 ) @ R5 )
% 0.25/0.53 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W4 @ Z6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_Suc_D2'
% 0.25/0.53 thf(fact_4980_relpow__0__E,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) )
% 0.25/0.53 => ( X = Y ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_0_E
% 0.25/0.53 thf(fact_4981_relpow__0__I,axiom,
% 0.25/0.53 ! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_0_I
% 0.25/0.53 thf(fact_4982_relpow__Suc__E,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
% 0.25/0.53 => ~ ! [Y4: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_Suc_E
% 0.25/0.53 thf(fact_4983_relpow__Suc__I,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Y: A,N: nat,R5: set @ ( product_prod @ A @ A ),Z2: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R5 )
% 0.25/0.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_Suc_I
% 0.25/0.53 thf(fact_4984_relpow__Suc__D2,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
% 0.25/0.53 => ? [Y4: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
% 0.25/0.53 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_Suc_D2
% 0.25/0.53 thf(fact_4985_relpow__Suc__E2,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
% 0.25/0.53 => ~ ! [Y4: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_Suc_E2
% 0.25/0.53 thf(fact_4986_relpow__Suc__I2,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_Suc_I2
% 0.25/0.53 thf(fact_4987_relpow__add,axiom,
% 0.25/0.53 ! [A: $tType,M2: nat,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M2 @ N ) @ R5 )
% 0.25/0.53 = ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M2 @ R5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_add
% 0.25/0.53 thf(fact_4988_transp__trans__eq,axiom,
% 0.25/0.53 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( transp @ A
% 0.25/0.53 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.53 = ( trans @ A @ R ) ) ).
% 0.25/0.53
% 0.25/0.53 % transp_trans_eq
% 0.25/0.53 thf(fact_4989_relpowp__relpow__eq,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( compow @ ( A > A > $o ) @ N
% 0.25/0.53 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) )
% 0.25/0.53 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpowp_relpow_eq
% 0.25/0.53 thf(fact_4990_relpow__E2,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 => ( ( ( N
% 0.25/0.53 = ( zero_zero @ nat ) )
% 0.25/0.53 => ( X != Z2 ) )
% 0.25/0.53 => ~ ! [Y4: A,M5: nat] :
% 0.25/0.53 ( ( N
% 0.25/0.53 = ( suc @ M5 ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R5 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_E2
% 0.25/0.53 thf(fact_4991_relpow__E,axiom,
% 0.25/0.53 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 0.25/0.53 => ( ( ( N
% 0.25/0.53 = ( zero_zero @ nat ) )
% 0.25/0.53 => ( X != Z2 ) )
% 0.25/0.53 => ~ ! [Y4: A,M5: nat] :
% 0.25/0.53 ( ( N
% 0.25/0.53 = ( suc @ M5 ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R5 ) )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R5 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpow_E
% 0.25/0.53 thf(fact_4992_nths__drop,axiom,
% 0.25/0.53 ! [A: $tType,N: nat,Xs: list @ A,I4: set @ nat] :
% 0.25/0.53 ( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I4 )
% 0.25/0.53 = ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I4 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nths_drop
% 0.25/0.53 thf(fact_4993_nths__append,axiom,
% 0.25/0.53 ! [A: $tType,L: list @ A,L4: list @ A,A4: set @ nat] :
% 0.25/0.53 ( ( nths @ A @ ( append @ A @ L @ L4 ) @ A4 )
% 0.25/0.53 = ( append @ A @ ( nths @ A @ L @ A4 )
% 0.25/0.53 @ ( nths @ A @ L4
% 0.25/0.53 @ ( collect @ nat
% 0.25/0.53 @ ^ [J2: nat] : ( member @ nat @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) ) @ A4 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nths_append
% 0.25/0.53 thf(fact_4994_asymp__asym__eq,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( asymp @ A
% 0.25/0.53 @ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) )
% 0.25/0.53 = ( asym @ A @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % asymp_asym_eq
% 0.25/0.53 thf(fact_4995_asym_Ocases,axiom,
% 0.25/0.53 ! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( asym @ A @ A3 )
% 0.25/0.53 => ! [A17: A,B8: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A17 @ B8 ) @ A3 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A17 ) @ A3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % asym.cases
% 0.25/0.53 thf(fact_4996_asym_Osimps,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( asym @ A )
% 0.25/0.53 = ( ^ [A7: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ? [R6: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( A7 = R6 )
% 0.25/0.53 & ! [X3: A,Y3: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % asym.simps
% 0.25/0.53 thf(fact_4997_asym_Ointros,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ! [A5: A,B3: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R5 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R5 ) )
% 0.25/0.53 => ( asym @ A @ R5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % asym.intros
% 0.25/0.53 thf(fact_4998_asymD,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.53 ( ( asym @ A @ R5 )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % asymD
% 0.25/0.53 thf(fact_4999_asym__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( asym @ A )
% 0.25/0.53 = ( ^ [R6: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ! [X3: A,Y3: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 )
% 0.25/0.53 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % asym_iff
% 0.25/0.53 thf(fact_5000_lexord__asymmetric,axiom,
% 0.25/0.53 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A3: list @ A,B2: list @ A] :
% 0.25/0.53 ( ( asym @ A @ R5 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A3 @ B2 ) @ ( lexord @ A @ R5 ) )
% 0.25/0.53 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A3 ) @ ( lexord @ A @ R5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % lexord_asymmetric
% 0.25/0.53 thf(fact_5001_ordLess__iff,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.53 = ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 & ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.53 & ~ ? [X8: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R @ X8 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLess_iff
% 0.25/0.53 thf(fact_5002_option_Oin__rel,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( rel_option @ A @ B )
% 0.25/0.53 = ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
% 0.25/0.53 ? [Z5: option @ ( product_prod @ A @ B )] :
% 0.25/0.53 ( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z5
% 0.25/0.53 @ ( collect @ ( option @ ( product_prod @ A @ B ) )
% 0.25/0.53 @ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
% 0.25/0.53 & ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
% 0.25/0.53 = A7 )
% 0.25/0.53 & ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
% 0.25/0.53 = B5 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.in_rel
% 0.25/0.53 thf(fact_5003_bit__double__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_semiring_bits @ A )
% 0.25/0.53 => ! [A3: A,N: nat] :
% 0.25/0.53 ( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
% 0.25/0.53 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 0.25/0.53 & ( N
% 0.25/0.53 != ( zero_zero @ nat ) )
% 0.25/0.53 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bit_double_iff
% 0.25/0.53 thf(fact_5004_rel__option__None2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,P: A > B > $o,X: option @ A] :
% 0.25/0.53 ( ( rel_option @ A @ B @ P @ X @ ( none @ B ) )
% 0.25/0.53 = ( X
% 0.25/0.53 = ( none @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % rel_option_None2
% 0.25/0.53 thf(fact_5005_rel__option__None1,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,P: A > B > $o,X: option @ B] :
% 0.25/0.53 ( ( rel_option @ A @ B @ P @ ( none @ A ) @ X )
% 0.25/0.53 = ( X
% 0.25/0.53 = ( none @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % rel_option_None1
% 0.25/0.53 thf(fact_5006_bit__minus__1__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.53 => ! [N: nat] :
% 0.25/0.53 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 0.25/0.53 = ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bit_minus_1_iff
% 0.25/0.53 thf(fact_5007_rel__option__iff,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( rel_option @ A @ B )
% 0.25/0.53 = ( ^ [R6: A > B > $o,X3: option @ A,Y3: option @ B] :
% 0.25/0.53 ( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
% 0.25/0.53 @ ^ [A7: option @ A,B5: option @ B] :
% 0.25/0.53 ( case_option @ $o @ A
% 0.25/0.53 @ ( case_option @ $o @ B @ $true
% 0.25/0.53 @ ^ [C3: B] : $false
% 0.25/0.53 @ B5 )
% 0.25/0.53 @ ^ [Z5: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z5 ) @ B5 )
% 0.25/0.53 @ A7 )
% 0.25/0.53 @ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X3 @ Y3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % rel_option_iff
% 0.25/0.53 thf(fact_5008_option_Orel__mono,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
% 0.25/0.53 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
% 0.25/0.53 => ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ A @ B @ Ra2 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_mono
% 0.25/0.53 thf(fact_5009_option_Orel__inject_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
% 0.25/0.53 ( ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) )
% 0.25/0.53 = ( R5 @ X22 @ Y2 ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_inject(2)
% 0.25/0.53 thf(fact_5010_option_Orel__intros_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
% 0.25/0.53 ( ( R5 @ X22 @ Y2 )
% 0.25/0.53 => ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_intros(2)
% 0.25/0.53 thf(fact_5011_option__rel__Some1,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: option @ B] :
% 0.25/0.53 ( ( rel_option @ A @ B @ A4 @ ( some @ A @ X ) @ Y )
% 0.25/0.53 = ( ? [Y8: B] :
% 0.25/0.53 ( ( Y
% 0.25/0.53 = ( some @ B @ Y8 ) )
% 0.25/0.53 & ( A4 @ X @ Y8 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option_rel_Some1
% 0.25/0.53 thf(fact_5012_option__rel__Some2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: A > B > $o,X: option @ A,Y: B] :
% 0.25/0.53 ( ( rel_option @ A @ B @ A4 @ X @ ( some @ B @ Y ) )
% 0.25/0.53 = ( ? [X9: A] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( some @ A @ X9 ) )
% 0.25/0.53 & ( A4 @ X9 @ Y ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option_rel_Some2
% 0.25/0.53 thf(fact_5013_option_Octr__transfer_I1_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o] : ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( none @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.ctr_transfer(1)
% 0.25/0.53 thf(fact_5014_rel__option__inf,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
% 0.25/0.53 ( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A4 ) @ ( rel_option @ A @ B @ B6 ) )
% 0.25/0.53 = ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A4 @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % rel_option_inf
% 0.25/0.53 thf(fact_5015_option_Orel__refl,axiom,
% 0.25/0.53 ! [B: $tType,Ra2: B > B > $o,X: option @ B] :
% 0.25/0.53 ( ! [X2: B] : ( Ra2 @ X2 @ X2 )
% 0.25/0.53 => ( rel_option @ B @ B @ Ra2 @ X @ X ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_refl
% 0.25/0.53 thf(fact_5016_option_Orel__eq,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( rel_option @ A @ A
% 0.25/0.53 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.25/0.53 = ( ^ [Y5: option @ A,Z4: option @ A] : ( Y5 = Z4 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_eq
% 0.25/0.53 thf(fact_5017_option_Orel__cong,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,X: option @ A,Ya: option @ A,Y: option @ B,Xa: option @ B,R5: A > B > $o,Ra2: A > B > $o] :
% 0.25/0.53 ( ( X = Ya )
% 0.25/0.53 => ( ( Y = Xa )
% 0.25/0.53 => ( ! [Z3: A,Yb: B] :
% 0.25/0.53 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 0.25/0.53 => ( ( member @ B @ Yb @ ( set_option @ B @ Xa ) )
% 0.25/0.53 => ( ( R5 @ Z3 @ Yb )
% 0.25/0.53 = ( Ra2 @ Z3 @ Yb ) ) ) )
% 0.25/0.53 => ( ( rel_option @ A @ B @ R5 @ X @ Y )
% 0.25/0.53 = ( rel_option @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_cong
% 0.25/0.53 thf(fact_5018_option_Orel__mono__strong,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y: option @ B,Ra2: A > B > $o] :
% 0.25/0.53 ( ( rel_option @ A @ B @ R5 @ X @ Y )
% 0.25/0.53 => ( ! [Z3: A,Yb: B] :
% 0.25/0.53 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 0.25/0.53 => ( ( member @ B @ Yb @ ( set_option @ B @ Y ) )
% 0.25/0.53 => ( ( R5 @ Z3 @ Yb )
% 0.25/0.53 => ( Ra2 @ Z3 @ Yb ) ) ) )
% 0.25/0.53 => ( rel_option @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_mono_strong
% 0.25/0.53 thf(fact_5019_option_Orel__refl__strong,axiom,
% 0.25/0.53 ! [A: $tType,X: option @ A,Ra2: A > A > $o] :
% 0.25/0.53 ( ! [Z3: A] :
% 0.25/0.53 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 0.25/0.53 => ( Ra2 @ Z3 @ Z3 ) )
% 0.25/0.53 => ( rel_option @ A @ A @ Ra2 @ X @ X ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_refl_strong
% 0.25/0.53 thf(fact_5020_rel__option__reflI,axiom,
% 0.25/0.53 ! [A: $tType,Y: option @ A,P: A > A > $o] :
% 0.25/0.53 ( ! [X2: A] :
% 0.25/0.53 ( ( member @ A @ X2 @ ( set_option @ A @ Y ) )
% 0.25/0.53 => ( P @ X2 @ X2 ) )
% 0.25/0.53 => ( rel_option @ A @ A @ P @ Y @ Y ) ) ).
% 0.25/0.53
% 0.25/0.53 % rel_option_reflI
% 0.25/0.53 thf(fact_5021_option_Orel__map_I1_J,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: option @ A,Y: option @ B] :
% 0.25/0.53 ( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X ) @ Y )
% 0.25/0.53 = ( rel_option @ A @ B
% 0.25/0.53 @ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
% 0.25/0.53 @ X
% 0.25/0.53 @ Y ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_map(1)
% 0.25/0.53 thf(fact_5022_option_Orel__map_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G: B > C,Y: option @ B] :
% 0.25/0.53 ( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G @ Y ) )
% 0.25/0.53 = ( rel_option @ A @ B
% 0.25/0.53 @ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
% 0.25/0.53 @ X
% 0.25/0.53 @ Y ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_map(2)
% 0.25/0.53 thf(fact_5023_option_Orel__flip,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ B,B2: option @ A] :
% 0.25/0.53 ( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
% 0.25/0.53 = ( rel_option @ A @ B @ R5 @ B2 @ A3 ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_flip
% 0.25/0.53 thf(fact_5024_option_Orel__conversep,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 0.25/0.53 ( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) )
% 0.25/0.53 = ( conversep @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_conversep
% 0.25/0.53 thf(fact_5025_option_Octr__transfer_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R5 @ ( rel_option @ A @ B @ R5 ) @ ( some @ A ) @ ( some @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.ctr_transfer(2)
% 0.25/0.53 thf(fact_5026_option_Odisc__transfer_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 0.25/0.53 ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.25/0.53 @ ^ [Option3: option @ A] :
% 0.25/0.53 ( Option3
% 0.25/0.53 != ( none @ A ) )
% 0.25/0.53 @ ^ [Option3: option @ B] :
% 0.25/0.53 ( Option3
% 0.25/0.53 != ( none @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.disc_transfer(2)
% 0.25/0.53 thf(fact_5027_option_Odisc__transfer_I1_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 0.25/0.53 ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 0.25/0.53 @ ^ [Option3: option @ A] :
% 0.25/0.53 ( Option3
% 0.25/0.53 = ( none @ A ) )
% 0.25/0.53 @ ^ [Option3: option @ B] :
% 0.25/0.53 ( Option3
% 0.25/0.53 = ( none @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.disc_transfer(1)
% 0.25/0.53 thf(fact_5028_option__bind__transfer,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A4 @ ( rel_option @ C @ D @ B6 ) ) @ ( rel_option @ C @ D @ B6 ) ) @ ( bind @ A @ C ) @ ( bind @ B @ D ) ) ).
% 0.25/0.53
% 0.25/0.53 % option_bind_transfer
% 0.25/0.53 thf(fact_5029_option_Orel__transp,axiom,
% 0.25/0.53 ! [A: $tType,R5: A > A > $o] :
% 0.25/0.53 ( ( transp @ A @ R5 )
% 0.25/0.53 => ( transp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_transp
% 0.25/0.53 thf(fact_5030_option_Orel__distinct_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,Y2: A] :
% 0.25/0.53 ~ ( rel_option @ A @ B @ R5 @ ( some @ A @ Y2 ) @ ( none @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_distinct(2)
% 0.25/0.53 thf(fact_5031_option_Orel__distinct_I1_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,Y2: B] :
% 0.25/0.53 ~ ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( some @ B @ Y2 ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_distinct(1)
% 0.25/0.53 thf(fact_5032_option_Orel__cases,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ A,B2: option @ B] :
% 0.25/0.53 ( ( rel_option @ A @ B @ R5 @ A3 @ B2 )
% 0.25/0.53 => ( ( ( A3
% 0.25/0.53 = ( none @ A ) )
% 0.25/0.53 => ( B2
% 0.25/0.53 != ( none @ B ) ) )
% 0.25/0.53 => ~ ! [X2: A] :
% 0.25/0.53 ( ( A3
% 0.25/0.53 = ( some @ A @ X2 ) )
% 0.25/0.53 => ! [Y4: B] :
% 0.25/0.53 ( ( B2
% 0.25/0.53 = ( some @ B @ Y4 ) )
% 0.25/0.53 => ~ ( R5 @ X2 @ Y4 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_cases
% 0.25/0.53 thf(fact_5033_option_Orel__induct,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
% 0.25/0.53 ( ( rel_option @ A @ B @ R5 @ X @ Y )
% 0.25/0.53 => ( ( Q @ ( none @ A ) @ ( none @ B ) )
% 0.25/0.53 => ( ! [A24: A,B24: B] :
% 0.25/0.53 ( ( R5 @ A24 @ B24 )
% 0.25/0.53 => ( Q @ ( some @ A @ A24 ) @ ( some @ B @ B24 ) ) )
% 0.25/0.53 => ( Q @ X @ Y ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_induct
% 0.25/0.53 thf(fact_5034_option_Omap__transfer,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,F5: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F5 ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E ) > ( option @ F5 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F5 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E ) @ ( option @ B ) @ ( option @ F5 ) @ ( rel_option @ A @ E @ Rb2 ) @ ( rel_option @ B @ F5 @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E @ F5 ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.map_transfer
% 0.25/0.53 thf(fact_5035_option_Ocase__transfer,axiom,
% 0.25/0.53 ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S2 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.case_transfer
% 0.25/0.53 thf(fact_5036_option_Orel__transfer,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.25/0.53 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
% 0.25/0.53 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.25/0.53 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 0.25/0.53 @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
% 0.25/0.53 @ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 0.25/0.53 @ ( rel_option @ A @ B )
% 0.25/0.53 @ ( rel_option @ C @ D ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_transfer
% 0.25/0.53 thf(fact_5037_bit__minus__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.53 => ! [A3: A,N: nat] :
% 0.25/0.53 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 0.25/0.53 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.25/0.53 & ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bit_minus_iff
% 0.25/0.53 thf(fact_5038_option_Orel__sel,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( rel_option @ A @ B )
% 0.25/0.53 = ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
% 0.25/0.53 ( ( ( A7
% 0.25/0.53 = ( none @ A ) )
% 0.25/0.53 = ( B5
% 0.25/0.53 = ( none @ B ) ) )
% 0.25/0.53 & ( ( A7
% 0.25/0.53 != ( none @ A ) )
% 0.25/0.53 => ( ( B5
% 0.25/0.53 != ( none @ B ) )
% 0.25/0.53 => ( R6 @ ( the2 @ A @ A7 ) @ ( the2 @ B @ B5 ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_sel
% 0.25/0.53 thf(fact_5039_ordLess__not__embed,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.53 => ~ ? [X_1: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R @ X_1 ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLess_not_embed
% 0.25/0.53 thf(fact_5040_option_Orec__transfer,axiom,
% 0.25/0.53 ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S2 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rec_transfer
% 0.25/0.53 thf(fact_5041_bit__2__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_semiring_bits @ A )
% 0.25/0.53 => ! [N: nat] :
% 0.25/0.53 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 0.25/0.53 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
% 0.25/0.53 & ( N
% 0.25/0.53 = ( one_one @ nat ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bit_2_iff
% 0.25/0.53 thf(fact_5042_BNF__Wellorder__Constructions_OordLess__Field,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ A @ B @ R1 @ R22 @ F )
% 0.25/0.53 => ( ( image2 @ A @ B @ F @ ( field2 @ A @ R1 ) )
% 0.25/0.53 != ( field2 @ B @ R22 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % BNF_Wellorder_Constructions.ordLess_Field
% 0.25/0.53 thf(fact_5043_bit__mask__sub__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bit_semiring_bits @ A )
% 0.25/0.53 => ! [M2: nat,N: nat] :
% 0.25/0.53 ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
% 0.25/0.53 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 0.25/0.53 & ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % bit_mask_sub_iff
% 0.25/0.53 thf(fact_5044_embed__ordLess__ofilterIncl,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),F13: A > C,F232: B > C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R22 @ R33 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ A @ C @ R1 @ R33 @ F13 )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ B @ C @ R22 @ R33 @ F232 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F13 @ ( field2 @ A @ R1 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R22 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R33 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % embed_ordLess_ofilterIncl
% 0.25/0.53 thf(fact_5045_comp__embed,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % comp_embed
% 0.25/0.53 thf(fact_5046_override__on__insert,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X10: set @ A] :
% 0.25/0.53 ( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X10 ) )
% 0.25/0.53 = ( fun_upd @ A @ B @ ( override_on @ A @ B @ F @ G @ X10 ) @ X @ ( G @ X ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % override_on_insert
% 0.25/0.53 thf(fact_5047_override__on__apply__in,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,A4: set @ A,F: A > B,G: A > B] :
% 0.25/0.53 ( ( member @ A @ A3 @ A4 )
% 0.25/0.53 => ( ( override_on @ A @ B @ F @ G @ A4 @ A3 )
% 0.25/0.53 = ( G @ A3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % override_on_apply_in
% 0.25/0.53 thf(fact_5048_override__on__apply__notin,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A3: A,A4: set @ A,F: A > B,G: A > B] :
% 0.25/0.53 ( ~ ( member @ A @ A3 @ A4 )
% 0.25/0.53 => ( ( override_on @ A @ B @ F @ G @ A4 @ A3 )
% 0.25/0.53 = ( F @ A3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % override_on_apply_notin
% 0.25/0.53 thf(fact_5049_override__on__emptyset,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.25/0.53 ( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 = F ) ).
% 0.25/0.53
% 0.25/0.53 % override_on_emptyset
% 0.25/0.53 thf(fact_5050_override__on__def,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( override_on @ A @ B )
% 0.25/0.53 = ( ^ [F2: A > B,G2: A > B,A9: set @ A,A7: A] : ( if @ B @ ( member @ A @ A7 @ A9 ) @ ( G2 @ A7 ) @ ( F2 @ A7 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % override_on_def
% 0.25/0.53 thf(fact_5051_override__on__insert_H,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X10: set @ A] :
% 0.25/0.53 ( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X10 ) )
% 0.25/0.53 = ( override_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ ( G @ X ) ) @ G @ X10 ) ) ).
% 0.25/0.53
% 0.25/0.53 % override_on_insert'
% 0.25/0.53 thf(fact_5052_embedS__comp__embed,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.53 => ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % embedS_comp_embed
% 0.25/0.53 thf(fact_5053_embed__comp__embedS,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_embedS @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % embed_comp_embedS
% 0.25/0.53 thf(fact_5054_pred__option__parametric,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: A > B > $o] :
% 0.25/0.53 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
% 0.25/0.53 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.53 @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A4 )
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.53 @ ( pred_option @ A )
% 0.25/0.53 @ ( pred_option @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % pred_option_parametric
% 0.25/0.53 thf(fact_5055_option_Opred__inject_I2_J,axiom,
% 0.25/0.53 ! [A: $tType,P: A > $o,A3: A] :
% 0.25/0.53 ( ( pred_option @ A @ P @ ( some @ A @ A3 ) )
% 0.25/0.53 = ( P @ A3 ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_inject(2)
% 0.25/0.53 thf(fact_5056_option_Opred__mono,axiom,
% 0.25/0.53 ! [A: $tType,P: A > $o,Pa: A > $o] :
% 0.25/0.53 ( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
% 0.25/0.53 => ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P ) @ ( pred_option @ A @ Pa ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_mono
% 0.25/0.53 thf(fact_5057_option_Opred__inject_I1_J,axiom,
% 0.25/0.53 ! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_inject(1)
% 0.25/0.53 thf(fact_5058_option_Opred__True,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( pred_option @ A
% 0.25/0.53 @ ^ [Uu: A] : $true )
% 0.25/0.53 = ( ^ [Uu: option @ A] : $true ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_True
% 0.25/0.53 thf(fact_5059_option_Opred__mono__strong,axiom,
% 0.25/0.53 ! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
% 0.25/0.53 ( ( pred_option @ A @ P @ X )
% 0.25/0.53 => ( ! [Z3: A] :
% 0.25/0.53 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 0.25/0.53 => ( ( P @ Z3 )
% 0.25/0.53 => ( Pa @ Z3 ) ) )
% 0.25/0.53 => ( pred_option @ A @ Pa @ X ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_mono_strong
% 0.25/0.53 thf(fact_5060_option_Opred__cong,axiom,
% 0.25/0.53 ! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
% 0.25/0.53 ( ( X = Ya )
% 0.25/0.53 => ( ! [Z3: A] :
% 0.25/0.53 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 0.25/0.53 => ( ( P @ Z3 )
% 0.25/0.53 = ( Pa @ Z3 ) ) )
% 0.25/0.53 => ( ( pred_option @ A @ P @ X )
% 0.25/0.53 = ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_cong
% 0.25/0.53 thf(fact_5061_option_Omap__cong__pred,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
% 0.25/0.53 ( ( X = Ya )
% 0.25/0.53 => ( ( pred_option @ A
% 0.25/0.53 @ ^ [Z5: A] :
% 0.25/0.53 ( ( F @ Z5 )
% 0.25/0.53 = ( G @ Z5 ) )
% 0.25/0.53 @ Ya )
% 0.25/0.53 => ( ( map_option @ A @ B @ F @ X )
% 0.25/0.53 = ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.map_cong_pred
% 0.25/0.53 thf(fact_5062_option_Opred__set,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( pred_option @ A )
% 0.25/0.53 = ( ^ [P4: A > $o,X3: option @ A] :
% 0.25/0.53 ! [Y3: A] :
% 0.25/0.53 ( ( member @ A @ Y3 @ ( set_option @ A @ X3 ) )
% 0.25/0.53 => ( P4 @ Y3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_set
% 0.25/0.53 thf(fact_5063_option_Opred__rel,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( pred_option @ A )
% 0.25/0.53 = ( ^ [P4: A > $o,X3: option @ A] : ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X3 @ X3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_rel
% 0.25/0.53 thf(fact_5064_option_Orel__eq__onp,axiom,
% 0.25/0.53 ! [A: $tType,P: A > $o] :
% 0.25/0.53 ( ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P ) )
% 0.25/0.53 = ( bNF_eq_onp @ ( option @ A ) @ ( pred_option @ A @ P ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_eq_onp
% 0.25/0.53 thf(fact_5065_option_Opred__map,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: option @ A] :
% 0.25/0.53 ( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F @ X ) )
% 0.25/0.53 = ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_map
% 0.25/0.53 thf(fact_5066_option_Opred__transfer,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 0.25/0.53 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
% 0.25/0.53 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.53 @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.53 @ ( pred_option @ A )
% 0.25/0.53 @ ( pred_option @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.pred_transfer
% 0.25/0.53 thf(fact_5067_iso__comp__embedS,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.53 => ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_embedS @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % iso_comp_embedS
% 0.25/0.53 thf(fact_5068_cut__def,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( cut @ A @ B )
% 0.25/0.53 = ( ^ [F2: A > B,R6: set @ ( product_prod @ A @ A ),X3: A,Y3: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) @ ( F2 @ Y3 ) @ ( undefined @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % cut_def
% 0.25/0.53 thf(fact_5069_above__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( order_above @ A )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 0.25/0.53 ( collect @ A
% 0.25/0.53 @ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % above_def
% 0.25/0.53 thf(fact_5070_comp__iso,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_iso @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % comp_iso
% 0.25/0.53 thf(fact_5071_iso__forward,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.53 => ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R4 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % iso_forward
% 0.25/0.53 thf(fact_5072_cut__apply,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,X: A,A3: A,R5: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R5 )
% 0.25/0.53 => ( ( cut @ A @ B @ F @ R5 @ A3 @ X )
% 0.25/0.53 = ( F @ X ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % cut_apply
% 0.25/0.53 thf(fact_5073_cuts__eq,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,R5: set @ ( product_prod @ A @ A ),X: A,G: A > B] :
% 0.25/0.53 ( ( ( cut @ A @ B @ F @ R5 @ X )
% 0.25/0.53 = ( cut @ A @ B @ G @ R5 @ X ) )
% 0.25/0.53 = ( ! [Y3: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R5 )
% 0.25/0.53 => ( ( F @ Y3 )
% 0.25/0.53 = ( G @ Y3 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % cuts_eq
% 0.25/0.53 thf(fact_5074_iso__comp__embed,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % iso_comp_embed
% 0.25/0.53 thf(fact_5075_embed__comp__iso,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % embed_comp_iso
% 0.25/0.53 thf(fact_5076_embedS__comp__iso,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 0.25/0.53 ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 0.25/0.53 => ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
% 0.25/0.53 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % embedS_comp_iso
% 0.25/0.53 thf(fact_5077_iso__iff2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( bNF_Wellorder_iso @ A @ B )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ B @ B ),F2: A > B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F2 @ ( field2 @ A @ R3 ) @ ( field2 @ B @ R11 ) )
% 0.25/0.53 & ! [X3: A] :
% 0.25/0.53 ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
% 0.25/0.53 => ! [Y3: A] :
% 0.25/0.53 ( ( member @ A @ Y3 @ ( field2 @ A @ R3 ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 0.25/0.53 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R11 ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % iso_iff2
% 0.25/0.53 thf(fact_5078_iso__backward,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),F: B > A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
% 0.25/0.53 => ( ( bNF_Wellorder_iso @ B @ A @ R @ R4 @ F )
% 0.25/0.53 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ X ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ Y ) ) @ R ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % iso_backward
% 0.25/0.53 thf(fact_5079_lists__of__len__fin1,axiom,
% 0.25/0.53 ! [A: $tType,P: set @ A,N: nat] :
% 0.25/0.53 ( ( finite_finite2 @ A @ P )
% 0.25/0.53 => ( finite_finite2 @ ( list @ A )
% 0.25/0.53 @ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
% 0.25/0.53 @ ( collect @ ( list @ A )
% 0.25/0.53 @ ^ [L3: list @ A] :
% 0.25/0.53 ( ( size_size @ ( list @ A ) @ L3 )
% 0.25/0.53 = N ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % lists_of_len_fin1
% 0.25/0.53 thf(fact_5080_lists__of__len__fin2,axiom,
% 0.25/0.53 ! [A: $tType,P: set @ A,N: nat] :
% 0.25/0.53 ( ( finite_finite2 @ A @ P )
% 0.25/0.53 => ( finite_finite2 @ ( list @ A )
% 0.25/0.53 @ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
% 0.25/0.53 @ ( collect @ ( list @ A )
% 0.25/0.53 @ ^ [L3: list @ A] :
% 0.25/0.53 ( N
% 0.25/0.53 = ( size_size @ ( list @ A ) @ L3 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % lists_of_len_fin2
% 0.25/0.53 thf(fact_5081_Random__Pred_Ounion__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( random_union @ A )
% 0.25/0.53 = ( ^ [R13: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R24: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S4: product_prod @ code_natural @ code_natural] :
% 0.25/0.53 ( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.53 @ ^ [P12: pred @ A,S8: product_prod @ code_natural @ code_natural] :
% 0.25/0.53 ( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.53 @ ^ [P23: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P12 @ P23 ) )
% 0.25/0.53 @ ( R24 @ S8 ) )
% 0.25/0.53 @ ( R13 @ S4 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Random_Pred.union_def
% 0.25/0.53 thf(fact_5082_lists__image__witness,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,X: list @ A,F: B > A,Q: set @ B] :
% 0.25/0.53 ( ( member @ ( list @ A ) @ X @ ( lists @ A @ ( image2 @ B @ A @ F @ Q ) ) )
% 0.25/0.53 => ~ ! [Xo2: list @ B] :
% 0.25/0.53 ( ( member @ ( list @ B ) @ Xo2 @ ( lists @ B @ Q ) )
% 0.25/0.53 => ( X
% 0.25/0.53 != ( map @ B @ A @ F @ Xo2 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % lists_image_witness
% 0.25/0.53 thf(fact_5083_Random__Pred_Oempty__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( random_empty @ A )
% 0.25/0.53 = ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Random_Pred.empty_def
% 0.25/0.53 thf(fact_5084_Random__Pred_Obind__def,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( random_bind @ A @ B )
% 0.25/0.53 = ( ^ [R6: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),F2: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S4: product_prod @ code_natural @ code_natural] :
% 0.25/0.53 ( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.53 @ ^ [P4: pred @ A,S8: product_prod @ code_natural @ code_natural] :
% 0.25/0.53 ( product_case_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
% 0.25/0.53 @ ^ [S12: product_prod @ code_natural @ code_natural] :
% 0.25/0.53 ( product_Pair @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural )
% 0.25/0.53 @ ( bind2 @ A @ B @ P4
% 0.25/0.53 @ ^ [A7: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F2 @ A7 @ S12 ) ) ) )
% 0.25/0.53 @ ( split_seed @ S8 ) )
% 0.25/0.53 @ ( R6 @ S4 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Random_Pred.bind_def
% 0.25/0.53 thf(fact_5085_card__Plus__conv__if,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 & ( finite_finite2 @ B @ B6 ) )
% 0.25/0.53 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 0.25/0.53 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) )
% 0.25/0.53 & ( ~ ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 & ( finite_finite2 @ B @ B6 ) )
% 0.25/0.53 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 0.25/0.53 = ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_Plus_conv_if
% 0.25/0.53 thf(fact_5086_card__Plus,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.53 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 0.25/0.53 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_Plus
% 0.25/0.53 thf(fact_5087_option_Orel__compp__Grp,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( rel_option @ A @ B )
% 0.25/0.53 = ( ^ [R6: A > B > $o] :
% 0.25/0.53 ( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
% 0.25/0.53 @ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
% 0.25/0.53 @ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
% 0.25/0.53 @ ( collect @ ( option @ ( product_prod @ A @ B ) )
% 0.25/0.53 @ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 0.25/0.53 @ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
% 0.25/0.53 @ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
% 0.25/0.53 @ ( collect @ ( option @ ( product_prod @ A @ B ) )
% 0.25/0.53 @ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 0.25/0.53 @ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_compp_Grp
% 0.25/0.53 thf(fact_5088_mult__one__div__unit__factor,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.53 => ! [A3: A,B2: A] :
% 0.25/0.53 ( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
% 0.25/0.53 = ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mult_one_div_unit_factor
% 0.25/0.53 thf(fact_5089_unit__factor__mult__unit__left,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semido2269285787275462019factor @ A )
% 0.25/0.53 => ! [A3: A,B2: A] :
% 0.25/0.53 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.25/0.53 = ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_mult_unit_left
% 0.25/0.53 thf(fact_5090_unit__factor__simps_I2_J,axiom,
% 0.25/0.53 ! [N: nat] :
% 0.25/0.53 ( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
% 0.25/0.53 = ( one_one @ nat ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_simps(2)
% 0.25/0.53 thf(fact_5091_unit__factor__1,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_1
% 0.25/0.53 thf(fact_5092_inv__unit__factor__eq__0__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.53 => ! [A3: A] :
% 0.25/0.53 ( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.25/0.53 = ( zero_zero @ A ) )
% 0.25/0.53 = ( A3
% 0.25/0.53 = ( zero_zero @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % inv_unit_factor_eq_0_iff
% 0.25/0.53 thf(fact_5093_unit__factor__mult__unit__right,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semido2269285787275462019factor @ A )
% 0.25/0.53 => ! [A3: A,B2: A] :
% 0.25/0.53 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
% 0.25/0.53 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_mult_unit_right
% 0.25/0.53 thf(fact_5094_option_Orel__compp,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S2: B > C > $o] :
% 0.25/0.53 ( ( rel_option @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S2 ) )
% 0.25/0.53 = ( relcompp @ ( option @ A ) @ ( option @ B ) @ ( option @ C ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ B @ C @ S2 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_compp
% 0.25/0.53 thf(fact_5095_OOO__quotient3,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: B > B > $o,Abs22: B > C,Rep22: C > B,R25: A > A > $o] :
% 0.25/0.53 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 0.25/0.53 => ( ( quotient3 @ B @ C @ R23 @ Abs22 @ Rep22 )
% 0.25/0.53 => ( ! [X2: A,Y4: A] :
% 0.25/0.53 ( ( R25 @ X2 @ Y4 )
% 0.25/0.53 => ( ( R12 @ X2 @ X2 )
% 0.25/0.53 => ( ( R12 @ Y4 @ Y4 )
% 0.25/0.53 => ( R23 @ ( Abs1 @ X2 ) @ ( Abs1 @ Y4 ) ) ) ) )
% 0.25/0.53 => ( ! [X2: B,Y4: B] :
% 0.25/0.53 ( ( R23 @ X2 @ Y4 )
% 0.25/0.53 => ( R25 @ ( Rep1 @ X2 ) @ ( Rep1 @ Y4 ) ) )
% 0.25/0.53 => ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R12 @ ( relcompp @ A @ A @ A @ R25 @ R12 ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % OOO_quotient3
% 0.25/0.53 thf(fact_5096_OOO__eq__quotient3,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,Abs22: B > C,Rep22: C > B] :
% 0.25/0.53 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 0.25/0.53 => ( ( quotient3 @ B @ C
% 0.25/0.53 @ ^ [Y5: B,Z4: B] : ( Y5 = Z4 )
% 0.25/0.53 @ Abs22
% 0.25/0.53 @ Rep22 )
% 0.25/0.53 => ( quotient3 @ A @ C
% 0.25/0.53 @ ( relcompp @ A @ A @ A @ R12
% 0.25/0.53 @ ( relcompp @ A @ A @ A
% 0.25/0.53 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.25/0.53 @ R12 ) )
% 0.25/0.53 @ ( comp @ B @ C @ A @ Abs22 @ Abs1 )
% 0.25/0.53 @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % OOO_eq_quotient3
% 0.25/0.53 thf(fact_5097_relpowp__add,axiom,
% 0.25/0.53 ! [A: $tType,M2: nat,N: nat,P: A > A > $o] :
% 0.25/0.53 ( ( compow @ ( A > A > $o ) @ ( plus_plus @ nat @ M2 @ N ) @ P )
% 0.25/0.53 = ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ M2 @ P ) @ ( compow @ ( A > A > $o ) @ N @ P ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relpowp_add
% 0.25/0.53 thf(fact_5098_relcompp__relcomp__eq,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
% 0.25/0.53 ( ( relcompp @ A @ B @ C
% 0.25/0.53 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 0.25/0.53 @ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S ) )
% 0.25/0.53 = ( ^ [X3: A,Y3: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relcompp_relcomp_eq
% 0.25/0.53 thf(fact_5099_is__unit__unit__factor,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semido2269285787275462019factor @ A )
% 0.25/0.53 => ! [A3: A] :
% 0.25/0.53 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ A3 )
% 0.25/0.53 = A3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % is_unit_unit_factor
% 0.25/0.53 thf(fact_5100_unit__factor__nat__def,axiom,
% 0.25/0.53 ( ( unit_f5069060285200089521factor @ nat )
% 0.25/0.53 = ( ^ [N5: nat] :
% 0.25/0.53 ( if @ nat
% 0.25/0.53 @ ( N5
% 0.25/0.53 = ( zero_zero @ nat ) )
% 0.25/0.53 @ ( zero_zero @ nat )
% 0.25/0.53 @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_nat_def
% 0.25/0.53 thf(fact_5101_option_Orel__Grp,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 0.25/0.53 ( ( rel_option @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F ) )
% 0.25/0.53 = ( bNF_Grp @ ( option @ A ) @ ( option @ B )
% 0.25/0.53 @ ( collect @ ( option @ A )
% 0.25/0.53 @ ^ [X3: option @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_option @ A @ X3 ) @ A4 ) )
% 0.25/0.53 @ ( map_option @ A @ B @ F ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % option.rel_Grp
% 0.25/0.53 thf(fact_5102_fun_Orel__Grp,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 0.25/0.53 ( ( bNF_rel_fun @ D @ D @ A @ B
% 0.25/0.53 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_Grp @ A @ B @ A4 @ F ) )
% 0.25/0.53 = ( bNF_Grp @ ( D > A ) @ ( D > B )
% 0.25/0.53 @ ( collect @ ( D > A )
% 0.25/0.53 @ ^ [X3: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) @ A4 ) )
% 0.25/0.53 @ ( comp @ A @ B @ D @ F ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % fun.rel_Grp
% 0.25/0.53 thf(fact_5103_relcomp__def,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.53 ( ( relcomp @ A @ B @ C )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
% 0.25/0.53 ( collect @ ( product_prod @ A @ C )
% 0.25/0.53 @ ( product_case_prod @ A @ C @ $o
% 0.25/0.53 @ ( relcompp @ A @ B @ C
% 0.25/0.53 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 0.25/0.53 @ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S4 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relcomp_def
% 0.25/0.53 thf(fact_5104_unit__factor__is__unit,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semido2269285787275462019factor @ A )
% 0.25/0.53 => ! [A3: A] :
% 0.25/0.53 ( ( A3
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 => ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_is_unit
% 0.25/0.53 thf(fact_5105_unit__factor__gcd,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_gcd @ A )
% 0.25/0.53 => ! [A3: A,B2: A] :
% 0.25/0.53 ( ( ( ( A3
% 0.25/0.53 = ( zero_zero @ A ) )
% 0.25/0.53 & ( B2
% 0.25/0.53 = ( zero_zero @ A ) ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.25/0.53 = ( zero_zero @ A ) ) )
% 0.25/0.53 & ( ~ ( ( A3
% 0.25/0.53 = ( zero_zero @ A ) )
% 0.25/0.53 & ( B2
% 0.25/0.53 = ( zero_zero @ A ) ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_gcd
% 0.25/0.53 thf(fact_5106_type__copy__vimage2p__Grp__Abs,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: A > B,Abs: B > A,G: D > C,P: C > $o,H3: C > A] :
% 0.25/0.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( ( bNF_vimage2p @ D @ C @ B @ A @ $o @ G @ Abs @ ( bNF_Grp @ C @ A @ ( collect @ C @ P ) @ H3 ) )
% 0.25/0.53 = ( bNF_Grp @ D @ B
% 0.25/0.53 @ ( collect @ D
% 0.25/0.53 @ ^ [X3: D] : ( P @ ( G @ X3 ) ) )
% 0.25/0.53 @ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H3 ) @ G ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % type_copy_vimage2p_Grp_Abs
% 0.25/0.53 thf(fact_5107_type__copy__vimage2p__Grp__Rep,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: A > B,Abs: B > A,F: C > D,P: D > $o,H3: D > B] :
% 0.25/0.53 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 0.25/0.53 => ( ( bNF_vimage2p @ C @ D @ A @ B @ $o @ F @ Rep @ ( bNF_Grp @ D @ B @ ( collect @ D @ P ) @ H3 ) )
% 0.25/0.53 = ( bNF_Grp @ C @ A
% 0.25/0.53 @ ( collect @ C
% 0.25/0.53 @ ^ [X3: C] : ( P @ ( F @ X3 ) ) )
% 0.25/0.53 @ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H3 ) @ F ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % type_copy_vimage2p_Grp_Rep
% 0.25/0.53 thf(fact_5108_Nat_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% 0.25/0.53
% 0.25/0.53 % Nat.transfer
% 0.25/0.53 thf(fact_5109_pcr__natural__def,axiom,
% 0.25/0.53 ( code_pcr_natural
% 0.25/0.53 = ( relcompp @ nat @ nat @ code_natural
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ code_cr_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % pcr_natural_def
% 0.25/0.53 thf(fact_5110_natural_Orep__transfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ^ [X3: nat] : X3
% 0.25/0.53 @ code_nat_of_natural ) ).
% 0.25/0.53
% 0.25/0.53 % natural.rep_transfer
% 0.25/0.53 thf(fact_5111_natural_Oid__abs__transfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ code_pcr_natural
% 0.25/0.53 @ ^ [X3: nat] : X3
% 0.25/0.53 @ code_natural_of_nat ) ).
% 0.25/0.53
% 0.25/0.53 % natural.id_abs_transfer
% 0.25/0.53 thf(fact_5112_natural_Opcr__cr__eq,axiom,
% 0.25/0.53 code_pcr_natural = code_cr_natural ).
% 0.25/0.53
% 0.25/0.53 % natural.pcr_cr_eq
% 0.25/0.53 thf(fact_5113_zero__natural_Otransfer,axiom,
% 0.25/0.53 code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % zero_natural.transfer
% 0.25/0.53 thf(fact_5114_one__natural_Otransfer,axiom,
% 0.25/0.53 code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % one_natural.transfer
% 0.25/0.53 thf(fact_5115_less__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.53 @ ( ord_less @ nat )
% 0.25/0.53 @ ( ord_less @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % less_natural.transfer
% 0.25/0.53 thf(fact_5116_vimage2p__comp,axiom,
% 0.25/0.53 ! [E: $tType,D: $tType,F5: $tType,A: $tType,C: $tType,B: $tType,G3: $tType,F1: F5 > A,F22: D > F5,G1: G3 > B,G22: E > G3] :
% 0.25/0.53 ( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F5 @ A @ D @ F1 @ F22 ) @ ( comp @ G3 @ B @ E @ G1 @ G22 ) )
% 0.25/0.53 = ( comp @ ( F5 > G3 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F5 @ E @ G3 @ C @ F22 @ G22 ) @ ( bNF_vimage2p @ F5 @ A @ G3 @ B @ C @ F1 @ G1 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % vimage2p_comp
% 0.25/0.53 thf(fact_5117_plus__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( plus_plus @ nat ) @ ( plus_plus @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % plus_natural.transfer
% 0.25/0.53 thf(fact_5118_Suc_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% 0.25/0.53
% 0.25/0.53 % Suc.transfer
% 0.25/0.53 thf(fact_5119_minus__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( minus_minus @ nat ) @ ( minus_minus @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % minus_natural.transfer
% 0.25/0.53 thf(fact_5120_pcr__integer__def,axiom,
% 0.25/0.53 ( code_pcr_integer
% 0.25/0.53 = ( relcompp @ int @ int @ code_integer
% 0.25/0.53 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.53 @ code_cr_integer ) ) ).
% 0.25/0.53
% 0.25/0.53 % pcr_integer_def
% 0.25/0.53 thf(fact_5121_less__eq__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 0.25/0.53 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.53 @ ( ord_less_eq @ nat )
% 0.25/0.53 @ ( ord_less_eq @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % less_eq_natural.transfer
% 0.25/0.53 thf(fact_5122_times__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( times_times @ nat ) @ ( times_times @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % times_natural.transfer
% 0.25/0.53 thf(fact_5123_divide__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( divide_divide @ nat ) @ ( divide_divide @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % divide_natural.transfer
% 0.25/0.53 thf(fact_5124_modulo__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( modulo_modulo @ nat ) @ ( modulo_modulo @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % modulo_natural.transfer
% 0.25/0.53 thf(fact_5125_take__bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.25/0.53 @ ( bit_se2584673776208193580ke_bit @ nat )
% 0.25/0.53 @ ( bit_se2584673776208193580ke_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % take_bit_natural.transfer
% 0.25/0.53 thf(fact_5126_xor__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se5824344971392196577ns_xor @ nat ) @ ( bit_se5824344971392196577ns_xor @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % xor_natural.transfer
% 0.25/0.53 thf(fact_5127_or__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se1065995026697491101ons_or @ nat ) @ ( bit_se1065995026697491101ons_or @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % or_natural.transfer
% 0.25/0.53 thf(fact_5128_and__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se5824344872417868541ns_and @ nat ) @ ( bit_se5824344872417868541ns_and @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % and_natural.transfer
% 0.25/0.53 thf(fact_5129_bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_natural
% 0.25/0.53 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ nat )
% 0.25/0.53 @ ( bit_se5641148757651400278ts_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % bit_natural.transfer
% 0.25/0.53 thf(fact_5130_push__bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.25/0.53 @ ( bit_se4730199178511100633sh_bit @ nat )
% 0.25/0.53 @ ( bit_se4730199178511100633sh_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % push_bit_natural.transfer
% 0.25/0.53 thf(fact_5131_mask__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ code_pcr_natural
% 0.25/0.53 @ ( bit_se2239418461657761734s_mask @ nat )
% 0.25/0.53 @ ( bit_se2239418461657761734s_mask @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % mask_natural.transfer
% 0.25/0.53 thf(fact_5132_drop__bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.25/0.53 @ ( bit_se4197421643247451524op_bit @ nat )
% 0.25/0.53 @ ( bit_se4197421643247451524op_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % drop_bit_natural.transfer
% 0.25/0.53 thf(fact_5133_division__segment__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ ( euclid7384307370059645450egment @ nat ) @ ( euclid7384307370059645450egment @ code_natural ) ).
% 0.25/0.53
% 0.25/0.53 % division_segment_natural.transfer
% 0.25/0.53 thf(fact_5134_flip__bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.25/0.53 @ ( bit_se8732182000553998342ip_bit @ nat )
% 0.25/0.53 @ ( bit_se8732182000553998342ip_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % flip_bit_natural.transfer
% 0.25/0.53 thf(fact_5135_set__bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.25/0.53 @ ( bit_se5668285175392031749et_bit @ nat )
% 0.25/0.53 @ ( bit_se5668285175392031749et_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % set_bit_natural.transfer
% 0.25/0.53 thf(fact_5136_unset__bit__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 0.25/0.53 @ ( bit_se2638667681897837118et_bit @ nat )
% 0.25/0.53 @ ( bit_se2638667681897837118et_bit @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % unset_bit_natural.transfer
% 0.25/0.53 thf(fact_5137_euclidean__size__natural_Otransfer,axiom,
% 0.25/0.53 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 0.25/0.53 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.53 @ ( euclid6346220572633701492n_size @ nat )
% 0.25/0.53 @ ( euclid6346220572633701492n_size @ code_natural ) ) ).
% 0.25/0.53
% 0.25/0.53 % euclidean_size_natural.transfer
% 0.25/0.53 thf(fact_5138_integer__of__natural_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ nat @ code_natural @ int @ code_integer @ code_pcr_natural @ code_pcr_integer @ ( semiring_1_of_nat @ int ) @ code_i5400310926305786745atural ).
% 0.25/0.53
% 0.25/0.53 % integer_of_natural.transfer
% 0.25/0.53 thf(fact_5139_natural__of__integer_Otransfer,axiom,
% 0.25/0.53 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% 0.25/0.53
% 0.25/0.53 % natural_of_integer.transfer
% 0.25/0.53 thf(fact_5140_fun_Orel__compp__Grp,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,A: $tType,R5: A > B > $o] :
% 0.25/0.53 ( ( bNF_rel_fun @ D @ D @ A @ B
% 0.25/0.53 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 0.25/0.53 @ R5 )
% 0.25/0.53 = ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
% 0.25/0.53 @ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
% 0.25/0.53 @ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
% 0.25/0.53 @ ( collect @ ( D > ( product_prod @ A @ B ) )
% 0.25/0.53 @ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
% 0.25/0.53 @ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
% 0.25/0.53 @ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
% 0.25/0.53 @ ( collect @ ( D > ( product_prod @ A @ B ) )
% 0.25/0.53 @ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
% 0.25/0.53 @ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % fun.rel_compp_Grp
% 0.25/0.53 thf(fact_5141_plus__num__def,axiom,
% 0.25/0.53 ( ( plus_plus @ num )
% 0.25/0.53 = ( ^ [M: num,N5: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % plus_num_def
% 0.25/0.53 thf(fact_5142_Lcm__no__units,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ( ( gcd_Lcm @ A )
% 0.25/0.53 = ( ^ [A9: set @ A] :
% 0.25/0.53 ( gcd_Lcm @ A
% 0.25/0.53 @ ( minus_minus @ ( set @ A ) @ A9
% 0.25/0.53 @ ( collect @ A
% 0.25/0.53 @ ^ [A7: A] : ( dvd_dvd @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Lcm_no_units
% 0.25/0.53 thf(fact_5143_remdups__adj__altdef,axiom,
% 0.25/0.53 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 0.25/0.53 ( ( ( remdups_adj @ A @ Xs )
% 0.25/0.53 = Ys2 )
% 0.25/0.53 = ( ? [F2: nat > nat] :
% 0.25/0.53 ( ( order_mono @ nat @ nat @ F2 )
% 0.25/0.53 & ( ( image2 @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 0.25/0.53 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
% 0.25/0.53 & ! [I3: nat] :
% 0.25/0.53 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.53 => ( ( nth @ A @ Xs @ I3 )
% 0.25/0.53 = ( nth @ A @ Ys2 @ ( F2 @ I3 ) ) ) )
% 0.25/0.53 & ! [I3: nat] :
% 0.25/0.53 ( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 0.25/0.53 => ( ( ( nth @ A @ Xs @ I3 )
% 0.25/0.53 = ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
% 0.25/0.53 = ( ( F2 @ I3 )
% 0.25/0.53 = ( F2 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % remdups_adj_altdef
% 0.25/0.53 thf(fact_5144_Lcm__empty,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Lcm_empty
% 0.25/0.53 thf(fact_5145_Lcm__1__iff,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ! [A4: set @ A] :
% 0.25/0.53 ( ( ( gcd_Lcm @ A @ A4 )
% 0.25/0.53 = ( one_one @ A ) )
% 0.25/0.53 = ( ! [X3: A] :
% 0.25/0.53 ( ( member @ A @ X3 @ A4 )
% 0.25/0.53 => ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Lcm_1_iff
% 0.25/0.53 thf(fact_5146_mono__add,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( ordere6658533253407199908up_add @ A )
% 0.25/0.53 => ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mono_add
% 0.25/0.53 thf(fact_5147_Lcm__nat__empty,axiom,
% 0.25/0.53 ( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.53 = ( one_one @ nat ) ) ).
% 0.25/0.53
% 0.25/0.53 % Lcm_nat_empty
% 0.25/0.53 thf(fact_5148_nat__of__num_Osimps_I2_J,axiom,
% 0.25/0.53 ! [X: num] :
% 0.25/0.53 ( ( nat_of_num @ ( bit0 @ X ) )
% 0.25/0.53 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nat_of_num.simps(2)
% 0.25/0.53 thf(fact_5149_nat__of__num__code_I1_J,axiom,
% 0.25/0.53 ( ( nat_of_num @ one2 )
% 0.25/0.53 = ( one_one @ nat ) ) ).
% 0.25/0.53
% 0.25/0.53 % nat_of_num_code(1)
% 0.25/0.53 thf(fact_5150_nat__of__num__add,axiom,
% 0.25/0.53 ! [X: num,Y: num] :
% 0.25/0.53 ( ( nat_of_num @ ( plus_plus @ num @ X @ Y ) )
% 0.25/0.53 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nat_of_num_add
% 0.25/0.53 thf(fact_5151_nat__of__num__code_I2_J,axiom,
% 0.25/0.53 ! [N: num] :
% 0.25/0.53 ( ( nat_of_num @ ( bit0 @ N ) )
% 0.25/0.53 = ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nat_of_num_code(2)
% 0.25/0.53 thf(fact_5152_mono__image__least,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType] :
% 0.25/0.53 ( ( ( order @ B )
% 0.25/0.53 & ( order @ A ) )
% 0.25/0.53 => ! [F: A > B,M2: A,N: A,M7: B,N2: B] :
% 0.25/0.53 ( ( order_mono @ A @ B @ F )
% 0.25/0.53 => ( ( ( image2 @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
% 0.25/0.53 = ( set_or7035219750837199246ssThan @ B @ M7 @ N2 ) )
% 0.25/0.53 => ( ( ord_less @ A @ M2 @ N )
% 0.25/0.53 => ( ( F @ M2 )
% 0.25/0.53 = M7 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % mono_image_least
% 0.25/0.53 thf(fact_5153_nat__of__num_Osimps_I3_J,axiom,
% 0.25/0.53 ! [X: num] :
% 0.25/0.53 ( ( nat_of_num @ ( bit1 @ X ) )
% 0.25/0.53 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nat_of_num.simps(3)
% 0.25/0.53 thf(fact_5154_unit__factor__Lcm,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ! [A4: set @ A] :
% 0.25/0.53 ( ( ( ( gcd_Lcm @ A @ A4 )
% 0.25/0.53 = ( zero_zero @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
% 0.25/0.53 = ( zero_zero @ A ) ) )
% 0.25/0.53 & ( ( ( gcd_Lcm @ A @ A4 )
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_Lcm
% 0.25/0.53 thf(fact_5155_nat__of__num__code_I3_J,axiom,
% 0.25/0.53 ! [N: num] :
% 0.25/0.53 ( ( nat_of_num @ ( bit1 @ N ) )
% 0.25/0.53 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % nat_of_num_code(3)
% 0.25/0.53 thf(fact_5156_remdups__adj_Opelims,axiom,
% 0.25/0.53 ! [A: $tType,X: list @ A,Y: list @ A] :
% 0.25/0.53 ( ( ( remdups_adj @ A @ X )
% 0.25/0.53 = Y )
% 0.25/0.53 => ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
% 0.25/0.53 => ( ( ( X
% 0.25/0.53 = ( nil @ A ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( nil @ A ) )
% 0.25/0.53 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
% 0.25/0.53 => ( ! [X2: A] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.53 => ( ( Y
% 0.25/0.53 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.53 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 0.25/0.53 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 0.25/0.53 ( ( X
% 0.25/0.53 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 0.25/0.53 => ( ( ( ( X2 = Y4 )
% 0.25/0.53 => ( Y
% 0.25/0.53 = ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 0.25/0.53 & ( ( X2 != Y4 )
% 0.25/0.53 => ( Y
% 0.25/0.53 = ( cons @ A @ X2 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) )
% 0.25/0.53 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % remdups_adj.pelims
% 0.25/0.53 thf(fact_5157_relChain__def,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] :
% 0.25/0.53 ( ( ord @ B )
% 0.25/0.53 => ( ( bNF_Ca3754400796208372196lChain @ A @ B )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ A ),As3: A > B] :
% 0.25/0.53 ! [I3: A,J2: A] :
% 0.25/0.53 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J2 ) @ R3 )
% 0.25/0.53 => ( ord_less_eq @ B @ ( As3 @ I3 ) @ ( As3 @ J2 ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % relChain_def
% 0.25/0.53 thf(fact_5158_unit__factor__Gcd,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ! [A4: set @ A] :
% 0.25/0.53 ( ( ( ( gcd_Gcd @ A @ A4 )
% 0.25/0.53 = ( zero_zero @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
% 0.25/0.53 = ( zero_zero @ A ) ) )
% 0.25/0.53 & ( ( ( gcd_Gcd @ A @ A4 )
% 0.25/0.53 != ( zero_zero @ A ) )
% 0.25/0.53 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % unit_factor_Gcd
% 0.25/0.53 thf(fact_5159_Gcd__UNIV,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
% 0.25/0.53 = ( one_one @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Gcd_UNIV
% 0.25/0.53 thf(fact_5160_Gcd__1,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ! [A4: set @ A] :
% 0.25/0.53 ( ( member @ A @ ( one_one @ A ) @ A4 )
% 0.25/0.53 => ( ( gcd_Gcd @ A @ A4 )
% 0.25/0.53 = ( one_one @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Gcd_1
% 0.25/0.53 thf(fact_5161_Gcd__nat__eq__one,axiom,
% 0.25/0.53 ! [N6: set @ nat] :
% 0.25/0.53 ( ( member @ nat @ ( one_one @ nat ) @ N6 )
% 0.25/0.53 => ( ( gcd_Gcd @ nat @ N6 )
% 0.25/0.53 = ( one_one @ nat ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Gcd_nat_eq_one
% 0.25/0.53 thf(fact_5162_Gcd__eq__1__I,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( semiring_Gcd @ A )
% 0.25/0.53 => ! [A3: A,A4: set @ A] :
% 0.25/0.53 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.53 => ( ( member @ A @ A3 @ A4 )
% 0.25/0.53 => ( ( gcd_Gcd @ A @ A4 )
% 0.25/0.53 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % Gcd_eq_1_I
% 0.25/0.53 thf(fact_5163_cofinal__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bNF_Ca7293521722713021262ofinal @ A )
% 0.25/0.53 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ! [X3: A] :
% 0.25/0.53 ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
% 0.25/0.53 => ? [Y3: A] :
% 0.25/0.53 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.53 & ( X3 != Y3 )
% 0.25/0.53 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % cofinal_def
% 0.25/0.53 thf(fact_5164_card__of__UNION__ordLeq__infinite,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,B6: set @ A,I4: set @ B,A4: B > ( set @ C )] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ B6 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( ! [X2: B] :
% 0.25/0.53 ( ( member @ B @ X2 @ I4 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_UNION_ordLeq_infinite
% 0.25/0.53 thf(fact_5165_card__of__ordLess,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( ~ ? [F2: A > B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.25/0.53 & ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B6 ) ) )
% 0.25/0.53 = ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLess
% 0.25/0.53 thf(fact_5166_card__of__Times3,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A] :
% 0.25/0.53 ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
% 0.25/0.53 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.53 @ ^ [Uu: A] : A4 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times3
% 0.25/0.53 thf(fact_5167_card__of__Sigma__mono1,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ C )] :
% 0.25/0.53 ( ! [X2: A] :
% 0.25/0.53 ( ( member @ A @ X2 @ I4 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B6 @ X2 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Sigma_mono1
% 0.25/0.53 thf(fact_5168_card__of__Times__mono1,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
% 0.25/0.53 @ ( product_Sigma @ A @ C @ A4
% 0.25/0.53 @ ^ [Uu: A] : C5 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
% 0.25/0.53 @ ( product_Sigma @ B @ C @ B6
% 0.25/0.53 @ ^ [Uu: B] : C5 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_mono1
% 0.25/0.53 thf(fact_5169_card__of__Times__mono2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
% 0.25/0.53 @ ( product_Sigma @ C @ A @ C5
% 0.25/0.53 @ ^ [Uu: C] : A4 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
% 0.25/0.53 @ ( product_Sigma @ C @ B @ C5
% 0.25/0.53 @ ^ [Uu: C] : B6 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_mono2
% 0.25/0.53 thf(fact_5170_infinite__iff__card__of__nat,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A] :
% 0.25/0.53 ( ( ~ ( finite_finite2 @ A @ A4 ) )
% 0.25/0.53 = ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % infinite_iff_card_of_nat
% 0.25/0.53 thf(fact_5171_card__of__UNION__Sigma,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: B > ( set @ A ),I4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I4 @ A4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_UNION_Sigma
% 0.25/0.53 thf(fact_5172_card__of__Times__commute,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.53 @ ( product_Sigma @ B @ A @ B6
% 0.25/0.53 @ ^ [Uu: B] : A4 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_commute
% 0.25/0.53 thf(fact_5173_card__of__Times__same__infinite,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
% 0.25/0.53 @ ( product_Sigma @ A @ A @ A4
% 0.25/0.53 @ ^ [Uu: A] : A4 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_same_infinite
% 0.25/0.53 thf(fact_5174_card__of__Times1,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( A4
% 0.25/0.53 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.53 @ ( product_Sigma @ B @ A @ B6
% 0.25/0.53 @ ^ [Uu: B] : A4 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times1
% 0.25/0.53 thf(fact_5175_card__of__Times2,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( A4
% 0.25/0.53 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times2
% 0.25/0.53 thf(fact_5176_card__of__bool,axiom,
% 0.25/0.53 ! [A: $tType,A1: A,A22: A] :
% 0.25/0.53 ( ( A1 != A22 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_bool
% 0.25/0.53 thf(fact_5177_card__of__Plus__empty1,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_empty1
% 0.25/0.53 thf(fact_5178_card__of__Plus__empty2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_empty2
% 0.25/0.53 thf(fact_5179_card__of__Plus__Times__bool,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A] :
% 0.25/0.53 ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ A4 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
% 0.25/0.53 @ ( product_Sigma @ A @ $o @ A4
% 0.25/0.53 @ ^ [Uu: A] : ( top_top @ ( set @ $o ) ) ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_Times_bool
% 0.25/0.53 thf(fact_5180_card__of__Plus__commute,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_commute
% 0.25/0.53 thf(fact_5181_card__of__Plus__cong2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_cong2
% 0.25/0.53 thf(fact_5182_card__of__Plus__cong1,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_cong1
% 0.25/0.53 thf(fact_5183_card__of__Plus__assoc,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_Plus @ ( sum_sum @ A @ B ) @ C @ ( sum_Plus @ A @ B @ A4 @ B6 ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A4 @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_assoc
% 0.25/0.53 thf(fact_5184_card__of__Plus__cong,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C,D4: set @ D] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_cong
% 0.25/0.53 thf(fact_5185_card__of__Times__Plus__distrib,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ ( sum_sum @ B @ C ) )
% 0.25/0.53 @ ( product_Sigma @ A @ ( sum_sum @ B @ C ) @ A4
% 0.25/0.53 @ ^ [Uu: A] : ( sum_Plus @ B @ C @ B6 @ C5 ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) )
% 0.25/0.53 @ ( sum_Plus @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 )
% 0.25/0.53 @ ( product_Sigma @ A @ C @ A4
% 0.25/0.53 @ ^ [Uu: A] : C5 ) ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_Plus_distrib
% 0.25/0.53 thf(fact_5186_card__of__Plus__mono2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_mono2
% 0.25/0.53 thf(fact_5187_card__of__Plus__mono1,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_mono1
% 0.25/0.53 thf(fact_5188_card__of__Plus__mono,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C,D4: set @ D] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_mono
% 0.25/0.53 thf(fact_5189_card__of__Plus2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,B6: set @ A,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus2
% 0.25/0.53 thf(fact_5190_card__of__Plus1,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus1
% 0.25/0.53 thf(fact_5191_card__of__Un__Plus__ordLeq,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A,B6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Un_Plus_ordLeq
% 0.25/0.53 thf(fact_5192_card__of__Plus__infinite,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_infinite
% 0.25/0.53 thf(fact_5193_card__of__Plus__infinite1,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_infinite1
% 0.25/0.53 thf(fact_5194_card__of__Plus__infinite2,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_infinite2
% 0.25/0.53 thf(fact_5195_card__of__Plus__ordLess__infinite,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,C5: set @ A,A4: set @ B,B6: set @ C] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ C5 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_ordLess_infinite
% 0.25/0.53 thf(fact_5196_card__of__Sigma__ordLeq__infinite,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,B6: set @ A,I4: set @ B,A4: B > ( set @ C )] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ B6 )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( ! [X2: B] :
% 0.25/0.53 ( ( member @ B @ X2 @ I4 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I4 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Sigma_ordLeq_infinite
% 0.25/0.53 thf(fact_5197_card__of__refl,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_refl
% 0.25/0.53 thf(fact_5198_card__of__Plus__Times__aux,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( ( A1 != A22 )
% 0.25/0.53 & ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_Times_aux
% 0.25/0.53 thf(fact_5199_card__of__Times__infinite__simps_I4_J,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( B6
% 0.25/0.53 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.53 @ ( product_Sigma @ B @ A @ B6
% 0.25/0.53 @ ^ [Uu: B] : A4 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_infinite_simps(4)
% 0.25/0.53 thf(fact_5200_card__of__Times__infinite__simps_I3_J,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( B6
% 0.25/0.53 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.53 @ ( product_Sigma @ B @ A @ B6
% 0.25/0.53 @ ^ [Uu: B] : A4 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_infinite_simps(3)
% 0.25/0.53 thf(fact_5201_card__of__Times__infinite__simps_I2_J,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( B6
% 0.25/0.53 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_infinite_simps(2)
% 0.25/0.53 thf(fact_5202_card__of__Times__infinite__simps_I1_J,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( B6
% 0.25/0.53 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_infinite_simps(1)
% 0.25/0.53 thf(fact_5203_card__of__Times__infinite,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ( ( B6
% 0.25/0.53 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.53 @ ( product_Sigma @ B @ A @ B6
% 0.25/0.53 @ ^ [Uu: B] : A4 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Times_infinite
% 0.25/0.53 thf(fact_5204_card__of__Plus__Times,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B1: B,B22: B,B6: set @ B] :
% 0.25/0.53 ( ( ( A1 != A22 )
% 0.25/0.53 & ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
% 0.25/0.53 => ( ( ( B1 != B22 )
% 0.25/0.53 & ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B1 @ ( insert2 @ B @ B22 @ ( bot_bot @ ( set @ B ) ) ) ) @ B6 ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.53 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.53 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Plus_Times
% 0.25/0.53 thf(fact_5205_card__of__image,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: B > A,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( image2 @ B @ A @ F @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_image
% 0.25/0.53 thf(fact_5206_card__of__empty,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_empty
% 0.25/0.53 thf(fact_5207_card__of__empty3,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( A4
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_empty3
% 0.25/0.53 thf(fact_5208_card__of__ordLeq__finite,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( ( finite_finite2 @ B @ B6 )
% 0.25/0.53 => ( finite_finite2 @ A @ A4 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLeq_finite
% 0.25/0.53 thf(fact_5209_card__of__ordLeq__infinite,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.53 => ~ ( finite_finite2 @ B @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLeq_infinite
% 0.25/0.53 thf(fact_5210_card__of__mono1,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.53 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_mono1
% 0.25/0.53 thf(fact_5211_card__of__empty__ordIso,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_empty_ordIso
% 0.25/0.53 thf(fact_5212_card__of__empty2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( A4
% 0.25/0.53 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_empty2
% 0.25/0.53 thf(fact_5213_card__of__ordIso__finite,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( ( finite_finite2 @ A @ A4 )
% 0.25/0.53 = ( finite_finite2 @ B @ B6 ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordIso_finite
% 0.25/0.53 thf(fact_5214_card__of__mono2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_mono2
% 0.25/0.53 thf(fact_5215_card__of__cong,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_cong
% 0.25/0.53 thf(fact_5216_card__of__ordLeqI,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.53 => ( ! [A5: A] :
% 0.25/0.53 ( ( member @ A @ A5 @ A4 )
% 0.25/0.53 => ( member @ B @ ( F @ A5 ) @ B6 ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLeqI
% 0.25/0.53 thf(fact_5217_ex__bij__betw,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ? [F4: B > A,B10: set @ B] : ( bij_betw @ B @ A @ F4 @ B10 @ A4 ) ) ).
% 0.25/0.53
% 0.25/0.53 % ex_bij_betw
% 0.25/0.53 thf(fact_5218_card__of__least,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( order_well_order_on @ A @ A4 @ R )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_least
% 0.25/0.53 thf(fact_5219_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % BNF_Cardinal_Order_Relation.ordLess_Field
% 0.25/0.53 thf(fact_5220_card__of__ordIsoI,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordIsoI
% 0.25/0.53 thf(fact_5221_card__of__ordIso,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B6 ) )
% 0.25/0.53 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordIso
% 0.25/0.53 thf(fact_5222_type__copy__set__bd,axiom,
% 0.25/0.53 ! [A: $tType,D: $tType,C: $tType,B: $tType,S2: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X: D] :
% 0.25/0.53 ( ! [Y4: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S2 @ Y4 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S2 @ Rep @ X ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % type_copy_set_bd
% 0.25/0.53 thf(fact_5223_card__of__ordLeq2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( A4
% 0.25/0.53 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( ( ? [G2: B > A] :
% 0.25/0.53 ( ( image2 @ B @ A @ G2 @ B6 )
% 0.25/0.53 = A4 ) )
% 0.25/0.53 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLeq2
% 0.25/0.53 thf(fact_5224_surj__imp__ordLeq,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,B6: set @ A,F: B > A,A4: set @ B] :
% 0.25/0.53 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A4 ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % surj_imp_ordLeq
% 0.25/0.53 thf(fact_5225_card__of__singl__ordLeq,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,B2: B] :
% 0.25/0.53 ( ( A4
% 0.25/0.53 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_singl_ordLeq
% 0.25/0.53 thf(fact_5226_card__of__ordLess2,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,B6: set @ A,A4: set @ B] :
% 0.25/0.53 ( ( B6
% 0.25/0.53 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.53 => ( ( ~ ? [F2: B > A] :
% 0.25/0.53 ( ( image2 @ B @ A @ F2 @ A4 )
% 0.25/0.53 = B6 ) )
% 0.25/0.53 = ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLess2
% 0.25/0.53 thf(fact_5227_internalize__card__of__ordLeq2,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,C5: set @ B] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 = ( ? [B9: set @ B] :
% 0.25/0.53 ( ( ord_less_eq @ ( set @ B ) @ B9 @ C5 )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % internalize_card_of_ordLeq2
% 0.25/0.53 thf(fact_5228_card__of__Field__ordLess,axiom,
% 0.25/0.53 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_Field_ordLess
% 0.25/0.53 thf(fact_5229_ordLeq__Plus__mono,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P5 @ P8 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLeq_Plus_mono
% 0.25/0.53 thf(fact_5230_ordLeq__Plus__mono1,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLeq_Plus_mono1
% 0.25/0.53 thf(fact_5231_ordLeq__Plus__mono2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A4 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLeq_Plus_mono2
% 0.25/0.53 thf(fact_5232_ordIso__Plus__cong,axiom,
% 0.25/0.53 ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P5 @ P8 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordIso_Plus_cong
% 0.25/0.53 thf(fact_5233_ordIso__Plus__cong1,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordIso_Plus_cong1
% 0.25/0.53 thf(fact_5234_ordIso__Plus__cong2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A4 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordIso_Plus_cong2
% 0.25/0.53 thf(fact_5235_ordLeq__Times__mono1,axiom,
% 0.25/0.53 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
% 0.25/0.53 @ ( product_Sigma @ A @ C @ ( field2 @ A @ R )
% 0.25/0.53 @ ^ [Uu: A] : C5 ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
% 0.25/0.53 @ ( product_Sigma @ B @ C @ ( field2 @ B @ R4 )
% 0.25/0.53 @ ^ [Uu: B] : C5 ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLeq_Times_mono1
% 0.25/0.53 thf(fact_5236_ordLeq__Times__mono2,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
% 0.25/0.53 @ ( product_Sigma @ C @ A @ A4
% 0.25/0.53 @ ^ [Uu: C] : ( field2 @ A @ R ) ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
% 0.25/0.53 @ ( product_Sigma @ C @ B @ A4
% 0.25/0.53 @ ^ [Uu: C] : ( field2 @ B @ R4 ) ) ) )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % ordLeq_Times_mono2
% 0.25/0.53 thf(fact_5237_card__of__ordLeq,axiom,
% 0.25/0.53 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 0.25/0.53 ( ( ? [F2: A > B] :
% 0.25/0.53 ( ( inj_on @ A @ B @ F2 @ A4 )
% 0.25/0.53 & ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B6 ) ) )
% 0.25/0.53 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordLeq
% 0.25/0.53 thf(fact_5238_internalize__card__of__ordLeq,axiom,
% 0.25/0.53 ! [A: $tType,B: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 = ( ? [B9: set @ B] :
% 0.25/0.53 ( ( ord_less_eq @ ( set @ B ) @ B9 @ ( field2 @ B @ R ) )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.53 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % internalize_card_of_ordLeq
% 0.25/0.53 thf(fact_5239_regularCard__def,axiom,
% 0.25/0.53 ! [A: $tType] :
% 0.25/0.53 ( ( bNF_Ca7133664381575040944arCard @ A )
% 0.25/0.53 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.53 ! [K3: set @ A] :
% 0.25/0.53 ( ( ( ord_less_eq @ ( set @ A ) @ K3 @ ( field2 @ A @ R3 ) )
% 0.25/0.53 & ( bNF_Ca7293521722713021262ofinal @ A @ K3 @ R3 ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K3 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % regularCard_def
% 0.25/0.53 thf(fact_5240_card__of__ordIso__subst,axiom,
% 0.25/0.53 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 0.25/0.53 ( ( A4 = B6 )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % card_of_ordIso_subst
% 0.25/0.53 thf(fact_5241_comp__set__bd__Union__o__collect,axiom,
% 0.25/0.53 ! [C: $tType,B: $tType,A: $tType,X: C,X10: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
% 0.25/0.53 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
% 0.25/0.53 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
% 0.25/0.53 @ ( bNF_Ca6860139660246222851ard_of @ A
% 0.25/0.53 @ ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.53 @ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
% 0.25/0.53 @ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
% 0.25/0.53 @ ^ [F2: C > ( set @ ( set @ A ) )] : ( F2 @ X )
% 0.25/0.53 @ X10 ) ) ) )
% 0.25/0.53 @ Hbd )
% 0.25/0.53 @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.53 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X10 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 0.25/0.53
% 0.25/0.53 % comp_set_bd_Union_o_collect
% 0.25/0.53 thf(fact_5242_card__of__Plus__ordLess__infinite__Field,axiom,
% 0.25/0.53 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
% 0.25/0.53 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.53 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.53 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Plus_ordLess_infinite_Field
% 0.25/0.54 thf(fact_5243_card__of__Times__ordLeq__infinite__Field,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
% 0.25/0.54 @ ( product_Sigma @ B @ C @ A4
% 0.25/0.54 @ ^ [Uu: B] : B6 ) )
% 0.25/0.54 @ R )
% 0.25/0.54 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Times_ordLeq_infinite_Field
% 0.25/0.54 thf(fact_5244_card__of__Sigma__ordLeq__infinite__Field,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ! [X2: B] :
% 0.25/0.54 ( ( member @ B @ X2 @ I4 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I4 @ A4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Sigma_ordLeq_infinite_Field
% 0.25/0.54 thf(fact_5245_Card__order__trans,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( X != Y )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.54 => ( ( Y != Z2 )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 0.25/0.54 => ( ( X != Z2 )
% 0.25/0.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_trans
% 0.25/0.54 thf(fact_5246_card__order__on__ordIso,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R4 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_order_on_ordIso
% 0.25/0.54 thf(fact_5247_infinite__Card__order__limit,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.54 => ? [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.54 & ( A3 != X2 )
% 0.25/0.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % infinite_Card_order_limit
% 0.25/0.54 thf(fact_5248_ordLeq__refl,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLeq_refl
% 0.25/0.54 thf(fact_5249_Card__order__ordIso,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
% 0.25/0.54 => ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_ordIso
% 0.25/0.54 thf(fact_5250_Card__order__ordIso2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_ordIso2
% 0.25/0.54 thf(fact_5251_ordIso__refl,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordIso_refl
% 0.25/0.54 thf(fact_5252_card__of__unique,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_unique
% 0.25/0.54 thf(fact_5253_card__order__on__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A )
% 0.25/0.54 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( order_well_order_on @ A @ A9 @ R3 )
% 0.25/0.54 & ! [R11: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( order_well_order_on @ A @ A9 @ R11 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_order_on_def
% 0.25/0.54 thf(fact_5254_Card__order__iff__ordLeq__card__of,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_iff_ordLeq_card_of
% 0.25/0.54 thf(fact_5255_ordIso__card__of__imp__Card__order,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordIso_card_of_imp_Card_order
% 0.25/0.54 thf(fact_5256_Card__order__iff__ordIso__card__of,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_iff_ordIso_card_of
% 0.25/0.54 thf(fact_5257_card__of__Field__ordIso,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Field_ordIso
% 0.25/0.54 thf(fact_5258_dir__image,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ! [X2: A,Y4: A] :
% 0.25/0.54 ( ( ( F @ X2 )
% 0.25/0.54 = ( F @ Y4 ) )
% 0.25/0.54 = ( X2 = Y4 ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % dir_image
% 0.25/0.54 thf(fact_5259_exists__minim__Card__order,axiom,
% 0.25/0.54 ! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
% 0.25/0.54 ( ( R5
% 0.25/0.54 != ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
% 0.25/0.54 => ( ! [X2: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 0.25/0.54 => ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
% 0.25/0.54 => ? [X2: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 0.25/0.54 & ! [Xa2: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % exists_minim_Card_order
% 0.25/0.54 thf(fact_5260_Card__order__empty,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_empty
% 0.25/0.54 thf(fact_5261_card__of__ordIso__finite__Field,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 = ( finite_finite2 @ B @ A4 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_ordIso_finite_Field
% 0.25/0.54 thf(fact_5262_card__of__underS,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R @ A3 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_underS
% 0.25/0.54 thf(fact_5263_Card__order__Plus1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Plus1
% 0.25/0.54 thf(fact_5264_Card__order__Plus2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ ( field2 @ A @ R ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Plus2
% 0.25/0.54 thf(fact_5265_Card__order__singl__ordLeq,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),B2: B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( ( field2 @ A @ R )
% 0.25/0.54 != ( bot_bot @ ( set @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_singl_ordLeq
% 0.25/0.54 thf(fact_5266_card__of__Un__ordLeq__infinite__Field,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ B] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Un_ordLeq_infinite_Field
% 0.25/0.54 thf(fact_5267_card__of__empty1,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 | ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_empty1
% 0.25/0.54 thf(fact_5268_Card__order__Times1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( B6
% 0.25/0.54 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.54 @ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
% 0.25/0.54 @ ^ [Uu: A] : B6 ) ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Times1
% 0.25/0.54 thf(fact_5269_Card__order__Times2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( A4
% 0.25/0.54 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.54 @ ( product_Sigma @ B @ A @ A4
% 0.25/0.54 @ ^ [Uu: B] : ( field2 @ A @ R ) ) ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Times2
% 0.25/0.54 thf(fact_5270_Card__order__Times__same__infinite,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
% 0.25/0.54 @ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
% 0.25/0.54 @ ^ [Uu: A] : ( field2 @ A @ R ) ) )
% 0.25/0.54 @ R )
% 0.25/0.54 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Times_same_infinite
% 0.25/0.54 thf(fact_5271_card__of__UNION__ordLeq__infinite__Field,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ! [X2: B] :
% 0.25/0.54 ( ( member @ B @ X2 @ I4 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_UNION_ordLeq_infinite_Field
% 0.25/0.54 thf(fact_5272_Card__order__Plus__infinite,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ ( field2 @ B @ P5 ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
% 0.25/0.54 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( field2 @ B @ P5 ) @ ( field2 @ A @ R ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Plus_infinite
% 0.25/0.54 thf(fact_5273_Card__order__iff__Restr__underS,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 = ( ! [X3: A] :
% 0.25/0.54 ( ( member @ A @ X3 @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.54 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 0.25/0.54 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
% 0.25/0.54 @ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) )
% 0.25/0.54 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_iff_Restr_underS
% 0.25/0.54 thf(fact_5274_regularCard__UNION,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: A > ( set @ B ),B6: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca7133664381575040944arCard @ A @ R )
% 0.25/0.54 => ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R @ As4 )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As4 @ ( field2 @ A @ R ) ) ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 0.25/0.54 => ? [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.54 & ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % regularCard_UNION
% 0.25/0.54 thf(fact_5275_card__of__Plus__ordLeq__infinite__Field,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Plus_ordLeq_infinite_Field
% 0.25/0.54 thf(fact_5276_Card__order__Times__infinite,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( ( field2 @ B @ P5 )
% 0.25/0.54 != ( bot_bot @ ( set @ B ) ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 0.25/0.54 @ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
% 0.25/0.54 @ ^ [Uu: A] : ( field2 @ B @ P5 ) ) )
% 0.25/0.54 @ R )
% 0.25/0.54 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
% 0.25/0.54 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 0.25/0.54 @ ( product_Sigma @ B @ A @ ( field2 @ B @ P5 )
% 0.25/0.54 @ ^ [Uu: B] : ( field2 @ A @ R ) ) )
% 0.25/0.54 @ R )
% 0.25/0.54 @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Times_infinite
% 0.25/0.54 thf(fact_5277_ex__toCard__pred,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 0.25/0.54 => ? [X_12: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R @ X_12 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ex_toCard_pred
% 0.25/0.54 thf(fact_5278_cardSuc__UNION,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B6: set @ B] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ? [X2: set @ A] :
% 0.25/0.54 ( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
% 0.25/0.54 & ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_UNION
% 0.25/0.54 thf(fact_5279_toCard__inj,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),X: A,Y: A] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 0.25/0.54 => ( ( member @ A @ X @ A4 )
% 0.25/0.54 => ( ( member @ A @ Y @ A4 )
% 0.25/0.54 => ( ( ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ X )
% 0.25/0.54 = ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ Y ) )
% 0.25/0.54 = ( X = Y ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % toCard_inj
% 0.25/0.54 thf(fact_5280_cardSuc__greater,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_greater
% 0.25/0.54 thf(fact_5281_cardSuc__ordLeq,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_ordLeq
% 0.25/0.54 thf(fact_5282_toCard__pred__toCard,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 0.25/0.54 => ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % toCard_pred_toCard
% 0.25/0.54 thf(fact_5283_cardSuc__least__aux,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R4 ) @ R4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_least_aux
% 0.25/0.54 thf(fact_5284_cardSuc__ordLeq__ordLess,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
% 0.25/0.54 = ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_ordLeq_ordLess
% 0.25/0.54 thf(fact_5285_cardSuc__ordLess__ordLeq,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.54 = ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_ordLess_ordLeq
% 0.25/0.54 thf(fact_5286_cardSuc__least,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_least
% 0.25/0.54 thf(fact_5287_cardSuc__mono__ordLeq,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ B ) ) )
% 0.25/0.54 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_mono_ordLeq
% 0.25/0.54 thf(fact_5288_cardSuc__invar__ordIso,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( set @ B ) ) )
% 0.25/0.54 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_invar_ordIso
% 0.25/0.54 thf(fact_5289_fromCard__toCard,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),B2: A] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 0.25/0.54 => ( ( member @ A @ B2 @ A4 )
% 0.25/0.54 => ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A4 @ R @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ B2 ) )
% 0.25/0.54 = B2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fromCard_toCard
% 0.25/0.54 thf(fact_5290_cardSuc__UNION__Cinfinite,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B6: set @ B] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ? [X2: set @ A] :
% 0.25/0.54 ( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
% 0.25/0.54 & ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cardSuc_UNION_Cinfinite
% 0.25/0.54 thf(fact_5291_isCardSuc__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( bNF_Ca6246979054910435723ardSuc @ A )
% 0.25/0.54 = ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R11 ) @ R11 )
% 0.25/0.54 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
% 0.25/0.54 & ! [R14: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
% 0.25/0.54 ( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R14 ) @ R14 )
% 0.25/0.54 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ R14 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R11 @ R14 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % isCardSuc_def
% 0.25/0.54 thf(fact_5292_cinfinite__mono,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
% 0.25/0.54 => ( bNF_Ca4139267488887388095finite @ B @ R22 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cinfinite_mono
% 0.25/0.54 thf(fact_5293_Cinfinite__limit2,axiom,
% 0.25/0.54 ! [A: $tType,X1: A,R: set @ ( product_prod @ A @ A ),X22: A] :
% 0.25/0.54 ( ( member @ A @ X1 @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( member @ A @ X22 @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ? [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.54 & ( X1 != X2 )
% 0.25/0.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X2 ) @ R )
% 0.25/0.54 & ( X22 != X2 )
% 0.25/0.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X2 ) @ R ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_limit2
% 0.25/0.54 thf(fact_5294_Cinfinite__limit,axiom,
% 0.25/0.54 ! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( member @ A @ X @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ? [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.54 & ( X != X2 )
% 0.25/0.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X2 ) @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_limit
% 0.25/0.54 thf(fact_5295_Cinfinite__cong,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 0.25/0.54 => ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_cong
% 0.25/0.54 thf(fact_5296_Cinfinite__limit__finite,axiom,
% 0.25/0.54 ! [A: $tType,X10: set @ A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( finite_finite2 @ A @ X10 )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ A ) @ X10 @ ( field2 @ A @ R ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ? [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 0.25/0.54 & ! [Xa2: A] :
% 0.25/0.54 ( ( member @ A @ Xa2 @ X10 )
% 0.25/0.54 => ( ( Xa2 != X2 )
% 0.25/0.54 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X2 ) @ R ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_limit_finite
% 0.25/0.54 thf(fact_5297_Un__Cinfinite__bound,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),B6: set @ A] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Un_Cinfinite_bound
% 0.25/0.54 thf(fact_5298_UNION__Cinfinite__bound,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,C: $tType,I4: set @ A,R: set @ ( product_prod @ B @ B ),A4: A > ( set @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ! [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ I4 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A4 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % UNION_Cinfinite_bound
% 0.25/0.54 thf(fact_5299_Cfinite__ordLess__Cinfinite,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ( bNF_Cardinal_cfinite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ S ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cfinite_ordLess_Cinfinite
% 0.25/0.54 thf(fact_5300_comp__single__set__bd,axiom,
% 0.25/0.54 ! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X: D] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
% 0.25/0.54 => ( ! [X2: B] : ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( Fset @ X2 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
% 0.25/0.54 => ( ! [X2: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X2 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % comp_single_set_bd
% 0.25/0.54 thf(fact_5301_cprod__dup,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P5 @ P8 ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P5 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_dup
% 0.25/0.54 thf(fact_5302_cprod__com,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_com
% 0.25/0.54 thf(fact_5303_Cfinite__cprod__Cinfinite,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ( bNF_Cardinal_cfinite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ S ) @ S ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cfinite_cprod_Cinfinite
% 0.25/0.54 thf(fact_5304_cprod__mono,axiom,
% 0.25/0.54 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_mono
% 0.25/0.54 thf(fact_5305_cprod__mono1,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_mono1
% 0.25/0.54 thf(fact_5306_cprod__mono2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_mono2
% 0.25/0.54 thf(fact_5307_cprod__cong,axiom,
% 0.25/0.54 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_cong
% 0.25/0.54 thf(fact_5308_cprod__cong1,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ P24 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_cong1
% 0.25/0.54 thf(fact_5309_cprod__cong2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_cong2
% 0.25/0.54 thf(fact_5310_cprod__infinite,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_infinite
% 0.25/0.54 thf(fact_5311_cprod__cinfinite__bound,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P5: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P5 ) @ P5 )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P5 @ Q3 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_cinfinite_bound
% 0.25/0.54 thf(fact_5312_card__of__Csum__Times_H,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ! [X2: B] :
% 0.25/0.54 ( ( member @ B @ X2 @ I4 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
% 0.25/0.54 @ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 )
% 0.25/0.54 @ ^ [I3: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ I3 ) ) )
% 0.25/0.54 @ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Csum_Times'
% 0.25/0.54 thf(fact_5313_card__of__Csum__Times,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: set @ C] :
% 0.25/0.54 ( ! [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ I4 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) )
% 0.25/0.54 @ ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 )
% 0.25/0.54 @ ^ [I3: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ I3 ) ) )
% 0.25/0.54 @ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Csum_Times
% 0.25/0.54 thf(fact_5314_cprod__infinite1_H,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P5 ) @ P5 ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ P5 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_infinite1'
% 0.25/0.54 thf(fact_5315_czero__ordIso,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_czero @ A ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% 0.25/0.54
% 0.25/0.54 % czero_ordIso
% 0.25/0.54 thf(fact_5316_cinfinite__not__czero,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 => ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cinfinite_not_czero
% 0.25/0.54 thf(fact_5317_czeroE,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( field2 @ A @ R )
% 0.25/0.54 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % czeroE
% 0.25/0.54 thf(fact_5318_card__of__ordIso__czero__iff__empty,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 = ( A4
% 0.25/0.54 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_ordIso_czero_iff_empty
% 0.25/0.54 thf(fact_5319_czeroI,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( ( field2 @ A @ R )
% 0.25/0.54 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % czeroI
% 0.25/0.54 thf(fact_5320_Cnotzero__imp__not__empty,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( field2 @ A @ R )
% 0.25/0.54 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cnotzero_imp_not_empty
% 0.25/0.54 thf(fact_5321_Cnotzero__mono,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ Q3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cnotzero_mono
% 0.25/0.54 thf(fact_5322_Cinfinite__Cnotzero,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_Cnotzero
% 0.25/0.54 thf(fact_5323_Cnotzero__UNIV,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cnotzero_UNIV
% 0.25/0.54 thf(fact_5324_cinfinite__cprod2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
% 0.25/0.54 => ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cinfinite_cprod2
% 0.25/0.54 thf(fact_5325_Cinfinite__cprod2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
% 0.25/0.54 => ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_cprod2
% 0.25/0.54 thf(fact_5326_ordLeq__cprod2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P24 ) @ P24 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ P24 @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLeq_cprod2
% 0.25/0.54 thf(fact_5327_cone__ordLeq__Cnotzero,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cone_ordLeq_Cnotzero
% 0.25/0.54 thf(fact_5328_csum__dup,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Cardinal_csum @ B @ C @ P5 @ P8 ) @ ( bNF_Cardinal_csum @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ ( sum_sum @ A @ A ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ C @ P5 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_dup
% 0.25/0.54 thf(fact_5329_csum__Cnotzero1,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 0.25/0.54 => ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_Cnotzero1
% 0.25/0.54 thf(fact_5330_csum__csum,axiom,
% 0.25/0.54 ! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_csum @ C @ D @ R33 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R1 @ R33 ) @ ( bNF_Cardinal_csum @ B @ D @ R22 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_csum
% 0.25/0.54 thf(fact_5331_csum__assoc,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B ),P32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ P32 ) @ ( bNF_Cardinal_csum @ A @ ( sum_sum @ B @ C ) @ P13 @ ( bNF_Cardinal_csum @ B @ C @ P24 @ P32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_assoc
% 0.25/0.54 thf(fact_5332_cprod__csum__distrib1,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cprod @ A @ C @ R1 @ R33 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R1 @ ( bNF_Cardinal_csum @ B @ C @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_csum_distrib1
% 0.25/0.54 thf(fact_5333_cone__not__czero,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ~ ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ product_unit @ A ) ) ).
% 0.25/0.54
% 0.25/0.54 % cone_not_czero
% 0.25/0.54 thf(fact_5334_csum__com,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_com
% 0.25/0.54 thf(fact_5335_Un__csum,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,B6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Un_csum
% 0.25/0.54 thf(fact_5336_ordLeq__csum1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ P13 @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLeq_csum1
% 0.25/0.54 thf(fact_5337_ordLeq__csum2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),P13: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ P24 @ ( bNF_Cardinal_csum @ B @ A @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLeq_csum2
% 0.25/0.54 thf(fact_5338_csum__cong,axiom,
% 0.25/0.54 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_cong
% 0.25/0.54 thf(fact_5339_csum__cong1,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_cong1
% 0.25/0.54 thf(fact_5340_csum__cong2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_cong2
% 0.25/0.54 thf(fact_5341_csum__mono,axiom,
% 0.25/0.54 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_mono
% 0.25/0.54 thf(fact_5342_csum__mono1,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_mono1
% 0.25/0.54 thf(fact_5343_csum__mono2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_mono2
% 0.25/0.54 thf(fact_5344_Plus__csum,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Plus_csum
% 0.25/0.54 thf(fact_5345_csum__absorb2_H,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
% 0.25/0.54 | ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ A @ R1 @ R22 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_absorb2'
% 0.25/0.54 thf(fact_5346_csum__absorb1_H,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
% 0.25/0.54 | ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_absorb1'
% 0.25/0.54 thf(fact_5347_csum__absorb1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R22 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_absorb1
% 0.25/0.54 thf(fact_5348_csum__cinfinite__bound,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P5: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P5 ) @ P5 )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_csum @ A @ C @ P5 @ Q3 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_cinfinite_bound
% 0.25/0.54 thf(fact_5349_natLeq__ordLeq__cinfinite,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ R ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % natLeq_ordLeq_cinfinite
% 0.25/0.54 thf(fact_5350_cexp__mono2__Cnotzero,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P24 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_mono2_Cnotzero
% 0.25/0.54 thf(fact_5351_cexp__mono2,axiom,
% 0.25/0.54 ! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ ( bNF_Cardinal_czero @ D ) ) @ ( bNF_Wellorder_ordIso @ A @ D ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_mono2
% 0.25/0.54 thf(fact_5352_cprod__cexp,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C ),T2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_cexp
% 0.25/0.54 thf(fact_5353_cexp__cprod,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ C @ C ),R33: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R1 @ ( bNF_Cardinal_cprod @ C @ B @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_cprod
% 0.25/0.54 thf(fact_5354_cprod__cexp__csum__cexp__Cinfinite,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R @ S ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_cexp_csum_cexp_Cinfinite
% 0.25/0.54 thf(fact_5355_card__of__Field__natLeq,axiom,
% 0.25/0.54 member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Field_natLeq
% 0.25/0.54 thf(fact_5356_cexp__cone,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R @ bNF_Cardinal_cone ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_cone
% 0.25/0.54 thf(fact_5357_finite__iff__ordLess__natLeq,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( finite_finite2 @ A )
% 0.25/0.54 = ( ^ [A9: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A9 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % finite_iff_ordLess_natLeq
% 0.25/0.54 thf(fact_5358_cexp__cprod__ordLeq,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
% 0.25/0.54 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R33 @ ( bNF_Cardinal_czero @ C ) ) @ ( bNF_Wellorder_ordIso @ C @ C ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ R33 ) @ R33 ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R33 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_cprod_ordLeq
% 0.25/0.54 thf(fact_5359_cexp__mono_H,axiom,
% 0.25/0.54 ! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 0.25/0.54 => ( ( ( ( field2 @ C @ P24 )
% 0.25/0.54 = ( bot_bot @ ( set @ C ) ) )
% 0.25/0.54 => ( ( field2 @ D @ R22 )
% 0.25/0.54 = ( bot_bot @ ( set @ D ) ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_mono'
% 0.25/0.54 thf(fact_5360_cexp__mono1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_mono1
% 0.25/0.54 thf(fact_5361_cexp__cong2,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_cong2
% 0.25/0.54 thf(fact_5362_cexp__cong1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_cong1
% 0.25/0.54 thf(fact_5363_cexp__cong,axiom,
% 0.25/0.54 ! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R22 ) @ R22 )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_cong
% 0.25/0.54 thf(fact_5364_card__of__nat,axiom,
% 0.25/0.54 member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_nat
% 0.25/0.54 thf(fact_5365_cexp__mono2_H,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 0.25/0.54 => ( ( ( ( field2 @ A @ P24 )
% 0.25/0.54 = ( bot_bot @ ( set @ A ) ) )
% 0.25/0.54 => ( ( field2 @ B @ R22 )
% 0.25/0.54 = ( bot_bot @ ( set @ B ) ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_mono2'
% 0.25/0.54 thf(fact_5366_ordLeq__cexp1,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q3 @ ( bNF_Cardinal_cexp @ B @ A @ Q3 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLeq_cexp1
% 0.25/0.54 thf(fact_5367_infinite__iff__natLeq__ordLeq,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A] :
% 0.25/0.54 ( ( finite_finite2 @ A @ A4 )
% 0.25/0.54 != ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % infinite_iff_natLeq_ordLeq
% 0.25/0.54 thf(fact_5368_cexp__mono,axiom,
% 0.25/0.54 ! [E: $tType,F5: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 0.25/0.54 => ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) @ P24 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F5 @ F5 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F5 @ F5 ) ) @ R22 @ ( bNF_Cardinal_czero @ F5 ) ) @ ( bNF_Wellorder_ordIso @ D @ F5 ) ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cexp_mono
% 0.25/0.54 thf(fact_5369_ordLeq__cexp2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
% 0.25/0.54 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLeq_cexp2
% 0.25/0.54 thf(fact_5370_Cfinite__cexp__Cinfinite,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( ( bNF_Cardinal_cfinite @ A @ S )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S ) @ S ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cfinite_cexp_Cinfinite
% 0.25/0.54 thf(fact_5371_cinfinite__cexp,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 0.25/0.54 => ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cinfinite_cexp
% 0.25/0.54 thf(fact_5372_ctwo__Cnotzero,axiom,
% 0.25/0.54 ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ $o ) ) @ ( bNF_Wellorder_ordIso @ $o @ $o ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ $o @ ( field2 @ $o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% 0.25/0.54
% 0.25/0.54 % ctwo_Cnotzero
% 0.25/0.54 thf(fact_5373_ctwo__ordLess__natLeq,axiom,
% 0.25/0.54 member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ $o @ nat ) ).
% 0.25/0.54
% 0.25/0.54 % ctwo_ordLess_natLeq
% 0.25/0.54 thf(fact_5374_ordLess__ctwo__cexp,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ordLess_ctwo_cexp
% 0.25/0.54 thf(fact_5375_cprod__csum__cexp,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % cprod_csum_cexp
% 0.25/0.54 thf(fact_5376_ctwo__not__czero,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ).
% 0.25/0.54
% 0.25/0.54 % ctwo_not_czero
% 0.25/0.54 thf(fact_5377_ctwo__ordLeq__Cinfinite,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ctwo_ordLeq_Cinfinite
% 0.25/0.54 thf(fact_5378_ctwo__ordLess__Cinfinite,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % ctwo_ordLess_Cinfinite
% 0.25/0.54 thf(fact_5379_csum__Cfinite__cexp__Cinfinite,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( ( ( bNF_Cardinal_cfinite @ B @ S )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % csum_Cfinite_cexp_Cinfinite
% 0.25/0.54 thf(fact_5380_Cinfinite__cexp,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 0.25/0.54 ( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
% 0.25/0.54 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 0.25/0.54 => ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) )
% 0.25/0.54 & ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cinfinite_cexp
% 0.25/0.54 thf(fact_5381_card__of__Func__UNIV,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,B6: set @ B] :
% 0.25/0.54 ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B6 ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
% 0.25/0.54 @ ( collect @ ( A > B )
% 0.25/0.54 @ ^ [F2: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) @ B6 ) ) ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Func_UNIV
% 0.25/0.54 thf(fact_5382_Card__order__Pow,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 0.25/0.54 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ ( field2 @ A @ R ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Card_order_Pow
% 0.25/0.54 thf(fact_5383_Func__Times__Range,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.54 ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
% 0.25/0.54 @ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A4
% 0.25/0.54 @ ( product_Sigma @ B @ C @ B6
% 0.25/0.54 @ ^ [Uu: B] : C5 ) ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
% 0.25/0.54 @ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A4 @ B6 )
% 0.25/0.54 @ ^ [Uu: A > B] : ( bNF_Wellorder_Func @ A @ C @ A4 @ C5 ) ) ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Func_Times_Range
% 0.25/0.54 thf(fact_5384_card__of__Func__Times,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 0.25/0.54 ( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
% 0.25/0.54 @ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
% 0.25/0.54 @ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
% 0.25/0.54 @ ( product_Sigma @ A @ B @ A4
% 0.25/0.54 @ ^ [Uu: A] : B6 )
% 0.25/0.54 @ C5 ) )
% 0.25/0.54 @ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B6 @ C5 ) ) ) )
% 0.25/0.54 @ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Func_Times
% 0.25/0.54 thf(fact_5385_card__of__Pow__Func,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A4 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Pow_Func
% 0.25/0.54 thf(fact_5386_Cantors__paradox,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A] :
% 0.25/0.54 ~ ? [F8: A > ( set @ A )] :
% 0.25/0.54 ( ( image2 @ A @ ( set @ A ) @ F8 @ A4 )
% 0.25/0.54 = ( pow @ A @ A4 ) ) ).
% 0.25/0.54
% 0.25/0.54 % Cantors_paradox
% 0.25/0.54 thf(fact_5387_inj__on__image__Pow,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 0.25/0.54 ( ( inj_on @ A @ B @ F @ A4 )
% 0.25/0.54 => ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( pow @ A @ A4 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % inj_on_image_Pow
% 0.25/0.54 thf(fact_5388_card__of__Pow,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A4 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % card_of_Pow
% 0.25/0.54 thf(fact_5389_lfp__induct2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,A3: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) )
% 0.25/0.54 => ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F )
% 0.25/0.54 => ( ! [A5: A,B3: B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( F @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
% 0.25/0.54 => ( P @ A5 @ B3 ) )
% 0.25/0.54 => ( P @ A3 @ B2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lfp_induct2
% 0.25/0.54 thf(fact_5390_sort__quicksort__by__rel,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( linorder @ A )
% 0.25/0.54 => ( ( linorder_sort_key @ A @ A
% 0.25/0.54 @ ^ [X3: A] : X3 )
% 0.25/0.54 = ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sort_quicksort_by_rel
% 0.25/0.54 thf(fact_5391_Gr__def,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( bNF_Gr @ A @ B )
% 0.25/0.54 = ( ^ [A9: set @ A,F2: A > B] :
% 0.25/0.54 ( collect @ ( product_prod @ A @ B )
% 0.25/0.54 @ ^ [Uu: product_prod @ A @ B] :
% 0.25/0.54 ? [A7: A] :
% 0.25/0.54 ( ( Uu
% 0.25/0.54 = ( product_Pair @ A @ B @ A7 @ ( F2 @ A7 ) ) )
% 0.25/0.54 & ( member @ A @ A7 @ A9 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Gr_def
% 0.25/0.54 thf(fact_5392_GrD2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: A,Fx: B,A4: set @ A,F: A > B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F ) )
% 0.25/0.54 => ( ( F @ X )
% 0.25/0.54 = Fx ) ) ).
% 0.25/0.54
% 0.25/0.54 % GrD2
% 0.25/0.54 thf(fact_5393_GrD1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: A,Fx: B,A4: set @ A,F: A > B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F ) )
% 0.25/0.54 => ( member @ A @ X @ A4 ) ) ).
% 0.25/0.54
% 0.25/0.54 % GrD1
% 0.25/0.54 thf(fact_5394_sort__mergesort,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( linorder @ A )
% 0.25/0.54 => ( ( linorder_sort_key @ A @ A
% 0.25/0.54 @ ^ [X3: A] : X3 )
% 0.25/0.54 = ( mergesort @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sort_mergesort
% 0.25/0.54 thf(fact_5395_sort__mergesort__by__rel,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( linorder @ A )
% 0.25/0.54 => ( ( linorder_sort_key @ A @ A
% 0.25/0.54 @ ^ [X3: A] : X3 )
% 0.25/0.54 = ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sort_mergesort_by_rel
% 0.25/0.54 thf(fact_5396_remdup__sort__mergesort__remdups,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( linorder @ A )
% 0.25/0.54 => ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
% 0.25/0.54 @ ( linorder_sort_key @ A @ A
% 0.25/0.54 @ ^ [X3: A] : X3 ) )
% 0.25/0.54 = ( mergesort_remdups @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % remdup_sort_mergesort_remdups
% 0.25/0.54 thf(fact_5397_prod_H__def,axiom,
% 0.25/0.54 ! [C: $tType,A: $tType] :
% 0.25/0.54 ( ( comm_monoid_mult @ A )
% 0.25/0.54 => ( ( groups1962203154675924110t_prod @ C @ A )
% 0.25/0.54 = ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % prod'_def
% 0.25/0.54 thf(fact_5398_prod_Ocomm__monoid__list__set__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_mult @ A )
% 0.25/0.54 => ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % prod.comm_monoid_list_set_axioms
% 0.25/0.54 thf(fact_5399_length__remdups__card,axiom,
% 0.25/0.54 ! [A: $tType,L: list @ A] :
% 0.25/0.54 ( ( size_size @ ( list @ A ) @ ( remdups @ A @ L ) )
% 0.25/0.54 = ( finite_card @ A @ ( set2 @ A @ L ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % length_remdups_card
% 0.25/0.54 thf(fact_5400_sum_H__def,axiom,
% 0.25/0.54 ! [C: $tType,A: $tType] :
% 0.25/0.54 ( ( comm_monoid_add @ A )
% 0.25/0.54 => ( ( groups1027152243600224163dd_sum @ C @ A )
% 0.25/0.54 = ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sum'_def
% 0.25/0.54 thf(fact_5401_sum_Ocomm__monoid__list__set__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_add @ A )
% 0.25/0.54 => ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sum.comm_monoid_list_set_axioms
% 0.25/0.54 thf(fact_5402_and_Ocomm__monoid__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( bit_ri3973907225187159222ations @ A )
% 0.25/0.54 => ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % and.comm_monoid_axioms
% 0.25/0.54 thf(fact_5403_bijective__alt,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( bijective @ A @ B )
% 0.25/0.54 = ( ^ [R6: set @ ( product_prod @ A @ B )] :
% 0.25/0.54 ( ( single_valued @ A @ B @ R6 )
% 0.25/0.54 & ( single_valued @ B @ A @ ( converse @ A @ B @ R6 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % bijective_alt
% 0.25/0.54 thf(fact_5404_Heap_Osize__gen__o__map,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: B > nat,G: A > B] :
% 0.25/0.54 ( ( comp @ ( heap_Time_Heap @ B ) @ nat @ ( heap_Time_Heap @ A ) @ ( heap_Time_size_Heap @ B @ F ) @ ( heap_Time_map_Heap @ A @ B @ G ) )
% 0.25/0.54 = ( heap_Time_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.size_gen_o_map
% 0.25/0.54 thf(fact_5405_single__valued__inter1,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.54 ( ( single_valued @ A @ B @ R5 )
% 0.25/0.54 => ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valued_inter1
% 0.25/0.54 thf(fact_5406_single__valued__inter2,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.25/0.54 ( ( single_valued @ A @ B @ R5 )
% 0.25/0.54 => ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 @ R5 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valued_inter2
% 0.25/0.54 thf(fact_5407_comm__monoid_Ocomm__neutral,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A,A3: A] :
% 0.25/0.54 ( ( comm_monoid @ A @ F @ Z2 )
% 0.25/0.54 => ( ( F @ A3 @ Z2 )
% 0.25/0.54 = A3 ) ) ).
% 0.25/0.54
% 0.25/0.54 % comm_monoid.comm_neutral
% 0.25/0.54 thf(fact_5408_Heap_Omap__ident,axiom,
% 0.25/0.54 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( heap_Time_map_Heap @ A @ A
% 0.25/0.54 @ ^ [X3: A] : X3
% 0.25/0.54 @ T2 )
% 0.25/0.54 = T2 ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_ident
% 0.25/0.54 thf(fact_5409_Heap_Omap__id0,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
% 0.25/0.54 = ( id @ ( heap_Time_Heap @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_id0
% 0.25/0.54 thf(fact_5410_Heap_Omap__id,axiom,
% 0.25/0.54 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) @ T2 )
% 0.25/0.54 = T2 ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_id
% 0.25/0.54 thf(fact_5411_Heap_Omap__comp,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( heap_Time_map_Heap @ B @ C @ G @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 0.25/0.54 = ( heap_Time_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_comp
% 0.25/0.54 thf(fact_5412_Heap_Omap,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: A > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( heap_Time_map_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 0.25/0.54 = ( heap_Time_Heap2 @ B @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map
% 0.25/0.54 thf(fact_5413_single__valuedD,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
% 0.25/0.54 ( ( single_valued @ A @ B @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ R )
% 0.25/0.54 => ( Y = Z2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valuedD
% 0.25/0.54 thf(fact_5414_single__valuedI,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.54 ( ! [X2: A,Y4: B,Z3: B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R )
% 0.25/0.54 => ( Y4 = Z3 ) ) )
% 0.25/0.54 => ( single_valued @ A @ B @ R ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valuedI
% 0.25/0.54 thf(fact_5415_single__valued__def,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( single_valued @ A @ B )
% 0.25/0.54 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.25/0.54 ! [X3: A,Y3: B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 0.25/0.54 => ! [Z5: B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R3 )
% 0.25/0.54 => ( Y3 = Z5 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valued_def
% 0.25/0.54 thf(fact_5416_add_Ocomm__monoid__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_add @ A )
% 0.25/0.54 => ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % add.comm_monoid_axioms
% 0.25/0.54 thf(fact_5417_mult_Ocomm__monoid__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_mult @ A )
% 0.25/0.54 => ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mult.comm_monoid_axioms
% 0.25/0.54 thf(fact_5418_Heap_Oinj__map,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.54 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.54 => ( inj_on @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_map_Heap @ A @ B @ F ) @ ( top_top @ ( set @ ( heap_Time_Heap @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.inj_map
% 0.25/0.54 thf(fact_5419_single__valued__confluent,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 0.25/0.54 ( ( single_valued @ A @ A @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.54 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valued_confluent
% 0.25/0.54 thf(fact_5420_single__valued__below__Id,axiom,
% 0.25/0.54 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) )
% 0.25/0.54 => ( single_valued @ A @ A @ R5 ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valued_below_Id
% 0.25/0.54 thf(fact_5421_Heap_Orec__o__map,axiom,
% 0.25/0.54 ! [B: $tType,C: $tType,A: $tType,G: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,F: A > B] :
% 0.25/0.54 ( ( comp @ ( heap_Time_Heap @ B ) @ C @ ( heap_Time_Heap @ A ) @ ( heap_Time_rec_Heap @ B @ C @ G ) @ ( heap_Time_map_Heap @ A @ B @ F ) )
% 0.25/0.54 = ( heap_Time_rec_Heap @ A @ C
% 0.25/0.54 @ ^ [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( G @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rec_o_map
% 0.25/0.54 thf(fact_5422_single__valuedp__single__valued__eq,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.54 ( ( single_valuedp @ A @ B
% 0.25/0.54 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.25/0.54 = ( single_valued @ A @ B @ R ) ) ).
% 0.25/0.54
% 0.25/0.54 % single_valuedp_single_valued_eq
% 0.25/0.54 thf(fact_5423_disjnt__equiv__class,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( disjnt @ A @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.54 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % disjnt_equiv_class
% 0.25/0.54 thf(fact_5424_equiv__class__eq__iff,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.54 = ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.54 = ( image @ A @ A @ R @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.54 & ( member @ A @ X @ A4 )
% 0.25/0.54 & ( member @ A @ Y @ A4 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % equiv_class_eq_iff
% 0.25/0.54 thf(fact_5425_eq__equiv__class__iff,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ A @ X @ A4 )
% 0.25/0.54 => ( ( member @ A @ Y @ A4 )
% 0.25/0.54 => ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.54 = ( image @ A @ A @ R @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.54 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % eq_equiv_class_iff
% 0.25/0.54 thf(fact_5426_equiv__class__eq,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.54 => ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.54 = ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % equiv_class_eq
% 0.25/0.54 thf(fact_5427_eq__equiv__class,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,A4: set @ A] :
% 0.25/0.54 ( ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.25/0.54 = ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.54 => ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ A @ B2 @ A4 )
% 0.25/0.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % eq_equiv_class
% 0.25/0.54 thf(fact_5428_Heap_Orec,axiom,
% 0.25/0.54 ! [C: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( heap_Time_rec_Heap @ A @ C @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 0.25/0.54 = ( F @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rec
% 0.25/0.54 thf(fact_5429_subset__equiv__class,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.25/0.54 => ( ( member @ A @ B2 @ A4 )
% 0.25/0.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % subset_equiv_class
% 0.25/0.54 thf(fact_5430_equiv__class__subset,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.54 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % equiv_class_subset
% 0.25/0.54 thf(fact_5431_equiv__class__nondisjoint,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.25/0.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % equiv_class_nondisjoint
% 0.25/0.54 thf(fact_5432_proj__iff,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 )
% 0.25/0.54 => ( ( ( equiv_proj @ A @ A @ R @ X )
% 0.25/0.54 = ( equiv_proj @ A @ A @ R @ Y ) )
% 0.25/0.54 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % proj_iff
% 0.25/0.54 thf(fact_5433_in__quotient__imp__in__rel,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X10 )
% 0.25/0.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % in_quotient_imp_in_rel
% 0.25/0.54 thf(fact_5434_congruent2__commuteI,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > A > B] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ! [Y4: A,Z3: A] :
% 0.25/0.54 ( ( member @ A @ Y4 @ A4 )
% 0.25/0.54 => ( ( member @ A @ Z3 @ A4 )
% 0.25/0.54 => ( ( F @ Y4 @ Z3 )
% 0.25/0.54 = ( F @ Z3 @ Y4 ) ) ) )
% 0.25/0.54 => ( ! [Y4: A,Z3: A,W4: A] :
% 0.25/0.54 ( ( member @ A @ W4 @ A4 )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.54 => ( ( F @ W4 @ Y4 )
% 0.25/0.54 = ( F @ W4 @ Z3 ) ) ) )
% 0.25/0.54 => ( equiv_congruent2 @ A @ A @ B @ R @ R @ F ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % congruent2_commuteI
% 0.25/0.54 thf(fact_5435_quotient__eqI,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,Y9: set @ A,X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( member @ A @ X @ X10 )
% 0.25/0.54 => ( ( member @ A @ Y @ Y9 )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.54 => ( X10 = Y9 ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % quotient_eqI
% 0.25/0.54 thf(fact_5436_quotient__eq__iff,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,Y9: set @ A,X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( member @ A @ X @ X10 )
% 0.25/0.54 => ( ( member @ A @ Y @ Y9 )
% 0.25/0.54 => ( ( X10 = Y9 )
% 0.25/0.54 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % quotient_eq_iff
% 0.25/0.54 thf(fact_5437_in__quotient__imp__closed,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( member @ A @ X @ X10 )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 0.25/0.54 => ( member @ A @ Y @ X10 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % in_quotient_imp_closed
% 0.25/0.54 thf(fact_5438_congruent2D,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z1: A,Y2: B,Z22: B] :
% 0.25/0.54 ( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
% 0.25/0.54 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R22 )
% 0.25/0.54 => ( ( F @ Y1 @ Y2 )
% 0.25/0.54 = ( F @ Z1 @ Z22 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % congruent2D
% 0.25/0.54 thf(fact_5439_congruent2I_H,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 0.25/0.54 ( ! [Y12: A,Z12: A,Y22: B,Z23: B] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z12 ) @ R1 )
% 0.25/0.54 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R22 )
% 0.25/0.54 => ( ( F @ Y12 @ Y22 )
% 0.25/0.54 = ( F @ Z12 @ Z23 ) ) ) )
% 0.25/0.54 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ).
% 0.25/0.54
% 0.25/0.54 % congruent2I'
% 0.25/0.54 thf(fact_5440_eq__equiv__class__iff2,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( member @ A @ X @ A4 )
% 0.25/0.54 => ( ( member @ A @ Y @ A4 )
% 0.25/0.54 => ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R )
% 0.25/0.54 = ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R ) )
% 0.25/0.54 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % eq_equiv_class_iff2
% 0.25/0.54 thf(fact_5441_congruent2I,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,A110: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A110 @ R1 )
% 0.25/0.54 => ( ( equiv_equiv @ B @ A26 @ R22 )
% 0.25/0.54 => ( ! [Y4: A,Z3: A,W4: B] :
% 0.25/0.54 ( ( member @ B @ W4 @ A26 )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R1 )
% 0.25/0.54 => ( ( F @ Y4 @ W4 )
% 0.25/0.54 = ( F @ Z3 @ W4 ) ) ) )
% 0.25/0.54 => ( ! [Y4: B,Z3: B,W4: A] :
% 0.25/0.54 ( ( member @ A @ W4 @ A110 )
% 0.25/0.54 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Z3 ) @ R22 )
% 0.25/0.54 => ( ( F @ W4 @ Y4 )
% 0.25/0.54 = ( F @ W4 @ Z3 ) ) ) )
% 0.25/0.54 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % congruent2I
% 0.25/0.54 thf(fact_5442_UN__equiv__class__inject,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X10: set @ A,Y9: set @ A] :
% 0.25/0.54 ( ( equiv_equiv @ A @ A4 @ R )
% 0.25/0.54 => ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
% 0.25/0.54 => ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X10 ) )
% 0.25/0.54 = ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ Y9 ) ) )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
% 0.25/0.54 => ( ! [X2: A,Y4: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ A4 )
% 0.25/0.54 => ( ( member @ A @ Y4 @ A4 )
% 0.25/0.54 => ( ( ( F @ X2 )
% 0.25/0.54 = ( F @ Y4 ) )
% 0.25/0.54 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R ) ) ) )
% 0.25/0.54 => ( X10 = Y9 ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % UN_equiv_class_inject
% 0.25/0.54 thf(fact_5443_equivp__equiv,axiom,
% 0.25/0.54 ! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A4 )
% 0.25/0.54 = ( equiv_equivp @ A
% 0.25/0.54 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ A4 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % equivp_equiv
% 0.25/0.54 thf(fact_5444_prod__mset__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_mult @ A )
% 0.25/0.54 => ( ( comm_m9189036328036947845d_mset @ A )
% 0.25/0.54 = ( comm_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % prod_mset_def
% 0.25/0.54 thf(fact_5445_congruentI,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
% 0.25/0.54 ( ! [Y4: A,Z3: A] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 0.25/0.54 => ( ( F @ Y4 )
% 0.25/0.54 = ( F @ Z3 ) ) )
% 0.25/0.54 => ( equiv_congruent @ A @ B @ R @ F ) ) ).
% 0.25/0.54
% 0.25/0.54 % congruentI
% 0.25/0.54 thf(fact_5446_congruentD,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B,Y: A,Z2: A] :
% 0.25/0.54 ( ( equiv_congruent @ A @ B @ R @ F )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 0.25/0.54 => ( ( F @ Y )
% 0.25/0.54 = ( F @ Z2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % congruentD
% 0.25/0.54 thf(fact_5447_sum__mset__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_add @ A )
% 0.25/0.54 => ( ( comm_m7189776963980413722m_mset @ A )
% 0.25/0.54 = ( comm_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sum_mset_def
% 0.25/0.54 thf(fact_5448_or__num_Opelims,axiom,
% 0.25/0.54 ! [X: num,Xa: num,Y: num] :
% 0.25/0.54 ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
% 0.25/0.54 = Y )
% 0.25/0.54 => ( ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 0.25/0.54 => ( ( ( X = one2 )
% 0.25/0.54 => ( ( Xa = one2 )
% 0.25/0.54 => ( ( Y = one2 )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 0.25/0.54 => ( ( ( X = one2 )
% 0.25/0.54 => ! [N3: num] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( bit0 @ N3 ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ N3 ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 0.25/0.54 => ( ( ( X = one2 )
% 0.25/0.54 => ! [N3: num] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( bit1 @ N3 ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ N3 ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 0.25/0.54 => ( ! [M5: num] :
% 0.25/0.54 ( ( X
% 0.25/0.54 = ( bit0 @ M5 ) )
% 0.25/0.54 => ( ( Xa = one2 )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ M5 ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 0.25/0.54 => ( ! [M5: num] :
% 0.25/0.54 ( ( X
% 0.25/0.54 = ( bit0 @ M5 ) )
% 0.25/0.54 => ! [N3: num] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( bit0 @ N3 ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit0 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.25/0.54 => ( ! [M5: num] :
% 0.25/0.54 ( ( X
% 0.25/0.54 = ( bit0 @ M5 ) )
% 0.25/0.54 => ! [N3: num] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( bit1 @ N3 ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 0.25/0.54 => ( ! [M5: num] :
% 0.25/0.54 ( ( X
% 0.25/0.54 = ( bit1 @ M5 ) )
% 0.25/0.54 => ( ( Xa = one2 )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ M5 ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 0.25/0.54 => ( ! [M5: num] :
% 0.25/0.54 ( ( X
% 0.25/0.54 = ( bit1 @ M5 ) )
% 0.25/0.54 => ! [N3: num] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( bit0 @ N3 ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 0.25/0.54 => ~ ! [M5: num] :
% 0.25/0.54 ( ( X
% 0.25/0.54 = ( bit1 @ M5 ) )
% 0.25/0.54 => ! [N3: num] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( bit1 @ N3 ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % or_num.pelims
% 0.25/0.54 thf(fact_5449_Heap_Oset,axiom,
% 0.25/0.54 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) )
% 0.25/0.54 = ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.54 @ ( image2 @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( set @ A )
% 0.25/0.54 @ ^ [Uu3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set @ A ) @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Uu3 ) ) )
% 0.25/0.54 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.set
% 0.25/0.54 thf(fact_5450_under__incr,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 0.25/0.54 ( ( trans @ A @ R )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.54 => ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A3 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % under_incr
% 0.25/0.54 thf(fact_5451_Heap_Oset__intros,axiom,
% 0.25/0.54 ! [A: $tType,Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ),Yb2: A] :
% 0.25/0.54 ( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Y @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
% 0.25/0.54 => ( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Ya @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Y ) )
% 0.25/0.54 => ( ( member @ A @ Yb2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Ya ) )
% 0.25/0.54 => ( member @ A @ Yb2 @ ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.set_intros
% 0.25/0.54 thf(fact_5452_Heap_Oset__cases,axiom,
% 0.25/0.54 ! [A: $tType,E2: A,A3: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( member @ A @ E2 @ ( heap_Time_set_Heap @ A @ A3 ) )
% 0.25/0.54 => ~ ! [Z3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( A3
% 0.25/0.54 = ( heap_Time_Heap2 @ A @ Z3 ) )
% 0.25/0.54 => ! [Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 0.25/0.54 ( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Xa3 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Z3 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
% 0.25/0.54 => ! [Xb3: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat )] :
% 0.25/0.54 ( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xb3 @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xa3 ) )
% 0.25/0.54 => ~ ( member @ A @ E2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Xb3 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.set_cases
% 0.25/0.54 thf(fact_5453_under__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( order_under @ A )
% 0.25/0.54 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 0.25/0.54 ( collect @ A
% 0.25/0.54 @ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % under_def
% 0.25/0.54 thf(fact_5454_Heap_Oinj__map__strong,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
% 0.25/0.54 ( ! [Z3: A,Za: A] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.25/0.54 => ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa ) )
% 0.25/0.54 => ( ( ( F @ Z3 )
% 0.25/0.54 = ( Fa @ Za ) )
% 0.25/0.54 => ( Z3 = Za ) ) ) )
% 0.25/0.54 => ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.25/0.54 = ( heap_Time_map_Heap @ A @ B @ Fa @ Xa ) )
% 0.25/0.54 => ( X = Xa ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.inj_map_strong
% 0.25/0.54 thf(fact_5455_Heap_Omap__cong0,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 0.25/0.54 ( ! [Z3: A] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.25/0.54 => ( ( F @ Z3 )
% 0.25/0.54 = ( G @ Z3 ) ) )
% 0.25/0.54 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.25/0.54 = ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_cong0
% 0.25/0.54 thf(fact_5456_Heap_Omap__cong,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 0.25/0.54 ( ( X = Ya )
% 0.25/0.54 => ( ! [Z3: A] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 0.25/0.54 => ( ( F @ Z3 )
% 0.25/0.54 = ( G @ Z3 ) ) )
% 0.25/0.54 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.25/0.54 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_cong
% 0.25/0.54 thf(fact_5457_Heap_Oset__map,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: A > B,V: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( heap_Time_set_Heap @ B @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 0.25/0.54 = ( image2 @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.set_map
% 0.25/0.54 thf(fact_5458_Heap_Orel__Grp,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F ) )
% 0.25/0.54 = ( bNF_Grp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
% 0.25/0.54 @ ( collect @ ( heap_Time_Heap @ A )
% 0.25/0.54 @ ^ [X3: heap_Time_Heap @ A] : ( ord_less_eq @ ( set @ A ) @ ( heap_Time_set_Heap @ A @ X3 ) @ A4 ) )
% 0.25/0.54 @ ( heap_Time_map_Heap @ A @ B @ F ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_Grp
% 0.25/0.54 thf(fact_5459_Heap_Orel__compp__Grp,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B )
% 0.25/0.54 = ( ^ [R6: A > B > $o] :
% 0.25/0.54 ( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
% 0.25/0.54 @ ( conversep @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
% 0.25/0.54 @ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
% 0.25/0.54 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 0.25/0.54 @ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 0.25/0.54 @ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
% 0.25/0.54 @ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
% 0.25/0.54 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 0.25/0.54 @ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 0.25/0.54 @ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_compp_Grp
% 0.25/0.54 thf(fact_5460_Heap_Oin__rel,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B )
% 0.25/0.54 = ( ^ [R6: A > B > $o,A7: heap_Time_Heap @ A,B5: heap_Time_Heap @ B] :
% 0.25/0.54 ? [Z5: heap_Time_Heap @ ( product_prod @ A @ B )] :
% 0.25/0.54 ( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z5
% 0.25/0.54 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 0.25/0.54 @ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
% 0.25/0.54 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
% 0.25/0.54 = A7 )
% 0.25/0.54 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
% 0.25/0.54 = B5 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.in_rel
% 0.25/0.54 thf(fact_5461_Heap_Orel__cong,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Xa: heap_Time_Heap @ B,R5: A > B > $o,Ra2: A > B > $o] :
% 0.25/0.54 ( ( X = Ya )
% 0.25/0.54 => ( ( Y = Xa )
% 0.25/0.54 => ( ! [Z3: A,Yb: B] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 0.25/0.54 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa ) )
% 0.25/0.54 => ( ( R5 @ Z3 @ Yb )
% 0.25/0.54 = ( Ra2 @ Z3 @ Yb ) ) ) )
% 0.25/0.54 => ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
% 0.25/0.54 = ( heap_Time_rel_Heap @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_cong
% 0.25/0.54 thf(fact_5462_Heap_Orel__mono__strong,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Ra2: A > B > $o] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
% 0.25/0.54 => ( ! [Z3: A,Yb: B] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.25/0.54 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y ) )
% 0.25/0.54 => ( ( R5 @ Z3 @ Yb )
% 0.25/0.54 => ( Ra2 @ Z3 @ Yb ) ) ) )
% 0.25/0.54 => ( heap_Time_rel_Heap @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_mono_strong
% 0.25/0.54 thf(fact_5463_Heap_Orel__refl__strong,axiom,
% 0.25/0.54 ! [A: $tType,X: heap_Time_Heap @ A,Ra2: A > A > $o] :
% 0.25/0.54 ( ! [Z3: A] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.25/0.54 => ( Ra2 @ Z3 @ Z3 ) )
% 0.25/0.54 => ( heap_Time_rel_Heap @ A @ A @ Ra2 @ X @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_refl_strong
% 0.25/0.54 thf(fact_5464_Heap_Orel__map_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: heap_Time_Heap @ A,G: B > C,Y: heap_Time_Heap @ B] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ C @ Sa @ X @ ( heap_Time_map_Heap @ B @ C @ G @ Y ) )
% 0.25/0.54 = ( heap_Time_rel_Heap @ A @ B
% 0.25/0.54 @ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
% 0.25/0.54 @ X
% 0.25/0.54 @ Y ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_map(2)
% 0.25/0.54 thf(fact_5465_Heap_Orel__map_I1_J,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ C @ B @ Sb @ ( heap_Time_map_Heap @ A @ C @ I @ X ) @ Y )
% 0.25/0.54 = ( heap_Time_rel_Heap @ A @ B
% 0.25/0.54 @ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
% 0.25/0.54 @ X
% 0.25/0.54 @ Y ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_map(1)
% 0.25/0.54 thf(fact_5466_Heap_Orel__mono,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
% 0.25/0.54 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
% 0.25/0.54 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_mono
% 0.25/0.54 thf(fact_5467_Heap_Orel__transp,axiom,
% 0.25/0.54 ! [A: $tType,R5: A > A > $o] :
% 0.25/0.54 ( ( transp @ A @ R5 )
% 0.25/0.54 => ( transp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_transp
% 0.25/0.54 thf(fact_5468_Heap_Orel__refl,axiom,
% 0.25/0.54 ! [B: $tType,Ra2: B > B > $o,X: heap_Time_Heap @ B] :
% 0.25/0.54 ( ! [X2: B] : ( Ra2 @ X2 @ X2 )
% 0.25/0.54 => ( heap_Time_rel_Heap @ B @ B @ Ra2 @ X @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_refl
% 0.25/0.54 thf(fact_5469_Heap_Orel__eq,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ A
% 0.25/0.54 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.25/0.54 = ( ^ [Y5: heap_Time_Heap @ A,Z4: heap_Time_Heap @ A] : ( Y5 = Z4 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_eq
% 0.25/0.54 thf(fact_5470_Heap_Orel__conversep,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) )
% 0.25/0.54 = ( conversep @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_conversep
% 0.25/0.54 thf(fact_5471_Heap_Orel__flip,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ B,B2: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
% 0.25/0.54 = ( heap_Time_rel_Heap @ A @ B @ R5 @ B2 @ A3 ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_flip
% 0.25/0.54 thf(fact_5472_Heap_Orel__compp,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S2: B > C > $o] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S2 ) )
% 0.25/0.54 = ( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ C ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ B @ C @ S2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_compp
% 0.25/0.54 thf(fact_5473_Heap_Orel__transfer,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 0.25/0.54 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ C ) > ( heap_Time_Heap @ D ) > $o )
% 0.25/0.54 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 0.25/0.54 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 0.25/0.54 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ C ) @ ( ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ D ) > $o ) @ ( heap_Time_rel_Heap @ A @ C @ Sa )
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ D ) @ $o @ $o @ ( heap_Time_rel_Heap @ B @ D @ Sc )
% 0.25/0.54 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ ( heap_Time_rel_Heap @ A @ B )
% 0.25/0.54 @ ( heap_Time_rel_Heap @ C @ D ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_transfer
% 0.25/0.54 thf(fact_5474_Heap_Omap__transfer,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,F5: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F5 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E ) > ( heap_Time_Heap @ F5 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F5 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F5 ) @ ( heap_Time_rel_Heap @ A @ E @ Rb2 ) @ ( heap_Time_rel_Heap @ B @ F5 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E @ F5 ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_transfer
% 0.25/0.54 thf(fact_5475_those_Osimps_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,X: option @ A,Xs: list @ ( option @ A )] :
% 0.25/0.54 ( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs ) )
% 0.25/0.54 = ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
% 0.25/0.54 @ ^ [Y3: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( those @ A @ Xs ) )
% 0.25/0.54 @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % those.simps(2)
% 0.25/0.54 thf(fact_5476_Quotient__natural,axiom,
% 0.25/0.54 ( quotient @ nat @ code_natural
% 0.25/0.54 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 0.25/0.54 @ code_natural_of_nat
% 0.25/0.54 @ code_nat_of_natural
% 0.25/0.54 @ code_cr_natural ) ).
% 0.25/0.54
% 0.25/0.54 % Quotient_natural
% 0.25/0.54 thf(fact_5477_Quotient__integer,axiom,
% 0.25/0.54 ( quotient @ int @ code_integer
% 0.25/0.54 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 0.25/0.54 @ code_integer_of_int
% 0.25/0.54 @ code_int_of_integer
% 0.25/0.54 @ code_cr_integer ) ).
% 0.25/0.54
% 0.25/0.54 % Quotient_integer
% 0.25/0.54 thf(fact_5478_Heap_OQuotient,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
% 0.25/0.54 ( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T6 )
% 0.25/0.54 => ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T6 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.Quotient
% 0.25/0.54 thf(fact_5479_option_OQuotient,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
% 0.25/0.54 ( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T6 )
% 0.25/0.54 => ( quotient @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ A @ R5 ) @ ( map_option @ A @ B @ Abs ) @ ( map_option @ B @ A @ Rep ) @ ( rel_option @ A @ B @ T6 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % option.Quotient
% 0.25/0.54 thf(fact_5480_Quotient__compose,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,T1: A > B > $o,R23: B > B > $o,Abs22: B > C,Rep22: C > B,T22: B > C > $o] :
% 0.25/0.54 ( ( quotient @ A @ B @ R12 @ Abs1 @ Rep1 @ T1 )
% 0.25/0.54 => ( ( quotient @ B @ C @ R23 @ Abs22 @ Rep22 @ T22 )
% 0.25/0.54 => ( quotient @ A @ C @ ( relcompp @ A @ B @ A @ T1 @ ( relcompp @ B @ B @ A @ R23 @ ( conversep @ A @ B @ T1 ) ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) @ ( relcompp @ A @ B @ C @ T1 @ T22 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Quotient_compose
% 0.25/0.54 thf(fact_5481_Heap_Opred__transfer,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 0.25/0.54 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Time_Heap @ A ) > $o ) @ ( ( heap_Time_Heap @ B ) > $o )
% 0.25/0.54 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 0.25/0.54 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R5 )
% 0.25/0.54 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 0.25/0.54 @ ( heap_Time_pred_Heap @ A )
% 0.25/0.54 @ ( heap_Time_pred_Heap @ B ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_transfer
% 0.25/0.54 thf(fact_5482_Heap_Orec__transfer,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S2: C > D > $o] :
% 0.25/0.54 ( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
% 0.25/0.54 @ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) ) )
% 0.25/0.54 @ S2 )
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S2 )
% 0.25/0.54 @ ( heap_Time_rec_Heap @ A @ C )
% 0.25/0.54 @ ( heap_Time_rec_Heap @ B @ D ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rec_transfer
% 0.25/0.54 thf(fact_5483_rel__prod__inject,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A3: A,B2: C,C2: B,D3: D] :
% 0.25/0.54 ( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
% 0.25/0.54 = ( ( R12 @ A3 @ C2 )
% 0.25/0.54 & ( R23 @ B2 @ D3 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % rel_prod_inject
% 0.25/0.54 thf(fact_5484_Heap_Opred__mono,axiom,
% 0.25/0.54 ! [A: $tType,P: A > $o,Pa: A > $o] :
% 0.25/0.54 ( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
% 0.25/0.54 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > $o ) @ ( heap_Time_pred_Heap @ A @ P ) @ ( heap_Time_pred_Heap @ A @ Pa ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_mono
% 0.25/0.54 thf(fact_5485_rel__prod_Ointros,axiom,
% 0.25/0.54 ! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A3: A,B2: B,R23: C > D > $o,C2: C,D3: D] :
% 0.25/0.54 ( ( R12 @ A3 @ B2 )
% 0.25/0.54 => ( ( R23 @ C2 @ D3 )
% 0.25/0.54 => ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( product_Pair @ B @ D @ B2 @ D3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % rel_prod.intros
% 0.25/0.54 thf(fact_5486_rel__prod_Osimps,axiom,
% 0.25/0.54 ! [C: $tType,D: $tType,B: $tType,A: $tType] :
% 0.25/0.54 ( ( basic_rel_prod @ A @ B @ C @ D )
% 0.25/0.54 = ( ^ [R13: A > B > $o,R24: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
% 0.25/0.54 ? [A7: A,B5: B,C3: C,D5: D] :
% 0.25/0.54 ( ( A12
% 0.25/0.54 = ( product_Pair @ A @ C @ A7 @ C3 ) )
% 0.25/0.54 & ( A23
% 0.25/0.54 = ( product_Pair @ B @ D @ B5 @ D5 ) )
% 0.25/0.54 & ( R13 @ A7 @ B5 )
% 0.25/0.54 & ( R24 @ C3 @ D5 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % rel_prod.simps
% 0.25/0.54 thf(fact_5487_rel__prod_Ocases,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
% 0.25/0.54 ( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ A1 @ A22 )
% 0.25/0.54 => ~ ! [A5: A,B3: B,C4: C] :
% 0.25/0.54 ( ( A1
% 0.25/0.54 = ( product_Pair @ A @ C @ A5 @ C4 ) )
% 0.25/0.54 => ! [D2: D] :
% 0.25/0.54 ( ( A22
% 0.25/0.54 = ( product_Pair @ B @ D @ B3 @ D2 ) )
% 0.25/0.54 => ( ( R12 @ A5 @ B3 )
% 0.25/0.54 => ~ ( R23 @ C4 @ D2 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % rel_prod.cases
% 0.25/0.54 thf(fact_5488_Heap_Opred__True,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap_Time_pred_Heap @ A
% 0.25/0.54 @ ^ [Uu: A] : $true )
% 0.25/0.54 = ( ^ [Uu: heap_Time_Heap @ A] : $true ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_True
% 0.25/0.54 thf(fact_5489_Heap_Opred__mono__strong,axiom,
% 0.25/0.54 ! [A: $tType,P: A > $o,X: heap_Time_Heap @ A,Pa: A > $o] :
% 0.25/0.54 ( ( heap_Time_pred_Heap @ A @ P @ X )
% 0.25/0.54 => ( ! [Z3: A] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 0.25/0.54 => ( ( P @ Z3 )
% 0.25/0.54 => ( Pa @ Z3 ) ) )
% 0.25/0.54 => ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_mono_strong
% 0.25/0.54 thf(fact_5490_Heap_Opred__cong,axiom,
% 0.25/0.54 ! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
% 0.25/0.54 ( ( X = Ya )
% 0.25/0.54 => ( ! [Z3: A] :
% 0.25/0.54 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 0.25/0.54 => ( ( P @ Z3 )
% 0.25/0.54 = ( Pa @ Z3 ) ) )
% 0.25/0.54 => ( ( heap_Time_pred_Heap @ A @ P @ X )
% 0.25/0.54 = ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_cong
% 0.25/0.54 thf(fact_5491_Heap_Octr__transfer,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 0.25/0.54 ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) ) )
% 0.25/0.54 @ ( heap_Time_rel_Heap @ A @ B @ R5 )
% 0.25/0.54 @ ( heap_Time_Heap2 @ A )
% 0.25/0.54 @ ( heap_Time_Heap2 @ B ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.ctr_transfer
% 0.25/0.54 thf(fact_5492_Heap_Orel__intros,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ X
% 0.25/0.54 @ Y )
% 0.25/0.54 => ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_intros
% 0.25/0.54 thf(fact_5493_Heap_Orel__inject,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y ) )
% 0.25/0.54 = ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ X
% 0.25/0.54 @ Y ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_inject
% 0.25/0.54 thf(fact_5494_Heap_Orel__induct,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Q: ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
% 0.25/0.54 => ( ! [A5: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),B3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ A5
% 0.25/0.54 @ B3 )
% 0.25/0.54 => ( Q @ ( heap_Time_Heap2 @ A @ A5 ) @ ( heap_Time_Heap2 @ B @ B3 ) ) )
% 0.25/0.54 => ( Q @ X @ Y ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_induct
% 0.25/0.54 thf(fact_5495_Heap_Orel__cases,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ A,B2: heap_Time_Heap @ B] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ A3 @ B2 )
% 0.25/0.54 => ~ ! [Xa3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( A3
% 0.25/0.54 = ( heap_Time_Heap2 @ A @ Xa3 ) )
% 0.25/0.54 => ! [Ya2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( B2
% 0.25/0.54 = ( heap_Time_Heap2 @ B @ Ya2 ) )
% 0.25/0.54 => ~ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ Xa3
% 0.25/0.54 @ Ya2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_cases
% 0.25/0.54 thf(fact_5496_Heap_Omap__cong__pred,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 0.25/0.54 ( ( X = Ya )
% 0.25/0.54 => ( ( heap_Time_pred_Heap @ A
% 0.25/0.54 @ ^ [Z5: A] :
% 0.25/0.54 ( ( F @ Z5 )
% 0.25/0.54 = ( G @ Z5 ) )
% 0.25/0.54 @ Ya )
% 0.25/0.54 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 0.25/0.54 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.map_cong_pred
% 0.25/0.54 thf(fact_5497_Heap_Orel__eq__onp,axiom,
% 0.25/0.54 ! [A: $tType,P: A > $o] :
% 0.25/0.54 ( ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P ) )
% 0.25/0.54 = ( bNF_eq_onp @ ( heap_Time_Heap @ A ) @ ( heap_Time_pred_Heap @ A @ P ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_eq_onp
% 0.25/0.54 thf(fact_5498_Heap_Opred__rel,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap_Time_pred_Heap @ A )
% 0.25/0.54 = ( ^ [P4: A > $o,X3: heap_Time_Heap @ A] : ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X3 @ X3 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_rel
% 0.25/0.54 thf(fact_5499_Heap_Opred__set,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap_Time_pred_Heap @ A )
% 0.25/0.54 = ( ^ [P4: A > $o,X3: heap_Time_Heap @ A] :
% 0.25/0.54 ! [Y3: A] :
% 0.25/0.54 ( ( member @ A @ Y3 @ ( heap_Time_set_Heap @ A @ X3 ) )
% 0.25/0.54 => ( P4 @ Y3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_set
% 0.25/0.54 thf(fact_5500_Pair__transfer,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A4 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B6 @ ( basic_rel_prod @ A @ B @ C @ D @ A4 @ B6 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% 0.25/0.54
% 0.25/0.54 % Pair_transfer
% 0.25/0.54 thf(fact_5501_Heap_Opred__map,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( heap_Time_pred_Heap @ B @ Q @ ( heap_Time_map_Heap @ A @ B @ F @ X ) )
% 0.25/0.54 = ( heap_Time_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_map
% 0.25/0.54 thf(fact_5502_Heap_Ocase__transfer,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S2: C > D > $o] :
% 0.25/0.54 ( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
% 0.25/0.54 @ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 0.25/0.54 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.54 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 0.25/0.54 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) ) )
% 0.25/0.54 @ S2 )
% 0.25/0.54 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S2 )
% 0.25/0.54 @ ( heap_Time_case_Heap @ A @ C )
% 0.25/0.54 @ ( heap_Time_case_Heap @ B @ D ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.case_transfer
% 0.25/0.54 thf(fact_5503_sndOp__def,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType] :
% 0.25/0.54 ( ( bNF_sndOp @ C @ A @ B )
% 0.25/0.54 = ( ^ [P4: C > A > $o,Q6: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q6 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sndOp_def
% 0.25/0.54 thf(fact_5504_fstOp__def,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.54 ( ( bNF_fstOp @ A @ B @ C )
% 0.25/0.54 = ( ^ [P4: A > B > $o,Q6: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q6 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fstOp_def
% 0.25/0.54 thf(fact_5505_Heap_Ocase__distrib,axiom,
% 0.25/0.54 ! [C: $tType,B: $tType,A: $tType,H3: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
% 0.25/0.54 ( ( H3 @ ( heap_Time_case_Heap @ A @ B @ F @ Heap ) )
% 0.25/0.54 = ( heap_Time_case_Heap @ A @ C
% 0.25/0.54 @ ^ [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( H3 @ ( F @ X3 ) )
% 0.25/0.54 @ Heap ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.case_distrib
% 0.25/0.54 thf(fact_5506_Heap_Ocase,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( heap_Time_case_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 0.25/0.54 = ( F @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.case
% 0.25/0.54 thf(fact_5507_trancl__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( transitive_trancl @ A )
% 0.25/0.54 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( collect @ ( product_prod @ A @ A )
% 0.25/0.54 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.54 @ ( transitive_tranclp @ A
% 0.25/0.54 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % trancl_def
% 0.25/0.54 thf(fact_5508_Nitpick_Otranclp__unfold,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( transitive_tranclp @ A )
% 0.25/0.54 = ( ^ [R3: A > A > $o,A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Nitpick.tranclp_unfold
% 0.25/0.54 thf(fact_5509_reflp__refl__eq,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( reflp @ A
% 0.25/0.54 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.54 = ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R ) ) ).
% 0.25/0.54
% 0.25/0.54 % reflp_refl_eq
% 0.25/0.54 thf(fact_5510_tranclp__induct2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
% 0.25/0.54 ( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 0.25/0.54 => ( ! [A5: A,B3: B] :
% 0.25/0.54 ( ( R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.25/0.54 => ( P @ A5 @ B3 ) )
% 0.25/0.54 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 0.25/0.54 ( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.25/0.54 => ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
% 0.25/0.54 => ( ( P @ A5 @ B3 )
% 0.25/0.54 => ( P @ Aa3 @ Ba ) ) ) )
% 0.25/0.54 => ( P @ Bx @ By ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % tranclp_induct2
% 0.25/0.54 thf(fact_5511_Heap_Orel__reflp,axiom,
% 0.25/0.54 ! [A: $tType,R5: A > A > $o] :
% 0.25/0.54 ( ( reflp @ A @ R5 )
% 0.25/0.54 => ( reflp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.rel_reflp
% 0.25/0.54 thf(fact_5512_option_Orel__reflp,axiom,
% 0.25/0.54 ! [A: $tType,R5: A > A > $o] :
% 0.25/0.54 ( ( reflp @ A @ R5 )
% 0.25/0.54 => ( reflp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % option.rel_reflp
% 0.25/0.54 thf(fact_5513_tranclp__trancl__eq,axiom,
% 0.25/0.54 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.54 ( ( transitive_tranclp @ A
% 0.25/0.54 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.54 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % tranclp_trancl_eq
% 0.25/0.54 thf(fact_5514_Lcm__fin__1__iff,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A4: set @ A] :
% 0.25/0.54 ( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
% 0.25/0.54 = ( one_one @ A ) )
% 0.25/0.54 = ( ! [X3: A] :
% 0.25/0.54 ( ( member @ A @ X3 @ A4 )
% 0.25/0.54 => ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) )
% 0.25/0.54 & ( finite_finite2 @ A @ A4 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_fin_1_iff
% 0.25/0.54 thf(fact_5515_heap__interpretation,axiom,
% 0.25/0.54 ! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ ( heap_Time_Heap_lub @ A ) ) ).
% 0.25/0.54
% 0.25/0.54 % heap_interpretation
% 0.25/0.54 thf(fact_5516_old_Orec__bool__def,axiom,
% 0.25/0.54 ! [T: $tType] :
% 0.25/0.54 ( ( product_rec_bool @ T )
% 0.25/0.54 = ( ^ [F12: T,F23: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % old.rec_bool_def
% 0.25/0.54 thf(fact_5517_old_Obool_Osimps_I6_J,axiom,
% 0.25/0.54 ! [T: $tType,F1: T,F22: T] :
% 0.25/0.54 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 0.25/0.54 = F22 ) ).
% 0.25/0.54
% 0.25/0.54 % old.bool.simps(6)
% 0.25/0.54 thf(fact_5518_old_Obool_Osimps_I5_J,axiom,
% 0.25/0.54 ! [T: $tType,F1: T,F22: T] :
% 0.25/0.54 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 0.25/0.54 = F1 ) ).
% 0.25/0.54
% 0.25/0.54 % old.bool.simps(5)
% 0.25/0.54 thf(fact_5519_Lcm__fin_Oempty,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
% 0.25/0.54 = ( one_one @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_fin.empty
% 0.25/0.54 thf(fact_5520_is__unit__Lcm__fin__iff,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A4: set @ A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) @ ( one_one @ A ) )
% 0.25/0.54 = ( ( semiring_gcd_Lcm_fin @ A @ A4 )
% 0.25/0.54 = ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_unit_Lcm_fin_iff
% 0.25/0.54 thf(fact_5521_Lcm__fin_Oeq__fold,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( ( semiring_gcd_Lcm_fin @ A )
% 0.25/0.54 = ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A9 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_fin.eq_fold
% 0.25/0.54 thf(fact_5522_Lcm__fin__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( ( semiring_gcd_Lcm_fin @ A )
% 0.25/0.54 = ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_fin_def
% 0.25/0.54 thf(fact_5523_fun_Opred__map,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
% 0.25/0.54 ( ( basic_pred_fun @ D @ B
% 0.25/0.54 @ ^ [Uu: D] : $true
% 0.25/0.54 @ Q
% 0.25/0.54 @ ( comp @ A @ B @ D @ F @ X ) )
% 0.25/0.54 = ( basic_pred_fun @ D @ A
% 0.25/0.54 @ ^ [Uu: D] : $true
% 0.25/0.54 @ ( comp @ B @ $o @ A @ Q @ F )
% 0.25/0.54 @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % fun.pred_map
% 0.25/0.54 thf(fact_5524_lcm__eq__1__iff,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A,B2: A] :
% 0.25/0.54 ( ( ( gcd_lcm @ A @ A3 @ B2 )
% 0.25/0.54 = ( one_one @ A ) )
% 0.25/0.54 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm_eq_1_iff
% 0.25/0.54 thf(fact_5525_unit__factor__lcm,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A,B2: A] :
% 0.25/0.54 ( ( ( ( A3
% 0.25/0.54 = ( zero_zero @ A ) )
% 0.25/0.54 | ( B2
% 0.25/0.54 = ( zero_zero @ A ) ) )
% 0.25/0.54 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.25/0.54 = ( zero_zero @ A ) ) )
% 0.25/0.54 & ( ~ ( ( A3
% 0.25/0.54 = ( zero_zero @ A ) )
% 0.25/0.54 | ( B2
% 0.25/0.54 = ( zero_zero @ A ) ) )
% 0.25/0.54 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 0.25/0.54 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % unit_factor_lcm
% 0.25/0.54 thf(fact_5526_fun_Omap__cong__pred,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
% 0.25/0.54 ( ( X = Ya )
% 0.25/0.54 => ( ( basic_pred_fun @ D @ A
% 0.25/0.54 @ ^ [Uu: D] : $true
% 0.25/0.54 @ ^ [Z5: A] :
% 0.25/0.54 ( ( F @ Z5 )
% 0.25/0.54 = ( G @ Z5 ) )
% 0.25/0.54 @ Ya )
% 0.25/0.54 => ( ( comp @ A @ B @ D @ F @ X )
% 0.25/0.54 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fun.map_cong_pred
% 0.25/0.54 thf(fact_5527_lcm__mult__unit2,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 0.25/0.54 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm_mult_unit2
% 0.25/0.54 thf(fact_5528_lcm__mult__unit1,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 0.25/0.54 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm_mult_unit1
% 0.25/0.54 thf(fact_5529_lcm__div__unit1,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 0.25/0.54 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm_div_unit1
% 0.25/0.54 thf(fact_5530_lcm__div__unit2,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A,B2: A,C2: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 0.25/0.54 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm_div_unit2
% 0.25/0.54 thf(fact_5531_Lcm__nat__def,axiom,
% 0.25/0.54 ( ( gcd_Lcm @ nat )
% 0.25/0.54 = ( ^ [M8: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M8 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M8 ) @ ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_nat_def
% 0.25/0.54 thf(fact_5532_Heap_Opred__inject,axiom,
% 0.25/0.54 ! [A: $tType,P: A > $o,A3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 0.25/0.54 ( ( heap_Time_pred_Heap @ A @ P @ ( heap_Time_Heap2 @ A @ A3 ) )
% 0.25/0.54 = ( basic_pred_fun @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 0.25/0.54 @ ^ [Uu: heap_ext @ product_unit] : $true
% 0.25/0.54 @ ( pred_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( basic_pred_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ P @ ( top_top @ ( ( product_prod @ ( heap_ext @ product_unit ) @ nat ) > $o ) ) ) )
% 0.25/0.54 @ A3 ) ) ).
% 0.25/0.54
% 0.25/0.54 % Heap.pred_inject
% 0.25/0.54 thf(fact_5533_Lcm__fin_Oset__eq__fold,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [Xs: list @ A] :
% 0.25/0.54 ( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
% 0.25/0.54 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_fin.set_eq_fold
% 0.25/0.54 thf(fact_5534_pred__prod__inject,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P14: A > $o,P25: B > $o,A3: A,B2: B] :
% 0.25/0.54 ( ( basic_pred_prod @ A @ B @ P14 @ P25 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 0.25/0.54 = ( ( P14 @ A3 )
% 0.25/0.54 & ( P25 @ B2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % pred_prod_inject
% 0.25/0.54 thf(fact_5535_fold__append,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,F: B > A > A,Xs: list @ B,Ys2: list @ B] :
% 0.25/0.54 ( ( fold @ B @ A @ F @ ( append @ B @ Xs @ Ys2 ) )
% 0.25/0.54 = ( comp @ A @ A @ A @ ( fold @ B @ A @ F @ Ys2 ) @ ( fold @ B @ A @ F @ Xs ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_append
% 0.25/0.54 thf(fact_5536_pred__prod__split,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,P: $o > $o,Q: A > $o,R5: B > $o,Xy2: product_prod @ A @ B] :
% 0.25/0.54 ( ( P @ ( basic_pred_prod @ A @ B @ Q @ R5 @ Xy2 ) )
% 0.25/0.54 = ( ! [X3: A,Y3: B] :
% 0.25/0.54 ( ( Xy2
% 0.25/0.54 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.25/0.54 => ( P
% 0.25/0.54 @ ( ( Q @ X3 )
% 0.25/0.54 & ( R5 @ Y3 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % pred_prod_split
% 0.25/0.54 thf(fact_5537_pred__prod_Ocases,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P14: A > $o,P25: B > $o,A3: product_prod @ A @ B] :
% 0.25/0.54 ( ( basic_pred_prod @ A @ B @ P14 @ P25 @ A3 )
% 0.25/0.54 => ~ ! [A5: A,B3: B] :
% 0.25/0.54 ( ( A3
% 0.25/0.54 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.25/0.54 => ( ( P14 @ A5 )
% 0.25/0.54 => ~ ( P25 @ B3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % pred_prod.cases
% 0.25/0.54 thf(fact_5538_pred__prod_Osimps,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( basic_pred_prod @ A @ B )
% 0.25/0.54 = ( ^ [P12: A > $o,P23: B > $o,A7: product_prod @ A @ B] :
% 0.25/0.54 ? [B5: A,C3: B] :
% 0.25/0.54 ( ( A7
% 0.25/0.54 = ( product_Pair @ A @ B @ B5 @ C3 ) )
% 0.25/0.54 & ( P12 @ B5 )
% 0.25/0.54 & ( P23 @ C3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % pred_prod.simps
% 0.25/0.54 thf(fact_5539_pred__prod_Ointros,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P14: A > $o,A3: A,P25: B > $o,B2: B] :
% 0.25/0.54 ( ( P14 @ A3 )
% 0.25/0.54 => ( ( P25 @ B2 )
% 0.25/0.54 => ( basic_pred_prod @ A @ B @ P14 @ P25 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % pred_prod.intros
% 0.25/0.54 thf(fact_5540_fold__Cons,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
% 0.25/0.54 ( ( fold @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
% 0.25/0.54 = ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ Xs ) @ ( F @ X ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_Cons
% 0.25/0.54 thf(fact_5541_fold__commute__apply,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H3: B > C,G: A > B > B,F: A > C > C,S: B] :
% 0.25/0.54 ( ! [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( comp @ B @ C @ B @ H3 @ ( G @ X2 ) )
% 0.25/0.54 = ( comp @ C @ C @ B @ ( F @ X2 ) @ H3 ) ) )
% 0.25/0.54 => ( ( H3 @ ( fold @ A @ B @ G @ Xs @ S ) )
% 0.25/0.54 = ( fold @ A @ C @ F @ Xs @ ( H3 @ S ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_commute_apply
% 0.25/0.54 thf(fact_5542_fold__commute,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H3: B > C,G: A > B > B,F: A > C > C] :
% 0.25/0.54 ( ! [X2: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( comp @ B @ C @ B @ H3 @ ( G @ X2 ) )
% 0.25/0.54 = ( comp @ C @ C @ B @ ( F @ X2 ) @ H3 ) ) )
% 0.25/0.54 => ( ( comp @ B @ C @ B @ H3 @ ( fold @ A @ B @ G @ Xs ) )
% 0.25/0.54 = ( comp @ C @ C @ B @ ( fold @ A @ C @ F @ Xs ) @ H3 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_commute
% 0.25/0.54 thf(fact_5543_Lcm__nat__set__eq__fold,axiom,
% 0.25/0.54 ! [Xs: list @ nat] :
% 0.25/0.54 ( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
% 0.25/0.54 = ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_nat_set_eq_fold
% 0.25/0.54 thf(fact_5544_fold__multiset__equiv,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,Ys2: list @ A] :
% 0.25/0.54 ( ! [X2: A,Y4: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) )
% 0.25/0.54 = ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) )
% 0.25/0.54 => ( ( ( mset @ A @ Xs )
% 0.25/0.54 = ( mset @ A @ Ys2 ) )
% 0.25/0.54 => ( ( fold @ A @ B @ F @ Xs )
% 0.25/0.54 = ( fold @ A @ B @ F @ Ys2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_multiset_equiv
% 0.25/0.54 thf(fact_5545_fold__rev,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
% 0.25/0.54 ( ! [X2: A,Y4: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 0.25/0.54 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) )
% 0.25/0.54 => ( ( fold @ A @ B @ F @ ( rev @ A @ Xs ) )
% 0.25/0.54 = ( fold @ A @ B @ F @ Xs ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_rev
% 0.25/0.54 thf(fact_5546_foldr__fold,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
% 0.25/0.54 ( ! [X2: A,Y4: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 0.25/0.54 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) )
% 0.25/0.54 => ( ( foldr @ A @ B @ F @ Xs )
% 0.25/0.54 = ( fold @ A @ B @ F @ Xs ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % foldr_fold
% 0.25/0.54 thf(fact_5547_fold__plus__sum__list__rev,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( monoid_add @ A )
% 0.25/0.54 => ! [Xs: list @ A] :
% 0.25/0.54 ( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
% 0.25/0.54 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_plus_sum_list_rev
% 0.25/0.54 thf(fact_5548_fold__remove1__split,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,X: A] :
% 0.25/0.54 ( ! [X2: A,Y4: A] :
% 0.25/0.54 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) )
% 0.25/0.54 = ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) )
% 0.25/0.54 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 0.25/0.54 => ( ( fold @ A @ B @ F @ Xs )
% 0.25/0.54 = ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ ( remove1 @ A @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % fold_remove1_split
% 0.25/0.54 thf(fact_5549_Lcm__set__eq__fold,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_Gcd @ A )
% 0.25/0.54 => ! [Xs: list @ A] :
% 0.25/0.54 ( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
% 0.25/0.54 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_set_eq_fold
% 0.25/0.54 thf(fact_5550_subset__mset_Omax__add__distrib__left,axiom,
% 0.25/0.54 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 0.25/0.54 ( ( plus_plus @ ( multiset @ A ) @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X @ Y ) @ Z2 )
% 0.25/0.54 = ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % subset_mset.max_add_distrib_left
% 0.25/0.54 thf(fact_5551_subset__mset_Omax__add__distrib__right,axiom,
% 0.25/0.54 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 0.25/0.54 ( ( plus_plus @ ( multiset @ A ) @ X @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
% 0.25/0.54 = ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % subset_mset.max_add_distrib_right
% 0.25/0.54 thf(fact_5552_prod__list_Ocomm__monoid__list__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_mult @ A )
% 0.25/0.54 => ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % prod_list.comm_monoid_list_axioms
% 0.25/0.54 thf(fact_5553_sum__list_Ocomm__monoid__list__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( comm_monoid_add @ A )
% 0.25/0.54 => ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sum_list.comm_monoid_list_axioms
% 0.25/0.54 thf(fact_5554_lazI,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
% 0.25/0.54 ( ( ( size_size @ ( list @ A ) @ A3 )
% 0.25/0.54 = ( size_size @ ( list @ B ) @ B2 ) )
% 0.25/0.54 => ( ! [I2: nat] :
% 0.25/0.54 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ B2 ) )
% 0.25/0.54 => ( P @ ( nth @ A @ A3 @ I2 ) @ ( nth @ B @ B2 @ I2 ) ) )
% 0.25/0.54 => ( list_all_zip @ A @ B @ P @ A3 @ B2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lazI
% 0.25/0.54 thf(fact_5555_list__all__zip__alt,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( list_all_zip @ A @ B )
% 0.25/0.54 = ( ^ [P4: A > B > $o,As5: list @ A,Bs3: list @ B] :
% 0.25/0.54 ( ( ( size_size @ ( list @ A ) @ As5 )
% 0.25/0.54 = ( size_size @ ( list @ B ) @ Bs3 ) )
% 0.25/0.54 & ! [I3: nat] :
% 0.25/0.54 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ As5 ) )
% 0.25/0.54 => ( P4 @ ( nth @ A @ As5 @ I3 ) @ ( nth @ B @ Bs3 @ I3 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip_alt
% 0.25/0.54 thf(fact_5556_is__none__bind,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A )] :
% 0.25/0.54 ( ( is_none @ A @ ( bind @ B @ A @ F @ G ) )
% 0.25/0.54 = ( ( is_none @ B @ F )
% 0.25/0.54 | ( is_none @ A @ ( G @ ( the2 @ B @ F ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_none_bind
% 0.25/0.54 thf(fact_5557_laz__conj,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,A3: list @ A,B2: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B
% 0.25/0.54 @ ^ [X3: A,Y3: B] :
% 0.25/0.54 ( ( P @ X3 @ Y3 )
% 0.25/0.54 & ( Q @ X3 @ Y3 ) )
% 0.25/0.54 @ A3
% 0.25/0.54 @ B2 )
% 0.25/0.54 = ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ Q @ A3 @ B2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % laz_conj
% 0.25/0.54 thf(fact_5558_is__none__code_I2_J,axiom,
% 0.25/0.54 ! [B: $tType,X: B] :
% 0.25/0.54 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_none_code(2)
% 0.25/0.54 thf(fact_5559_is__none__code_I1_J,axiom,
% 0.25/0.54 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_none_code(1)
% 0.25/0.54 thf(fact_5560_is__none__map__option,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,F: B > A,X: option @ B] :
% 0.25/0.54 ( ( is_none @ A @ ( map_option @ B @ A @ F @ X ) )
% 0.25/0.54 = ( is_none @ B @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_none_map_option
% 0.25/0.54 thf(fact_5561_laz__weak__Pb,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: B > $o,A4: list @ A,B6: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B
% 0.25/0.54 @ ^ [A7: A] : P
% 0.25/0.54 @ A4
% 0.25/0.54 @ B6 )
% 0.25/0.54 = ( ( ( size_size @ ( list @ A ) @ A4 )
% 0.25/0.54 = ( size_size @ ( list @ B ) @ B6 ) )
% 0.25/0.54 & ! [X3: B] :
% 0.25/0.54 ( ( member @ B @ X3 @ ( set2 @ B @ B6 ) )
% 0.25/0.54 => ( P @ X3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % laz_weak_Pb
% 0.25/0.54 thf(fact_5562_laz__weak__Pa,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,P: A > $o,A4: list @ A,B6: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B
% 0.25/0.54 @ ^ [A7: A,B5: B] : ( P @ A7 )
% 0.25/0.54 @ A4
% 0.25/0.54 @ B6 )
% 0.25/0.54 = ( ( ( size_size @ ( list @ A ) @ A4 )
% 0.25/0.54 = ( size_size @ ( list @ B ) @ B6 ) )
% 0.25/0.54 & ! [X3: A] :
% 0.25/0.54 ( ( member @ A @ X3 @ ( set2 @ A @ A4 ) )
% 0.25/0.54 => ( P @ X3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % laz_weak_Pa
% 0.25/0.54 thf(fact_5563_list__all__zip_Osimps_I1_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: A > B > $o] : ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.simps(1)
% 0.25/0.54 thf(fact_5564_laz__eq,axiom,
% 0.25/0.54 ! [A: $tType,A3: list @ A,B2: list @ A] :
% 0.25/0.54 ( ( list_all_zip @ A @ A
% 0.25/0.54 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 0.25/0.54 @ A3
% 0.25/0.54 @ B2 )
% 0.25/0.54 = ( A3 = B2 ) ) ).
% 0.25/0.54
% 0.25/0.54 % laz_eq
% 0.25/0.54 thf(fact_5565_laz__len,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: A > B > $o,A3: list @ A,B2: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
% 0.25/0.54 => ( ( size_size @ ( list @ A ) @ A3 )
% 0.25/0.54 = ( size_size @ ( list @ B ) @ B2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % laz_len
% 0.25/0.54 thf(fact_5566_laz__swap__ex,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A4: list @ A,B6: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B
% 0.25/0.54 @ ^ [A7: A,B5: B] :
% 0.25/0.54 ? [X8: C] : ( P @ A7 @ B5 @ X8 )
% 0.25/0.54 @ A4
% 0.25/0.54 @ B6 )
% 0.25/0.54 => ~ ! [C8: list @ C] :
% 0.25/0.54 ( ( list_all_zip @ A @ C
% 0.25/0.54 @ ^ [A7: A,C3: C] :
% 0.25/0.54 ? [B5: B] : ( P @ A7 @ B5 @ C3 )
% 0.25/0.54 @ A4
% 0.25/0.54 @ C8 )
% 0.25/0.54 => ~ ( list_all_zip @ B @ C
% 0.25/0.54 @ ^ [B5: B,C3: C] :
% 0.25/0.54 ? [A7: A] : ( P @ A7 @ B5 @ C3 )
% 0.25/0.54 @ B6
% 0.25/0.54 @ C8 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % laz_swap_ex
% 0.25/0.54 thf(fact_5567_is__none__simps_I2_J,axiom,
% 0.25/0.54 ! [B: $tType,X: B] :
% 0.25/0.54 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_none_simps(2)
% 0.25/0.54 thf(fact_5568_list__all__zip__map2,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F: C > B,Bs2: list @ C] :
% 0.25/0.54 ( ( list_all_zip @ A @ B @ P @ As2 @ ( map @ C @ B @ F @ Bs2 ) )
% 0.25/0.54 = ( list_all_zip @ A @ C
% 0.25/0.54 @ ^ [A7: A,B5: C] : ( P @ A7 @ ( F @ B5 ) )
% 0.25/0.54 @ As2
% 0.25/0.54 @ Bs2 ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip_map2
% 0.25/0.54 thf(fact_5569_list__all__zip__map1,axiom,
% 0.25/0.54 ! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As2: list @ C,Bs2: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F @ As2 ) @ Bs2 )
% 0.25/0.54 = ( list_all_zip @ C @ B
% 0.25/0.54 @ ^ [A7: C] : ( P @ ( F @ A7 ) )
% 0.25/0.54 @ As2
% 0.25/0.54 @ Bs2 ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip_map1
% 0.25/0.54 thf(fact_5570_is__none__simps_I1_J,axiom,
% 0.25/0.54 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_none_simps(1)
% 0.25/0.54 thf(fact_5571_Option_Ois__none__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( is_none @ A )
% 0.25/0.54 = ( ^ [X3: option @ A] :
% 0.25/0.54 ( X3
% 0.25/0.54 = ( none @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Option.is_none_def
% 0.25/0.54 thf(fact_5572_list__all__zip_Osimps_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: A > B > $o,A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B @ P @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
% 0.25/0.54 = ( ( P @ A3 @ B2 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ P @ As2 @ Bs2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.simps(2)
% 0.25/0.54 thf(fact_5573_list__all__zip_Oelims_I3_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 0.25/0.54 ( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 0.25/0.54 => ( ! [A5: A,As: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ A5 @ As ) )
% 0.25/0.54 => ! [B3: B,Bs: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.54 => ( ( X @ A5 @ B3 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) )
% 0.25/0.54 => ( ( ? [V4: A,Va: list @ A] :
% 0.25/0.54 ( Xa
% 0.25/0.54 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.54 => ( Xb
% 0.25/0.54 != ( nil @ B ) ) )
% 0.25/0.54 => ~ ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ! [V4: B,Va: list @ B] :
% 0.25/0.54 ( Xb
% 0.25/0.54 != ( cons @ B @ V4 @ Va ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.elims(3)
% 0.25/0.54 thf(fact_5574_list__all__zip_Oelims_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 0.25/0.54 => ( ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ( Xb
% 0.25/0.54 != ( nil @ B ) ) )
% 0.25/0.54 => ~ ! [A5: A,As: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ A5 @ As ) )
% 0.25/0.54 => ! [B3: B,Bs: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.54 => ~ ( ( X @ A5 @ B3 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.elims(2)
% 0.25/0.54 thf(fact_5575_list__all__zip_Oelims_I1_J,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
% 0.25/0.54 ( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 0.25/0.54 = Y )
% 0.25/0.54 => ( ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ( ( Xb
% 0.25/0.54 = ( nil @ B ) )
% 0.25/0.54 => ~ Y ) )
% 0.25/0.54 => ( ! [A5: A,As: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ A5 @ As ) )
% 0.25/0.54 => ! [B3: B,Bs: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.54 => ( Y
% 0.25/0.54 = ( ~ ( ( X @ A5 @ B3 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) )
% 0.25/0.54 => ( ( ? [V4: A,Va: list @ A] :
% 0.25/0.54 ( Xa
% 0.25/0.54 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.54 => ( ( Xb
% 0.25/0.54 = ( nil @ B ) )
% 0.25/0.54 => Y ) )
% 0.25/0.54 => ~ ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ( ? [V4: B,Va: list @ B] :
% 0.25/0.54 ( Xb
% 0.25/0.54 = ( cons @ B @ V4 @ Va ) )
% 0.25/0.54 => Y ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.elims(1)
% 0.25/0.54 thf(fact_5576_list__all__zip_Osimps_I4_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: A > B > $o,V: B,Va2: list @ B] :
% 0.25/0.54 ~ ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ V @ Va2 ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.simps(4)
% 0.25/0.54 thf(fact_5577_list__all__zip_Osimps_I3_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,P: A > B > $o,V: A,Va2: list @ A] :
% 0.25/0.54 ~ ( list_all_zip @ A @ B @ P @ ( cons @ A @ V @ Va2 ) @ ( nil @ B ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.simps(3)
% 0.25/0.54 thf(fact_5578_the__map__option,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType,X: option @ A,F: A > B] :
% 0.25/0.54 ( ~ ( is_none @ A @ X )
% 0.25/0.54 => ( ( the2 @ B @ ( map_option @ A @ B @ F @ X ) )
% 0.25/0.54 = ( F @ ( the2 @ A @ X ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % the_map_option
% 0.25/0.54 thf(fact_5579_rel__optionI,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: option @ A,Y: option @ B,P: A > B > $o] :
% 0.25/0.54 ( ( ( is_none @ A @ X )
% 0.25/0.54 = ( is_none @ B @ Y ) )
% 0.25/0.54 => ( ( ~ ( is_none @ A @ X )
% 0.25/0.54 => ( ~ ( is_none @ B @ Y )
% 0.25/0.54 => ( P @ ( the2 @ A @ X ) @ ( the2 @ B @ Y ) ) ) )
% 0.25/0.54 => ( rel_option @ A @ B @ P @ X @ Y ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % rel_optionI
% 0.25/0.54 thf(fact_5580_rel__option__unfold,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( rel_option @ A @ B )
% 0.25/0.54 = ( ^ [R6: A > B > $o,X3: option @ A,Y3: option @ B] :
% 0.25/0.54 ( ( ( is_none @ A @ X3 )
% 0.25/0.54 = ( is_none @ B @ Y3 ) )
% 0.25/0.54 & ( ~ ( is_none @ A @ X3 )
% 0.25/0.54 => ( ~ ( is_none @ B @ Y3 )
% 0.25/0.54 => ( R6 @ ( the2 @ A @ X3 ) @ ( the2 @ B @ Y3 ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % rel_option_unfold
% 0.25/0.54 thf(fact_5581_list__all__zip_Opelims_I1_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
% 0.25/0.54 ( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 0.25/0.54 = Y )
% 0.25/0.54 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 0.25/0.54 => ( ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ( ( Xb
% 0.25/0.54 = ( nil @ B ) )
% 0.25/0.54 => ( Y
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
% 0.25/0.54 => ( ! [A5: A,As: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ A5 @ As ) )
% 0.25/0.54 => ! [B3: B,Bs: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.54 => ( ( Y
% 0.25/0.54 = ( ( X @ A5 @ B3 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) ) ) ) )
% 0.25/0.54 => ( ! [V4: A,Va: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.54 => ( ( Xb
% 0.25/0.54 = ( nil @ B ) )
% 0.25/0.54 => ( ~ Y
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
% 0.25/0.54 => ~ ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ! [V4: B,Va: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ V4 @ Va ) )
% 0.25/0.54 => ( ~ Y
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.pelims(1)
% 0.25/0.54 thf(fact_5582_list__all__zip_Opelims_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 0.25/0.54 ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 0.25/0.54 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 0.25/0.54 => ( ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ( ( Xb
% 0.25/0.54 = ( nil @ B ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) )
% 0.25/0.54 => ~ ! [A5: A,As: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ A5 @ As ) )
% 0.25/0.54 => ! [B3: B,Bs: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.54 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 0.25/0.54 => ~ ( ( X @ A5 @ B3 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.pelims(2)
% 0.25/0.54 thf(fact_5583_list__all__zip_Opelims_I3_J,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 0.25/0.54 ( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 0.25/0.54 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 0.25/0.54 => ( ! [A5: A,As: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ A5 @ As ) )
% 0.25/0.54 => ! [B3: B,Bs: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ B3 @ Bs ) )
% 0.25/0.54 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 0.25/0.54 => ( ( X @ A5 @ B3 )
% 0.25/0.54 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) )
% 0.25/0.54 => ( ! [V4: A,Va: list @ A] :
% 0.25/0.54 ( ( Xa
% 0.25/0.54 = ( cons @ A @ V4 @ Va ) )
% 0.25/0.54 => ( ( Xb
% 0.25/0.54 = ( nil @ B ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) )
% 0.25/0.54 => ~ ( ( Xa
% 0.25/0.54 = ( nil @ A ) )
% 0.25/0.54 => ! [V4: B,Va: list @ B] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( cons @ B @ V4 @ Va ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % list_all_zip.pelims(3)
% 0.25/0.54 thf(fact_5584_plus__multiset_Otransfer,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
% 0.25/0.54 @ ( pcr_multiset @ A @ A
% 0.25/0.54 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.25/0.54 @ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
% 0.25/0.54 @ ( pcr_multiset @ A @ A
% 0.25/0.54 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 0.25/0.54 @ ( pcr_multiset @ A @ A
% 0.25/0.54 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) ) )
% 0.25/0.54 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) )
% 0.25/0.54 @ ( plus_plus @ ( multiset @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % plus_multiset.transfer
% 0.25/0.54 thf(fact_5585_map__option__o__case__sum,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: D > C,G: A > ( option @ D ),H3: B > ( option @ D )] :
% 0.25/0.54 ( ( comp @ ( option @ D ) @ ( option @ C ) @ ( sum_sum @ A @ B ) @ ( map_option @ D @ C @ F ) @ ( sum_case_sum @ A @ ( option @ D ) @ B @ G @ H3 ) )
% 0.25/0.54 = ( sum_case_sum @ A @ ( option @ C ) @ B @ ( comp @ ( option @ D ) @ ( option @ C ) @ A @ ( map_option @ D @ C @ F ) @ G ) @ ( comp @ ( option @ D ) @ ( option @ C ) @ B @ ( map_option @ D @ C @ F ) @ H3 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % map_option_o_case_sum
% 0.25/0.54 thf(fact_5586_semilattice__set_Oeq__fold_H,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,A4: set @ A] :
% 0.25/0.54 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.25/0.54 => ( ( lattic1715443433743089157tice_F @ A @ F @ A4 )
% 0.25/0.54 = ( the2 @ A
% 0.25/0.54 @ ( finite_fold @ A @ ( option @ A )
% 0.25/0.54 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( F @ X3 ) @ Y3 ) )
% 0.25/0.54 @ ( none @ A )
% 0.25/0.54 @ A4 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % semilattice_set.eq_fold'
% 0.25/0.54 thf(fact_5587_o__case__sum,axiom,
% 0.25/0.54 ! [A: $tType,C: $tType,D: $tType,B: $tType,H3: D > C,F: A > D,G: B > D] :
% 0.25/0.54 ( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H3 @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
% 0.25/0.54 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H3 @ F ) @ ( comp @ D @ C @ B @ H3 @ G ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % o_case_sum
% 0.25/0.54 thf(fact_5588_semilattice__set_Oinfinite,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,A4: set @ A] :
% 0.25/0.54 ( ( lattic149705377957585745ce_set @ A @ F )
% 0.25/0.54 => ( ~ ( finite_finite2 @ A @ A4 )
% 0.25/0.54 => ( ( lattic1715443433743089157tice_F @ A @ F @ A4 )
% 0.25/0.54 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % semilattice_set.infinite
% 0.25/0.54 thf(fact_5589_group_Oaxioms_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 0.25/0.54 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.25/0.54 => ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ).
% 0.25/0.54
% 0.25/0.54 % group.axioms(2)
% 0.25/0.54 thf(fact_5590_normalize__div,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
% 0.25/0.54 = ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_div
% 0.25/0.54 thf(fact_5591_strict__mono__inv,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType] :
% 0.25/0.54 ( ( ( linorder @ B )
% 0.25/0.54 & ( linorder @ A ) )
% 0.25/0.54 => ! [F: A > B,G: B > A] :
% 0.25/0.54 ( ( order_strict_mono @ A @ B @ F )
% 0.25/0.54 => ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.25/0.54 = ( top_top @ ( set @ B ) ) )
% 0.25/0.54 => ( ! [X2: A] :
% 0.25/0.54 ( ( G @ ( F @ X2 ) )
% 0.25/0.54 = X2 )
% 0.25/0.54 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % strict_mono_inv
% 0.25/0.54 thf(fact_5592_gcd_Onormalize__bottom,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 0.25/0.54 = ( one_one @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % gcd.normalize_bottom
% 0.25/0.54 thf(fact_5593_normalize__1,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 0.25/0.54 = ( one_one @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_1
% 0.25/0.54 thf(fact_5594_lcm_Otop__left__normalize,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
% 0.25/0.54 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm.top_left_normalize
% 0.25/0.54 thf(fact_5595_lcm_Otop__right__normalize,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm.top_right_normalize
% 0.25/0.54 thf(fact_5596_normalize__mult__unit__right,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [B2: A,A3: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.25/0.54 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_mult_unit_right
% 0.25/0.54 thf(fact_5597_normalize__mult__unit__left,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A,B2: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 0.25/0.54 = ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_mult_unit_left
% 0.25/0.54 thf(fact_5598_unit__factor__normalize,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( A3
% 0.25/0.54 != ( zero_zero @ A ) )
% 0.25/0.54 => ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
% 0.25/0.54 = ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % unit_factor_normalize
% 0.25/0.54 thf(fact_5599_normalize__unit__factor,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( A3
% 0.25/0.54 != ( zero_zero @ A ) )
% 0.25/0.54 => ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 0.25/0.54 = ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_unit_factor
% 0.25/0.54 thf(fact_5600_associated__unit,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A,B2: A] :
% 0.25/0.54 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.25/0.54 = ( normal6383669964737779283malize @ A @ B2 ) )
% 0.25/0.54 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % associated_unit
% 0.25/0.54 thf(fact_5601_normalize__1__iff,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.25/0.54 = ( one_one @ A ) )
% 0.25/0.54 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_1_iff
% 0.25/0.54 thf(fact_5602_is__unit__normalize,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 => ( ( normal6383669964737779283malize @ A @ A3 )
% 0.25/0.54 = ( one_one @ A ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % is_unit_normalize
% 0.25/0.54 thf(fact_5603_normalize__idem__imp__is__unit__iff,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( normal8620421768224518004emidom @ A )
% 0.25/0.54 => ! [A3: A] :
% 0.25/0.54 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 0.25/0.54 = A3 )
% 0.25/0.54 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 0.25/0.54 = ( A3
% 0.25/0.54 = ( one_one @ A ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % normalize_idem_imp_is_unit_iff
% 0.25/0.54 thf(fact_5604_group__axioms__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( group_axioms @ A )
% 0.25/0.54 = ( ^ [F2: A > A > A,Z5: A,Inverse2: A > A] :
% 0.25/0.54 ( ! [A7: A] :
% 0.25/0.54 ( ( F2 @ Z5 @ A7 )
% 0.25/0.54 = A7 )
% 0.25/0.54 & ! [A7: A] :
% 0.25/0.54 ( ( F2 @ ( Inverse2 @ A7 ) @ A7 )
% 0.25/0.54 = Z5 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % group_axioms_def
% 0.25/0.54 thf(fact_5605_group__axioms_Ointro,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 0.25/0.54 ( ! [A5: A] :
% 0.25/0.54 ( ( F @ Z2 @ A5 )
% 0.25/0.54 = A5 )
% 0.25/0.54 => ( ! [A5: A] :
% 0.25/0.54 ( ( F @ ( Inverse @ A5 ) @ A5 )
% 0.25/0.54 = Z2 )
% 0.25/0.54 => ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % group_axioms.intro
% 0.25/0.54 thf(fact_5606_strict__mono__imp__inj__on,axiom,
% 0.25/0.54 ! [B: $tType,A: $tType] :
% 0.25/0.54 ( ( ( linorder @ A )
% 0.25/0.54 & ( order @ B ) )
% 0.25/0.54 => ! [F: A > B,A4: set @ A] :
% 0.25/0.54 ( ( order_strict_mono @ A @ B @ F )
% 0.25/0.54 => ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % strict_mono_imp_inj_on
% 0.25/0.54 thf(fact_5607_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Lcm_fin.bounded_quasi_semilattice_set_axioms
% 0.25/0.54 thf(fact_5608_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Gcd_fin.bounded_quasi_semilattice_set_axioms
% 0.25/0.54 thf(fact_5609_lcm_Obounded__quasi__semilattice__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % lcm.bounded_quasi_semilattice_axioms
% 0.25/0.54 thf(fact_5610_gcd_Obounded__quasi__semilattice__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semiring_gcd @ A )
% 0.25/0.54 => ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % gcd.bounded_quasi_semilattice_axioms
% 0.25/0.54 thf(fact_5611_group__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( group @ A )
% 0.25/0.54 = ( ^ [F2: A > A > A,Z5: A,Inverse2: A > A] :
% 0.25/0.54 ( ( semigroup @ A @ F2 )
% 0.25/0.54 & ( group_axioms @ A @ F2 @ Z5 @ Inverse2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % group_def
% 0.25/0.54 thf(fact_5612_group_Ointro,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 0.25/0.54 ( ( semigroup @ A @ F )
% 0.25/0.54 => ( ( group_axioms @ A @ F @ Z2 @ Inverse )
% 0.25/0.54 => ( group @ A @ F @ Z2 @ Inverse ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % group.intro
% 0.25/0.54 thf(fact_5613_Predicate_Oiterate__upto_Opsimps,axiom,
% 0.25/0.54 ! [A: $tType,F: code_natural > A,N: code_natural,M2: code_natural] :
% 0.25/0.54 ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F @ ( product_Pair @ code_natural @ code_natural @ N @ M2 ) ) )
% 0.25/0.54 => ( ( iterate_upto @ A @ F @ N @ M2 )
% 0.25/0.54 = ( seq2 @ A
% 0.25/0.54 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M2 @ N ) @ ( empty @ A ) @ ( insert @ A @ ( F @ N ) @ ( iterate_upto @ A @ F @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M2 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Predicate.iterate_upto.psimps
% 0.25/0.54 thf(fact_5614_group_Oaxioms_I1_J,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 0.25/0.54 ( ( group @ A @ F @ Z2 @ Inverse )
% 0.25/0.54 => ( semigroup @ A @ F ) ) ).
% 0.25/0.54
% 0.25/0.54 % group.axioms(1)
% 0.25/0.54 thf(fact_5615_monoid_Oaxioms_I1_J,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.54 ( ( monoid @ A @ F @ Z2 )
% 0.25/0.54 => ( semigroup @ A @ F ) ) ).
% 0.25/0.54
% 0.25/0.54 % monoid.axioms(1)
% 0.25/0.54 thf(fact_5616_semigroup__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semigroup @ A )
% 0.25/0.54 = ( ^ [F2: A > A > A] :
% 0.25/0.54 ! [A7: A,B5: A,C3: A] :
% 0.25/0.54 ( ( F2 @ ( F2 @ A7 @ B5 ) @ C3 )
% 0.25/0.54 = ( F2 @ A7 @ ( F2 @ B5 @ C3 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % semigroup_def
% 0.25/0.54 thf(fact_5617_semigroup_Ointro,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A] :
% 0.25/0.54 ( ! [A5: A,B3: A,C4: A] :
% 0.25/0.54 ( ( F @ ( F @ A5 @ B3 ) @ C4 )
% 0.25/0.54 = ( F @ A5 @ ( F @ B3 @ C4 ) ) )
% 0.25/0.54 => ( semigroup @ A @ F ) ) ).
% 0.25/0.54
% 0.25/0.54 % semigroup.intro
% 0.25/0.54 thf(fact_5618_semigroup_Oassoc,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,A3: A,B2: A,C2: A] :
% 0.25/0.54 ( ( semigroup @ A @ F )
% 0.25/0.54 => ( ( F @ ( F @ A3 @ B2 ) @ C2 )
% 0.25/0.54 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % semigroup.assoc
% 0.25/0.54 thf(fact_5619_mult_Osemigroup__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semigroup_mult @ A )
% 0.25/0.54 => ( semigroup @ A @ ( times_times @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mult.semigroup_axioms
% 0.25/0.54 thf(fact_5620_Predicate_Oiterate__upto_Osimps,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( iterate_upto @ A )
% 0.25/0.54 = ( ^ [F2: code_natural > A,N5: code_natural,M: code_natural] :
% 0.25/0.54 ( seq2 @ A
% 0.25/0.54 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M @ N5 ) @ ( empty @ A ) @ ( insert @ A @ ( F2 @ N5 ) @ ( iterate_upto @ A @ F2 @ ( plus_plus @ code_natural @ N5 @ ( one_one @ code_natural ) ) @ M ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Predicate.iterate_upto.simps
% 0.25/0.54 thf(fact_5621_Predicate_Oiterate__upto_Oelims,axiom,
% 0.25/0.54 ! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
% 0.25/0.54 ( ( ( iterate_upto @ A @ X @ Xa @ Xb )
% 0.25/0.54 = Y )
% 0.25/0.54 => ( Y
% 0.25/0.54 = ( seq2 @ A
% 0.25/0.54 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Predicate.iterate_upto.elims
% 0.25/0.54 thf(fact_5622_add_Osemigroup__axioms,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( semigroup_add @ A )
% 0.25/0.54 => ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % add.semigroup_axioms
% 0.25/0.54 thf(fact_5623_Predicate_Oiterate__upto_Opelims,axiom,
% 0.25/0.54 ! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
% 0.25/0.54 ( ( ( iterate_upto @ A @ X @ Xa @ Xb )
% 0.25/0.54 = Y )
% 0.25/0.54 => ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) )
% 0.25/0.54 => ~ ( ( Y
% 0.25/0.54 = ( seq2 @ A
% 0.25/0.54 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) )
% 0.25/0.54 => ~ ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Predicate.iterate_upto.pelims
% 0.25/0.54 thf(fact_5624_Random__Pred_Oiterate__upto__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( random_iterate_upto @ A )
% 0.25/0.54 = ( ^ [F2: code_natural > A,N5: code_natural,M: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F2 @ N5 @ M ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % Random_Pred.iterate_upto_def
% 0.25/0.54 thf(fact_5625_monoid_Ointro,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.54 ( ( semigroup @ A @ F )
% 0.25/0.54 => ( ( monoid_axioms @ A @ F @ Z2 )
% 0.25/0.54 => ( monoid @ A @ F @ Z2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % monoid.intro
% 0.25/0.54 thf(fact_5626_monoid__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( monoid @ A )
% 0.25/0.54 = ( ^ [F2: A > A > A,Z5: A] :
% 0.25/0.54 ( ( semigroup @ A @ F2 )
% 0.25/0.54 & ( monoid_axioms @ A @ F2 @ Z5 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % monoid_def
% 0.25/0.54 thf(fact_5627_monoid__axioms_Ointro,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.54 ( ! [A5: A] :
% 0.25/0.54 ( ( F @ Z2 @ A5 )
% 0.25/0.54 = A5 )
% 0.25/0.54 => ( ! [A5: A] :
% 0.25/0.54 ( ( F @ A5 @ Z2 )
% 0.25/0.54 = A5 )
% 0.25/0.54 => ( monoid_axioms @ A @ F @ Z2 ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % monoid_axioms.intro
% 0.25/0.54 thf(fact_5628_monoid__axioms__def,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( monoid_axioms @ A )
% 0.25/0.54 = ( ^ [F2: A > A > A,Z5: A] :
% 0.25/0.54 ( ! [A7: A] :
% 0.25/0.54 ( ( F2 @ Z5 @ A7 )
% 0.25/0.54 = A7 )
% 0.25/0.54 & ! [A7: A] :
% 0.25/0.54 ( ( F2 @ A7 @ Z5 )
% 0.25/0.54 = A7 ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % monoid_axioms_def
% 0.25/0.54 thf(fact_5629_monoid_Oaxioms_I2_J,axiom,
% 0.25/0.54 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.54 ( ( monoid @ A @ F @ Z2 )
% 0.25/0.54 => ( monoid_axioms @ A @ F @ Z2 ) ) ).
% 0.25/0.54
% 0.25/0.54 % monoid.axioms(2)
% 0.25/0.54 thf(fact_5630_mod__h__bot__normalize,axiom,
% 0.25/0.54 ! [A: $tType,H3: heap_ext @ product_unit,P: assn] :
% 0.25/0.54 ( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H3 )
% 0.25/0.54 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_normalize
% 0.25/0.54 thf(fact_5631_image__bind,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,C: $tType,F: B > A,A4: set @ C,G: C > ( set @ B )] :
% 0.25/0.54 ( ( image2 @ B @ A @ F @ ( bind3 @ C @ B @ A4 @ G ) )
% 0.25/0.54 = ( bind3 @ C @ A @ A4 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image2 @ B @ A @ F ) @ G ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % image_bind
% 0.25/0.54 thf(fact_5632_pure__assn__raw_Ocases,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
% 0.25/0.54 ~ ! [B3: $o,H2: A,As: set @ B] :
% 0.25/0.54 ( X
% 0.25/0.54 != ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % pure_assn_raw.cases
% 0.25/0.54 thf(fact_5633_mod__h__bot__iff_I7_J,axiom,
% 0.25/0.54 ! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
% 0.25/0.54 ( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 | ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_iff(7)
% 0.25/0.54 thf(fact_5634_mod__h__bot__iff_I5_J,axiom,
% 0.25/0.54 ! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
% 0.25/0.54 ( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 & ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_iff(5)
% 0.25/0.54 thf(fact_5635_mod__h__bot__iff_I6_J,axiom,
% 0.25/0.54 ! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
% 0.25/0.54 ( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 & ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_iff(6)
% 0.25/0.54 thf(fact_5636_mod__h__bot__iff_I2_J,axiom,
% 0.25/0.54 ! [H3: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_iff(2)
% 0.25/0.54 thf(fact_5637_times__assn__raw_Ocases,axiom,
% 0.25/0.54 ! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
% 0.25/0.54 ~ ! [P7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.54 ( X
% 0.25/0.54 != ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P7 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % times_assn_raw.cases
% 0.25/0.54 thf(fact_5638_sngr__assn__raw_Ocases,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap @ A )
% 0.25/0.54 => ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
% 0.25/0.54 ~ ! [R2: ref @ A,X2: A,H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.54 ( X
% 0.25/0.54 != ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R2 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % sngr_assn_raw.cases
% 0.25/0.54 thf(fact_5639_snga__assn__raw_Ocases,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap @ A )
% 0.25/0.54 => ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
% 0.25/0.54 ~ ! [R2: array @ A,X2: list @ A,H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.54 ( X
% 0.25/0.54 != ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R2 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % snga_assn_raw.cases
% 0.25/0.54 thf(fact_5640_one__assn__raw_Ocases,axiom,
% 0.25/0.54 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.54 ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.54 ( X
% 0.25/0.54 != ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ).
% 0.25/0.54
% 0.25/0.54 % one_assn_raw.cases
% 0.25/0.54 thf(fact_5641_mod__star__conv,axiom,
% 0.25/0.54 ! [A4: assn,B6: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.54 ( ( rep_assn @ ( times_times @ assn @ A4 @ B6 ) @ H3 )
% 0.25/0.54 = ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
% 0.25/0.54 ( ( H3
% 0.25/0.54 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) ) )
% 0.25/0.54 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 0.25/0.54 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.54 & ( rep_assn @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
% 0.25/0.54 & ( rep_assn @ B6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_star_conv
% 0.25/0.54 thf(fact_5642_star__assnI,axiom,
% 0.25/0.54 ! [P: assn,H3: heap_ext @ product_unit,As2: set @ nat,Q: assn,As6: set @ nat] :
% 0.25/0.54 ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.54 => ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) )
% 0.25/0.54 => ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As6 )
% 0.25/0.54 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.54 => ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % star_assnI
% 0.25/0.54 thf(fact_5643_mod__emp__simp,axiom,
% 0.25/0.54 ! [H3: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_emp_simp
% 0.25/0.54 thf(fact_5644_mod__h__bot__indep,axiom,
% 0.25/0.54 ! [P: assn,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 0.25/0.54 ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.54 = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_indep
% 0.25/0.54 thf(fact_5645_bind__image,axiom,
% 0.25/0.54 ! [A: $tType,B: $tType,C: $tType,F: C > B,A4: set @ C,G: B > ( set @ A )] :
% 0.25/0.54 ( ( bind3 @ B @ A @ ( image2 @ C @ B @ F @ A4 ) @ G )
% 0.25/0.54 = ( bind3 @ C @ A @ A4 @ ( comp @ B @ ( set @ A ) @ C @ G @ F ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % bind_image
% 0.25/0.54 thf(fact_5646_mod__h__bot__iff_I3_J,axiom,
% 0.25/0.54 ! [A: $tType] :
% 0.25/0.54 ( ( heap @ A )
% 0.25/0.54 => ! [P5: ref @ A,X: A,H3: heap_ext @ product_unit] :
% 0.25/0.54 ~ ( rep_assn @ ( sngr_assn @ A @ P5 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_iff(3)
% 0.25/0.54 thf(fact_5647_mod__h__bot__iff_I4_J,axiom,
% 0.25/0.54 ! [B: $tType] :
% 0.25/0.54 ( ( heap @ B )
% 0.25/0.54 => ! [Q3: array @ B,Y: list @ B,H3: heap_ext @ product_unit] :
% 0.25/0.54 ~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % mod_h_bot_iff(4)
% 0.25/0.54 thf(fact_5648_times__assn__raw_Osimps,axiom,
% 0.25/0.54 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.54 ( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.54 = ( ? [As1: set @ nat,As22: set @ nat] :
% 0.25/0.54 ( ( As2
% 0.25/0.54 = ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
% 0.25/0.54 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 0.25/0.54 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.54 & ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As1 ) )
% 0.25/0.54 & ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As22 ) ) ) ) ) ).
% 0.25/0.54
% 0.25/0.54 % times_assn_raw.simps
% 0.25/0.54 thf(fact_5649_times__assn__raw_Oelims_I3_J,axiom,
% 0.25/0.54 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.54 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 0.25/0.54 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.54 ( ( Xb
% 0.25/0.54 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.54 => ? [As12: set @ nat,As23: set @ nat] :
% 0.25/0.54 ( ( As
% 0.25/0.54 = ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
% 0.25/0.54 & ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
% 0.25/0.54 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.54 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As12 ) )
% 0.25/0.54 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As23 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % times_assn_raw.elims(3)
% 0.25/0.55 thf(fact_5650_times__assn__raw_Oelims_I2_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ~ ? [As13: set @ nat,As24: set @ nat] :
% 0.25/0.55 ( ( As
% 0.25/0.55 = ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
% 0.25/0.55 & ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As13 ) )
% 0.25/0.55 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As24 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % times_assn_raw.elims(2)
% 0.25/0.55 thf(fact_5651_times__assn__raw_Oelims_I1_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( ~ ? [As1: set @ nat,As22: set @ nat] :
% 0.25/0.55 ( ( As
% 0.25/0.55 = ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
% 0.25/0.55 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
% 0.25/0.55 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % times_assn_raw.elims(1)
% 0.25/0.55 thf(fact_5652_times__assn__raw_Opelims_I1_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ? [As1: set @ nat,As22: set @ nat] :
% 0.25/0.55 ( ( As
% 0.25/0.55 = ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
% 0.25/0.55 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
% 0.25/0.55 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % times_assn_raw.pelims(1)
% 0.25/0.55 thf(fact_5653_times__assn__raw_Opelims_I2_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( times_assn_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ~ ? [As13: set @ nat,As24: set @ nat] :
% 0.25/0.55 ( ( As
% 0.25/0.55 = ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
% 0.25/0.55 & ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As13 ) )
% 0.25/0.55 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As24 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % times_assn_raw.pelims(2)
% 0.25/0.55 thf(fact_5654_times__assn__raw_Opelims_I3_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ? [As12: set @ nat,As23: set @ nat] :
% 0.25/0.55 ( ( As
% 0.25/0.55 = ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
% 0.25/0.55 & ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As12 ) )
% 0.25/0.55 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As23 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % times_assn_raw.pelims(3)
% 0.25/0.55 thf(fact_5655_mod__h__bot__iff_I8_J,axiom,
% 0.25/0.55 ! [C: $tType,R5: C > assn,H3: heap_ext @ product_unit] :
% 0.25/0.55 ( ( rep_assn @ ( ex_assn @ C @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 = ( ? [X3: C] : ( rep_assn @ ( R5 @ X3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mod_h_bot_iff(8)
% 0.25/0.55 thf(fact_5656_make__rule,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [N: nat,F: nat > A] :
% 0.25/0.55 ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F )
% 0.25/0.55 @ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % make_rule
% 0.25/0.55 thf(fact_5657_mod__h__bot__iff_I1_J,axiom,
% 0.25/0.55 ! [B2: $o,H3: heap_ext @ product_unit] :
% 0.25/0.55 ( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 = B2 ) ).
% 0.25/0.55
% 0.25/0.55 % mod_h_bot_iff(1)
% 0.25/0.55 thf(fact_5658_bind__rule,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap @ A,R5: A > assn,G: A > ( heap_Time_Heap @ B ),Q: B > assn] :
% 0.25/0.55 ( ( hoare_hoare_triple @ A @ P @ F @ R5 )
% 0.25/0.55 => ( ! [X2: A] : ( hoare_hoare_triple @ B @ ( R5 @ X2 ) @ ( G @ X2 ) @ Q )
% 0.25/0.55 => ( hoare_hoare_triple @ B @ P @ ( heap_Time_bind @ A @ B @ F @ G ) @ Q ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % bind_rule
% 0.25/0.55 thf(fact_5659_raise__iff,axiom,
% 0.25/0.55 ! [A: $tType,P: assn,S: list @ char,Q: A > assn] :
% 0.25/0.55 ( ( hoare_hoare_triple @ A @ P @ ( heap_Time_raise @ A @ S ) @ Q )
% 0.25/0.55 = ( P
% 0.25/0.55 = ( bot_bot @ assn ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % raise_iff
% 0.25/0.55 thf(fact_5660_raise__rule,axiom,
% 0.25/0.55 ! [A: $tType,S: list @ char,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ ( heap_Time_raise @ A @ S ) @ Q ) ).
% 0.25/0.55
% 0.25/0.55 % raise_rule
% 0.25/0.55 thf(fact_5661_bind__rule_H,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap @ B,R5: B > assn,G: B > ( heap_Time_Heap @ C ),Q: C > assn] :
% 0.25/0.55 ( ! [R2: A] :
% 0.25/0.55 ( hoare_hoare_triple @ B @ P @ F
% 0.25/0.55 @ ^ [S4: B] : ( times_times @ assn @ P @ ( R5 @ S4 ) ) )
% 0.25/0.55 => ( ! [X2: B] : ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( R5 @ X2 ) @ P ) @ ( G @ X2 ) @ Q )
% 0.25/0.55 => ( hoare_hoare_triple @ C @ P @ ( heap_Time_bind @ B @ C @ F @ G ) @ Q ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % bind_rule'
% 0.25/0.55 thf(fact_5662_case__prod__rule,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,C: $tType,X: product_prod @ A @ B,P: assn,F: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
% 0.25/0.55 ( ! [A5: A,B3: B] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.25/0.55 => ( hoare_hoare_triple @ C @ P @ ( F @ A5 @ B3 ) @ Q ) )
% 0.25/0.55 => ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ X ) @ Q ) ) ).
% 0.25/0.55
% 0.25/0.55 % case_prod_rule
% 0.25/0.55 thf(fact_5663_hoare__triple__success,axiom,
% 0.25/0.55 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
% 0.25/0.55 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( heap_Time_success @ A @ C2 @ H3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % hoare_triple_success
% 0.25/0.55 thf(fact_5664_case__option__rule,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,V: option @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fs: A > ( heap_Time_Heap @ B )] :
% 0.25/0.55 ( ( ( V
% 0.25/0.55 = ( none @ A ) )
% 0.25/0.55 => ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( V
% 0.25/0.55 = ( some @ A @ X2 ) )
% 0.25/0.55 => ( hoare_hoare_triple @ B @ P @ ( Fs @ X2 ) @ Q ) )
% 0.25/0.55 => ( hoare_hoare_triple @ B @ P @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fs @ V ) @ Q ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % case_option_rule
% 0.25/0.55 thf(fact_5665_of__list__rule,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [Xs: list @ A] :
% 0.25/0.55 ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs )
% 0.25/0.55 @ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ Xs ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % of_list_rule
% 0.25/0.55 thf(fact_5666_new__rule,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [N: nat,X: A] :
% 0.25/0.55 ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X )
% 0.25/0.55 @ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ ( replicate @ A @ N @ X ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % new_rule
% 0.25/0.55 thf(fact_5667_hoare__triple__effect,axiom,
% 0.25/0.55 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
% 0.25/0.55 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ? [H6: heap_ext @ product_unit,R2: A,T7: nat] :
% 0.25/0.55 ( ( heap_Time_effect @ A @ C2 @ H3 @ H6 @ R2 @ T7 )
% 0.25/0.55 & ( rep_assn @ ( Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( hoare_new_addrs @ H3 @ As2 @ H6 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % hoare_triple_effect
% 0.25/0.55 thf(fact_5668_hoare__triple__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( hoare_hoare_triple @ A )
% 0.25/0.55 = ( ^ [P4: assn,C3: heap_Time_Heap @ A,Q6: A > assn] :
% 0.25/0.55 ! [H: heap_ext @ product_unit,As5: set @ nat] :
% 0.25/0.55 ( ( rep_assn @ P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 0.25/0.55 => ? [R3: A,H7: heap_ext @ product_unit] :
% 0.25/0.55 ( ? [T3: nat] :
% 0.25/0.55 ( ( heap_Time_execute @ A @ C3 @ H )
% 0.25/0.55 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T3 ) ) ) )
% 0.25/0.55 & ( rep_assn @ ( Q6 @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( hoare_new_addrs @ H @ As5 @ H7 ) ) )
% 0.25/0.55 & ( relH
% 0.25/0.55 @ ( collect @ nat
% 0.25/0.55 @ ^ [A7: nat] :
% 0.25/0.55 ( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H ) )
% 0.25/0.55 & ~ ( member @ nat @ A7 @ As5 ) ) )
% 0.25/0.55 @ H
% 0.25/0.55 @ H7 )
% 0.25/0.55 & ( ord_less_eq @ nat @ ( lim @ product_unit @ H ) @ ( lim @ product_unit @ H7 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % hoare_triple_def
% 0.25/0.55 thf(fact_5669_hoare__tripleI,axiom,
% 0.25/0.55 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
% 0.25/0.55 ( ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ? [R15: A,H9: heap_ext @ product_unit,T8: nat] :
% 0.25/0.55 ( ( ( heap_Time_execute @ A @ C2 @ H2 )
% 0.25/0.55 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R15 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ T8 ) ) ) )
% 0.25/0.55 & ( rep_assn @ ( Q @ R15 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( hoare_new_addrs @ H2 @ As @ H9 ) ) )
% 0.25/0.55 & ( relH
% 0.25/0.55 @ ( collect @ nat
% 0.25/0.55 @ ^ [A7: nat] :
% 0.25/0.55 ( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H2 ) )
% 0.25/0.55 & ~ ( member @ nat @ A7 @ As ) ) )
% 0.25/0.55 @ H2
% 0.25/0.55 @ H9 )
% 0.25/0.55 & ( ord_less_eq @ nat @ ( lim @ product_unit @ H2 ) @ ( lim @ product_unit @ H9 ) ) ) )
% 0.25/0.55 => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% 0.25/0.55
% 0.25/0.55 % hoare_tripleI
% 0.25/0.55 thf(fact_5670_mod__relH,axiom,
% 0.25/0.55 ! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,P: assn] :
% 0.25/0.55 ( ( relH @ As2 @ H3 @ H4 )
% 0.25/0.55 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mod_relH
% 0.25/0.55 thf(fact_5671_hoare__tripleE,axiom,
% 0.25/0.55 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
% 0.25/0.55 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ~ ! [R2: A,H6: heap_ext @ product_unit] :
% 0.25/0.55 ( ? [T7: nat] :
% 0.25/0.55 ( ( heap_Time_execute @ A @ C2 @ H3 )
% 0.25/0.55 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ T7 ) ) ) )
% 0.25/0.55 => ( ( rep_assn @ ( Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( hoare_new_addrs @ H3 @ As2 @ H6 ) ) )
% 0.25/0.55 => ( ( relH
% 0.25/0.55 @ ( collect @ nat
% 0.25/0.55 @ ^ [A7: nat] :
% 0.25/0.55 ( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H3 ) )
% 0.25/0.55 & ~ ( member @ nat @ A7 @ As2 ) ) )
% 0.25/0.55 @ H3
% 0.25/0.55 @ H6 )
% 0.25/0.55 => ~ ( ord_less_eq @ nat @ ( lim @ product_unit @ H3 ) @ ( lim @ product_unit @ H6 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % hoare_tripleE
% 0.25/0.55 thf(fact_5672_Array__Time_Oalloc__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( ( array_alloc @ A )
% 0.25/0.55 = ( ^ [Xs3: list @ A,H: heap_ext @ product_unit] :
% 0.25/0.55 ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H ) )
% 0.25/0.55 @ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H ) ) @ Xs3
% 0.25/0.55 @ ( lim_update @ product_unit
% 0.25/0.55 @ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
% 0.25/0.55 @ H ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Array_Time.alloc_def
% 0.25/0.55 thf(fact_5673_snga__assn__raw_Opelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ( ( array_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.pelims(1)
% 0.25/0.55 thf(fact_5674_snga__assn__raw_Opelims_I2_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ~ ( ( ( array_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.pelims(2)
% 0.25/0.55 thf(fact_5675_Array__Time_Opresent__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( ( array_present @ A )
% 0.25/0.55 = ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A7 ) @ ( lim @ product_unit @ H ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Array_Time.present_def
% 0.25/0.55 thf(fact_5676_snga__assn__raw_Osimps,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: array @ A,X: list @ A,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( snga_assn_raw @ A @ R @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( ( ( array_get @ A @ H3 @ R )
% 0.25/0.55 = X )
% 0.25/0.55 & ( As2
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ R ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ R ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.simps
% 0.25/0.55 thf(fact_5677_snga__assn__raw_Oelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( ~ ( ( ( array_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.elims(1)
% 0.25/0.55 thf(fact_5678_snga__assn__raw_Oelims_I2_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ~ ( ( ( array_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.elims(2)
% 0.25/0.55 thf(fact_5679_snga__assn__raw_Oelims_I3_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( ( array_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.elims(3)
% 0.25/0.55 thf(fact_5680_snga__assn__raw_Opelims_I3_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ( ( ( array_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % snga_assn_raw.pelims(3)
% 0.25/0.55 thf(fact_5681_relH__set__array,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: array @ A,As2: set @ nat,H3: heap_ext @ product_unit,X: list @ A] :
% 0.25/0.55 ( ~ ( member @ nat @ ( addr_of_array @ A @ R ) @ As2 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( relH @ As2 @ H3 @ ( array_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % relH_set_array
% 0.25/0.55 thf(fact_5682_sngr__assn__raw_Opelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ( ( ref_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.pelims(1)
% 0.25/0.55 thf(fact_5683_sngr__assn__raw_Opelims_I2_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ~ ( ( ( ref_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.pelims(2)
% 0.25/0.55 thf(fact_5684_in__range__empty,axiom,
% 0.25/0.55 ! [H3: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range_empty
% 0.25/0.55 thf(fact_5685_in__range__dist__union,axiom,
% 0.25/0.55 ! [H3: heap_ext @ product_unit,As2: set @ nat,As6: set @ nat] :
% 0.25/0.55 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) )
% 0.25/0.55 = ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range_dist_union
% 0.25/0.55 thf(fact_5686_relH__set__ref,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: ref @ A,As2: set @ nat,H3: heap_ext @ product_unit,X: A] :
% 0.25/0.55 ( ~ ( member @ nat @ ( addr_of_ref @ A @ R ) @ As2 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( relH @ As2 @ H3 @ ( ref_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % relH_set_ref
% 0.25/0.55 thf(fact_5687_relH__refl,axiom,
% 0.25/0.55 ! [H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( relH @ As2 @ H3 @ H3 ) ) ).
% 0.25/0.55
% 0.25/0.55 % relH_refl
% 0.25/0.55 thf(fact_5688_relH__in__rangeI_I1_J,axiom,
% 0.25/0.55 ! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 0.25/0.55 ( ( relH @ As2 @ H3 @ H4 )
% 0.25/0.55 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % relH_in_rangeI(1)
% 0.25/0.55 thf(fact_5689_relH__in__rangeI_I2_J,axiom,
% 0.25/0.55 ! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 0.25/0.55 ( ( relH @ As2 @ H3 @ H4 )
% 0.25/0.55 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % relH_in_rangeI(2)
% 0.25/0.55 thf(fact_5690_in__range__subset,axiom,
% 0.25/0.55 ! [As2: set @ nat,As6: set @ nat,H3: heap_ext @ product_unit] :
% 0.25/0.55 ( ( ord_less_eq @ ( set @ nat ) @ As2 @ As6 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) )
% 0.25/0.55 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range_subset
% 0.25/0.55 thf(fact_5691_in__range_Oelims_I3_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( in_range @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ! [X2: nat] :
% 0.25/0.55 ( ( member @ nat @ X2 @ As )
% 0.25/0.55 => ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.elims(3)
% 0.25/0.55 thf(fact_5692_in__range_Oelims_I2_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( in_range @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ~ ! [X7: nat] :
% 0.25/0.55 ( ( member @ nat @ X7 @ As )
% 0.25/0.55 => ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.elims(2)
% 0.25/0.55 thf(fact_5693_in__range_Oelims_I1_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( in_range @ X )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( ~ ! [X3: nat] :
% 0.25/0.55 ( ( member @ nat @ X3 @ As )
% 0.25/0.55 => ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.elims(1)
% 0.25/0.55 thf(fact_5694_in__range_Osimps,axiom,
% 0.25/0.55 ! [H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( ! [X3: nat] :
% 0.25/0.55 ( ( member @ nat @ X3 @ As2 )
% 0.25/0.55 => ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.simps
% 0.25/0.55 thf(fact_5695_sngr__assn__raw_Oelims_I3_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( ( ref_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.elims(3)
% 0.25/0.55 thf(fact_5696_sngr__assn__raw_Oelims_I2_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ~ ( ( ( ref_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.elims(2)
% 0.25/0.55 thf(fact_5697_sngr__assn__raw_Oelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( ~ ( ( ( ref_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.elims(1)
% 0.25/0.55 thf(fact_5698_sngr__assn__raw_Osimps,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: ref @ A,X: A,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( sngr_assn_raw @ A @ R @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( ( ( ref_get @ A @ H3 @ R )
% 0.25/0.55 = X )
% 0.25/0.55 & ( As2
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ R ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ R ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.simps
% 0.25/0.55 thf(fact_5699_sngr__assn__raw_Opelims_I3_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ( ( ( ref_get @ A @ H2 @ X )
% 0.25/0.55 = Xa )
% 0.25/0.55 & ( As
% 0.25/0.55 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sngr_assn_raw.pelims(3)
% 0.25/0.55 thf(fact_5700_wand__assnI,axiom,
% 0.25/0.55 ! [H3: heap_ext @ product_unit,As2: set @ nat,Q: assn,R5: assn] :
% 0.25/0.55 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( ! [H6: heap_ext @ product_unit,As7: set @ nat] :
% 0.25/0.55 ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As7 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 => ( ( relH @ As2 @ H3 @ H6 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
% 0.25/0.55 => ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) )
% 0.25/0.55 => ( rep_assn @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As2 @ As7 ) ) ) ) ) ) )
% 0.25/0.55 => ( rep_assn @ ( wand_assn @ Q @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_assnI
% 0.25/0.55 thf(fact_5701_wand__raw_Osimps,axiom,
% 0.25/0.55 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 & ! [H7: heap_ext @ product_unit,As8: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As8 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As2 @ H3 @ H7 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As2 ) )
% 0.25/0.55 & ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
% 0.25/0.55 => ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As2 @ As8 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.simps
% 0.25/0.55 thf(fact_5702_wand__raw_Oelims_I1_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( wand_raw @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 & ! [H7: heap_ext @ product_unit,As8: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As @ H2 @ H7 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
% 0.25/0.55 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.elims(1)
% 0.25/0.55 thf(fact_5703_wand__raw_Oelims_I3_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 & ! [H6: heap_ext @ product_unit,As7: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As @ H2 @ H6 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) ) )
% 0.25/0.55 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.elims(3)
% 0.25/0.55 thf(fact_5704_wand__raw_Oelims_I2_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( wand_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 & ! [H9: heap_ext @ product_unit,As9: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As @ H2 @ H9 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
% 0.25/0.55 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.elims(2)
% 0.25/0.55 thf(fact_5705_wand__raw_Opelims_I3_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 & ! [H6: heap_ext @ product_unit,As7: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As @ H2 @ H6 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) ) )
% 0.25/0.55 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.pelims(3)
% 0.25/0.55 thf(fact_5706_wand__raw_Opelims_I2_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( wand_raw @ X @ Xa @ Xb )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 0.25/0.55 => ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 & ! [H9: heap_ext @ product_unit,As9: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As @ H2 @ H9 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
% 0.25/0.55 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.pelims(2)
% 0.25/0.55 thf(fact_5707_wand__raw_Opelims_I1_J,axiom,
% 0.25/0.55 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( wand_raw @ X @ Xa @ Xb )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( Xb
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 & ! [H7: heap_ext @ product_unit,As8: set @ nat] :
% 0.25/0.55 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) )
% 0.25/0.55 & ( relH @ As @ H2 @ H7 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
% 0.25/0.55 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
% 0.25/0.55 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wand_raw.pelims(1)
% 0.25/0.55 thf(fact_5708_in__range_Opelims_I3_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( in_range @ X )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ! [X2: nat] :
% 0.25/0.55 ( ( member @ nat @ X2 @ As )
% 0.25/0.55 => ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.pelims(3)
% 0.25/0.55 thf(fact_5709_in__range_Opelims_I2_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( in_range @ X )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ~ ! [X7: nat] :
% 0.25/0.55 ( ( member @ nat @ X7 @ As )
% 0.25/0.55 => ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.pelims(2)
% 0.25/0.55 thf(fact_5710_in__range_Opelims_I1_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( in_range @ X )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ! [X3: nat] :
% 0.25/0.55 ( ( member @ nat @ X3 @ As )
% 0.25/0.55 => ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H2 ) ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % in_range.pelims(1)
% 0.25/0.55 thf(fact_5711_Ref__Time_Oalloc__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( ( ref_alloc @ A )
% 0.25/0.55 = ( ^ [X3: A,H: heap_ext @ product_unit] :
% 0.25/0.55 ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( ref2 @ A @ ( lim @ product_unit @ H ) )
% 0.25/0.55 @ ( ref_set @ A @ ( ref2 @ A @ ( lim @ product_unit @ H ) ) @ X3
% 0.25/0.55 @ ( lim_update @ product_unit
% 0.25/0.55 @ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
% 0.25/0.55 @ H ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Ref_Time.alloc_def
% 0.25/0.55 thf(fact_5712_pure__assn__raw_Osimps,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,B2: $o,H3: A,As2: set @ B] :
% 0.25/0.55 ( ( pure_assn_raw @ A @ B @ B2 @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As2 ) )
% 0.25/0.55 = ( ( As2
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & B2 ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.simps
% 0.25/0.55 thf(fact_5713_pure__assn__raw_Oelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
% 0.25/0.55 ( ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: A,As: set @ B] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( ~ ( ( As
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & X ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.elims(1)
% 0.25/0.55 thf(fact_5714_pure__assn__raw_Oelims_I3_J,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 0.25/0.55 ( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
% 0.25/0.55 => ~ ! [H2: A,As: set @ B] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 0.25/0.55 => ( ( As
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & X ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.elims(3)
% 0.25/0.55 thf(fact_5715_pure__assn__raw_Oelims_I2_J,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 0.25/0.55 ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 0.25/0.55 => ~ ! [H2: A,As: set @ B] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 0.25/0.55 => ~ ( ( As
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & X ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.elims(2)
% 0.25/0.55 thf(fact_5716_pure__assn__raw_Opelims_I3_J,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 0.25/0.55 ( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
% 0.25/0.55 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
% 0.25/0.55 => ~ ! [H2: A,As: set @ B] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) )
% 0.25/0.55 => ( ( As
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & X ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.pelims(3)
% 0.25/0.55 thf(fact_5717_pure__assn__raw_Opelims_I2_J,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 0.25/0.55 ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 0.25/0.55 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
% 0.25/0.55 => ~ ! [H2: A,As: set @ B] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) )
% 0.25/0.55 => ~ ( ( As
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & X ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.pelims(2)
% 0.25/0.55 thf(fact_5718_pure__assn__raw_Opelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
% 0.25/0.55 ( ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
% 0.25/0.55 => ~ ! [H2: A,As: set @ B] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ( As
% 0.25/0.55 = ( bot_bot @ ( set @ B ) ) )
% 0.25/0.55 & X ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % pure_assn_raw.pelims(1)
% 0.25/0.55 thf(fact_5719_effect__refI,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: ref @ A,H4: heap_ext @ product_unit,V: A,H3: heap_ext @ product_unit,N: nat] :
% 0.25/0.55 ( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
% 0.25/0.55 = ( ref_alloc @ A @ V @ H3 ) )
% 0.25/0.55 => ( ( N
% 0.25/0.55 = ( one_one @ nat ) )
% 0.25/0.55 => ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 @ H4 @ R @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % effect_refI
% 0.25/0.55 thf(fact_5720_ref__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( ( ref_ref @ A )
% 0.25/0.55 = ( ^ [V2: A] :
% 0.25/0.55 ( heap_Time_heap @ ( ref @ A )
% 0.25/0.55 @ ^ [H: heap_ext @ product_unit] :
% 0.25/0.55 ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.55 @ ^ [R3: ref @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( one_one @ nat ) ) )
% 0.25/0.55 @ ( ref_alloc @ A @ V2 @ H ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % ref_def
% 0.25/0.55 thf(fact_5721_execute__ref,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [V: A,H3: heap_ext @ product_unit] :
% 0.25/0.55 ( ( heap_Time_execute @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 )
% 0.25/0.55 = ( some @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.55 @ ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 0.25/0.55 @ ^ [R3: ref @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( one_one @ nat ) ) )
% 0.25/0.55 @ ( ref_alloc @ A @ V @ H3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % execute_ref
% 0.25/0.55 thf(fact_5722_effect__refE,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [V: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: ref @ A,N: nat] :
% 0.25/0.55 ( ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 @ H4 @ R @ N )
% 0.25/0.55 => ~ ( ( ( ref_get @ A @ H4 @ R )
% 0.25/0.55 = V )
% 0.25/0.55 => ( ( ref_present @ A @ H4 @ R )
% 0.25/0.55 => ( ~ ( ref_present @ A @ H3 @ R )
% 0.25/0.55 => ( N
% 0.25/0.55 != ( one_one @ nat ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % effect_refE
% 0.25/0.55 thf(fact_5723_next__fresh,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: ref @ A,H4: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit] :
% 0.25/0.55 ( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
% 0.25/0.55 = ( ref_alloc @ A @ X @ H3 ) )
% 0.25/0.55 => ~ ( ref_present @ A @ H3 @ R ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % next_fresh
% 0.25/0.55 thf(fact_5724_next__present,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ! [R: ref @ A,H4: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit] :
% 0.25/0.55 ( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
% 0.25/0.55 = ( ref_alloc @ A @ X @ H3 ) )
% 0.25/0.55 => ( ref_present @ A @ H4 @ R ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % next_present
% 0.25/0.55 thf(fact_5725_one__assn__raw_Oelims_I3_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( one_assn_raw @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( As
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.elims(3)
% 0.25/0.55 thf(fact_5726_one__assn__raw_Oelims_I2_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( one_assn_raw @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( As
% 0.25/0.55 != ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.elims(2)
% 0.25/0.55 thf(fact_5727_one__assn__raw_Oelims_I1_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( one_assn_raw @ X )
% 0.25/0.55 = Y )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 = ( As
% 0.25/0.55 != ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.elims(1)
% 0.25/0.55 thf(fact_5728_one__assn__raw_Osimps,axiom,
% 0.25/0.55 ! [H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( As2
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.simps
% 0.25/0.55 thf(fact_5729_one__assn__raw_Opelims_I3_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ~ ( one_assn_raw @ X )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( As
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.pelims(3)
% 0.25/0.55 thf(fact_5730_one__assn__raw_Opelims_I2_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 0.25/0.55 ( ( one_assn_raw @ X )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( As
% 0.25/0.55 != ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.pelims(2)
% 0.25/0.55 thf(fact_5731_one__assn__raw_Opelims_I1_J,axiom,
% 0.25/0.55 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 0.25/0.55 ( ( ( one_assn_raw @ X )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
% 0.25/0.55 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( As
% 0.25/0.55 = ( bot_bot @ ( set @ nat ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % one_assn_raw.pelims(1)
% 0.25/0.55 thf(fact_5732_proper__iff,axiom,
% 0.25/0.55 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 0.25/0.55 ( ( proper @ P )
% 0.25/0.55 => ( ( relH @ As2 @ H3 @ H4 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
% 0.25/0.55 => ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 = ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % proper_iff
% 0.25/0.55 thf(fact_5733_proper__def,axiom,
% 0.25/0.55 ( proper
% 0.25/0.55 = ( ^ [P4: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
% 0.25/0.55 ! [H: heap_ext @ product_unit,H7: heap_ext @ product_unit,As5: set @ nat] :
% 0.25/0.55 ( ( ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 0.25/0.55 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) ) )
% 0.25/0.55 & ( ( ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 0.25/0.55 & ( relH @ As5 @ H @ H7 )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) ) )
% 0.25/0.55 => ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % proper_def
% 0.25/0.55 thf(fact_5734_properD2,axiom,
% 0.25/0.55 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat,H4: heap_ext @ product_unit] :
% 0.25/0.55 ( ( proper @ P )
% 0.25/0.55 => ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( ( relH @ As2 @ H3 @ H4 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
% 0.25/0.55 => ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % properD2
% 0.25/0.55 thf(fact_5735_properD1,axiom,
% 0.25/0.55 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
% 0.25/0.55 ( ( proper @ P )
% 0.25/0.55 => ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 0.25/0.55 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % properD1
% 0.25/0.55 thf(fact_5736_properI,axiom,
% 0.25/0.55 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
% 0.25/0.55 ( ! [As: set @ nat,H2: heap_ext @ product_unit] :
% 0.25/0.55 ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) )
% 0.25/0.55 => ( ! [As: set @ nat,H2: heap_ext @ product_unit,H6: heap_ext @ product_unit] :
% 0.25/0.55 ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 0.25/0.55 => ( ( relH @ As @ H2 @ H6 )
% 0.25/0.55 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
% 0.25/0.55 => ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) ) ) ) )
% 0.25/0.55 => ( proper @ P ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % properI
% 0.25/0.55 thf(fact_5737_Array__Time_Onoteq__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( heap @ A )
% 0.25/0.55 & ( heap @ B ) )
% 0.25/0.55 => ( ( array_noteq @ A @ B )
% 0.25/0.55 = ( ^ [R3: array @ A,S4: array @ B] :
% 0.25/0.55 ( ( ( typerep_typerep @ A @ ( type2 @ A ) )
% 0.25/0.55 != ( typerep_typerep @ B @ ( type2 @ B ) ) )
% 0.25/0.55 | ( ( addr_of_array @ A @ R3 )
% 0.25/0.55 != ( addr_of_array @ B @ S4 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Array_Time.noteq_def
% 0.25/0.55 thf(fact_5738_comm__monoid__mset_Ounion,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.55 ( ( comm_monoid_mset @ A @ F @ Z2 )
% 0.25/0.55 => ( ( comm_monoid_F @ A @ F @ Z2 @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.55 = ( F @ ( comm_monoid_F @ A @ F @ Z2 @ M3 ) @ ( comm_monoid_F @ A @ F @ Z2 @ N6 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_mset.union
% 0.25/0.55 thf(fact_5739_prod__list_Omonoid__list__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( monoid_mult @ A )
% 0.25/0.55 => ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % prod_list.monoid_list_axioms
% 0.25/0.55 thf(fact_5740_sum__mset_Ocomm__monoid__mset__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comm_monoid_add @ A )
% 0.25/0.55 => ( comm_monoid_mset @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum_mset.comm_monoid_mset_axioms
% 0.25/0.55 thf(fact_5741_prod__mset_Ocomm__monoid__mset__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comm_monoid_mult @ A )
% 0.25/0.55 => ( comm_monoid_mset @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % prod_mset.comm_monoid_mset_axioms
% 0.25/0.55 thf(fact_5742_sum__list_Omonoid__list__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( monoid_add @ A )
% 0.25/0.55 => ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum_list.monoid_list_axioms
% 0.25/0.55 thf(fact_5743_Array__Time_Oset__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( ( array_set @ A )
% 0.25/0.55 = ( ^ [A7: array @ A,X3: list @ A] :
% 0.25/0.55 ( arrays_update @ product_unit
% 0.25/0.55 @ ^ [H: typerep > nat > ( list @ nat )] : ( fun_upd @ typerep @ ( nat > ( list @ nat ) ) @ H @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( fun_upd @ nat @ ( list @ nat ) @ ( H @ ( typerep_typerep @ A @ ( type2 @ A ) ) ) @ ( addr_of_array @ A @ A7 ) @ ( map @ A @ nat @ ( to_nat @ A ) @ X3 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Array_Time.set_def
% 0.25/0.55 thf(fact_5744_Array__Time_Oget__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( ( array_get @ A )
% 0.25/0.55 = ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A7 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Array_Time.get_def
% 0.25/0.55 thf(fact_5745_Int_Osub__code_I8_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 0.25/0.55 = ( plus_plus @ int @ ( dup @ ( sub @ M2 @ N ) ) @ ( one_one @ int ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Int.sub_code(8)
% 0.25/0.55 thf(fact_5746_Int_Odup__def,axiom,
% 0.25/0.55 ( dup
% 0.25/0.55 = ( ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Int.dup_def
% 0.25/0.55 thf(fact_5747_relH__def,axiom,
% 0.25/0.55 ( relH
% 0.25/0.55 = ( ^ [As5: set @ nat,H: heap_ext @ product_unit,H7: heap_ext @ product_unit] :
% 0.25/0.55 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 0.25/0.55 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) )
% 0.25/0.55 & ! [T3: typerep,X3: nat] :
% 0.25/0.55 ( ( member @ nat @ X3 @ As5 )
% 0.25/0.55 => ( ( ( refs @ product_unit @ H @ T3 @ X3 )
% 0.25/0.55 = ( refs @ product_unit @ H7 @ T3 @ X3 ) )
% 0.25/0.55 & ( ( arrays @ product_unit @ H @ T3 @ X3 )
% 0.25/0.55 = ( arrays @ product_unit @ H7 @ T3 @ X3 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % relH_def
% 0.25/0.55 thf(fact_5748_case__sum__o__map__sum__id,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,C: $tType,G: B > A,F: C > A,X: sum_sum @ C @ B] :
% 0.25/0.55 ( ( comp @ ( sum_sum @ A @ B ) @ A @ ( sum_sum @ C @ B ) @ ( sum_case_sum @ A @ A @ B @ ( id @ A ) @ G ) @ ( sum_map_sum @ C @ A @ B @ B @ F @ ( id @ B ) ) @ X )
% 0.25/0.55 = ( sum_case_sum @ C @ A @ B @ ( comp @ C @ A @ C @ F @ ( id @ C ) ) @ G @ X ) ) ).
% 0.25/0.55
% 0.25/0.55 % case_sum_o_map_sum_id
% 0.25/0.55 thf(fact_5749_comm__monoid_Oaxioms_I2_J,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.55 ( ( comm_monoid @ A @ F @ Z2 )
% 0.25/0.55 => ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid.axioms(2)
% 0.25/0.55 thf(fact_5750_sum_Omap__comp,axiom,
% 0.25/0.55 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F5,F1: A > C,F22: B > D,V: sum_sum @ A @ B] :
% 0.25/0.55 ( ( sum_map_sum @ C @ E @ D @ F5 @ G1 @ G22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 0.25/0.55 = ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F5 @ B @ G22 @ F22 ) @ V ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum.map_comp
% 0.25/0.55 thf(fact_5751_comm__monoid__axioms_Ointro,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.55 ( ! [A5: A] :
% 0.25/0.55 ( ( F @ A5 @ Z2 )
% 0.25/0.55 = A5 )
% 0.25/0.55 => ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_axioms.intro
% 0.25/0.55 thf(fact_5752_comm__monoid__axioms__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comm_monoid_axioms @ A )
% 0.25/0.55 = ( ^ [F2: A > A > A,Z5: A] :
% 0.25/0.55 ! [A7: A] :
% 0.25/0.55 ( ( F2 @ A7 @ Z5 )
% 0.25/0.55 = A7 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_axioms_def
% 0.25/0.55 thf(fact_5753_case__sum__map__sum,axiom,
% 0.25/0.55 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: B > A,R: C > A,F: D > B,G: E > C,X: sum_sum @ D @ E] :
% 0.25/0.55 ( ( sum_case_sum @ B @ A @ C @ L @ R @ ( sum_map_sum @ D @ B @ E @ C @ F @ G @ X ) )
% 0.25/0.55 = ( sum_case_sum @ D @ A @ E @ ( comp @ B @ A @ D @ L @ F ) @ ( comp @ C @ A @ E @ R @ G ) @ X ) ) ).
% 0.25/0.55
% 0.25/0.55 % case_sum_map_sum
% 0.25/0.55 thf(fact_5754_case__sum__o__map__sum,axiom,
% 0.25/0.55 ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: D > C,G: E > C,H1: A > D,H22: B > E] :
% 0.25/0.55 ( ( comp @ ( sum_sum @ D @ E ) @ C @ ( sum_sum @ A @ B ) @ ( sum_case_sum @ D @ C @ E @ F @ G ) @ ( sum_map_sum @ A @ D @ B @ E @ H1 @ H22 ) )
% 0.25/0.55 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ F @ H1 ) @ ( comp @ E @ C @ B @ G @ H22 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % case_sum_o_map_sum
% 0.25/0.55 thf(fact_5755_map__sum_Ocomp,axiom,
% 0.25/0.55 ! [A: $tType,C: $tType,E: $tType,F5: $tType,D: $tType,B: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D] :
% 0.25/0.55 ( ( comp @ ( sum_sum @ C @ D ) @ ( sum_sum @ E @ F5 ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ C @ E @ D @ F5 @ F @ G ) @ ( sum_map_sum @ A @ C @ B @ D @ H3 @ I ) )
% 0.25/0.55 = ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % map_sum.comp
% 0.25/0.55 thf(fact_5756_map__sum_Ocompositionality,axiom,
% 0.25/0.55 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D,Sum: sum_sum @ A @ B] :
% 0.25/0.55 ( ( sum_map_sum @ C @ E @ D @ F5 @ F @ G @ ( sum_map_sum @ A @ C @ B @ D @ H3 @ I @ Sum ) )
% 0.25/0.55 = ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Sum ) ) ).
% 0.25/0.55
% 0.25/0.55 % map_sum.compositionality
% 0.25/0.55 thf(fact_5757_comm__monoid__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comm_monoid @ A )
% 0.25/0.55 = ( ^ [F2: A > A > A,Z5: A] :
% 0.25/0.55 ( ( abel_semigroup @ A @ F2 )
% 0.25/0.55 & ( comm_monoid_axioms @ A @ F2 @ Z5 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_def
% 0.25/0.55 thf(fact_5758_mult_Oabel__semigroup__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ab_semigroup_mult @ A )
% 0.25/0.55 => ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mult.abel_semigroup_axioms
% 0.25/0.55 thf(fact_5759_comm__monoid_Oaxioms_I1_J,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.55 ( ( comm_monoid @ A @ F @ Z2 )
% 0.25/0.55 => ( abel_semigroup @ A @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid.axioms(1)
% 0.25/0.55 thf(fact_5760_abel__semigroup_Oleft__commute,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,B2: A,A3: A,C2: A] :
% 0.25/0.55 ( ( abel_semigroup @ A @ F )
% 0.25/0.55 => ( ( F @ B2 @ ( F @ A3 @ C2 ) )
% 0.25/0.55 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup.left_commute
% 0.25/0.55 thf(fact_5761_abel__semigroup_Ocommute,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 0.25/0.55 ( ( abel_semigroup @ A @ F )
% 0.25/0.55 => ( ( F @ A3 @ B2 )
% 0.25/0.55 = ( F @ B2 @ A3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup.commute
% 0.25/0.55 thf(fact_5762_add_Oabel__semigroup__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ab_semigroup_add @ A )
% 0.25/0.55 => ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % add.abel_semigroup_axioms
% 0.25/0.55 thf(fact_5763_abel__semigroup_Oaxioms_I1_J,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A] :
% 0.25/0.55 ( ( abel_semigroup @ A @ F )
% 0.25/0.55 => ( semigroup @ A @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup.axioms(1)
% 0.25/0.55 thf(fact_5764_comm__monoid_Ointro,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A] :
% 0.25/0.55 ( ( abel_semigroup @ A @ F )
% 0.25/0.55 => ( ( comm_monoid_axioms @ A @ F @ Z2 )
% 0.25/0.55 => ( comm_monoid @ A @ F @ Z2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid.intro
% 0.25/0.55 thf(fact_5765_abel__semigroup_Ointro,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A] :
% 0.25/0.55 ( ( semigroup @ A @ F )
% 0.25/0.55 => ( ( abel_s757365448890700780axioms @ A @ F )
% 0.25/0.55 => ( abel_semigroup @ A @ F ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup.intro
% 0.25/0.55 thf(fact_5766_abel__semigroup__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( abel_semigroup @ A )
% 0.25/0.55 = ( ^ [F2: A > A > A] :
% 0.25/0.55 ( ( semigroup @ A @ F2 )
% 0.25/0.55 & ( abel_s757365448890700780axioms @ A @ F2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup_def
% 0.25/0.55 thf(fact_5767_curr__def,axiom,
% 0.25/0.55 ! [B: $tType,C: $tType,A: $tType] :
% 0.25/0.55 ( ( bNF_Wellorder_curr @ A @ B @ C )
% 0.25/0.55 = ( ^ [A9: set @ A,F2: ( product_prod @ A @ B ) > C,A7: A] :
% 0.25/0.55 ( if @ ( B > C ) @ ( member @ A @ A7 @ A9 )
% 0.25/0.55 @ ^ [B5: B] : ( F2 @ ( product_Pair @ A @ B @ A7 @ B5 ) )
% 0.25/0.55 @ ( undefined @ ( B > C ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % curr_def
% 0.25/0.55 thf(fact_5768_abel__semigroup__axioms_Ointro,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A] :
% 0.25/0.55 ( ! [A5: A,B3: A] :
% 0.25/0.55 ( ( F @ A5 @ B3 )
% 0.25/0.55 = ( F @ B3 @ A5 ) )
% 0.25/0.55 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup_axioms.intro
% 0.25/0.55 thf(fact_5769_abel__semigroup__axioms__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( abel_s757365448890700780axioms @ A )
% 0.25/0.55 = ( ^ [F2: A > A > A] :
% 0.25/0.55 ! [A7: A,B5: A] :
% 0.25/0.55 ( ( F2 @ A7 @ B5 )
% 0.25/0.55 = ( F2 @ B5 @ A7 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup_axioms_def
% 0.25/0.55 thf(fact_5770_abel__semigroup_Oaxioms_I2_J,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A] :
% 0.25/0.55 ( ( abel_semigroup @ A @ F )
% 0.25/0.55 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % abel_semigroup.axioms(2)
% 0.25/0.55 thf(fact_5771_comm__monoid__set_Ozero__middle,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,P5: nat,K: nat,G: nat > A,H3: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ K @ P5 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ Z2 @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ P5 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.zero_middle
% 0.25/0.55 thf(fact_5772_iterate__add__diff__add__eq2,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.55 => ( ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( minus_minus @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_diff_add_eq2
% 0.25/0.55 thf(fact_5773_iterate__add__add__eq1,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ! [J: nat,I: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ J @ I )
% 0.25/0.55 => ( ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_add_eq1
% 0.25/0.55 thf(fact_5774_iterate__add__simps_I2_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ! [N: nat,A3: A] :
% 0.25/0.55 ( ( iterate_add @ A @ ( suc @ N ) @ A3 )
% 0.25/0.55 = ( plus_plus @ A @ A3 @ ( iterate_add @ A @ N @ A3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_simps(2)
% 0.25/0.55 thf(fact_5775_iterate__add__distrib,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ! [M2: nat,N: nat,A3: A] :
% 0.25/0.55 ( ( iterate_add @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 )
% 0.25/0.55 = ( plus_plus @ A @ ( iterate_add @ A @ M2 @ A3 ) @ ( iterate_add @ A @ N @ A3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_distrib
% 0.25/0.55 thf(fact_5776_iterate__add__1,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 & ( semiring_1 @ A ) )
% 0.25/0.55 => ! [N: nat] :
% 0.25/0.55 ( ( iterate_add @ A @ N @ ( one_one @ A ) )
% 0.25/0.55 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_1
% 0.25/0.55 thf(fact_5777_prod_Ocomm__monoid__set__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comm_monoid_mult @ A )
% 0.25/0.55 => ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % prod.comm_monoid_set_axioms
% 0.25/0.55 thf(fact_5778_comm__monoid__set_Onat__group,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,K: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [M: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.nat_group
% 0.25/0.55 thf(fact_5779_comm__monoid__set_Oub__add__nat,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A,P5: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.ub_add_nat
% 0.25/0.55 thf(fact_5780_comm__monoid__set_Oivl__cong,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType] :
% 0.25/0.55 ( ( ord @ B )
% 0.25/0.55 => ! [F: A > A > A,Z2: A,A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( A3 = C2 )
% 0.25/0.55 => ( ( B2 = D3 )
% 0.25/0.55 => ( ! [X2: B] :
% 0.25/0.55 ( ( ord_less_eq @ B @ C2 @ X2 )
% 0.25/0.55 => ( ( ord_less @ B @ X2 @ D3 )
% 0.25/0.55 => ( ( G @ X2 )
% 0.25/0.55 = ( H3 @ X2 ) ) ) )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.ivl_cong
% 0.25/0.55 thf(fact_5781_comm__monoid__set_Ohead,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.55 = ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.head
% 0.25/0.55 thf(fact_5782_comm__monoid__set_Olast__plus,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.55 = ( F @ ( G @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.last_plus
% 0.25/0.55 thf(fact_5783_comm__monoid__set_OatLeast__Suc__atMost,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.55 = ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_Suc_atMost
% 0.25/0.55 thf(fact_5784_comm__monoid__set_Onat__ivl__Suc_H,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( G @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.nat_ivl_Suc'
% 0.25/0.55 thf(fact_5785_comm__monoid__set_OatLeastLessThan__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,A3: nat,B2: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ A3 @ B2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_Suc
% 0.25/0.55 thf(fact_5786_comm__monoid__set_OatLeastLessThan__concat,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,P5: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ N @ P5 )
% 0.25/0.55 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_concat
% 0.25/0.55 thf(fact_5787_comm__monoid__set_OSuc__reindex__ivl,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.55 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( G @ M2 )
% 0.25/0.55 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.Suc_reindex_ivl
% 0.25/0.55 thf(fact_5788_comm__monoid__set_OatLeast1__atMost__eq,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast1_atMost_eq
% 0.25/0.55 thf(fact_5789_comm__monoid__set_OatLeastLessThan__rev,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
% 0.25/0.55 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_rev
% 0.25/0.55 thf(fact_5790_comm__monoid__set_Onested__swap,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,A3: nat > nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [J2: nat] :
% 0.25/0.55 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 0.25/0.55 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.nested_swap
% 0.25/0.55 thf(fact_5791_comm__monoid__set_Onat__diff__reindex,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.nat_diff_reindex
% 0.25/0.55 thf(fact_5792_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.shift_bounds_cl_Suc_ivl
% 0.25/0.55 thf(fact_5793_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.55 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.shift_bounds_Suc_ivl
% 0.25/0.55 thf(fact_5794_comm__monoid__set_OlessThan__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.lessThan_Suc
% 0.25/0.55 thf(fact_5795_comm__monoid__set_OatMost__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atMost_Suc
% 0.25/0.55 thf(fact_5796_comm__monoid__set_Ohead__if,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ( ord_less @ nat @ N @ M2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.55 = Z2 ) )
% 0.25/0.55 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.head_if
% 0.25/0.55 thf(fact_5797_comm__monoid__set_Ocl__ivl__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.55 = Z2 ) )
% 0.25/0.55 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.cl_ivl_Suc
% 0.25/0.55 thf(fact_5798_comm__monoid__set_Oop__ivl__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ( ord_less @ nat @ N @ M2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.55 = Z2 ) )
% 0.25/0.55 & ( ~ ( ord_less @ nat @ N @ M2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.op_ivl_Suc
% 0.25/0.55 thf(fact_5799_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less @ nat @ M2 @ N )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.55 = ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_Suc_lessThan
% 0.25/0.55 thf(fact_5800_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast0_lessThan_Suc
% 0.25/0.55 thf(fact_5801_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast0_atMost_Suc
% 0.25/0.55 thf(fact_5802_comm__monoid__set_OatMost__Suc__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( G @ ( zero_zero @ nat ) )
% 0.25/0.55 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atMost_Suc_shift
% 0.25/0.55 thf(fact_5803_comm__monoid__set_OatLeastAtMost__rev,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastAtMost_rev
% 0.25/0.55 thf(fact_5804_comm__monoid__set_OlessThan__Suc__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( G @ ( zero_zero @ nat ) )
% 0.25/0.55 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.lessThan_Suc_shift
% 0.25/0.55 thf(fact_5805_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.shift_bounds_cl_nat_ivl
% 0.25/0.55 thf(fact_5806_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 0.25/0.55 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.shift_bounds_nat_ivl
% 0.25/0.55 thf(fact_5807_sum_Ocomm__monoid__set__axioms,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comm_monoid_add @ A )
% 0.25/0.55 => ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum.comm_monoid_set_axioms
% 0.25/0.55 thf(fact_5808_comm__monoid__set_OatMost__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ N ) )
% 0.25/0.55 = ( F @ ( G @ ( zero_zero @ nat ) )
% 0.25/0.55 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atMost_shift
% 0.25/0.55 thf(fact_5809_comm__monoid__set_Onested__swap_H,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,A3: nat > nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [J2: nat] :
% 0.25/0.55 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.nested_swap'
% 0.25/0.55 thf(fact_5810_comm__monoid__set_Oreindex__nontrivial,axiom,
% 0.25/0.55 ! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z2: A,A4: set @ B,H3: B > C,G: C > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( finite_finite2 @ B @ A4 )
% 0.25/0.55 => ( ! [X2: B,Y4: B] :
% 0.25/0.55 ( ( member @ B @ X2 @ A4 )
% 0.25/0.55 => ( ( member @ B @ Y4 @ A4 )
% 0.25/0.55 => ( ( X2 != Y4 )
% 0.25/0.55 => ( ( ( H3 @ X2 )
% 0.25/0.55 = ( H3 @ Y4 ) )
% 0.25/0.55 => ( ( G @ ( H3 @ X2 ) )
% 0.25/0.55 = Z2 ) ) ) ) )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.reindex_nontrivial
% 0.25/0.55 thf(fact_5811_comm__monoid__set_Oreindex,axiom,
% 0.25/0.55 ! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z2: A,H3: B > C,A4: set @ B,G: C > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( inj_on @ B @ C @ H3 @ A4 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.reindex
% 0.25/0.55 thf(fact_5812_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
% 0.25/0.55 thf(fact_5813_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_Suc_atMost_Suc_shift
% 0.25/0.55 thf(fact_5814_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastAtMost_shift_bounds
% 0.25/0.55 thf(fact_5815_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_shift_bounds
% 0.25/0.55 thf(fact_5816_comm__monoid__set_OatLeastLessThan__reindex,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ord @ B )
% 0.25/0.55 => ! [F: A > A > A,Z2: A,H3: nat > B,M2: nat,N: nat,G: B > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_reindex
% 0.25/0.55 thf(fact_5817_comm__monoid__set_OatLeastAtMost__reindex,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ord @ B )
% 0.25/0.55 => ! [F: A > A > A,Z2: A,H3: nat > B,M2: nat,N: nat,G: B > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastAtMost_reindex
% 0.25/0.55 thf(fact_5818_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast0_lessThan_Suc_shift
% 0.25/0.55 thf(fact_5819_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast0_atMost_Suc_shift
% 0.25/0.55 thf(fact_5820_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_shift_0
% 0.25/0.55 thf(fact_5821_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
% 0.25/0.55 thf(fact_5822_iterate__add__eq__add__iff1,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.55 => ( ( ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ M2 )
% 0.25/0.55 = ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ N ) )
% 0.25/0.55 = ( ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ M2 )
% 0.25/0.55 = N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_eq_add_iff1
% 0.25/0.55 thf(fact_5823_iterate__add__eq__add__iff2,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.55 => ( ( ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 )
% 0.25/0.55 = ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( M2
% 0.25/0.55 = ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_eq_add_iff2
% 0.25/0.55 thf(fact_5824_iterate__add__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 => ( ( iterate_add @ A )
% 0.25/0.55 = ( ^ [N5: nat,A7: A] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_def
% 0.25/0.55 thf(fact_5825_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ( comp @ nat @ A @ nat @ G
% 0.25/0.55 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_atMost_pred_shift
% 0.25/0.55 thf(fact_5826_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ( comp @ nat @ A @ nat @ G
% 0.25/0.55 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 0.25/0.55 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_lessThan_pred_shift
% 0.25/0.55 thf(fact_5827_comm__monoid__add__class_Osum__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( comm_monoid_add @ A )
% 0.25/0.55 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_add_class.sum_def
% 0.25/0.55 thf(fact_5828_comm__monoid__mult__class_Oprod__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( comm_monoid_mult @ A )
% 0.25/0.55 => ( ( groups7121269368397514597t_prod @ B @ A )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_mult_class.prod_def
% 0.25/0.55 thf(fact_5829_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( ord_less_eq @ nat @ M2 @ N )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeastAtMost_shift_0
% 0.25/0.55 thf(fact_5830_comm__monoid__set_Oin__pairs,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 0.25/0.55 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.in_pairs
% 0.25/0.55 thf(fact_5831_comm__monoid__set_Oin__pairs__0,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.in_pairs_0
% 0.25/0.55 thf(fact_5832_iterate__add__less__eq__iff2,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 & ( ordere2412721322843649153imp_le @ A ) )
% 0.25/0.55 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.55 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( ord_less_eq @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_less_eq_iff2
% 0.25/0.55 thf(fact_5833_iterate__add__less__eq__iff1,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 & ( ordere2412721322843649153imp_le @ A ) )
% 0.25/0.55 => ! [J: nat,I: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ J @ I )
% 0.25/0.55 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_less_eq_iff1
% 0.25/0.55 thf(fact_5834_iterate__add__less__iff1,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 & ( ordere2412721322843649153imp_le @ A ) )
% 0.25/0.55 => ! [J: nat,I: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ J @ I )
% 0.25/0.55 => ( ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_less_iff1
% 0.25/0.55 thf(fact_5835_iterate__add__less__iff2,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( ( cancel1802427076303600483id_add @ A )
% 0.25/0.55 & ( ordere2412721322843649153imp_le @ A ) )
% 0.25/0.55 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 0.25/0.55 ( ( ord_less_eq @ nat @ I @ J )
% 0.25/0.55 => ( ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 0.25/0.55 = ( ord_less @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % iterate_add_less_iff2
% 0.25/0.55 thf(fact_5836_comm__monoid__set_Otriangle__reindex,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z2 @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.25/0.55 @ ( collect @ ( product_prod @ nat @ nat )
% 0.25/0.55 @ ( product_case_prod @ nat @ nat @ $o
% 0.25/0.55 @ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [K4: nat] :
% 0.25/0.55 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ K4 ) )
% 0.25/0.55 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.triangle_reindex
% 0.25/0.55 thf(fact_5837_comm__monoid__set_Otriangle__reindex__eq,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z2 @ ( product_case_prod @ nat @ nat @ A @ G )
% 0.25/0.55 @ ( collect @ ( product_prod @ nat @ nat )
% 0.25/0.55 @ ( product_case_prod @ nat @ nat @ $o
% 0.25/0.55 @ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [K4: nat] :
% 0.25/0.55 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 0.25/0.55 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ K4 ) )
% 0.25/0.55 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.triangle_reindex_eq
% 0.25/0.55 thf(fact_5838_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: int > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_int_lessThan_int_shift
% 0.25/0.55 thf(fact_5839_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
% 0.25/0.55 ! [A: $tType,F: A > A > A,Z2: A,G: int > A,M2: nat,N: nat] :
% 0.25/0.55 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 0.25/0.55 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 0.25/0.55 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comm_monoid_set.atLeast_int_atMost_int_shift
% 0.25/0.55 thf(fact_5840_old_Ounit_Ocase,axiom,
% 0.25/0.55 ! [A: $tType,F: A] :
% 0.25/0.55 ( ( product_case_unit @ A @ F @ product_Unity )
% 0.25/0.55 = F ) ).
% 0.25/0.55
% 0.25/0.55 % old.unit.case
% 0.25/0.55 thf(fact_5841_wfP__wf__eq,axiom,
% 0.25/0.55 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ( wfP @ A
% 0.25/0.55 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.55 = ( wf @ A @ R ) ) ).
% 0.25/0.55
% 0.25/0.55 % wfP_wf_eq
% 0.25/0.55 thf(fact_5842_subset__mset_Omin__add__distrib__left,axiom,
% 0.25/0.55 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 0.25/0.55 ( ( plus_plus @ ( multiset @ A ) @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X @ Y ) @ Z2 )
% 0.25/0.55 = ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % subset_mset.min_add_distrib_left
% 0.25/0.55 thf(fact_5843_subset__mset_Omin__arg__le_I1_J,axiom,
% 0.25/0.55 ! [A: $tType,N: multiset @ A,M2: multiset @ A] :
% 0.25/0.55 ( ( subseteq_mset @ A @ N @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N ) )
% 0.25/0.55 = ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N )
% 0.25/0.55 = N ) ) ).
% 0.25/0.55
% 0.25/0.55 % subset_mset.min_arg_le(1)
% 0.25/0.55 thf(fact_5844_subset__mset_Omin__arg__le_I2_J,axiom,
% 0.25/0.55 ! [A: $tType,M2: multiset @ A,N: multiset @ A] :
% 0.25/0.55 ( ( subseteq_mset @ A @ M2 @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N ) )
% 0.25/0.55 = ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N )
% 0.25/0.55 = M2 ) ) ).
% 0.25/0.55
% 0.25/0.55 % subset_mset.min_arg_le(2)
% 0.25/0.55 thf(fact_5845_wfP__accp__iff,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( wfP @ A )
% 0.25/0.55 = ( ^ [R3: A > A > $o] :
% 0.25/0.55 ! [X8: A] : ( accp @ A @ R3 @ X8 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % wfP_accp_iff
% 0.25/0.55 thf(fact_5846_accp__wfPI,axiom,
% 0.25/0.55 ! [A: $tType,R: A > A > $o] :
% 0.25/0.55 ( ! [X_12: A] : ( accp @ A @ R @ X_12 )
% 0.25/0.55 => ( wfP @ A @ R ) ) ).
% 0.25/0.55
% 0.25/0.55 % accp_wfPI
% 0.25/0.55 thf(fact_5847_accp__wfPD,axiom,
% 0.25/0.55 ! [A: $tType,R: A > A > $o,X: A] :
% 0.25/0.55 ( ( wfP @ A @ R )
% 0.25/0.55 => ( accp @ A @ R @ X ) ) ).
% 0.25/0.55
% 0.25/0.55 % accp_wfPD
% 0.25/0.55 thf(fact_5848_unit_Ocase__distrib,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,H3: A > B,F: A,Unit: product_unit] :
% 0.25/0.55 ( ( H3 @ ( product_case_unit @ A @ F @ Unit ) )
% 0.25/0.55 = ( product_case_unit @ B @ ( H3 @ F ) @ Unit ) ) ).
% 0.25/0.55
% 0.25/0.55 % unit.case_distrib
% 0.25/0.55 thf(fact_5849_subset__mset_Omin__add__distrib__right,axiom,
% 0.25/0.55 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 0.25/0.55 ( ( plus_plus @ ( multiset @ A ) @ X @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
% 0.25/0.55 = ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % subset_mset.min_add_distrib_right
% 0.25/0.55 thf(fact_5850_accp__eq__acc,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( accp @ A )
% 0.25/0.55 = ( ^ [R3: A > A > $o,X3: A] : ( member @ A @ X3 @ ( acc @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % accp_eq_acc
% 0.25/0.55 thf(fact_5851_rtrancl__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( transitive_rtrancl @ A )
% 0.25/0.55 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( collect @ ( product_prod @ A @ A )
% 0.25/0.55 @ ( product_case_prod @ A @ A @ $o
% 0.25/0.55 @ ( transitive_rtranclp @ A
% 0.25/0.55 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % rtrancl_def
% 0.25/0.55 thf(fact_5852_Enum_Ortranclp__rtrancl__eq,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( transitive_rtranclp @ A )
% 0.25/0.55 = ( ^ [R3: A > A > $o,X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Enum.rtranclp_rtrancl_eq
% 0.25/0.55 thf(fact_5853_acc_Ocases,axiom,
% 0.25/0.55 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 0.25/0.55 => ! [Y6: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ A3 ) @ R )
% 0.25/0.55 => ( member @ A @ Y6 @ ( acc @ A @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc.cases
% 0.25/0.55 thf(fact_5854_acc_Osimps,axiom,
% 0.25/0.55 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 0.25/0.55 = ( ? [X3: A] :
% 0.25/0.55 ( ( A3 = X3 )
% 0.25/0.55 & ! [Y3: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R )
% 0.25/0.55 => ( member @ A @ Y3 @ ( acc @ A @ R ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc.simps
% 0.25/0.55 thf(fact_5855_acc_Ointros,axiom,
% 0.25/0.55 ! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ! [Y4: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R )
% 0.25/0.55 => ( member @ A @ Y4 @ ( acc @ A @ R ) ) )
% 0.25/0.55 => ( member @ A @ X @ ( acc @ A @ R ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc.intros
% 0.25/0.55 thf(fact_5856_acc__induct,axiom,
% 0.25/0.55 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.25/0.55 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( ! [Y6: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 0.25/0.55 => ( P @ Y6 ) )
% 0.25/0.55 => ( P @ X2 ) ) )
% 0.25/0.55 => ( P @ A3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_induct
% 0.25/0.55 thf(fact_5857_acc__downward,axiom,
% 0.25/0.55 ! [A: $tType,B2: A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 0.25/0.55 ( ( member @ A @ B2 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 0.25/0.55 => ( member @ A @ A3 @ ( acc @ A @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_downward
% 0.25/0.55 thf(fact_5858_not__acc__down,axiom,
% 0.25/0.55 ! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ~ ( member @ A @ X @ ( acc @ A @ R5 ) )
% 0.25/0.55 => ~ ! [Z3: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X ) @ R5 )
% 0.25/0.55 => ( member @ A @ Z3 @ ( acc @ A @ R5 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % not_acc_down
% 0.25/0.55 thf(fact_5859_acc__induct__rule,axiom,
% 0.25/0.55 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 0.25/0.55 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( ! [Y6: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 0.25/0.55 => ( P @ Y6 ) )
% 0.25/0.55 => ( P @ X2 ) ) )
% 0.25/0.55 => ( P @ A3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_induct_rule
% 0.25/0.55 thf(fact_5860_rtranclp__induct2,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
% 0.25/0.55 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 0.25/0.55 => ( ( P @ Ax @ Ay )
% 0.25/0.55 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 0.25/0.55 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.25/0.55 => ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
% 0.25/0.55 => ( ( P @ A5 @ B3 )
% 0.25/0.55 => ( P @ Aa3 @ Ba ) ) ) )
% 0.25/0.55 => ( P @ Bx @ By ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % rtranclp_induct2
% 0.25/0.55 thf(fact_5861_converse__rtranclpE2,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za2: A,Zb: B] :
% 0.25/0.55 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) )
% 0.25/0.55 => ( ( ( product_Pair @ A @ B @ Xa @ Xb )
% 0.25/0.55 != ( product_Pair @ A @ B @ Za2 @ Zb ) )
% 0.25/0.55 => ~ ! [A5: A,B3: B] :
% 0.25/0.55 ( ( R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 0.25/0.55 => ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % converse_rtranclpE2
% 0.25/0.55 thf(fact_5862_converse__rtranclp__induct2,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
% 0.25/0.55 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 0.25/0.55 => ( ( P @ Bx @ By )
% 0.25/0.55 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 0.25/0.55 ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
% 0.25/0.55 => ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 0.25/0.55 => ( ( P @ Aa3 @ Ba )
% 0.25/0.55 => ( P @ A5 @ B3 ) ) ) )
% 0.25/0.55 => ( P @ Ax @ Ay ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % converse_rtranclp_induct2
% 0.25/0.55 thf(fact_5863_accp__downwards__aux,axiom,
% 0.25/0.55 ! [A: $tType,R: A > A > $o,B2: A,A3: A] :
% 0.25/0.55 ( ( transitive_rtranclp @ A @ R @ B2 @ A3 )
% 0.25/0.55 => ( ( accp @ A @ R @ A3 )
% 0.25/0.55 => ( accp @ A @ R @ B2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % accp_downwards_aux
% 0.25/0.55 thf(fact_5864_accp__downwards,axiom,
% 0.25/0.55 ! [A: $tType,R: A > A > $o,A3: A,B2: A] :
% 0.25/0.55 ( ( accp @ A @ R @ A3 )
% 0.25/0.55 => ( ( transitive_rtranclp @ A @ R @ B2 @ A3 )
% 0.25/0.55 => ( accp @ A @ R @ B2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % accp_downwards
% 0.25/0.55 thf(fact_5865_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
% 0.25/0.55 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ( transitive_rtranclp @ A
% 0.25/0.55 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.55 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Transitive_Closure.rtranclp_rtrancl_eq
% 0.25/0.55 thf(fact_5866_acc__subset__induct,axiom,
% 0.25/0.55 ! [A: $tType,D4: set @ A,R5: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
% 0.25/0.55 ( ( ord_less_eq @ ( set @ A ) @ D4 @ ( acc @ A @ R5 ) )
% 0.25/0.55 => ( ! [X2: A,Z3: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ D4 )
% 0.25/0.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X2 ) @ R5 )
% 0.25/0.55 => ( member @ A @ Z3 @ D4 ) ) )
% 0.25/0.55 => ( ( member @ A @ X @ D4 )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ D4 )
% 0.25/0.55 => ( ! [Z6: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X2 ) @ R5 )
% 0.25/0.55 => ( P @ Z6 ) )
% 0.25/0.55 => ( P @ X2 ) ) )
% 0.25/0.55 => ( P @ X ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_subset_induct
% 0.25/0.55 thf(fact_5867_acc__downwards,axiom,
% 0.25/0.55 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),B2: A] :
% 0.25/0.55 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.55 => ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_downwards
% 0.25/0.55 thf(fact_5868_acc__downwards__aux,axiom,
% 0.25/0.55 ! [A: $tType,B2: A,A3: A,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
% 0.25/0.55 => ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 0.25/0.55 => ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_downwards_aux
% 0.25/0.55 thf(fact_5869_accp__acc__eq,axiom,
% 0.25/0.55 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( ( accp @ A
% 0.25/0.55 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 0.25/0.55 = ( ^ [X3: A] : ( member @ A @ X3 @ ( acc @ A @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % accp_acc_eq
% 0.25/0.55 thf(fact_5870_acc__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( acc @ A )
% 0.25/0.55 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 0.25/0.55 ( collect @ A
% 0.25/0.55 @ ( accp @ A
% 0.25/0.55 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % acc_def
% 0.25/0.55 thf(fact_5871_scomp__fcomp,axiom,
% 0.25/0.55 ! [A: $tType,C: $tType,B: $tType,E: $tType,D: $tType,F: A > ( product_prod @ D @ E ),G: D > E > C,H3: C > B] :
% 0.25/0.55 ( ( fcomp @ A @ C @ B @ ( product_scomp @ A @ D @ E @ C @ F @ G ) @ H3 )
% 0.25/0.55 = ( product_scomp @ A @ D @ E @ B @ F
% 0.25/0.55 @ ^ [X3: D] : ( fcomp @ E @ C @ B @ ( G @ X3 ) @ H3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % scomp_fcomp
% 0.25/0.55 thf(fact_5872_successively_Opelims_I2_J,axiom,
% 0.25/0.55 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 0.25/0.55 ( ( successively @ A @ X @ Xa )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.55 => ( ( ( Xa
% 0.25/0.55 = ( nil @ A ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 0.25/0.55 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
% 0.25/0.55 => ~ ( ( X @ X2 @ Y4 )
% 0.25/0.55 & ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % successively.pelims(2)
% 0.25/0.55 thf(fact_5873_successively_Opelims_I1_J,axiom,
% 0.25/0.55 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
% 0.25/0.55 ( ( ( successively @ A @ X @ Xa )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.55 => ( ( ( Xa
% 0.25/0.55 = ( nil @ A ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 0.25/0.55 => ( Y
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 0.25/0.55 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( ( X @ X2 @ Y4 )
% 0.25/0.55 & ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
% 0.25/0.55 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % successively.pelims(1)
% 0.25/0.55 thf(fact_5874_fcomp__apply,axiom,
% 0.25/0.55 ! [A: $tType,C: $tType,B: $tType] :
% 0.25/0.55 ( ( fcomp @ B @ C @ A )
% 0.25/0.55 = ( ^ [F2: B > C,G2: C > A,X3: B] : ( G2 @ ( F2 @ X3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fcomp_apply
% 0.25/0.55 thf(fact_5875_id__fcomp,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,G: A > B] :
% 0.25/0.55 ( ( fcomp @ A @ A @ B @ ( id @ A ) @ G )
% 0.25/0.55 = G ) ).
% 0.25/0.55
% 0.25/0.55 % id_fcomp
% 0.25/0.55 thf(fact_5876_fcomp__id,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,F: A > B] :
% 0.25/0.55 ( ( fcomp @ A @ B @ B @ F @ ( id @ B ) )
% 0.25/0.55 = F ) ).
% 0.25/0.55
% 0.25/0.55 % fcomp_id
% 0.25/0.55 thf(fact_5877_fcomp__scomp,axiom,
% 0.25/0.55 ! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F: A > E,G: E > ( product_prod @ C @ D ),H3: C > D > B] :
% 0.25/0.55 ( ( product_scomp @ A @ C @ D @ B @ ( fcomp @ A @ E @ ( product_prod @ C @ D ) @ F @ G ) @ H3 )
% 0.25/0.55 = ( fcomp @ A @ E @ B @ F @ ( product_scomp @ E @ C @ D @ B @ G @ H3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fcomp_scomp
% 0.25/0.55 thf(fact_5878_fcomp__assoc,axiom,
% 0.25/0.55 ! [A: $tType,D: $tType,B: $tType,C: $tType,F: A > D,G: D > C,H3: C > B] :
% 0.25/0.55 ( ( fcomp @ A @ C @ B @ ( fcomp @ A @ D @ C @ F @ G ) @ H3 )
% 0.25/0.55 = ( fcomp @ A @ D @ B @ F @ ( fcomp @ D @ C @ B @ G @ H3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fcomp_assoc
% 0.25/0.55 thf(fact_5879_fcomp__def,axiom,
% 0.25/0.55 ! [C: $tType,B: $tType,A: $tType] :
% 0.25/0.55 ( ( fcomp @ A @ B @ C )
% 0.25/0.55 = ( ^ [F2: A > B,G2: B > C,X3: A] : ( G2 @ ( F2 @ X3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fcomp_def
% 0.25/0.55 thf(fact_5880_fcomp__comp,axiom,
% 0.25/0.55 ! [B: $tType,C: $tType,A: $tType] :
% 0.25/0.55 ( ( fcomp @ A @ C @ B )
% 0.25/0.55 = ( ^ [F2: A > C,G2: C > B] : ( comp @ C @ B @ A @ G2 @ F2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fcomp_comp
% 0.25/0.55 thf(fact_5881_successively_Opelims_I3_J,axiom,
% 0.25/0.55 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 0.25/0.55 ( ~ ( successively @ A @ X @ Xa )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 0.25/0.55 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 0.25/0.55 ( ( Xa
% 0.25/0.55 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 0.25/0.55 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
% 0.25/0.55 => ( ( X @ X2 @ Y4 )
% 0.25/0.55 & ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % successively.pelims(3)
% 0.25/0.55 thf(fact_5882_folding__idem__axioms_Ointro,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.25/0.55 ( ! [X2: A] :
% 0.25/0.55 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 0.25/0.55 = ( F @ X2 ) )
% 0.25/0.55 => ( finite7837460588564673216axioms @ A @ B @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_idem_axioms.intro
% 0.25/0.55 thf(fact_5883_comp__fun__idem__axioms_Ointro,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.25/0.55 ( ! [X2: A] :
% 0.25/0.55 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 0.25/0.55 = ( F @ X2 ) )
% 0.25/0.55 => ( finite1981172056906289455axioms @ A @ B @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % comp_fun_idem_axioms.intro
% 0.25/0.55 thf(fact_5884_folding__idem__on__axioms_Ointro,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,S2: set @ A,F: A > B > B] :
% 0.25/0.55 ( ! [X2: A,Y4: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ S2 )
% 0.25/0.55 => ( ( member @ A @ Y4 @ S2 )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 0.25/0.55 = ( F @ X2 ) ) ) )
% 0.25/0.55 => ( finite6916993218817215295axioms @ A @ B @ S2 @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_idem_on_axioms.intro
% 0.25/0.55 thf(fact_5885_folding__idem__on__axioms__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( finite6916993218817215295axioms @ A @ B )
% 0.25/0.55 = ( ^ [S5: set @ A,F2: A > B > B] :
% 0.25/0.55 ! [X3: A,Y3: A] :
% 0.25/0.55 ( ( member @ A @ X3 @ S5 )
% 0.25/0.55 => ( ( member @ A @ Y3 @ S5 )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 0.25/0.55 = ( F2 @ X3 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_idem_on_axioms_def
% 0.25/0.55 thf(fact_5886_comp__fun__idem__axioms__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( finite1981172056906289455axioms @ A @ B )
% 0.25/0.55 = ( ^ [F2: A > B > B] :
% 0.25/0.55 ! [X3: A] :
% 0.25/0.55 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 0.25/0.55 = ( F2 @ X3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comp_fun_idem_axioms_def
% 0.25/0.55 thf(fact_5887_folding__idem__axioms__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( finite7837460588564673216axioms @ A @ B )
% 0.25/0.55 = ( ^ [F2: A > B > B] :
% 0.25/0.55 ! [X3: A] :
% 0.25/0.55 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 0.25/0.55 = ( F2 @ X3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_idem_axioms_def
% 0.25/0.55 thf(fact_5888_comp__fun__idem__on__axioms__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( finite4980608107308702382axioms @ A @ B )
% 0.25/0.55 = ( ^ [S5: set @ A,F2: A > B > B] :
% 0.25/0.55 ! [X3: A] :
% 0.25/0.55 ( ( member @ A @ X3 @ S5 )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 0.25/0.55 = ( F2 @ X3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % comp_fun_idem_on_axioms_def
% 0.25/0.55 thf(fact_5889_comp__fun__idem__on__axioms_Ointro,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,S2: set @ A,F: A > B > B] :
% 0.25/0.55 ( ! [X2: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ S2 )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 0.25/0.55 = ( F @ X2 ) ) )
% 0.25/0.55 => ( finite4980608107308702382axioms @ A @ B @ S2 @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % comp_fun_idem_on_axioms.intro
% 0.25/0.55 thf(fact_5890_image__mset__If,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,A4: multiset @ B] :
% 0.25/0.55 ( ( image_mset @ B @ A
% 0.25/0.55 @ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
% 0.25/0.55 @ A4 )
% 0.25/0.55 = ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ ( filter_mset @ B @ P @ A4 ) )
% 0.25/0.55 @ ( image_mset @ B @ A @ G
% 0.25/0.55 @ ( filter_mset @ B
% 0.25/0.55 @ ^ [X3: B] :
% 0.25/0.55 ~ ( P @ X3 )
% 0.25/0.55 @ A4 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % image_mset_If
% 0.25/0.55 thf(fact_5891_union__filter__mset__complement,axiom,
% 0.25/0.55 ! [A: $tType,P: A > $o,Q: A > $o,M3: multiset @ A] :
% 0.25/0.55 ( ! [X2: A] :
% 0.25/0.55 ( ( P @ X2 )
% 0.25/0.55 = ( ~ ( Q @ X2 ) ) )
% 0.25/0.55 => ( ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ Q @ M3 ) )
% 0.25/0.55 = M3 ) ) ).
% 0.25/0.55
% 0.25/0.55 % union_filter_mset_complement
% 0.25/0.55 thf(fact_5892_filter__union__mset,axiom,
% 0.25/0.55 ! [A: $tType,P: A > $o,M3: multiset @ A,N6: multiset @ A] :
% 0.25/0.55 ( ( filter_mset @ A @ P @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 0.25/0.55 = ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ P @ N6 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % filter_union_mset
% 0.25/0.55 thf(fact_5893_multiset__partition,axiom,
% 0.25/0.55 ! [A: $tType,M3: multiset @ A,P: A > $o] :
% 0.25/0.55 ( M3
% 0.25/0.55 = ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 )
% 0.25/0.55 @ ( filter_mset @ A
% 0.25/0.55 @ ^ [X3: A] :
% 0.25/0.55 ~ ( P @ X3 )
% 0.25/0.55 @ M3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % multiset_partition
% 0.25/0.55 thf(fact_5894_heap_Olub__upper,axiom,
% 0.25/0.55 ! [A: $tType,A4: set @ ( heap_Time_Heap @ A ),X: heap_Time_Heap @ A] :
% 0.25/0.55 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A4 )
% 0.25/0.55 => ( ( member @ ( heap_Time_Heap @ A ) @ X @ A4 )
% 0.25/0.55 => ( heap_Time_Heap_ord @ A @ X @ ( heap_Time_Heap_lub @ A @ A4 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % heap.lub_upper
% 0.25/0.55 thf(fact_5895_heap_Olub__least,axiom,
% 0.25/0.55 ! [A: $tType,A4: set @ ( heap_Time_Heap @ A ),Z2: heap_Time_Heap @ A] :
% 0.25/0.55 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A4 )
% 0.25/0.55 => ( ! [X2: heap_Time_Heap @ A] :
% 0.25/0.55 ( ( member @ ( heap_Time_Heap @ A ) @ X2 @ A4 )
% 0.25/0.55 => ( heap_Time_Heap_ord @ A @ X2 @ Z2 ) )
% 0.25/0.55 => ( heap_Time_Heap_ord @ A @ ( heap_Time_Heap_lub @ A @ A4 ) @ Z2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % heap.lub_least
% 0.25/0.55 thf(fact_5896_plus__int__code_I6_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( plus_plus @ int @ ( neg @ M2 ) @ ( neg @ N ) )
% 0.25/0.55 = ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % plus_int_code(6)
% 0.25/0.55 thf(fact_5897_ccpo__Sup__mono,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comple9053668089753744459l_ccpo @ A )
% 0.25/0.55 => ! [A4: set @ A,B6: set @ A] :
% 0.25/0.55 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
% 0.25/0.55 => ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B6 )
% 0.25/0.55 => ( ! [X2: A] :
% 0.25/0.55 ( ( member @ A @ X2 @ A4 )
% 0.25/0.55 => ? [Xa2: A] :
% 0.25/0.55 ( ( member @ A @ Xa2 @ B6 )
% 0.25/0.55 & ( ord_less_eq @ A @ X2 @ Xa2 ) ) )
% 0.25/0.55 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % ccpo_Sup_mono
% 0.25/0.55 thf(fact_5898_plus__int__code_I5_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( plus_plus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
% 0.25/0.55 = ( sub @ N @ M2 ) ) ).
% 0.25/0.55
% 0.25/0.55 % plus_int_code(5)
% 0.25/0.55 thf(fact_5899_plus__int__code_I4_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( plus_plus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
% 0.25/0.55 = ( sub @ M2 @ N ) ) ).
% 0.25/0.55
% 0.25/0.55 % plus_int_code(4)
% 0.25/0.55 thf(fact_5900_minus__int__code_I4_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( minus_minus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
% 0.25/0.55 = ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % minus_int_code(4)
% 0.25/0.55 thf(fact_5901_plus__int__code_I3_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( plus_plus @ int @ ( pos @ M2 ) @ ( pos @ N ) )
% 0.25/0.55 = ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % plus_int_code(3)
% 0.25/0.55 thf(fact_5902_minus__int__code_I5_J,axiom,
% 0.25/0.55 ! [M2: num,N: num] :
% 0.25/0.55 ( ( minus_minus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
% 0.25/0.55 = ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % minus_int_code(5)
% 0.25/0.55 thf(fact_5903_Unity__def,axiom,
% 0.25/0.55 ( product_Unity
% 0.25/0.55 = ( product_Abs_unit @ $true ) ) ).
% 0.25/0.55
% 0.25/0.55 % Unity_def
% 0.25/0.55 thf(fact_5904_Domainp__Domain__eq,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 0.25/0.55 ( ( domainp @ A @ B
% 0.25/0.55 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 0.25/0.55 = ( ^ [X3: A] : ( member @ A @ X3 @ ( domain @ A @ B @ R ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Domainp_Domain_eq
% 0.25/0.55 thf(fact_5905_Domain__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( domain @ A @ B )
% 0.25/0.55 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 0.25/0.55 ( collect @ A
% 0.25/0.55 @ ( domainp @ A @ B
% 0.25/0.55 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Domain_def
% 0.25/0.55 thf(fact_5906_Abs__unit__inject,axiom,
% 0.25/0.55 ! [X: $o,Y: $o] :
% 0.25/0.55 ( ( member @ $o @ X @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 0.25/0.55 => ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 0.25/0.55 => ( ( ( product_Abs_unit @ X )
% 0.25/0.55 = ( product_Abs_unit @ Y ) )
% 0.25/0.55 = ( X = Y ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Abs_unit_inject
% 0.25/0.55 thf(fact_5907_Abs__unit__induct,axiom,
% 0.25/0.55 ! [P: product_unit > $o,X: product_unit] :
% 0.25/0.55 ( ! [Y4: $o] :
% 0.25/0.55 ( ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 0.25/0.55 => ( P @ ( product_Abs_unit @ Y4 ) ) )
% 0.25/0.55 => ( P @ X ) ) ).
% 0.25/0.55
% 0.25/0.55 % Abs_unit_induct
% 0.25/0.55 thf(fact_5908_Abs__unit__cases,axiom,
% 0.25/0.55 ! [X: product_unit] :
% 0.25/0.55 ~ ! [Y4: $o] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( product_Abs_unit @ Y4 ) )
% 0.25/0.55 => ~ ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Abs_unit_cases
% 0.25/0.55 thf(fact_5909_Heap_ODomainp__rel,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 0.25/0.55 ( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) )
% 0.25/0.55 = ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Heap.Domainp_rel
% 0.25/0.55 thf(fact_5910_integer_Odomain,axiom,
% 0.25/0.55 ! [P: int > $o] :
% 0.25/0.55 ( ( ( domainp @ int @ int
% 0.25/0.55 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 0.25/0.55 = P )
% 0.25/0.55 => ( ( domainp @ int @ code_integer @ code_pcr_integer )
% 0.25/0.55 = P ) ) ).
% 0.25/0.55
% 0.25/0.55 % integer.domain
% 0.25/0.55 thf(fact_5911_natural_Odomain,axiom,
% 0.25/0.55 ! [P: nat > $o] :
% 0.25/0.55 ( ( ( domainp @ nat @ nat
% 0.25/0.55 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 0.25/0.55 = P )
% 0.25/0.55 => ( ( domainp @ nat @ code_natural @ code_pcr_natural )
% 0.25/0.55 = P ) ) ).
% 0.25/0.55
% 0.25/0.55 % natural.domain
% 0.25/0.55 thf(fact_5912_option_ODomainp__rel,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 0.25/0.55 ( ( domainp @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) )
% 0.25/0.55 = ( pred_option @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.Domainp_rel
% 0.25/0.55 thf(fact_5913_Abs__unit__inverse,axiom,
% 0.25/0.55 ! [Y: $o] :
% 0.25/0.55 ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 0.25/0.55 => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
% 0.25/0.55 = Y ) ) ).
% 0.25/0.55
% 0.25/0.55 % Abs_unit_inverse
% 0.25/0.55 thf(fact_5914_type__definition__unit,axiom,
% 0.25/0.55 type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % type_definition_unit
% 0.25/0.55 thf(fact_5915_Rep__unit__inverse,axiom,
% 0.25/0.55 ! [X: product_unit] :
% 0.25/0.55 ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
% 0.25/0.55 = X ) ).
% 0.25/0.55
% 0.25/0.55 % Rep_unit_inverse
% 0.25/0.55 thf(fact_5916_Rep__unit__inject,axiom,
% 0.25/0.55 ! [X: product_unit,Y: product_unit] :
% 0.25/0.55 ( ( ( product_Rep_unit @ X )
% 0.25/0.55 = ( product_Rep_unit @ Y ) )
% 0.25/0.55 = ( X = Y ) ) ).
% 0.25/0.55
% 0.25/0.55 % Rep_unit_inject
% 0.25/0.55 thf(fact_5917_Rep__unit,axiom,
% 0.25/0.55 ! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Rep_unit
% 0.25/0.55 thf(fact_5918_Rep__unit__cases,axiom,
% 0.25/0.55 ! [Y: $o] :
% 0.25/0.55 ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 0.25/0.55 => ~ ! [X2: product_unit] :
% 0.25/0.55 ( Y
% 0.25/0.55 = ( ~ ( product_Rep_unit @ X2 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Rep_unit_cases
% 0.25/0.55 thf(fact_5919_Rep__unit__induct,axiom,
% 0.25/0.55 ! [Y: $o,P: $o > $o] :
% 0.25/0.55 ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 0.25/0.55 => ( ! [X2: product_unit] : ( P @ ( product_Rep_unit @ X2 ) )
% 0.25/0.55 => ( P @ Y ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Rep_unit_induct
% 0.25/0.55 thf(fact_5920_adm__wf__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( adm_wf @ A @ B )
% 0.25/0.55 = ( ^ [R6: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
% 0.25/0.55 ! [F2: A > B,G2: A > B,X3: A] :
% 0.25/0.55 ( ! [Z5: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X3 ) @ R6 )
% 0.25/0.55 => ( ( F2 @ Z5 )
% 0.25/0.55 = ( G2 @ Z5 ) ) )
% 0.25/0.55 => ( ( F7 @ F2 @ X3 )
% 0.25/0.55 = ( F7 @ G2 @ X3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % adm_wf_def
% 0.25/0.55 thf(fact_5921_folding__idem_Ocomp__fun__idem,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,F: A > B > B,X: A] :
% 0.25/0.55 ( ( finite_folding_idem @ A @ B @ F )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 0.25/0.55 = ( F @ X ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_idem.comp_fun_idem
% 0.25/0.55 thf(fact_5922_compat__def,axiom,
% 0.25/0.55 ! [A2: $tType,A: $tType] :
% 0.25/0.55 ( ( bNF_Wellorder_compat @ A @ A2 )
% 0.25/0.55 = ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
% 0.25/0.55 ! [A7: A,B5: A] :
% 0.25/0.55 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 )
% 0.25/0.55 => ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F2 @ A7 ) @ ( F2 @ B5 ) ) @ R11 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % compat_def
% 0.25/0.55 thf(fact_5923_positive__add,axiom,
% 0.25/0.55 ! [X: rat,Y: rat] :
% 0.25/0.55 ( ( positive @ X )
% 0.25/0.55 => ( ( positive @ Y )
% 0.25/0.55 => ( positive @ ( plus_plus @ rat @ X @ Y ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % positive_add
% 0.25/0.55 thf(fact_5924_inj__on__iff__Uniq,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( inj_on @ A @ B )
% 0.25/0.55 = ( ^ [F2: A > B,A9: set @ A] :
% 0.25/0.55 ! [X3: A] :
% 0.25/0.55 ( ( member @ A @ X3 @ A9 )
% 0.25/0.55 => ( uniq @ A
% 0.25/0.55 @ ^ [Y3: A] :
% 0.25/0.55 ( ( member @ A @ Y3 @ A9 )
% 0.25/0.55 & ( ( F2 @ X3 )
% 0.25/0.55 = ( F2 @ Y3 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % inj_on_iff_Uniq
% 0.25/0.55 thf(fact_5925_mono__imp__mono__on,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( order @ A )
% 0.25/0.55 & ( order @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A] :
% 0.25/0.55 ( ( order_mono @ A @ B @ F )
% 0.25/0.55 => ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mono_imp_mono_on
% 0.25/0.55 thf(fact_5926_mono__on__subset,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A,B6: set @ A] :
% 0.25/0.55 ( ( mono_on @ A @ B @ F @ A4 )
% 0.25/0.55 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 0.25/0.55 => ( mono_on @ A @ B @ F @ B6 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mono_on_subset
% 0.25/0.55 thf(fact_5927_mono__on__greaterD,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( linorder @ A )
% 0.25/0.55 & ( linorder @ B ) )
% 0.25/0.55 => ! [G: A > B,A4: set @ A,X: A,Y: A] :
% 0.25/0.55 ( ( mono_on @ A @ B @ G @ A4 )
% 0.25/0.55 => ( ( member @ A @ X @ A4 )
% 0.25/0.55 => ( ( member @ A @ Y @ A4 )
% 0.25/0.55 => ( ( ord_less @ B @ ( G @ Y ) @ ( G @ X ) )
% 0.25/0.55 => ( ord_less @ A @ Y @ X ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mono_on_greaterD
% 0.25/0.55 thf(fact_5928_mono__onD,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A,R: A,S: A] :
% 0.25/0.55 ( ( mono_on @ A @ B @ F @ A4 )
% 0.25/0.55 => ( ( member @ A @ R @ A4 )
% 0.25/0.55 => ( ( member @ A @ S @ A4 )
% 0.25/0.55 => ( ( ord_less_eq @ A @ R @ S )
% 0.25/0.55 => ( ord_less_eq @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mono_onD
% 0.25/0.55 thf(fact_5929_mono__onI,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ! [A4: set @ A,F: A > B] :
% 0.25/0.55 ( ! [R2: A,S6: A] :
% 0.25/0.55 ( ( member @ A @ R2 @ A4 )
% 0.25/0.55 => ( ( member @ A @ S6 @ A4 )
% 0.25/0.55 => ( ( ord_less_eq @ A @ R2 @ S6 )
% 0.25/0.55 => ( ord_less_eq @ B @ ( F @ R2 ) @ ( F @ S6 ) ) ) ) )
% 0.25/0.55 => ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mono_onI
% 0.25/0.55 thf(fact_5930_mono__on__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ( ( mono_on @ A @ B )
% 0.25/0.55 = ( ^ [F2: A > B,A9: set @ A] :
% 0.25/0.55 ! [R3: A,S4: A] :
% 0.25/0.55 ( ( ( member @ A @ R3 @ A9 )
% 0.25/0.55 & ( member @ A @ S4 @ A9 )
% 0.25/0.55 & ( ord_less_eq @ A @ R3 @ S4 ) )
% 0.25/0.55 => ( ord_less_eq @ B @ ( F2 @ R3 ) @ ( F2 @ S4 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % mono_on_def
% 0.25/0.55 thf(fact_5931_strict__mono__on__imp__mono__on,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( linorder @ A )
% 0.25/0.55 & ( preorder @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A] :
% 0.25/0.55 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 0.25/0.55 => ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_on_imp_mono_on
% 0.25/0.55 thf(fact_5932_folding_Ointro,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,F: A > B > B] :
% 0.25/0.55 ( ! [Y4: A,X2: A] :
% 0.25/0.55 ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 0.25/0.55 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.25/0.55 => ( finite_folding @ A @ B @ F ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding.intro
% 0.25/0.55 thf(fact_5933_folding_Ocomp__fun__commute,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 0.25/0.55 ( ( finite_folding @ A @ B @ F )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 0.25/0.55 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding.comp_fun_commute
% 0.25/0.55 thf(fact_5934_strict__mono__on__eqD,axiom,
% 0.25/0.55 ! [D: $tType,C: $tType] :
% 0.25/0.55 ( ( ( linorder @ C )
% 0.25/0.55 & ( preorder @ D ) )
% 0.25/0.55 => ! [F: C > D,A4: set @ C,X: C,Y: C] :
% 0.25/0.55 ( ( strict_mono_on @ C @ D @ F @ A4 )
% 0.25/0.55 => ( ( ( F @ X )
% 0.25/0.55 = ( F @ Y ) )
% 0.25/0.55 => ( ( member @ C @ X @ A4 )
% 0.25/0.55 => ( ( member @ C @ Y @ A4 )
% 0.25/0.55 => ( Y = X ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_on_eqD
% 0.25/0.55 thf(fact_5935_strict__mono__on__leD,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( linorder @ A )
% 0.25/0.55 & ( preorder @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A,X: A,Y: A] :
% 0.25/0.55 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 0.25/0.55 => ( ( member @ A @ X @ A4 )
% 0.25/0.55 => ( ( member @ A @ Y @ A4 )
% 0.25/0.55 => ( ( ord_less_eq @ A @ X @ Y )
% 0.25/0.55 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_on_leD
% 0.25/0.55 thf(fact_5936_strict__mono__on__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ( ( strict_mono_on @ A @ B )
% 0.25/0.55 = ( ^ [F2: A > B,A9: set @ A] :
% 0.25/0.55 ! [R3: A,S4: A] :
% 0.25/0.55 ( ( ( member @ A @ R3 @ A9 )
% 0.25/0.55 & ( member @ A @ S4 @ A9 )
% 0.25/0.55 & ( ord_less @ A @ R3 @ S4 ) )
% 0.25/0.55 => ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S4 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_on_def
% 0.25/0.55 thf(fact_5937_strict__mono__onI,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ! [A4: set @ A,F: A > B] :
% 0.25/0.55 ( ! [R2: A,S6: A] :
% 0.25/0.55 ( ( member @ A @ R2 @ A4 )
% 0.25/0.55 => ( ( member @ A @ S6 @ A4 )
% 0.25/0.55 => ( ( ord_less @ A @ R2 @ S6 )
% 0.25/0.55 => ( ord_less @ B @ ( F @ R2 ) @ ( F @ S6 ) ) ) ) )
% 0.25/0.55 => ( strict_mono_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_onI
% 0.25/0.55 thf(fact_5938_strict__mono__onD,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( ord @ A )
% 0.25/0.55 & ( ord @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A,R: A,S: A] :
% 0.25/0.55 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 0.25/0.55 => ( ( member @ A @ R @ A4 )
% 0.25/0.55 => ( ( member @ A @ S @ A4 )
% 0.25/0.55 => ( ( ord_less @ A @ R @ S )
% 0.25/0.55 => ( ord_less @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_onD
% 0.25/0.55 thf(fact_5939_strict__mono__on__imp__inj__on,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( ( linorder @ A )
% 0.25/0.55 & ( preorder @ B ) )
% 0.25/0.55 => ! [F: A > B,A4: set @ A] :
% 0.25/0.55 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 0.25/0.55 => ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % strict_mono_on_imp_inj_on
% 0.25/0.55 thf(fact_5940_folding__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( finite_folding @ A @ B )
% 0.25/0.55 = ( ^ [F2: A > B > B] :
% 0.25/0.55 ! [Y3: A,X3: A] :
% 0.25/0.55 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 0.25/0.55 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_def
% 0.25/0.55 thf(fact_5941_folding__idem__on_Ocomp__fun__idem__on,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,S2: set @ A,F: A > B > B,X: A,Y: A] :
% 0.25/0.55 ( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F )
% 0.25/0.55 => ( ( member @ A @ X @ S2 )
% 0.25/0.55 => ( ( member @ A @ Y @ S2 )
% 0.25/0.55 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 0.25/0.55 = ( F @ X ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % folding_idem_on.comp_fun_idem_on
% 0.25/0.55 thf(fact_5942_sum_OPlus,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,C: $tType] :
% 0.25/0.55 ( ( comm_monoid_add @ A )
% 0.25/0.55 => ! [A4: set @ B,B6: set @ C,G: ( sum_sum @ B @ C ) > A] :
% 0.25/0.55 ( ( finite_finite2 @ B @ A4 )
% 0.25/0.55 => ( ( finite_finite2 @ C @ B6 )
% 0.25/0.55 => ( ( groups7311177749621191930dd_sum @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A4 @ B6 ) )
% 0.25/0.55 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A4 ) @ ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum.Plus
% 0.25/0.55 thf(fact_5943_map__sum__Inl__conv,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,D: $tType,C: $tType,Fl: C > A,Fr: D > B,S: sum_sum @ C @ D,Y: A] :
% 0.25/0.55 ( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S )
% 0.25/0.55 = ( sum_Inl @ A @ B @ Y ) )
% 0.25/0.55 = ( ? [X3: C] :
% 0.25/0.55 ( ( S
% 0.25/0.55 = ( sum_Inl @ C @ D @ X3 ) )
% 0.25/0.55 & ( Y
% 0.25/0.55 = ( Fl @ X3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % map_sum_Inl_conv
% 0.25/0.55 thf(fact_5944_map__sum__Inr__conv,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,C: $tType,D: $tType,Fl: C > A,Fr: D > B,S: sum_sum @ C @ D,Y: B] :
% 0.25/0.55 ( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S )
% 0.25/0.55 = ( sum_Inr @ B @ A @ Y ) )
% 0.25/0.55 = ( ? [X3: D] :
% 0.25/0.55 ( ( S
% 0.25/0.55 = ( sum_Inr @ D @ C @ X3 ) )
% 0.25/0.55 & ( Y
% 0.25/0.55 = ( Fr @ X3 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % map_sum_Inr_conv
% 0.25/0.55 thf(fact_5945_Union__plus,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,C: $tType,F: ( sum_sum @ B @ C ) > ( set @ A ),A4: set @ B,B6: set @ C] :
% 0.25/0.55 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( sum_sum @ B @ C ) @ ( set @ A ) @ F @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) )
% 0.25/0.55 = ( sup_sup @ ( set @ A )
% 0.25/0.55 @ ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.55 @ ( image2 @ B @ ( set @ A )
% 0.25/0.55 @ ^ [A7: B] : ( F @ ( sum_Inl @ B @ C @ A7 ) )
% 0.25/0.55 @ A4 ) )
% 0.25/0.55 @ ( complete_Sup_Sup @ ( set @ A )
% 0.25/0.55 @ ( image2 @ C @ ( set @ A )
% 0.25/0.55 @ ^ [B5: C] : ( F @ ( sum_Inr @ C @ B @ B5 ) )
% 0.25/0.55 @ B6 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Union_plus
% 0.25/0.55 thf(fact_5946_Union__sum,axiom,
% 0.25/0.55 ! [C: $tType,A: $tType,B: $tType,F: ( sum_sum @ A @ B ) > ( set @ C )] :
% 0.25/0.55 ( ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ ( sum_sum @ A @ B ) @ ( set @ C ) @ F @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) )
% 0.25/0.55 = ( sup_sup @ ( set @ C )
% 0.25/0.55 @ ( complete_Sup_Sup @ ( set @ C )
% 0.25/0.55 @ ( image2 @ A @ ( set @ C )
% 0.25/0.55 @ ^ [L3: A] : ( F @ ( sum_Inl @ A @ B @ L3 ) )
% 0.25/0.55 @ ( top_top @ ( set @ A ) ) ) )
% 0.25/0.55 @ ( complete_Sup_Sup @ ( set @ C )
% 0.25/0.55 @ ( image2 @ B @ ( set @ C )
% 0.25/0.55 @ ^ [R3: B] : ( F @ ( sum_Inr @ B @ A @ R3 ) )
% 0.25/0.55 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Union_sum
% 0.25/0.55 thf(fact_5947_sum_Osize__gen_I1_J,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,Xa: A > nat,X: B > nat,X1: A] :
% 0.25/0.55 ( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inl @ A @ B @ X1 ) )
% 0.25/0.55 = ( plus_plus @ nat @ ( Xa @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum.size_gen(1)
% 0.25/0.55 thf(fact_5948_sum_Osize__gen_I2_J,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,Xa: A > nat,X: B > nat,X22: B] :
% 0.25/0.55 ( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inr @ B @ A @ X22 ) )
% 0.25/0.55 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % sum.size_gen(2)
% 0.25/0.55 thf(fact_5949_Node__def,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType] :
% 0.25/0.55 ( ( old_Node @ B @ A )
% 0.25/0.55 = ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
% 0.25/0.55 @ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
% 0.25/0.55 ? [F2: nat > ( sum_sum @ B @ nat ),X3: sum_sum @ A @ nat,K4: nat] :
% 0.25/0.55 ( ( P6
% 0.25/0.55 = ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F2 @ X3 ) )
% 0.25/0.55 & ( ( F2 @ K4 )
% 0.25/0.55 = ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Node_def
% 0.25/0.55 thf(fact_5950_Node__K0__I,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
% 0.25/0.55 ( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
% 0.25/0.55 @ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
% 0.25/0.55 @ ^ [K4: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
% 0.25/0.55 @ A3 )
% 0.25/0.55 @ ( old_Node @ A @ B ) ) ).
% 0.25/0.55
% 0.25/0.55 % Node_K0_I
% 0.25/0.55 thf(fact_5951_nth__item_Opinduct,axiom,
% 0.25/0.55 ! [A0: nat,P: nat > $o] :
% 0.25/0.55 ( ( accp @ nat @ nth_item_rel @ A0 )
% 0.25/0.55 => ( ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
% 0.25/0.55 => ( P @ ( zero_zero @ nat ) ) )
% 0.25/0.55 => ( ! [N3: nat] :
% 0.25/0.55 ( ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) )
% 0.25/0.55 => ( ! [A17: nat,Aa4: nat] :
% 0.25/0.55 ( ( ( nat_sum_decode @ N3 )
% 0.25/0.55 = ( sum_Inl @ nat @ nat @ A17 ) )
% 0.25/0.55 => ( ( ( nat_sum_decode @ A17 )
% 0.25/0.55 = ( sum_Inl @ nat @ nat @ Aa4 ) )
% 0.25/0.55 => ( P @ Aa4 ) ) )
% 0.25/0.55 => ( ! [A17: nat,B8: nat] :
% 0.25/0.55 ( ( ( nat_sum_decode @ N3 )
% 0.25/0.55 = ( sum_Inl @ nat @ nat @ A17 ) )
% 0.25/0.55 => ( ( ( nat_sum_decode @ A17 )
% 0.25/0.55 = ( sum_Inr @ nat @ nat @ B8 ) )
% 0.25/0.55 => ( P @ B8 ) ) )
% 0.25/0.55 => ( ! [B8: nat,Ba2: nat,X7: nat,Y6: nat] :
% 0.25/0.55 ( ( ( nat_sum_decode @ N3 )
% 0.25/0.55 = ( sum_Inr @ nat @ nat @ B8 ) )
% 0.25/0.55 => ( ( ( nat_sum_decode @ B8 )
% 0.25/0.55 = ( sum_Inr @ nat @ nat @ Ba2 ) )
% 0.25/0.55 => ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
% 0.25/0.55 = ( nat_prod_decode @ Ba2 ) )
% 0.25/0.55 => ( P @ X7 ) ) ) )
% 0.25/0.55 => ( ! [B8: nat,Ba2: nat,X7: nat,Y6: nat] :
% 0.25/0.55 ( ( ( nat_sum_decode @ N3 )
% 0.25/0.55 = ( sum_Inr @ nat @ nat @ B8 ) )
% 0.25/0.55 => ( ( ( nat_sum_decode @ B8 )
% 0.25/0.55 = ( sum_Inr @ nat @ nat @ Ba2 ) )
% 0.25/0.55 => ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
% 0.25/0.55 = ( nat_prod_decode @ Ba2 ) )
% 0.25/0.55 => ( P @ Y6 ) ) ) )
% 0.25/0.55 => ( P @ ( suc @ N3 ) ) ) ) ) ) )
% 0.25/0.55 => ( P @ A0 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % nth_item.pinduct
% 0.25/0.55 thf(fact_5952_ndepth__K0,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
% 0.25/0.55 ( ( old_ndepth @ A @ B
% 0.25/0.55 @ ( old_Abs_Node @ B @ A
% 0.25/0.55 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 0.25/0.55 @ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 0.25/0.55 @ X ) ) )
% 0.25/0.55 = ( zero_zero @ nat ) ) ).
% 0.25/0.55
% 0.25/0.55 % ndepth_K0
% 0.25/0.55 thf(fact_5953_Atom__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( old_Atom @ A @ B )
% 0.25/0.55 = ( ^ [X3: sum_sum @ A @ nat] :
% 0.25/0.55 ( insert2 @ ( old_node @ A @ B )
% 0.25/0.55 @ ( old_Abs_Node @ B @ A
% 0.25/0.55 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 0.25/0.55 @ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 0.25/0.55 @ X3 ) )
% 0.25/0.55 @ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Atom_def
% 0.25/0.55 thf(fact_5954_nth__item_Opsimps_I1_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( countable @ A )
% 0.25/0.55 => ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
% 0.25/0.55 => ( ( nth_item @ A @ ( zero_zero @ nat ) )
% 0.25/0.55 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % nth_item.psimps(1)
% 0.25/0.55 thf(fact_5955_nth__item_Opelims,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( countable @ A )
% 0.25/0.55 => ! [X: nat,Y: set @ ( old_node @ A @ product_unit )] :
% 0.25/0.55 ( ( ( nth_item @ A @ X )
% 0.25/0.55 = Y )
% 0.25/0.55 => ( ( accp @ nat @ nth_item_rel @ X )
% 0.25/0.55 => ( ( ( X
% 0.25/0.55 = ( zero_zero @ nat ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
% 0.25/0.55 => ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
% 0.25/0.55 => ~ ! [N3: nat] :
% 0.25/0.55 ( ( X
% 0.25/0.55 = ( suc @ N3 ) )
% 0.25/0.55 => ( ( Y
% 0.25/0.55 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.25/0.55 @ ^ [I3: nat] :
% 0.25/0.55 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.25/0.55 @ ^ [J2: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 0.25/0.55 @ ^ [J2: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 0.25/0.55 @ ( nat_sum_decode @ I3 ) )
% 0.25/0.55 @ ^ [I3: nat] :
% 0.25/0.55 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.25/0.55 @ ^ [J2: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J2 ) )
% 0.25/0.55 @ ^ [J2: nat] :
% 0.25/0.55 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 0.25/0.55 @ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
% 0.25/0.55 @ ( nat_prod_decode @ J2 ) )
% 0.25/0.55 @ ( nat_sum_decode @ I3 ) )
% 0.25/0.55 @ ( nat_sum_decode @ N3 ) ) )
% 0.25/0.55 => ~ ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % nth_item.pelims
% 0.25/0.55 thf(fact_5956_nth__item_Opsimps_I2_J,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( countable @ A )
% 0.25/0.55 => ! [N: nat] :
% 0.25/0.55 ( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
% 0.25/0.55 => ( ( nth_item @ A @ ( suc @ N ) )
% 0.25/0.55 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.25/0.55 @ ^ [I3: nat] :
% 0.25/0.55 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.25/0.55 @ ^ [J2: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 0.25/0.55 @ ^ [J2: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 0.25/0.55 @ ( nat_sum_decode @ I3 ) )
% 0.25/0.55 @ ^ [I3: nat] :
% 0.25/0.55 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 0.25/0.55 @ ^ [J2: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J2 ) )
% 0.25/0.55 @ ^ [J2: nat] :
% 0.25/0.55 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 0.25/0.55 @ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
% 0.25/0.55 @ ( nat_prod_decode @ J2 ) )
% 0.25/0.55 @ ( nat_sum_decode @ I3 ) )
% 0.25/0.55 @ ( nat_sum_decode @ N ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % nth_item.psimps(2)
% 0.25/0.55 thf(fact_5957_Scons__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( old_Scons @ A @ B )
% 0.25/0.55 = ( ^ [M8: set @ ( old_node @ A @ B ),N9: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M8 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N9 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Scons_def
% 0.25/0.55 thf(fact_5958_In1__def,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType] :
% 0.25/0.55 ( ( old_In1 @ A @ B )
% 0.25/0.55 = ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % In1_def
% 0.25/0.55 thf(fact_5959_dsum__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( old_dsum @ A @ B )
% 0.25/0.55 = ( ^ [R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( sup_sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X3 ) @ ( old_In0 @ A @ B @ X9 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
% 0.25/0.55 @ R3 ) )
% 0.25/0.55 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y3 ) @ ( old_In1 @ A @ B @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
% 0.25/0.55 @ S4 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dsum_def
% 0.25/0.55 thf(fact_5960_dsumE,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,W: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ W @ ( old_dsum @ A @ B @ R @ S ) )
% 0.25/0.55 => ( ! [X2: set @ ( old_node @ A @ B ),X5: set @ ( old_node @ A @ B )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X5 ) @ R )
% 0.25/0.55 => ( W
% 0.25/0.55 != ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X2 ) @ ( old_In0 @ A @ B @ X5 ) ) ) )
% 0.25/0.55 => ~ ! [Y4: set @ ( old_node @ A @ B ),Y14: set @ ( old_node @ A @ B )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y14 ) @ S )
% 0.25/0.55 => ( W
% 0.25/0.55 != ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y4 ) @ ( old_In1 @ A @ B @ Y14 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dsumE
% 0.25/0.55 thf(fact_5961_dsum__In0I,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,M3: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M3 @ M10 ) @ R )
% 0.25/0.55 => ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ M3 ) @ ( old_In0 @ A @ B @ M10 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dsum_In0I
% 0.25/0.55 thf(fact_5962_dsum__In1I,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,N6: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N11 ) @ S )
% 0.25/0.55 => ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ N6 ) @ ( old_In1 @ A @ B @ N11 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dsum_In1I
% 0.25/0.55 thf(fact_5963_dprod__def,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType] :
% 0.25/0.55 ( ( old_dprod @ A @ B )
% 0.25/0.55 = ( ^ [R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] :
% 0.25/0.55 ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 0.25/0.55 @ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X3 @ Y3 ) @ ( old_Scons @ A @ B @ X9 @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
% 0.25/0.55 @ S4 ) ) )
% 0.25/0.55 @ R3 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dprod_def
% 0.25/0.55 thf(fact_5964_dprodI,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,M3: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),N6: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M3 @ M10 ) @ R )
% 0.25/0.55 => ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N11 ) @ S )
% 0.25/0.55 => ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ M3 @ N6 ) @ ( old_Scons @ A @ B @ M10 @ N11 ) ) @ ( old_dprod @ A @ B @ R @ S ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dprodI
% 0.25/0.55 thf(fact_5965_dprodE,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,C2: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ C2 @ ( old_dprod @ A @ B @ R @ S ) )
% 0.25/0.55 => ~ ! [X2: set @ ( old_node @ A @ B ),Y4: set @ ( old_node @ A @ B ),X5: set @ ( old_node @ A @ B )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X5 ) @ R )
% 0.25/0.55 => ! [Y14: set @ ( old_node @ A @ B )] :
% 0.25/0.55 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y14 ) @ S )
% 0.25/0.55 => ( C2
% 0.25/0.55 != ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y4 ) @ ( old_Scons @ A @ B @ X5 @ Y14 ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % dprodE
% 0.25/0.55 thf(fact_5966_prop__matchD,axiom,
% 0.25/0.55 ! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,E6: A,Bl2: list @ A,P: A > $o] :
% 0.25/0.55 ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
% 0.25/0.55 = ( append @ A @ Al2 @ ( cons @ A @ E6 @ Bl2 ) ) )
% 0.25/0.55 => ( ( list_all @ A @ P @ Al )
% 0.25/0.55 => ( ~ ( P @ E2 )
% 0.25/0.55 => ( ~ ( P @ E6 )
% 0.25/0.55 => ( ( list_all @ A @ P @ Bl )
% 0.25/0.55 => ( ( Al = Al2 )
% 0.25/0.55 & ( E2 = E6 )
% 0.25/0.55 & ( Bl = Bl2 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % prop_matchD
% 0.25/0.55 thf(fact_5967_prop__match,axiom,
% 0.25/0.55 ! [A: $tType,P: A > $o,Al: list @ A,E2: A,E6: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
% 0.25/0.55 ( ( list_all @ A @ P @ Al )
% 0.25/0.55 => ( ~ ( P @ E2 )
% 0.25/0.55 => ( ~ ( P @ E6 )
% 0.25/0.55 => ( ( list_all @ A @ P @ Bl )
% 0.25/0.55 => ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
% 0.25/0.55 = ( append @ A @ Al2 @ ( cons @ A @ E6 @ Bl2 ) ) )
% 0.25/0.55 = ( ( Al = Al2 )
% 0.25/0.55 & ( E2 = E6 )
% 0.25/0.55 & ( Bl = Bl2 ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % prop_match
% 0.25/0.55 thf(fact_5968_Heap__lub__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap_Time_Heap_lub @ A )
% 0.25/0.55 = ( partial_img_lub @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_execute @ A ) @ ( heap_Time_Heap2 @ A ) @ ( partial_fun_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Heap_lub_def
% 0.25/0.55 thf(fact_5969_heap__step__admissible,axiom,
% 0.25/0.55 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: E > A > C > B > D > $o,X: E] :
% 0.25/0.55 ( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) )
% 0.25/0.55 @ ^ [F2: A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) )] :
% 0.25/0.55 ! [H: A,H7: C,R3: B,N5: D] :
% 0.25/0.55 ( ( ( F2 @ H )
% 0.25/0.55 = ( some @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ R3 @ ( product_Pair @ C @ D @ H7 @ N5 ) ) ) )
% 0.25/0.55 => ( P @ X @ H @ H7 @ R3 @ N5 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % heap_step_admissible
% 0.25/0.55 thf(fact_5970_option__admissible,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,P: A > B > $o] :
% 0.25/0.55 ( comple1908693960933563346ssible @ ( A > ( option @ B ) ) @ ( partial_fun_lub @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_lub @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) )
% 0.25/0.55 @ ^ [F2: A > ( option @ B )] :
% 0.25/0.55 ! [X3: A,Y3: B] :
% 0.25/0.55 ( ( ( F2 @ X3 )
% 0.25/0.55 = ( some @ B @ Y3 ) )
% 0.25/0.55 => ( P @ X3 @ Y3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option_admissible
% 0.25/0.55 thf(fact_5971_option_Olub__upper,axiom,
% 0.25/0.55 ! [A: $tType,A4: set @ ( option @ A ),X: option @ A] :
% 0.25/0.55 ( ( comple1602240252501008431_chain @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ A4 )
% 0.25/0.55 => ( ( member @ ( option @ A ) @ X @ A4 )
% 0.25/0.55 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) @ A4 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.lub_upper
% 0.25/0.55 thf(fact_5972_option_Oleq__antisym,axiom,
% 0.25/0.55 ! [A: $tType,X: option @ A,Y: option @ A] :
% 0.25/0.55 ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Y )
% 0.25/0.55 => ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y @ X )
% 0.25/0.55 => ( X = Y ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.leq_antisym
% 0.25/0.55 thf(fact_5973_option_Oleq__trans,axiom,
% 0.25/0.55 ! [A: $tType,X: option @ A,Y: option @ A,Z2: option @ A] :
% 0.25/0.55 ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Y )
% 0.25/0.55 => ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y @ Z2 )
% 0.25/0.55 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Z2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.leq_trans
% 0.25/0.55 thf(fact_5974_option_Oleq__refl,axiom,
% 0.25/0.55 ! [A: $tType,X: option @ A] : ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ X ) ).
% 0.25/0.55
% 0.25/0.55 % option.leq_refl
% 0.25/0.55 thf(fact_5975_option_Olub__least,axiom,
% 0.25/0.55 ! [A: $tType,A4: set @ ( option @ A ),Z2: option @ A] :
% 0.25/0.55 ( ( comple1602240252501008431_chain @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ A4 )
% 0.25/0.55 => ( ! [X2: option @ A] :
% 0.25/0.55 ( ( member @ ( option @ A ) @ X2 @ A4 )
% 0.25/0.55 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ Z2 ) )
% 0.25/0.55 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) @ A4 ) @ Z2 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.lub_least
% 0.25/0.55 thf(fact_5976_Heap__ord__def,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap_Time_Heap_ord @ A )
% 0.25/0.55 = ( partial_img_ord @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ $o @ ( heap_Time_execute @ A ) @ ( partial_fun_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Heap_ord_def
% 0.25/0.55 thf(fact_5977_Partial__Function_Obind__mono,axiom,
% 0.25/0.55 ! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( option @ B ) ) > ( option @ C ),C5: C > ( A > ( option @ B ) ) > ( option @ D )] :
% 0.25/0.55 ( ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ C ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ C ) @ ( none @ C ) ) @ B6 )
% 0.25/0.55 => ( ! [Y4: C] : ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ D ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ D ) @ ( none @ D ) ) @ ( C5 @ Y4 ) )
% 0.25/0.55 => ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ D ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ D ) @ ( none @ D ) )
% 0.25/0.55 @ ^ [F2: A > ( option @ B )] :
% 0.25/0.55 ( bind @ C @ D @ ( B6 @ F2 )
% 0.25/0.55 @ ^ [Y3: C] : ( C5 @ Y3 @ F2 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Partial_Function.bind_mono
% 0.25/0.55 thf(fact_5978_option_Oconst__mono,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,Ord: B > B > $o,C2: option @ A] :
% 0.25/0.55 ( comple7038119648293358887notone @ B @ ( option @ A ) @ Ord @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 0.25/0.55 @ ^ [F2: B] : C2 ) ).
% 0.25/0.55
% 0.25/0.55 % option.const_mono
% 0.25/0.55 thf(fact_5979_option_Ofixp__induct__uc,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: ( B > ( option @ A ) ) > $o] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 0.25/0.55 => ( ( F
% 0.25/0.55 = ( C5
% 0.25/0.55 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 0.25/0.55 => ( ! [F4: B > ( option @ A )] :
% 0.25/0.55 ( ( U4 @ ( C5 @ F4 ) )
% 0.25/0.55 = F4 )
% 0.25/0.55 => ( ( comple1908693960933563346ssible @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ P )
% 0.25/0.55 => ( ( P
% 0.25/0.55 @ ^ [Uu: B] : ( none @ A ) )
% 0.25/0.55 => ( ! [F4: C] :
% 0.25/0.55 ( ( P @ ( U4 @ F4 ) )
% 0.25/0.55 => ( P @ ( U4 @ ( F6 @ F4 ) ) ) )
% 0.25/0.55 => ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.fixp_induct_uc
% 0.25/0.55 thf(fact_5980_fixp__induct__option,axiom,
% 0.25/0.55 ! [C: $tType,B: $tType,A: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: B > A > $o,X: B,Y: A] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 0.25/0.55 => ( ( F
% 0.25/0.55 = ( C5
% 0.25/0.55 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 0.25/0.55 => ( ! [F4: B > ( option @ A )] :
% 0.25/0.55 ( ( U4 @ ( C5 @ F4 ) )
% 0.25/0.55 = F4 )
% 0.25/0.55 => ( ! [F4: C,X2: B,Y4: A] :
% 0.25/0.55 ( ! [Xa2: B,Ya3: A] :
% 0.25/0.55 ( ( ( U4 @ F4 @ Xa2 )
% 0.25/0.55 = ( some @ A @ Ya3 ) )
% 0.25/0.55 => ( P @ Xa2 @ Ya3 ) )
% 0.25/0.55 => ( ( ( U4 @ ( F6 @ F4 ) @ X2 )
% 0.25/0.55 = ( some @ A @ Y4 ) )
% 0.25/0.55 => ( P @ X2 @ Y4 ) ) )
% 0.25/0.55 => ( ( ( U4 @ F @ X )
% 0.25/0.55 = ( some @ A @ Y ) )
% 0.25/0.55 => ( P @ X @ Y ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fixp_induct_option
% 0.25/0.55 thf(fact_5981_option_Ofixp__rule__uc,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 0.25/0.55 => ( ( F
% 0.25/0.55 = ( C5
% 0.25/0.55 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 0.25/0.55 => ( ! [F4: C] :
% 0.25/0.55 ( ( C5 @ ( U4 @ F4 ) )
% 0.25/0.55 = F4 )
% 0.25/0.55 => ( F
% 0.25/0.55 = ( F6 @ F ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.fixp_rule_uc
% 0.25/0.55 thf(fact_5982_option_Omono__body__fixp,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,F6: ( B > ( option @ A ) ) > B > ( option @ A )] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( option @ A )] : ( F6 @ F2 @ X2 ) )
% 0.25/0.55 => ( ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F6 )
% 0.25/0.55 = ( F6 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F6 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.mono_body_fixp
% 0.25/0.55 thf(fact_5983_option_Opartial__function__definitions__axioms,axiom,
% 0.25/0.55 ! [A: $tType] : ( partia7178651479351089652itions @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % option.partial_function_definitions_axioms
% 0.25/0.55 thf(fact_5984_fixp__mono,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( comple9053668089753744459l_ccpo @ A )
% 0.25/0.55 => ! [F: A > A,G: A > A] :
% 0.25/0.55 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 0.25/0.55 => ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
% 0.25/0.55 => ( ! [Z11: A] : ( ord_less_eq @ A @ ( F @ Z11 ) @ ( G @ Z11 ) )
% 0.25/0.55 => ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple115746919287870866o_fixp @ A @ G ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fixp_mono
% 0.25/0.55 thf(fact_5985_heap_Oconst__mono,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
% 0.25/0.55 ( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
% 0.25/0.55 @ ^ [F2: B] : C2 ) ).
% 0.25/0.55
% 0.25/0.55 % heap.const_mono
% 0.25/0.55 thf(fact_5986_Heap__Time__Monad_Obind__mono,axiom,
% 0.25/0.55 ! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C5: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
% 0.25/0.55 ( ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ C ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ C ) @ B6 )
% 0.25/0.55 => ( ! [Y4: C] : ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D ) @ ( C5 @ Y4 ) )
% 0.25/0.55 => ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D )
% 0.25/0.55 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 0.25/0.55 ( heap_Time_bind @ C @ D @ ( B6 @ F2 )
% 0.25/0.55 @ ^ [Y3: C] : ( C5 @ Y3 @ F2 ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % Heap_Time_Monad.bind_mono
% 0.25/0.55 thf(fact_5987_fixp__induct__heap,axiom,
% 0.25/0.55 ! [C: $tType,A: $tType,B: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: B > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o,X: B,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 0.25/0.55 => ( ( F
% 0.25/0.55 = ( C5
% 0.25/0.55 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 0.25/0.55 => ( ! [F4: B > ( heap_Time_Heap @ A )] :
% 0.25/0.55 ( ( U4 @ ( C5 @ F4 ) )
% 0.25/0.55 = F4 )
% 0.25/0.55 => ( ! [F4: C,X2: B,H2: heap_ext @ product_unit,H6: heap_ext @ product_unit,R2: A,N3: nat] :
% 0.25/0.55 ( ! [Xa2: B,Ha: heap_ext @ product_unit,H_a: heap_ext @ product_unit,Ra3: A,Na: nat] :
% 0.25/0.55 ( ( heap_Time_effect @ A @ ( U4 @ F4 @ Xa2 ) @ Ha @ H_a @ Ra3 @ Na )
% 0.25/0.55 => ( P @ Xa2 @ Ha @ H_a @ Ra3 @ Na ) )
% 0.25/0.55 => ( ( heap_Time_effect @ A @ ( U4 @ ( F6 @ F4 ) @ X2 ) @ H2 @ H6 @ R2 @ N3 )
% 0.25/0.55 => ( P @ X2 @ H2 @ H6 @ R2 @ N3 ) ) )
% 0.25/0.55 => ( ( heap_Time_effect @ A @ ( U4 @ F @ X ) @ H3 @ H4 @ R @ N )
% 0.25/0.55 => ( P @ X @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % fixp_induct_heap
% 0.25/0.55 thf(fact_5988_refines__adm,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,T2: A > ( heap_Time_Heap @ B )] :
% 0.25/0.55 ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
% 0.25/0.55 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 0.25/0.55 ! [X3: A] : ( refine_Imp_refines @ B @ ( T2 @ X3 ) @ ( F2 @ X3 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % refines_adm
% 0.25/0.55 thf(fact_5989_heap_Ofixp__induct__uc,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 0.25/0.55 => ( ( F
% 0.25/0.55 = ( C5
% 0.25/0.55 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 0.25/0.55 => ( ! [F4: B > ( heap_Time_Heap @ A )] :
% 0.25/0.55 ( ( U4 @ ( C5 @ F4 ) )
% 0.25/0.55 = F4 )
% 0.25/0.55 => ( ( comple1908693960933563346ssible @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ P )
% 0.25/0.55 => ( ( P
% 0.25/0.55 @ ^ [Uu: B] :
% 0.25/0.55 ( heap_Time_Heap2 @ A
% 0.25/0.55 @ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
% 0.25/0.55 => ( ! [F4: C] :
% 0.25/0.55 ( ( P @ ( U4 @ F4 ) )
% 0.25/0.55 => ( P @ ( U4 @ ( F6 @ F4 ) ) ) )
% 0.25/0.55 => ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % heap.fixp_induct_uc
% 0.25/0.55 thf(fact_5990_heap_Omono__body__fixp,axiom,
% 0.25/0.55 ! [A: $tType,B: $tType,F6: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( F6 @ F2 @ X2 ) )
% 0.25/0.55 => ( ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F6 )
% 0.25/0.55 = ( F6 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F6 ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % heap.mono_body_fixp
% 0.25/0.55 thf(fact_5991_heap_Ofixp__rule__uc,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
% 0.25/0.55 ( ! [X2: B] :
% 0.25/0.55 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 0.25/0.55 => ( ( F
% 0.25/0.55 = ( C5
% 0.25/0.55 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
% 0.25/0.55 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 0.25/0.55 => ( ! [F4: C] :
% 0.25/0.55 ( ( C5 @ ( U4 @ F4 ) )
% 0.25/0.55 = F4 )
% 0.25/0.55 => ( F
% 0.25/0.55 = ( F6 @ F ) ) ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % heap.fixp_rule_uc
% 0.25/0.55 thf(fact_5992_admissible__heap,axiom,
% 0.25/0.55 ! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
% 0.25/0.55 ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
% 0.25/0.55 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 0.25/0.55 ! [X3: A,H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R3: B,N5: nat] :
% 0.25/0.55 ( ( heap_Time_effect @ B @ ( F2 @ X3 ) @ H @ H7 @ R3 @ N5 )
% 0.25/0.55 => ( P @ X3 @ H @ H7 @ R3 @ N5 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 % admissible_heap
% 0.25/0.55
% 0.25/0.55 % Subclasses (3)
% 0.25/0.55 thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( typerep2 @ A ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(subcl_Heap_Oheap___HOL_Otype,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( type @ A ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
% 0.25/0.55 ! [A: $tType] :
% 0.25/0.55 ( ( heap @ A )
% 0.25/0.55 => ( countable @ A ) ) ).
% 0.25/0.55
% 0.25/0.55 % Type constructors (697)
% 0.25/0.55 thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( heap_Time_Heap @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( heap_Time_Heap @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
% 0.25/0.55 code_term_of @ code_natural ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
% 0.25/0.55 typerep2 @ code_natural ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
% 0.25/0.55 code_term_of @ code_integer ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
% 0.25/0.55 typerep2 @ code_integer ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
% 0.25/0.55 code_term_of @ code_term ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
% 0.25/0.55 typerep2 @ code_term ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( heap_ext @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( heap_ext @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
% 0.25/0.55 code_term_of @ product_unit ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
% 0.25/0.55 enum @ product_unit ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
% 0.25/0.55 typerep2 @ product_unit ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
% 0.25/0.55 bounded_lattice_top @ product_unit ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( typerep2 @ A20 )
% 0.25/0.55 & ( typerep2 @ A21 ) )
% 0.25/0.55 => ( code_term_of @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( enum @ A20 )
% 0.25/0.55 & ( enum @ A21 ) )
% 0.25/0.55 => ( enum @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( typerep2 @ A20 )
% 0.25/0.55 & ( typerep2 @ A21 ) )
% 0.25/0.55 => ( typerep2 @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_14,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( typerep2 @ A20 )
% 0.25/0.55 & ( typerep2 @ A21 ) )
% 0.25/0.55 => ( typerep2 @ ( old_node @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( multiset @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Multiset_Omultiset___Typerep_Otyperep_16,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( multiset @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17,axiom,
% 0.25/0.55 code_term_of @ typerep ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Typerep_Otyperep___Typerep_Otyperep_18,axiom,
% 0.25/0.55 typerep2 @ typerep ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
% 0.25/0.55 typerep2 @ assn ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_20,axiom,
% 0.25/0.55 bounded_lattice_top @ assn ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_21,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( pred @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Predicate_Opred___Typerep_Otyperep_22,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( pred @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_23,axiom,
% 0.25/0.55 ! [A20: $tType] : ( bounded_lattice_top @ ( pred @ A20 ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_24,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( seq @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Predicate_Oseq___Typerep_Otyperep_25,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( seq @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_26,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( option @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Option_Ooption___Enum_Oenum_27,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( enum @ A20 )
% 0.25/0.55 => ( enum @ ( option @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Option_Ooption___Typerep_Otyperep_28,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( option @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_29,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( bounded_lattice_top @ A20 )
% 0.25/0.55 => ( bounded_lattice_top @ ( option @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_30,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( typerep2 @ A20 )
% 0.25/0.55 & ( typerep2 @ A21 ) )
% 0.25/0.55 => ( code_term_of @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Sum__Type_Osum___Enum_Oenum_31,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( enum @ A20 )
% 0.25/0.55 & ( enum @ A21 ) )
% 0.25/0.55 => ( enum @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Sum__Type_Osum___Typerep_Otyperep_32,axiom,
% 0.25/0.55 ! [A20: $tType,A21: $tType] :
% 0.25/0.55 ( ( ( typerep2 @ A20 )
% 0.25/0.55 & ( typerep2 @ A21 ) )
% 0.25/0.55 => ( typerep2 @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_33,axiom,
% 0.25/0.55 code_term_of @ char ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_String_Ochar___Enum_Oenum_34,axiom,
% 0.25/0.55 enum @ char ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_String_Ochar___Typerep_Otyperep_35,axiom,
% 0.25/0.55 typerep2 @ char ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_36,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( array @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap_Oarray___Typerep_Otyperep_37,axiom,
% 0.25/0.55 ! [A20: $tType] : ( typerep2 @ ( array @ A20 ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_List_Olist___Code__Evaluation_Oterm__of_38,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( list @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_List_Olist___Typerep_Otyperep_39,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( typerep2 @ ( list @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_40,axiom,
% 0.25/0.55 ! [A20: $tType] :
% 0.25/0.55 ( ( typerep2 @ A20 )
% 0.25/0.55 => ( code_term_of @ ( ref @ A20 ) ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_Heap_Oref___Typerep_Otyperep_41,axiom,
% 0.25/0.55 ! [A20: $tType] : ( typerep2 @ ( ref @ A20 ) ) ).
% 0.25/0.55
% 0.25/0.55 thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_42,axiom,
% 0.25/0.56 code_term_of @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Enum_Oenum_43,axiom,
% 0.25/0.56 enum @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Typerep_Otyperep_44,axiom,
% 0.25/0.56 typerep2 @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_45,axiom,
% 0.25/0.56 bounded_lattice_top @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_46,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( code_term_of @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Enum_Oenum_47,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( enum @ A20 )
% 0.25/0.56 => ( enum @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Typerep_Otyperep_48,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( typerep2 @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_49,axiom,
% 0.25/0.56 ! [A20: $tType] : ( bounded_lattice_top @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_50,axiom,
% 0.25/0.56 code_term_of @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Typerep_Otyperep_51,axiom,
% 0.25/0.56 typerep2 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_52,axiom,
% 0.25/0.56 code_term_of @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Typerep_Otyperep_53,axiom,
% 0.25/0.56 typerep2 @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_54,axiom,
% 0.25/0.56 code_term_of @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Typerep_Otyperep_55,axiom,
% 0.25/0.56 typerep2 @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_56,axiom,
% 0.25/0.56 code_term_of @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Typerep_Otyperep_57,axiom,
% 0.25/0.56 typerep2 @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_itself___Code__Evaluation_Oterm__of_58,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( code_term_of @ ( itself @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_itself___Typerep_Otyperep_59,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( typerep2 @ ( itself @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Code__Evaluation_Oterm__of_60,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( typerep2 @ A20 )
% 0.25/0.56 & ( typerep2 @ A21 ) )
% 0.25/0.56 => ( code_term_of @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Enum_Oenum_61,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( enum @ A20 )
% 0.25/0.56 & ( enum @ A21 ) )
% 0.25/0.56 => ( enum @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Typerep_Otyperep_62,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( typerep2 @ A20 )
% 0.25/0.56 & ( typerep2 @ A21 ) )
% 0.25/0.56 => ( typerep2 @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Lattices_Obounded__lattice__top_63,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( bounded_lattice @ A21 )
% 0.25/0.56 => ( bounded_lattice_top @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( cl_HOL_Oequal @ A20 )
% 0.25/0.56 & ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 & ( quickc3360725361186068524ustive @ A21 ) )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( comple6319245703460814977attice @ A21 )
% 0.25/0.56 => ( comple6319245703460814977attice @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( cl_HOL_Oequal @ A20 )
% 0.25/0.56 & ( quickc658316121487927005ustive @ A20 )
% 0.25/0.56 & ( quickc658316121487927005ustive @ A21 ) )
% 0.25/0.56 => ( quickc658316121487927005ustive @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( boolea8198339166811842893lgebra @ A21 )
% 0.25/0.56 => ( boolea8198339166811842893lgebra @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( comple6319245703460814977attice @ A21 )
% 0.25/0.56 => ( comple9053668089753744459l_ccpo @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( code_term_of @ A20 )
% 0.25/0.56 & ( cl_HOL_Oequal @ A20 )
% 0.25/0.56 & ( quickcheck_random @ A21 ) )
% 0.25/0.56 => ( quickcheck_random @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( semilattice_sup @ A21 )
% 0.25/0.56 => ( semilattice_sup @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( semilattice_inf @ A21 )
% 0.25/0.56 => ( semilattice_inf @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Lattices_Obounded__lattice,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( bounded_lattice @ A21 )
% 0.25/0.56 => ( bounded_lattice @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( complete_Sup @ A21 )
% 0.25/0.56 => ( complete_Sup @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Complete__Lattices_OInf,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( complete_Inf @ A21 )
% 0.25/0.56 => ( complete_Inf @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( order_top @ A21 )
% 0.25/0.56 => ( order_top @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( order_bot @ A21 )
% 0.25/0.56 => ( order_bot @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Countable_Ocountable,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( finite_finite @ A20 )
% 0.25/0.56 & ( countable @ A21 ) )
% 0.25/0.56 => ( countable @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( preorder @ A21 )
% 0.25/0.56 => ( preorder @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( finite_finite @ A20 )
% 0.25/0.56 & ( finite_finite @ A21 ) )
% 0.25/0.56 => ( finite_finite @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( order @ A21 )
% 0.25/0.56 => ( order @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Orderings_Oord,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ord @ A21 )
% 0.25/0.56 => ( ord @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( uminus @ A21 )
% 0.25/0.56 => ( uminus @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Lattices_Osup,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( semilattice_sup @ A21 )
% 0.25/0.56 => ( sup @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___Lattices_Oinf,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( semilattice_inf @ A21 )
% 0.25/0.56 => ( inf @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_fun___HOL_Oequal,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( enum @ A20 )
% 0.25/0.56 & ( cl_HOL_Oequal @ A21 ) )
% 0.25/0.56 => ( cl_HOL_Oequal @ ( A20 > A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_itself___Quickcheck__Random_Orandom_64,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( itself @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_itself___HOL_Oequal_65,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( itself @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
% 0.25/0.56 bit_un5681908812861735899ations @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 0.25/0.56 semiri1453513574482234551roduct @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
% 0.25/0.56 euclid5411537665997757685th_nat @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
% 0.25/0.56 euclid8789492081693882211th_nat @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 0.25/0.56 ordere1937475149494474687imp_le @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
% 0.25/0.56 euclid3128863361964157862miring @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
% 0.25/0.56 euclid4440199948858584721cancel @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 0.25/0.56 unique1627219031080169319umeral @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
% 0.25/0.56 euclid8851590272496341667cancel @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 0.25/0.56 strict9044650504122735259up_add @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 0.25/0.56 ordere580206878836729694up_add @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 0.25/0.56 ordere2412721322843649153imp_le @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
% 0.25/0.56 bit_se359711467146920520ations @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_66,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 0.25/0.56 strict7427464778891057005id_add @ int ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 0.25/0.56 ordere8940638589300402666id_add @ int ).
% 0.25/0.56
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% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Parity_Osemiring__parity_117,axiom,
% 0.25/0.56 semiring_parity @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_118,axiom,
% 0.25/0.56 comm_monoid_add @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_119,axiom,
% 0.25/0.56 semiring_modulo @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_120,axiom,
% 0.25/0.56 comm_semiring_1 @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_121,axiom,
% 0.25/0.56 comm_semiring_0 @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_122,axiom,
% 0.25/0.56 semigroup_mult @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Complete__Lattices_OSup_123,axiom,
% 0.25/0.56 complete_Sup @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Complete__Lattices_OInf_124,axiom,
% 0.25/0.56 complete_Inf @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_125,axiom,
% 0.25/0.56 semidom_modulo @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Osemidom__divide_126,axiom,
% 0.25/0.56 semidom_divide @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Num_Osemiring__numeral_127,axiom,
% 0.25/0.56 semiring_numeral @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Osemigroup__add_128,axiom,
% 0.25/0.56 semigroup_add @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Ozero__less__one_129,axiom,
% 0.25/0.56 zero_less_one @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_130,axiom,
% 0.25/0.56 comm_semiring @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Orderings_Oorder__bot_131,axiom,
% 0.25/0.56 order_bot @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_132,axiom,
% 0.25/0.56 semiring_char_0 @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Countable_Ocountable_133,axiom,
% 0.25/0.56 countable @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_134,axiom,
% 0.25/0.56 zero_neq_one @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Orderings_Opreorder_135,axiom,
% 0.25/0.56 preorder @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Orderings_Olinorder_136,axiom,
% 0.25/0.56 linorder @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Omonoid__mult_137,axiom,
% 0.25/0.56 monoid_mult @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Omonoid__add_138,axiom,
% 0.25/0.56 monoid_add @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Osemiring__1_139,axiom,
% 0.25/0.56 semiring_1 @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Osemiring__0_140,axiom,
% 0.25/0.56 semiring_0 @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Orderings_Ono__top_141,axiom,
% 0.25/0.56 no_top @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_142,axiom,
% 0.25/0.56 semiring_gcd @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_143,axiom,
% 0.25/0.56 semiring_Gcd @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Orderings_Oorder_144,axiom,
% 0.25/0.56 order @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Osemiring_145,axiom,
% 0.25/0.56 semiring @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Orderings_Oord_146,axiom,
% 0.25/0.56 ord @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Lattices_Osup_147,axiom,
% 0.25/0.56 sup @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Lattices_Oinf_148,axiom,
% 0.25/0.56 inf @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Power_Opower_149,axiom,
% 0.25/0.56 power @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Num_Onumeral_150,axiom,
% 0.25/0.56 numeral @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Ozero_151,axiom,
% 0.25/0.56 zero @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Oplus_152,axiom,
% 0.25/0.56 plus @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Groups_Oone_153,axiom,
% 0.25/0.56 one @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Rings_Odvd_154,axiom,
% 0.25/0.56 dvd @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___Heap_Oheap_155,axiom,
% 0.25/0.56 heap @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Nat_Onat___HOL_Oequal_156,axiom,
% 0.25/0.56 cl_HOL_Oequal @ nat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_157,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Quickcheck__Random_Orandom_158,axiom,
% 0.25/0.56 quickcheck_random @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Orderings_Opreorder_159,axiom,
% 0.25/0.56 preorder @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Orderings_Olinorder_160,axiom,
% 0.25/0.56 linorder @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Orderings_Oorder_161,axiom,
% 0.25/0.56 order @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Orderings_Oord_162,axiom,
% 0.25/0.56 ord @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___Groups_Oplus_163,axiom,
% 0.25/0.56 plus @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Num_Onum___HOL_Oequal_164,axiom,
% 0.25/0.56 cl_HOL_Oequal @ num ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_165,axiom,
% 0.25/0.56 semiri1453513574482234551roduct @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_166,axiom,
% 0.25/0.56 ordere1937475149494474687imp_le @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167,axiom,
% 0.25/0.56 strict9044650504122735259up_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168,axiom,
% 0.25/0.56 ordere580206878836729694up_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169,axiom,
% 0.25/0.56 ordere2412721322843649153imp_le @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_170,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171,axiom,
% 0.25/0.56 strict7427464778891057005id_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172,axiom,
% 0.25/0.56 ordere8940638589300402666id_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
% 0.25/0.56 archim462609752435547400_field @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_173,axiom,
% 0.25/0.56 linord715952674999750819strict @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_174,axiom,
% 0.25/0.56 linord4140545234300271783up_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_175,axiom,
% 0.25/0.56 linord181362715937106298miring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_176,axiom,
% 0.25/0.56 quickc658316121487927005ustive @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_177,axiom,
% 0.25/0.56 semiri3467727345109120633visors @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_178,axiom,
% 0.25/0.56 ordere6658533253407199908up_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_179,axiom,
% 0.25/0.56 ordere166539214618696060dd_abs @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
% 0.25/0.56 archim2362893244070406136eiling @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_180,axiom,
% 0.25/0.56 ordere6911136660526730532id_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_181,axiom,
% 0.25/0.56 linord5086331880401160121up_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_182,axiom,
% 0.25/0.56 cancel2418104881723323429up_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_183,axiom,
% 0.25/0.56 ring_15535105094025558882visors @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_184,axiom,
% 0.25/0.56 cancel1802427076303600483id_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_185,axiom,
% 0.25/0.56 linord4710134922213307826strict @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_186,axiom,
% 0.25/0.56 comm_s4317794764714335236cancel @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_187,axiom,
% 0.25/0.56 linord6961819062388156250ring_1 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_188,axiom,
% 0.25/0.56 ordered_ab_group_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_189,axiom,
% 0.25/0.56 cancel_semigroup_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_190,axiom,
% 0.25/0.56 linordered_semidom @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_191,axiom,
% 0.25/0.56 quickcheck_random @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
% 0.25/0.56 dense_linorder @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_192,axiom,
% 0.25/0.56 semilattice_sup @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_193,axiom,
% 0.25/0.56 semilattice_inf @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_194,axiom,
% 0.25/0.56 ab_semigroup_mult @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_195,axiom,
% 0.25/0.56 comm_monoid_mult @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_196,axiom,
% 0.25/0.56 ab_semigroup_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
% 0.25/0.56 linordered_field @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_197,axiom,
% 0.25/0.56 comm_monoid_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__ring_198,axiom,
% 0.25/0.56 linordered_ring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Olinordered__idom_199,axiom,
% 0.25/0.56 linordered_idom @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_200,axiom,
% 0.25/0.56 comm_semiring_1 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_201,axiom,
% 0.25/0.56 comm_semiring_0 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_202,axiom,
% 0.25/0.56 semigroup_mult @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Osemidom__divide_203,axiom,
% 0.25/0.56 semidom_divide @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Num_Osemiring__numeral_204,axiom,
% 0.25/0.56 semiring_numeral @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Osemigroup__add_205,axiom,
% 0.25/0.56 semigroup_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
% 0.25/0.56 division_ring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ozero__less__one_206,axiom,
% 0.25/0.56 zero_less_one @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_207,axiom,
% 0.25/0.56 comm_semiring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_208,axiom,
% 0.25/0.56 semiring_char_0 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oab__group__add_209,axiom,
% 0.25/0.56 ab_group_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
% 0.25/0.56 field_char_0 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Countable_Ocountable_210,axiom,
% 0.25/0.56 countable @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_211,axiom,
% 0.25/0.56 zero_neq_one @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Oordered__ring_212,axiom,
% 0.25/0.56 ordered_ring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_213,axiom,
% 0.25/0.56 idom_abs_sgn @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Opreorder_214,axiom,
% 0.25/0.56 preorder @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Olinorder_215,axiom,
% 0.25/0.56 linorder @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Omonoid__mult_216,axiom,
% 0.25/0.56 monoid_mult @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_217,axiom,
% 0.25/0.56 comm_ring_1 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Omonoid__add_218,axiom,
% 0.25/0.56 monoid_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Osemiring__1_219,axiom,
% 0.25/0.56 semiring_1 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Osemiring__0_220,axiom,
% 0.25/0.56 semiring_0 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Ono__top_221,axiom,
% 0.25/0.56 no_top @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Ono__bot_222,axiom,
% 0.25/0.56 no_bot @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ogroup__add_223,axiom,
% 0.25/0.56 group_add @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Ocomm__ring_224,axiom,
% 0.25/0.56 comm_ring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Oorder_225,axiom,
% 0.25/0.56 order @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Num_Oneg__numeral_226,axiom,
% 0.25/0.56 neg_numeral @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Nat_Oring__char__0_227,axiom,
% 0.25/0.56 ring_char_0 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Osemiring_228,axiom,
% 0.25/0.56 semiring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
% 0.25/0.56 inverse @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Orderings_Oord_229,axiom,
% 0.25/0.56 ord @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ouminus_230,axiom,
% 0.25/0.56 uminus @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Oring__1_231,axiom,
% 0.25/0.56 ring_1 @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Lattices_Osup_232,axiom,
% 0.25/0.56 sup @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Lattices_Oinf_233,axiom,
% 0.25/0.56 inf @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Fields_Ofield,axiom,
% 0.25/0.56 field @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Power_Opower_234,axiom,
% 0.25/0.56 power @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Num_Onumeral_235,axiom,
% 0.25/0.56 numeral @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Ozero_236,axiom,
% 0.25/0.56 zero @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oplus_237,axiom,
% 0.25/0.56 plus @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Oring_238,axiom,
% 0.25/0.56 ring @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Groups_Oone_239,axiom,
% 0.25/0.56 one @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___Rings_Odvd_240,axiom,
% 0.25/0.56 dvd @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Rat_Orat___HOL_Oequal_241,axiom,
% 0.25/0.56 cl_HOL_Oequal @ rat ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_242,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243,axiom,
% 0.25/0.56 ! [A20: $tType] : ( comple6319245703460814977attice @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_244,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickc658316121487927005ustive @ A20 )
% 0.25/0.56 => ( quickc658316121487927005ustive @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_245,axiom,
% 0.25/0.56 ! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_246,axiom,
% 0.25/0.56 ! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Quickcheck__Random_Orandom_247,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickcheck_random @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_248,axiom,
% 0.25/0.56 ! [A20: $tType] : ( semilattice_sup @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_249,axiom,
% 0.25/0.56 ! [A20: $tType] : ( semilattice_inf @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Lattices_Obounded__lattice_250,axiom,
% 0.25/0.56 ! [A20: $tType] : ( bounded_lattice @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Complete__Lattices_OSup_251,axiom,
% 0.25/0.56 ! [A20: $tType] : ( complete_Sup @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Complete__Lattices_OInf_252,axiom,
% 0.25/0.56 ! [A20: $tType] : ( complete_Inf @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Orderings_Oorder__top_253,axiom,
% 0.25/0.56 ! [A20: $tType] : ( order_top @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Orderings_Oorder__bot_254,axiom,
% 0.25/0.56 ! [A20: $tType] : ( order_bot @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Countable_Ocountable_255,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( finite_finite @ A20 )
% 0.25/0.56 => ( countable @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Orderings_Opreorder_256,axiom,
% 0.25/0.56 ! [A20: $tType] : ( preorder @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Finite__Set_Ofinite_257,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( finite_finite @ A20 )
% 0.25/0.56 => ( finite_finite @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Orderings_Oorder_258,axiom,
% 0.25/0.56 ! [A20: $tType] : ( order @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Orderings_Oord_259,axiom,
% 0.25/0.56 ! [A20: $tType] : ( ord @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Groups_Ouminus_260,axiom,
% 0.25/0.56 ! [A20: $tType] : ( uminus @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Lattices_Osup_261,axiom,
% 0.25/0.56 ! [A20: $tType] : ( sup @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___Lattices_Oinf_262,axiom,
% 0.25/0.56 ! [A20: $tType] : ( inf @ ( set @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Set_Oset___HOL_Oequal_263,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( cl_HOL_Oequal @ A20 )
% 0.25/0.56 => ( cl_HOL_Oequal @ ( set @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_265,axiom,
% 0.25/0.56 comple6319245703460814977attice @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_266,axiom,
% 0.25/0.56 boolea8198339166811842893lgebra @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_267,axiom,
% 0.25/0.56 comple9053668089753744459l_ccpo @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_268,axiom,
% 0.25/0.56 quickcheck_random @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_269,axiom,
% 0.25/0.56 semilattice_sup @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_270,axiom,
% 0.25/0.56 semilattice_inf @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_271,axiom,
% 0.25/0.56 bounded_lattice @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Complete__Lattices_OSup_272,axiom,
% 0.25/0.56 complete_Sup @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Complete__Lattices_OInf_273,axiom,
% 0.25/0.56 complete_Inf @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Orderings_Oorder__top_274,axiom,
% 0.25/0.56 order_top @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Orderings_Oorder__bot_275,axiom,
% 0.25/0.56 order_bot @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Countable_Ocountable_276,axiom,
% 0.25/0.56 countable @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Orderings_Opreorder_277,axiom,
% 0.25/0.56 preorder @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Orderings_Olinorder_278,axiom,
% 0.25/0.56 linorder @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Finite__Set_Ofinite_279,axiom,
% 0.25/0.56 finite_finite @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Orderings_Oorder_280,axiom,
% 0.25/0.56 order @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Orderings_Oord_281,axiom,
% 0.25/0.56 ord @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Groups_Ouminus_282,axiom,
% 0.25/0.56 uminus @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Lattices_Osup_283,axiom,
% 0.25/0.56 sup @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Lattices_Oinf_284,axiom,
% 0.25/0.56 inf @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___Heap_Oheap_285,axiom,
% 0.25/0.56 heap @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_HOL_Obool___HOL_Oequal_286,axiom,
% 0.25/0.56 cl_HOL_Oequal @ $o ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_287,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( ref @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_288,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( ref @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oref___Countable_Ocountable_289,axiom,
% 0.25/0.56 ! [A20: $tType] : ( countable @ ( ref @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oref___Heap_Oheap_290,axiom,
% 0.25/0.56 ! [A20: $tType] : ( heap @ ( ref @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oref___HOL_Oequal_291,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( ref @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_292,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( list @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_List_Olist___Quickcheck__Random_Orandom_293,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickcheck_random @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( list @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_List_Olist___Countable_Ocountable_294,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( countable @ A20 )
% 0.25/0.56 => ( countable @ ( list @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_List_Olist___Heap_Oheap_295,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( heap @ A20 )
% 0.25/0.56 => ( heap @ ( list @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_List_Olist___HOL_Oequal_296,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( list @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_297,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( array @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_298,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( typerep2 @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( array @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oarray___Countable_Ocountable_299,axiom,
% 0.25/0.56 ! [A20: $tType] : ( countable @ ( array @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oarray___Heap_Oheap_300,axiom,
% 0.25/0.56 ! [A20: $tType] : ( heap @ ( array @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oarray___HOL_Oequal_301,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( array @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_302,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ char ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_String_Ochar___Quickcheck__Random_Orandom_303,axiom,
% 0.25/0.56 quickcheck_random @ char ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_String_Ochar___Countable_Ocountable_304,axiom,
% 0.25/0.56 countable @ char ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_String_Ochar___Finite__Set_Ofinite_305,axiom,
% 0.25/0.56 finite_finite @ char ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_String_Ochar___Heap_Oheap_306,axiom,
% 0.25/0.56 heap @ char ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_String_Ochar___HOL_Oequal_307,axiom,
% 0.25/0.56 cl_HOL_Oequal @ char ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_308,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 & ( quickc3360725361186068524ustive @ A21 ) )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_309,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( quickcheck_random @ A20 )
% 0.25/0.56 & ( quickcheck_random @ A21 ) )
% 0.25/0.56 => ( quickcheck_random @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Sum__Type_Osum___Countable_Ocountable_310,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( countable @ A20 )
% 0.25/0.56 & ( countable @ A21 ) )
% 0.25/0.56 => ( countable @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_311,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( finite_finite @ A20 )
% 0.25/0.56 & ( finite_finite @ A21 ) )
% 0.25/0.56 => ( finite_finite @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Sum__Type_Osum___Heap_Oheap_312,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( heap @ A20 )
% 0.25/0.56 & ( heap @ A21 ) )
% 0.25/0.56 => ( heap @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Sum__Type_Osum___HOL_Oequal_313,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A20 @ A21 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_314,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_315,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( comple6319245703460814977attice @ A20 )
% 0.25/0.56 => ( comple6319245703460814977attice @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_316,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( comple6319245703460814977attice @ A20 )
% 0.25/0.56 => ( comple9053668089753744459l_ccpo @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_317,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickcheck_random @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_318,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( semilattice_sup @ A20 )
% 0.25/0.56 => ( semilattice_sup @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_319,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( semilattice_inf @ A20 )
% 0.25/0.56 => ( semilattice_inf @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Lattices_Obounded__lattice_320,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( bounded_lattice_top @ A20 )
% 0.25/0.56 => ( bounded_lattice @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Complete__Lattices_OSup_321,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( comple6319245703460814977attice @ A20 )
% 0.25/0.56 => ( complete_Sup @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Complete__Lattices_OInf_322,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( comple6319245703460814977attice @ A20 )
% 0.25/0.56 => ( complete_Inf @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Orderings_Oorder__top_323,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( order_top @ A20 )
% 0.25/0.56 => ( order_top @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Orderings_Oorder__bot_324,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( order @ A20 )
% 0.25/0.56 => ( order_bot @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Countable_Ocountable_325,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( countable @ A20 )
% 0.25/0.56 => ( countable @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Orderings_Opreorder_326,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( preorder @ A20 )
% 0.25/0.56 => ( preorder @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Orderings_Olinorder_327,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( linorder @ A20 )
% 0.25/0.56 => ( linorder @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Finite__Set_Ofinite_328,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( finite_finite @ A20 )
% 0.25/0.56 => ( finite_finite @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Orderings_Oorder_329,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( order @ A20 )
% 0.25/0.56 => ( order @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Orderings_Oord_330,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( preorder @ A20 )
% 0.25/0.56 => ( ord @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Lattices_Osup_331,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( sup @ A20 )
% 0.25/0.56 => ( sup @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Lattices_Oinf_332,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( inf @ A20 )
% 0.25/0.56 => ( inf @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___Heap_Oheap_333,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( heap @ A20 )
% 0.25/0.56 => ( heap @ ( option @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Option_Ooption___HOL_Oequal_334,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( option @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Oseq___HOL_Oequal_335,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( seq @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_336,axiom,
% 0.25/0.56 ! [A20: $tType] : ( comple6319245703460814977attice @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_337,axiom,
% 0.25/0.56 ! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_338,axiom,
% 0.25/0.56 ! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_339,axiom,
% 0.25/0.56 ! [A20: $tType] : ( semilattice_sup @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_340,axiom,
% 0.25/0.56 ! [A20: $tType] : ( semilattice_inf @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_341,axiom,
% 0.25/0.56 ! [A20: $tType] : ( bounded_lattice @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Complete__Lattices_OSup_342,axiom,
% 0.25/0.56 ! [A20: $tType] : ( complete_Sup @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Complete__Lattices_OInf_343,axiom,
% 0.25/0.56 ! [A20: $tType] : ( complete_Inf @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Orderings_Oorder__top_344,axiom,
% 0.25/0.56 ! [A20: $tType] : ( order_top @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Orderings_Oorder__bot_345,axiom,
% 0.25/0.56 ! [A20: $tType] : ( order_bot @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Orderings_Opreorder_346,axiom,
% 0.25/0.56 ! [A20: $tType] : ( preorder @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Orderings_Oorder_347,axiom,
% 0.25/0.56 ! [A20: $tType] : ( order @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Orderings_Oord_348,axiom,
% 0.25/0.56 ! [A20: $tType] : ( ord @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Groups_Ouminus_349,axiom,
% 0.25/0.56 ! [A20: $tType] : ( uminus @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Lattices_Osup_350,axiom,
% 0.25/0.56 ! [A20: $tType] : ( sup @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___Lattices_Oinf_351,axiom,
% 0.25/0.56 ! [A20: $tType] : ( inf @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Predicate_Opred___HOL_Oequal_352,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( pred @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_353,axiom,
% 0.25/0.56 boolea8198339166811842893lgebra @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_354,axiom,
% 0.25/0.56 semilattice_sup @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_355,axiom,
% 0.25/0.56 semilattice_inf @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_356,axiom,
% 0.25/0.56 bounded_lattice @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_357,axiom,
% 0.25/0.56 ab_semigroup_mult @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_358,axiom,
% 0.25/0.56 comm_monoid_mult @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_359,axiom,
% 0.25/0.56 semigroup_mult @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Orderings_Oorder__top_360,axiom,
% 0.25/0.56 order_top @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_361,axiom,
% 0.25/0.56 order_bot @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Orderings_Opreorder_362,axiom,
% 0.25/0.56 preorder @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_363,axiom,
% 0.25/0.56 monoid_mult @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Orderings_Oorder_364,axiom,
% 0.25/0.56 order @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Orderings_Oord_365,axiom,
% 0.25/0.56 ord @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Groups_Ouminus_366,axiom,
% 0.25/0.56 uminus @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Lattices_Osup_367,axiom,
% 0.25/0.56 sup @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Lattices_Oinf_368,axiom,
% 0.25/0.56 inf @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Power_Opower_369,axiom,
% 0.25/0.56 power @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Groups_Oone_370,axiom,
% 0.25/0.56 one @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Assertions_Oassn___Rings_Odvd_371,axiom,
% 0.25/0.56 dvd @ assn ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Typerep_Otyperep___Countable_Ocountable_372,axiom,
% 0.25/0.56 countable @ typerep ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Typerep_Otyperep___Heap_Oheap_373,axiom,
% 0.25/0.56 heap @ typerep ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Typerep_Otyperep___HOL_Oequal_374,axiom,
% 0.25/0.56 cl_HOL_Oequal @ typerep ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_375,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_376,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( preorder @ A20 )
% 0.25/0.56 => ( ordere6658533253407199908up_add @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_377,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_378,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_379,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cancel_semigroup_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_380,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickcheck_random @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_381,axiom,
% 0.25/0.56 ! [A20: $tType] : ( comm_monoid_diff @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_382,axiom,
% 0.25/0.56 ! [A20: $tType] : ( ab_semigroup_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_383,axiom,
% 0.25/0.56 ! [A20: $tType] : ( comm_monoid_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_384,axiom,
% 0.25/0.56 ! [A20: $tType] : ( complete_Sup @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_385,axiom,
% 0.25/0.56 ! [A20: $tType] : ( complete_Inf @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_386,axiom,
% 0.25/0.56 ! [A20: $tType] : ( semigroup_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_387,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( preorder @ A20 )
% 0.25/0.56 => ( preorder @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_388,axiom,
% 0.25/0.56 ! [A20: $tType] : ( monoid_add @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Orderings_Oorder_389,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( preorder @ A20 )
% 0.25/0.56 => ( order @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Orderings_Oord_390,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( preorder @ A20 )
% 0.25/0.56 => ( ord @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Ozero_391,axiom,
% 0.25/0.56 ! [A20: $tType] : ( zero @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___Groups_Oplus_392,axiom,
% 0.25/0.56 ! [A20: $tType] : ( plus @ ( multiset @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Multiset_Omultiset___HOL_Oequal_393,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( cl_HOL_Oequal @ A20 )
% 0.25/0.56 => ( cl_HOL_Oequal @ ( multiset @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_394,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( quickc3360725361186068524ustive @ A20 )
% 0.25/0.56 & ( quickc3360725361186068524ustive @ A21 ) )
% 0.25/0.56 => ( quickc3360725361186068524ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_395,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( quickc658316121487927005ustive @ A20 )
% 0.25/0.56 & ( quickc658316121487927005ustive @ A21 ) )
% 0.25/0.56 => ( quickc658316121487927005ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_396,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( quickcheck_random @ A20 )
% 0.25/0.56 & ( quickcheck_random @ A21 ) )
% 0.25/0.56 => ( quickcheck_random @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___Countable_Ocountable_397,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( countable @ A20 )
% 0.25/0.56 & ( countable @ A21 ) )
% 0.25/0.56 => ( countable @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_398,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( finite_finite @ A20 )
% 0.25/0.56 & ( finite_finite @ A21 ) )
% 0.25/0.56 => ( finite_finite @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___Heap_Oheap_399,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] :
% 0.25/0.56 ( ( ( heap @ A20 )
% 0.25/0.56 & ( heap @ A21 ) )
% 0.25/0.56 => ( heap @ ( product_prod @ A20 @ A21 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Oprod___HOL_Oequal_400,axiom,
% 0.25/0.56 ! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A20 @ A21 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_401,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_402,axiom,
% 0.25/0.56 comple6319245703460814977attice @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_403,axiom,
% 0.25/0.56 boolea8198339166811842893lgebra @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_404,axiom,
% 0.25/0.56 comple9053668089753744459l_ccpo @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_405,axiom,
% 0.25/0.56 quickcheck_random @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_406,axiom,
% 0.25/0.56 semilattice_sup @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_407,axiom,
% 0.25/0.56 semilattice_inf @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_408,axiom,
% 0.25/0.56 bounded_lattice @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_409,axiom,
% 0.25/0.56 complete_Sup @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_410,axiom,
% 0.25/0.56 complete_Inf @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_411,axiom,
% 0.25/0.56 order_top @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_412,axiom,
% 0.25/0.56 order_bot @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Countable_Ocountable_413,axiom,
% 0.25/0.56 countable @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_414,axiom,
% 0.25/0.56 preorder @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_415,axiom,
% 0.25/0.56 linorder @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_416,axiom,
% 0.25/0.56 finite_finite @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Orderings_Oorder_417,axiom,
% 0.25/0.56 order @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Orderings_Oord_418,axiom,
% 0.25/0.56 ord @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Groups_Ouminus_419,axiom,
% 0.25/0.56 uminus @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Lattices_Osup_420,axiom,
% 0.25/0.56 sup @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Lattices_Oinf_421,axiom,
% 0.25/0.56 inf @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___Heap_Oheap_422,axiom,
% 0.25/0.56 heap @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Product__Type_Ounit___HOL_Oequal_423,axiom,
% 0.25/0.56 cl_HOL_Oequal @ product_unit ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_424,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickcheck_random @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( heap_ext @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_425,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_426,axiom,
% 0.25/0.56 bit_un5681908812861735899ations @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_427,axiom,
% 0.25/0.56 semiri1453513574482234551roduct @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_428,axiom,
% 0.25/0.56 euclid5411537665997757685th_nat @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_429,axiom,
% 0.25/0.56 euclid8789492081693882211th_nat @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_430,axiom,
% 0.25/0.56 ordere1937475149494474687imp_le @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_431,axiom,
% 0.25/0.56 euclid3128863361964157862miring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_432,axiom,
% 0.25/0.56 euclid4440199948858584721cancel @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_433,axiom,
% 0.25/0.56 unique1627219031080169319umeral @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_434,axiom,
% 0.25/0.56 euclid8851590272496341667cancel @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_435,axiom,
% 0.25/0.56 strict9044650504122735259up_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_436,axiom,
% 0.25/0.56 ordere580206878836729694up_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_437,axiom,
% 0.25/0.56 ordere2412721322843649153imp_le @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_438,axiom,
% 0.25/0.56 bit_se359711467146920520ations @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_439,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_440,axiom,
% 0.25/0.56 strict7427464778891057005id_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_441,axiom,
% 0.25/0.56 ordere8940638589300402666id_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_442,axiom,
% 0.25/0.56 euclid3725896446679973847miring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_443,axiom,
% 0.25/0.56 linord715952674999750819strict @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_444,axiom,
% 0.25/0.56 linord4140545234300271783up_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_445,axiom,
% 0.25/0.56 bit_ri3973907225187159222ations @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_446,axiom,
% 0.25/0.56 linord181362715937106298miring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_447,axiom,
% 0.25/0.56 quickc658316121487927005ustive @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_448,axiom,
% 0.25/0.56 semiri3467727345109120633visors @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_449,axiom,
% 0.25/0.56 ordere6658533253407199908up_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_450,axiom,
% 0.25/0.56 ordere166539214618696060dd_abs @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_451,axiom,
% 0.25/0.56 ordere6911136660526730532id_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_452,axiom,
% 0.25/0.56 linord5086331880401160121up_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_453,axiom,
% 0.25/0.56 cancel2418104881723323429up_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_454,axiom,
% 0.25/0.56 ring_15535105094025558882visors @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_455,axiom,
% 0.25/0.56 cancel1802427076303600483id_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_456,axiom,
% 0.25/0.56 linord4710134922213307826strict @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_457,axiom,
% 0.25/0.56 comm_s4317794764714335236cancel @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_458,axiom,
% 0.25/0.56 bit_semiring_bits @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_459,axiom,
% 0.25/0.56 linord6961819062388156250ring_1 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_460,axiom,
% 0.25/0.56 ordered_ab_group_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_461,axiom,
% 0.25/0.56 cancel_semigroup_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_462,axiom,
% 0.25/0.56 linordered_semidom @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_463,axiom,
% 0.25/0.56 quickcheck_random @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_464,axiom,
% 0.25/0.56 ab_semigroup_mult @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_465,axiom,
% 0.25/0.56 algebraic_semidom @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_466,axiom,
% 0.25/0.56 comm_monoid_mult @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_467,axiom,
% 0.25/0.56 ab_semigroup_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_468,axiom,
% 0.25/0.56 semiring_parity @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_469,axiom,
% 0.25/0.56 comm_monoid_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_470,axiom,
% 0.25/0.56 semiring_modulo @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_471,axiom,
% 0.25/0.56 linordered_ring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_472,axiom,
% 0.25/0.56 linordered_idom @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_473,axiom,
% 0.25/0.56 comm_semiring_1 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_474,axiom,
% 0.25/0.56 comm_semiring_0 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_475,axiom,
% 0.25/0.56 semigroup_mult @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_476,axiom,
% 0.25/0.56 semidom_modulo @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_477,axiom,
% 0.25/0.56 semidom_divide @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_478,axiom,
% 0.25/0.56 semiring_numeral @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_479,axiom,
% 0.25/0.56 semigroup_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_480,axiom,
% 0.25/0.56 zero_less_one @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_481,axiom,
% 0.25/0.56 comm_semiring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_482,axiom,
% 0.25/0.56 semiring_char_0 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_483,axiom,
% 0.25/0.56 ab_group_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_484,axiom,
% 0.25/0.56 zero_neq_one @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_485,axiom,
% 0.25/0.56 ordered_ring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_486,axiom,
% 0.25/0.56 idom_abs_sgn @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_487,axiom,
% 0.25/0.56 ring_parity @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_488,axiom,
% 0.25/0.56 preorder @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_489,axiom,
% 0.25/0.56 linorder @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_490,axiom,
% 0.25/0.56 monoid_mult @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_491,axiom,
% 0.25/0.56 comm_ring_1 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_492,axiom,
% 0.25/0.56 monoid_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_493,axiom,
% 0.25/0.56 semiring_1 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_494,axiom,
% 0.25/0.56 semiring_0 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_495,axiom,
% 0.25/0.56 group_add @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_496,axiom,
% 0.25/0.56 comm_ring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_497,axiom,
% 0.25/0.56 order @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_498,axiom,
% 0.25/0.56 neg_numeral @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_499,axiom,
% 0.25/0.56 ring_char_0 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_500,axiom,
% 0.25/0.56 semiring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_501,axiom,
% 0.25/0.56 ord @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_502,axiom,
% 0.25/0.56 uminus @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_503,axiom,
% 0.25/0.56 ring_1 @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Power_Opower_504,axiom,
% 0.25/0.56 power @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_505,axiom,
% 0.25/0.56 numeral @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_506,axiom,
% 0.25/0.56 zero @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_507,axiom,
% 0.25/0.56 plus @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_508,axiom,
% 0.25/0.56 ring @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_509,axiom,
% 0.25/0.56 one @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_510,axiom,
% 0.25/0.56 dvd @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_511,axiom,
% 0.25/0.56 cl_HOL_Oequal @ code_integer ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_512,axiom,
% 0.25/0.56 bit_un5681908812861735899ations @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_513,axiom,
% 0.25/0.56 euclid5411537665997757685th_nat @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_514,axiom,
% 0.25/0.56 ordere1937475149494474687imp_le @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_515,axiom,
% 0.25/0.56 euclid3128863361964157862miring @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_516,axiom,
% 0.25/0.56 euclid4440199948858584721cancel @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_517,axiom,
% 0.25/0.56 strict9044650504122735259up_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_518,axiom,
% 0.25/0.56 ordere580206878836729694up_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_519,axiom,
% 0.25/0.56 ordere2412721322843649153imp_le @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_520,axiom,
% 0.25/0.56 bit_se359711467146920520ations @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_521,axiom,
% 0.25/0.56 quickc3360725361186068524ustive @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_522,axiom,
% 0.25/0.56 strict7427464778891057005id_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_523,axiom,
% 0.25/0.56 ordere8940638589300402666id_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_524,axiom,
% 0.25/0.56 euclid3725896446679973847miring @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_525,axiom,
% 0.25/0.56 linord4140545234300271783up_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_526,axiom,
% 0.25/0.56 linord181362715937106298miring @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_527,axiom,
% 0.25/0.56 quickc658316121487927005ustive @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_528,axiom,
% 0.25/0.56 semiri3467727345109120633visors @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_529,axiom,
% 0.25/0.56 ordere6658533253407199908up_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_530,axiom,
% 0.25/0.56 ordere6911136660526730532id_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_531,axiom,
% 0.25/0.56 cancel2418104881723323429up_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_532,axiom,
% 0.25/0.56 cancel1802427076303600483id_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_533,axiom,
% 0.25/0.56 comm_s4317794764714335236cancel @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_534,axiom,
% 0.25/0.56 bit_semiring_bits @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_535,axiom,
% 0.25/0.56 cancel_semigroup_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_536,axiom,
% 0.25/0.56 linordered_semidom @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_537,axiom,
% 0.25/0.56 quickcheck_random @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_538,axiom,
% 0.25/0.56 ab_semigroup_mult @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_539,axiom,
% 0.25/0.56 algebraic_semidom @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_540,axiom,
% 0.25/0.56 comm_monoid_mult @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_541,axiom,
% 0.25/0.56 comm_monoid_diff @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_542,axiom,
% 0.25/0.56 ab_semigroup_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_543,axiom,
% 0.25/0.56 semiring_parity @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_544,axiom,
% 0.25/0.56 comm_monoid_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_545,axiom,
% 0.25/0.56 semiring_modulo @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_546,axiom,
% 0.25/0.56 comm_semiring_1 @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_547,axiom,
% 0.25/0.56 comm_semiring_0 @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_548,axiom,
% 0.25/0.56 semigroup_mult @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_549,axiom,
% 0.25/0.56 semidom_modulo @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_550,axiom,
% 0.25/0.56 semidom_divide @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_551,axiom,
% 0.25/0.56 semiring_numeral @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_552,axiom,
% 0.25/0.56 semigroup_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_553,axiom,
% 0.25/0.56 zero_less_one @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_554,axiom,
% 0.25/0.56 comm_semiring @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_555,axiom,
% 0.25/0.56 semiring_char_0 @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_556,axiom,
% 0.25/0.56 zero_neq_one @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_557,axiom,
% 0.25/0.56 preorder @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_558,axiom,
% 0.25/0.56 linorder @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_559,axiom,
% 0.25/0.56 monoid_mult @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_560,axiom,
% 0.25/0.56 monoid_add @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_561,axiom,
% 0.25/0.56 semiring_1 @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_562,axiom,
% 0.25/0.56 semiring_0 @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_563,axiom,
% 0.25/0.56 order @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_564,axiom,
% 0.25/0.56 semiring @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_565,axiom,
% 0.25/0.56 ord @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Power_Opower_566,axiom,
% 0.25/0.56 power @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_567,axiom,
% 0.25/0.56 numeral @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_568,axiom,
% 0.25/0.56 zero @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oplus_569,axiom,
% 0.25/0.56 plus @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Groups_Oone_570,axiom,
% 0.25/0.56 one @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___Rings_Odvd_571,axiom,
% 0.25/0.56 dvd @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Code__Numeral_Onatural___HOL_Oequal_572,axiom,
% 0.25/0.56 cl_HOL_Oequal @ code_natural ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_573,axiom,
% 0.25/0.56 ! [A20: $tType] :
% 0.25/0.56 ( ( quickcheck_random @ A20 )
% 0.25/0.56 => ( quickcheck_random @ ( heap_Time_Heap @ A20 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_574,axiom,
% 0.25/0.56 ! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A20 ) ) ).
% 0.25/0.56
% 0.25/0.56 % Helper facts (4)
% 0.25/0.56 thf(help_If_3_1_T,axiom,
% 0.25/0.56 ! [P: $o] :
% 0.25/0.56 ( ( P = $true )
% 0.25/0.56 | ( P = $false ) ) ).
% 0.25/0.56
% 0.25/0.56 thf(help_If_2_1_T,axiom,
% 0.25/0.56 ! [A: $tType,X: A,Y: A] :
% 0.25/0.56 ( ( if @ A @ $false @ X @ Y )
% 0.25/0.56 = Y ) ).
% 0.25/0.56
% 0.25/0.56 thf(help_If_1_1_T,axiom,
% 0.25/0.56 ! [A: $tType,X: A,Y: A] :
% 0.25/0.56 ( ( if @ A @ $true @ X @ Y )
% 0.25/0.56 = X ) ).
% 0.25/0.56
% 0.25/0.56 thf(help_fChoice_1_1_T,axiom,
% 0.25/0.56 ! [A: $tType,P: A > $o] :
% 0.25/0.56 ( ( P @ ( fChoice @ A @ P ) )
% 0.25/0.56 = ( ? [X8: A] : ( P @ X8 ) ) ) ).
% 0.25/0.56
% 0.25/0.56 % Free types (1)
% 0.25/0.56 thf(tfree_0,hypothesis,
% 0.25/0.56 heap @ a ).
% 0.25/0.56
% 0.25/0.56 % Conjectures (1)
% 0.25/0.56 thf(conj_0,conjecture,
% 0.25/0.56 ( ( heap_Time_execute @ ( array @ a ) @ ( array_of_list @ a @ xs ) @ h )
% 0.25/0.56 != ( none @ ( product_prod @ ( array @ a ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 0.25/0.56
% 0.25/0.56 %------------------------------------------------------------------------------
% 0.25/0.56 ------------------------
% 9.43/3.43 ---- Embedded in THF ---
% 9.43/3.43 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 9.43/3.43 %%% Generated on Thu Feb 26 05:04:12 EST 2026
% 9.43/3.43 %%% using '$$dhol' embedding, version 1.3.0.
% 9.43/3.43 %%% Logic specification used:
% 9.43/3.43 %%% thf(spec, logic, $$dhol).
% 9.43/3.43
% 9.43/3.43 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EbIqTwV4kM/DTF2DTF24760.p
% 9.43/3.43 thf(ty_t_Heap__Time__Monad_OHeap, type, heap_Time_Heap: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Code__Numeral_Onatural, type, code_natural: $tType).
% 9.43/3.43 thf(ty_t_Code__Numeral_Ointeger, type, code_integer: $tType).
% 9.43/3.43 thf(ty_t_Code__Evaluation_Oterm, type, code_term: $tType).
% 9.43/3.43 thf(ty_t_Heap_Oheap_Oheap__ext, type, heap_ext: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Product__Type_Ounit, type, product_unit: $tType).
% 9.43/3.43 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 9.43/3.43 thf(ty_t_Old__Datatype_Onode, type, old_node: ($tType > ($tType > $tType))).
% 9.43/3.43 thf(ty_t_Multiset_Omultiset, type, multiset: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Typerep_Otyperep, type, typerep: $tType).
% 9.43/3.43 thf(ty_t_Assertions_Oassn, type, assn: $tType).
% 9.43/3.43 thf(ty_t_Predicate_Opred, type, pred: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 9.43/3.43 thf(ty_t_Predicate_Oseq, type, seq: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_String_Ochar, type, char: $tType).
% 9.43/3.43 thf(ty_t_Heap_Oarray, type, array: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_List_Olist, type, list: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Heap_Oref, type, ref: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 9.43/3.43 thf(ty_t_Rat_Orat, type, rat: $tType).
% 9.43/3.43 thf(ty_t_Num_Onum, type, num: $tType).
% 9.43/3.43 thf(ty_t_Nat_Onat, type, nat: $tType).
% 9.43/3.43 thf(ty_t_Int_Oint, type, int: $tType).
% 9.43/3.43 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 9.43/3.43 thf(ty_tf_a, type, a: $tType).
% 9.43/3.43 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Typerep_Otyperep, type, typerep2: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Enum_Oenum, type, enum: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Code__Evaluation_Oterm__of, type, code_term_of: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_HOL_Oequal, type, cl_HOL_Oequal: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Heap_Oheap, type, heap: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Odvd, type, dvd: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oone, type, one: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Oring, type, ring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oplus, type, plus: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ozero, type, zero: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Num_Onumeral, type, numeral: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Power_Opower, type, power: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Fields_Ofield, type, field: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_GCD_Oring__gcd, type, ring_gcd: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Lattices_Oinf, type, inf: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Lattices_Osup, type, sup: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Oring__1, type, ring_1: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Fields_Oinverse, type, inverse: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemiring, type, semiring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Nat_Oring__char__0, type, ring_char_0: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Num_Oneg__numeral, type, neg_numeral: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ocomm__ring, type, comm_ring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_GCD_Osemiring__Gcd, type, semiring_Gcd: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_GCD_Osemiring__gcd, type, semiring_gcd: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Ono__bot, type, no_bot: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Ono__top, type, no_top: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemiring__0, type, semiring_0: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemiring__1, type, semiring_1: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Omonoid__add, type, monoid_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ocomm__ring__1, type, comm_ring_1: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Omonoid__mult, type, monoid_mult: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Parity_Oring__parity, type, ring_parity: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Oidom__abs__sgn, type, idom_abs_sgn: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Oordered__ring, type, ordered_ring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ozero__neq__one, type, zero_neq_one: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Countable_Ocountable, type, countable: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Fields_Ofield__char__0, type, field_char_0: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oab__group__add, type, ab_group_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ocomm__semiring, type, comm_semiring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ozero__less__one, type, zero_less_one: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Fields_Odivision__ring, type, division_ring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Osemigroup__add, type, semigroup_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Num_Osemiring__numeral, type, semiring_numeral: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemidom__divide, type, semidom_divide: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemidom__modulo, type, semidom_modulo: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Complete__Lattices_OInf, type, complete_Inf: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Complete__Lattices_OSup, type, complete_Sup: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Osemigroup__mult, type, semigroup_mult: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ocomm__semiring__0, type, comm_semiring_0: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ocomm__semiring__1, type, comm_semiring_1: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__idom, type, linordered_idom: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__ring, type, linordered_ring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemiring__modulo, type, semiring_modulo: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocomm__monoid__add, type, comm_monoid_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Parity_Osemiring__parity, type, semiring_parity: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Fields_Olinordered__field, type, linordered_field: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oab__semigroup__add, type, ab_semigroup_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocomm__monoid__diff, type, comm_monoid_diff: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocomm__monoid__mult, type, comm_monoid_mult: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Oalgebraic__semidom, type, algebraic_semidom: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oab__semigroup__mult, type, ab_semigroup_mult: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Lattices_Osemilattice__inf, type, semilattice_inf: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Orderings_Odense__linorder, type, dense_linorder: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Quickcheck__Random_Orandom, type, quickcheck_random: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__semidom, type, linordered_semidom: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Complete__Partial__Order_Occpo, type, comple9053668089753744459l_ccpo: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocancel__semigroup__add, type, cancel_semigroup_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__semiring__1, type, linord6961819062388156250ring_1: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Onormalization__semidom, type, normal8620421768224518004emidom: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Bit__Operations_Osemiring__bits, type, bit_semiring_bits: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Ocomm__semiring__1__cancel, type, comm_s4317794764714335236cancel: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__ring__strict, type, linord4710134922213307826strict: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocancel__comm__monoid__add, type, cancel1802427076303600483id_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Oring__1__no__zero__divisors, type, ring_15535105094025558882visors: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel2418104881723323429up_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Olinordered__ab__group__add, type, linord5086331880401160121up_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__comm__monoid__add, type, ordere6911136660526730532id_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Archimedean__Field_Ofloor__ceiling, type, archim2362893244070406136eiling: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__ab__group__add__abs, type, ordere166539214618696060dd_abs: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__ab__semigroup__add, type, ordere6658533253407199908up_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemiring__no__zero__divisors, type, semiri3467727345109120633visors: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Boolean__Algebras_Oboolean__algebra, type, boolea8198339166811842893lgebra: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Quickcheck__Exhaustive_Oexhaustive, type, quickc658316121487927005ustive: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Osemidom__divide__unit__factor, type, semido2269285787275462019factor: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__nonzero__semiring, type, linord181362715937106298miring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Bit__Operations_Oring__bit__operations, type, bit_ri3973907225187159222ations: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Complete__Lattices_Ocomplete__lattice, type, comple6319245703460814977attice: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Olinordered__ab__semigroup__add, type, linord4140545234300271783up_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Rings_Olinordered__semiring__1__strict, type, linord715952674999750819strict: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Archimedean__Field_Oarchimedean__field, type, archim462609752435547400_field: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring, type, euclid3725896446679973847miring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add, type, canoni5634975068530333245id_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add, type, ordere8940638589300402666id_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add, type, strict7427464778891057005id_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive, type, quickc3360725361186068524ustive: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Bit__Operations_Osemiring__bit__operations, type, bit_se359711467146920520ations: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le, type, ordere2412721322843649153imp_le: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add, type, ordere580206878836729694up_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff, type, ordere1170586879665033532d_diff: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add, type, strict9044650504122735259up_add: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel, type, euclid8851590272496341667cancel: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral, type, unique1627219031080169319umeral: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel, type, euclid4440199948858584721cancel: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring, type, euclid3128863361964157862miring: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le, type, ordere1937475149494474687imp_le: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat, type, euclid8789492081693882211th_nat: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat, type, euclid5411537665997757685th_nat: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, type, semiri1453513574482234551roduct: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations, type, bit_un5681908812861735899ations: (!> [A:$tType]: ($o))).
% 9.43/3.43 thf(sy_c_Archimedean__Field_Oceiling, type, archimedean_ceiling: (!> [A:$tType]: ((A > int)))).
% 9.43/3.43 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor, type, archim6421214686448440834_floor: (!> [A:$tType]: ((A > int)))).
% 9.43/3.43 thf(sy_c_Archimedean__Field_Ofrac, type, archimedean_frac: (!> [A:$tType]: ((A > A)))).
% 9.43/3.43 thf(sy_c_Archimedean__Field_Oround, type, archimedean_round: (!> [A:$tType]: ((A > int)))).
% 9.43/3.43 thf(sy_c_Array__Time_Oalloc, type, array_alloc: (!> [A:$tType]: (((list @ A) > ((heap_ext @ product_unit) > ((product_prod @ (array @ A)) @ (heap_ext @ product_unit))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Ofreeze, type, array_freeze: (!> [A:$tType]: (((array @ A) > (heap_Time_Heap @ (list @ A)))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oget, type, array_get: (!> [A:$tType]: (((heap_ext @ product_unit) > ((array @ A) > (list @ A)))))).
% 9.43/3.43 thf(sy_c_Array__Time_Olen, type, array_len: (!> [A:$tType]: (((array @ A) > (heap_Time_Heap @ nat))))).
% 9.43/3.43 thf(sy_c_Array__Time_Olen_H, type, array_len2: (!> [B:$tType,A:$tType]: (((array @ B) > (heap_Time_Heap @ A))))).
% 9.43/3.43 thf(sy_c_Array__Time_Olength, type, array_length: (!> [A:$tType]: (((heap_ext @ product_unit) > ((array @ A) > nat))))).
% 9.43/3.43 thf(sy_c_Array__Time_Omake, type, array_make: (!> [A:$tType]: ((nat > ((nat > A) > (heap_Time_Heap @ (array @ A))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Omake_H, type, array_make2: (!> [B:$tType,A:$tType]: ((code_integer > ((B > A) > (heap_Time_Heap @ (array @ A))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Omap__entry, type, array_map_entry: (!> [A:$tType]: ((nat > ((A > A) > ((array @ A) > (heap_Time_Heap @ (array @ A)))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Onew, type, array_new: (!> [A:$tType]: ((nat > (A > (heap_Time_Heap @ (array @ A))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Onew_H, type, array_new2: (!> [A:$tType]: ((code_integer > (A > (heap_Time_Heap @ (array @ A))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Onoteq, type, array_noteq: (!> [A:$tType,B:$tType]: (((array @ A) > ((array @ B) > $o))))).
% 9.43/3.43 thf(sy_c_Array__Time_Onth, type, array_nth: (!> [A:$tType]: (((array @ A) > (nat > (heap_Time_Heap @ A)))))).
% 9.43/3.43 thf(sy_c_Array__Time_Onth_H, type, array_nth2: (!> [A:$tType]: (((array @ A) > (code_integer > (heap_Time_Heap @ A)))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oof__list, type, array_of_list: (!> [A:$tType]: (((list @ A) > (heap_Time_Heap @ (array @ A)))))).
% 9.43/3.43 thf(sy_c_Array__Time_Opresent, type, array_present: (!> [A:$tType]: (((heap_ext @ product_unit) > ((array @ A) > $o))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oset, type, array_set: (!> [A:$tType]: (((array @ A) > ((list @ A) > ((heap_ext @ product_unit) > (heap_ext @ product_unit))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oswap, type, array_swap: (!> [A:$tType]: ((nat > (A > ((array @ A) > (heap_Time_Heap @ A))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oupd, type, array_upd: (!> [A:$tType]: ((nat > (A > ((array @ A) > (heap_Time_Heap @ (array @ A)))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oupd_H, type, array_upd2: (!> [A:$tType]: (((array @ A) > (code_integer > (A > (heap_Time_Heap @ product_unit))))))).
% 9.43/3.43 thf(sy_c_Array__Time_Oupdate, type, array_update: (!> [A:$tType]: (((array @ A) > (nat > (A > ((heap_ext @ product_unit) > (heap_ext @ product_unit)))))))).
% 9.43/3.43 thf(sy_c_Assertions_Oassn_ORep__assn, type, rep_assn: (assn > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o))).
% 9.43/3.43 thf(sy_c_Assertions_Oex__assn, type, ex_assn: (!> [A:$tType]: (((A > assn) > assn)))).
% 9.43/3.43 thf(sy_c_Assertions_Oin__range, type, in_range: (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)).
% 9.43/3.43 thf(sy_c_Assertions_Oin__range__rel, type, in_range_rel: (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o))).
% 9.43/3.43 thf(sy_c_Assertions_Oone__assn__raw, type, one_assn_raw: (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)).
% 9.43/3.43 thf(sy_c_Assertions_Oone__assn__raw__rel, type, one_assn_raw_rel: (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o))).
% 9.43/3.43 thf(sy_c_Assertions_Oproper, type, proper: ((((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o) > $o)).
% 9.43/3.43 thf(sy_c_Assertions_Opure__assn, type, pure_assn: ($o > assn)).
% 9.43/3.43 thf(sy_c_Assertions_Opure__assn__raw, type, pure_assn_raw: (!> [A:$tType,B:$tType]: (($o > (((product_prod @ A) @ (set @ B)) > $o))))).
% 9.43/3.43 thf(sy_c_Assertions_Opure__assn__raw__rel, type, pure_assn_raw_rel: (!> [A:$tType,B:$tType]: ((((product_prod @ $o) @ ((product_prod @ A) @ (set @ B))) > (((product_prod @ $o) @ ((product_prod @ A) @ (set @ B))) > $o))))).
% 9.43/3.43 thf(sy_c_Assertions_OrelH, type, relH: ((set @ nat) > ((heap_ext @ product_unit) > ((heap_ext @ product_unit) > $o)))).
% 9.43/3.43 thf(sy_c_Assertions_Osnga__assn, type, snga_assn: (!> [A:$tType]: (((array @ A) > ((list @ A) > assn))))).
% 9.43/3.43 thf(sy_c_Assertions_Osnga__assn__raw, type, snga_assn_raw: (!> [A:$tType]: (((array @ A) > ((list @ A) > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)))))).
% 9.43/3.43 thf(sy_c_Assertions_Osnga__assn__raw__rel, type, snga_assn_raw_rel: (!> [A:$tType]: ((((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > (((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > $o))))).
% 9.43/3.43 thf(sy_c_Assertions_Osngr__assn, type, sngr_assn: (!> [A:$tType]: (((ref @ A) > (A > assn))))).
% 9.43/3.43 thf(sy_c_Assertions_Osngr__assn__raw, type, sngr_assn_raw: (!> [A:$tType]: (((ref @ A) > (A > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)))))).
% 9.43/3.43 thf(sy_c_Assertions_Osngr__assn__raw__rel, type, sngr_assn_raw_rel: (!> [A:$tType]: ((((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > (((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > $o))))).
% 9.43/3.43 thf(sy_c_Assertions_Otimes__assn__raw, type, times_assn_raw: ((((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o) > ((((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o) > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)))).
% 9.43/3.43 thf(sy_c_Assertions_Otimes__assn__raw__rel, type, times_assn_raw_rel: (((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > (((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > $o))).
% 9.43/3.43 thf(sy_c_Assertions_Owand__assn, type, wand_assn: (assn > (assn > assn))).
% 9.43/3.43 thf(sy_c_Assertions_Owand__raw, type, wand_raw: ((((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o) > ((((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o) > (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)))).
% 9.43/3.43 thf(sy_c_Assertions_Owand__raw__rel, type, wand_raw_rel: (((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > (((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) > $o))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_OCsum, type, bNF_Cardinal_Csum: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp, type, bNF_Cardinal_cexp: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (A > B)) @ (A > B)))))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite, type, bNF_Cardinal_cfinite: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite, type, bNF_Ca4139267488887388095finite: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Ocone, type, bNF_Cardinal_cone: (set @ ((product_prod @ product_unit) @ product_unit))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod, type, bNF_Cardinal_cprod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum, type, bNF_Cardinal_csum: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Octwo, type, bNF_Cardinal_ctwo: (set @ ((product_prod @ $o) @ $o))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Arithmetic_Oczero, type, bNF_Cardinal_czero: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc, type, bNF_Ca8387033319878233205ardSuc: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of, type, bNF_Ca6860139660246222851ard_of: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on, type, bNF_Ca8970107618336181345der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal, type, bNF_Ca7293521722713021262ofinal: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc, type, bNF_Ca6246979054910435723ardSuc: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ (set @ A)) @ (set @ A))) > $o))))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq, type, bNF_Ca8665028551170535155natLeq: (set @ ((product_prod @ nat) @ nat))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard, type, bNF_Ca7133664381575040944arCard: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.43 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain, type, bNF_Ca3754400796208372196lChain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 9.43/3.43 thf(sy_c_BNF__Def_OGr, type, bNF_Gr: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_OGrp, type, bNF_Grp: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (A > (B > $o))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_Ocollect, type, bNF_collect: (!> [B:$tType,A:$tType]: (((set @ (B > (set @ A))) > (B > (set @ A)))))).
% 9.43/3.43 thf(sy_c_BNF__Def_Oconvol, type, bNF_convol: (!> [A:$tType,B:$tType,C:$tType]: (((A > B) > ((A > C) > (A > ((product_prod @ B) @ C))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_Oeq__onp, type, bNF_eq_onp: (!> [A:$tType]: (((A > $o) > (A > (A > $o)))))).
% 9.43/3.43 thf(sy_c_BNF__Def_OfstOp, type, bNF_fstOp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (((product_prod @ A) @ C) > ((product_prod @ A) @ B))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_Opick__middlep, type, bNF_pick_middlep: (!> [B:$tType,A:$tType,C:$tType]: (((B > (A > $o)) > ((A > (C > $o)) > (B > (C > A))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_Orel__fun, type, bNF_rel_fun: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > (C > $o)) > ((B > (D > $o)) > ((A > B) > ((C > D) > $o))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_OsndOp, type, bNF_sndOp: (!> [C:$tType,A:$tType,B:$tType]: (((C > (A > $o)) > ((A > (B > $o)) > (((product_prod @ C) @ B) > ((product_prod @ A) @ B))))))).
% 9.43/3.43 thf(sy_c_BNF__Def_Ovimage2p, type, bNF_vimage2p: (!> [A:$tType,D:$tType,B:$tType,E:$tType,C:$tType]: (((A > D) > ((B > E) > ((D > (E > C)) > (A > (B > C)))))))).
% 9.43/3.44 thf(sy_c_BNF__Greatest__Fixpoint_OfromCard, type, bNF_Gr5436034075474128252omCard: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > (B > A)))))).
% 9.43/3.44 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2, type, bNF_Greatest_image2: (!> [C:$tType,A:$tType,B:$tType]: (((set @ C) > ((C > A) > ((C > B) > (set @ ((product_prod @ A) @ B)))))))).
% 9.43/3.44 thf(sy_c_BNF__Greatest__Fixpoint_OrelImage, type, bNF_Gr4221423524335903396lImage: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((B > A) > (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage, type, bNF_Gr7122648621184425601vImage: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.44 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard, type, bNF_Greatest_toCard: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > (A > B)))))).
% 9.43/3.44 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred, type, bNF_Gr1419584066657907630d_pred: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ ((product_prod @ B) @ B)) > ((A > B) > $o)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_OFunc, type, bNF_Wellorder_Func: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ (A > B))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_Obsqr, type, bNF_Wellorder_bsqr: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_Ocurr, type, bNF_Wellorder_curr: (!> [A:$tType,B:$tType,C:$tType]: (((set @ A) > ((((product_prod @ A) @ B) > C) > (A > (B > C))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_Odir__image, type, bNF_We2720479622203943262_image: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > A2) > (set @ ((product_prod @ A2) @ A2))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl, type, bNF_We413866401316099525erIncl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_OordIso, type, bNF_Wellorder_ordIso: (!> [A:$tType,A2:$tType]: ((set @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A2) @ A2))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_OordLeq, type, bNF_Wellorder_ordLeq: (!> [A:$tType,A2:$tType]: ((set @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A2) @ A2))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Constructions_OordLess, type, bNF_We4044943003108391690rdLess: (!> [A:$tType,A2:$tType]: ((set @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A2) @ A2))))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Embedding_Ocompat, type, bNF_Wellorder_compat: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Embedding_Oembed, type, bNF_Wellorder_embed: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Embedding_OembedS, type, bNF_Wellorder_embedS: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Embedding_Oiso, type, bNF_Wellorder_iso: (!> [A:$tType,A2:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ A2) @ A2)) > ((A > A2) > $o)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Relation_Owo__rel, type, bNF_Wellorder_wo_rel: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim, type, bNF_We4791949203932849705sMinim: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (A > $o)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2, type, bNF_We1388413361240627857o_max2: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (A > A)))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim, type, bNF_We6954850376910717587_minim: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > A))))).
% 9.43/3.44 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc, type, bNF_Wellorder_wo_suc: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > A))))).
% 9.43/3.44 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod, type, basic_BNF_size_prod: (!> [A:$tType,B:$tType]: (((A > nat) > ((B > nat) > (((product_prod @ A) @ B) > nat)))))).
% 9.43/3.44 thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum, type, basic_BNF_size_sum: (!> [A:$tType,B:$tType]: (((A > nat) > ((B > nat) > (((sum_sum @ A) @ B) > nat)))))).
% 9.43/3.44 thf(sy_c_Basic__BNFs_Ofsts, type, basic_fsts: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Basic__BNFs_Opred__fun, type, basic_pred_fun: (!> [A:$tType,B:$tType]: (((A > $o) > ((B > $o) > ((A > B) > $o)))))).
% 9.43/3.44 thf(sy_c_Basic__BNFs_Opred__prod, type, basic_pred_prod: (!> [A:$tType,B:$tType]: (((A > $o) > ((B > $o) > (((product_prod @ A) @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_Basic__BNFs_Orel__prod, type, basic_rel_prod: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > (B > $o)) > ((C > (D > $o)) > (((product_prod @ A) @ C) > (((product_prod @ B) @ D) > $o))))))).
% 9.43/3.44 thf(sy_c_Basic__BNFs_Osnds, type, basic_snds: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > (set @ B))))).
% 9.43/3.44 thf(sy_c_Binomial_Obinomial, type, binomial: (nat > (nat > nat))).
% 9.43/3.44 thf(sy_c_Binomial_Ogbinomial, type, gbinomial: (!> [A:$tType]: ((A > (nat > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oand__int__rel, type, bit_and_int_rel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oand__not__num, type, bit_and_not_num: (num > (num > (option @ num)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oand__not__num__rel, type, bit_and_not_num_rel: (((product_prod @ num) @ num) > (((product_prod @ num) @ num) > $o))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oconcat__bit, type, bit_concat_bit: (nat > (int > (int > int)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oor__not__num__neg, type, bit_or_not_num_neg: (num > (num > num))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel, type, bit_or3848514188828904588eg_rel: (((product_prod @ num) @ num) > (((product_prod @ num) @ num) > $o))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot, type, bit_ri4277139882892585799ns_not: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit, type, bit_ri4674362597316999326ke_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand, type, bit_se5824344872417868541ns_and: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit, type, bit_se4197421643247451524op_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit, type, bit_se8732182000553998342ip_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask, type, bit_se2239418461657761734s_mask: (!> [A:$tType]: ((nat > A)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor, type, bit_se1065995026697491101ons_or: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit, type, bit_se4730199178511100633sh_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit, type, bit_se5668285175392031749et_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit, type, bit_se2584673776208193580ke_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit, type, bit_se2638667681897837118et_bit: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor, type, bit_se5824344971392196577ns_xor: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit, type, bit_se5641148757651400278ts_bit: (!> [A:$tType]: ((A > (nat > $o))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit, type, bit_se6407376104438227557le_bit: (!> [A:$tType]: (((itself @ A) > (nat > $o))))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Otake__bit__num, type, bit_take_bit_num: (nat > (num > (option @ num)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num, type, bit_un7362597486090784418nd_num: (num > (num > (option @ num)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel, type, bit_un4731106466462545111um_rel: (((product_prod @ num) @ num) > (((product_prod @ num) @ num) > $o))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num, type, bit_un6697907153464112080or_num: (num > (num > num))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel, type, bit_un4773296044027857193um_rel: (((product_prod @ num) @ num) > (((product_prod @ num) @ num) > $o))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num, type, bit_un2480387367778600638or_num: (num > (num > (option @ num)))).
% 9.43/3.44 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel, type, bit_un2901131394128224187um_rel: (((product_prod @ num) @ num) > (((product_prod @ num) @ num) > $o))).
% 9.43/3.44 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra, type, boolea2506097494486148201lgebra: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > $o)))))))).
% 9.43/3.44 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff, type, boolea3799213064322606851m_diff: (!> [A:$tType]: (((A > (A > A)) > ((A > (A > A)) > ((A > A) > (A > (A > ((A > (A > A)) > $o))))))))).
% 9.43/3.44 thf(sy_c_Cancellation_Oiterate__add, type, iterate_add: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Code__Numeral_ONat, type, code_Nat: (code_integer > code_natural)).
% 9.43/3.44 thf(sy_c_Code__Numeral_ONeg, type, code_Neg: (num > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_OPos, type, code_Pos: (num > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_OSuc, type, code_Suc: (code_natural > code_natural)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Obit__cut__integer, type, code_bit_cut_integer: (code_integer > ((product_prod @ code_integer) @ $o))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ocr__integer, type, code_cr_integer: (int > (code_integer > $o))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ocr__natural, type, code_cr_natural: (nat > (code_natural > $o))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Odivmod__abs, type, code_divmod_abs: (code_integer > (code_integer > ((product_prod @ code_integer) @ code_integer)))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Odivmod__integer, type, code_divmod_integer: (code_integer > (code_integer > ((product_prod @ code_integer) @ code_integer)))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Odup, type, code_dup: (code_integer > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer, type, code_int_of_integer: (code_integer > int)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int, type, code_integer_of_int: (int > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ointeger__of__nat, type, code_integer_of_nat: (nat > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ointeger__of__natural, type, code_i5400310926305786745atural: (code_natural > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Ointeger__of__num, type, code_integer_of_num: (num > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onat__of__integer, type, code_nat_of_integer: (code_integer > nat)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onatural_Ocase__natural, type, code_case_natural: (!> [T:$tType]: ((T > ((code_natural > T) > (code_natural > T)))))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onatural_Onat__of__natural, type, code_nat_of_natural: (code_natural > nat)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat, type, code_natural_of_nat: (nat > code_natural)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onatural_Orec__natural, type, code_rec_natural: (!> [T:$tType]: ((T > ((code_natural > (T > T)) > (code_natural > T)))))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onatural_Orec__set__natural, type, code_rec_set_natural: (!> [T:$tType]: ((T > ((code_natural > (T > T)) > (code_natural > (T > $o))))))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onatural__of__integer, type, code_n4118661773612635043nteger: (code_integer > code_natural)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onegative, type, code_negative: (num > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Onum__of__integer, type, code_num_of_integer: (code_integer > num)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Opcr__integer, type, code_pcr_integer: (int > (code_integer > $o))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Opcr__natural, type, code_pcr_natural: (nat > (code_natural > $o))).
% 9.43/3.44 thf(sy_c_Code__Numeral_Opositive, type, code_positive: (num > code_integer)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Osize__nat, type, code_size_nat: (code_natural > nat)).
% 9.43/3.44 thf(sy_c_Code__Numeral_Osub, type, code_sub: (num > (num > code_integer))).
% 9.43/3.44 thf(sy_c_Complete__Lattices_OInf__class_OInf, type, complete_Inf_Inf: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Complete__Lattices_OSup__class_OSup, type, complete_Sup_Sup: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible, type, comple1908693960933563346ssible: (!> [A:$tType]: ((((set @ A) > A) > ((A > (A > $o)) > ((A > $o) > $o)))))).
% 9.43/3.44 thf(sy_c_Complete__Partial__Order_Occpo_Ofixp, type, comple187402453842119260l_fixp: (!> [A:$tType]: ((((set @ A) > A) > ((A > (A > $o)) > ((A > A) > A)))))).
% 9.43/3.44 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp, type, comple115746919287870866o_fixp: (!> [A:$tType]: (((A > A) > A)))).
% 9.43/3.44 thf(sy_c_Complete__Partial__Order_Ochain, type, comple1602240252501008431_chain: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Complete__Partial__Order_Omonotone, type, comple7038119648293358887notone: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((B > (B > $o)) > ((A > B) > $o)))))).
% 9.43/3.44 thf(sy_c_Countable_Ofrom__nat, type, from_nat: (!> [A:$tType]: ((nat > A)))).
% 9.43/3.44 thf(sy_c_Countable_Onth__item, type, nth_item: (!> [A:$tType]: ((nat > (set @ ((old_node @ A) @ product_unit)))))).
% 9.43/3.44 thf(sy_c_Countable_Onth__item__rel, type, nth_item_rel: (nat > (nat > $o))).
% 9.43/3.44 thf(sy_c_Countable_Oto__nat, type, to_nat: (!> [A:$tType]: ((A > nat)))).
% 9.43/3.44 thf(sy_c_Divides_Oadjust__div, type, adjust_div: (((product_prod @ int) @ int) > int)).
% 9.43/3.44 thf(sy_c_Divides_Odivmod__nat, type, divmod_nat: (nat > (nat > ((product_prod @ nat) @ nat)))).
% 9.43/3.44 thf(sy_c_Divides_Oeucl__rel__int, type, eucl_rel_int: (int > (int > (((product_prod @ int) @ int) > $o)))).
% 9.43/3.44 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux, type, unique5940410009612947441es_aux: (!> [A:$tType]: ((((product_prod @ A) @ A) > $o)))).
% 9.43/3.44 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod, type, unique8689654367752047608divmod: (!> [A:$tType]: ((num > (num > ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step, type, unique1321980374590559556d_step: (!> [A:$tType]: ((num > (((product_prod @ A) @ A) > ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_Equiv__Relations_Ocongruent, type, equiv_congruent: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((A > B) > $o))))).
% 9.43/3.44 thf(sy_c_Equiv__Relations_Ocongruent2, type, equiv_congruent2: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > ((A > (B > C)) > $o)))))).
% 9.43/3.44 thf(sy_c_Equiv__Relations_Oequiv, type, equiv_equiv: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Equiv__Relations_Oequivp, type, equiv_equivp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Equiv__Relations_Oproj, type, equiv_proj: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Equiv__Relations_Oquotient, type, equiv_quotient: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > (set @ (set @ A))))))).
% 9.43/3.44 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size, type, euclid6346220572633701492n_size: (!> [A:$tType]: ((A > nat)))).
% 9.43/3.44 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment, type, euclid7384307370059645450egment: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer, type, comm_s3205402744901411588hammer: (!> [A:$tType]: ((A > (nat > A))))).
% 9.43/3.44 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact, type, semiring_char_0_fact: (!> [A:$tType]: ((nat > A)))).
% 9.43/3.44 thf(sy_c_Fields_Oinverse__class_Oinverse, type, inverse_inverse: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ocard, type, finite_card: (!> [B:$tType]: (((set @ B) > nat)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ocomp__fun__commute, type, finite6289374366891150609ommute: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms, type, finite1981172056906289455axioms: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms, type, finite4980608107308702382axioms: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofold, type, finite_fold: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((set @ A) > B)))))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofolding, type, finite_folding: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofolding__idem, type, finite_folding_idem: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofolding__idem__axioms, type, finite7837460588564673216axioms: (!> [A:$tType,B:$tType]: (((A > (B > B)) > $o)))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofolding__idem__on, type, finite1890593828518410140dem_on: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))).
% 9.43/3.44 thf(sy_c_Finite__Set_Ofolding__idem__on__axioms, type, finite6916993218817215295axioms: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (B > B)) > $o))))).
% 9.43/3.44 thf(sy_c_Fun_Obij__betw, type, bij_betw: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > ((set @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 9.43/3.44 thf(sy_c_Fun_Ofcomp, type, fcomp: (!> [A:$tType,B:$tType,C:$tType]: (((A > B) > ((B > C) > (A > C)))))).
% 9.43/3.44 thf(sy_c_Fun_Ofun__upd, type, fun_upd: (!> [A:$tType,B:$tType]: (((A > B) > (A > (B > (A > B))))))).
% 9.43/3.44 thf(sy_c_Fun_Oid, type, id: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Fun_Omap__fun, type, map_fun: (!> [C:$tType,A:$tType,B:$tType,D:$tType]: (((C > A) > ((B > D) > ((A > B) > (C > D))))))).
% 9.43/3.44 thf(sy_c_Fun_Omono__on, type, mono_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Fun_Ooverride__on, type, override_on: (!> [A:$tType,B:$tType]: (((A > B) > ((A > B) > ((set @ A) > (A > B))))))).
% 9.43/3.44 thf(sy_c_Fun_Ostrict__mono__on, type, strict_mono_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 9.43/3.44 thf(sy_c_Fun__Def_Omax__strict, type, fun_max_strict: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 9.43/3.44 thf(sy_c_Fun__Def_Omax__weak, type, fun_max_weak: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 9.43/3.44 thf(sy_c_Fun__Def_Omin__strict, type, fun_min_strict: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 9.43/3.44 thf(sy_c_Fun__Def_Omin__weak, type, fun_min_weak: (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))).
% 9.43/3.44 thf(sy_c_Fun__Def_Opair__leq, type, fun_pair_leq: (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))).
% 9.43/3.44 thf(sy_c_Fun__Def_Opair__less, type, fun_pair_less: (set @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)))).
% 9.43/3.44 thf(sy_c_Fun__Def_Oreduction__pair, type, fun_reduction_pair: (!> [A:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > $o)))).
% 9.43/3.44 thf(sy_c_Fun__Def_Orp__inv__image, type, fun_rp_inv_image: (!> [A:$tType,B:$tType]: ((((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) > ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))))))).
% 9.43/3.44 thf(sy_c_GCD_OGcd__class_OGcd, type, gcd_Gcd: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_GCD_OGcd__class_OLcm, type, gcd_Lcm: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_GCD_Obezw, type, bezw: (nat > (nat > ((product_prod @ int) @ int)))).
% 9.43/3.44 thf(sy_c_GCD_Obezw__rel, type, bezw_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 9.43/3.44 thf(sy_c_GCD_Obounded__quasi__semilattice, type, bounde8507323023520639062attice: (!> [A:$tType]: (((A > (A > A)) > (A > (A > ((A > A) > $o))))))).
% 9.43/3.44 thf(sy_c_GCD_Obounded__quasi__semilattice__set, type, bounde6485984586167503788ce_set: (!> [A:$tType]: (((A > (A > A)) > (A > (A > ((A > A) > $o))))))).
% 9.43/3.44 thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF, type, bounde2362111253966948842tice_F: (!> [A:$tType]: (((A > (A > A)) > (A > (A > ((set @ A) > A))))))).
% 9.43/3.44 thf(sy_c_GCD_Ogcd__class_Ogcd, type, gcd_gcd: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_GCD_Ogcd__class_Olcm, type, gcd_lcm: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_GCD_Ogcd__nat__rel, type, gcd_nat_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 9.43/3.44 thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin, type, semiring_gcd_Gcd_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin, type, semiring_gcd_Lcm_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Groups_Oabel__semigroup, type, abel_semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 9.43/3.44 thf(sy_c_Groups_Oabel__semigroup__axioms, type, abel_s757365448890700780axioms: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 9.43/3.44 thf(sy_c_Groups_Oabs__class_Oabs, type, abs_abs: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Groups_Ocomm__monoid, type, comm_monoid: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups_Ocomm__monoid__axioms, type, comm_monoid_axioms: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups_Ogroup, type, group: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > A) > $o)))))).
% 9.43/3.44 thf(sy_c_Groups_Ogroup__axioms, type, group_axioms: (!> [A:$tType]: (((A > (A > A)) > (A > ((A > A) > $o)))))).
% 9.43/3.44 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Groups_Omonoid, type, monoid: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups_Omonoid__axioms, type, monoid_axioms: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups_Oone__class_Oone, type, one_one: (!> [A:$tType]: (A))).
% 9.43/3.44 thf(sy_c_Groups_Oplus__class_Oplus, type, plus_plus: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Groups_Osemigroup, type, semigroup: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 9.43/3.44 thf(sy_c_Groups_Osgn__class_Osgn, type, sgn_sgn: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Groups_Otimes__class_Otimes, type, times_times: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Groups_Ozero__class_Ozero, type, zero_zero: (!> [A:$tType]: (A))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum, type, groups3894954378712506084id_sum: (!> [A:$tType,B:$tType]: (((A > (A > A)) > (A > ((B > A) > ((set @ B) > A))))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum, type, groups7311177749621191930dd_sum: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H, type, groups1027152243600224163dd_sum: (!> [C:$tType,A:$tType]: (((C > A) > ((set @ C) > A))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod, type, groups7121269368397514597t_prod: (!> [B:$tType,A:$tType]: (((B > A) > ((set @ B) > A))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H, type, groups1962203154675924110t_prod: (!> [C:$tType,A:$tType]: (((C > A) > ((set @ C) > A))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__set, type, groups778175481326437816id_set: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__set_OF, type, groups_comm_monoid_F: (!> [A:$tType,B:$tType]: (((A > (A > A)) > (A > ((B > A) > ((set @ B) > A))))))).
% 9.43/3.44 thf(sy_c_Groups__Big_Ocomm__monoid__set_OG, type, groups_comm_monoid_G: (!> [A:$tType,B:$tType]: (((A > (A > A)) > (A > ((B > A) > ((set @ B) > A))))))).
% 9.43/3.44 thf(sy_c_Groups__List_Ocomm__monoid__list, type, groups1828464146339083142d_list: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups__List_Ocomm__monoid__list__set, type, groups4802862169904069756st_set: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum, type, groups4207007520872428315er_sum: (!> [B:$tType,A:$tType]: (((B > A) > (A > ((list @ B) > A)))))).
% 9.43/3.44 thf(sy_c_Groups__List_Omonoid__add_Osum__list, type, groups4543113879258116180m_list: (!> [A:$tType]: (((A > (A > A)) > (A > ((list @ A) > A)))))).
% 9.43/3.44 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list, type, groups8242544230860333062m_list: (!> [A:$tType]: (((list @ A) > A)))).
% 9.43/3.44 thf(sy_c_Groups__List_Omonoid__list, type, groups_monoid_list: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Groups__List_Omonoid__list_OF, type, groups_monoid_F: (!> [A:$tType]: (((A > (A > A)) > (A > ((list @ A) > A)))))).
% 9.43/3.44 thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list, type, groups5270119922927024881d_list: (!> [A:$tType]: (((list @ A) > A)))).
% 9.43/3.44 thf(sy_c_HOL_ONO__MATCH, type, nO_MATCH: (!> [A:$tType,B:$tType]: ((A > (B > $o))))).
% 9.43/3.44 thf(sy_c_HOL_OThe, type, the: (!> [A:$tType]: (((A > $o) > A)))).
% 9.43/3.44 thf(sy_c_HOL_OUniq, type, uniq: (!> [A:$tType]: (((A > $o) > $o)))).
% 9.43/3.44 thf(sy_c_HOL_Odefault__class_Odefault, type, default_default: (!> [A:$tType]: (A))).
% 9.43/3.44 thf(sy_c_HOL_Oundefined, type, undefined: (!> [A:$tType]: (A))).
% 9.43/3.44 thf(sy_c_Heap_Oaddr__of__array, type, addr_of_array: (!> [A:$tType]: (((array @ A) > nat)))).
% 9.43/3.44 thf(sy_c_Heap_Oaddr__of__ref, type, addr_of_ref: (!> [A:$tType]: (((ref @ A) > nat)))).
% 9.43/3.44 thf(sy_c_Heap_Oarray_OArray, type, array2: (!> [A:$tType]: ((nat > (array @ A))))).
% 9.43/3.44 thf(sy_c_Heap_Oheap_Oarrays, type, arrays: (!> [Z:$tType]: (((heap_ext @ Z) > (typerep > (nat > (list @ nat))))))).
% 9.43/3.44 thf(sy_c_Heap_Oheap_Oarrays__update, type, arrays_update: (!> [Z:$tType]: ((((typerep > (nat > (list @ nat))) > (typerep > (nat > (list @ nat)))) > ((heap_ext @ Z) > (heap_ext @ Z)))))).
% 9.43/3.44 thf(sy_c_Heap_Oheap_Olim, type, lim: (!> [Z:$tType]: (((heap_ext @ Z) > nat)))).
% 9.43/3.44 thf(sy_c_Heap_Oheap_Olim__update, type, lim_update: (!> [Z:$tType]: (((nat > nat) > ((heap_ext @ Z) > (heap_ext @ Z)))))).
% 9.43/3.44 thf(sy_c_Heap_Oheap_Orefs, type, refs: (!> [Z:$tType]: (((heap_ext @ Z) > (typerep > (nat > nat)))))).
% 9.43/3.44 thf(sy_c_Heap_Oref_ORef, type, ref2: (!> [A:$tType]: ((nat > (ref @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_OHeap, type, heap_Time_Heap2: (!> [A:$tType]: ((((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap, type, heap_Time_case_Heap: (!> [A:$tType,B:$tType]: (((((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > B) > ((heap_Time_Heap @ A) > B))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap, type, heap_Time_map_Heap: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((heap_Time_Heap @ A) > (heap_Time_Heap @ Aa)))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap, type, heap_Time_pred_Heap: (!> [A:$tType]: (((A > $o) > ((heap_Time_Heap @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap, type, heap_Time_rec_Heap: (!> [A:$tType,C:$tType]: (((((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > C) > ((heap_Time_Heap @ A) > C))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap, type, heap_Time_rel_Heap: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((heap_Time_Heap @ A) > ((heap_Time_Heap @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap, type, heap_Time_set_Heap: (!> [A:$tType]: (((heap_Time_Heap @ A) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap, type, heap_Time_size_Heap: (!> [A:$tType]: (((A > nat) > ((heap_Time_Heap @ A) > nat))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap__lub, type, heap_Time_Heap_lub: (!> [A:$tType]: (((set @ (heap_Time_Heap @ A)) > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OHeap__ord, type, heap_Time_Heap_ord: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_Time_Heap @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oassert, type, heap_Time_assert: (!> [A:$tType]: (((A > $o) > (A > (heap_Time_Heap @ A)))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Obind, type, heap_Time_bind: (!> [A:$tType,B:$tType]: (((heap_Time_Heap @ A) > ((A > (heap_Time_Heap @ B)) > (heap_Time_Heap @ B)))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oeffect, type, heap_Time_effect: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > ((heap_ext @ product_unit) > (A > (nat > $o)))))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oexecute, type, heap_Time_execute: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oguard, type, heap_Time_guard: (!> [A:$tType]: ((((heap_ext @ product_unit) > $o) > (((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) > (heap_Time_Heap @ A)))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oheap, type, heap_Time_heap: (!> [A:$tType]: ((((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Olift, type, heap_Time_lift: (!> [A:$tType,B:$tType]: (((A > B) > (A > (heap_Time_Heap @ B)))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oraise, type, heap_Time_raise: (!> [A:$tType]: (((list @ char) > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oreturn, type, heap_Time_return: (!> [A:$tType]: ((A > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Osuccess, type, heap_Time_success: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > $o))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Otap, type, heap_Time_tap: (!> [A:$tType]: ((((heap_ext @ product_unit) > A) > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OtimeFrame, type, heap_Time_timeFrame: (!> [A:$tType]: ((nat > ((option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_OtimeFrame__rel, type, heap_T5500966940807335491me_rel: (!> [A:$tType]: ((((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > (((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > $o))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Oureturn, type, heap_Time_ureturn: (!> [A:$tType]: ((A > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Heap__Time__Monad_Owait, type, heap_Time_wait: (nat > (heap_Time_Heap @ product_unit))).
% 9.43/3.44 thf(sy_c_Hilbert__Choice_Obijection, type, hilbert_bijection: (!> [A:$tType]: (((A > A) > $o)))).
% 9.43/3.44 thf(sy_c_Hilbert__Choice_Oinv__into, type, hilbert_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 9.43/3.44 thf(sy_c_Hoare__Triple_Ohoare__triple, type, hoare_hoare_triple: (!> [A:$tType]: ((assn > ((heap_Time_Heap @ A) > ((A > assn) > $o)))))).
% 9.43/3.44 thf(sy_c_Hoare__Triple_Onew__addrs, type, hoare_new_addrs: ((heap_ext @ product_unit) > ((set @ nat) > ((heap_ext @ product_unit) > (set @ nat))))).
% 9.43/3.44 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 9.43/3.44 thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp, type, complete_lattice_lfp: (!> [A:$tType]: (((A > A) > A)))).
% 9.43/3.44 thf(sy_c_Int_OAbs__Integ, type, abs_Integ: (((product_prod @ nat) @ nat) > int)).
% 9.43/3.44 thf(sy_c_Int_ONeg, type, neg: (num > int)).
% 9.43/3.44 thf(sy_c_Int_OPos, type, pos: (num > int)).
% 9.43/3.44 thf(sy_c_Int_ORep__Integ, type, rep_Integ: (int > ((product_prod @ nat) @ nat))).
% 9.43/3.44 thf(sy_c_Int_Odup, type, dup: (int > int)).
% 9.43/3.44 thf(sy_c_Int_Ointrel, type, intrel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 9.43/3.44 thf(sy_c_Int_Onat, type, nat2: (int > nat)).
% 9.43/3.44 thf(sy_c_Int_Opcr__int, type, pcr_int: (((product_prod @ nat) @ nat) > (int > $o))).
% 9.43/3.44 thf(sy_c_Int_Opower__int, type, power_int: (!> [A:$tType]: ((A > (int > A))))).
% 9.43/3.44 thf(sy_c_Int_Oring__1__class_OInts, type, ring_1_Ints: (!> [A:$tType]: ((set @ A)))).
% 9.43/3.44 thf(sy_c_Int_Oring__1__class_Oof__int, type, ring_1_of_int: (!> [A:$tType]: ((int > A)))).
% 9.43/3.44 thf(sy_c_Int_Osub, type, sub: (num > (num > int))).
% 9.43/3.44 thf(sy_c_Lattices_Oinf__class_Oinf, type, inf_inf: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Lattices_Osemilattice__neutr, type, semilattice_neutr: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Olinorder__class_OMax, type, lattic643756798349783984er_Max: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Olinorder__class_OMin, type, lattic643756798350308766er_Min: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin, type, lattic7752659483105999362nf_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF, type, lattic5214292709420241887eutr_F: (!> [A:$tType]: (((A > (A > A)) > (A > ((set @ A) > A)))))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Osemilattice__set, type, lattic149705377957585745ce_set: (!> [A:$tType]: (((A > (A > A)) > $o)))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Osemilattice__set_OF, type, lattic1715443433743089157tice_F: (!> [A:$tType]: (((A > (A > A)) > ((set @ A) > A))))).
% 9.43/3.44 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin, type, lattic5882676163264333800up_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 9.43/3.44 thf(sy_c_Lifting_OQuotient, type, quotient: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > ((A > (B > $o)) > $o))))))).
% 9.43/3.44 thf(sy_c_List_Oappend, type, append: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Oarg__min__list, type, arg_min_list: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > A))))).
% 9.43/3.44 thf(sy_c_List_Oarg__min__list__rel, type, arg_min_list_rel: (!> [A:$tType,B:$tType]: ((((product_prod @ (A > B)) @ (list @ A)) > (((product_prod @ (A > B)) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_List_Obutlast, type, butlast: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Oconcat, type, concat: (!> [A:$tType]: (((list @ (list @ A)) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Ocount__list, type, count_list: (!> [A:$tType]: (((list @ A) > (A > nat))))).
% 9.43/3.44 thf(sy_c_List_Odistinct, type, distinct: (!> [A:$tType]: (((list @ A) > $o)))).
% 9.43/3.44 thf(sy_c_List_Odrop, type, drop: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_OdropWhile, type, dropWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Oenumerate, type, enumerate: (!> [A:$tType]: ((nat > ((list @ A) > (list @ ((product_prod @ nat) @ A))))))).
% 9.43/3.44 thf(sy_c_List_Oextract, type, extract: (!> [A:$tType]: (((A > $o) > ((list @ A) > (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))))))).
% 9.43/3.44 thf(sy_c_List_Ofilter, type, filter: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Ofind, type, find: (!> [A:$tType]: (((A > $o) > ((list @ A) > (option @ A)))))).
% 9.43/3.44 thf(sy_c_List_Ofold, type, fold: (!> [A:$tType,B:$tType]: (((A > (B > B)) > ((list @ A) > (B > B)))))).
% 9.43/3.44 thf(sy_c_List_Ofoldl, type, foldl: (!> [B:$tType,A:$tType]: (((B > (A > B)) > (B > ((list @ A) > B)))))).
% 9.43/3.44 thf(sy_c_List_Ofoldr, type, foldr: (!> [A:$tType,B:$tType]: (((A > (B > B)) > ((list @ A) > (B > B)))))).
% 9.43/3.44 thf(sy_c_List_Ogen__length, type, gen_length: (!> [A:$tType]: ((nat > ((list @ A) > nat))))).
% 9.43/3.44 thf(sy_c_List_Olast, type, last: (!> [A:$tType]: (((list @ A) > A)))).
% 9.43/3.44 thf(sy_c_List_Olenlex, type, lenlex: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Olex, type, lex: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Olexn, type, lexn: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (nat > (set @ ((product_prod @ (list @ A)) @ (list @ A)))))))).
% 9.43/3.44 thf(sy_c_List_Olexord, type, lexord: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Olexordp, type, lexordp: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_List_Olinorder__class_Osort__key, type, linorder_sort_key: (!> [B:$tType,A:$tType]: (((B > A) > ((list @ B) > (list @ B)))))).
% 9.43/3.44 thf(sy_c_List_Olinorder__class_Osorted__list__of__set, type, linord4507533701916653071of_set: (!> [A:$tType]: (((set @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Olist_OCons, type, cons: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Olist_ONil, type, nil: (!> [A:$tType]: ((list @ A)))).
% 9.43/3.44 thf(sy_c_List_Olist_Ocase__list, type, case_list: (!> [B:$tType,A:$tType]: ((B > ((A > ((list @ A) > B)) > ((list @ A) > B)))))).
% 9.43/3.44 thf(sy_c_List_Olist_Ohd, type, hd: (!> [A:$tType]: (((list @ A) > A)))).
% 9.43/3.44 thf(sy_c_List_Olist_Olist__all, type, list_all: (!> [A:$tType]: (((A > $o) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_List_Olist_Olist__all2, type, list_all2: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((list @ A) > ((list @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_List_Olist_Omap, type, map: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((list @ A) > (list @ Aa)))))).
% 9.43/3.44 thf(sy_c_List_Olist_Oset, type, set2: (!> [A:$tType]: (((list @ A) > (set @ A))))).
% 9.43/3.44 thf(sy_c_List_Olist_Osize__list, type, size_list: (!> [A:$tType]: (((A > nat) > ((list @ A) > nat))))).
% 9.43/3.44 thf(sy_c_List_Olist_Otl, type, tl: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Olist__update, type, list_update: (!> [A:$tType]: (((list @ A) > (nat > (A > (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Olistrel, type, listrel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ (list @ A)) @ (list @ B))))))).
% 9.43/3.44 thf(sy_c_List_Olistrel1, type, listrel1: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (list @ A)) @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Olistrel1p, type, listrel1p: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_List_Olistrelp, type, listrelp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((list @ A) > ((list @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_List_Olists, type, lists: (!> [A:$tType]: (((set @ A) > (set @ (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Omap__filter, type, map_filter: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((list @ A) > (list @ B)))))).
% 9.43/3.44 thf(sy_c_List_Omap__project, type, map_project: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((set @ A) > (set @ B)))))).
% 9.43/3.44 thf(sy_c_List_Omap__tailrec__rev, type, map_tailrec_rev: (!> [A:$tType,B:$tType]: (((A > B) > ((list @ A) > ((list @ B) > (list @ B))))))).
% 9.43/3.44 thf(sy_c_List_Omap__tailrec__rev__rel, type, map_tailrec_rev_rel: (!> [A:$tType,B:$tType]: ((((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) > (((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) > $o))))).
% 9.43/3.44 thf(sy_c_List_Omeasures, type, measures: (!> [A:$tType]: (((list @ (A > nat)) > (set @ ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_List_Omin__list, type, min_list: (!> [A:$tType]: (((list @ A) > A)))).
% 9.43/3.44 thf(sy_c_List_Omin__list__rel, type, min_list_rel: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_List_Onth, type, nth: (!> [A:$tType]: (((list @ A) > (nat > A))))).
% 9.43/3.44 thf(sy_c_List_Onths, type, nths: (!> [A:$tType]: (((list @ A) > ((set @ nat) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Oord__class_Olexordp, type, ord_lexordp: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_List_Opartition, type, partition: (!> [A:$tType]: (((A > $o) > ((list @ A) > ((product_prod @ (list @ A)) @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Oproduct, type, product: (!> [A:$tType,B:$tType]: (((list @ A) > ((list @ B) > (list @ ((product_prod @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_List_Oproduct__lists, type, product_lists: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Oremdups, type, remdups: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Oremdups__adj, type, remdups_adj: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Oremdups__adj__rel, type, remdups_adj_rel: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_List_Oremove1, type, remove1: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Oreplicate, type, replicate: (!> [A:$tType]: ((nat > (A > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Orev, type, rev: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Orotate, type, rotate: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Orotate1, type, rotate1: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_List_Oshuffles, type, shuffles: (!> [A:$tType]: (((list @ A) > ((list @ A) > (set @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_List_Oshuffles__rel, type, shuffles_rel: (!> [A:$tType]: ((((product_prod @ (list @ A)) @ (list @ A)) > (((product_prod @ (list @ A)) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_List_Osorted__wrt, type, sorted_wrt: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_List_Osorted__wrt__rel, type, sorted_wrt_rel: (!> [A:$tType]: ((((product_prod @ (A > (A > $o))) @ (list @ A)) > (((product_prod @ (A > (A > $o))) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_List_Osplice, type, splice: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Osplice__rel, type, splice_rel: (!> [A:$tType]: ((((product_prod @ (list @ A)) @ (list @ A)) > (((product_prod @ (list @ A)) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_List_Osuccessively, type, successively: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_List_Osuccessively__rel, type, successively_rel: (!> [A:$tType]: ((((product_prod @ (A > (A > $o))) @ (list @ A)) > (((product_prod @ (A > (A > $o))) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_List_Otake, type, take: (!> [A:$tType]: ((nat > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_OtakeWhile, type, takeWhile: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Othose, type, those: (!> [A:$tType]: (((list @ (option @ A)) > (option @ (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Otranspose, type, transpose: (!> [A:$tType]: (((list @ (list @ A)) > (list @ (list @ A)))))).
% 9.43/3.44 thf(sy_c_List_Otranspose__rel, type, transpose_rel: (!> [A:$tType]: (((list @ (list @ A)) > ((list @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_List_Oupt, type, upt: (nat > (nat > (list @ nat)))).
% 9.43/3.44 thf(sy_c_List_Oupto, type, upto: (int > (int > (list @ int)))).
% 9.43/3.44 thf(sy_c_List_Oupto__rel, type, upto_rel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))).
% 9.43/3.44 thf(sy_c_List_Ozip, type, zip: (!> [A:$tType,B:$tType]: (((list @ A) > ((list @ B) > (list @ ((product_prod @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_Map_Odom, type, dom: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Map_Ograph, type, graph: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ ((product_prod @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Map_Omap__add, type, map_add: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((A > (option @ B)) > (A > (option @ B))))))).
% 9.43/3.44 thf(sy_c_Map_Omap__comp, type, map_comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > (option @ C)) > ((A > (option @ B)) > (A > (option @ C))))))).
% 9.43/3.44 thf(sy_c_Map_Omap__le, type, map_le: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((A > (option @ B)) > $o))))).
% 9.43/3.44 thf(sy_c_Map_Omap__of, type, map_of: (!> [A:$tType,B:$tType]: (((list @ ((product_prod @ A) @ B)) > (A > (option @ B)))))).
% 9.43/3.44 thf(sy_c_Map_Omap__upds, type, map_upds: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((list @ A) > ((list @ B) > (A > (option @ B)))))))).
% 9.43/3.44 thf(sy_c_Map_Oran, type, ran: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ B))))).
% 9.43/3.44 thf(sy_c_Map_Orestrict__map, type, restrict_map: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((set @ A) > (A > (option @ B))))))).
% 9.43/3.44 thf(sy_c_Misc_OCODE__ABORT, type, cODE_ABORT: (!> [A:$tType]: (((product_unit > A) > A)))).
% 9.43/3.44 thf(sy_c_Misc_OEps__Opt, type, eps_Opt: (!> [A:$tType]: (((A > $o) > (option @ A))))).
% 9.43/3.44 thf(sy_c_Misc_Obijective, type, bijective: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > $o)))).
% 9.43/3.44 thf(sy_c_Misc_Obrk__rel, type, brk_rel: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B))))))).
% 9.43/3.44 thf(sy_c_Misc_Odflt__None__set, type, dflt_None_set: (!> [A:$tType]: (((set @ A) > (option @ (set @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Ofilter__rev, type, filter_rev: (!> [A:$tType]: (((A > $o) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Ofilter__rev__aux, type, filter_rev_aux: (!> [A:$tType]: (((list @ A) > ((A > $o) > ((list @ A) > (list @ A))))))).
% 9.43/3.44 thf(sy_c_Misc_Ofun__of__rel, type, fun_of_rel: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > A))))).
% 9.43/3.44 thf(sy_c_Misc_Oinv__on, type, inv_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (B > A)))))).
% 9.43/3.44 thf(sy_c_Misc_Olist__all__zip, type, list_all_zip: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((list @ A) > ((list @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_Misc_Olist__all__zip__rel, type, list_all_zip_rel: (!> [A:$tType,B:$tType]: ((((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) > (((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Olist__collect__set, type, list_collect_set: (!> [B:$tType,A:$tType]: (((B > (set @ A)) > ((list @ B) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Omap__mmupd, type, map_mmupd: (!> [B:$tType,A:$tType]: (((B > (option @ A)) > ((set @ B) > (A > (B > (option @ A)))))))).
% 9.43/3.44 thf(sy_c_Misc_Omap__to__set, type, map_to_set: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > (set @ ((product_prod @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Misc_Omerge, type, merge: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Omerge__list, type, merge_list: (!> [A:$tType]: (((list @ (list @ A)) > ((list @ (list @ A)) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Omerge__list__rel, type, merge_list_rel: (!> [A:$tType]: ((((product_prod @ (list @ (list @ A))) @ (list @ (list @ A))) > (((product_prod @ (list @ (list @ A))) @ (list @ (list @ A))) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Omerge__rel, type, merge_rel: (!> [A:$tType]: ((((product_prod @ (list @ A)) @ (list @ A)) > (((product_prod @ (list @ A)) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort, type, mergesort: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__by__rel, type, mergesort_by_rel: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__by__rel__merge, type, merges9089515139780605204_merge: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > (list @ A))))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__by__rel__merge__rel, type, merges2244889521215249637ge_rel: (!> [A:$tType]: ((((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) > (((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__by__rel__rel, type, mergesort_by_rel_rel: (!> [A:$tType]: ((((product_prod @ (A > (A > $o))) @ (list @ A)) > (((product_prod @ (A > (A > $o))) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__by__rel__split, type, merges295452479951948502_split: (!> [A:$tType]: ((((product_prod @ (list @ A)) @ (list @ A)) > ((list @ A) > ((product_prod @ (list @ A)) @ (list @ A))))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__by__rel__split__rel, type, merges7066485432131860899it_rel: (!> [A:$tType]: ((((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) > (((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Omergesort__remdups, type, mergesort_remdups: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_Misc_Opairself, type, pairself: (!> [A:$tType,B:$tType]: (((A > B) > (((product_prod @ A) @ A) > ((product_prod @ B) @ B)))))).
% 9.43/3.44 thf(sy_c_Misc_Opairself__rel, type, pairself_rel: (!> [A:$tType,B:$tType]: ((((product_prod @ (A > B)) @ ((product_prod @ A) @ A)) > (((product_prod @ (A > B)) @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Opartition__rev, type, partition_rev: (!> [A:$tType]: (((A > $o) > (((product_prod @ (list @ A)) @ (list @ A)) > ((list @ A) > ((product_prod @ (list @ A)) @ (list @ A)))))))).
% 9.43/3.44 thf(sy_c_Misc_Opartition__rev__rel, type, partition_rev_rel: (!> [A:$tType]: ((((product_prod @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) > (((product_prod @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Oquicksort__by__rel, type, quicksort_by_rel: (!> [A:$tType]: (((A > (A > $o)) > ((list @ A) > ((list @ A) > (list @ A))))))).
% 9.43/3.44 thf(sy_c_Misc_Oquicksort__by__rel__rel, type, quicksort_by_rel_rel: (!> [A:$tType]: ((((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) > (((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Orel__of, type, rel_of: (!> [A:$tType,B:$tType]: (((A > (option @ B)) > ((((product_prod @ A) @ B) > $o) > (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_Misc_Orel__restrict, type, rel_restrict: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_Misc_Oremove__rev, type, remove_rev: (!> [A:$tType]: ((A > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Orevg, type, revg: (!> [A:$tType]: (((list @ A) > ((list @ A) > (list @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Orevg__rel, type, revg_rel: (!> [A:$tType]: ((((product_prod @ (list @ A)) @ (list @ A)) > (((product_prod @ (list @ A)) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Oset__to__map, type, set_to_map: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ A)) > (B > (option @ A)))))).
% 9.43/3.44 thf(sy_c_Misc_Oslice, type, slice: (!> [A:$tType]: ((nat > (nat > ((list @ A) > (list @ A))))))).
% 9.43/3.44 thf(sy_c_Misc_Osu__rel__fun, type, su_rel_fun: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((A > B) > $o))))).
% 9.43/3.44 thf(sy_c_Misc_Othe__default, type, the_default: (!> [A:$tType]: ((A > ((option @ A) > A))))).
% 9.43/3.44 thf(sy_c_Misc_Ouncurry, type, uncurry: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 9.43/3.44 thf(sy_c_Misc_Ozipf, type, zipf: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > ((list @ A) > ((list @ B) > (list @ C))))))).
% 9.43/3.44 thf(sy_c_Misc_Ozipf__rel, type, zipf_rel: (!> [A:$tType,B:$tType,C:$tType]: ((((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) > (((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) > $o))))).
% 9.43/3.44 thf(sy_c_Multiset_Oadd__mset, type, add_mset: (!> [A:$tType]: ((A > ((multiset @ A) > (multiset @ A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset, type, comm_monoid_sum_mset: (!> [A:$tType]: (((A > (A > A)) > (A > ((multiset @ A) > A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset, type, comm_m7189776963980413722m_mset: (!> [A:$tType]: (((multiset @ A) > A)))).
% 9.43/3.44 thf(sy_c_Multiset_Ocomm__monoid__mset, type, comm_monoid_mset: (!> [A:$tType]: (((A > (A > A)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Multiset_Ocomm__monoid__mset_OF, type, comm_monoid_F: (!> [A:$tType]: (((A > (A > A)) > (A > ((multiset @ A) > A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset, type, comm_m9189036328036947845d_mset: (!> [A:$tType]: (((multiset @ A) > A)))).
% 9.43/3.44 thf(sy_c_Multiset_Ofilter__mset, type, filter_mset: (!> [A:$tType]: (((A > $o) > ((multiset @ A) > (multiset @ A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Ofold__mset, type, fold_mset: (!> [A:$tType,B:$tType]: (((A > (B > B)) > (B > ((multiset @ A) > B)))))).
% 9.43/3.44 thf(sy_c_Multiset_Oimage__mset, type, image_mset: (!> [A:$tType,B:$tType]: (((A > B) > ((multiset @ A) > (multiset @ B)))))).
% 9.43/3.44 thf(sy_c_Multiset_Ointer__mset, type, inter_mset: (!> [A:$tType]: (((multiset @ A) > ((multiset @ A) > (multiset @ A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Olinorder__class_Opart, type, linorder_part: (!> [B:$tType,A:$tType]: (((B > A) > (A > ((list @ B) > ((product_prod @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))))))))).
% 9.43/3.44 thf(sy_c_Multiset_Oms__strict, type, ms_strict: (set @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))))).
% 9.43/3.44 thf(sy_c_Multiset_Oms__weak, type, ms_weak: (set @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))))).
% 9.43/3.44 thf(sy_c_Multiset_Omset, type, mset: (!> [A:$tType]: (((list @ A) > (multiset @ A))))).
% 9.43/3.44 thf(sy_c_Multiset_Omset__set, type, mset_set: (!> [B:$tType]: (((set @ B) > (multiset @ B))))).
% 9.43/3.44 thf(sy_c_Multiset_Omult, type, mult: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (multiset @ A)) @ (multiset @ A))))))).
% 9.43/3.44 thf(sy_c_Multiset_Omult1, type, mult1: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (multiset @ A)) @ (multiset @ A))))))).
% 9.43/3.44 thf(sy_c_Multiset_Omulteqp__code, type, multeqp_code: (!> [A:$tType]: (((A > (A > $o)) > ((multiset @ A) > ((multiset @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_Multiset_Omultiset_OAbs__multiset, type, abs_multiset: (!> [A:$tType]: (((A > nat) > (multiset @ A))))).
% 9.43/3.44 thf(sy_c_Multiset_Omultiset_Ocount, type, count: (!> [A:$tType]: (((multiset @ A) > (A > nat))))).
% 9.43/3.44 thf(sy_c_Multiset_Omultp, type, multp: (!> [A:$tType]: (((A > (A > $o)) > ((multiset @ A) > ((multiset @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_Multiset_Omultp__code, type, multp_code: (!> [A:$tType]: (((A > (A > $o)) > ((multiset @ A) > ((multiset @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_Multiset_Opcr__multiset, type, pcr_multiset: (!> [C:$tType,B:$tType]: (((C > (B > $o)) > ((C > nat) > ((multiset @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_Multiset_Opw__leq, type, pw_leq: ((multiset @ ((product_prod @ nat) @ nat)) > ((multiset @ ((product_prod @ nat) @ nat)) > $o))).
% 9.43/3.44 thf(sy_c_Multiset_Orepeat__mset, type, repeat_mset: (!> [A:$tType]: ((nat > ((multiset @ A) > (multiset @ A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Oreplicate__mset, type, replicate_mset: (!> [A:$tType]: ((nat > (A > (multiset @ A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Oset__mset, type, set_mset: (!> [A:$tType]: (((multiset @ A) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Multiset_Osize__multiset, type, size_multiset: (!> [A:$tType]: (((A > nat) > ((multiset @ A) > nat))))).
% 9.43/3.44 thf(sy_c_Multiset_Osubset__eq__mset__impl, type, subset_eq_mset_impl: (!> [A:$tType]: (((list @ A) > ((list @ A) > (option @ $o)))))).
% 9.43/3.44 thf(sy_c_Multiset_Osubset__eq__mset__impl__rel, type, subset751672762298770561pl_rel: (!> [A:$tType]: ((((product_prod @ (list @ A)) @ (list @ A)) > (((product_prod @ (list @ A)) @ (list @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Multiset_Osubset__mset, type, subset_mset: (!> [A:$tType]: (((multiset @ A) > ((multiset @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Multiset_Osubseteq__mset, type, subseteq_mset: (!> [A:$tType]: (((multiset @ A) > ((multiset @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Multiset_Ounion__mset, type, union_mset: (!> [A:$tType]: (((multiset @ A) > ((multiset @ A) > (multiset @ A)))))).
% 9.43/3.44 thf(sy_c_Multiset_Owcount, type, wcount: (!> [A:$tType]: (((A > nat) > ((multiset @ A) > (A > nat)))))).
% 9.43/3.44 thf(sy_c_Nat_OSuc, type, suc: (nat > nat)).
% 9.43/3.44 thf(sy_c_Nat_Ocompow, type, compow: (!> [A:$tType]: ((nat > (A > A))))).
% 9.43/3.44 thf(sy_c_Nat_Onat_Ocase__nat, type, case_nat: (!> [A:$tType]: ((A > ((nat > A) > (nat > A)))))).
% 9.43/3.44 thf(sy_c_Nat_Oold_Onat_Orec__nat, type, rec_nat: (!> [T:$tType]: ((T > ((nat > (T > T)) > (nat > T)))))).
% 9.43/3.44 thf(sy_c_Nat_Osemiring__1__class_Oof__nat, type, semiring_1_of_nat: (!> [A:$tType]: ((nat > A)))).
% 9.43/3.44 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux, type, semiri8178284476397505188at_aux: (!> [A:$tType]: (((A > A) > (nat > (A > A)))))).
% 9.43/3.44 thf(sy_c_Nat_Osize__class_Osize, type, size_size: (!> [A:$tType]: ((A > nat)))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Olist__decode, type, nat_list_decode: (nat > (list @ nat))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Olist__decode__rel, type, nat_list_decode_rel: (nat > (nat > $o))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Olist__encode, type, nat_list_encode: ((list @ nat) > nat)).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Olist__encode__rel, type, nat_list_encode_rel: ((list @ nat) > ((list @ nat) > $o))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Oprod__decode, type, nat_prod_decode: (nat > ((product_prod @ nat) @ nat))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Oprod__decode__aux, type, nat_prod_decode_aux: (nat > (nat > ((product_prod @ nat) @ nat)))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel, type, nat_pr5047031295181774490ux_rel: (((product_prod @ nat) @ nat) > (((product_prod @ nat) @ nat) > $o))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Oprod__encode, type, nat_prod_encode: (((product_prod @ nat) @ nat) > nat)).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Oset__decode, type, nat_set_decode: (nat > (set @ nat))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Oset__encode, type, nat_set_encode: ((set @ nat) > nat)).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Osum__decode, type, nat_sum_decode: (nat > ((sum_sum @ nat) @ nat))).
% 9.43/3.44 thf(sy_c_Nat__Bijection_Otriangle, type, nat_triangle: (nat > nat)).
% 9.43/3.44 thf(sy_c_Num_OBitM, type, bitM: (num > num)).
% 9.43/3.44 thf(sy_c_Num_Oinc, type, inc: (num > num)).
% 9.43/3.44 thf(sy_c_Num_Onat__of__num, type, nat_of_num: (num > nat)).
% 9.43/3.44 thf(sy_c_Num_Oneg__numeral__class_Odbl, type, neg_numeral_dbl: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec, type, neg_numeral_dbl_dec: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc, type, neg_numeral_dbl_inc: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Num_Oneg__numeral__class_Ois__num, type, neg_numeral_is_num: (!> [A:$tType]: ((A > $o)))).
% 9.43/3.44 thf(sy_c_Num_Oneg__numeral__class_Osub, type, neg_numeral_sub: (!> [A:$tType]: ((num > (num > A))))).
% 9.43/3.44 thf(sy_c_Num_Onum_OBit0, type, bit0: (num > num)).
% 9.43/3.44 thf(sy_c_Num_Onum_OBit1, type, bit1: (num > num)).
% 9.43/3.44 thf(sy_c_Num_Onum_OOne, type, one2: num).
% 9.43/3.44 thf(sy_c_Num_Onum_Ocase__num, type, case_num: (!> [A:$tType]: ((A > ((num > A) > ((num > A) > (num > A))))))).
% 9.43/3.44 thf(sy_c_Num_Onum_Osize__num, type, size_num: (num > nat)).
% 9.43/3.44 thf(sy_c_Num_Onum__of__nat, type, num_of_nat: (nat > num)).
% 9.43/3.44 thf(sy_c_Num_Onumeral__class_Onumeral, type, numeral_numeral: (!> [A:$tType]: ((num > A)))).
% 9.43/3.44 thf(sy_c_Num_Opred__numeral, type, pred_numeral: (num > nat)).
% 9.43/3.44 thf(sy_c_Num_Oring__1__class_Oiszero, type, ring_1_iszero: (!> [A:$tType]: ((A > $o)))).
% 9.43/3.44 thf(sy_c_Num_Osqr, type, sqr: (num > num)).
% 9.43/3.44 thf(sy_c_Old__Datatype_OAtom, type, old_Atom: (!> [A:$tType,B:$tType]: ((((sum_sum @ A) @ nat) > (set @ ((old_node @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_OIn0, type, old_In0: (!> [A:$tType,B:$tType]: (((set @ ((old_node @ A) @ B)) > (set @ ((old_node @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_OIn1, type, old_In1: (!> [A:$tType,B:$tType]: (((set @ ((old_node @ A) @ B)) > (set @ ((old_node @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_OLeaf, type, old_Leaf: (!> [A:$tType,B:$tType]: ((A > (set @ ((old_node @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_ONode, type, old_Node: (!> [B:$tType,A:$tType]: ((set @ ((product_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_ONumb, type, old_Numb: (!> [A:$tType,B:$tType]: ((nat > (set @ ((old_node @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_OPush__Node, type, old_Push_Node: (!> [B:$tType,A:$tType]: ((((sum_sum @ B) @ nat) > (((old_node @ A) @ B) > ((old_node @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_OScons, type, old_Scons: (!> [A:$tType,B:$tType]: (((set @ ((old_node @ A) @ B)) > ((set @ ((old_node @ A) @ B)) > (set @ ((old_node @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_Odprod, type, old_dprod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) > ((set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) > (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_Odsum, type, old_dsum: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) > ((set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) > (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))))))).
% 9.43/3.44 thf(sy_c_Old__Datatype_Ondepth, type, old_ndepth: (!> [A:$tType,B:$tType]: ((((old_node @ A) @ B) > nat)))).
% 9.43/3.44 thf(sy_c_Old__Datatype_Onode_OAbs__Node, type, old_Abs_Node: (!> [B:$tType,A:$tType]: ((((product_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) > ((old_node @ A) @ B))))).
% 9.43/3.44 thf(sy_c_Option_Obind, type, bind: (!> [A:$tType,B:$tType]: (((option @ A) > ((A > (option @ B)) > (option @ B)))))).
% 9.43/3.44 thf(sy_c_Option_Ocombine__options, type, combine_options: (!> [A:$tType]: (((A > (A > A)) > ((option @ A) > ((option @ A) > (option @ A))))))).
% 9.43/3.44 thf(sy_c_Option_Ois__none, type, is_none: (!> [A:$tType]: (((option @ A) > $o)))).
% 9.43/3.44 thf(sy_c_Option_Ooption_ONone, type, none: (!> [A:$tType]: ((option @ A)))).
% 9.43/3.44 thf(sy_c_Option_Ooption_OSome, type, some: (!> [A:$tType]: ((A > (option @ A))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Ocase__option, type, case_option: (!> [B:$tType,A:$tType]: ((B > ((A > B) > ((option @ A) > B)))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Omap__option, type, map_option: (!> [A:$tType,Aa:$tType]: (((A > Aa) > ((option @ A) > (option @ Aa)))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Opred__option, type, pred_option: (!> [A:$tType]: (((A > $o) > ((option @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Orec__option, type, rec_option: (!> [C:$tType,A:$tType]: ((C > ((A > C) > ((option @ A) > C)))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Orel__option, type, rel_option: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > ((option @ A) > ((option @ B) > $o)))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Oset__option, type, set_option: (!> [A:$tType]: (((option @ A) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Osize__option, type, size_option: (!> [A:$tType]: (((A > nat) > ((option @ A) > nat))))).
% 9.43/3.44 thf(sy_c_Option_Ooption_Othe, type, the2: (!> [A:$tType]: (((option @ A) > A)))).
% 9.43/3.44 thf(sy_c_Option_Othese, type, these: (!> [A:$tType]: (((set @ (option @ A)) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Order__Relation_OAbove, type, order_Above: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_OAboveS, type, order_AboveS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_OUnder, type, order_Under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_OUnderS, type, order_UnderS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Oabove, type, order_above: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_OaboveS, type, order_aboveS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Olinear__order__on, type, order_679001287576687338der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Oofilter, type, order_ofilter: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Opartial__order__on, type, order_7125193373082350890der_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Opreorder__on, type, order_preorder_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Orelation__of, type, order_relation_of: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Ounder, type, order_under: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_OunderS, type, order_underS: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Order__Relation_Owell__order__on, type, order_well_order_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 9.43/3.44 thf(sy_c_Orderings_Oord_Omax, type, max: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 9.43/3.44 thf(sy_c_Orderings_Oord_Omin, type, min: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > A)))))).
% 9.43/3.44 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 9.43/3.44 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 9.43/3.44 thf(sy_c_Orderings_Oord__class_Omax, type, ord_max: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Orderings_Oord__class_Omin, type, ord_min: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Orderings_Oorder__class_Omono, type, order_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 9.43/3.44 thf(sy_c_Orderings_Oorder__class_Ostrict__mono, type, order_strict_mono: (!> [A:$tType,B:$tType]: (((A > B) > $o)))).
% 9.43/3.44 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 9.43/3.44 thf(sy_c_Partial__Function_Oflat__lub, type, partial_flat_lub: (!> [A:$tType]: ((A > ((set @ A) > A))))).
% 9.43/3.44 thf(sy_c_Partial__Function_Oflat__ord, type, partial_flat_ord: (!> [A:$tType]: ((A > (A > (A > $o)))))).
% 9.43/3.44 thf(sy_c_Partial__Function_Ofun__lub, type, partial_fun_lub: (!> [C:$tType,B:$tType,A:$tType]: ((((set @ C) > B) > ((set @ (A > C)) > (A > B)))))).
% 9.43/3.44 thf(sy_c_Partial__Function_Ofun__ord, type, partial_fun_ord: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((C > A) > ((C > B) > $o)))))).
% 9.43/3.44 thf(sy_c_Partial__Function_Oimg__lub, type, partial_img_lub: (!> [A:$tType,C:$tType,D:$tType,B:$tType]: (((A > C) > ((D > B) > (((set @ C) > D) > ((set @ A) > B))))))).
% 9.43/3.44 thf(sy_c_Partial__Function_Oimg__ord, type, partial_img_ord: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > ((C > (C > B)) > (A > (A > B))))))).
% 9.43/3.44 thf(sy_c_Partial__Function_Opartial__function__definitions, type, partia7178651479351089652itions: (!> [A:$tType]: (((A > (A > $o)) > (((set @ A) > A) > $o))))).
% 9.43/3.44 thf(sy_c_Power_Opower_Opower, type, power2: (!> [A:$tType]: ((A > ((A > (A > A)) > (A > (nat > A))))))).
% 9.43/3.44 thf(sy_c_Power_Opower__class_Opower, type, power_power: (!> [A:$tType]: ((A > (nat > A))))).
% 9.43/3.44 thf(sy_c_Predicate_OSeq, type, seq2: (!> [A:$tType]: (((product_unit > (seq @ A)) > (pred @ A))))).
% 9.43/3.44 thf(sy_c_Predicate_Obind, type, bind2: (!> [A:$tType,B:$tType]: (((pred @ A) > ((A > (pred @ B)) > (pred @ B)))))).
% 9.43/3.44 thf(sy_c_Predicate_Oiterate__upto, type, iterate_upto: (!> [A:$tType]: (((code_natural > A) > (code_natural > (code_natural > (pred @ A))))))).
% 9.43/3.44 thf(sy_c_Predicate_Oiterate__upto__rel, type, iterate_upto_rel: (!> [A:$tType]: ((((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) > (((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) > $o))))).
% 9.43/3.44 thf(sy_c_Predicate_Oseq_OEmpty, type, empty: (!> [A:$tType]: ((seq @ A)))).
% 9.43/3.44 thf(sy_c_Predicate_Oseq_OInsert, type, insert: (!> [A:$tType]: ((A > ((pred @ A) > (seq @ A)))))).
% 9.43/3.44 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 9.43/3.44 thf(sy_c_Product__Type_OSigma, type, product_Sigma: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_Product__Type_OUnity, type, product_Unity: product_unit).
% 9.43/3.44 thf(sy_c_Product__Type_Oapfst, type, product_apfst: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((product_prod @ A) @ B) > ((product_prod @ C) @ B)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oapsnd, type, product_apsnd: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((product_prod @ A) @ B) > ((product_prod @ A) @ C)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Ocurry, type, product_curry: (!> [A:$tType,B:$tType,C:$tType]: (((((product_prod @ A) @ B) > C) > (A > (B > C)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc5280177257484947105e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 9.43/3.44 thf(sy_c_Product__Type_Omap__prod, type, product_map_prod: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((product_prod @ A) @ B) > ((product_prod @ C) @ D))))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oold_Obool_Orec__bool, type, product_rec_bool: (!> [T:$tType]: ((T > (T > ($o > T)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool, type, product_rec_set_bool: (!> [T:$tType]: ((T > (T > ($o > (T > $o))))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod, type, product_rec_set_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > (T > $o)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit, type, product_rec_set_unit: (!> [T:$tType]: ((T > (product_unit > (T > $o)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit, type, product_rec_unit: (!> [T:$tType]: ((T > (product_unit > T))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oprod_Ocase__prod, type, product_case_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oprod_Ofst, type, product_fst: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > A)))).
% 9.43/3.44 thf(sy_c_Product__Type_Oprod_Osnd, type, product_snd: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > B)))).
% 9.43/3.44 thf(sy_c_Product__Type_Oprod_Oswap, type, product_swap: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > ((product_prod @ B) @ A))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oproduct, type, product_product: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_Product__Type_Oscomp, type, product_scomp: (!> [A:$tType,B:$tType,C:$tType,D:$tType]: (((A > ((product_prod @ B) @ C)) > ((B > (C > D)) > (A > D)))))).
% 9.43/3.44 thf(sy_c_Product__Type_Ounit_OAbs__unit, type, product_Abs_unit: ($o > product_unit)).
% 9.43/3.44 thf(sy_c_Product__Type_Ounit_ORep__unit, type, product_Rep_unit: (product_unit > $o)).
% 9.43/3.44 thf(sy_c_Product__Type_Ounit_Ocase__unit, type, product_case_unit: (!> [A:$tType]: ((A > (product_unit > A))))).
% 9.43/3.44 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 9.43/3.44 thf(sy_c_Quicksort_Olinorder__class_Oquicksort, type, linorder_quicksort: (!> [A:$tType]: (((list @ A) > (list @ A))))).
% 9.43/3.44 thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel, type, linord6200660962353139674rt_rel: (!> [A:$tType]: (((list @ A) > ((list @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Quotient_OQuotient3, type, quotient3: (!> [A:$tType,B:$tType]: (((A > (A > $o)) > ((A > B) > ((B > A) > $o)))))).
% 9.43/3.44 thf(sy_c_Random_Oinc__shift, type, inc_shift: (code_natural > (code_natural > code_natural))).
% 9.43/3.44 thf(sy_c_Random_Oiterate, type, iterate: (!> [B:$tType,A:$tType]: ((code_natural > ((B > (A > ((product_prod @ B) @ A))) > (B > (A > ((product_prod @ B) @ A)))))))).
% 9.43/3.44 thf(sy_c_Random_Oiterate__rel, type, iterate_rel: (!> [B:$tType,A:$tType]: ((((product_prod @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B)) > (((product_prod @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B)) > $o))))).
% 9.43/3.44 thf(sy_c_Random_Olog, type, log: (code_natural > (code_natural > code_natural))).
% 9.43/3.44 thf(sy_c_Random_Olog__rel, type, log_rel: (((product_prod @ code_natural) @ code_natural) > (((product_prod @ code_natural) @ code_natural) > $o))).
% 9.43/3.44 thf(sy_c_Random_Ominus__shift, type, minus_shift: (code_natural > (code_natural > (code_natural > code_natural)))).
% 9.43/3.44 thf(sy_c_Random_Onext, type, next: (((product_prod @ code_natural) @ code_natural) > ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural)))).
% 9.43/3.44 thf(sy_c_Random_Opick, type, pick: (!> [A:$tType]: (((list @ ((product_prod @ code_natural) @ A)) > (code_natural > A))))).
% 9.43/3.44 thf(sy_c_Random_Orange, type, range: (code_natural > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural))))).
% 9.43/3.44 thf(sy_c_Random_Oselect, type, select: (!> [A:$tType]: (((list @ A) > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ A) @ ((product_prod @ code_natural) @ code_natural))))))).
% 9.43/3.44 thf(sy_c_Random_Oselect__weight, type, select_weight: (!> [A:$tType]: (((list @ ((product_prod @ code_natural) @ A)) > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ A) @ ((product_prod @ code_natural) @ code_natural))))))).
% 9.43/3.44 thf(sy_c_Random_Osplit__seed, type, split_seed: (((product_prod @ code_natural) @ code_natural) > ((product_prod @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ code_natural)))).
% 9.43/3.44 thf(sy_c_Random__Pred_Obind, type, random_bind: (!> [A:$tType,B:$tType]: (((((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))) > ((A > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural)))) > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural)))))))).
% 9.43/3.44 thf(sy_c_Random__Pred_Oempty, type, random_empty: (!> [A:$tType]: ((((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)))))).
% 9.43/3.44 thf(sy_c_Random__Pred_Oiterate__upto, type, random_iterate_upto: (!> [A:$tType]: (((code_natural > A) > (code_natural > (code_natural > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))))))))).
% 9.43/3.44 thf(sy_c_Random__Pred_Ounion, type, random_union: (!> [A:$tType]: (((((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))) > ((((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))) > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)))))))).
% 9.43/3.44 thf(sy_c_Rat_OAbs__Rat, type, abs_Rat: (((product_prod @ int) @ int) > rat)).
% 9.43/3.44 thf(sy_c_Rat_OFract, type, fract: (int > (int > rat))).
% 9.43/3.44 thf(sy_c_Rat_OFrct, type, frct: (((product_prod @ int) @ int) > rat)).
% 9.43/3.44 thf(sy_c_Rat_ORep__Rat, type, rep_Rat: (rat > ((product_prod @ int) @ int))).
% 9.43/3.44 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat, type, field_char_0_of_rat: (!> [A:$tType]: ((rat > A)))).
% 9.43/3.44 thf(sy_c_Rat_Onormalize, type, normalize: (((product_prod @ int) @ int) > ((product_prod @ int) @ int))).
% 9.43/3.44 thf(sy_c_Rat_Oof__int, type, of_int: (int > rat)).
% 9.43/3.44 thf(sy_c_Rat_Opcr__rat, type, pcr_rat: (((product_prod @ int) @ int) > (rat > $o))).
% 9.43/3.44 thf(sy_c_Rat_Opositive, type, positive: (rat > $o)).
% 9.43/3.44 thf(sy_c_Rat_Oquotient__of, type, quotient_of: (rat > ((product_prod @ int) @ int))).
% 9.43/3.44 thf(sy_c_Rat_Oratrel, type, ratrel: (((product_prod @ int) @ int) > (((product_prod @ int) @ int) > $o))).
% 9.43/3.44 thf(sy_c_Ref__Time_Oalloc, type, ref_alloc: (!> [A:$tType]: ((A > ((heap_ext @ product_unit) > ((product_prod @ (ref @ A)) @ (heap_ext @ product_unit))))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Ochange, type, ref_change: (!> [A:$tType]: (((A > A) > ((ref @ A) > (heap_Time_Heap @ A)))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Oget, type, ref_get: (!> [A:$tType]: (((heap_ext @ product_unit) > ((ref @ A) > A))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Olookup, type, ref_lookup: (!> [A:$tType]: (((ref @ A) > (heap_Time_Heap @ A))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Opresent, type, ref_present: (!> [A:$tType]: (((heap_ext @ product_unit) > ((ref @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Oref, type, ref_ref: (!> [A:$tType]: ((A > (heap_Time_Heap @ (ref @ A)))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Oset, type, ref_set: (!> [A:$tType]: (((ref @ A) > (A > ((heap_ext @ product_unit) > (heap_ext @ product_unit))))))).
% 9.43/3.44 thf(sy_c_Ref__Time_Oupdate, type, ref_update: (!> [A:$tType]: (((ref @ A) > (A > (heap_Time_Heap @ product_unit)))))).
% 9.43/3.44 thf(sy_c_Refine__Imp__Hol_Oassert_H, type, refine_Imp_assert: ($o > (heap_Time_Heap @ product_unit))).
% 9.43/3.44 thf(sy_c_Refine__Imp__Hol_Orefines, type, refine_Imp_refines: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_Time_Heap @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Relation_ODomain, type, domain: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Relation_ODomainp, type, domainp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Relation_OField, type, field2: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Relation_OId, type, id2: (!> [A:$tType]: ((set @ ((product_prod @ A) @ A))))).
% 9.43/3.44 thf(sy_c_Relation_OId__on, type, id_on: (!> [A:$tType]: (((set @ A) > (set @ ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_Relation_OImage, type, image: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ A) > (set @ B)))))).
% 9.43/3.44 thf(sy_c_Relation_ORange, type, range2: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ B))))).
% 9.43/3.44 thf(sy_c_Relation_ORangep, type, rangep: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (B > $o))))).
% 9.43/3.44 thf(sy_c_Relation_Oantisym, type, antisym: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Oantisymp, type, antisymp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Oasym, type, asym: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Oasymp, type, asymp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Oconverse, type, converse: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ B) @ A)))))).
% 9.43/3.44 thf(sy_c_Relation_Oconversep, type, conversep: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > (B > (A > $o)))))).
% 9.43/3.44 thf(sy_c_Relation_Oinv__image, type, inv_image: (!> [B:$tType,A:$tType]: (((set @ ((product_prod @ B) @ B)) > ((A > B) > (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_Relation_Oirrefl, type, irrefl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Oirreflp, type, irreflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Orefl__on, type, refl_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Relation_Oreflp, type, reflp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Orelcomp, type, relcomp: (!> [A:$tType,B:$tType,C:$tType]: (((set @ ((product_prod @ A) @ B)) > ((set @ ((product_prod @ B) @ C)) > (set @ ((product_prod @ A) @ C))))))).
% 9.43/3.44 thf(sy_c_Relation_Orelcompp, type, relcompp: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > $o)) > ((B > (C > $o)) > (A > (C > $o))))))).
% 9.43/3.44 thf(sy_c_Relation_Osingle__valued, type, single_valued: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ B)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Osingle__valuedp, type, single_valuedp: (!> [A:$tType,B:$tType]: (((A > (B > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Ototal__on, type, total_on: (!> [A:$tType]: (((set @ A) > ((set @ ((product_prod @ A) @ A)) > $o))))).
% 9.43/3.44 thf(sy_c_Relation_Otrans, type, trans: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_Relation_Otransp, type, transp: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime, type, algebr8660921524188924756oprime: (!> [A:$tType]: ((A > (A > $o))))).
% 9.43/3.44 thf(sy_c_Rings_Odivide__class_Odivide, type, divide_divide: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Rings_Odvd__class_Odvd, type, dvd_dvd: (!> [A:$tType]: ((A > (A > $o))))).
% 9.43/3.44 thf(sy_c_Rings_Omodulo__class_Omodulo, type, modulo_modulo: (!> [A:$tType]: ((A > (A > A))))).
% 9.43/3.44 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize, type, normal6383669964737779283malize: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor, type, unit_f5069060285200089521factor: (!> [A:$tType]: ((A > A)))).
% 9.43/3.44 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool, type, zero_neq_one_of_bool: (!> [A:$tType]: (($o > A)))).
% 9.43/3.44 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Set_OPow, type, pow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set_Obind, type, bind3: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > (set @ B)) > (set @ B)))))).
% 9.43/3.44 thf(sy_c_Set_Odisjnt, type, disjnt: (!> [A:$tType]: (((set @ A) > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_c_Set_Oimage, type, image2: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 9.43/3.44 thf(sy_c_Set_Oinsert, type, insert2: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 9.43/3.44 thf(sy_c_Set_Ovimage, type, vimage: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ B) > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat, type, set_fo6178422350223883121st_nat: (!> [A:$tType]: (((nat > (A > A)) > (nat > (nat > (A > A))))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel, type, set_fo1817059534552279752at_rel: (!> [A:$tType]: ((((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) > (((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) > $o))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OatLeast, type, set_ord_atLeast: (!> [A:$tType]: ((A > (set @ A))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost, type, set_or1337092689740270186AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan, type, set_or7035219750837199246ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OatMost, type, set_ord_atMost: (!> [A:$tType]: ((A > (set @ A))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OgreaterThan, type, set_ord_greaterThan: (!> [A:$tType]: ((A > (set @ A))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost, type, set_or3652927894154168847AtMost: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan, type, set_or5935395276787703475ssThan: (!> [A:$tType]: ((A > (A > (set @ A)))))).
% 9.43/3.44 thf(sy_c_Set__Interval_Oord__class_OlessThan, type, set_ord_lessThan: (!> [A:$tType]: ((A > (set @ A))))).
% 9.43/3.44 thf(sy_c_String_Ochar_OChar, type, char2: ($o > ($o > ($o > ($o > ($o > ($o > ($o > ($o > char))))))))).
% 9.43/3.44 thf(sy_c_String_Ointeger__of__char, type, integer_of_char: (char > code_integer)).
% 9.43/3.44 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of, type, unique5772411509450598832har_of: (!> [A:$tType]: ((A > char)))).
% 9.43/3.44 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 9.43/3.44 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 9.43/3.44 thf(sy_c_Sum__Type_OPlus, type, sum_Plus: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((sum_sum @ A) @ B))))))).
% 9.43/3.44 thf(sy_c_Sum__Type_Omap__sum, type, sum_map_sum: (!> [A:$tType,C:$tType,B:$tType,D:$tType]: (((A > C) > ((B > D) > (((sum_sum @ A) @ B) > ((sum_sum @ C) @ D))))))).
% 9.43/3.44 thf(sy_c_Sum__Type_Osum_Ocase__sum, type, sum_case_sum: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > ((B > C) > (((sum_sum @ A) @ B) > C)))))).
% 9.43/3.44 thf(sy_c_Syntax__Match_Osyntax__fo__nomatch, type, syntax7388354845996824322omatch: (!> [A:$tType,B:$tType]: ((A > (B > $o))))).
% 9.43/3.44 thf(sy_c_Time__Reasoning_Ofails, type, time_fails: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > $o))))).
% 9.43/3.44 thf(sy_c_Time__Reasoning_Othe__heap, type, time_the_heap: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > (heap_ext @ product_unit)))))).
% 9.43/3.44 thf(sy_c_Time__Reasoning_Othe__res, type, time_the_res: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > A))))).
% 9.43/3.44 thf(sy_c_Time__Reasoning_Otime, type, time_time: (!> [A:$tType]: (((heap_Time_Heap @ A) > ((heap_ext @ product_unit) > nat))))).
% 9.43/3.44 thf(sy_c_Transitive__Closure_Oacyclic, type, transitive_acyclic: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_Transitive__Closure_Ontrancl, type, transitive_ntrancl: (!> [A:$tType]: ((nat > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_Transitive__Closure_Ortrancl, type, transitive_rtrancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_Transitive__Closure_Ortranclp, type, transitive_rtranclp: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > $o)))))).
% 9.43/3.44 thf(sy_c_Transitive__Closure_Otrancl, type, transitive_trancl: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_Transitive__Closure_Otranclp, type, transitive_tranclp: (!> [A:$tType]: (((A > (A > $o)) > (A > (A > $o)))))).
% 9.43/3.44 thf(sy_c_Typedef_Otype__definition, type, type_definition: (!> [B:$tType,A:$tType]: (((B > A) > ((A > B) > ((set @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_Typerep_Otyperep__class_Otyperep, type, typerep_typerep: (!> [A:$tType]: (((itself @ A) > typerep)))).
% 9.43/3.44 thf(sy_c_Wellfounded_Oacc, type, acc: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ A))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Oaccp, type, accp: (!> [A:$tType]: (((A > (A > $o)) > (A > $o))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Ofinite__psubset, type, finite_psubset: (!> [A:$tType]: ((set @ ((product_prod @ (set @ A)) @ (set @ A)))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Oless__than, type, less_than: (set @ ((product_prod @ nat) @ nat))).
% 9.43/3.44 thf(sy_c_Wellfounded_Olex__prod, type, lex_prod: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > ((set @ ((product_prod @ B) @ B)) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Omax__ext, type, max_ext: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Omax__extp, type, max_extp: (!> [A:$tType]: (((A > (A > $o)) > ((set @ A) > ((set @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Omeasure, type, measure: (!> [A:$tType]: (((A > nat) > (set @ ((product_prod @ A) @ A)))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Omin__ext, type, min_ext: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ (set @ A)) @ (set @ A))))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Omlex__prod, type, mlex_prod: (!> [A:$tType]: (((A > nat) > ((set @ ((product_prod @ A) @ A)) > (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_Wellfounded_Opred__nat, type, pred_nat: (set @ ((product_prod @ nat) @ nat))).
% 9.43/3.44 thf(sy_c_Wellfounded_Owf, type, wf: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > $o)))).
% 9.43/3.44 thf(sy_c_Wellfounded_OwfP, type, wfP: (!> [A:$tType]: (((A > (A > $o)) > $o)))).
% 9.43/3.44 thf(sy_c_Wfrec_Oadm__wf, type, adm_wf: (!> [A:$tType,B:$tType]: (((set @ ((product_prod @ A) @ A)) > (((A > B) > (A > B)) > $o))))).
% 9.43/3.44 thf(sy_c_Wfrec_Ocut, type, cut: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ ((product_prod @ A) @ A)) > (A > (A > B))))))).
% 9.43/3.44 thf(sy_c_Wfrec_Osame__fst, type, same_fst: (!> [A:$tType,B:$tType]: (((A > $o) > ((A > (set @ ((product_prod @ B) @ B))) > (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.44 thf(sy_c_Zorn_OChains, type, chains: (!> [A:$tType]: (((set @ ((product_prod @ A) @ A)) > (set @ (set @ A)))))).
% 9.43/3.44 thf(sy_c_Zorn_Oinit__seg__of, type, init_seg_of: (!> [A:$tType]: ((set @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.44 thf(sy_c_Zorn_Opred__on_Ochain, type, pred_chain: (!> [A:$tType]: (((set @ A) > ((A > (A > $o)) > ((set @ A) > $o)))))).
% 9.43/3.44 thf(sy_c_fChoice, type, fChoice: (!> [A:$tType]: (((A > $o) > A)))).
% 9.43/3.44 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 9.43/3.44 thf(sy_v_h, type, h: (heap_ext @ product_unit)).
% 9.43/3.44 thf(sy_v_xs, type, xs: (list @ a)).
% 9.43/3.44 thf(fact_0_fails__def, axiom, (! [A:$tType]: (((time_fails @ A) = (^ [M:(heap_Time_Heap @ A),H:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ M) @ H) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 9.43/3.44 thf(fact_1_Heap__eqI, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),G:(heap_Time_Heap @ A)]: (((! [H2:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ F) @ H2) = (((heap_Time_execute @ A) @ G) @ H2)))) => (F = G))))).
% 9.43/3.44 thf(fact_2_execute__raise, axiom, (! [A:$tType,S:(list @ char)]: ((((heap_Time_execute @ A) @ ((heap_Time_raise @ A) @ S)) = (^ [Uu:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))).
% 9.43/3.44 thf(fact_3_execute__guard_I1_J, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H3:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((~ ((P @ H3))) => ((((heap_Time_execute @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H3) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 9.43/3.44 thf(fact_4_execute__assert_I2_J, axiom, (! [A:$tType,P:(A > $o),X:A,H3:(heap_ext @ product_unit)]: (((~ ((P @ X))) => ((((heap_Time_execute @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H3) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 9.43/3.44 thf(fact_5_successI, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: ((((((heap_Time_execute @ A) @ F) @ H3) != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => (((heap_Time_success @ A) @ F) @ H3))))).
% 9.43/3.44 thf(fact_6_success__def, axiom, (! [A:$tType]: (((heap_Time_success @ A) = (^ [F2:(heap_Time_Heap @ A),H:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ F2) @ H) != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 9.43/3.44 thf(fact_7_Heap__ordE, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_Heap_ord @ A) @ X) @ Y) => (((((heap_Time_execute @ A) @ X) @ H3) != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => ((((heap_Time_execute @ A) @ X) @ H3) = (((heap_Time_execute @ A) @ Y) @ H3))))))).
% 9.43/3.44 thf(fact_8_Heap__ordI, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ A)]: (((! [H2:(heap_ext @ product_unit)]: ((((((heap_Time_execute @ A) @ X) @ H2) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) | ((((heap_Time_execute @ A) @ X) @ H2) = (((heap_Time_execute @ A) @ Y) @ H2))))) => (((heap_Time_Heap_ord @ A) @ X) @ Y))))).
% 9.43/3.44 thf(fact_9_execute__bind_I2_J, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),G:(A > (heap_Time_Heap @ B))]: ((((((heap_Time_execute @ A) @ F) @ H3) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3) = (none @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 9.43/3.44 thf(fact_10_the__default_Osimps_I2_J, axiom, (! [A:$tType,X:A]: (((((the_default @ A) @ X) @ (none @ A)) = X)))).
% 9.43/3.44 thf(fact_11_raise__bind, axiom, (! [B:$tType,A:$tType,E2:(list @ char),F:(B > (heap_Time_Heap @ A))]: ((((((heap_Time_bind @ B) @ A) @ ((heap_Time_raise @ B) @ E2)) @ F) = ((heap_Time_raise @ A) @ E2))))).
% 9.43/3.44 thf(fact_12_heap_Oleq__refl, axiom, (! [A:$tType,X:(heap_Time_Heap @ A)]: ((((heap_Time_Heap_ord @ A) @ X) @ X)))).
% 9.43/3.44 thf(fact_13_heap_Oleq__trans, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ A),Z2:(heap_Time_Heap @ A)]: (((((heap_Time_Heap_ord @ A) @ X) @ Y) => ((((heap_Time_Heap_ord @ A) @ Y) @ Z2) => (((heap_Time_Heap_ord @ A) @ X) @ Z2)))))).
% 9.43/3.44 thf(fact_14_heap_Oleq__antisym, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ A)]: (((((heap_Time_Heap_ord @ A) @ X) @ Y) => ((((heap_Time_Heap_ord @ A) @ Y) @ X) => (X = Y)))))).
% 9.43/3.44 thf(fact_15_assert__cong, axiom, (! [B:$tType,A:$tType,P:(A > $o),P2:(A > $o),F:(A > (heap_Time_Heap @ B)),F3:(A > (heap_Time_Heap @ B)),X:A]: (((P = P2) => ((! [X2:A]: (((P2 @ X2) => ((F @ X2) = (F3 @ X2))))) => (((((heap_Time_bind @ A) @ B) @ (((heap_Time_assert @ A) @ P) @ X)) @ F) = ((((heap_Time_bind @ A) @ B) @ (((heap_Time_assert @ A) @ P2) @ X)) @ F3))))))).
% 9.43/3.44 thf(fact_16_success__ifI, axiom, (! [A:$tType,C2:$o,T2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),E2:(heap_Time_Heap @ A)]: (((C2 => (((heap_Time_success @ A) @ T2) @ H3)) => (((~ (C2)) => (((heap_Time_success @ A) @ E2) @ H3)) => (((heap_Time_success @ A) @ ((((if @ (heap_Time_Heap @ A)) @ C2) @ T2) @ E2)) @ H3)))))).
% 9.43/3.44 thf(fact_17_success__guardI, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H3:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((P @ H3) => (((heap_Time_success @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H3))))).
% 9.43/3.44 thf(fact_18_success__assertI, axiom, (! [A:$tType,P:(A > $o),X:A,H3:(heap_ext @ product_unit)]: (((P @ X) => (((heap_Time_success @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H3))))).
% 9.43/3.44 thf(fact_19_fails__bind, axiom, (! [A:$tType,B:$tType,M2:(heap_Time_Heap @ B),F:(B > (heap_Time_Heap @ A)),H3:(heap_ext @ product_unit)]: (((((time_fails @ A) @ ((((heap_Time_bind @ B) @ A) @ M2) @ F)) @ H3) = ((~ ((((time_fails @ B) @ M2) @ H3))) => (((time_fails @ A) @ (F @ (((time_the_res @ B) @ M2) @ H3))) @ (((time_the_heap @ B) @ M2) @ H3))))))).
% 9.43/3.44 thf(fact_20_success__of__listI, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H3))))))).
% 9.43/3.44 thf(fact_21_success__tapI, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_tap @ A) @ F)) @ H3)))).
% 9.43/3.44 thf(fact_22_success__heapI, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_heap @ A) @ F)) @ H3)))).
% 9.43/3.44 thf(fact_23_success__ureturnI, axiom, (! [A:$tType,X:A,H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_ureturn @ A) @ X)) @ H3)))).
% 9.43/3.44 thf(fact_24_bind__ureturn, axiom, (! [A:$tType,F:(heap_Time_Heap @ A)]: ((((((heap_Time_bind @ A) @ A) @ F) @ (heap_Time_ureturn @ A)) = F)))).
% 9.43/3.44 thf(fact_25_ureturn__bind, axiom, (! [A:$tType,B:$tType,X:B,F:(B > (heap_Time_Heap @ A))]: ((((((heap_Time_bind @ B) @ A) @ ((heap_Time_ureturn @ B) @ X)) @ F) = (F @ X))))).
% 9.43/3.44 thf(fact_26_option_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,F1:C,F22:(A > C)]: (((((((rec_option @ C) @ A) @ F1) @ F22) @ (none @ A)) = F1)))).
% 9.43/3.44 thf(fact_27_success__bind__effectI, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),X:A,N:nat,G:(A > (heap_Time_Heap @ B))]: ((((((((heap_Time_effect @ A) @ F) @ H3) @ H4) @ X) @ N) => ((((heap_Time_success @ B) @ (G @ X)) @ H4) => (((heap_Time_success @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3)))))).
% 9.43/3.44 thf(fact_28_execute__guard_I2_J, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H3:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((P @ H3) => ((((heap_Time_execute @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (F @ H3))))))).
% 9.43/3.44 thf(fact_29_combine__options__simps_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Y:(option @ A)]: ((((((combine_options @ A) @ F) @ (none @ A)) @ Y) = Y)))).
% 9.43/3.44 thf(fact_30_combine__options__simps_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),X:(option @ A)]: ((((((combine_options @ A) @ F) @ X) @ (none @ A)) = X)))).
% 9.43/3.44 thf(fact_31_option_Oinject, axiom, (! [A:$tType,X22:A,Y2:A]: (((((some @ A) @ X22) = ((some @ A) @ Y2)) = (X22 = Y2))))).
% 9.43/3.44 thf(fact_32_not__None__eq, axiom, (! [A:$tType,X:(option @ A)]: (((X != (none @ A)) = (? [Y3:A]: ((X = ((some @ A) @ Y3)))))))).
% 9.43/3.44 thf(fact_33_not__Some__eq, axiom, (! [A:$tType,X:(option @ A)]: (((! [Y3:A]: ((X != ((some @ A) @ Y3)))) = (X = (none @ A)))))).
% 9.43/3.44 thf(fact_34_combine__options__simps_I3_J, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A]: ((((((combine_options @ A) @ F) @ ((some @ A) @ A3)) @ ((some @ A) @ B2)) = ((some @ A) @ ((F @ A3) @ B2)))))).
% 9.43/3.44 thf(fact_35_option_Osimps_I7_J, axiom, (! [C:$tType,A:$tType,F1:C,F22:(A > C),X22:A]: (((((((rec_option @ C) @ A) @ F1) @ F22) @ ((some @ A) @ X22)) = (F22 @ X22))))).
% 9.43/3.44 thf(fact_36_effect__deterministic_I3_J, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),A3:A,N:nat,H5:(heap_ext @ product_unit),B2:A,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H3) @ H4) @ A3) @ N) => (((((((heap_Time_effect @ A) @ F) @ H3) @ H5) @ B2) @ N2) => (N = N2)))))).
% 9.43/3.44 thf(fact_37_effect__deterministic_I2_J, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),A3:A,N:nat,H5:(heap_ext @ product_unit),B2:A,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H3) @ H4) @ A3) @ N) => (((((((heap_Time_effect @ A) @ F) @ H3) @ H5) @ B2) @ N2) => (H4 = H5)))))).
% 9.43/3.44 thf(fact_38_effect__deterministic_I1_J, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),A3:A,N:nat,H5:(heap_ext @ product_unit),B2:A,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H3) @ H4) @ A3) @ N) => (((((((heap_Time_effect @ A) @ F) @ H3) @ H5) @ B2) @ N2) => (A3 = B2)))))).
% 9.43/3.44 thf(fact_39_effect__ifE, axiom, (! [A:$tType,C2:$o,T2:(heap_Time_Heap @ A),E2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((((if @ (heap_Time_Heap @ A)) @ C2) @ T2) @ E2)) @ H3) @ H4) @ R) @ N) => ((C2 => (~ (((((((heap_Time_effect @ A) @ T2) @ H3) @ H4) @ R) @ N)))) => (~ (((~ (C2)) => (~ (((((((heap_Time_effect @ A) @ E2) @ H3) @ H4) @ R) @ N))))))))))).
% 9.43/3.44 thf(fact_40_effect__ifI, axiom, (! [A:$tType,C2:$o,T2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat,E2:(heap_Time_Heap @ A)]: (((C2 => ((((((heap_Time_effect @ A) @ T2) @ H3) @ H4) @ R) @ N)) => (((~ (C2)) => ((((((heap_Time_effect @ A) @ E2) @ H3) @ H4) @ R) @ N)) => ((((((heap_Time_effect @ A) @ ((((if @ (heap_Time_Heap @ A)) @ C2) @ T2) @ E2)) @ H3) @ H4) @ R) @ N)))))).
% 9.43/3.44 thf(fact_41_combine__options__assoc, axiom, (! [A:$tType,F:(A > (A > A)),X:(option @ A),Y:(option @ A),Z2:(option @ A)]: (((! [X2:A,Y4:A,Z3:A]: ((((F @ ((F @ X2) @ Y4)) @ Z3) = ((F @ X2) @ ((F @ Y4) @ Z3))))) => (((((combine_options @ A) @ F) @ ((((combine_options @ A) @ F) @ X) @ Y)) @ Z2) = ((((combine_options @ A) @ F) @ X) @ ((((combine_options @ A) @ F) @ Y) @ Z2))))))).
% 9.43/3.44 thf(fact_42_combine__options__commute, axiom, (! [A:$tType,F:(A > (A > A)),X:(option @ A),Y:(option @ A)]: (((! [X2:A,Y4:A]: ((((F @ X2) @ Y4) = ((F @ Y4) @ X2)))) => (((((combine_options @ A) @ F) @ X) @ Y) = ((((combine_options @ A) @ F) @ Y) @ X)))))).
% 9.43/3.44 thf(fact_43_combine__options__left__commute, axiom, (! [A:$tType,F:(A > (A > A)),Y:(option @ A),X:(option @ A),Z2:(option @ A)]: (((! [X2:A,Y4:A]: ((((F @ X2) @ Y4) = ((F @ Y4) @ X2)))) => ((! [X2:A,Y4:A,Z3:A]: ((((F @ ((F @ X2) @ Y4)) @ Z3) = ((F @ X2) @ ((F @ Y4) @ Z3))))) => (((((combine_options @ A) @ F) @ Y) @ ((((combine_options @ A) @ F) @ X) @ Z2)) = ((((combine_options @ A) @ F) @ X) @ ((((combine_options @ A) @ F) @ Y) @ Z2)))))))).
% 9.43/3.44 thf(fact_44_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 9.43/3.44 thf(fact_45_Collect__mem__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((collect @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ A4)))) = A4)))).
% 9.43/3.44 thf(fact_46_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 9.43/3.44 thf(fact_47_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X2:A]: (((F @ X2) = (G @ X2)))) => (F = G))))).
% 9.43/3.44 thf(fact_48_option_Odistinct_I1_J, axiom, (! [A:$tType,X22:A]: (((none @ A) != ((some @ A) @ X22))))).
% 9.43/3.44 thf(fact_49_option_OdiscI, axiom, (! [A:$tType,Option:(option @ A),X22:A]: (((Option = ((some @ A) @ X22)) => (Option != (none @ A)))))).
% 9.43/3.44 thf(fact_50_option_Oexhaust, axiom, (! [A:$tType,Y:(option @ A)]: (((Y != (none @ A)) => (~ ((! [X23:A]: ((Y != ((some @ A) @ X23)))))))))).
% 9.43/3.44 thf(fact_51_split__option__ex, axiom, (! [A:$tType]: (((^ [P3:((option @ A) > $o)]: ((? [X4:(option @ A)]: ((P3 @ X4))))) = (^ [P4:((option @ A) > $o)]: (((P4 @ (none @ A)) | (? [X3:A]: ((P4 @ ((some @ A) @ X3))))))))))).
% 9.43/3.44 thf(fact_52_split__option__all, axiom, (! [A:$tType]: (((^ [P3:((option @ A) > $o)]: ((! [X4:(option @ A)]: ((P3 @ X4))))) = (^ [P4:((option @ A) > $o)]: (((P4 @ (none @ A)) & (! [X3:A]: ((P4 @ ((some @ A) @ X3))))))))))).
% 9.43/3.44 thf(fact_53_combine__options__cases, axiom, (! [A:$tType,B:$tType,X:(option @ A),P:((option @ A) > ((option @ B) > $o)),Y:(option @ B)]: ((((X = (none @ A)) => ((P @ X) @ Y)) => (((Y = (none @ B)) => ((P @ X) @ Y)) => ((! [A5:A,B3:B]: (((X = ((some @ A) @ A5)) => ((Y = ((some @ B) @ B3)) => ((P @ X) @ Y))))) => ((P @ X) @ Y))))))).
% 9.43/3.44 thf(fact_54_success__effectE, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_success @ A) @ C2) @ H3) => (~ ((! [R2:A,H6:(heap_ext @ product_unit),N3:nat]: ((~ (((((((heap_Time_effect @ A) @ C2) @ H3) @ H6) @ R2) @ N3))))))))))).
% 9.43/3.44 thf(fact_55_effect__success, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ C2) @ H3) @ H4) @ R) @ N) => (((heap_Time_success @ A) @ C2) @ H3))))).
% 9.43/3.44 thf(fact_56_effect__raiseE, axiom, (! [A:$tType,X:(list @ char),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((~ (((((((heap_Time_effect @ A) @ ((heap_Time_raise @ A) @ X)) @ H3) @ H4) @ R) @ N)))))).
% 9.43/3.44 thf(fact_57_the__default_Osimps_I1_J, axiom, (! [A:$tType,Uu2:A,X:A]: (((((the_default @ A) @ Uu2) @ ((some @ A) @ X)) = X)))).
% 9.43/3.44 thf(fact_58_Heap__cases, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: (((! [X2:A,H6:((product_prod @ (heap_ext @ product_unit)) @ nat)]: (((((heap_Time_execute @ A) @ F) @ H3) != ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X2) @ H6))))) => ((((heap_Time_execute @ A) @ F) @ H3) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))).
% 9.43/3.44 thf(fact_59_successE, axiom, (! [A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_success @ A) @ F) @ H3) => (~ ((! [R2:A,H6:((product_prod @ (heap_ext @ product_unit)) @ nat)]: (((((heap_Time_execute @ A) @ F) @ H3) != ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ H6))))))))))).
% 9.43/3.44 thf(fact_60_execute__heap, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: ((((heap_Time_execute @ A) @ ((heap_Time_heap @ A) @ F)) = (((((comp @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ F))))).
% 9.43/3.44 thf(fact_61_some__opt__sym__eq__trivial, axiom, (! [A:$tType,X:A]: ((((eps_Opt @ A) @ ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) @ X)) = ((some @ A) @ X))))).
% 9.43/3.44 thf(fact_62_success__newI, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H3))))))).
% 9.43/3.44 thf(fact_63_success__makeI, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H3))))))).
% 9.43/3.44 thf(fact_64_mmupd__in__upd, axiom, (! [A:$tType,B:$tType,K:A,K2:(set @ A),M2:(A > (option @ B)),V:B]: (((((member @ A) @ K) @ K2) => (((((((map_mmupd @ A) @ B) @ M2) @ K2) @ V) @ K) = ((some @ B) @ V)))))).
% 9.43/3.44 thf(fact_65_option_Ocollapse, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (((some @ A) @ ((the2 @ A) @ Option)) = Option))))).
% 9.43/3.44 thf(fact_66_not__Some__eq2, axiom, (! [B:$tType,A:$tType,V:(option @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B]: ((V != ((some @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) = (V = (none @ ((product_prod @ A) @ B))))))).
% 9.43/3.44 thf(fact_67_effect__tapE, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_tap @ A) @ F)) @ H3) @ H4) @ R) @ N) => (~ (((H4 = H3) => ((R = (F @ H3)) => (N != (one_one @ nat)))))))))).
% 9.43/3.44 thf(fact_68_effect__tapI, axiom, (! [A:$tType,H4:(heap_ext @ product_unit),H3:(heap_ext @ product_unit),R:A,F:((heap_ext @ product_unit) > A)]: (((H4 = H3) => ((R = (F @ H3)) => ((((((heap_Time_effect @ A) @ ((heap_Time_tap @ A) @ F)) @ H3) @ H4) @ R) @ (one_one @ nat))))))).
% 9.43/3.44 thf(fact_69_effect__assertI, axiom, (! [A:$tType,P:(A > $o),X:A,H4:(heap_ext @ product_unit),H3:(heap_ext @ product_unit),R:A,N:nat]: (((P @ X) => ((H4 = H3) => ((R = X) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H3) @ H4) @ R) @ N)))))))).
% 9.43/3.44 thf(fact_70_mmupd__notin__upd, axiom, (! [B:$tType,A:$tType,K:A,K2:(set @ A),M2:(A > (option @ B)),V:B]: (((~ ((((member @ A) @ K) @ K2))) => (((((((map_mmupd @ A) @ B) @ M2) @ K2) @ V) @ K) = (M2 @ K)))))).
% 9.43/3.44 thf(fact_71_fun__comp__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),G:(A > C),Fg:(A > B)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = Fg) = (! [X3:A]: (((F @ (G @ X3)) = (Fg @ X3)))))))).
% 9.43/3.44 thf(fact_72_comp__cong__right, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > B),Y:(A > B),F:(B > C)]: (((X = Y) => ((((((comp @ B) @ C) @ A) @ F) @ X) = (((((comp @ B) @ C) @ A) @ F) @ Y)))))).
% 9.43/3.44 thf(fact_73_comp__cong__left, axiom, (! [B:$tType,A:$tType,C:$tType,X:(A > B),Y:(A > B),F:(C > A)]: (((X = Y) => ((((((comp @ A) @ B) @ C) @ X) @ F) = (((((comp @ A) @ B) @ C) @ Y) @ F)))))).
% 9.43/3.44 thf(fact_74_bex2I, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,S2:(set @ ((product_prod @ A) @ B)),P:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ S2) => (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ S2) => ((P @ A3) @ B2)) => (? [A5:A,B3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ S2) & ((P @ A5) @ B3))))))))).
% 9.43/3.44 thf(fact_75_pairself_Ocases, axiom, (! [B:$tType,A:$tType,X:((product_prod @ (A > B)) @ ((product_prod @ A) @ A))]: ((~ ((! [F4:(A > B),A5:A,B3:A]: ((X != ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ F4) @ ((((product_Pair @ A) @ A) @ A5) @ B3)))))))))).
% 9.43/3.44 thf(fact_76_fails__array__new, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H3:(heap_ext @ product_unit)]: ((~ ((((time_fails @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H3))))))))).
% 9.43/3.44 thf(fact_77_timeFrame_Ocases, axiom, (! [A:$tType,X:((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((! [N3:nat,R2:A,H2:(heap_ext @ product_unit),N4:nat]: ((X != ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ N3) @ ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ N4))))))) => (~ ((! [N3:nat]: ((X != ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ N3) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))))).
% 9.43/3.44 thf(fact_78_execute__assert_I1_J, axiom, (! [A:$tType,P:(A > $o),X:A,H3:(heap_ext @ product_unit)]: (((P @ X) => ((((heap_Time_execute @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (one_one @ nat))))))))).
% 9.43/3.44 thf(fact_79_execute__tap, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ ((heap_Time_tap @ A) @ F)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (one_one @ nat)))))))).
% 9.43/3.44 thf(fact_80_option_Osel, axiom, (! [A:$tType,X22:A]: ((((the2 @ A) @ ((some @ A) @ X22)) = X22)))).
% 9.43/3.44 thf(fact_81_option_Oexpand, axiom, (! [A:$tType,Option:(option @ A),Option2:(option @ A)]: ((((Option = (none @ A)) = (Option2 = (none @ A))) => (((Option != (none @ A)) => ((Option2 != (none @ A)) => (((the2 @ A) @ Option) = ((the2 @ A) @ Option2)))) => (Option = Option2)))))).
% 9.43/3.44 thf(fact_82_effect__def, axiom, (! [A:$tType]: (((heap_Time_effect @ A) = (^ [C3:(heap_Time_Heap @ A),H:(heap_ext @ product_unit),H7:(heap_ext @ product_unit),R3:A,N5:nat]: (((((heap_Time_execute @ A) @ C3) @ H) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ N5)))))))))).
% 9.43/3.44 thf(fact_83_effectI, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),R:A,H4:(heap_ext @ product_unit),N:nat]: ((((((heap_Time_execute @ A) @ C2) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ N)))) => ((((((heap_Time_effect @ A) @ C2) @ H3) @ H4) @ R) @ N))))).
% 9.43/3.44 thf(fact_84_map__mmupd__def, axiom, (! [A:$tType,B:$tType]: ((((map_mmupd @ B) @ A) = (^ [M:(B > (option @ A)),K3:(set @ B),V2:A,K4:B]: (((((if @ (option @ A)) @ (((member @ B) @ K4) @ K3)) @ ((some @ A) @ V2)) @ (M @ K4)))))))).
% 9.43/3.44 thf(fact_85_map__mmupdE, axiom, (! [B:$tType,A:$tType,M2:(B > (option @ A)),K2:(set @ B),V:A,K:B,X:A]: (((((((((map_mmupd @ B) @ A) @ M2) @ K2) @ V) @ K) = ((some @ A) @ X)) => (((~ ((((member @ B) @ K) @ K2))) => ((M2 @ K) != ((some @ A) @ X))) => (~ (((((member @ B) @ K) @ K2) => (X != V))))))))).
% 9.43/3.44 thf(fact_86_success__bind__executeI, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),X:A,H4:(heap_ext @ product_unit),N:nat,G:(A > (heap_Time_Heap @ B))]: ((((((heap_Time_execute @ A) @ F) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ N)))) => ((((heap_Time_success @ B) @ (G @ X)) @ H4) => (((heap_Time_success @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3)))))).
% 9.43/3.44 thf(fact_87_Eps__Opt__eq__Some__implies, axiom, (! [A:$tType,P:(A > $o),X:A]: (((((eps_Opt @ A) @ P) = ((some @ A) @ X)) => (P @ X))))).
% 9.43/3.44 thf(fact_88_Eps__Opt__eq__Some, axiom, (! [A:$tType,P:(A > $o),X:A]: (((! [X5:A]: (((P @ X) => ((P @ X5) => (X5 = X))))) => ((((eps_Opt @ A) @ P) = ((some @ A) @ X)) = (P @ X)))))).
% 9.43/3.44 thf(fact_89_option_Oexhaust__sel, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) => (Option = ((some @ A) @ ((the2 @ A) @ Option))))))).
% 9.43/3.44 thf(fact_90_effect__assertE, axiom, (! [A:$tType,P:(A > $o),X:A,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ (((heap_Time_assert @ A) @ P) @ X)) @ H3) @ H4) @ R) @ N) => (~ (((P @ X) => ((R = X) => ((H4 = H3) => (N != (one_one @ nat))))))))))).
% 9.43/3.44 thf(fact_91_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F2:(B > A),G2:(C > B),X3:C]: ((F2 @ (G2 @ X3)))))))).
% 9.43/3.44 thf(fact_92_prod_Oinject, axiom, (! [A:$tType,B:$tType,X1:A,X22:B,Y1:A,Y2:B]: (((((((product_Pair @ A) @ B) @ X1) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y2)) = ((X1 = Y1) & (X22 = Y2)))))).
% 9.43/3.44 thf(fact_93_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,A6:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B4)) = ((A3 = A6) & (B2 = B4)))))).
% 9.43/3.44 thf(fact_94_execute__bind_I1_J, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),X:A,H4:(heap_ext @ product_unit),N:nat,G:(A > (heap_Time_Heap @ B))]: ((((((heap_Time_execute @ A) @ F) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ N)))) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3) = (((heap_Time_timeFrame @ B) @ N) @ (((heap_Time_execute @ B) @ (G @ X)) @ H4))))))).
% 9.43/3.44 thf(fact_95_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B3:B,C4:C]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B3) @ C4)))))))))).
% 9.43/3.44 thf(fact_96_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: ((~ ((! [A5:A,B3:B,C4:C,D2:D]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C4) @ D2))))))))))).
% 9.43/3.44 thf(fact_97_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: ((~ ((! [A5:A,B3:B,C4:C,D2:D,E3:E]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C4) @ ((((product_Pair @ D) @ E) @ D2) @ E3)))))))))))).
% 9.43/3.44 thf(fact_98_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F5:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5)))))]: ((~ ((! [A5:A,B3:B,C4:C,D2:D,E3:E,F4:F5]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F5)) @ D2) @ ((((product_Pair @ E) @ F5) @ E3) @ F4))))))))))))).
% 9.43/3.44 thf(fact_99_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,E:$tType,F5:$tType,G3:$tType,Y:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))))]: ((~ ((! [A5:A,B3:B,C4:C,D2:D,E3:E,F4:F5,G4:G3]: ((Y != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F5) @ G3)) @ E3) @ ((((product_Pair @ F5) @ G3) @ F4) @ G4)))))))))))))).
% 9.43/3.44 thf(fact_100_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B3:B,C4:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B3) @ C4))))) => (P @ X))))).
% 9.43/3.44 thf(fact_101_prod__induct4, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D)))]: (((! [A5:A,B3:B,C4:C,D2:D]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ B3) @ ((((product_Pair @ C) @ D) @ C4) @ D2)))))) => (P @ X))))).
% 9.43/3.44 thf(fact_102_timeFrame_Osimps_I2_J, axiom, (! [A:$tType,N:nat]: (((((heap_Time_timeFrame @ A) @ N) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))).
% 9.43/3.44 thf(fact_103_Pair__inject, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,A6:A,B4:B]: (((((((product_Pair @ A) @ B) @ A3) @ B2) = ((((product_Pair @ A) @ B) @ A6) @ B4)) => (~ (((A3 = A6) => (B2 != B4)))))))).
% 9.43/3.44 thf(fact_104_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P5:((product_prod @ A) @ B)]: (((! [A5:A,B3:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B3)))) => (P @ P5))))).
% 9.43/3.44 thf(fact_105_surj__pair, axiom, (! [A:$tType,B:$tType,P5:((product_prod @ A) @ B)]: ((? [X2:A,Y4:B]: ((P5 = ((((product_Pair @ A) @ B) @ X2) @ Y4))))))).
% 9.43/3.44 thf(fact_106_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B3:B]: ((Y != ((((product_Pair @ A) @ B) @ A5) @ B3))))))))).
% 9.43/3.44 thf(fact_107_one__reorient, axiom, (! [A:$tType]: (((one @ A) => (! [X:A]: ((((one_one @ A) = X) = (X = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_108_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = C2) => ((A3 @ (B2 @ V)) = (C2 @ V)))))).
% 9.43/3.44 thf(fact_109_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(D > B),D3:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D3)) => (! [V3:A]: (((A3 @ (B2 @ V3)) = (C2 @ (D3 @ V3))))))))).
% 9.43/3.44 thf(fact_110_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(D > B),D3:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = (((((comp @ D) @ B) @ A) @ C2) @ D3)) => ((A3 @ (B2 @ V)) = (C2 @ (D3 @ V))))))).
% 9.43/3.44 thf(fact_111_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G:(C > D),H3:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G)) @ H3) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G) @ H3)))))).
% 9.43/3.44 thf(fact_112_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F2:(B > C),G2:(A > B),X3:A]: ((F2 @ (G2 @ X3)))))))).
% 9.43/3.44 thf(fact_113_prod__induct7, axiom, (! [G3:$tType,F5:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))))]: (((! [A5:A,B3:B,C4:C,D2:D,E3:E,F4:F5,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3)))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ ((product_prod @ F5) @ G3))) @ D2) @ ((((product_Pair @ E) @ ((product_prod @ F5) @ G3)) @ E3) @ ((((product_Pair @ F5) @ G3) @ F4) @ G4))))))))) => (P @ X))))).
% 9.43/3.44 thf(fact_114_prod__induct6, axiom, (! [F5:$tType,E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5))))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5)))))]: (((! [A5:A,B3:B,C4:C,D2:D,E3:E,F4:F5]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5)))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ ((product_prod @ E) @ F5))) @ C4) @ ((((product_Pair @ D) @ ((product_prod @ E) @ F5)) @ D2) @ ((((product_Pair @ E) @ F5) @ E3) @ F4)))))))) => (P @ X))))).
% 9.43/3.44 thf(fact_115_prod__induct5, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) > $o),X:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))))]: (((! [A5:A,B3:B,C4:C,D2:D,E3:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D) @ E))) @ B3) @ ((((product_Pair @ C) @ ((product_prod @ D) @ E)) @ C4) @ ((((product_Pair @ D) @ E) @ D2) @ E3))))))) => (P @ X))))).
% 9.43/3.44 thf(fact_116_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A3:A,B2:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) = ((F1 @ A3) @ B2))))).
% 9.43/3.44 thf(fact_117_timeFrame_Oelims, axiom, (! [A:$tType,X:nat,Xa:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),Y:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: ((((((heap_Time_timeFrame @ A) @ X) @ Xa) = Y) => ((! [R2:A,H2:(heap_ext @ product_unit),N4:nat]: (((Xa = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ N4)))) => (Y != ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ (((plus_plus @ nat) @ X) @ N4)))))))) => (~ (((Xa = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => (Y != (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))))).
% 9.43/3.44 thf(fact_118_execute__bind__eq__SomeI, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),X:A,H4:(heap_ext @ product_unit),N:nat,G:(A > (heap_Time_Heap @ B)),Y:B,H5:(heap_ext @ product_unit),N2:nat]: ((((((heap_Time_execute @ A) @ F) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H4) @ N)))) => (((((heap_Time_execute @ B) @ (G @ X)) @ H4) = ((some @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ Y) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ N2)))) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3) = ((some @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ Y) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H5) @ (((plus_plus @ nat) @ N) @ N2)))))))))).
% 9.43/3.44 thf(fact_119_uncurry__apply, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A3:B,B2:C]: (((((((uncurry @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A3) @ B2)) = ((F @ A3) @ B2))))).
% 9.43/3.44 thf(fact_120_timeFrame_Osimps_I1_J, axiom, (! [A:$tType,N:nat,R:A,H3:(heap_ext @ product_unit),N2:nat]: (((((heap_Time_timeFrame @ A) @ N) @ ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ N2)))) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (((plus_plus @ nat) @ N) @ N2)))))))).
% 9.43/3.44 thf(fact_121_heap__def, axiom, (! [A:$tType]: (((heap_Time_heap @ A) = (^ [F2:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((heap_Time_Heap2 @ A) @ (((((comp @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ F2)))))))).
% 9.43/3.44 thf(fact_122_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A3:B,B2:C]: (((((((produc5280177257484947105e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B2)) = ((C2 @ A3) @ B2))))).
% 9.43/3.44 thf(fact_123_option_Osplit__sel, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (((Option = (none @ A)) => (P @ F1)) & ((Option = ((some @ A) @ ((the2 @ A) @ Option))) => (P @ (F22 @ ((the2 @ A) @ Option))))))))).
% 9.43/3.44 thf(fact_124_option_Osplit__sel__asm, axiom, (! [B:$tType,A:$tType,P:(B > $o),F1:B,F22:(A > B),Option:(option @ A)]: (((P @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (~ ((((Option = (none @ A)) & (~ ((P @ F1)))) | ((Option = ((some @ A) @ ((the2 @ A) @ Option))) & (~ ((P @ (F22 @ ((the2 @ A) @ Option))))))))))))).
% 9.43/3.44 thf(fact_125_fold__atLeastAtMost__nat_Ocases, axiom, (! [A:$tType,X:((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))]: ((~ ((! [F4:(nat > (A > A)),A5:nat,B3:nat,Acc:A]: ((X != ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ F4) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A5) @ ((((product_Pair @ nat) @ A) @ B3) @ Acc))))))))))).
% 9.43/3.44 thf(fact_126_one__natural_Orsp, axiom, ((one_one @ nat) = (one_one @ nat))).
% 9.43/3.44 thf(fact_127_timeFrame_Opelims, axiom, (! [A:$tType,X:nat,Xa:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),Y:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: ((((((heap_Time_timeFrame @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ (heap_T5500966940807335491me_rel @ A)) @ ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ X) @ Xa)) => ((! [R2:A,H2:(heap_ext @ product_unit),N4:nat]: (((Xa = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ N4)))) => ((Y = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ (((plus_plus @ nat) @ X) @ N4))))) => (~ ((((accp @ ((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ (heap_T5500966940807335491me_rel @ A)) @ ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ X) @ ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H2) @ N4))))))))))) => (~ (((Xa = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => ((Y = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) => (~ ((((accp @ ((product_prod @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ (heap_T5500966940807335491me_rel @ A)) @ ((((product_Pair @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ X) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))))))))).
% 9.43/3.44 thf(fact_128_add__right__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A3:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C2) @ A3)) = (B2 = C2)))))))).
% 9.43/3.44 thf(fact_129_add__left__cancel, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C2)) = (B2 = C2)))))))).
% 9.43/3.44 thf(fact_130_Heap_Oinject, axiom, (! [A:$tType,X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))),Ya:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((((heap_Time_Heap2 @ A) @ X) = ((heap_Time_Heap2 @ A) @ Ya)) = (X = Ya))))).
% 9.43/3.44 thf(fact_131_Heap__execute, axiom, (! [A:$tType,F:(heap_Time_Heap @ A)]: ((((heap_Time_Heap2 @ A) @ ((heap_Time_execute @ A) @ F)) = F)))).
% 9.43/3.44 thf(fact_132_timeFrame__assoc, axiom, (! [A:$tType,N:nat,N2:nat,F:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((((heap_Time_timeFrame @ A) @ N) @ (((heap_Time_timeFrame @ A) @ N2) @ F)) = (((heap_Time_timeFrame @ A) @ (((plus_plus @ nat) @ N) @ N2)) @ F))))).
% 9.43/3.44 thf(fact_133_add__right__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [B2:A,A3:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C2) @ A3)) => (B2 = C2)))))))).
% 9.43/3.44 thf(fact_134_add__left__imp__eq, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C2)) => (B2 = C2)))))))).
% 9.43/3.44 thf(fact_135_ab__semigroup__add__class_Oadd_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [B2:A,A3:A,C2:A]: (((((plus_plus @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_136_ab__semigroup__add__class_Oadd_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((plus_plus @ A) = (^ [A7:A,B5:A]: ((((plus_plus @ A) @ B5) @ A7)))))))).
% 9.43/3.44 thf(fact_137_add_Oright__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [B2:A,A3:A,C2:A]: ((((((plus_plus @ A) @ B2) @ A3) = (((plus_plus @ A) @ C2) @ A3)) = (B2 = C2)))))))).
% 9.43/3.44 thf(fact_138_add_Oleft__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ A3) @ C2)) = (B2 = C2)))))))).
% 9.43/3.44 thf(fact_139_add_Oassoc, axiom, (! [A:$tType]: (((semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_140_group__cancel_Oadd2, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [B6:A,K:A,B2:A,A3:A]: (((B6 = (((plus_plus @ A) @ K) @ B2)) => ((((plus_plus @ A) @ A3) @ B6) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_141_group__cancel_Oadd1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A4:A,K:A,A3:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A3)) => ((((plus_plus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_142_add__mono__thms__linordered__semiring_I4_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (K = L)) => ((((plus_plus @ A) @ I) @ K) = (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_143_Heap_Oexhaust, axiom, (! [A:$tType,Y:(heap_Time_Heap @ A)]: ((~ ((! [X2:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((Y != ((heap_Time_Heap2 @ A) @ X2))))))))).
% 9.43/3.44 thf(fact_144_execute_Osimps, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((heap_Time_execute @ A) @ ((heap_Time_Heap2 @ A) @ F)) = F)))).
% 9.43/3.44 thf(fact_145_option_Osimps_I5_J, axiom, (! [B:$tType,A:$tType,F1:B,F22:(A > B),X22:A]: (((((((case_option @ B) @ A) @ F1) @ F22) @ ((some @ A) @ X22)) = (F22 @ X22))))).
% 9.43/3.44 thf(fact_146_option_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,F1:B,F22:(A > B)]: (((((((case_option @ B) @ A) @ F1) @ F22) @ (none @ A)) = F1)))).
% 9.43/3.44 thf(fact_147_effect__bindE, axiom, (! [B:$tType,A:$tType,F:(heap_Time_Heap @ B),G:(B > (heap_Time_Heap @ A)),H3:(heap_ext @ product_unit),H5:(heap_ext @ product_unit),R4:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((((heap_Time_bind @ B) @ A) @ F) @ G)) @ H3) @ H5) @ R4) @ N) => (~ ((! [H6:(heap_ext @ product_unit),R2:B,N1:nat]: ((((((((heap_Time_effect @ B) @ F) @ H3) @ H6) @ R2) @ N1) => (! [N22:nat]: ((((((((heap_Time_effect @ A) @ (G @ R2)) @ H6) @ H5) @ R4) @ N22) => (N != (((plus_plus @ nat) @ N1) @ N22)))))))))))))).
% 9.43/3.44 thf(fact_148_effect__bindI, axiom, (! [B:$tType,A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat,G:(A > (heap_Time_Heap @ B)),H5:(heap_ext @ product_unit),R4:B,N2:nat]: ((((((((heap_Time_effect @ A) @ F) @ H3) @ H4) @ R) @ N) => (((((((heap_Time_effect @ B) @ (G @ R)) @ H4) @ H5) @ R4) @ N2) => ((((((heap_Time_effect @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3) @ H5) @ R4) @ (((plus_plus @ nat) @ N) @ N2))))))).
% 9.43/3.44 thf(fact_149_option_Ocase__eq__if, axiom, (! [A:$tType,B:$tType]: ((((case_option @ B) @ A) = (^ [F12:B,F23:(A > B),Option3:(option @ A)]: (((((if @ B) @ (Option3 = (none @ A))) @ F12) @ (F23 @ ((the2 @ A) @ Option3))))))))).
% 9.43/3.44 thf(fact_150_uncurry__curry__id, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((uncurry @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 9.43/3.44 thf(fact_151_curry__uncurry__id, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((uncurry @ A) @ B) @ C) @ F)) = F)))).
% 9.43/3.44 thf(fact_152_accp_Ocases, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A]: (((((accp @ A) @ R) @ A3) => (! [Y6:A]: ((((R @ Y6) @ A3) => (((accp @ A) @ R) @ Y6)))))))).
% 9.43/3.44 thf(fact_153_accp_Osimps, axiom, (! [A:$tType]: (((accp @ A) = (^ [R3:(A > (A > $o)),A7:A]: ((? [X3:A]: (((A7 = X3) & (! [Y3:A]: ((((R3 @ Y3) @ X3) => (((accp @ A) @ R3) @ Y3))))))))))))).
% 9.43/3.44 thf(fact_154_accpI, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: (((! [Y4:A]: ((((R @ Y4) @ X) => (((accp @ A) @ R) @ Y4)))) => (((accp @ A) @ R) @ X))))).
% 9.43/3.44 thf(fact_155_accp__induct, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,P:(A > $o)]: (((((accp @ A) @ R) @ A3) => ((! [X2:A]: (((((accp @ A) @ R) @ X2) => ((! [Y6:A]: ((((R @ Y6) @ X2) => (P @ Y6)))) => (P @ X2))))) => (P @ A3)))))).
% 9.43/3.44 thf(fact_156_accp__downward, axiom, (! [A:$tType,R:(A > (A > $o)),B2:A,A3:A]: (((((accp @ A) @ R) @ B2) => (((R @ A3) @ B2) => (((accp @ A) @ R) @ A3)))))).
% 9.43/3.44 thf(fact_157_not__accp__down, axiom, (! [A:$tType,R5:(A > (A > $o)),X:A]: (((~ ((((accp @ A) @ R5) @ X))) => (~ ((! [Z3:A]: ((((R5 @ Z3) @ X) => (((accp @ A) @ R5) @ Z3)))))))))).
% 9.43/3.44 thf(fact_158_accp__induct__rule, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,P:(A > $o)]: (((((accp @ A) @ R) @ A3) => ((! [X2:A]: (((((accp @ A) @ R) @ X2) => ((! [Y6:A]: ((((R @ Y6) @ X2) => (P @ Y6)))) => (P @ X2))))) => (P @ A3)))))).
% 9.43/3.44 thf(fact_159_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G:(B > C),F:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G) @ (((((comp @ A) @ B) @ D) @ F) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 9.43/3.44 thf(fact_160_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F:(B > A),G:(C > B),X:C,F3:(D > A),G5:(E > D),X6:E]: ((((F @ (G @ X)) = (F3 @ (G5 @ X6))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ E) @ F3) @ G5) @ X6)))))).
% 9.43/3.44 thf(fact_161_curryI, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B2:B]: (((F @ ((((product_Pair @ A) @ B) @ A3) @ B2)) => ((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B2))))).
% 9.43/3.44 thf(fact_162_curry__conv, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_curry @ B) @ C) @ A) = (^ [F2:(((product_prod @ B) @ C) > A),A7:B,B5:C]: ((F2 @ ((((product_Pair @ B) @ C) @ A7) @ B5)))))))).
% 9.43/3.44 thf(fact_163_curryE, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A3) @ B2)))))).
% 9.43/3.44 thf(fact_164_curryD, axiom, (! [A:$tType,B:$tType,F:(((product_prod @ A) @ B) > $o),A3:A,B2:B]: ((((((((product_curry @ A) @ B) @ $o) @ F) @ A3) @ B2) => (F @ ((((product_Pair @ A) @ B) @ A3) @ B2)))))).
% 9.43/3.44 thf(fact_165_fold__atLeastAtMost__nat_Opinduct, axiom, (! [A:$tType,A0:(nat > (A > A)),A1:nat,A22:nat,A32:A,P:((nat > (A > A)) > (nat > (nat > (A > $o))))]: (((((accp @ ((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))) @ (set_fo1817059534552279752at_rel @ A)) @ ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ A0) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A1) @ ((((product_Pair @ nat) @ A) @ A22) @ A32)))) => ((! [F4:(nat > (A > A)),A5:nat,B3:nat,Acc:A]: (((((accp @ ((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))) @ (set_fo1817059534552279752at_rel @ A)) @ ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ F4) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A5) @ ((((product_Pair @ nat) @ A) @ B3) @ Acc)))) => (((~ ((((ord_less @ nat) @ B3) @ A5))) => ((((P @ F4) @ (((plus_plus @ nat) @ A5) @ (one_one @ nat))) @ B3) @ ((F4 @ A5) @ Acc))) => ((((P @ F4) @ A5) @ B3) @ Acc))))) => ((((P @ A0) @ A1) @ A22) @ A32)))))).
% 9.43/3.44 thf(fact_166_in__lex__prod, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,A6:A,B4:B,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Pair @ A) @ B) @ A6) @ B4))) @ ((((lex_prod @ A) @ B) @ R) @ S)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A6)) @ R) | ((A3 = A6) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B2) @ B4)) @ S))))))).
% 9.43/3.44 thf(fact_167_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G:(C > B),H3:(A > C),R1:(D > B),R22:(A > D),F:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G) @ H3) = (((((comp @ D) @ B) @ A) @ R1) @ R22)) => (((((((comp @ B) @ E) @ D) @ F) @ R1) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F) @ G)) @ H3) = (((((comp @ D) @ E) @ A) @ L) @ R22))))))).
% 9.43/3.44 thf(fact_168_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F:(C > B),G:(A > C),L1:(D > B),L2:(A > D),H3:(E > A),R:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H3) = R) => ((((((comp @ C) @ B) @ E) @ F) @ (((((comp @ A) @ C) @ E) @ G) @ H3)) = (((((comp @ D) @ B) @ E) @ L1) @ R))))))).
% 9.43/3.44 thf(fact_169_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G:(C > B),H3:(A > C),R:(A > B),F:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G) @ H3) = R) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F) @ G)) @ H3) = (((((comp @ B) @ D) @ A) @ F) @ R)))))).
% 9.43/3.44 thf(fact_170_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(C > B),G:(A > C),L:(A > B),H3:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = L) => ((((((comp @ C) @ B) @ D) @ F) @ (((((comp @ A) @ C) @ D) @ G) @ H3)) = (((((comp @ A) @ B) @ D) @ L) @ H3)))))).
% 9.43/3.44 thf(fact_171_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M3:(B > A),G:(C > B),X:C,N6:(D > A),H3:(C > D),F:(A > E)]: ((((M3 @ (G @ X)) = (N6 @ (H3 @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F) @ M3)) @ G) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F) @ N6)) @ H3) @ X)))))).
% 9.43/3.44 thf(fact_172_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),X:C,H3:(D > A),K:(C > D)]: ((((F @ (G @ X)) = (H3 @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ C) @ H3) @ K) @ X)))))).
% 9.43/3.44 thf(fact_173_effect__returnE, axiom, (! [A:$tType,X:A,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_return @ A) @ X)) @ H3) @ H4) @ R) @ N) => (~ (((R = X) => ((H4 = H3) => (N != (one_one @ nat)))))))))).
% 9.43/3.44 thf(fact_174_effect__returnI, axiom, (! [A:$tType,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),X:A]: (((H3 = H4) => ((((((heap_Time_effect @ A) @ ((heap_Time_return @ A) @ X)) @ H3) @ H4) @ X) @ (one_one @ nat)))))).
% 9.43/3.44 thf(fact_175_add_Oright__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2)))))))).
% 9.43/3.44 thf(fact_176_add__less__cancel__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_177_add__less__cancel__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_178_fold__if__return, axiom, (! [A:$tType,B2:$o,C2:A,D3:A]: (((B2 => (((heap_Time_return @ A) @ C2) = ((heap_Time_return @ A) @ ((((if @ A) @ B2) @ C2) @ D3)))) & ((~ (B2)) => (((heap_Time_return @ A) @ D3) = ((heap_Time_return @ A) @ ((((if @ A) @ B2) @ C2) @ D3)))))))).
% 9.43/3.44 thf(fact_179_add__less__imp__less__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_180_add__less__imp__less__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_181_add__strict__right__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_182_add__strict__left__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_183_add__strict__mono, axiom, (! [A:$tType]: (((strict9044650504122735259up_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D3) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_184_add__mono__thms__linordered__field_I1_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (K = L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_185_add__mono__thms__linordered__field_I2_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_186_add__mono__thms__linordered__field_I5_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_187_success__returnI, axiom, (! [A:$tType,X:A,H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((heap_Time_return @ A) @ X)) @ H3)))).
% 9.43/3.44 thf(fact_188_fails__return, axiom, (! [A:$tType,X:A,H3:(heap_ext @ product_unit)]: ((~ ((((time_fails @ A) @ ((heap_Time_return @ A) @ X)) @ H3)))))).
% 9.43/3.44 thf(fact_189_add_Oright__assoc, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_190_nat__add__left__cancel__less, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less @ nat) @ M2) @ N))))).
% 9.43/3.44 thf(fact_191_less__option__Some, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less @ (option @ A)) @ ((some @ A) @ X)) @ ((some @ A) @ Y)) = (((ord_less @ A) @ X) @ Y)))))))).
% 9.43/3.44 thf(fact_192_fold__atLeastAtMost__nat_Opelims, axiom, (! [A:$tType,X:(nat > (A > A)),Xa:nat,Xb:nat,Xc:A,Y:A]: ((((((((set_fo6178422350223883121st_nat @ A) @ X) @ Xa) @ Xb) @ Xc) = Y) => ((((accp @ ((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))) @ (set_fo1817059534552279752at_rel @ A)) @ ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ X) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ Xa) @ ((((product_Pair @ nat) @ A) @ Xb) @ Xc)))) => (~ (((((((ord_less @ nat) @ Xb) @ Xa) => (Y = Xc)) & ((~ ((((ord_less @ nat) @ Xb) @ Xa))) => (Y = (((((set_fo6178422350223883121st_nat @ A) @ X) @ (((plus_plus @ nat) @ Xa) @ (one_one @ nat))) @ Xb) @ ((X @ Xa) @ Xc))))) => (~ ((((accp @ ((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))) @ (set_fo1817059534552279752at_rel @ A)) @ ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ X) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ Xa) @ ((((product_Pair @ nat) @ A) @ Xb) @ Xc)))))))))))))).
% 9.43/3.44 thf(fact_193_fold__atLeastAtMost__nat_Opsimps, axiom, (! [A:$tType,F:(nat > (A > A)),A3:nat,B2:nat,Acc2:A]: (((((accp @ ((product_prod @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A)))) @ (set_fo1817059534552279752at_rel @ A)) @ ((((product_Pair @ (nat > (A > A))) @ ((product_prod @ nat) @ ((product_prod @ nat) @ A))) @ F) @ ((((product_Pair @ nat) @ ((product_prod @ nat) @ A)) @ A3) @ ((((product_Pair @ nat) @ A) @ B2) @ Acc2)))) => (((((ord_less @ nat) @ B2) @ A3) => ((((((set_fo6178422350223883121st_nat @ A) @ F) @ A3) @ B2) @ Acc2) = Acc2)) & ((~ ((((ord_less @ nat) @ B2) @ A3))) => ((((((set_fo6178422350223883121st_nat @ A) @ F) @ A3) @ B2) @ Acc2) = (((((set_fo6178422350223883121st_nat @ A) @ F) @ (((plus_plus @ nat) @ A3) @ (one_one @ nat))) @ B2) @ ((F @ A3) @ Acc2))))))))).
% 9.43/3.44 thf(fact_194_in__measure, axiom, (! [A:$tType,X:A,Y:A,F:(A > nat)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measure @ A) @ F)) = (((ord_less @ nat) @ (F @ X)) @ (F @ Y)))))).
% 9.43/3.44 thf(fact_195_less__add__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A]: ((((ord_less @ A) @ A3) @ (((plus_plus @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_196_add__mono1, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (((plus_plus @ A) @ B2) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_197_lift__collapse, axiom, (! [A:$tType,B:$tType]: ((((heap_Time_lift @ B) @ A) = (^ [F2:(B > A),X3:B]: (((heap_Time_return @ A) @ (F2 @ X3)))))))).
% 9.43/3.44 thf(fact_198_add__lessD1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) => (((ord_less @ nat) @ I) @ K))))).
% 9.43/3.44 thf(fact_199_add__less__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ K) @ L) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 9.43/3.44 thf(fact_200_not__add__less1, axiom, (! [I:nat,J:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ I)))))).
% 9.43/3.44 thf(fact_201_less__option__None, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:(option @ A)]: ((~ ((((ord_less @ (option @ A)) @ X) @ (none @ A)))))))))).
% 9.43/3.44 thf(fact_202_less__option__None__Some__code, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less @ (option @ A)) @ (none @ A)) @ ((some @ A) @ X)))))))).
% 9.43/3.44 thf(fact_203_less__option__None__Some, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((((ord_less @ (option @ A)) @ (none @ A)) @ ((some @ A) @ X)))))))).
% 9.43/3.44 thf(fact_204_less__option__None__is__Some, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:(option @ A)]: (((((ord_less @ (option @ A)) @ (none @ A)) @ X) => (? [Z3:A]: ((X = ((some @ A) @ Z3))))))))))).
% 9.43/3.44 thf(fact_205_lift__def, axiom, (! [A:$tType,B:$tType]: ((((heap_Time_lift @ A) @ B) = ((((comp @ B) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_return @ B)))))).
% 9.43/3.44 thf(fact_206_fold__atLeastAtMost__nat_Osimps, axiom, (! [A:$tType]: (((set_fo6178422350223883121st_nat @ A) = (^ [F2:(nat > (A > A)),A7:nat,B5:nat,Acc3:A]: (((((if @ A) @ (((ord_less @ nat) @ B5) @ A7)) @ Acc3) @ (((((set_fo6178422350223883121st_nat @ A) @ F2) @ (((plus_plus @ nat) @ A7) @ (one_one @ nat))) @ B5) @ ((F2 @ A7) @ Acc3))))))))).
% 9.43/3.44 thf(fact_207_fold__atLeastAtMost__nat_Oelims, axiom, (! [A:$tType,X:(nat > (A > A)),Xa:nat,Xb:nat,Xc:A,Y:A]: ((((((((set_fo6178422350223883121st_nat @ A) @ X) @ Xa) @ Xb) @ Xc) = Y) => (((((ord_less @ nat) @ Xb) @ Xa) => (Y = Xc)) & ((~ ((((ord_less @ nat) @ Xb) @ Xa))) => (Y = (((((set_fo6178422350223883121st_nat @ A) @ X) @ (((plus_plus @ nat) @ Xa) @ (one_one @ nat))) @ Xb) @ ((X @ Xa) @ Xc))))))))).
% 9.43/3.44 thf(fact_208_less__add__eq__less, axiom, (! [K:nat,L:nat,M2:nat,N:nat]: (((((ord_less @ nat) @ K) @ L) => (((((plus_plus @ nat) @ M2) @ L) = (((plus_plus @ nat) @ K) @ N)) => (((ord_less @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_209_trans__less__add2, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ M2) @ J)))))).
% 9.43/3.44 thf(fact_210_trans__less__add1, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M2)))))).
% 9.43/3.44 thf(fact_211_add__less__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ J) => (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 9.43/3.44 thf(fact_212_not__add__less2, axiom, (! [J:nat,I:nat]: ((~ ((((ord_less @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ I)))))).
% 9.43/3.44 thf(fact_213_less__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_214_mlex__iff, axiom, (! [A:$tType,X:A,Y:A,F:(A > nat),R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((mlex_prod @ A) @ F) @ R5)) = ((((ord_less @ nat) @ (F @ X)) @ (F @ Y)) | (((F @ X) = (F @ Y)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R5))))))).
% 9.43/3.44 thf(fact_215_mlex__less, axiom, (! [A:$tType,F:(A > nat),X:A,Y:A,R5:(set @ ((product_prod @ A) @ A))]: (((((ord_less @ nat) @ (F @ X)) @ (F @ Y)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((mlex_prod @ A) @ F) @ R5)))))).
% 9.43/3.44 thf(fact_216_pairself_Opelims, axiom, (! [B:$tType,A:$tType,X:(A > B),Xa:((product_prod @ A) @ A),Y:((product_prod @ B) @ B)]: (((((((pairself @ A) @ B) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (A > B)) @ ((product_prod @ A) @ A))) @ ((pairself_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ X) @ Xa)) => (~ ((! [A5:A,B3:A]: (((Xa = ((((product_Pair @ A) @ A) @ A5) @ B3)) => ((Y = ((((product_Pair @ B) @ B) @ (X @ A5)) @ (X @ B3))) => (~ ((((accp @ ((product_prod @ (A > B)) @ ((product_prod @ A) @ A))) @ ((pairself_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ ((product_prod @ A) @ A)) @ X) @ ((((product_Pair @ A) @ A) @ A5) @ B3)))))))))))))))).
% 9.43/3.44 thf(fact_217_execute__return, axiom, (! [A:$tType,X:A]: ((((heap_Time_execute @ A) @ ((heap_Time_return @ A) @ X)) = (((((comp @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (one_one @ nat)))))))))).
% 9.43/3.44 thf(fact_218_same__fstI, axiom, (! [B:$tType,A:$tType,P:(A > $o),X:A,Y7:B,Y:B,R5:(A > (set @ ((product_prod @ B) @ B)))]: (((P @ X) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y7) @ Y)) @ (R5 @ X)) => (((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y7)) @ ((((product_Pair @ A) @ B) @ X) @ Y))) @ ((((same_fst @ A) @ B) @ P) @ R5))))))).
% 9.43/3.44 thf(fact_219_zero__less__two, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_220_is__num__normalize_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_221_add_Osafe__commute, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => (! [X:A,Y:A,A3:A,B2:A]: ((((((syntax7388354845996824322omatch @ A) @ A) @ (((plus_plus @ A) @ X) @ Y)) @ A3) => ((((plus_plus @ A) @ A3) @ B2) = (((plus_plus @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_222_bijective__def, axiom, (! [B:$tType,A:$tType]: ((((bijective @ A) @ B) = (^ [R6:(set @ ((product_prod @ A) @ B))]: (((! [X3:A,Y3:B,Z5:B]: ((((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R6) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Z5)) @ R6)) => (Y3 = Z5)))) & (! [X3:A,Y3:A,Z5:B]: ((((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Z5)) @ R6) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y3) @ Z5)) @ R6)) => (X3 = Y3))))))))))).
% 9.43/3.44 thf(fact_223_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S:B,R5:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S)) @ R5) => ((S3 = S) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S3)) @ R5)))))).
% 9.43/3.44 thf(fact_224_add__is__0, axiom, (! [M2:nat,N:nat]: ((((((plus_plus @ nat) @ M2) @ N) = (zero_zero @ nat)) = ((M2 = (zero_zero @ nat)) & (N = (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_225_Nat_Oadd__0__right, axiom, (! [M2:nat]: (((((plus_plus @ nat) @ M2) @ (zero_zero @ nat)) = M2)))).
% 9.43/3.44 thf(fact_226_timeFrame__zero, axiom, (! [A:$tType,H3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((((heap_Time_timeFrame @ A) @ (zero_zero @ nat)) @ H3) = H3)))).
% 9.43/3.44 thf(fact_227_Heap__Time__Monad_Obind__bind, axiom, (! [C:$tType,B:$tType,A:$tType,F:(heap_Time_Heap @ A),G:(A > (heap_Time_Heap @ C)),K:(C > (heap_Time_Heap @ B))]: ((((((heap_Time_bind @ C) @ B) @ ((((heap_Time_bind @ A) @ C) @ F) @ G)) @ K) = ((((heap_Time_bind @ A) @ B) @ F) @ (^ [X3:A]: (((((heap_Time_bind @ C) @ B) @ (G @ X3)) @ K)))))))).
% 9.43/3.44 thf(fact_228_not__gr__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((~ ((((ord_less @ A) @ (zero_zero @ A)) @ N))) = (N = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_229_add__0, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 9.43/3.44 thf(fact_230_zero__eq__add__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A,Y:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ X) @ Y)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_231_add__eq__0__iff__both__eq__0, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A,Y:A]: ((((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_232_add__cancel__right__right, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A,B2:A]: (((A3 = (((plus_plus @ A) @ A3) @ B2)) = (B2 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_233_add__cancel__right__left, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A,B2:A]: (((A3 = (((plus_plus @ A) @ B2) @ A3)) = (B2 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_234_add__cancel__left__right, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = A3) = (B2 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_235_add__cancel__left__left, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [B2:A,A3:A]: ((((((plus_plus @ A) @ B2) @ A3) = A3) = (B2 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_236_double__zero__sym, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: ((((zero_zero @ A) = (((plus_plus @ A) @ A3) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_237_add_Oright__neutral, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 9.43/3.44 thf(fact_238_add__gr__0, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ M2) @ N)) = ((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) | (((ord_less @ nat) @ (zero_zero @ nat)) @ N)))))).
% 9.43/3.44 thf(fact_239_less__one, axiom, (! [N:nat]: (((((ord_less @ nat) @ N) @ (one_one @ nat)) = (N = (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_240_some__opt__eq__trivial, axiom, (! [A:$tType,X:A]: ((((eps_Opt @ A) @ (^ [Y3:A]: ((Y3 = X)))) = ((some @ A) @ X))))).
% 9.43/3.44 thf(fact_241_zero__less__double__add__iff__zero__less__single__add, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_242_double__add__less__zero__iff__single__add__less__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_243_less__add__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (((plus_plus @ A) @ B2) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 9.43/3.44 thf(fact_244_less__add__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((ord_less @ A) @ (zero_zero @ A)) @ B2)))))))).
% 9.43/3.44 thf(fact_245_add__less__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_246_add__less__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_247_some__opt__false__trivial, axiom, (! [A:$tType]: ((((eps_Opt @ A) @ (^ [Uu:A]: ($false))) = (none @ A))))).
% 9.43/3.44 thf(fact_248_zero__reorient, axiom, (! [A:$tType]: (((zero @ A) => (! [X:A]: ((((zero_zero @ A) = X) = (X = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_249_fun_Omap__ident, axiom, (! [A:$tType,D:$tType,T2:(D > A)]: (((((((comp @ A) @ A) @ D) @ (^ [X3:A]: (X3))) @ T2) = T2)))).
% 9.43/3.44 thf(fact_250_option_Odisc__eq__case_I2_J, axiom, (! [A:$tType,Option:(option @ A)]: (((Option != (none @ A)) = (((((case_option @ $o) @ A) @ $false) @ (^ [Uu:A]: ($true))) @ Option))))).
% 9.43/3.44 thf(fact_251_option_Odisc__eq__case_I1_J, axiom, (! [A:$tType,Option:(option @ A)]: (((Option = (none @ A)) = (((((case_option @ $o) @ A) @ $true) @ (^ [Uu:A]: ($false))) @ Option))))).
% 9.43/3.44 thf(fact_252_option_Ocase__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,H3:(B > C),F1:B,F22:(A > B),Option:(option @ A)]: (((H3 @ (((((case_option @ B) @ A) @ F1) @ F22) @ Option)) = (((((case_option @ C) @ A) @ (H3 @ F1)) @ (^ [X3:A]: ((H3 @ (F22 @ X3))))) @ Option))))).
% 9.43/3.44 thf(fact_253_effect__LetI, axiom, (! [B:$tType,A:$tType,X:A,T2:A,F:(A > (heap_Time_Heap @ B)),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:B,N:nat]: (((X = T2) => (((((((heap_Time_effect @ B) @ (F @ X)) @ H3) @ H4) @ R) @ N) => ((((((heap_Time_effect @ B) @ (F @ T2)) @ H3) @ H4) @ R) @ N)))))).
% 9.43/3.44 thf(fact_254_distrib__if__bind, axiom, (! [A:$tType,B:$tType,B2:$o,C2:(heap_Time_Heap @ B),D3:(heap_Time_Heap @ B),F:(B > (heap_Time_Heap @ A))]: (((B2 => (((((heap_Time_bind @ B) @ A) @ ((((if @ (heap_Time_Heap @ B)) @ B2) @ C2) @ D3)) @ F) = ((((heap_Time_bind @ B) @ A) @ C2) @ F))) & ((~ (B2)) => (((((heap_Time_bind @ B) @ A) @ ((((if @ (heap_Time_Heap @ B)) @ B2) @ C2) @ D3)) @ F) = ((((heap_Time_bind @ B) @ A) @ D3) @ F))))))).
% 9.43/3.44 thf(fact_255_success__LetI, axiom, (! [A:$tType,B:$tType,X:A,T2:A,F:(A > (heap_Time_Heap @ B)),H3:(heap_ext @ product_unit)]: (((X = T2) => ((((heap_Time_success @ B) @ (F @ X)) @ H3) => (((heap_Time_success @ B) @ (F @ T2)) @ H3)))))).
% 9.43/3.44 thf(fact_256_curry__K, axiom, (! [B:$tType,C:$tType,A:$tType,C2:C]: ((((((product_curry @ A) @ B) @ C) @ (^ [X3:((product_prod @ A) @ B)]: (C2))) = (^ [X3:A,Y3:B]: (C2)))))).
% 9.43/3.44 thf(fact_257_case__optionE, axiom, (! [A:$tType,P:$o,Q:(A > $o),X:(option @ A)]: (((((((case_option @ $o) @ A) @ P) @ Q) @ X) => (((X = (none @ A)) => (~ (P))) => (~ ((! [Y4:A]: (((X = ((some @ A) @ Y4)) => (~ ((Q @ Y4))))))))))))).
% 9.43/3.44 thf(fact_258_curry__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_curry @ A) @ B) @ C) = (^ [C3:(((product_prod @ A) @ B) > C),X3:A,Y3:B]: ((C3 @ ((((product_Pair @ A) @ B) @ X3) @ Y3)))))))).
% 9.43/3.44 thf(fact_259_combine__options__def, axiom, (! [A:$tType]: (((combine_options @ A) = (^ [F2:(A > (A > A)),X3:(option @ A),Y3:(option @ A)]: ((((((case_option @ (option @ A)) @ A) @ Y3) @ (^ [Z5:A]: ((((((case_option @ (option @ A)) @ A) @ ((some @ A) @ Z5)) @ (^ [Aa2:A]: (((some @ A) @ ((F2 @ Z5) @ Aa2))))) @ Y3)))) @ X3))))))).
% 9.43/3.44 thf(fact_260_less__option__def, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less @ (option @ A)) = (^ [X3:(option @ A)]: (((((case_option @ $o) @ A) @ $false) @ (^ [Y3:A]: ((((((case_option @ $o) @ A) @ $true) @ (^ [Z5:A]: ((((ord_less @ A) @ Z5) @ Y3)))) @ X3))))))))))).
% 9.43/3.44 thf(fact_261_zero__less__iff__neq__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ N) = (N != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_262_gr__implies__not__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [M2:A,N:A]: (((((ord_less @ A) @ M2) @ N) => (N != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_263_not__less__zero, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: ((~ ((((ord_less @ A) @ N) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_264_gr__zeroI, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((N != (zero_zero @ A)) => (((ord_less @ A) @ (zero_zero @ A)) @ N)))))))).
% 9.43/3.44 thf(fact_265_add_Ogroup__left__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 9.43/3.44 thf(fact_266_add_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 9.43/3.44 thf(fact_267_comm__monoid__add__class_Oadd__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ (zero_zero @ A)) @ A3) = A3))))))).
% 9.43/3.44 thf(fact_268_zero__neq__one, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_zero @ A) != (one_one @ A)))))).
% 9.43/3.44 thf(fact_269_array__make, axiom, (! [A:$tType]: (((heap @ A) => ((array_new @ A) = (^ [N5:nat,X3:A]: ((((array_make @ A) @ N5) @ (^ [Uu:nat]: (X3)))))))))).
% 9.43/3.44 thf(fact_270_plus__nat_Oadd__0, axiom, (! [N:nat]: (((((plus_plus @ nat) @ (zero_zero @ nat)) @ N) = N)))).
% 9.43/3.44 thf(fact_271_add__eq__self__zero, axiom, (! [M2:nat,N:nat]: ((((((plus_plus @ nat) @ M2) @ N) = M2) => (N = (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_272_less__numeral__extra_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_273_ureturn__def, axiom, (! [A:$tType]: (((heap_Time_ureturn @ A) = (^ [X3:A]: (((heap_Time_heap @ A) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (zero_zero @ nat)))))))))))).
% 9.43/3.44 thf(fact_274_raise__def, axiom, (! [A:$tType]: (((heap_Time_raise @ A) = (^ [S4:(list @ char)]: (((heap_Time_Heap2 @ A) @ (^ [Uu:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))))).
% 9.43/3.44 thf(fact_275_bind__lift, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ B),G:(B > A)]: ((((((heap_Time_bind @ B) @ A) @ F) @ (((heap_Time_lift @ B) @ A) @ G)) = ((((heap_Time_bind @ B) @ A) @ F) @ (^ [X3:B]: (((heap_Time_return @ A) @ (G @ X3))))))))).
% 9.43/3.44 thf(fact_276_guard__def, axiom, (! [A:$tType]: (((heap_Time_guard @ A) = (^ [P4:((heap_ext @ product_unit) > $o),F2:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((heap_Time_Heap2 @ A) @ (^ [H:(heap_ext @ product_unit)]: (((((if @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (P4 @ H)) @ ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (F2 @ H))) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))))).
% 9.43/3.44 thf(fact_277_pos__add__strict, axiom, (! [A:$tType]: (((strict7427464778891057005id_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)))))))))).
% 9.43/3.44 thf(fact_278_canonically__ordered__monoid__add__class_OlessE, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ ((! [C4:A]: (((B2 = (((plus_plus @ A) @ A3) @ C4)) => (C4 = (zero_zero @ A)))))))))))))).
% 9.43/3.44 thf(fact_279_add__pos__pos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_280_add__neg__neg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_281_add__less__zeroD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ X) @ Y)) @ (zero_zero @ A)) => ((((ord_less @ A) @ X) @ (zero_zero @ A)) | (((ord_less @ A) @ Y) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_282_zero__less__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_283_not__one__less__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_284_execute__ureturn, axiom, (! [A:$tType,X:A]: ((((heap_Time_execute @ A) @ ((heap_Time_ureturn @ A) @ X)) = (((((comp @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_285_less__imp__add__positive, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (? [K5:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K5) & ((((plus_plus @ nat) @ I) @ K5) = J)))))))).
% 9.43/3.44 thf(fact_286_pairself_Osimps, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:A,B2:A]: ((((((pairself @ A) @ B) @ F) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) = ((((product_Pair @ B) @ B) @ (F @ A3)) @ (F @ B2)))))).
% 9.43/3.44 thf(fact_287_pairself_Oelims, axiom, (! [B:$tType,A:$tType,X:(A > B),Xa:((product_prod @ A) @ A),Y:((product_prod @ B) @ B)]: (((((((pairself @ A) @ B) @ X) @ Xa) = Y) => (~ ((! [A5:A,B3:A]: (((Xa = ((((product_Pair @ A) @ A) @ A5) @ B3)) => (Y != ((((product_Pair @ B) @ B) @ (X @ A5)) @ (X @ B3)))))))))))).
% 9.43/3.44 thf(fact_288_return__def, axiom, (! [A:$tType]: (((heap_Time_return @ A) = (^ [X3:A]: (((heap_Time_heap @ A) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ X3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_289_effect__ureturnI, axiom, (! [A:$tType,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),X:A]: (((H3 = H4) => ((((((heap_Time_effect @ A) @ ((heap_Time_ureturn @ A) @ X)) @ H3) @ H4) @ X) @ (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_290_effect__ureturnE, axiom, (! [A:$tType,X:A,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_ureturn @ A) @ X)) @ H3) @ H4) @ R) @ N) => (~ (((R = X) => ((H4 = H3) => (N != (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_291_tap__def, axiom, (! [A:$tType]: (((heap_Time_tap @ A) = (^ [F2:((heap_ext @ product_unit) > A)]: (((heap_Time_Heap2 @ A) @ (^ [H:(heap_ext @ product_unit)]: (((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F2 @ H)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (one_one @ nat))))))))))))).
% 9.43/3.44 thf(fact_292_double__eq__0__iff, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: ((((((plus_plus @ A) @ A3) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_293_return__bind, axiom, (! [B:$tType,A:$tType,X:B,F:(B > (heap_Time_Heap @ A))]: ((((((heap_Time_bind @ B) @ A) @ ((heap_Time_return @ B) @ X)) @ F) = ((((heap_Time_bind @ product_unit) @ A) @ (heap_Time_wait @ (one_one @ nat))) @ (^ [Uu:product_unit]: ((F @ X)))))))).
% 9.43/3.44 thf(fact_294_bind__return, axiom, (! [A:$tType,F:(heap_Time_Heap @ A)]: ((((((heap_Time_bind @ A) @ A) @ F) @ (heap_Time_return @ A)) = ((((heap_Time_bind @ product_unit) @ A) @ (heap_Time_wait @ (one_one @ nat))) @ (^ [Uu:product_unit]: (F))))))).
% 9.43/3.44 thf(fact_295_add__0__iff, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [B2:A,A3:A]: (((B2 = (((plus_plus @ A) @ B2) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_296_verit__sum__simplify, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 9.43/3.44 thf(fact_297_dbl__inc__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ (zero_zero @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_298_disjE__realizer2, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > $o),X:(option @ A),R5:(B > $o),F:B,G:(A > B)]: (((((((case_option @ $o) @ A) @ P) @ Q) @ X) => ((P => (R5 @ F)) => ((! [Q2:A]: (((Q @ Q2) => (R5 @ (G @ Q2))))) => (R5 @ (((((case_option @ B) @ A) @ F) @ G) @ X)))))))).
% 9.43/3.44 thf(fact_299_set__to__map__def, axiom, (! [A:$tType,B:$tType]: ((((set_to_map @ B) @ A) = (^ [S5:(set @ ((product_prod @ B) @ A)),K4:B]: (((eps_Opt @ A) @ (^ [V2:A]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ K4) @ V2)) @ S5)))))))))).
% 9.43/3.44 thf(fact_300_K__record__comp, axiom, (! [C:$tType,B:$tType,A:$tType,C2:B,F:(A > C)]: (((((((comp @ C) @ B) @ A) @ (^ [X3:C]: (C2))) @ F) = (^ [X3:A]: (C2)))))).
% 9.43/3.44 thf(fact_301_pred__equals__eq2, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: ((((^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R5))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S2)))) = (R5 = S2))))).
% 9.43/3.44 thf(fact_302_dbl__inc__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl_inc @ A) = (^ [X3:A]: ((((plus_plus @ A) @ (((plus_plus @ A) @ X3) @ X3)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_303_zero__natural_Orsp, axiom, ((zero_zero @ nat) = (zero_zero @ nat))).
% 9.43/3.44 thf(fact_304_fails__wait, axiom, (! [X:nat,H3:(heap_ext @ product_unit)]: ((~ ((((time_fails @ product_unit) @ (heap_Time_wait @ X)) @ H3)))))).
% 9.43/3.44 thf(fact_305_Euclid__induct, axiom, (! [P:(nat > (nat > $o)),A3:nat,B2:nat]: (((! [A5:nat,B3:nat]: ((((P @ A5) @ B3) = ((P @ B3) @ A5)))) => ((! [A5:nat]: (((P @ A5) @ (zero_zero @ nat)))) => ((! [A5:nat,B3:nat]: ((((P @ A5) @ B3) => ((P @ A5) @ (((plus_plus @ nat) @ A5) @ B3))))) => ((P @ A3) @ B2))))))).
% 9.43/3.44 thf(fact_306_lookup__chain, axiom, (! [B:$tType,A:$tType]: (((heap @ B) => (! [R:(ref @ B),F:(heap_Time_Heap @ A)]: ((((((heap_Time_bind @ B) @ A) @ ((ref_lookup @ B) @ R)) @ (^ [Uu:B]: (F))) = ((((heap_Time_bind @ product_unit) @ A) @ (heap_Time_wait @ (one_one @ nat))) @ (^ [Uu:product_unit]: (F)))))))))).
% 9.43/3.44 thf(fact_307_Ints__odd__less__0, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => ((((ord_less @ A) @ (((plus_plus @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_308_num__of__nat__plus__distrib, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((num_of_nat @ (((plus_plus @ nat) @ M2) @ N)) = (((plus_plus @ num) @ (num_of_nat @ M2)) @ (num_of_nat @ N)))))))).
% 9.43/3.44 thf(fact_309_divides__aux__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [Q3:A,R:A]: ((((unique5940410009612947441es_aux @ A) @ ((((product_Pair @ A) @ A) @ Q3) @ R)) = (R = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_310_sgn__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((sgn_sgn @ A) @ A3) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_311_abs__add__one__gt__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: ((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X))))))))).
% 9.43/3.44 thf(fact_312_map__option__case, axiom, (! [A:$tType,B:$tType]: ((((map_option @ B) @ A) = (^ [F2:(B > A)]: (((((case_option @ (option @ A)) @ B) @ (none @ A)) @ (^ [X3:B]: (((some @ A) @ (F2 @ X3))))))))))).
% 9.43/3.44 thf(fact_313_add__neg__numeral__special_I7_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_314_add__neg__numeral__special_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_315_dbl__dec__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_316_neg__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((uminus_uminus @ A) @ A3) = ((uminus_uminus @ A) @ B2)) = (A3 = B2)))))))).
% 9.43/3.44 thf(fact_317_add_Oinverse__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A3)) = A3))))))).
% 9.43/3.44 thf(fact_318_abs__idempotent, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((abs_abs @ A) @ A3)) = ((abs_abs @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_319_neg__equal__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((uminus_uminus @ A) @ A3) = A3) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_320_equal__neg__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((A3 = ((uminus_uminus @ A) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_321_neg__equal__0__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((uminus_uminus @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_322_neg__0__equal__iff__equal, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: ((((zero_zero @ A) = ((uminus_uminus @ A) @ A3)) = ((zero_zero @ A) = A3)))))))).
% 9.43/3.44 thf(fact_323_add_Oinverse__neutral, axiom, (! [A:$tType]: (((group_add @ A) => (((uminus_uminus @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_324_neg__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_325_minus__add__distrib, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ ((uminus_uminus @ A) @ B2))))))))).
% 9.43/3.44 thf(fact_326_minus__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ (((plus_plus @ A) @ A3) @ B2)) = B2))))))).
% 9.43/3.44 thf(fact_327_add__minus__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)) = B2))))))).
% 9.43/3.44 thf(fact_328_abs__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (((abs_abs @ A) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_329_abs__eq__0, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((abs_abs @ A) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_330_abs__0__eq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((zero_zero @ A) = ((abs_abs @ A) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_331_abs__add__abs, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((abs_abs @ A) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))) = (((plus_plus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))))))))).
% 9.43/3.44 thf(fact_332_abs__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_333_abs__minus__cancel, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((abs_abs @ A) @ ((uminus_uminus @ A) @ A3)) = ((abs_abs @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_334_sgn__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_335_map__option__eq__Some, axiom, (! [A:$tType,B:$tType,F:(B > A),Xo:(option @ B),Y:A]: (((((((map_option @ B) @ A) @ F) @ Xo) = ((some @ A) @ Y)) = (? [Z5:B]: (((Xo = ((some @ B) @ Z5)) & ((F @ Z5) = Y)))))))).
% 9.43/3.44 thf(fact_336_None__eq__map__option__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),X:(option @ B)]: ((((none @ A) = ((((map_option @ B) @ A) @ F) @ X)) = (X = (none @ B)))))).
% 9.43/3.44 thf(fact_337_map__option__is__None, axiom, (! [A:$tType,B:$tType,F:(B > A),Opt:(option @ B)]: (((((((map_option @ B) @ A) @ F) @ Opt) = (none @ A)) = (Opt = (none @ B)))))).
% 9.43/3.44 thf(fact_338_option_Omap__disc__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:(option @ A)]: (((((((map_option @ A) @ B) @ F) @ A3) = (none @ B)) = (A3 = (none @ A)))))).
% 9.43/3.44 thf(fact_339_neg__less__0__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_340_neg__0__less__iff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_341_neg__less__pos, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_342_less__neg__neg, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_343_ab__left__minus, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_344_add_Oright__inverse, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_345_zero__less__abs__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A3)) = (A3 != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_346_abs__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_347_case__map__option, axiom, (! [B:$tType,A:$tType,C:$tType,G:A,H3:(B > A),F:(C > B),X:(option @ C)]: (((((((case_option @ A) @ B) @ G) @ H3) @ ((((map_option @ C) @ B) @ F) @ X)) = (((((case_option @ A) @ C) @ G) @ (((((comp @ B) @ A) @ C) @ H3) @ F)) @ X))))).
% 9.43/3.44 thf(fact_348_dbl__inc__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_349_abs__sgn__eq__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_350_dbl__dec__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ (zero_zero @ A)) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_351_sgn__neg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => (((sgn_sgn @ A) @ A3) = ((uminus_uminus @ A) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_352_minus__equation__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((uminus_uminus @ A) @ A3) = B2) = (((uminus_uminus @ A) @ B2) = A3)))))))).
% 9.43/3.44 thf(fact_353_equation__minus__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((A3 = ((uminus_uminus @ A) @ B2)) = (B2 = ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_354_same__sgn__abs__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:A,A3:A]: (((((sgn_sgn @ A) @ B2) = ((sgn_sgn @ A) @ A3)) => (((abs_abs @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2)))))))))).
% 9.43/3.44 thf(fact_355_sgn__minus__1, axiom, (! [A:$tType]: (((idom_abs_sgn @ A) => (((sgn_sgn @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_356_option_Omap__ident, axiom, (! [A:$tType,T2:(option @ A)]: ((((((map_option @ A) @ A) @ (^ [X3:A]: (X3))) @ T2) = T2)))).
% 9.43/3.44 thf(fact_357_abs__of__neg, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => (((abs_abs @ A) @ A3) = ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_358_abs__sgn__eq, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: ((((A3 = (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = (zero_zero @ A))) & ((A3 != (zero_zero @ A)) => (((abs_abs @ A) @ ((sgn_sgn @ A) @ A3)) = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_359_abs__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((abs_abs @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_360_map__option_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > C),G:(A > B),Option:(option @ A)]: ((((((map_option @ B) @ C) @ F) @ ((((map_option @ A) @ B) @ G) @ Option)) = ((((map_option @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)) @ Option))))).
% 9.43/3.44 thf(fact_361_option_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(option @ A)]: ((((((map_option @ B) @ C) @ G) @ ((((map_option @ A) @ B) @ F) @ V)) = ((((map_option @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 9.43/3.44 thf(fact_362_map__option_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C),G:(A > B)]: (((((((comp @ (option @ B)) @ (option @ C)) @ (option @ A)) @ (((map_option @ B) @ C) @ F)) @ (((map_option @ A) @ B) @ G)) = (((map_option @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)))))).
% 9.43/3.44 thf(fact_363_minus__less__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((ord_less @ A) @ ((uminus_uminus @ A) @ B2)) @ A3)))))))).
% 9.43/3.44 thf(fact_364_less__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((ord_less @ A) @ B2) @ ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_365_is__num__normalize_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_366_add_Oinverse__distrib__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_367_group__cancel_Oneg1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,K:A,A3:A]: (((A4 = (((plus_plus @ A) @ K) @ A3)) => (((uminus_uminus @ A) @ A4) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ ((uminus_uminus @ A) @ A3)))))))))).
% 9.43/3.44 thf(fact_368_one__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => ((one_one @ A) != ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_369_map__option__cong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > B),G:(A > B)]: (((X = Y) => ((! [A5:A]: (((Y = ((some @ A) @ A5)) => ((F @ A5) = (G @ A5))))) => (((((map_option @ A) @ B) @ F) @ X) = ((((map_option @ A) @ B) @ G) @ Y))))))).
% 9.43/3.44 thf(fact_370_option_Osimps_I9_J, axiom, (! [B:$tType,A:$tType,F:(A > B),X22:A]: ((((((map_option @ A) @ B) @ F) @ ((some @ A) @ X22)) = ((some @ B) @ (F @ X22)))))).
% 9.43/3.44 thf(fact_371_option_Osimps_I8_J, axiom, (! [A:$tType,B:$tType,F:(A > B)]: ((((((map_option @ A) @ B) @ F) @ (none @ A)) = (none @ B))))).
% 9.43/3.44 thf(fact_372_same__sgn__sgn__add, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [B2:A,A3:A]: (((((sgn_sgn @ A) @ B2) = ((sgn_sgn @ A) @ A3)) => (((sgn_sgn @ A) @ (((plus_plus @ A) @ A3) @ B2)) = ((sgn_sgn @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_373_Ints__add, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,B2:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => ((((member @ A) @ B2) @ (ring_1_Ints @ A)) => (((member @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (ring_1_Ints @ A)))))))))).
% 9.43/3.44 thf(fact_374_success__lookupI, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ ((ref_lookup @ A) @ R)) @ H3))))))).
% 9.43/3.44 thf(fact_375_Ints__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((member @ A) @ (one_one @ A)) @ (ring_1_Ints @ A)))))).
% 9.43/3.44 thf(fact_376_Ints__nonzero__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((((ord_less @ A) @ ((abs_abs @ A) @ X)) @ (one_one @ A)) => (X = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_377_sgn__if, axiom, (! [A:$tType]: (((linordered_idom @ A) => ((sgn_sgn @ A) = (^ [X3:A]: (((((if @ A) @ (X3 = (zero_zero @ A))) @ (zero_zero @ A)) @ ((((if @ A) @ (((ord_less @ A) @ (zero_zero @ A)) @ X3)) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_378_sgn__1__neg, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((sgn_sgn @ A) @ A3) = ((uminus_uminus @ A) @ (one_one @ A))) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_379_abs__not__less__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((~ ((((ord_less @ A) @ ((abs_abs @ A) @ A3)) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_380_abs__of__pos, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => (((abs_abs @ A) @ A3) = A3)))))))).
% 9.43/3.44 thf(fact_381_option_Orec__o__map, axiom, (! [B:$tType,C:$tType,A:$tType,G:C,Ga:(B > C),F:(A > B)]: (((((((comp @ (option @ B)) @ C) @ (option @ A)) @ ((((rec_option @ C) @ B) @ G) @ Ga)) @ (((map_option @ A) @ B) @ F)) = ((((rec_option @ C) @ A) @ G) @ (^ [X3:A]: ((Ga @ (F @ X3))))))))).
% 9.43/3.44 thf(fact_382_neg__eq__iff__add__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((uminus_uminus @ A) @ A3) = B2) = ((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_383_eq__neg__iff__add__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((A3 = ((uminus_uminus @ A) @ B2)) = ((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_384_add_Oinverse__unique, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A)) => (((uminus_uminus @ A) @ A3) = B2)))))))).
% 9.43/3.44 thf(fact_385_ab__group__add__class_Oab__left__minus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_386_add__eq__0__iff, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((((plus_plus @ A) @ A3) @ B2) = (zero_zero @ A)) = (B2 = ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_387_zero__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => ((zero_zero @ A) != ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_388_less__minus__one__simps_I4_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_389_less__minus__one__simps_I2_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_390_option_Omap__sel, axiom, (! [B:$tType,A:$tType,A3:(option @ A),F:(A > B)]: (((A3 != (none @ A)) => (((the2 @ B) @ ((((map_option @ A) @ B) @ F) @ A3)) = (F @ ((the2 @ A) @ A3))))))).
% 9.43/3.44 thf(fact_391_Ints__double__eq__0__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [A3:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => (((((plus_plus @ A) @ A3) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_392_less__minus__one__simps_I3_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_393_less__minus__one__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_394_sgn__1__pos, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A]: (((((sgn_sgn @ A) @ A3) = (one_one @ A)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_395_Ints__odd__nonzero, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [A3:A]: (((((member @ A) @ A3) @ (ring_1_Ints @ A)) => ((((plus_plus @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ A3) != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_396_map__option__o__empty, axiom, (! [C:$tType,B:$tType,A:$tType,F:(C > B)]: (((((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F)) @ (^ [X3:A]: ((none @ C)))) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.44 thf(fact_397_execute__lookup, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ ((ref_lookup @ A) @ R)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((ref_get @ A) @ H3) @ R)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (one_one @ nat))))))))))).
% 9.43/3.44 thf(fact_398_change__def, axiom, (! [A:$tType]: (((heap @ A) => ((ref_change @ A) = (^ [F2:(A > A),R3:(ref @ A)]: (((((heap_Time_bind @ A) @ A) @ ((ref_lookup @ A) @ R3)) @ (^ [X3:A]: (((((heap_Time_bind @ product_unit) @ A) @ (((ref_update @ A) @ R3) @ (F2 @ X3))) @ (^ [Uu:product_unit]: (((heap_Time_return @ A) @ (F2 @ X3))))))))))))))).
% 9.43/3.44 thf(fact_399_fails__assert_H, axiom, (! [P:$o,H3:(heap_ext @ product_unit)]: (((((time_fails @ product_unit) @ (refine_Imp_assert @ P)) @ H3) = (~ (P)))))).
% 9.43/3.44 thf(fact_400_effect__lookupI, axiom, (! [A:$tType]: (((heap @ A) => (! [H4:(heap_ext @ product_unit),H3:(heap_ext @ product_unit),X:A,R:(ref @ A),N:nat]: (((H4 = H3) => ((X = (((ref_get @ A) @ H3) @ R)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ A) @ ((ref_lookup @ A) @ R)) @ H3) @ H4) @ X) @ N)))))))))).
% 9.43/3.44 thf(fact_401_effect__lookupE, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),X:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((ref_lookup @ A) @ R)) @ H3) @ H4) @ X) @ N) => (~ (((H4 = H3) => ((X = (((ref_get @ A) @ H3) @ R)) => (N != (one_one @ nat))))))))))))).
% 9.43/3.44 thf(fact_402_add_Ogroup__axioms, axiom, (! [A:$tType]: (((group_add @ A) => ((((group @ A) @ (plus_plus @ A)) @ (zero_zero @ A)) @ (uminus_uminus @ A)))))).
% 9.43/3.44 thf(fact_403_num__of__nat__double, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((num_of_nat @ (((plus_plus @ nat) @ N) @ N)) = (bit0 @ (num_of_nat @ N))))))).
% 9.43/3.44 thf(fact_404_Ints__nonzero__abs__ge1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((X != (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((abs_abs @ A) @ X)))))))))).
% 9.43/3.44 thf(fact_405_lookup__def, axiom, (! [A:$tType]: (((heap @ A) => ((ref_lookup @ A) = (^ [R3:(ref @ A)]: (((heap_Time_tap @ A) @ (^ [H:(heap_ext @ product_unit)]: ((((ref_get @ A) @ H) @ R3))))))))))).
% 9.43/3.44 thf(fact_406_map__conv__bind__option, axiom, (! [A:$tType,B:$tType]: ((((map_option @ B) @ A) = (^ [F2:(B > A),X3:(option @ B)]: (((((bind @ B) @ A) @ X3) @ (((((comp @ A) @ (option @ A)) @ B) @ (some @ A)) @ F2)))))))).
% 9.43/3.44 thf(fact_407_nat__add__left__cancel__le, axiom, (! [K:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N)) = (((ord_less_eq @ nat) @ M2) @ N))))).
% 9.43/3.44 thf(fact_408_less__eq__option__None__code, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:(option @ A)]: ((((ord_less_eq @ (option @ A)) @ (none @ A)) @ X))))))).
% 9.43/3.44 thf(fact_409_bind__assoc, axiom, (! [B:$tType,A:$tType,C:$tType,X:(option @ C),F:(C > (option @ B)),G:(B > (option @ A))]: ((((((bind @ B) @ A) @ ((((bind @ C) @ B) @ X) @ F)) @ G) = ((((bind @ C) @ A) @ X) @ (^ [Y3:C]: (((((bind @ B) @ A) @ (F @ Y3)) @ G)))))))).
% 9.43/3.44 thf(fact_410_le__zero__eq, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [N:A]: (((((ord_less_eq @ A) @ N) @ (zero_zero @ A)) = (N = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_411_add__le__cancel__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_412_add__le__cancel__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_413_neg__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_414_less__eq__option__Some, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ (option @ A)) @ ((some @ A) @ X)) @ ((some @ A) @ Y)) = (((ord_less_eq @ A) @ X) @ Y)))))))).
% 9.43/3.44 thf(fact_415_less__eq__option__Some__None, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A]: ((~ ((((ord_less_eq @ (option @ A)) @ ((some @ A) @ X)) @ (none @ A)))))))))).
% 9.43/3.44 thf(fact_416_bind__runit, axiom, (! [A:$tType,X:(option @ A)]: ((((((bind @ A) @ A) @ X) @ (some @ A)) = X)))).
% 9.43/3.44 thf(fact_417_bind__rzero, axiom, (! [B:$tType,A:$tType,X:(option @ B)]: ((((((bind @ B) @ A) @ X) @ (^ [X3:B]: ((none @ A)))) = (none @ A))))).
% 9.43/3.44 thf(fact_418_add__le__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_419_add__le__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_420_le__add__same__cancel1, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 9.43/3.44 thf(fact_421_le__add__same__cancel2, axiom, (! [A:$tType]: (((ordere1937475149494474687imp_le @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (((plus_plus @ A) @ B2) @ A3)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ B2)))))))).
% 9.43/3.44 thf(fact_422_double__add__le__zero__iff__single__add__le__zero, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_423_zero__le__double__add__iff__zero__le__single__add, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ A3)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_424_neg__0__le__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_425_neg__le__0__iff__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_426_less__eq__neg__nonpos, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ A3)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_427_neg__less__eq__nonneg, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ A3) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_428_abs__of__nonneg, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((abs_abs @ A) @ A3) = A3)))))))).
% 9.43/3.44 thf(fact_429_abs__le__self__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ A3) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_430_abs__le__zero__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_431_abs__of__nonpos, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => (((abs_abs @ A) @ A3) = ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_432_ord__eq__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((A3 = B2) => ((((ord_less_eq @ A) @ B2) @ C2) => ((C2 = D3) => (((ord_less_eq @ A) @ A3) @ D3)))))))))).
% 9.43/3.44 thf(fact_433_bind__option__cong__code, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > (option @ B))]: (((X = Y) => (((((bind @ A) @ B) @ X) @ F) = ((((bind @ A) @ B) @ Y) @ F)))))).
% 9.43/3.44 thf(fact_434_group_Oleft__cancel, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A,B2:A,C2:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((((F @ A3) @ B2) = ((F @ A3) @ C2)) = (B2 = C2)))))).
% 9.43/3.44 thf(fact_435_group_Oleft__inverse, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => (((F @ (Inverse @ A3)) @ A3) = Z2))))).
% 9.43/3.44 thf(fact_436_group_Oright__cancel, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),B2:A,A3:A,C2:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((((F @ B2) @ A3) = ((F @ C2) @ A3)) = (B2 = C2)))))).
% 9.43/3.44 thf(fact_437_group_Oright__inverse, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => (((F @ A3) @ (Inverse @ A3)) = Z2))))).
% 9.43/3.44 thf(fact_438_group_Oinverse__unique, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A,B2:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((((F @ A3) @ B2) = Z2) => ((Inverse @ A3) = B2)))))).
% 9.43/3.44 thf(fact_439_group_Oinverse__inverse, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((Inverse @ (Inverse @ A3)) = A3))))).
% 9.43/3.44 thf(fact_440_group_Oinverse__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A)]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((Inverse @ Z2) = Z2))))).
% 9.43/3.44 thf(fact_441_group_Ogroup__left__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => (((F @ Z2) @ A3) = A3))))).
% 9.43/3.44 thf(fact_442_group_Oinverse__distrib__swap, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A),A3:A,B2:A]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((Inverse @ ((F @ A3) @ B2)) = ((F @ (Inverse @ B2)) @ (Inverse @ A3))))))).
% 9.43/3.44 thf(fact_443_accp__subset, axiom, (! [A:$tType,R12:(A > (A > $o)),R23:(A > (A > $o))]: (((((ord_less_eq @ (A > (A > $o))) @ R12) @ R23) => (((ord_less_eq @ (A > $o)) @ ((accp @ A) @ R23)) @ ((accp @ A) @ R12)))))).
% 9.43/3.44 thf(fact_444_le__some__optE, axiom, (! [A:$tType]: (((preorder @ A) => (! [M2:A,X:(option @ A)]: (((((ord_less_eq @ (option @ A)) @ ((some @ A) @ M2)) @ X) => (~ ((! [M4:A]: (((X = ((some @ A) @ M4)) => (~ ((((ord_less_eq @ A) @ M2) @ M4))))))))))))))).
% 9.43/3.44 thf(fact_445_pred__subset__eq2, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((((ord_less_eq @ (A > (B > $o))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R5)))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S2)))) = (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R5) @ S2))))).
% 9.43/3.44 thf(fact_446_less__eq__option__def, axiom, (! [A:$tType]: (((preorder @ A) => ((ord_less_eq @ (option @ A)) = (^ [X3:(option @ A),Y3:(option @ A)]: ((((((case_option @ $o) @ A) @ $true) @ (^ [Z5:A]: ((((((case_option @ $o) @ A) @ $false) @ ((ord_less_eq @ A) @ Z5)) @ Y3)))) @ X3)))))))).
% 9.43/3.44 thf(fact_447_zero__le, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X))))))).
% 9.43/3.44 thf(fact_448_add__mono__thms__linordered__semiring_I3_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (K = L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_449_add__mono__thms__linordered__semiring_I2_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((I = J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_450_add__mono__thms__linordered__semiring_I1_J, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_451_add__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D3) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_452_add__left__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_453_less__eqE, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (~ ((! [C4:A]: ((B2 != (((plus_plus @ A) @ A3) @ C4))))))))))))).
% 9.43/3.44 thf(fact_454_add__right__mono, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_455_le__iff__add, axiom, (! [A:$tType]: (((canoni5634975068530333245id_add @ A) => ((ord_less_eq @ A) = (^ [A7:A,B5:A]: ((? [C3:A]: ((B5 = (((plus_plus @ A) @ A7) @ C3))))))))))).
% 9.43/3.44 thf(fact_456_add__le__imp__le__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2)) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_457_add__le__imp__le__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_458_accp__subset__induct, axiom, (! [A:$tType,D4:(A > $o),R5:(A > (A > $o)),X:A,P:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ D4) @ ((accp @ A) @ R5)) => ((! [X2:A,Z3:A]: (((D4 @ X2) => (((R5 @ Z3) @ X2) => (D4 @ Z3))))) => ((D4 @ X) => ((! [X2:A]: (((D4 @ X2) => ((! [Z6:A]: ((((R5 @ Z6) @ X2) => (P @ Z6)))) => (P @ X2))))) => (P @ X)))))))).
% 9.43/3.44 thf(fact_459_le__numeral__extra_I4_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (one_one @ A)) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_460_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ A) @ B))]: (((! [X2:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ S)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S))))).
% 9.43/3.44 thf(fact_461_le__minus__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((ord_less_eq @ A) @ B2) @ ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_462_minus__le__iff, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ A3)))))))).
% 9.43/3.44 thf(fact_463_le__imp__neg__le, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B2)) @ ((uminus_uminus @ A) @ A3))))))))).
% 9.43/3.44 thf(fact_464_exists__leI, axiom, (! [N:nat,P:(nat > $o)]: ((((! [N7:nat]: (((((ord_less @ nat) @ N7) @ N) => (~ ((P @ N7)))))) => (P @ N)) => (? [N4:nat]: (((((ord_less_eq @ nat) @ N4) @ N) & (P @ N4)))))))).
% 9.43/3.44 thf(fact_465_nat__le__iff__add, axiom, ((ord_less_eq @ nat) = (^ [M:nat,N5:nat]: ((? [K4:nat]: ((N5 = (((plus_plus @ nat) @ M) @ K4)))))))).
% 9.43/3.44 thf(fact_466_trans__le__add2, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ M2) @ J)))))).
% 9.43/3.44 thf(fact_467_trans__le__add1, axiom, (! [I:nat,J:nat,M2:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ I) @ (((plus_plus @ nat) @ J) @ M2)))))).
% 9.43/3.44 thf(fact_468_add__le__mono1, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ K)))))).
% 9.43/3.44 thf(fact_469_add__le__mono, axiom, (! [I:nat,J:nat,K:nat,L:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ K) @ L) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (((plus_plus @ nat) @ J) @ L))))))).
% 9.43/3.44 thf(fact_470_le__Suc__ex, axiom, (! [K:nat,L:nat]: (((((ord_less_eq @ nat) @ K) @ L) => (? [N3:nat]: ((L = (((plus_plus @ nat) @ K) @ N3)))))))).
% 9.43/3.44 thf(fact_471_add__leD2, axiom, (! [M2:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ N) => (((ord_less_eq @ nat) @ K) @ N))))).
% 9.43/3.44 thf(fact_472_add__leD1, axiom, (! [M2:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ N) => (((ord_less_eq @ nat) @ M2) @ N))))).
% 9.43/3.44 thf(fact_473_le__add2, axiom, (! [N:nat,M2:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ M2) @ N))))).
% 9.43/3.44 thf(fact_474_le__add1, axiom, (! [N:nat,M2:nat]: ((((ord_less_eq @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M2))))).
% 9.43/3.44 thf(fact_475_add__leE, axiom, (! [M2:nat,K:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ N) => (~ (((((ord_less_eq @ nat) @ M2) @ N) => (~ ((((ord_less_eq @ nat) @ K) @ N)))))))))).
% 9.43/3.44 thf(fact_476_abs__le__D1, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2) => (((ord_less_eq @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_477_abs__ge__self, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ ((abs_abs @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_478_less__eq__option__None, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:(option @ A)]: ((((ord_less_eq @ (option @ A)) @ (none @ A)) @ X))))))).
% 9.43/3.44 thf(fact_479_less__eq__option__None__is__None, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:(option @ A)]: (((((ord_less_eq @ (option @ A)) @ X) @ (none @ A)) => (X = (none @ A))))))))).
% 9.43/3.44 thf(fact_480_bind_Obind__lunit, axiom, (! [B:$tType,A:$tType,X:A,F:(A > (option @ B))]: ((((((bind @ A) @ B) @ ((some @ A) @ X)) @ F) = (F @ X))))).
% 9.43/3.44 thf(fact_481_Option_Obind__cong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > (option @ B)),G:(A > (option @ B))]: (((X = Y) => ((! [A5:A]: (((Y = ((some @ A) @ A5)) => ((F @ A5) = (G @ A5))))) => (((((bind @ A) @ B) @ X) @ F) = ((((bind @ A) @ B) @ Y) @ G))))))).
% 9.43/3.44 thf(fact_482_bind__eq__Some__conv, axiom, (! [A:$tType,B:$tType,F:(option @ B),G:(B > (option @ A)),X:A]: (((((((bind @ B) @ A) @ F) @ G) = ((some @ A) @ X)) = (? [Y3:B]: (((F = ((some @ B) @ Y3)) & ((G @ Y3) = ((some @ A) @ X))))))))).
% 9.43/3.44 thf(fact_483_bind_Obind__lzero, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B))]: ((((((bind @ A) @ B) @ (none @ A)) @ F) = (none @ B))))).
% 9.43/3.44 thf(fact_484_map__option__bind, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),X:(option @ C),G:(C > (option @ B))]: ((((((map_option @ B) @ A) @ F) @ ((((bind @ C) @ B) @ X) @ G)) = ((((bind @ C) @ A) @ X) @ (((((comp @ (option @ B)) @ (option @ A)) @ C) @ (((map_option @ B) @ A) @ F)) @ G)))))).
% 9.43/3.44 thf(fact_485_add__decreasing, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ C2) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2))))))))).
% 9.43/3.44 thf(fact_486_add__increasing, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)))))))))).
% 9.43/3.44 thf(fact_487_add__decreasing2, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2))))))))).
% 9.43/3.44 thf(fact_488_add__increasing2, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [C2:A,B2:A,A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((ord_less_eq @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)))))))))).
% 9.43/3.44 thf(fact_489_add__nonneg__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_490_add__nonpos__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_491_add__nonneg__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 9.43/3.44 thf(fact_492_add__nonpos__eq__0__iff, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ Y) @ (zero_zero @ A)) => (((((plus_plus @ A) @ X) @ Y) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))))).
% 9.43/3.44 thf(fact_493_not__one__le__zero, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_494_linordered__nonzero__semiring__class_Ozero__le__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_495_zero__less__one__class_Ozero__le__one, axiom, (! [A:$tType]: (((zero_less_one @ A) => (((ord_less_eq @ A) @ (zero_zero @ A)) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_496_add__mono__thms__linordered__field_I4_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less_eq @ A) @ I) @ J) & (((ord_less @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_497_add__mono__thms__linordered__field_I3_J, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [I:A,J:A,K:A,L:A]: ((((((ord_less @ A) @ I) @ J) & (((ord_less_eq @ A) @ K) @ L)) => (((ord_less @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ L))))))))).
% 9.43/3.44 thf(fact_498_add__le__less__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D3) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_499_add__less__le__mono, axiom, (! [A:$tType]: (((ordere580206878836729694up_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ D3) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_500_le__minus__one__simps_I2_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_501_le__minus__one__simps_I4_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_502_abs__ge__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((abs_abs @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_503_abs__triangle__ineq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((plus_plus @ A) @ A3) @ B2))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))))))))).
% 9.43/3.44 thf(fact_504_abs__leI, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2))))))))).
% 9.43/3.44 thf(fact_505_abs__le__D2, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)))))))).
% 9.43/3.44 thf(fact_506_abs__le__iff, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ A3)) @ B2) = ((((ord_less_eq @ A) @ A3) @ B2) & (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ B2))))))))).
% 9.43/3.44 thf(fact_507_abs__ge__minus__self, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A3)) @ ((abs_abs @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_508_mono__nat__linear__lb, axiom, (! [F:(nat > nat),M2:nat,K:nat]: (((! [M5:nat,N3:nat]: (((((ord_less @ nat) @ M5) @ N3) => (((ord_less @ nat) @ (F @ M5)) @ (F @ N3))))) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (F @ M2)) @ K)) @ (F @ (((plus_plus @ nat) @ M2) @ K))))))).
% 9.43/3.44 thf(fact_509_nat__geq__1__eq__neqz, axiom, (! [X:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ X) = (X != (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_510_bind__split__asm, axiom, (! [A:$tType,B:$tType,P:((option @ A) > $o),M2:(option @ B),F:(B > (option @ A))]: (((P @ ((((bind @ B) @ A) @ M2) @ F)) = (~ ((((M2 = (none @ B)) & (~ ((P @ (none @ A))))) | (? [X3:B]: (((M2 = ((some @ B) @ X3)) & (~ ((P @ (F @ X3)))))))))))))).
% 9.43/3.44 thf(fact_511_bind__split, axiom, (! [A:$tType,B:$tType,P:((option @ A) > $o),M2:(option @ B),F:(B > (option @ A))]: (((P @ ((((bind @ B) @ A) @ M2) @ F)) = (((M2 = (none @ B)) => (P @ (none @ A))) & (! [V2:B]: (((M2 = ((some @ B) @ V2)) => (P @ (F @ V2)))))))))).
% 9.43/3.44 thf(fact_512_mlex__leq, axiom, (! [A:$tType,F:(A > nat),X:A,Y:A,R5:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ nat) @ (F @ X)) @ (F @ Y)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((mlex_prod @ A) @ F) @ R5))))))).
% 9.43/3.44 thf(fact_513_bind__eq__None__conv, axiom, (! [B:$tType,A:$tType,A3:(option @ B),F:(B > (option @ A))]: (((((((bind @ B) @ A) @ A3) @ F) = (none @ A)) = ((A3 = (none @ B)) | ((F @ ((the2 @ B) @ A3)) = (none @ A))))))).
% 9.43/3.44 thf(fact_514_bind__map__option, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),X:(option @ C),G:(B > (option @ A))]: ((((((bind @ B) @ A) @ ((((map_option @ C) @ B) @ F) @ X)) @ G) = ((((bind @ C) @ A) @ X) @ (((((comp @ B) @ (option @ A)) @ C) @ G) @ F)))))).
% 9.43/3.44 thf(fact_515_add__neg__nonpos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_516_add__nonneg__pos, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_517_add__nonpos__neg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ B2) @ (zero_zero @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_518_add__pos__nonneg, axiom, (! [A:$tType]: (((ordere6911136660526730532id_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => (((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_519_add__strict__increasing, axiom, (! [A:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)))))))))).
% 9.43/3.44 thf(fact_520_add__strict__increasing2, axiom, (! [A:$tType]: (((ordere8940638589300402666id_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ B2) @ C2) => (((ord_less @ A) @ B2) @ (((plus_plus @ A) @ A3) @ C2)))))))))).
% 9.43/3.44 thf(fact_521_discrete, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((ord_less @ A) = (^ [A7:A]: (((ord_less_eq @ A) @ (((plus_plus @ A) @ A7) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_522_le__minus__one__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_523_le__minus__one__simps_I3_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_524_dense__eq0__I, axiom, (! [A:$tType]: ((((ordere166539214618696060dd_abs @ A) & (dense_linorder @ A)) => (! [X:A]: (((! [E3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E3) => (((ord_less_eq @ A) @ ((abs_abs @ A) @ X)) @ E3)))) => (X = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_525_abs__minus__le__zero, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((abs_abs @ A) @ A3))) @ (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_526_success__changeI, axiom, (! [A:$tType]: (((heap @ A) => (! [F:(A > A),R:(ref @ A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ A) @ (((ref_change @ A) @ F) @ R)) @ H3))))))).
% 9.43/3.44 thf(fact_527_semiring__norm_I6_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ num) @ (bit0 @ M2)) @ (bit0 @ N)) = (bit0 @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_528_field__le__epsilon, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((! [E3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E3) => (((ord_less_eq @ A) @ X) @ (((plus_plus @ A) @ Y) @ E3))))) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 9.43/3.44 thf(fact_529_ex__has__greatest__nat__lemma, axiom, (! [A:$tType,P:(A > $o),K:A,F:(A > nat),N:nat]: (((P @ K) => ((! [X2:A]: (((P @ X2) => (? [Y6:A]: (((P @ Y6) & (~ ((((ord_less_eq @ nat) @ (F @ Y6)) @ (F @ X2)))))))))) => (? [Y4:A]: (((P @ Y4) & (~ ((((ord_less @ nat) @ (F @ Y4)) @ (((plus_plus @ nat) @ (F @ K)) @ N)))))))))))).
% 9.43/3.44 thf(fact_530_map__comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((map_comp @ B) @ C) @ A) = (^ [F2:(B > (option @ C)),G2:(A > (option @ B)),K4:A]: ((((((case_option @ (option @ C)) @ B) @ (none @ C)) @ F2) @ (G2 @ K4)))))))).
% 9.43/3.44 thf(fact_531_convex__bound__lt, axiom, (! [A:$tType]: (((linord715952674999750819strict @ A) => (! [X:A,A3:A,Y:A,U:A,V:A]: (((((ord_less @ A) @ X) @ A3) => ((((ord_less @ A) @ Y) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ U) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ V) => (((((plus_plus @ A) @ U) @ V) = (one_one @ A)) => (((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ U) @ X)) @ (((times_times @ A) @ V) @ Y))) @ A3)))))))))))).
% 9.43/3.44 thf(fact_532_map__comp__empty_I1_J, axiom, (! [C:$tType,B:$tType,A:$tType,M2:(C > (option @ B))]: (((((((map_comp @ C) @ B) @ A) @ M2) @ (^ [X3:A]: ((none @ C)))) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.44 thf(fact_533_map__comp__empty_I2_J, axiom, (! [B:$tType,D:$tType,C:$tType,M2:(C > (option @ B))]: (((((((map_comp @ B) @ D) @ C) @ (^ [X3:B]: ((none @ D)))) @ M2) = (^ [X3:C]: ((none @ D))))))).
% 9.43/3.44 thf(fact_534_map__comp__None__iff, axiom, (! [A:$tType,C:$tType,B:$tType,M1:(B > (option @ A)),M22:(C > (option @ B)),K:C]: (((((((((map_comp @ B) @ A) @ C) @ M1) @ M22) @ K) = (none @ A)) = (((M22 @ K) = (none @ B)) | (? [K6:B]: ((((M22 @ K) = ((some @ B) @ K6)) & ((M1 @ K6) = (none @ A)))))))))).
% 9.43/3.44 thf(fact_535_Ints__eq__abs__less1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((member @ A) @ X) @ (ring_1_Ints @ A)) => ((((member @ A) @ Y) @ (ring_1_Ints @ A)) => ((X = Y) = (((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ Y))) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_536_nat0__intermed__int__val, axiom, (! [N:nat,F:(nat > int),K:int]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ N) => (((ord_less_eq @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (F @ (((plus_plus @ nat) @ I2) @ (one_one @ nat)))) @ (F @ I2)))) @ (one_one @ int))))) => ((((ord_less_eq @ int) @ (F @ (zero_zero @ nat))) @ K) => ((((ord_less_eq @ int) @ K) @ (F @ N)) => (? [I2:nat]: (((((ord_less_eq @ nat) @ I2) @ N) & ((F @ I2) = K)))))))))).
% 9.43/3.44 thf(fact_537_diff__diff__left, axiom, (! [I:nat,J:nat,K:nat]: (((((minus_minus @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ K) = (((minus_minus @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)))))).
% 9.43/3.44 thf(fact_538_nat__1__eq__mult__iff, axiom, (! [M2:nat,N:nat]: ((((one_one @ nat) = (((times_times @ nat) @ M2) @ N)) = ((M2 = (one_one @ nat)) & (N = (one_one @ nat))))))).
% 9.43/3.44 thf(fact_539_nat__mult__eq__1__iff, axiom, (! [M2:nat,N:nat]: ((((((times_times @ nat) @ M2) @ N) = (one_one @ nat)) = ((M2 = (one_one @ nat)) & (N = (one_one @ nat))))))).
% 9.43/3.44 thf(fact_540_cancel__comm__monoid__add__class_Odiff__cancel, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_541_diff__zero, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 9.43/3.44 thf(fact_542_zero__diff, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_543_diff__0__right, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ (zero_zero @ A)) = A3))))))).
% 9.43/3.44 thf(fact_544_diff__self, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ A3) @ A3) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_545_mult_Oright__neutral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 9.43/3.44 thf(fact_546_mult__1, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ (one_one @ A)) @ A3) = A3))))))).
% 9.43/3.44 thf(fact_547_add__diff__cancel__right_H, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = A3))))))).
% 9.43/3.44 thf(fact_548_add__diff__cancel__right, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_549_add__diff__cancel__left_H, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ A3) = B2))))))).
% 9.43/3.44 thf(fact_550_add__diff__cancel__left, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [C2:A,A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2)) = (((minus_minus @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_551_diff__add__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ B2) = A3))))))).
% 9.43/3.44 thf(fact_552_add__diff__cancel, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = A3))))))).
% 9.43/3.44 thf(fact_553_minus__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A3) @ B2)) = (((minus_minus @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_554_Nat_Oadd__diff__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K)))))).
% 9.43/3.44 thf(fact_555_Nat_Oadd__diff__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K)))))).
% 9.43/3.44 thf(fact_556_Nat_Odiff__diff__right, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))).
% 9.43/3.44 thf(fact_557_zle__add1__eq__le, axiom, (! [W:int,Z2:int]: (((((ord_less @ int) @ W) @ (((plus_plus @ int) @ Z2) @ (one_one @ int))) = (((ord_less_eq @ int) @ W) @ Z2))))).
% 9.43/3.44 thf(fact_558_map__comp__simps_I2_J, axiom, (! [B:$tType,C:$tType,A:$tType,M22:(B > (option @ A)),K:B,K7:A,M1:(A > (option @ C))]: ((((M22 @ K) = ((some @ A) @ K7)) => (((((((map_comp @ A) @ C) @ B) @ M1) @ M22) @ K) = (M1 @ K7)))))).
% 9.43/3.44 thf(fact_559_map__comp__simps_I1_J, axiom, (! [B:$tType,A:$tType,C:$tType,M22:(B > (option @ A)),K:B,M1:(A > (option @ C))]: ((((M22 @ K) = (none @ A)) => (((((((map_comp @ A) @ C) @ B) @ M1) @ M22) @ K) = (none @ C)))))).
% 9.43/3.44 thf(fact_560_diff__ge__0__iff__ge, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_561_zero__comp__diff__simps_I1_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_562_diff__gt__0__iff__gt, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_563_zero__comp__diff__simps_I2_J, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((minus_minus @ A) @ A3) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_564_le__add__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A3) @ B2)) = A3)))))))).
% 9.43/3.44 thf(fact_565_le__add__diff__inverse2, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ B2) = A3)))))))).
% 9.43/3.44 thf(fact_566_mult__cancel__left1, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,B2:A]: (((C2 = (((times_times @ A) @ C2) @ B2)) = ((C2 = (zero_zero @ A)) | (B2 = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_567_mult__cancel__left2, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,A3:A]: ((((((times_times @ A) @ C2) @ A3) = C2) = ((C2 = (zero_zero @ A)) | (A3 = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_568_mult__cancel__right1, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [C2:A,B2:A]: (((C2 = (((times_times @ A) @ B2) @ C2)) = ((C2 = (zero_zero @ A)) | (B2 = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_569_mult__cancel__right2, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [A3:A,C2:A]: ((((((times_times @ A) @ A3) @ C2) = C2) = ((C2 = (zero_zero @ A)) | (A3 = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_570_diff__add__zero, axiom, (! [A:$tType]: (((comm_monoid_diff @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_571_diff__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_572_diff__0, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A]: (((((minus_minus @ A) @ (zero_zero @ A)) @ A3) = ((uminus_uminus @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_573_mult__minus1, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:A]: (((((times_times @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ Z2) = ((uminus_uminus @ A) @ Z2)))))))).
% 9.43/3.44 thf(fact_574_mult__minus1__right, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Z2:A]: (((((times_times @ A) @ Z2) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ Z2)))))))).
% 9.43/3.44 thf(fact_575_uminus__add__conv__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ B2) = (((minus_minus @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_576_diff__minus__eq__add, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A]: (((((minus_minus @ A) @ A3) @ ((uminus_uminus @ A) @ B2)) = (((plus_plus @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_577_diff__numeral__special_I12_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_578_le__imp__0__less, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => (((ord_less @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (one_one @ int)) @ Z2)))))).
% 9.43/3.44 thf(fact_579_q__pos__lemma, axiom, (! [B4:int,Q4:int,R4:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q4)) @ R4)) => ((((ord_less @ int) @ R4) @ B4) => ((((ord_less @ int) @ (zero_zero @ int)) @ B4) => (((ord_less_eq @ int) @ (zero_zero @ int)) @ Q4))))))).
% 9.43/3.44 thf(fact_580_zdiv__mono2__lemma, axiom, (! [B2:int,Q3:int,R:int,B4:int,Q4:int,R4:int]: ((((((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R) = (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q4)) @ R4)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q4)) @ R4)) => ((((ord_less @ int) @ R4) @ B4) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R) => ((((ord_less @ int) @ (zero_zero @ int)) @ B4) => ((((ord_less_eq @ int) @ B4) @ B2) => (((ord_less_eq @ int) @ Q3) @ Q4)))))))))).
% 9.43/3.44 thf(fact_581_zdiv__mono2__neg__lemma, axiom, (! [B2:int,Q3:int,R:int,B4:int,Q4:int,R4:int]: ((((((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R) = (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q4)) @ R4)) => ((((ord_less @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B4) @ Q4)) @ R4)) @ (zero_zero @ int)) => ((((ord_less @ int) @ R) @ B2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R4) => ((((ord_less @ int) @ (zero_zero @ int)) @ B4) => ((((ord_less_eq @ int) @ B4) @ B2) => (((ord_less_eq @ int) @ Q4) @ Q3)))))))))).
% 9.43/3.44 thf(fact_582_unique__quotient__lemma, axiom, (! [B2:int,Q4:int,R4:int,Q3:int,R:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R4)) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R4) => ((((ord_less @ int) @ R4) @ B2) => ((((ord_less @ int) @ R) @ B2) => (((ord_less_eq @ int) @ Q4) @ Q3)))))))).
% 9.43/3.44 thf(fact_583_unique__quotient__lemma__neg, axiom, (! [B2:int,Q4:int,R4:int,Q3:int,R:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q4)) @ R4)) @ (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R)) => ((((ord_less_eq @ int) @ R) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R) => ((((ord_less @ int) @ B2) @ R4) => (((ord_less_eq @ int) @ Q3) @ Q4)))))))).
% 9.43/3.44 thf(fact_584_int__ge__induct, axiom, (! [K:int,I:int,P:(int > $o)]: (((((ord_less_eq @ int) @ K) @ I) => ((P @ K) => ((! [I2:int]: (((((ord_less_eq @ int) @ K) @ I2) => ((P @ I2) => (P @ (((plus_plus @ int) @ I2) @ (one_one @ int))))))) => (P @ I))))))).
% 9.43/3.44 thf(fact_585_int__induct, axiom, (! [P:(int > $o),K:int,I:int]: (((P @ K) => ((! [I2:int]: (((((ord_less_eq @ int) @ K) @ I2) => ((P @ I2) => (P @ (((plus_plus @ int) @ I2) @ (one_one @ int))))))) => ((! [I2:int]: (((((ord_less_eq @ int) @ I2) @ K) => ((P @ I2) => (P @ (((minus_minus @ int) @ I2) @ (one_one @ int))))))) => (P @ I))))))).
% 9.43/3.44 thf(fact_586_zless__imp__add1__zle, axiom, (! [W:int,Z2:int]: (((((ord_less @ int) @ W) @ Z2) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ W) @ (one_one @ int))) @ Z2))))).
% 9.43/3.44 thf(fact_587_add1__zle__eq, axiom, (! [W:int,Z2:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ W) @ (one_one @ int))) @ Z2) = (((ord_less @ int) @ W) @ Z2))))).
% 9.43/3.44 thf(fact_588_nat__eq__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2) = (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = ((((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2) = N)))))).
% 9.43/3.44 thf(fact_589_nat__eq__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2) = (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (M2 = (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 9.43/3.44 thf(fact_590_nat__le__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N)))))).
% 9.43/3.44 thf(fact_591_nat__le__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less_eq @ nat) @ M2) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 9.43/3.44 thf(fact_592_nat__diff__add__eq1, axiom, (! [J:nat,I:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N)))))).
% 9.43/3.44 thf(fact_593_nat__diff__add__eq2, axiom, (! [I:nat,J:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((minus_minus @ nat) @ M2) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 9.43/3.44 thf(fact_594_bounded__Max__nat, axiom, (! [P:(nat > $o),X:nat,M3:nat]: (((P @ X) => ((! [X2:nat]: (((P @ X2) => (((ord_less_eq @ nat) @ X2) @ M3)))) => (~ ((! [M5:nat]: (((P @ M5) => (~ ((! [X7:nat]: (((P @ X7) => (((ord_less_eq @ nat) @ X7) @ M5)))))))))))))))).
% 9.43/3.44 thf(fact_595_subset__Collect__conv, axiom, (! [A:$tType,S2:(set @ A),P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ S2) @ ((collect @ A) @ P)) = (! [X3:A]: (((((member @ A) @ X3) @ S2) => (P @ X3)))))))).
% 9.43/3.44 thf(fact_596_cancel__ab__semigroup__add__class_Odiff__right__commute, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,C2:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_597_ab__semigroup__mult__class_Omult_Oleft__commute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => (! [B2:A,A3:A,C2:A]: (((((times_times @ A) @ B2) @ (((times_times @ A) @ A3) @ C2)) = (((times_times @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_598_ab__semigroup__mult__class_Omult_Ocommute, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((times_times @ A) = (^ [A7:A,B5:A]: ((((times_times @ A) @ B5) @ A7)))))))).
% 9.43/3.44 thf(fact_599_mult_Oassoc, axiom, (! [A:$tType]: (((semigroup_mult @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (((times_times @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_600_diff__eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D3)) => ((A3 = B2) = (C2 = D3))))))))).
% 9.43/3.44 thf(fact_601_one__integer_Orsp, axiom, ((one_one @ int) = (one_one @ int))).
% 9.43/3.44 thf(fact_602_int__gr__induct, axiom, (! [K:int,I:int,P:(int > $o)]: (((((ord_less @ int) @ K) @ I) => ((P @ (((plus_plus @ int) @ K) @ (one_one @ int))) => ((! [I2:int]: (((((ord_less @ int) @ K) @ I2) => ((P @ I2) => (P @ (((plus_plus @ int) @ I2) @ (one_one @ int))))))) => (P @ I))))))).
% 9.43/3.44 thf(fact_603_zless__add1__eq, axiom, (! [W:int,Z2:int]: (((((ord_less @ int) @ W) @ (((plus_plus @ int) @ Z2) @ (one_one @ int))) = ((((ord_less @ int) @ W) @ Z2) | (W = Z2)))))).
% 9.43/3.44 thf(fact_604_odd__nonzero, axiom, (! [Z2:int]: (((((plus_plus @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ Z2)) @ Z2) != (zero_zero @ int))))).
% 9.43/3.44 thf(fact_605_odd__less__0__iff, axiom, (! [Z2:int]: (((((ord_less @ int) @ (((plus_plus @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ Z2)) @ Z2)) @ (zero_zero @ int)) = (((ord_less @ int) @ Z2) @ (zero_zero @ int)))))).
% 9.43/3.44 thf(fact_606_eq__add__iff1, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,E2:A,C2:A,B2:A,D3:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ D3)) = ((((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ E2)) @ C2) = D3)))))))).
% 9.43/3.44 thf(fact_607_eq__add__iff2, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,E2:A,C2:A,B2:A,D3:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ D3)) = (C2 = (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ E2)) @ D3))))))))).
% 9.43/3.44 thf(fact_608_square__diff__square__factored, axiom, (! [A:$tType]: (((comm_ring @ A) => (! [X:A,Y:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y) @ Y)) = (((times_times @ A) @ (((plus_plus @ A) @ X) @ Y)) @ (((minus_minus @ A) @ X) @ Y))))))))).
% 9.43/3.44 thf(fact_609_left__add__mult__distrib, axiom, (! [I:nat,U:nat,J:nat,K:nat]: (((((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ K)) = (((plus_plus @ nat) @ (((times_times @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ U)) @ K))))).
% 9.43/3.44 thf(fact_610_nat__less__add__iff1, axiom, (! [J:nat,I:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N)))))).
% 9.43/3.44 thf(fact_611_nat__less__add__iff2, axiom, (! [I:nat,J:nat,U:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ I) @ U)) @ M2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ J) @ U)) @ N)) = (((ord_less @ nat) @ M2) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 9.43/3.44 thf(fact_612_le__add__iff1, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E2:A,C2:A,B2:A,D3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ D3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ E2)) @ C2)) @ D3)))))))).
% 9.43/3.44 thf(fact_613_le__add__iff2, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E2:A,C2:A,B2:A,D3:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ D3)) = (((ord_less_eq @ A) @ C2) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ E2)) @ D3))))))))).
% 9.43/3.44 thf(fact_614_less__add__iff1, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E2:A,C2:A,B2:A,D3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ D3)) = (((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ E2)) @ C2)) @ D3)))))))).
% 9.43/3.44 thf(fact_615_less__add__iff2, axiom, (! [A:$tType]: (((ordered_ring @ A) => (! [A3:A,E2:A,C2:A,B2:A,D3:A]: (((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ D3)) = (((ord_less @ A) @ C2) @ (((plus_plus @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ E2)) @ D3))))))))).
% 9.43/3.44 thf(fact_616_square__diff__one__factored, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A]: (((((minus_minus @ A) @ (((times_times @ A) @ X) @ X)) @ (one_one @ A)) = (((times_times @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A))) @ (((minus_minus @ A) @ X) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_617_diff__eq__diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D3)) => ((((ord_less_eq @ A) @ A3) @ B2) = (((ord_less_eq @ A) @ C2) @ D3))))))))).
% 9.43/3.44 thf(fact_618_diff__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_619_diff__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B2) @ A3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A3)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_620_diff__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,D3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ D3) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_621_eq__iff__diff__eq__0, axiom, (! [A:$tType]: (((group_add @ A) => ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) = (^ [A7:A,B5:A]: (((((minus_minus @ A) @ A7) @ B5) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_622_diff__strict__right__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_623_diff__strict__left__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [B2:A,A3:A,C2:A]: (((((ord_less @ A) @ B2) @ A3) => (((ord_less @ A) @ (((minus_minus @ A) @ C2) @ A3)) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_624_diff__eq__diff__less, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((minus_minus @ A) @ A3) @ B2) = (((minus_minus @ A) @ C2) @ D3)) => ((((ord_less @ A) @ A3) @ B2) = (((ord_less @ A) @ C2) @ D3))))))))).
% 9.43/3.44 thf(fact_625_diff__strict__mono, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,D3:A,C2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ D3) @ C2) => (((ord_less @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ (((minus_minus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_626_diff__diff__eq, axiom, (! [A:$tType]: (((cancel2418104881723323429up_add @ A) => (! [A3:A,B2:A,C2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_627_add__implies__diff, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [C2:A,B2:A,A3:A]: ((((((plus_plus @ A) @ C2) @ B2) = A3) => (C2 = (((minus_minus @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_628_diff__add__eq__diff__diff__swap, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((minus_minus @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((minus_minus @ A) @ A3) @ C2)) @ B2)))))))).
% 9.43/3.44 thf(fact_629_diff__add__eq, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2)))))))).
% 9.43/3.44 thf(fact_630_diff__diff__eq2, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((minus_minus @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ C2)) @ B2)))))))).
% 9.43/3.44 thf(fact_631_add__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ A3) @ (((minus_minus @ A) @ B2) @ C2)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_632_eq__diff__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,C2:A,B2:A]: (((A3 = (((minus_minus @ A) @ C2) @ B2)) = ((((plus_plus @ A) @ A3) @ B2) = C2)))))))).
% 9.43/3.44 thf(fact_633_diff__eq__eq, axiom, (! [A:$tType]: (((group_add @ A) => (! [A3:A,B2:A,C2:A]: ((((((minus_minus @ A) @ A3) @ B2) = C2) = (A3 = (((plus_plus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_634_group__cancel_Osub1, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A4:A,K:A,A3:A,B2:A]: (((A4 = (((plus_plus @ A) @ K) @ A3)) => ((((minus_minus @ A) @ A4) @ B2) = (((plus_plus @ A) @ K) @ (((minus_minus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_635_minus__diff__commute, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B2:A,A3:A]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ B2)) @ A3) = (((minus_minus @ A) @ ((uminus_uminus @ A) @ A3)) @ B2)))))))).
% 9.43/3.44 thf(fact_636_ring__class_Oring__distribs_I2_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_637_ring__class_Oring__distribs_I1_J, axiom, (! [A:$tType]: (((ring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_638_comm__semiring__class_Odistrib, axiom, (! [A:$tType]: (((comm_semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_639_distrib__left, axiom, (! [A:$tType]: (((semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_640_distrib__right, axiom, (! [A:$tType]: (((semiring @ A) => (! [A3:A,B2:A,C2:A]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_641_combine__common__factor, axiom, (! [A:$tType]: (((semiring @ A) => (! [A3:A,E2:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ A3) @ E2)) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ E2)) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ E2)) @ C2)))))))).
% 9.43/3.44 thf(fact_642_crossproduct__eq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [W:A,Y:A,X:A,Z2:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ W) @ Y)) @ (((times_times @ A) @ X) @ Z2)) = (((plus_plus @ A) @ (((times_times @ A) @ W) @ Z2)) @ (((times_times @ A) @ X) @ Y))) = ((W = X) | (Y = Z2))))))))).
% 9.43/3.44 thf(fact_643_crossproduct__noteq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((A3 != B2) & (C2 != D3)) = ((((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ D3)) != (((plus_plus @ A) @ (((times_times @ A) @ A3) @ D3)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 9.43/3.44 thf(fact_644_comm__monoid__mult__class_Omult__1, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ (one_one @ A)) @ A3) = A3))))))).
% 9.43/3.44 thf(fact_645_mult_Ocomm__neutral, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: (((((times_times @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 9.43/3.44 thf(fact_646_abs__minus__commute, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ B2)) = ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_647_Nat_Odiff__cancel, axiom, (! [K:nat,M2:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ K) @ M2)) @ (((plus_plus @ nat) @ K) @ N)) = (((minus_minus @ nat) @ M2) @ N))))).
% 9.43/3.44 thf(fact_648_diff__cancel2, axiom, (! [M2:nat,K:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K)) = (((minus_minus @ nat) @ M2) @ N))))).
% 9.43/3.44 thf(fact_649_diff__add__inverse, axiom, (! [N:nat,M2:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ M2)) @ N) = M2)))).
% 9.43/3.44 thf(fact_650_diff__add__inverse2, axiom, (! [M2:nat,N:nat]: (((((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ N) = M2)))).
% 9.43/3.44 thf(fact_651_le__diff__conv, axiom, (! [J:nat,K:nat,I:nat]: (((((ord_less_eq @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((ord_less_eq @ nat) @ J) @ (((plus_plus @ nat) @ I) @ K)))))).
% 9.43/3.44 thf(fact_652_Nat_Ole__diff__conv2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((ord_less_eq @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J)))))).
% 9.43/3.44 thf(fact_653_Nat_Odiff__add__assoc, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ J)) @ K) = (((plus_plus @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K))))))).
% 9.43/3.44 thf(fact_654_Nat_Odiff__add__assoc2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (((plus_plus @ nat) @ J) @ I)) @ K) = (((plus_plus @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I)))))).
% 9.43/3.44 thf(fact_655_Nat_Ole__imp__diff__is__add, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((((minus_minus @ nat) @ J) @ I) = K) = (J = (((plus_plus @ nat) @ K) @ I))))))).
% 9.43/3.44 thf(fact_656_add__mult__distrib, axiom, (! [M2:nat,N:nat,K:nat]: (((((times_times @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ K) = (((plus_plus @ nat) @ (((times_times @ nat) @ M2) @ K)) @ (((times_times @ nat) @ N) @ K)))))).
% 9.43/3.44 thf(fact_657_add__mult__distrib2, axiom, (! [K:nat,M2:nat,N:nat]: (((((times_times @ nat) @ K) @ (((plus_plus @ nat) @ M2) @ N)) = (((plus_plus @ nat) @ (((times_times @ nat) @ K) @ M2)) @ (((times_times @ nat) @ K) @ N)))))).
% 9.43/3.44 thf(fact_658_mlex__leI, axiom, (! [A3:nat,A6:nat,B2:nat,B4:nat,N6:nat]: (((((ord_less_eq @ nat) @ A3) @ A6) => ((((ord_less_eq @ nat) @ B2) @ B4) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ N6)) @ B2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A6) @ N6)) @ B4))))))).
% 9.43/3.44 thf(fact_659_nat__mult__1, axiom, (! [N:nat]: (((((times_times @ nat) @ (one_one @ nat)) @ N) = N)))).
% 9.43/3.44 thf(fact_660_nat__mult__1__right, axiom, (! [N:nat]: (((((times_times @ nat) @ N) @ (one_one @ nat)) = N)))).
% 9.43/3.44 thf(fact_661_mult__eq__if, axiom, ((times_times @ nat) = (^ [M:nat,N5:nat]: (((((if @ nat) @ (M = (zero_zero @ nat))) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N5) @ (((times_times @ nat) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))) @ N5))))))).
% 9.43/3.44 thf(fact_662_lambda__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((^ [X3:A]: (X3)) = ((times_times @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_663_is__num__normalize_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A,Y:A]: ((((neg_numeral_is_num @ A) @ X) => (((neg_numeral_is_num @ A) @ Y) => ((neg_numeral_is_num @ A) @ (((plus_plus @ A) @ X) @ Y)))))))))).
% 9.43/3.44 thf(fact_664_is__num__add__commute, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A,Y:A]: ((((neg_numeral_is_num @ A) @ X) => (((neg_numeral_is_num @ A) @ Y) => ((((plus_plus @ A) @ X) @ Y) = (((plus_plus @ A) @ Y) @ X)))))))))).
% 9.43/3.44 thf(fact_665_is__num__add__left__commute, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [X:A,Y:A,Z2:A]: ((((neg_numeral_is_num @ A) @ X) => (((neg_numeral_is_num @ A) @ Y) => ((((plus_plus @ A) @ X) @ (((plus_plus @ A) @ Y) @ Z2)) = (((plus_plus @ A) @ Y) @ (((plus_plus @ A) @ X) @ Z2))))))))))).
% 9.43/3.44 thf(fact_666_is__num__normalize_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_is_num @ A) @ (one_one @ A)))))).
% 9.43/3.44 thf(fact_667_le__iff__diff__le__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less_eq @ A) = (^ [A7:A,B5:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A7) @ B5)) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_668_less__iff__diff__less__0, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => ((ord_less @ A) = (^ [A7:A,B5:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ A7) @ B5)) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_669_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ A3) @ B2) => (((((minus_minus @ A) @ B2) @ A3) = C2) = (B2 = (((plus_plus @ A) @ C2) @ A3))))))))))).
% 9.43/3.44 thf(fact_670_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((plus_plus @ A) @ A3) @ (((minus_minus @ A) @ B2) @ A3)) = B2)))))))).
% 9.43/3.44 thf(fact_671_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((minus_minus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A3)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ B2))))))))).
% 9.43/3.44 thf(fact_672_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A3) = (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ C2))))))))).
% 9.43/3.44 thf(fact_673_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ C2) = (((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A3))))))))).
% 9.43/3.44 thf(fact_674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A3) = (((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A3)))))))))).
% 9.43/3.44 thf(fact_675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((plus_plus @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A3)) = (((minus_minus @ A) @ (((plus_plus @ A) @ C2) @ B2)) @ A3))))))))).
% 9.43/3.44 thf(fact_676_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ B2) @ A3)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ B2))))))))).
% 9.43/3.44 thf(fact_677_le__add__diff, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => (((ord_less_eq @ A) @ C2) @ (((minus_minus @ A) @ (((plus_plus @ A) @ B2) @ C2)) @ A3))))))))).
% 9.43/3.44 thf(fact_678_ordered__cancel__comm__monoid__diff__class_Odiff__add, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ B2) => ((((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ A3)) @ A3) = B2)))))))).
% 9.43/3.44 thf(fact_679_le__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (((minus_minus @ A) @ C2) @ B2)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_680_diff__le__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((ord_less_eq @ A) @ A3) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_681_add__le__imp__le__diff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => (((ord_less_eq @ A) @ I) @ (((minus_minus @ A) @ N) @ K))))))))).
% 9.43/3.44 thf(fact_682_add__le__add__imp__diff__le, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [I:A,K:A,N:A,J:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => ((((ord_less_eq @ A) @ N) @ (((plus_plus @ A) @ J) @ K)) => ((((ord_less_eq @ A) @ (((plus_plus @ A) @ I) @ K)) @ N) => ((((ord_less_eq @ A) @ N) @ (((plus_plus @ A) @ J) @ K)) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ N) @ K)) @ J))))))))))).
% 9.43/3.44 thf(fact_683_less__diff__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,C2:A,B2:A]: (((((ord_less @ A) @ A3) @ (((minus_minus @ A) @ C2) @ B2)) = (((ord_less @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_684_diff__less__eq, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A3:A,B2:A,C2:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ A3) @ B2)) @ C2) = (((ord_less @ A) @ A3) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 9.43/3.44 thf(fact_685_linordered__semidom__class_Oadd__diff__inverse, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,B2:A]: (((~ ((((ord_less @ A) @ A3) @ B2))) => ((((plus_plus @ A) @ B2) @ (((minus_minus @ A) @ A3) @ B2)) = A3)))))))).
% 9.43/3.44 thf(fact_686_ab__group__add__class_Oab__diff__conv__add__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => ((minus_minus @ A) = (^ [A7:A,B5:A]: ((((plus_plus @ A) @ A7) @ ((uminus_uminus @ A) @ B5))))))))).
% 9.43/3.44 thf(fact_687_diff__conv__add__uminus, axiom, (! [A:$tType]: (((group_add @ A) => ((minus_minus @ A) = (^ [A7:A,B5:A]: ((((plus_plus @ A) @ A7) @ ((uminus_uminus @ A) @ B5))))))))).
% 9.43/3.44 thf(fact_688_group__cancel_Osub2, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [B6:A,K:A,B2:A,A3:A]: (((B6 = (((plus_plus @ A) @ K) @ B2)) => ((((minus_minus @ A) @ A3) @ B6) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ K)) @ (((minus_minus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_689_abs__triangle__ineq2__sym, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_690_abs__triangle__ineq3, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2)))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_691_abs__triangle__ineq2, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_692_add__scale__eq__noteq, axiom, (! [A:$tType]: (((semiri1453513574482234551roduct @ A) => (! [R:A,A3:A,B2:A,C2:A,D3:A]: (((R != (zero_zero @ A)) => (((A3 = B2) & (C2 != D3)) => ((((plus_plus @ A) @ A3) @ (((times_times @ A) @ R) @ C2)) != (((plus_plus @ A) @ B2) @ (((times_times @ A) @ R) @ D3))))))))))).
% 9.43/3.44 thf(fact_693_less__1__mult, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [M2:A,N:A]: (((((ord_less @ A) @ (one_one @ A)) @ M2) => ((((ord_less @ A) @ (one_one @ A)) @ N) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ M2) @ N)))))))))).
% 9.43/3.44 thf(fact_694_square__eq__1__iff, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [X:A]: ((((((times_times @ A) @ X) @ X) = (one_one @ A)) = ((X = (one_one @ A)) | (X = ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_695_diff__add__0, axiom, (! [N:nat,M2:nat]: (((((minus_minus @ nat) @ N) @ (((plus_plus @ nat) @ N) @ M2)) = (zero_zero @ nat))))).
% 9.43/3.44 thf(fact_696_less__diff__conv, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ J) @ K)) = (((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J))))).
% 9.43/3.44 thf(fact_697_add__diff__inverse__nat, axiom, (! [M2:nat,N:nat]: (((~ ((((ord_less @ nat) @ M2) @ N))) => ((((plus_plus @ nat) @ N) @ (((minus_minus @ nat) @ M2) @ N)) = M2))))).
% 9.43/3.44 thf(fact_698_less__diff__conv2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((ord_less @ nat) @ (((minus_minus @ nat) @ J) @ K)) @ I) = (((ord_less @ nat) @ J) @ (((plus_plus @ nat) @ I) @ K))))))).
% 9.43/3.44 thf(fact_699_mlex__snd__decrI, axiom, (! [A3:nat,A6:nat,B2:nat,B4:nat,N6:nat]: (((A3 = A6) => ((((ord_less @ nat) @ B2) @ B4) => (((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ N6)) @ B2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A6) @ N6)) @ B4))))))).
% 9.43/3.44 thf(fact_700_mlex__fst__decrI, axiom, (! [A3:nat,A6:nat,B2:nat,N6:nat,B4:nat]: (((((ord_less @ nat) @ A3) @ A6) => ((((ord_less @ nat) @ B2) @ N6) => ((((ord_less @ nat) @ B4) @ N6) => (((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ N6)) @ B2)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A6) @ N6)) @ B4)))))))).
% 9.43/3.44 thf(fact_701_mlex__bound, axiom, (! [A3:nat,A4:nat,B2:nat,N6:nat]: (((((ord_less @ nat) @ A3) @ A4) => ((((ord_less @ nat) @ B2) @ N6) => (((ord_less @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ N6)) @ B2)) @ (((times_times @ nat) @ A4) @ N6))))))).
% 9.43/3.44 thf(fact_702_mult__eq__self__implies__10, axiom, (! [M2:nat,N:nat]: (((M2 = (((times_times @ nat) @ M2) @ N)) => ((N = (one_one @ nat)) | (M2 = (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_703_map__comp__Some__iff, axiom, (! [A:$tType,C:$tType,B:$tType,M1:(B > (option @ A)),M22:(C > (option @ B)),K:C,V:A]: (((((((((map_comp @ B) @ A) @ C) @ M1) @ M22) @ K) = ((some @ A) @ V)) = (? [K6:B]: ((((M22 @ K) = ((some @ B) @ K6)) & ((M1 @ K6) = ((some @ A) @ V))))))))).
% 9.43/3.44 thf(fact_704_field__le__mult__one__interval, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A]: (((! [Z3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Z3) => ((((ord_less @ A) @ Z3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Z3) @ X)) @ Y))))) => (((ord_less_eq @ A) @ X) @ Y)))))))).
% 9.43/3.44 thf(fact_705_abs__diff__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,A3:A,R:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ A3))) @ R) = ((((ord_less_eq @ A) @ (((minus_minus @ A) @ A3) @ R)) @ X) & (((ord_less_eq @ A) @ X) @ (((plus_plus @ A) @ A3) @ R)))))))))).
% 9.43/3.44 thf(fact_706_abs__diff__triangle__ineq, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (((plus_plus @ A) @ C2) @ D3)))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ C2))) @ ((abs_abs @ A) @ (((minus_minus @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_707_abs__triangle__ineq4, axiom, (! [A:$tType]: (((ordere166539214618696060dd_abs @ A) => (! [A3:A,B2:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ A3) @ B2))) @ (((plus_plus @ A) @ ((abs_abs @ A) @ A3)) @ ((abs_abs @ A) @ B2))))))))).
% 9.43/3.44 thf(fact_708_abs__diff__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,A3:A,R:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ A3))) @ R) = ((((ord_less @ A) @ (((minus_minus @ A) @ A3) @ R)) @ X) & (((ord_less @ A) @ X) @ (((plus_plus @ A) @ A3) @ R)))))))))).
% 9.43/3.44 thf(fact_709_sum__squares__ge__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y) @ Y))))))))).
% 9.43/3.44 thf(fact_710_mult__left__le, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [C2:A,A3:A]: (((((ord_less_eq @ A) @ C2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ A3))))))))).
% 9.43/3.44 thf(fact_711_mult__le__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ B2)) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_712_mult__right__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ Y) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ X) @ Y)) @ X)))))))))).
% 9.43/3.44 thf(fact_713_mult__left__le__one__le, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ Y) => ((((ord_less_eq @ A) @ Y) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((times_times @ A) @ Y) @ X)) @ X)))))))))).
% 9.43/3.44 thf(fact_714_not__sum__squares__lt__zero, axiom, (! [A:$tType]: (((linordered_ring @ A) => (! [X:A,Y:A]: ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y) @ Y))) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_715_nat__diff__split, axiom, (! [P:(nat > $o),A3:nat,B2:nat]: (((P @ (((minus_minus @ nat) @ A3) @ B2)) = (((((ord_less @ nat) @ A3) @ B2) => (P @ (zero_zero @ nat))) & (! [D5:nat]: (((A3 = (((plus_plus @ nat) @ B2) @ D5)) => (P @ D5))))))))).
% 9.43/3.44 thf(fact_716_nat__diff__split__asm, axiom, (! [P:(nat > $o),A3:nat,B2:nat]: (((P @ (((minus_minus @ nat) @ A3) @ B2)) = (~ ((((((ord_less @ nat) @ A3) @ B2) & (~ ((P @ (zero_zero @ nat))))) | (? [D5:nat]: (((A3 = (((plus_plus @ nat) @ B2) @ D5)) & (~ ((P @ D5))))))))))))).
% 9.43/3.44 thf(fact_717_dbl__dec__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl_dec @ A) = (^ [X3:A]: ((((minus_minus @ A) @ (((plus_plus @ A) @ X3) @ X3)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_718_mult__le__cancel__left1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (one_one @ A)) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_719_mult__le__cancel__left2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ C2) @ A3)) @ C2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ (one_one @ A))) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ A3)))))))))).
% 9.43/3.44 thf(fact_720_mult__le__cancel__right1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less_eq @ A) @ C2) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ (one_one @ A)) @ B2)) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ B2) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_721_mult__le__cancel__right2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,C2:A]: (((((ord_less_eq @ A) @ (((times_times @ A) @ A3) @ C2)) @ C2) = (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((ord_less_eq @ A) @ A3) @ (one_one @ A))) & ((((ord_less @ A) @ C2) @ (zero_zero @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ A3)))))))))).
% 9.43/3.44 thf(fact_722_mult__less__cancel__left1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less @ A) @ C2) @ (((times_times @ A) @ C2) @ B2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (one_one @ A)) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_723_mult__less__cancel__left2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A]: (((((ord_less @ A) @ (((times_times @ A) @ C2) @ A3)) @ C2) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ (one_one @ A))) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (one_one @ A)) @ A3)))))))))).
% 9.43/3.44 thf(fact_724_mult__less__cancel__right1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: (((((ord_less @ A) @ C2) @ (((times_times @ A) @ B2) @ C2)) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ (one_one @ A)) @ B2)) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ B2) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_725_mult__less__cancel__right2, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [A3:A,C2:A]: (((((ord_less @ A) @ (((times_times @ A) @ A3) @ C2)) @ C2) = (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => (((ord_less @ A) @ A3) @ (one_one @ A))) & ((((ord_less_eq @ A) @ C2) @ (zero_zero @ A)) => (((ord_less @ A) @ (one_one @ A)) @ A3)))))))))).
% 9.43/3.44 thf(fact_726_convex__bound__le, axiom, (! [A:$tType]: (((linord6961819062388156250ring_1 @ A) => (! [X:A,A3:A,Y:A,U:A,V:A]: (((((ord_less_eq @ A) @ X) @ A3) => ((((ord_less_eq @ A) @ Y) @ A3) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ U) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ V) => (((((plus_plus @ A) @ U) @ V) = (one_one @ A)) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ U) @ X)) @ (((times_times @ A) @ V) @ Y))) @ A3)))))))))))).
% 9.43/3.44 thf(fact_727_is__num_Ocases, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [A3:A]: ((((neg_numeral_is_num @ A) @ A3) => ((A3 != (one_one @ A)) => ((! [X2:A]: (((A3 = ((uminus_uminus @ A) @ X2)) => (~ (((neg_numeral_is_num @ A) @ X2)))))) => (~ ((! [X2:A,Y4:A]: (((A3 = (((plus_plus @ A) @ X2) @ Y4)) => (((neg_numeral_is_num @ A) @ X2) => (~ (((neg_numeral_is_num @ A) @ Y4)))))))))))))))))).
% 9.43/3.44 thf(fact_728_is__num_Osimps, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_is_num @ A) = (^ [A7:A]: (((A7 = (one_one @ A)) | ((? [X3:A]: (((A7 = ((uminus_uminus @ A) @ X3)) & ((neg_numeral_is_num @ A) @ X3)))) | (? [X3:A,Y3:A]: (((A7 = (((plus_plus @ A) @ X3) @ Y3)) & (((neg_numeral_is_num @ A) @ X3) & ((neg_numeral_is_num @ A) @ Y3)))))))))))))).
% 9.43/3.44 thf(fact_729_sum__squares__gt__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y) @ Y))) = ((X != (zero_zero @ A)) | (Y != (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_730_sum__squares__le__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y) @ Y))) @ (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_731_incr__lemma, axiom, (! [D3:int,Z2:int,X:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D3) => (((ord_less @ int) @ Z2) @ (((plus_plus @ int) @ X) @ (((times_times @ int) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ X) @ Z2))) @ (one_one @ int))) @ D3))))))).
% 9.43/3.44 thf(fact_732_decr__lemma, axiom, (! [D3:int,X:int,Z2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D3) => (((ord_less @ int) @ (((minus_minus @ int) @ X) @ (((times_times @ int) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ X) @ Z2))) @ (one_one @ int))) @ D3))) @ Z2))))).
% 9.43/3.44 thf(fact_733_sum__squares__eq__zero__iff, axiom, (! [A:$tType]: (((linord4710134922213307826strict @ A) => (! [X:A,Y:A]: ((((((plus_plus @ A) @ (((times_times @ A) @ X) @ X)) @ (((times_times @ A) @ Y) @ Y)) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_734_frac__unique__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,A3:A]: (((((archimedean_frac @ A) @ X) = A3) = ((((member @ A) @ (((minus_minus @ A) @ X) @ A3)) @ (ring_1_Ints @ A)) & ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ A3) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_735_scaling__mono, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [U:A,V:A,R:A,S:A]: (((((ord_less_eq @ A) @ U) @ V) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ R) => ((((ord_less_eq @ A) @ R) @ S) => (((ord_less_eq @ A) @ (((plus_plus @ A) @ U) @ (((divide_divide @ A) @ (((times_times @ A) @ R) @ (((minus_minus @ A) @ V) @ U))) @ S))) @ V)))))))))).
% 9.43/3.44 thf(fact_736_divide__le__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_737_divide__le__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_738_div__by__1, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 9.43/3.44 thf(fact_739_div__self, axiom, (! [A:$tType]: (((semidom_divide @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ A3) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_740_zero__eq__1__divide__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: ((((zero_zero @ A) = (((divide_divide @ A) @ (one_one @ A)) @ A3)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_741_one__divide__eq__0__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ A3) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_742_eq__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: ((((one_one @ A) = (((divide_divide @ A) @ B2) @ A3)) = ((A3 != (zero_zero @ A)) & (A3 = B2))))))))).
% 9.43/3.44 thf(fact_743_divide__eq__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: ((((((divide_divide @ A) @ B2) @ A3) = (one_one @ A)) = ((A3 != (zero_zero @ A)) & (A3 = B2))))))))).
% 9.43/3.44 thf(fact_744_divide__self__if, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: ((((A3 = (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ A3) = (zero_zero @ A))) & ((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ A3) = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_745_divide__self, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ A3) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_746_one__eq__divide__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((one_one @ A) = (((divide_divide @ A) @ A3) @ B2)) = ((B2 != (zero_zero @ A)) & (A3 = B2))))))))).
% 9.43/3.44 thf(fact_747_divide__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((((divide_divide @ A) @ A3) @ B2) = (one_one @ A)) = ((B2 != (zero_zero @ A)) & (A3 = B2))))))))).
% 9.43/3.44 thf(fact_748_divide__minus1, axiom, (! [A:$tType]: (((field @ A) => (! [X:A]: (((((divide_divide @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ X)))))))).
% 9.43/3.44 thf(fact_749_zero__le__divide__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) = (((ord_less_eq @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_750_divide__le__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ (zero_zero @ A)) = (((ord_less_eq @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_751_zero__less__divide__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) = (((ord_less @ A) @ (zero_zero @ A)) @ A3)))))))).
% 9.43/3.44 thf(fact_752_less__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((ord_less @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_753_less__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((ord_less @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_754_divide__less__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((ord_less @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_755_divide__less__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((ord_less @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_756_divide__less__0__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ (zero_zero @ A)) = (((ord_less @ A) @ A3) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_757_nonzero__divide__mult__cancel__right, axiom, (! [A:$tType]: (((field @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ B2) @ (((times_times @ A) @ A3) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ A3))))))))).
% 9.43/3.44 thf(fact_758_nonzero__divide__mult__cancel__left, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ A3) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2))))))))).
% 9.43/3.44 thf(fact_759_le__divide__eq__1__pos, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((ord_less_eq @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_760_le__divide__eq__1__neg, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ (zero_zero @ A)) => ((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((ord_less_eq @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_761_int__distrib_I2_J, axiom, (! [W:int,Z1:int,Z22:int]: (((((times_times @ int) @ W) @ (((plus_plus @ int) @ Z1) @ Z22)) = (((plus_plus @ int) @ (((times_times @ int) @ W) @ Z1)) @ (((times_times @ int) @ W) @ Z22)))))).
% 9.43/3.44 thf(fact_762_int__distrib_I1_J, axiom, (! [Z1:int,Z22:int,W:int]: (((((times_times @ int) @ (((plus_plus @ int) @ Z1) @ Z22)) @ W) = (((plus_plus @ int) @ (((times_times @ int) @ Z1) @ W)) @ (((times_times @ int) @ Z22) @ W)))))).
% 9.43/3.44 thf(fact_763_plus__int__code_I2_J, axiom, (! [L:int]: (((((plus_plus @ int) @ (zero_zero @ int)) @ L) = L)))).
% 9.43/3.44 thf(fact_764_plus__int__code_I1_J, axiom, (! [K:int]: (((((plus_plus @ int) @ K) @ (zero_zero @ int)) = K)))).
% 9.43/3.44 thf(fact_765_zero__integer_Orsp, axiom, ((zero_zero @ int) = (zero_zero @ int))).
% 9.43/3.44 thf(fact_766_add__divide__distrib, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_767_int__div__pos__eq, axiom, (! [A3:int,B2:int,Q3:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R) => ((((ord_less @ int) @ R) @ B2) => ((((divide_divide @ int) @ A3) @ B2) = Q3))))))).
% 9.43/3.44 thf(fact_768_int__div__neg__eq, axiom, (! [A3:int,B2:int,Q3:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R)) => ((((ord_less_eq @ int) @ R) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R) => ((((divide_divide @ int) @ A3) @ B2) = Q3))))))).
% 9.43/3.44 thf(fact_769_split__zdiv, axiom, (! [P:(int > $o),N:int,K:int]: (((P @ (((divide_divide @ int) @ N) @ K)) = (((K = (zero_zero @ int)) => (P @ (zero_zero @ int))) & (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (! [I3:int,J2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J2) & ((((ord_less @ int) @ J2) @ K) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J2)))) => (P @ I3))))) & ((((ord_less @ int) @ K) @ (zero_zero @ int)) => (! [I3:int,J2:int]: ((((((ord_less @ int) @ K) @ J2) & ((((ord_less_eq @ int) @ J2) @ (zero_zero @ int)) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J2)))) => (P @ I3))))))))))).
% 9.43/3.44 thf(fact_770_frac__lt__1, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less @ A) @ ((archimedean_frac @ A) @ X)) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_771_frac__1__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archimedean_frac @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A))) = ((archimedean_frac @ A) @ X)))))))).
% 9.43/3.44 thf(fact_772_right__inverse__eq, axiom, (! [A:$tType]: (((division_ring @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => (((((divide_divide @ A) @ A3) @ B2) = (one_one @ A)) = (A3 = B2))))))))).
% 9.43/3.44 thf(fact_773_less__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ A3) @ B2)) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ B2) @ A3)))))))))).
% 9.43/3.44 thf(fact_774_divide__less__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less @ A) @ B2) @ A3)) | (((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less @ A) @ A3) @ B2)) | (A3 = (zero_zero @ A))))))))))).
% 9.43/3.44 thf(fact_775_divide__add__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ Z2)) @ Y) = (((divide_divide @ A) @ (((plus_plus @ A) @ X) @ (((times_times @ A) @ Y) @ Z2))) @ Z2))))))))).
% 9.43/3.44 thf(fact_776_add__divide__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ X) @ (((divide_divide @ A) @ Y) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ Z2)) @ Y)) @ Z2))))))))).
% 9.43/3.44 thf(fact_777_add__num__frac, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,Z2:A,X:A]: (((Y != (zero_zero @ A)) => ((((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ X) @ Y)) = (((divide_divide @ A) @ (((plus_plus @ A) @ X) @ (((times_times @ A) @ Z2) @ Y))) @ Y))))))))).
% 9.43/3.44 thf(fact_778_add__frac__num, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,X:A,Z2:A]: (((Y != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ Y)) @ Z2) = (((divide_divide @ A) @ (((plus_plus @ A) @ X) @ (((times_times @ A) @ Z2) @ Y))) @ Y))))))))).
% 9.43/3.44 thf(fact_779_add__frac__eq, axiom, (! [A:$tType]: (((field @ A) => (! [Y:A,Z2:A,X:A,W:A]: (((Y != (zero_zero @ A)) => ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ X) @ Y)) @ (((divide_divide @ A) @ W) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ X) @ Z2)) @ (((times_times @ A) @ W) @ Y))) @ (((times_times @ A) @ Y) @ Z2))))))))))).
% 9.43/3.44 thf(fact_780_add__divide__eq__if__simps_I1_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A3:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((plus_plus @ A) @ A3) @ (((divide_divide @ A) @ B2) @ Z2)) = A3)) & ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ A3) @ (((divide_divide @ A) @ B2) @ Z2)) = (((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ A3) @ Z2)) @ B2)) @ Z2)))))))))).
% 9.43/3.44 thf(fact_781_add__divide__eq__if__simps_I2_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A3:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ Z2)) @ B2) = B2)) & ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ Z2)) @ B2) = (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ B2) @ Z2))) @ Z2)))))))))).
% 9.43/3.44 thf(fact_782_less__half__sum, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ A3) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_783_gt__half__sum, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (((ord_less @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ (((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A)))) @ B2)))))))).
% 9.43/3.44 thf(fact_784_divide__eq__minus__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: ((((((divide_divide @ A) @ A3) @ B2) = ((uminus_uminus @ A) @ (one_one @ A))) = ((B2 != (zero_zero @ A)) & (A3 = ((uminus_uminus @ A) @ B2)))))))))).
% 9.43/3.44 thf(fact_785_div__pos__geq, axiom, (! [L:int,K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ L) => ((((ord_less_eq @ int) @ L) @ K) => ((((divide_divide @ int) @ K) @ L) = (((plus_plus @ int) @ (((divide_divide @ int) @ (((minus_minus @ int) @ K) @ L)) @ L)) @ (one_one @ int)))))))).
% 9.43/3.44 thf(fact_786_frac__eq, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((archimedean_frac @ A) @ X) = X) = ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) & (((ord_less @ A) @ X) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_787_frac__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y:A]: ((((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)) => (((archimedean_frac @ A) @ (((plus_plus @ A) @ X) @ Y)) = (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y)))) & ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)))) => (((archimedean_frac @ A) @ (((plus_plus @ A) @ X) @ Y)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_788_incr__mult__lemma, axiom, (! [D3:int,P:(int > $o),K:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D3) => ((! [X2:int]: (((P @ X2) => (P @ (((plus_plus @ int) @ X2) @ D3))))) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ K) => (! [X7:int]: (((P @ X7) => (P @ (((plus_plus @ int) @ X7) @ (((times_times @ int) @ K) @ D3)))))))))))).
% 9.43/3.44 thf(fact_789_divide__le__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ (one_one @ A)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ B2) @ A3)) | (((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ A3) @ B2)) | (A3 = (zero_zero @ A))))))))))).
% 9.43/3.44 thf(fact_790_le__divide__eq__1, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ B2) @ A3)) = (((((ord_less @ A) @ (zero_zero @ A)) @ A3) & (((ord_less_eq @ A) @ A3) @ B2)) | ((((ord_less @ A) @ A3) @ (zero_zero @ A)) & (((ord_less_eq @ A) @ B2) @ A3)))))))))).
% 9.43/3.44 thf(fact_791_minus__divide__add__eq__iff, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,X:A,Y:A]: (((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ X) @ Z2))) @ Y) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ X)) @ (((times_times @ A) @ Y) @ Z2))) @ Z2))))))))).
% 9.43/3.44 thf(fact_792_add__divide__eq__if__simps_I3_J, axiom, (! [A:$tType]: (((division_ring @ A) => (! [Z2:A,A3:A,B2:A]: ((((Z2 = (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ Z2))) @ B2) = B2)) & ((Z2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (((divide_divide @ A) @ A3) @ Z2))) @ B2) = (((divide_divide @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ A3)) @ (((times_times @ A) @ B2) @ Z2))) @ Z2)))))))))).
% 9.43/3.44 thf(fact_793_frac__neg, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((((member @ A) @ X) @ (ring_1_Ints @ A)) => (((archimedean_frac @ A) @ ((uminus_uminus @ A) @ X)) = (zero_zero @ A))) & ((~ ((((member @ A) @ X) @ (ring_1_Ints @ A)))) => (((archimedean_frac @ A) @ ((uminus_uminus @ A) @ X)) = (((minus_minus @ A) @ (one_one @ A)) @ ((archimedean_frac @ A) @ X))))))))))).
% 9.43/3.44 thf(fact_794_div__mult__self4, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ C2)) @ A3)) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_795_div__mult__self3, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ B2)) @ A3)) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_796_div__mult__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_797_div__mult__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A,C2:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ C2) @ B2))) @ B2) = (((plus_plus @ A) @ C2) @ (((divide_divide @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_798_div__minus1__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ A3)))))))).
% 9.43/3.44 thf(fact_799_div__pos__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ L)) @ (zero_zero @ int)) => ((((divide_divide @ int) @ K) @ L) = ((uminus_uminus @ int) @ (one_one @ int)))))))).
% 9.43/3.44 thf(fact_800_bits__div__by__1, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((divide_divide @ A) @ A3) @ (one_one @ A)) = A3))))))).
% 9.43/3.44 thf(fact_801_dividend__less__times__div, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ N) @ (((divide_divide @ nat) @ M2) @ N)))))))).
% 9.43/3.44 thf(fact_802_dividend__less__div__times, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ (((divide_divide @ nat) @ M2) @ N)) @ N))))))).
% 9.43/3.44 thf(fact_803_split__div, axiom, (! [P:(nat > $o),M2:nat,N:nat]: (((P @ (((divide_divide @ nat) @ M2) @ N)) = (((N = (zero_zero @ nat)) => (P @ (zero_zero @ nat))) & ((N != (zero_zero @ nat)) => (! [I3:nat,J2:nat]: (((((ord_less @ nat) @ J2) @ N) => ((M2 = (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ I3)) @ J2)) => (P @ I3))))))))))).
% 9.43/3.44 thf(fact_804_div__less__dividend, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ (one_one @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) => (((ord_less @ nat) @ (((divide_divide @ nat) @ M2) @ N)) @ M2)))))).
% 9.43/3.44 thf(fact_805_div__eq__dividend__iff, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) => (((((divide_divide @ nat) @ M2) @ N) = M2) = (N = (one_one @ nat))))))).
% 9.43/3.44 thf(fact_806_div__add__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_807_div__add__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_808_upto_Opinduct, axiom, (! [A0:int,A1:int,P:(int > (int > $o))]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ A0) @ A1)) => ((! [I2:int,J3:int]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ I2) @ J3)) => (((((ord_less_eq @ int) @ I2) @ J3) => ((P @ (((plus_plus @ int) @ I2) @ (one_one @ int))) @ J3)) => ((P @ I2) @ J3))))) => ((P @ A0) @ A1)))))).
% 9.43/3.44 thf(fact_809_eucl__rel__int_Osimps, axiom, (eucl_rel_int = (^ [A12:int,A23:int,A33:((product_prod @ int) @ int)]: (((? [K4:int]: (((A12 = K4) & ((A23 = (zero_zero @ int)) & (A33 = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ K4)))))) | ((? [L3:int,K4:int,Q5:int]: (((A12 = K4) & ((A23 = L3) & ((A33 = ((((product_Pair @ int) @ int) @ Q5) @ (zero_zero @ int))) & ((L3 != (zero_zero @ int)) & (K4 = (((times_times @ int) @ Q5) @ L3)))))))) | (? [R3:int,L3:int,K4:int,Q5:int]: (((A12 = K4) & ((A23 = L3) & ((A33 = ((((product_Pair @ int) @ int) @ Q5) @ R3)) & ((((sgn_sgn @ int) @ R3) = ((sgn_sgn @ int) @ L3)) & ((((ord_less @ int) @ ((abs_abs @ int) @ R3)) @ ((abs_abs @ int) @ L3)) & (K4 = (((plus_plus @ int) @ (((times_times @ int) @ Q5) @ L3)) @ R3))))))))))))))).
% 9.43/3.44 thf(fact_810_eucl__rel__int_Ocases, axiom, (! [A1:int,A22:int,A32:((product_prod @ int) @ int)]: (((((eucl_rel_int @ A1) @ A22) @ A32) => (((A22 = (zero_zero @ int)) => (A32 != ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ A1))) => ((! [Q2:int]: (((A32 = ((((product_Pair @ int) @ int) @ Q2) @ (zero_zero @ int))) => ((A22 != (zero_zero @ int)) => (A1 != (((times_times @ int) @ Q2) @ A22)))))) => (~ ((! [R2:int,Q2:int]: (((A32 = ((((product_Pair @ int) @ int) @ Q2) @ R2)) => ((((sgn_sgn @ int) @ R2) = ((sgn_sgn @ int) @ A22)) => ((((ord_less @ int) @ ((abs_abs @ int) @ R2)) @ ((abs_abs @ int) @ A22)) => (A1 != (((plus_plus @ int) @ (((times_times @ int) @ Q2) @ A22)) @ R2))))))))))))))).
% 9.43/3.44 thf(fact_811_eucl__rel__int__iff, axiom, (! [K:int,L:int,Q3:int,R:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) = ((K = (((plus_plus @ int) @ (((times_times @ int) @ L) @ Q3)) @ R)) & (((((ord_less @ int) @ (zero_zero @ int)) @ L) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R) & (((ord_less @ int) @ R) @ L))) & ((~ ((((ord_less @ int) @ (zero_zero @ int)) @ L))) => (((((ord_less @ int) @ L) @ (zero_zero @ int)) => ((((ord_less @ int) @ L) @ R) & (((ord_less_eq @ int) @ R) @ (zero_zero @ int)))) & ((~ ((((ord_less @ int) @ L) @ (zero_zero @ int)))) => (Q3 = (zero_zero @ int))))))))))).
% 9.43/3.44 thf(fact_812_power_Opower__eq__if, axiom, (! [A:$tType]: (((power2 @ A) = (^ [One:A,Times:(A > (A > A)),P6:A,M:nat]: (((((if @ A) @ (M = (zero_zero @ nat))) @ One) @ ((Times @ P6) @ (((((power2 @ A) @ One) @ Times) @ P6) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))))))))))).
% 9.43/3.44 thf(fact_813_split__neg__lemma, axiom, (! [K:int,P:(int > (int > $o)),N:int]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (((P @ (((divide_divide @ int) @ N) @ K)) @ (((modulo_modulo @ int) @ N) @ K)) = (! [I3:int,J2:int]: ((((((ord_less @ int) @ K) @ J2) & ((((ord_less_eq @ int) @ J2) @ (zero_zero @ int)) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J2)))) => ((P @ I3) @ J2))))))))).
% 9.43/3.44 thf(fact_814_split__pos__lemma, axiom, (! [K:int,P:(int > (int > $o)),N:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (((P @ (((divide_divide @ int) @ N) @ K)) @ (((modulo_modulo @ int) @ N) @ K)) = (! [I3:int,J2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J2) & ((((ord_less @ int) @ J2) @ K) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J2)))) => ((P @ I3) @ J2))))))))).
% 9.43/3.44 thf(fact_815_mod__add__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_816_mod__add__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,A3:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_817_mod__by__1, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ (one_one @ A)) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_818_bits__mod__by__1, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ (one_one @ A)) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_819_mod__mult__self1, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ C2) @ B2))) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_820_mod__mult__self2, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ B2) @ C2))) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_821_mod__mult__self3, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A3:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ B2)) @ A3)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_822_mod__mult__self4, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [B2:A,C2:A,A3:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ C2)) @ A3)) @ B2) = (((modulo_modulo @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_823_mod__minus1__right, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A]: (((((modulo_modulo @ A) @ A3) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_824_mod__int__unique, axiom, (! [K:int,L:int,Q3:int,R:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => ((((modulo_modulo @ int) @ K) @ L) = R))))).
% 9.43/3.44 thf(fact_825_unique__quotient, axiom, (! [A3:int,B2:int,Q3:int,R:int,Q4:int,R4:int]: (((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => ((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R4)) => (Q3 = Q4)))))).
% 9.43/3.44 thf(fact_826_unique__remainder, axiom, (! [A3:int,B2:int,Q3:int,R:int,Q4:int,R4:int]: (((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => ((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q4) @ R4)) => (R = R4)))))).
% 9.43/3.44 thf(fact_827_full__exhaustive__int_H_Ocases, axiom, (! [X:((product_prod @ (((product_prod @ int) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int))]: ((~ ((! [F4:(((product_prod @ int) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term)))),D2:int,I2:int]: ((X != ((((product_Pair @ (((product_prod @ int) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int)) @ F4) @ ((((product_Pair @ int) @ int) @ D2) @ I2)))))))))).
% 9.43/3.44 thf(fact_828_exhaustive__int_H_Ocases, axiom, (! [X:((product_prod @ (int > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int))]: ((~ ((! [F4:(int > (option @ ((product_prod @ $o) @ (list @ code_term)))),D2:int,I2:int]: ((X != ((((product_Pair @ (int > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ int) @ int)) @ F4) @ ((((product_Pair @ int) @ int) @ D2) @ I2)))))))))).
% 9.43/3.44 thf(fact_829_small__lazy_H_Ocases, axiom, (! [X:((product_prod @ int) @ int)]: ((~ ((! [D2:int,I2:int]: ((X != ((((product_Pair @ int) @ int) @ D2) @ I2))))))))).
% 9.43/3.44 thf(fact_830_eucl__rel__int, axiom, (! [K:int,L:int]: ((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ K) @ L)) @ (((modulo_modulo @ int) @ K) @ L)))))).
% 9.43/3.44 thf(fact_831_mod__add__right__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_832_mod__add__left__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_833_mod__add__cong, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,A6:A,B2:A,B4:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ A6) @ C2)) => (((((modulo_modulo @ A) @ B2) @ C2) = (((modulo_modulo @ A) @ B4) @ C2)) => ((((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A6) @ B4)) @ C2)))))))))).
% 9.43/3.44 thf(fact_834_mod__add__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((modulo_modulo @ A) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2) = (((modulo_modulo @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2)))))))).
% 9.43/3.44 thf(fact_835_eucl__rel__int__by0, axiom, (! [K:int]: ((((eucl_rel_int @ K) @ (zero_zero @ int)) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ K))))).
% 9.43/3.44 thf(fact_836_div__int__unique, axiom, (! [K:int,L:int,Q3:int,R:int]: (((((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => ((((divide_divide @ int) @ K) @ L) = Q3))))).
% 9.43/3.44 thf(fact_837_mod__eqE, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,C2:A,B2:A]: ((((((modulo_modulo @ A) @ A3) @ C2) = (((modulo_modulo @ A) @ B2) @ C2)) => (~ ((! [D2:A]: ((B2 != (((plus_plus @ A) @ A3) @ (((times_times @ A) @ C2) @ D2)))))))))))))).
% 9.43/3.44 thf(fact_838_div__add1__eq, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))) @ (((divide_divide @ A) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ C2)) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2))))))))).
% 9.43/3.44 thf(fact_839_mult__div__mod__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [B2:A,A3:A]: (((((plus_plus @ A) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A3) @ B2))) @ (((modulo_modulo @ A) @ A3) @ B2)) = A3))))))).
% 9.43/3.44 thf(fact_840_mod__mult__div__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A3) @ B2))) = A3))))))).
% 9.43/3.44 thf(fact_841_mod__div__mult__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ B2)) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) = A3))))))).
% 9.43/3.44 thf(fact_842_div__mult__mod__eq, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A3) @ B2)) = A3))))))).
% 9.43/3.44 thf(fact_843_mod__div__decomp, axiom, (! [A:$tType]: (((semiring_modulo @ A) => (! [A3:A,B2:A]: ((A3 = (((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_844_cancel__div__mod__rules_I1_J, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [A3:A,B2:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)) @ (((modulo_modulo @ A) @ A3) @ B2))) @ C2) = (((plus_plus @ A) @ A3) @ C2)))))))).
% 9.43/3.44 thf(fact_845_cancel__div__mod__rules_I2_J, axiom, (! [A:$tType]: (((semidom_modulo @ A) => (! [B2:A,A3:A,C2:A]: (((((plus_plus @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ B2) @ (((divide_divide @ A) @ A3) @ B2))) @ (((modulo_modulo @ A) @ A3) @ B2))) @ C2) = (((plus_plus @ A) @ A3) @ C2)))))))).
% 9.43/3.44 thf(fact_846_div__mult1__eq, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A,B2:A,C2:A]: (((((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2))) @ (((divide_divide @ A) @ (((times_times @ A) @ A3) @ (((modulo_modulo @ A) @ B2) @ C2))) @ C2))))))))).
% 9.43/3.44 thf(fact_847_eucl__rel__int__dividesI, axiom, (! [L:int,K:int,Q3:int]: (((L != (zero_zero @ int)) => ((K = (((times_times @ int) @ Q3) @ L)) => (((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q3) @ (zero_zero @ int)))))))).
% 9.43/3.44 thf(fact_848_div__mod__decomp__int, axiom, (! [A4:int,N:int]: ((A4 = (((plus_plus @ int) @ (((times_times @ int) @ (((divide_divide @ int) @ A4) @ N)) @ N)) @ (((modulo_modulo @ int) @ A4) @ N)))))).
% 9.43/3.44 thf(fact_849_mod__pos__neg__trivial, axiom, (! [K:int,L:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ K) => ((((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ L)) @ (zero_zero @ int)) => ((((modulo_modulo @ int) @ K) @ L) = (((plus_plus @ int) @ K) @ L))))))).
% 9.43/3.44 thf(fact_850_unique__euclidean__semiring__numeral__class_Omod__mult2__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [C2:A,A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ C2) => ((((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ B2) @ (((modulo_modulo @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2))) @ (((modulo_modulo @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_851_int__mod__pos__eq, axiom, (! [A3:int,B2:int,Q3:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ R) => ((((ord_less @ int) @ R) @ B2) => ((((modulo_modulo @ int) @ A3) @ B2) = R))))))).
% 9.43/3.44 thf(fact_852_int__mod__neg__eq, axiom, (! [A3:int,B2:int,Q3:int,R:int]: (((A3 = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ Q3)) @ R)) => ((((ord_less_eq @ int) @ R) @ (zero_zero @ int)) => ((((ord_less @ int) @ B2) @ R) => ((((modulo_modulo @ int) @ A3) @ B2) = R))))))).
% 9.43/3.44 thf(fact_853_split__zmod, axiom, (! [P:(int > $o),N:int,K:int]: (((P @ (((modulo_modulo @ int) @ N) @ K)) = (((K = (zero_zero @ int)) => (P @ N)) & (((((ord_less @ int) @ (zero_zero @ int)) @ K) => (! [I3:int,J2:int]: ((((((ord_less_eq @ int) @ (zero_zero @ int)) @ J2) & ((((ord_less @ int) @ J2) @ K) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J2)))) => (P @ J2))))) & ((((ord_less @ int) @ K) @ (zero_zero @ int)) => (! [I3:int,J2:int]: ((((((ord_less @ int) @ K) @ J2) & ((((ord_less_eq @ int) @ J2) @ (zero_zero @ int)) & (N = (((plus_plus @ int) @ (((times_times @ int) @ K) @ I3)) @ J2)))) => (P @ J2))))))))))).
% 9.43/3.44 thf(fact_854_zmod__zmult2__eq, axiom, (! [C2:int,A3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ C2) => ((((modulo_modulo @ int) @ A3) @ (((times_times @ int) @ B2) @ C2)) = (((plus_plus @ int) @ (((times_times @ int) @ B2) @ (((modulo_modulo @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ C2))) @ (((modulo_modulo @ int) @ A3) @ B2))))))).
% 9.43/3.44 thf(fact_855_zminus1__lemma, axiom, (! [A3:int,B2:int,Q3:int,R:int]: (((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => ((B2 != (zero_zero @ int)) => (((eucl_rel_int @ ((uminus_uminus @ int) @ A3)) @ B2) @ ((((product_Pair @ int) @ int) @ ((((if @ int) @ (R = (zero_zero @ int))) @ ((uminus_uminus @ int) @ Q3)) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ Q3)) @ (one_one @ int)))) @ ((((if @ int) @ (R = (zero_zero @ int))) @ (zero_zero @ int)) @ (((minus_minus @ int) @ B2) @ R))))))))).
% 9.43/3.44 thf(fact_856_eucl__rel__int__remainderI, axiom, (! [R:int,L:int,K:int,Q3:int]: (((((sgn_sgn @ int) @ R) = ((sgn_sgn @ int) @ L)) => ((((ord_less @ int) @ ((abs_abs @ int) @ R)) @ ((abs_abs @ int) @ L)) => ((K = (((plus_plus @ int) @ (((times_times @ int) @ Q3) @ L)) @ R)) => (((eucl_rel_int @ K) @ L) @ ((((product_Pair @ int) @ int) @ Q3) @ R)))))))).
% 9.43/3.44 thf(fact_857_verit__le__mono__div__int, axiom, (! [A4:int,B6:int,N:int]: (((((ord_less @ int) @ A4) @ B6) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ (((divide_divide @ int) @ A4) @ N)) @ ((((if @ int) @ ((((modulo_modulo @ int) @ B6) @ N) = (zero_zero @ int))) @ (one_one @ int)) @ (zero_zero @ int)))) @ (((divide_divide @ int) @ B6) @ N))))))).
% 9.43/3.44 thf(fact_858_bezw__0, axiom, (! [X:nat]: ((((bezw @ X) @ (zero_zero @ nat)) = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))))).
% 9.43/3.44 thf(fact_859_verit__le__mono__div, axiom, (! [A4:nat,B6:nat,N:nat]: (((((ord_less @ nat) @ A4) @ B6) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ nat) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ A4) @ N)) @ ((((if @ nat) @ ((((modulo_modulo @ nat) @ B6) @ N) = (zero_zero @ nat))) @ (one_one @ nat)) @ (zero_zero @ nat)))) @ (((divide_divide @ nat) @ B6) @ N))))))).
% 9.43/3.44 thf(fact_860_zero__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_861_ceiling__less__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ (zero_zero @ int)) = (((ord_less_eq @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_862_floor__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y:A]: ((((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y)) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y)))) & ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ ((archimedean_frac @ A) @ X)) @ ((archimedean_frac @ A) @ Y))) @ (one_one @ A)))) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y)) = (((plus_plus @ int) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y))) @ (one_one @ int))))))))))).
% 9.43/3.44 thf(fact_863_cpmi, axiom, (! [D4:int,P:(int > $o),P2:(int > $o),B6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((? [Z6:int]: ((! [X2:int]: (((((ord_less @ int) @ X2) @ Z6) => ((P @ X2) = (P2 @ X2))))))) => ((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B6) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((P @ X2) => (P @ (((minus_minus @ int) @ X2) @ D4)))))) => ((! [X2:int,K5:int]: (((P2 @ X2) = (P2 @ (((minus_minus @ int) @ X2) @ (((times_times @ int) @ K5) @ D4)))))) => ((? [X8:int]: ((P @ X8))) = ((? [X3:int]: (((((member @ int) @ X3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (P2 @ X3)))) | (? [X3:int]: (((((member @ int) @ X3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (? [Y3:int]: (((((member @ int) @ Y3) @ B6) & (P @ (((plus_plus @ int) @ Y3) @ X3))))))))))))))))).
% 9.43/3.44 thf(fact_864_cppi, axiom, (! [D4:int,P:(int > $o),P2:(int > $o),A4:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((? [Z6:int]: ((! [X2:int]: (((((ord_less @ int) @ Z6) @ X2) => ((P @ X2) = (P2 @ X2))))))) => ((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A4) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((P @ X2) => (P @ (((plus_plus @ int) @ X2) @ D4)))))) => ((! [X2:int,K5:int]: (((P2 @ X2) = (P2 @ (((minus_minus @ int) @ X2) @ (((times_times @ int) @ K5) @ D4)))))) => ((? [X8:int]: ((P @ X8))) = ((? [X3:int]: (((((member @ int) @ X3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (P2 @ X3)))) | (? [X3:int]: (((((member @ int) @ X3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) & (? [Y3:int]: (((((member @ int) @ Y3) @ A4) & (P @ (((minus_minus @ int) @ Y3) @ X3))))))))))))))))).
% 9.43/3.44 thf(fact_865_bset_I6_J, axiom, (! [D4:int,B6:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less_eq @ int) @ X7) @ T2) => (((ord_less_eq @ int) @ (((minus_minus @ int) @ X7) @ D4)) @ T2))))))))).
% 9.43/3.44 thf(fact_866_bset_I8_J, axiom, (! [D4:int,T2:int,B6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((minus_minus @ int) @ T2) @ (one_one @ int))) @ B6) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less_eq @ int) @ T2) @ X7) => (((ord_less_eq @ int) @ T2) @ (((minus_minus @ int) @ X7) @ D4))))))))))).
% 9.43/3.44 thf(fact_867_aset_I6_J, axiom, (! [D4:int,T2:int,A4:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((plus_plus @ int) @ T2) @ (one_one @ int))) @ A4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less_eq @ int) @ X7) @ T2) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ X7) @ D4)) @ T2)))))))))).
% 9.43/3.44 thf(fact_868_atLeastAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))).
% 9.43/3.44 thf(fact_869_Icc__eq__Icc, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,H3:A,L4:A,H4:A]: ((((((set_or1337092689740270186AtMost @ A) @ L) @ H3) = (((set_or1337092689740270186AtMost @ A) @ L4) @ H4)) = (((L = L4) & (H3 = H4)) | ((~ ((((ord_less_eq @ A) @ L) @ H3))) & (~ ((((ord_less_eq @ A) @ L4) @ H4)))))))))))).
% 9.43/3.44 thf(fact_870_atLeastatMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = ((~ ((((ord_less_eq @ A) @ A3) @ B2))) | ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_871_floor__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archim6421214686448440834_floor @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 9.43/3.44 thf(fact_872_ceiling__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_ceiling @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 9.43/3.44 thf(fact_873_floor__add2, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y:A]: ((((((member @ A) @ X) @ (ring_1_Ints @ A)) | (((member @ A) @ Y) @ (ring_1_Ints @ A))) => (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y)) = (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y)))))))))).
% 9.43/3.44 thf(fact_874_zero__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ (zero_zero @ int)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (one_one @ A)) @ X)))))))).
% 9.43/3.44 thf(fact_875_floor__le__zero, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (zero_zero @ int)) = (((ord_less @ A) @ X) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_876_one__le__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ (one_one @ int)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (one_one @ A)) @ X)))))))).
% 9.43/3.44 thf(fact_877_floor__less__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int)) = (((ord_less @ A) @ X) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_878_ceiling__le__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ X)) @ (one_one @ int)) = (((ord_less_eq @ A) @ X) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_879_one__less__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ int) @ (one_one @ int)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (one_one @ A)) @ X)))))))).
% 9.43/3.44 thf(fact_880_ceiling__add__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ (one_one @ int))))))))).
% 9.43/3.44 thf(fact_881_floor__diff__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archim6421214686448440834_floor @ A) @ (((minus_minus @ A) @ X) @ (one_one @ A))) = (((minus_minus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int))))))))).
% 9.43/3.44 thf(fact_882_ceiling__diff__one, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((archimedean_ceiling @ A) @ (((minus_minus @ A) @ X) @ (one_one @ A))) = (((minus_minus @ int) @ ((archimedean_ceiling @ A) @ X)) @ (one_one @ int))))))))).
% 9.43/3.44 thf(fact_883_atLeastatMost__psubset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = (((~ ((((ord_less_eq @ A) @ A3) @ B2))) | ((((ord_less_eq @ A) @ C2) @ A3) & ((((ord_less_eq @ A) @ B2) @ D3) & ((((ord_less @ A) @ C2) @ A3) | (((ord_less @ A) @ B2) @ D3))))) & (((ord_less_eq @ A) @ C2) @ D3))))))))).
% 9.43/3.44 thf(fact_884_nat__mod__eq__iff, axiom, (! [X:nat,N:nat,Y:nat]: ((((((modulo_modulo @ nat) @ X) @ N) = (((modulo_modulo @ nat) @ Y) @ N)) = (? [Q1:nat,Q22:nat]: (((((plus_plus @ nat) @ X) @ (((times_times @ nat) @ N) @ Q1)) = (((plus_plus @ nat) @ Y) @ (((times_times @ nat) @ N) @ Q22))))))))).
% 9.43/3.44 thf(fact_885_le__floor__add, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ int) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((archim6421214686448440834_floor @ A) @ Y))) @ ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ Y))))))))).
% 9.43/3.44 thf(fact_886_bset_I1_J, axiom, (! [D4:int,B6:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B6) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((P @ X2) => (P @ (((minus_minus @ int) @ X2) @ D4)))))) => ((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B6) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((Q @ X2) => (Q @ (((minus_minus @ int) @ X2) @ D4)))))) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => (((P @ X7) & (Q @ X7)) => ((P @ (((minus_minus @ int) @ X7) @ D4)) & (Q @ (((minus_minus @ int) @ X7) @ D4)))))))))))).
% 9.43/3.44 thf(fact_887_bset_I2_J, axiom, (! [D4:int,B6:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B6) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((P @ X2) => (P @ (((minus_minus @ int) @ X2) @ D4)))))) => ((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ B6) => (X2 != (((plus_plus @ int) @ Xb2) @ Xa2)))))))) => ((Q @ X2) => (Q @ (((minus_minus @ int) @ X2) @ D4)))))) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => (((P @ X7) | (Q @ X7)) => ((P @ (((minus_minus @ int) @ X7) @ D4)) | (Q @ (((minus_minus @ int) @ X7) @ D4)))))))))))).
% 9.43/3.44 thf(fact_888_aset_I1_J, axiom, (! [D4:int,A4:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A4) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((P @ X2) => (P @ (((plus_plus @ int) @ X2) @ D4)))))) => ((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A4) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((Q @ X2) => (Q @ (((plus_plus @ int) @ X2) @ D4)))))) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => (((P @ X7) & (Q @ X7)) => ((P @ (((plus_plus @ int) @ X7) @ D4)) & (Q @ (((plus_plus @ int) @ X7) @ D4)))))))))))).
% 9.43/3.44 thf(fact_889_aset_I2_J, axiom, (! [D4:int,A4:(set @ int),P:(int > $o),Q:(int > $o)]: (((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A4) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((P @ X2) => (P @ (((plus_plus @ int) @ X2) @ D4)))))) => ((! [X2:int]: (((! [Xa2:int]: (((((member @ int) @ Xa2) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb2:int]: (((((member @ int) @ Xb2) @ A4) => (X2 != (((minus_minus @ int) @ Xb2) @ Xa2)))))))) => ((Q @ X2) => (Q @ (((plus_plus @ int) @ X2) @ D4)))))) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => (((P @ X7) | (Q @ X7)) => ((P @ (((plus_plus @ int) @ X7) @ D4)) | (Q @ (((plus_plus @ int) @ X7) @ D4)))))))))))).
% 9.43/3.44 thf(fact_890_nat__mod__eq__lemma, axiom, (! [X:nat,N:nat,Y:nat]: ((((((modulo_modulo @ nat) @ X) @ N) = (((modulo_modulo @ nat) @ Y) @ N)) => ((((ord_less_eq @ nat) @ Y) @ X) => (? [Q2:nat]: ((X = (((plus_plus @ nat) @ Y) @ (((times_times @ nat) @ N) @ Q2)))))))))).
% 9.43/3.44 thf(fact_891_mod__eq__nat2E, axiom, (! [M2:nat,Q3:nat,N:nat]: ((((((modulo_modulo @ nat) @ M2) @ Q3) = (((modulo_modulo @ nat) @ N) @ Q3)) => ((((ord_less_eq @ nat) @ M2) @ N) => (~ ((! [S6:nat]: ((N != (((plus_plus @ nat) @ M2) @ (((times_times @ nat) @ Q3) @ S6)))))))))))).
% 9.43/3.44 thf(fact_892_mod__eq__nat1E, axiom, (! [M2:nat,Q3:nat,N:nat]: ((((((modulo_modulo @ nat) @ M2) @ Q3) = (((modulo_modulo @ nat) @ N) @ Q3)) => ((((ord_less_eq @ nat) @ N) @ M2) => (~ ((! [S6:nat]: ((M2 != (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ Q3) @ S6)))))))))))).
% 9.43/3.44 thf(fact_893_ceiling__add__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ int) @ ((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ Y))) @ (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((archimedean_ceiling @ A) @ Y))))))))).
% 9.43/3.44 thf(fact_894_mod__mult2__eq, axiom, (! [M2:nat,N:nat,Q3:nat]: (((((modulo_modulo @ nat) @ M2) @ (((times_times @ nat) @ N) @ Q3)) = (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ (((modulo_modulo @ nat) @ (((divide_divide @ nat) @ M2) @ N)) @ Q3))) @ (((modulo_modulo @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_895_div__mod__decomp, axiom, (! [A4:nat,N:nat]: ((A4 = (((plus_plus @ nat) @ (((times_times @ nat) @ (((divide_divide @ nat) @ A4) @ N)) @ N)) @ (((modulo_modulo @ nat) @ A4) @ N)))))).
% 9.43/3.44 thf(fact_896_one__add__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int)) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_897_split__mod, axiom, (! [P:(nat > $o),M2:nat,N:nat]: (((P @ (((modulo_modulo @ nat) @ M2) @ N)) = (((N = (zero_zero @ nat)) => (P @ M2)) & ((N != (zero_zero @ nat)) => (! [I3:nat,J2:nat]: (((((ord_less @ nat) @ J2) @ N) => ((M2 = (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ I3)) @ J2)) => (P @ J2))))))))))).
% 9.43/3.44 thf(fact_898_bset_I3_J, axiom, (! [D4:int,T2:int,B6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((minus_minus @ int) @ T2) @ (one_one @ int))) @ B6) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((X7 = T2) => ((((minus_minus @ int) @ X7) @ D4) = T2)))))))))).
% 9.43/3.44 thf(fact_899_bset_I4_J, axiom, (! [D4:int,T2:int,B6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ B6) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((X7 != T2) => ((((minus_minus @ int) @ X7) @ D4) != T2)))))))))).
% 9.43/3.44 thf(fact_900_bset_I5_J, axiom, (! [D4:int,B6:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less @ int) @ X7) @ T2) => (((ord_less @ int) @ (((minus_minus @ int) @ X7) @ D4)) @ T2))))))))).
% 9.43/3.44 thf(fact_901_bset_I7_J, axiom, (! [D4:int,T2:int,B6:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ B6) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less @ int) @ T2) @ X7) => (((ord_less @ int) @ T2) @ (((minus_minus @ int) @ X7) @ D4))))))))))).
% 9.43/3.44 thf(fact_902_aset_I3_J, axiom, (! [D4:int,T2:int,A4:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ (((plus_plus @ int) @ T2) @ (one_one @ int))) @ A4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((X7 = T2) => ((((plus_plus @ int) @ X7) @ D4) = T2)))))))))).
% 9.43/3.44 thf(fact_903_aset_I4_J, axiom, (! [D4:int,T2:int,A4:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ A4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((X7 != T2) => ((((plus_plus @ int) @ X7) @ D4) != T2)))))))))).
% 9.43/3.44 thf(fact_904_aset_I5_J, axiom, (! [D4:int,T2:int,A4:(set @ int)]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => ((((member @ int) @ T2) @ A4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less @ int) @ X7) @ T2) => (((ord_less @ int) @ (((plus_plus @ int) @ X7) @ D4)) @ T2)))))))))).
% 9.43/3.44 thf(fact_905_aset_I7_J, axiom, (! [D4:int,A4:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less @ int) @ T2) @ X7) => (((ord_less @ int) @ T2) @ (((plus_plus @ int) @ X7) @ D4)))))))))).
% 9.43/3.44 thf(fact_906_aset_I8_J, axiom, (! [D4:int,A4:(set @ int),T2:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((((ord_less_eq @ int) @ T2) @ X7) => (((ord_less_eq @ int) @ T2) @ (((plus_plus @ int) @ X7) @ D4)))))))))).
% 9.43/3.44 thf(fact_907_ceiling__divide__lower, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q3:A,P5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q3) => (((ord_less @ A) @ (((times_times @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ (((divide_divide @ A) @ P5) @ Q3)))) @ (one_one @ A))) @ Q3)) @ P5)))))))).
% 9.43/3.44 thf(fact_908_floor__divide__upper, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Q3:A,P5:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ Q3) => (((ord_less @ A) @ P5) @ (((times_times @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ (((divide_divide @ A) @ P5) @ Q3)))) @ (one_one @ A))) @ Q3))))))))).
% 9.43/3.44 thf(fact_909_power__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,M2:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ B2) @ (one_one @ A)) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B2) @ M2)) @ (((power_power @ A) @ B2) @ N)) = (((ord_less_eq @ nat) @ N) @ M2)))))))))).
% 9.43/3.44 thf(fact_910_divmod__nat__def, axiom, (divmod_nat = (^ [M:nat,N5:nat]: (((((product_Pair @ nat) @ nat) @ (((divide_divide @ nat) @ M) @ N5)) @ (((modulo_modulo @ nat) @ M) @ N5)))))).
% 9.43/3.44 thf(fact_911_distrib__left__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((semiring @ A) => (! [X:B,Y:B,A3:A,B2:A,C2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X) @ Y)) @ A3) => ((((times_times @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ B2)) @ (((times_times @ A) @ A3) @ C2)))))))))).
% 9.43/3.44 thf(fact_912_distrib__right__NO__MATCH, axiom, (! [B:$tType,A:$tType]: (((semiring @ A) => (! [X:B,Y:B,C2:A,A3:A,B2:A]: ((((((nO_MATCH @ B) @ A) @ (((divide_divide @ B) @ X) @ Y)) @ C2) => ((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)))))))))).
% 9.43/3.44 thf(fact_913_Suc__times__mod__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (suc @ (zero_zero @ nat))) @ M2) => ((((modulo_modulo @ nat) @ (suc @ (((times_times @ nat) @ M2) @ N))) @ M2) = (one_one @ nat)))))).
% 9.43/3.44 thf(fact_914_div__add__self1__no__field, axiom, (! [B:$tType,A:$tType]: ((((euclid4440199948858584721cancel @ A) & (field @ B)) => (! [X:B,B2:A,A3:A]: ((((((nO_MATCH @ B) @ A) @ X) @ B2) => ((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ B2) @ A3)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_915_div__add__self2__no__field, axiom, (! [B:$tType,A:$tType]: ((((euclid4440199948858584721cancel @ A) & (field @ B)) => (! [X:B,B2:A,A3:A]: ((((((nO_MATCH @ B) @ A) @ X) @ B2) => ((B2 != (zero_zero @ A)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ B2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_916_power__one, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N:nat]: (((((power_power @ A) @ (one_one @ A)) @ N) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_917_add__Suc__right, axiom, (! [M2:nat,N:nat]: (((((plus_plus @ nat) @ M2) @ (suc @ N)) = (suc @ (((plus_plus @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_918_power__one__right, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ (one_one @ nat)) = A3))))))).
% 9.43/3.44 thf(fact_919_power__inject__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,M2:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => (((((power_power @ A) @ A3) @ M2) = (((power_power @ A) @ A3) @ N)) = (M2 = N))))))))).
% 9.43/3.44 thf(fact_920_mult__Suc__right, axiom, (! [M2:nat,N:nat]: (((((times_times @ nat) @ M2) @ (suc @ N)) = (((plus_plus @ nat) @ M2) @ (((times_times @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_921_diff__Suc__1, axiom, (! [N:nat]: (((((minus_minus @ nat) @ (suc @ N)) @ (one_one @ nat)) = N)))).
% 9.43/3.44 thf(fact_922_of__int__eq__1__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [Z2:int]: (((((ring_1_of_int @ A) @ Z2) = (one_one @ A)) = (Z2 = (one_one @ int))))))))).
% 9.43/3.44 thf(fact_923_of__int__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (((ring_1_of_int @ A) @ (one_one @ int)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_924_of__int__add, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [W:int,Z2:int]: ((((ring_1_of_int @ A) @ (((plus_plus @ int) @ W) @ Z2)) = (((plus_plus @ A) @ ((ring_1_of_int @ A) @ W)) @ ((ring_1_of_int @ A) @ Z2))))))))).
% 9.43/3.44 thf(fact_925_power__strict__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,X:nat,Y:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B2) => ((((ord_less @ A) @ (((power_power @ A) @ B2) @ X)) @ (((power_power @ A) @ B2) @ Y)) = (((ord_less @ nat) @ X) @ Y))))))))).
% 9.43/3.44 thf(fact_926_left__minus__one__mult__self, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat,A3:A]: (((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ A3)) = A3))))))).
% 9.43/3.44 thf(fact_927_minus__one__mult__self, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_928_diff__Suc__diff__eq2, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ (suc @ (((minus_minus @ nat) @ J) @ K))) @ I) = (((minus_minus @ nat) @ (suc @ J)) @ (((plus_plus @ nat) @ K) @ I))))))).
% 9.43/3.44 thf(fact_929_diff__Suc__diff__eq1, axiom, (! [K:nat,J:nat,I:nat]: (((((ord_less_eq @ nat) @ K) @ J) => ((((minus_minus @ nat) @ I) @ (suc @ (((minus_minus @ nat) @ J) @ K))) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ (suc @ J))))))).
% 9.43/3.44 thf(fact_930_Suc__diff, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ M2) => ((suc @ (((minus_minus @ nat) @ N) @ M2)) = (((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))))))))).
% 9.43/3.44 thf(fact_931_Suc__mod__mult__self4, axiom, (! [N:nat,K:nat,M2:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ (((times_times @ nat) @ N) @ K)) @ M2))) @ N) = (((modulo_modulo @ nat) @ (suc @ M2)) @ N))))).
% 9.43/3.44 thf(fact_932_Suc__mod__mult__self3, axiom, (! [K:nat,N:nat,M2:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ (((times_times @ nat) @ K) @ N)) @ M2))) @ N) = (((modulo_modulo @ nat) @ (suc @ M2)) @ N))))).
% 9.43/3.44 thf(fact_933_Suc__mod__mult__self2, axiom, (! [M2:nat,N:nat,K:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ M2) @ (((times_times @ nat) @ N) @ K)))) @ N) = (((modulo_modulo @ nat) @ (suc @ M2)) @ N))))).
% 9.43/3.44 thf(fact_934_Suc__mod__mult__self1, axiom, (! [M2:nat,K:nat,N:nat]: (((((modulo_modulo @ nat) @ (suc @ (((plus_plus @ nat) @ M2) @ (((times_times @ nat) @ K) @ N)))) @ N) = (((modulo_modulo @ nat) @ (suc @ M2)) @ N))))).
% 9.43/3.44 thf(fact_935_ceiling__add__of__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z2:int]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ ((ring_1_of_int @ A) @ Z2))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z2)))))))).
% 9.43/3.44 thf(fact_936_power__strict__decreasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,M2:nat,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less @ A) @ B2) @ (one_one @ A)) => ((((ord_less @ A) @ (((power_power @ A) @ B2) @ M2)) @ (((power_power @ A) @ B2) @ N)) = (((ord_less @ nat) @ N) @ M2)))))))))).
% 9.43/3.44 thf(fact_937_power__increasing__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [B2:A,X:nat,Y:nat]: (((((ord_less @ A) @ (one_one @ A)) @ B2) => ((((ord_less_eq @ A) @ (((power_power @ A) @ B2) @ X)) @ (((power_power @ A) @ B2) @ Y)) = (((ord_less_eq @ nat) @ X) @ Y))))))))).
% 9.43/3.44 thf(fact_938_Suc__diff__1, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((suc @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) = N))))).
% 9.43/3.44 thf(fact_939_of__int__1__le__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less_eq @ int) @ (one_one @ int)) @ Z2)))))))).
% 9.43/3.44 thf(fact_940_of__int__le__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)) = (((ord_less_eq @ int) @ Z2) @ (one_one @ int))))))))).
% 9.43/3.44 thf(fact_941_of__int__less__1__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)) = (((ord_less @ int) @ Z2) @ (one_one @ int))))))))).
% 9.43/3.44 thf(fact_942_of__int__1__less__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [Z2:int]: (((((ord_less @ A) @ (one_one @ A)) @ ((ring_1_of_int @ A) @ Z2)) = (((ord_less @ int) @ (one_one @ int)) @ Z2)))))))).
% 9.43/3.44 thf(fact_943_power__gt1, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_944_add__Suc__shift, axiom, (! [M2:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M2)) @ N) = (((plus_plus @ nat) @ M2) @ (suc @ N)))))).
% 9.43/3.44 thf(fact_945_add__Suc, axiom, (! [M2:nat,N:nat]: (((((plus_plus @ nat) @ (suc @ M2)) @ N) = (suc @ (((plus_plus @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_946_nat__arith_Osuc1, axiom, (! [A4:nat,K:nat,A3:nat]: (((A4 = (((plus_plus @ nat) @ K) @ A3)) => ((suc @ A4) = (((plus_plus @ nat) @ K) @ (suc @ A3))))))).
% 9.43/3.44 thf(fact_947_power__Suc__le__self, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ (suc @ N))) @ A3))))))))).
% 9.43/3.44 thf(fact_948_power__Suc__less__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ (one_one @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ (suc @ N))) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_949_power__minus_H, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:A,N:nat]: ((((((nO_MATCH @ A) @ A) @ (one_one @ A)) @ X) => ((((power_power @ A) @ ((uminus_uminus @ A) @ X)) @ N) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((power_power @ A) @ X) @ N)))))))))).
% 9.43/3.44 thf(fact_950_one__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A3) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ N))))))))).
% 9.43/3.44 thf(fact_951_left__right__inverse__power, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [X:A,Y:A,N:nat]: ((((((times_times @ A) @ X) @ Y) = (one_one @ A)) => ((((times_times @ A) @ (((power_power @ A) @ X) @ N)) @ (((power_power @ A) @ Y) @ N)) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_952_power__0, axiom, (! [A:$tType]: (((power @ A) => (! [A3:A]: (((((power_power @ A) @ A3) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_953_power__one__over, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ N) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ N))))))))).
% 9.43/3.44 thf(fact_954_power__add, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M2:nat,N:nat]: (((((power_power @ A) @ A3) @ (((plus_plus @ nat) @ M2) @ N)) = (((times_times @ A) @ (((power_power @ A) @ A3) @ M2)) @ (((power_power @ A) @ A3) @ N))))))))).
% 9.43/3.44 thf(fact_955_nat__compl__induct_H, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((! [Nn:nat]: (((((ord_less_eq @ nat) @ Nn) @ N3) => (P @ Nn)))) => (P @ (suc @ N3))))) => (P @ N)))))).
% 9.43/3.44 thf(fact_956_nat__compl__induct, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((! [N3:nat]: (((! [Nn:nat]: (((((ord_less_eq @ nat) @ Nn) @ N3) => (P @ Nn)))) => (P @ (suc @ N3))))) => (P @ N)))))).
% 9.43/3.44 thf(fact_957_one__is__add, axiom, (! [M2:nat,N:nat]: ((((suc @ (zero_zero @ nat)) = (((plus_plus @ nat) @ M2) @ N)) = (((M2 = (suc @ (zero_zero @ nat))) & (N = (zero_zero @ nat))) | ((M2 = (zero_zero @ nat)) & (N = (suc @ (zero_zero @ nat))))))))).
% 9.43/3.44 thf(fact_958_add__is__1, axiom, (! [M2:nat,N:nat]: ((((((plus_plus @ nat) @ M2) @ N) = (suc @ (zero_zero @ nat))) = (((M2 = (suc @ (zero_zero @ nat))) & (N = (zero_zero @ nat))) | ((M2 = (zero_zero @ nat)) & (N = (suc @ (zero_zero @ nat))))))))).
% 9.43/3.44 thf(fact_959_nat__in__between__eq_I2_J, axiom, (! [A3:nat,B2:nat]: ((((((ord_less_eq @ nat) @ A3) @ B2) & (((ord_less @ nat) @ B2) @ (suc @ A3))) = (B2 = A3))))).
% 9.43/3.44 thf(fact_960_nat__in__between__eq_I1_J, axiom, (! [A3:nat,B2:nat]: ((((((ord_less @ nat) @ A3) @ B2) & (((ord_less_eq @ nat) @ B2) @ (suc @ A3))) = (B2 = (suc @ A3)))))).
% 9.43/3.44 thf(fact_961_less__imp__Suc__add, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (? [K5:nat]: ((N = (suc @ (((plus_plus @ nat) @ M2) @ K5))))))))).
% 9.43/3.44 thf(fact_962_less__iff__Suc__add, axiom, ((ord_less @ nat) = (^ [M:nat,N5:nat]: ((? [K4:nat]: ((N5 = (suc @ (((plus_plus @ nat) @ M) @ K4))))))))).
% 9.43/3.44 thf(fact_963_less__add__Suc2, axiom, (! [I:nat,M2:nat]: ((((ord_less @ nat) @ I) @ (suc @ (((plus_plus @ nat) @ M2) @ I)))))).
% 9.43/3.44 thf(fact_964_less__add__Suc1, axiom, (! [I:nat,M2:nat]: ((((ord_less @ nat) @ I) @ (suc @ (((plus_plus @ nat) @ I) @ M2)))))).
% 9.43/3.44 thf(fact_965_less__natE, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ M2) @ N) => (~ ((! [Q2:nat]: ((N != (suc @ (((plus_plus @ nat) @ M2) @ Q2))))))))))).
% 9.43/3.44 thf(fact_966_Suc__to__right, axiom, (! [N:nat,M2:nat]: ((((suc @ N) = M2) => (N = (((minus_minus @ nat) @ M2) @ (suc @ (zero_zero @ nat)))))))).
% 9.43/3.44 thf(fact_967_One__nat__def, axiom, ((one_one @ nat) = (suc @ (zero_zero @ nat)))).
% 9.43/3.44 thf(fact_968_mult__Suc, axiom, (! [M2:nat,N:nat]: (((((times_times @ nat) @ (suc @ M2)) @ N) = (((plus_plus @ nat) @ N) @ (((times_times @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_969_Suc__eq__plus1__left, axiom, (suc = ((plus_plus @ nat) @ (one_one @ nat)))).
% 9.43/3.44 thf(fact_970_plus__1__eq__Suc, axiom, (((plus_plus @ nat) @ (one_one @ nat)) = suc)).
% 9.43/3.44 thf(fact_971_Suc__eq__plus1, axiom, (suc = (^ [N5:nat]: ((((plus_plus @ nat) @ N5) @ (one_one @ nat)))))).
% 9.43/3.44 thf(fact_972_diff__Suc__eq__diff__pred, axiom, (! [M2:nat,N:nat]: (((((minus_minus @ nat) @ M2) @ (suc @ N)) = (((minus_minus @ nat) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))) @ N))))).
% 9.43/3.44 thf(fact_973_power__le__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_974_power__less__power__Suc, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ N)) @ (((times_times @ A) @ A3) @ (((power_power @ A) @ A3) @ N)))))))))).
% 9.43/3.44 thf(fact_975_power__gt1__lemma, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (one_one @ A)) @ (((times_times @ A) @ A3) @ (((power_power @ A) @ A3) @ N)))))))))).
% 9.43/3.44 thf(fact_976_power__0__left, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (one_one @ A))) & ((N != (zero_zero @ nat)) => ((((power_power @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_977_power__strict__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N6:nat,A3:A]: (((((ord_less @ nat) @ N) @ N6) => ((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ A3) @ N6)))))))))).
% 9.43/3.44 thf(fact_978_power__less__imp__less__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,M2:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less @ A) @ (((power_power @ A) @ A3) @ M2)) @ (((power_power @ A) @ A3) @ N)) => (((ord_less @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_979_power__minus, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [A3:A,N:nat]: (((((power_power @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((power_power @ A) @ A3) @ N))))))))).
% 9.43/3.44 thf(fact_980_power__increasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N6:nat,A3:A]: (((((ord_less_eq @ nat) @ N) @ N6) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A3) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N)) @ (((power_power @ A) @ A3) @ N6)))))))))).
% 9.43/3.44 thf(fact_981_floor__add__int, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z2:int]: (((((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ Z2) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X) @ ((ring_1_of_int @ A) @ Z2)))))))))).
% 9.43/3.44 thf(fact_982_int__add__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X:A]: (((((plus_plus @ int) @ Z2) @ ((archim6421214686448440834_floor @ A) @ X)) = ((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X))))))))).
% 9.43/3.44 thf(fact_983_nat__induct__non__zero, axiom, (! [N:nat,P:(nat > $o)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((P @ (one_one @ nat)) => ((! [N3:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((P @ N3) => (P @ (suc @ N3)))))) => (P @ N))))))).
% 9.43/3.44 thf(fact_984_of__int__leD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:int,X:A]: (((((ord_less_eq @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N))) @ X) => ((N = (zero_zero @ int)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X))))))))).
% 9.43/3.44 thf(fact_985_of__int__lessD, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [N:int,X:A]: (((((ord_less @ A) @ ((abs_abs @ A) @ ((ring_1_of_int @ A) @ N))) @ X) => ((N = (zero_zero @ int)) | (((ord_less @ A) @ (one_one @ A)) @ X))))))))).
% 9.43/3.44 thf(fact_986_floor__exists, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [Z3:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z3)) @ X) & (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ Z3) @ (one_one @ int))))))))))))).
% 9.43/3.44 thf(fact_987_floor__exists1, axiom, (! [A:$tType]: (((archim462609752435547400_field @ A) => (! [X:A]: ((? [X2:int]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ X2)) @ X) & ((((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ X2) @ (one_one @ int)))) & (! [Y6:int]: ((((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Y6)) @ X) & (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ Y6) @ (one_one @ int))))) => (Y6 = X2)))))))))))))).
% 9.43/3.44 thf(fact_988_power__Suc__less, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ (one_one @ A)) => (((ord_less @ A) @ (((times_times @ A) @ A3) @ (((power_power @ A) @ A3) @ N))) @ (((power_power @ A) @ A3) @ N)))))))))).
% 9.43/3.44 thf(fact_989_power__strict__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N6:nat,A3:A]: (((((ord_less @ nat) @ N) @ N6) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ (one_one @ A)) => (((ord_less @ A) @ (((power_power @ A) @ A3) @ N6)) @ (((power_power @ A) @ A3) @ N))))))))))).
% 9.43/3.44 thf(fact_990_power__decreasing, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat,N6:nat,A3:A]: (((((ord_less_eq @ nat) @ N) @ N6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ N6)) @ (((power_power @ A) @ A3) @ N))))))))))).
% 9.43/3.44 thf(fact_991_power__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,M2:nat,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less_eq @ A) @ (((power_power @ A) @ A3) @ M2)) @ (((power_power @ A) @ A3) @ N)) => (((ord_less_eq @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_992_self__le__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less_eq @ A) @ A3) @ (((power_power @ A) @ A3) @ N)))))))))).
% 9.43/3.44 thf(fact_993_one__less__power, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [A3:A,N:nat]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((ord_less @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ N)))))))))).
% 9.43/3.44 thf(fact_994_nz__le__conv__less, axiom, (! [K:nat,M2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((ord_less_eq @ nat) @ K) @ M2) => (((ord_less @ nat) @ (((minus_minus @ nat) @ K) @ (suc @ (zero_zero @ nat)))) @ M2)))))).
% 9.43/3.44 thf(fact_995_ceiling__altdef, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_ceiling @ A) = (^ [X3:A]: (((((if @ int) @ (X3 = ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X3)))) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X3)) @ (one_one @ int)))))))))).
% 9.43/3.44 thf(fact_996_Suc__diff__eq__diff__pred, axiom, (! [N:nat,M2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((minus_minus @ nat) @ (suc @ M2)) @ N) = (((minus_minus @ nat) @ M2) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 9.43/3.44 thf(fact_997_Suc__pred_H, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (N = (suc @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 9.43/3.44 thf(fact_998_add__eq__if, axiom, ((plus_plus @ nat) = (^ [M:nat,N5:nat]: (((((if @ nat) @ (M = (zero_zero @ nat))) @ N5) @ (suc @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ M) @ (one_one @ nat))) @ N5))))))).
% 9.43/3.44 thf(fact_999_Suc__n__minus__m__eq, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less @ nat) @ (one_one @ nat)) @ M2) => ((suc @ (((minus_minus @ nat) @ N) @ M2)) = (((minus_minus @ nat) @ N) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))))))))).
% 9.43/3.44 thf(fact_1000_floor__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X:A]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Z2)) @ X) => ((((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) => (((archim6421214686448440834_floor @ A) @ X) = Z2))))))))).
% 9.43/3.44 thf(fact_1001_floor__eq__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,A3:int]: (((((archim6421214686448440834_floor @ A) @ X) = A3) = ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ A3)) @ X) & (((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ A3)) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1002_floor__split, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [P:(int > $o),T2:A]: (((P @ ((archim6421214686448440834_floor @ A) @ T2)) = (! [I3:int]: ((((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ I3)) @ T2) & (((ord_less @ A) @ T2) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ I3)) @ (one_one @ A)))) => (P @ I3))))))))))).
% 9.43/3.44 thf(fact_1003_ceiling__correct, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X))) @ (one_one @ A))) @ X) & (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ ((archimedean_ceiling @ A) @ X)))))))))).
% 9.43/3.44 thf(fact_1004_ceiling__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X:A]: (((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) @ X) => ((((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ Z2)) => (((archimedean_ceiling @ A) @ X) = Z2))))))))).
% 9.43/3.44 thf(fact_1005_ceiling__eq__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,A3:int]: (((((archimedean_ceiling @ A) @ X) = A3) = ((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ A3)) @ (one_one @ A))) @ X) & (((ord_less_eq @ A) @ X) @ ((ring_1_of_int @ A) @ A3)))))))))).
% 9.43/3.44 thf(fact_1006_ceiling__split, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [P:(int > $o),T2:A]: (((P @ ((archimedean_ceiling @ A) @ T2)) = (! [I3:int]: ((((((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ I3)) @ (one_one @ A))) @ T2) & (((ord_less_eq @ A) @ T2) @ ((ring_1_of_int @ A) @ I3))) => (P @ I3))))))))))).
% 9.43/3.44 thf(fact_1007_less__floor__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X:A]: (((((ord_less @ int) @ Z2) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_1008_floor__le__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z2:int]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ Z2) = (((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1009_ceiling__less__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Z2:int]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ Z2) = (((ord_less_eq @ A) @ X) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1010_le__ceiling__iff, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [Z2:int,X:A]: (((((ord_less_eq @ int) @ Z2) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ Z2)) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_1011_floor__correct, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: (((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archim6421214686448440834_floor @ A) @ X))) @ X) & (((ord_less @ A) @ X) @ ((ring_1_of_int @ A) @ (((plus_plus @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ (one_one @ int))))))))))).
% 9.43/3.44 thf(fact_1012_power__diff__power__eq, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A3:A,N:nat,M2:nat]: (((A3 != (zero_zero @ A)) => (((((ord_less_eq @ nat) @ N) @ M2) => ((((divide_divide @ A) @ (((power_power @ A) @ A3) @ M2)) @ (((power_power @ A) @ A3) @ N)) = (((power_power @ A) @ A3) @ (((minus_minus @ nat) @ M2) @ N)))) & ((~ ((((ord_less_eq @ nat) @ N) @ M2))) => ((((divide_divide @ A) @ (((power_power @ A) @ A3) @ M2)) @ (((power_power @ A) @ A3) @ N)) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ A3) @ (((minus_minus @ nat) @ N) @ M2))))))))))))).
% 9.43/3.44 thf(fact_1013_power__eq__if, axiom, (! [A:$tType]: (((power @ A) => ((power_power @ A) = (^ [P6:A,M:nat]: (((((if @ A) @ (M = (zero_zero @ nat))) @ (one_one @ A)) @ (((times_times @ A) @ P6) @ (((power_power @ A) @ P6) @ (((minus_minus @ nat) @ M) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1014_power__minus__mult, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [N:nat,A3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((times_times @ A) @ (((power_power @ A) @ A3) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))) @ A3) = (((power_power @ A) @ A3) @ N))))))))).
% 9.43/3.44 thf(fact_1015_neg__one__power__add__eq__neg__one__power__diff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [K:nat,N:nat]: (((((ord_less_eq @ nat) @ K) @ N) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((plus_plus @ nat) @ N) @ K)) = (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((minus_minus @ nat) @ N) @ K)))))))))).
% 9.43/3.44 thf(fact_1016_option_Osize__gen_I2_J, axiom, (! [A:$tType,X:(A > nat),X22:A]: (((((size_option @ A) @ X) @ ((some @ A) @ X22)) = (((plus_plus @ nat) @ (X @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_1017_Heap_Osize__gen, axiom, (! [A:$tType,Xa:(A > nat),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((((heap_Time_size_Heap @ A) @ Xa) @ ((heap_Time_Heap2 @ A) @ X)) = (suc @ (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_1018_num_Osize__gen_I2_J, axiom, (! [X22:num]: (((size_num @ (bit0 @ X22)) = (((plus_plus @ nat) @ (size_num @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_1019_normalize__negative, axiom, (! [Q3:int,P5:int]: (((((ord_less @ int) @ Q3) @ (zero_zero @ int)) => ((normalize @ ((((product_Pair @ int) @ int) @ P5) @ Q3)) = (normalize @ ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ P5)) @ ((uminus_uminus @ int) @ Q3)))))))).
% 9.43/3.44 thf(fact_1020_option_Osize__gen_I1_J, axiom, (! [A:$tType,X:(A > nat)]: (((((size_option @ A) @ X) @ (none @ A)) = (suc @ (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_1021_Heap_Osize_I2_J, axiom, (! [A:$tType,X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((size_size @ (heap_Time_Heap @ A)) @ ((heap_Time_Heap2 @ A) @ X)) = (suc @ (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_1022_num_Osize_I5_J, axiom, (! [X22:num]: ((((size_size @ num) @ (bit0 @ X22)) = (((plus_plus @ nat) @ ((size_size @ num) @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_1023_option_Osize_I3_J, axiom, (! [A:$tType]: ((((size_size @ (option @ A)) @ (none @ A)) = (suc @ (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_1024_normalize__denom__zero, axiom, (! [P5:int]: (((normalize @ ((((product_Pair @ int) @ int) @ P5) @ (zero_zero @ int))) = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1025_option_Osize__neq, axiom, (! [A:$tType,X:(option @ A)]: ((((size_size @ (option @ A)) @ X) != (zero_zero @ nat))))).
% 9.43/3.44 thf(fact_1026_Heap_Osize__neq, axiom, (! [A:$tType,X:(heap_Time_Heap @ A)]: ((((size_size @ (heap_Time_Heap @ A)) @ X) != (zero_zero @ nat))))).
% 9.43/3.44 thf(fact_1027_option_Osize__gen__o__map, axiom, (! [B:$tType,A:$tType,F:(B > nat),G:(A > B)]: (((((((comp @ (option @ B)) @ nat) @ (option @ A)) @ ((size_option @ B) @ F)) @ (((map_option @ A) @ B) @ G)) = ((size_option @ A) @ (((((comp @ B) @ nat) @ A) @ F) @ G)))))).
% 9.43/3.44 thf(fact_1028_normalize__denom__pos, axiom, (! [R:((product_prod @ int) @ int),P5:int,Q3:int]: ((((normalize @ R) = ((((product_Pair @ int) @ int) @ P5) @ Q3)) => (((ord_less @ int) @ (zero_zero @ int)) @ Q3))))).
% 9.43/3.44 thf(fact_1029_normalize__crossproduct, axiom, (! [Q3:int,S:int,P5:int,R:int]: (((Q3 != (zero_zero @ int)) => ((S != (zero_zero @ int)) => (((normalize @ ((((product_Pair @ int) @ int) @ P5) @ Q3)) = (normalize @ ((((product_Pair @ int) @ int) @ R) @ S))) => ((((times_times @ int) @ P5) @ S) = (((times_times @ int) @ R) @ Q3)))))))).
% 9.43/3.44 thf(fact_1030_option_Osize_I4_J, axiom, (! [A:$tType,X22:A]: ((((size_size @ (option @ A)) @ ((some @ A) @ X22)) = (suc @ (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_1031_divmod__nat__if, axiom, (divmod_nat = (^ [M:nat,N5:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ ((N5 = (zero_zero @ nat)) | (((ord_less @ nat) @ M) @ N5))) @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ M)) @ (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [Q5:nat]: ((((product_Pair @ nat) @ nat) @ (suc @ Q5))))) @ ((divmod_nat @ (((minus_minus @ nat) @ M) @ N5)) @ N5))))))).
% 9.43/3.44 thf(fact_1032_sum__gp__multiplied, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [M2:nat,N:nat,X:A]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) = (((minus_minus @ A) @ (((power_power @ A) @ X) @ M2)) @ (((power_power @ A) @ X) @ (suc @ N))))))))))).
% 9.43/3.44 thf(fact_1033_pair__lessI2, axiom, (! [A3:nat,B2:nat,S:nat,T2:nat]: (((((ord_less_eq @ nat) @ A3) @ B2) => ((((ord_less @ nat) @ S) @ T2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A3) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_less)))))).
% 9.43/3.44 thf(fact_1034_pair__less__iff1, axiom, (! [X:nat,Y:nat,Z2:nat]: (((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ X) @ Y)) @ ((((product_Pair @ nat) @ nat) @ X) @ Z2))) @ fun_pair_less) = (((ord_less @ nat) @ Y) @ Z2))))).
% 9.43/3.44 thf(fact_1035_Suc__times__numeral__mod__eq, axiom, (! [K:num,N:nat]: (((((numeral_numeral @ nat) @ K) != (one_one @ nat)) => ((((modulo_modulo @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ K)) @ N))) @ ((numeral_numeral @ nat) @ K)) = (one_one @ nat)))))).
% 9.43/3.44 thf(fact_1036_one__int_Otransfer, axiom, ((pcr_int @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat))) @ (one_one @ int))).
% 9.43/3.44 thf(fact_1037_less__than__iff, axiom, (! [X:nat,Y:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ X) @ Y)) @ less_than) = (((ord_less @ nat) @ X) @ Y))))).
% 9.43/3.44 thf(fact_1038_one__int__def, axiom, ((one_one @ int) = (abs_Integ @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat))))).
% 9.43/3.44 thf(fact_1039_sum__telescope_H_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M2:nat,N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((minus_minus @ A) @ (F @ K4)) @ (F @ (((minus_minus @ nat) @ K4) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ N)) = (((minus_minus @ A) @ (F @ N)) @ (F @ M2)))))))))).
% 9.43/3.44 thf(fact_1040_case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > (C > A)),A3:B,B2:C]: (((((((product_case_prod @ B) @ C) @ A) @ F) @ ((((product_Pair @ B) @ C) @ A3) @ B2)) = ((F @ A3) @ B2))))).
% 9.43/3.44 thf(fact_1041_case__prod__curry, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ ((((product_curry @ A) @ B) @ C) @ F)) = F)))).
% 9.43/3.44 thf(fact_1042_curry__case__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C))]: ((((((product_curry @ A) @ B) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) = F)))).
% 9.43/3.44 thf(fact_1043_distrib__right__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (semiring @ A)) => (! [A3:A,B2:A,V:num]: (((((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ ((numeral_numeral @ A) @ V)) = (((plus_plus @ A) @ (((times_times @ A) @ A3) @ ((numeral_numeral @ A) @ V))) @ (((times_times @ A) @ B2) @ ((numeral_numeral @ A) @ V)))))))))).
% 9.43/3.44 thf(fact_1044_distrib__left__numeral, axiom, (! [A:$tType]: ((((numeral @ A) & (semiring @ A)) => (! [V:num,B2:A,C2:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ B2) @ C2)) = (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ B2)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ V)) @ C2))))))))).
% 9.43/3.44 thf(fact_1045_add__neg__numeral__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num,N:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((uminus_uminus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N)))))))))).
% 9.43/3.44 thf(fact_1046_add__numeral__left, axiom, (! [A:$tType]: (((numeral @ A) => (! [V:num,W:num,Z2:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W)) @ Z2)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V) @ W))) @ Z2)))))))).
% 9.43/3.44 thf(fact_1047_numeral__plus__numeral, axiom, (! [A:$tType]: (((numeral @ A) => (! [M2:num,N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1048_semiring__norm_I168_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V:num,W:num,Y:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W))) @ Y)) = (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ V) @ W)))) @ Y)))))))).
% 9.43/3.44 thf(fact_1049_diff__numeral__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num,N:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((numeral_numeral @ A) @ N)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ M2) @ N)))))))))).
% 9.43/3.44 thf(fact_1050_diff__numeral__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num,N:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1051_ceiling__add__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: ((((archimedean_ceiling @ A) @ (((plus_plus @ A) @ X) @ ((numeral_numeral @ A) @ V))) = (((plus_plus @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V))))))))).
% 9.43/3.44 thf(fact_1052_power__add__numeral, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M2:num,N:num]: (((((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ M2))) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ N))) = (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (((plus_plus @ num) @ M2) @ N)))))))))).
% 9.43/3.44 thf(fact_1053_power__add__numeral2, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [A3:A,M2:num,N:num,B2:A]: (((((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ M2))) @ (((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ N))) @ B2)) = (((times_times @ A) @ (((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (((plus_plus @ num) @ M2) @ N)))) @ B2)))))))).
% 9.43/3.44 thf(fact_1054_numeral__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less @ int) @ ((numeral_numeral @ int) @ V)) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_1055_floor__le__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((numeral_numeral @ int) @ V)) = (((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1056_sum_Ocl__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N:nat,M2:nat,G:(nat > A)]: ((((((ord_less @ nat) @ (suc @ N)) @ M2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ (suc @ N)) @ M2))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ (G @ (suc @ N)))))))))))).
% 9.43/3.44 thf(fact_1057_ceiling__less__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((numeral_numeral @ int) @ V)) = (((ord_less_eq @ A) @ X) @ (((minus_minus @ A) @ ((numeral_numeral @ A) @ V)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1058_numeral__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ V)) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((numeral_numeral @ A) @ V)) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_1059_neg__numeral__less__floor, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archim6421214686448440834_floor @ A) @ X)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_1060_floor__le__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less_eq @ int) @ ((archim6421214686448440834_floor @ A) @ X)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) = (((ord_less @ A) @ X) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1061_ceiling__less__neg__numeral, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,V:num]: (((((ord_less @ int) @ ((archimedean_ceiling @ A) @ X)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) = (((ord_less_eq @ A) @ X) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1062_neg__numeral__le__ceiling, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [V:num,X:A]: (((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ V))) @ ((archimedean_ceiling @ A) @ X)) = (((ord_less @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (one_one @ A))) @ X)))))))).
% 9.43/3.44 thf(fact_1063_nested__case__prod__simp, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ (D > A)) = (^ [F2:(B > (C > (D > A))),X3:((product_prod @ B) @ C),Y3:D]: ((((((product_case_prod @ B) @ C) @ A) @ (^ [A7:B,B5:C]: ((((F2 @ A7) @ B5) @ Y3)))) @ X3))))))).
% 9.43/3.44 thf(fact_1064_prod_Ocase__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,H3:(C > D),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((H3 @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (((((product_case_prod @ A) @ B) @ D) @ (^ [X12:A,X24:B]: ((H3 @ ((F @ X12) @ X24))))) @ Prod))))).
% 9.43/3.44 thf(fact_1065_sum_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1066_sum_OatLeastAtMost__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1067_split__cong, axiom, (! [C:$tType,B:$tType,A:$tType,Q3:((product_prod @ A) @ B),F:(A > (B > C)),G:(A > (B > C)),P5:((product_prod @ A) @ B)]: (((! [X2:A,Y4:B]: (((((((product_Pair @ A) @ B) @ X2) @ Y4) = Q3) => (((F @ X2) @ Y4) = ((G @ X2) @ Y4))))) => ((P5 = Q3) => ((((((product_case_prod @ A) @ B) @ C) @ F) @ P5) = (((((product_case_prod @ A) @ B) @ C) @ G) @ Q3))))))).
% 9.43/3.44 thf(fact_1068_old_Oprod_Ocase, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),X1:A,X22:B]: (((((((product_case_prod @ A) @ B) @ C) @ F) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = ((F @ X1) @ X22))))).
% 9.43/3.44 thf(fact_1069_uminus__int_Oabs__eq, axiom, (! [X:((product_prod @ nat) @ nat)]: ((((uminus_uminus @ int) @ (abs_Integ @ X)) = (abs_Integ @ (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X3:nat,Y3:nat]: (((((product_Pair @ nat) @ nat) @ Y3) @ X3)))) @ X)))))).
% 9.43/3.44 thf(fact_1070_case__prod__Pair__iden, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ P5) = P5)))).
% 9.43/3.44 thf(fact_1071_sum_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1072_sum_Oshift__bounds__cl__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1073_cond__case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > (B > C)),G:(((product_prod @ A) @ B) > C)]: (((! [X2:A,Y4:B]: ((((F @ X2) @ Y4) = (G @ ((((product_Pair @ A) @ B) @ X2) @ Y4))))) => (((((product_case_prod @ A) @ B) @ C) @ F) = G))))).
% 9.43/3.44 thf(fact_1074_case__prod__eta, axiom, (! [C:$tType,B:$tType,A:$tType,F:(((product_prod @ A) @ B) > C)]: ((((((product_case_prod @ A) @ B) @ C) @ (^ [X3:A,Y3:B]: ((F @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) = F)))).
% 9.43/3.44 thf(fact_1075_case__prodE2, axiom, (! [B:$tType,A:$tType,C:$tType,Q:(A > $o),P:(B > (C > A)),Z2:((product_prod @ B) @ C)]: (((Q @ (((((product_case_prod @ B) @ C) @ A) @ P) @ Z2)) => (~ ((! [X2:B,Y4:C]: (((Z2 = ((((product_Pair @ B) @ C) @ X2) @ Y4)) => (~ ((Q @ ((P @ X2) @ Y4))))))))))))).
% 9.43/3.44 thf(fact_1076_eq__Abs__Integ, axiom, (! [Z2:int]: ((~ ((! [X2:nat,Y4:nat]: ((Z2 != (abs_Integ @ ((((product_Pair @ nat) @ nat) @ X2) @ Y4)))))))))).
% 9.43/3.44 thf(fact_1077_one__le__numeral, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)))))))).
% 9.43/3.44 thf(fact_1078_not__numeral__less__one, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1079_one__plus__numeral__commute, axiom, (! [A:$tType]: (((numeral @ A) => (! [X:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ X)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1080_numeral__Bit0, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bit0 @ N)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1081_one__neq__neg__numeral, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((one_one @ A) != ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1082_numeral__neq__neg__one, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: ((((numeral_numeral @ A) @ N) != ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1083_uncurry__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((uncurry @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 9.43/3.44 thf(fact_1084_internal__case__prod__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((produc5280177257484947105e_prod @ A) @ B) @ C) = (((product_case_prod @ A) @ B) @ C))))).
% 9.43/3.44 thf(fact_1085_sum_OatLeastAtMost__rev, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ N) @ M2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ N) @ M2))))))))).
% 9.43/3.44 thf(fact_1086_numeral__code_I2_J, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bit0 @ N)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1087_sum__shift__lb__Suc0__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),K:nat]: ((((F @ (zero_zero @ nat)) = (zero_zero @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ K)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ K)))))))))).
% 9.43/3.44 thf(fact_1088_sum_OatLeast0__atMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_1089_sum_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))))))).
% 9.43/3.44 thf(fact_1090_sum_OatLeast__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((plus_plus @ A) @ (G @ M2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ N))))))))))).
% 9.43/3.44 thf(fact_1091_sum_Onat__ivl__Suc_H, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ (suc @ N)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (suc @ N))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.44 thf(fact_1092_neg__numeral__le__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1093_neg__one__le__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ M2)))))))).
% 9.43/3.44 thf(fact_1094_neg__numeral__le__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1095_not__numeral__le__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1096_not__one__le__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((~ ((((ord_less_eq @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))))))))))).
% 9.43/3.44 thf(fact_1097_not__neg__one__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((~ ((((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))))))))))).
% 9.43/3.44 thf(fact_1098_not__one__less__neg__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((~ ((((ord_less @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))))))))))).
% 9.43/3.44 thf(fact_1099_not__numeral__less__neg__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((~ ((((ord_less @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1100_neg__one__less__numeral, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ M2)))))))).
% 9.43/3.44 thf(fact_1101_neg__numeral__less__one, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: ((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1102_sum_OSuc__reindex__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ (G @ (suc @ N))) = (((plus_plus @ A) @ (G @ M2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.44 thf(fact_1103_sum__Suc__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M2:nat,N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ (suc @ N)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ (F @ (suc @ I3))) @ (F @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((minus_minus @ A) @ (F @ (suc @ N))) @ (F @ M2)))))))))).
% 9.43/3.44 thf(fact_1104_sum__power__add, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,M2:nat,I4:(set @ nat)]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((power_power @ A) @ X) @ (((plus_plus @ nat) @ M2) @ I3))))) @ I4) = (((times_times @ A) @ (((power_power @ A) @ X) @ M2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ I4))))))))).
% 9.43/3.44 thf(fact_1105_sum_OatLeast__atMost__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N5:nat]: ((((minus_minus @ nat) @ N5) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1106_sum__atLeastAtMost__code, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),A3:nat,B2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ A3) @ B2)) = (((((set_fo6178422350223883121st_nat @ A) @ (^ [A7:nat]: (((plus_plus @ A) @ (F @ A7))))) @ A3) @ B2) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1107_sum_Oub__add__nat, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A),P5:nat]: (((((ord_less_eq @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ P5))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ P5)))))))))))).
% 9.43/3.44 thf(fact_1108_sum_OatLeastAtMost__shift__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M2))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M2))))))))))).
% 9.43/3.44 thf(fact_1109_zero__int__def, axiom, ((zero_zero @ int) = (abs_Integ @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))))).
% 9.43/3.44 thf(fact_1110_pair__lessI1, axiom, (! [A3:nat,B2:nat,S:nat,T2:nat]: (((((ord_less @ nat) @ A3) @ B2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A3) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_less))))).
% 9.43/3.44 thf(fact_1111_zero__int_Otransfer, axiom, ((pcr_int @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))) @ (zero_zero @ int))).
% 9.43/3.44 thf(fact_1112_sum__natinterval__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M2:nat,N:nat,F:(nat > A)]: ((((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((minus_minus @ A) @ (F @ K4)) @ (F @ (((plus_plus @ nat) @ K4) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((minus_minus @ A) @ (F @ M2)) @ (F @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))) & ((~ ((((ord_less_eq @ nat) @ M2) @ N))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((minus_minus @ A) @ (F @ K4)) @ (F @ (((plus_plus @ nat) @ K4) @ (one_one @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1113_convex__sum__bound__le, axiom, (! [A:$tType,B:$tType]: (((linordered_idom @ B) => (! [I4:(set @ A),X:(A > B),A3:(A > B),B2:B,Delta:B]: (((! [I2:A]: (((((member @ A) @ I2) @ I4) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (X @ I2))))) => ((((((groups7311177749621191930dd_sum @ A) @ B) @ X) @ I4) = (one_one @ B)) => ((! [I2:A]: (((((member @ A) @ I2) @ I4) => (((ord_less_eq @ B) @ ((abs_abs @ B) @ (((minus_minus @ B) @ (A3 @ I2)) @ B2))) @ Delta)))) => (((ord_less_eq @ B) @ ((abs_abs @ B) @ (((minus_minus @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ (^ [I3:A]: ((((times_times @ B) @ (A3 @ I3)) @ (X @ I3))))) @ I4)) @ B2))) @ Delta)))))))))).
% 9.43/3.44 thf(fact_1114_sum__comp__morphism, axiom, (! [A:$tType,B:$tType,C:$tType]: ((((comm_monoid_add @ B) & (comm_monoid_add @ A)) => (! [H3:(B > A),G:(C > B),A4:(set @ C)]: ((((H3 @ (zero_zero @ B)) = (zero_zero @ A)) => ((! [X2:B,Y4:B]: (((H3 @ (((plus_plus @ B) @ X2) @ Y4)) = (((plus_plus @ A) @ (H3 @ X2)) @ (H3 @ Y4))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ (((((comp @ B) @ A) @ C) @ H3) @ G)) @ A4) = (H3 @ ((((groups7311177749621191930dd_sum @ C) @ B) @ G) @ A4))))))))))).
% 9.43/3.44 thf(fact_1115_sum__gp, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [N:nat,M2:nat,X:A]: ((((((ord_less @ nat) @ N) @ M2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((X = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ M2)))) & ((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ X) @ M2)) @ (((power_power @ A) @ X) @ (suc @ N)))) @ (((minus_minus @ A) @ (one_one @ A)) @ X))))))))))))).
% 9.43/3.44 thf(fact_1116_divmod__digit__1_I1_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ (zero_zero @ A)) @ B2) => ((((ord_less_eq @ A) @ B2) @ (((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) => ((((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)))) @ (one_one @ A)) = (((divide_divide @ A) @ A3) @ B2))))))))))).
% 9.43/3.44 thf(fact_1117_sum__gp__offset, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [X:A,M2:nat,N:nat]: ((((X = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (((plus_plus @ nat) @ M2) @ N))) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A)))) & ((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (((plus_plus @ nat) @ M2) @ N))) = (((divide_divide @ A) @ (((times_times @ A) @ (((power_power @ A) @ X) @ M2)) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ (suc @ N))))) @ (((minus_minus @ A) @ (one_one @ A)) @ X))))))))))).
% 9.43/3.44 thf(fact_1118_Sum__Icc__nat, axiom, (! [M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((divide_divide @ nat) @ (((minus_minus @ nat) @ (((times_times @ nat) @ N) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))) @ (((times_times @ nat) @ M2) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1119_arith__series__nat, axiom, (! [A3:nat,D3:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I3:nat]: ((((plus_plus @ nat) @ A3) @ (((times_times @ nat) @ I3) @ D3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((divide_divide @ nat) @ (((times_times @ nat) @ (suc @ N)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ A3)) @ (((times_times @ nat) @ N) @ D3)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1120_mod__double__modulus, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M2:A,X:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ M2) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => (((((modulo_modulo @ A) @ X) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) = (((modulo_modulo @ A) @ X) @ M2)) | ((((modulo_modulo @ A) @ X) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ X) @ M2)) @ M2))))))))))).
% 9.43/3.44 thf(fact_1121_sum_Odistrib, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),H3:(B > A),A4:(set @ B)]: ((((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X3:B]: ((((plus_plus @ A) @ (G @ X3)) @ (H3 @ X3))))) @ A4) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H3) @ A4))))))))).
% 9.43/3.44 thf(fact_1122_case__prodI, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A3:A,B2:B]: ((((F @ A3) @ B2) => (((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A3) @ B2)))))).
% 9.43/3.44 thf(fact_1123_case__prodI2, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B),C2:(A > (B > $o))]: (((! [A5:A,B3:B]: (((P5 = ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((C2 @ A5) @ B3)))) => (((((product_case_prod @ A) @ B) @ $o) @ C2) @ P5))))).
% 9.43/3.44 thf(fact_1124_mem__case__prodI, axiom, (! [A:$tType,B:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),A3:B,B2:C]: (((((member @ A) @ Z2) @ ((C2 @ A3) @ B2)) => (((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ ((((product_Pair @ B) @ C) @ A3) @ B2))))))).
% 9.43/3.44 thf(fact_1125_mem__case__prodI2, axiom, (! [C:$tType,B:$tType,A:$tType,P5:((product_prod @ A) @ B),Z2:C,C2:(A > (B > (set @ C)))]: (((! [A5:A,B3:B]: (((P5 = ((((product_Pair @ A) @ B) @ A5) @ B3)) => (((member @ C) @ Z2) @ ((C2 @ A5) @ B3))))) => (((member @ C) @ Z2) @ (((((product_case_prod @ A) @ B) @ (set @ C)) @ C2) @ P5)))))).
% 9.43/3.44 thf(fact_1126_case__prodI2_H, axiom, (! [A:$tType,B:$tType,C:$tType,P5:((product_prod @ A) @ B),C2:(A > (B > (C > $o))),X:C]: (((! [A5:A,B3:B]: (((((((product_Pair @ A) @ B) @ A5) @ B3) = P5) => (((C2 @ A5) @ B3) @ X)))) => ((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P5) @ X))))).
% 9.43/3.44 thf(fact_1127_one__eq__numeral__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: ((((one_one @ A) = ((numeral_numeral @ A) @ N)) = (one2 = N)))))))).
% 9.43/3.44 thf(fact_1128_numeral__eq__one__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:num]: (((((numeral_numeral @ A) @ N) = (one_one @ A)) = (N = one2)))))))).
% 9.43/3.44 thf(fact_1129_of__nat__add, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M2:nat,N:nat]: ((((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M2) @ N)) = (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_1_of_nat @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1130_of__nat__1, axiom, (! [A:$tType]: (((semiring_1 @ A) => (((semiring_1_of_nat @ A) @ (one_one @ nat)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1131_of__nat__1__eq__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: ((((one_one @ A) = ((semiring_1_of_nat @ A) @ N)) = (N = (one_one @ nat))))))))).
% 9.43/3.44 thf(fact_1132_of__nat__eq__1__iff, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: (((((semiring_1_of_nat @ A) @ N) = (one_one @ A)) = (N = (one_one @ nat))))))))).
% 9.43/3.44 thf(fact_1133_semiring__norm_I2_J, axiom, ((((plus_plus @ num) @ one2) @ one2) = (bit0 @ one2))).
% 9.43/3.44 thf(fact_1134_numeral__le__one__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: (((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = (((ord_less_eq @ num) @ N) @ one2)))))))).
% 9.43/3.44 thf(fact_1135_one__less__numeral__iff, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [N:num]: (((((ord_less @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((ord_less @ num) @ one2) @ N)))))))).
% 9.43/3.44 thf(fact_1136_numeral__eq__neg__one__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N)) = ((uminus_uminus @ A) @ (one_one @ A))) = (N = one2)))))))).
% 9.43/3.44 thf(fact_1137_neg__one__eq__numeral__iff, axiom, (! [A:$tType]: (((ring_char_0 @ A) => (! [N:num]: (((((uminus_uminus @ A) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (N = one2)))))))).
% 9.43/3.44 thf(fact_1138_of__nat__Suc, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [M2:nat]: ((((semiring_1_of_nat @ A) @ (suc @ M2)) = (((plus_plus @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ M2))))))))).
% 9.43/3.44 thf(fact_1139_Suc__numeral, axiom, (! [N:num]: (((suc @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ nat) @ (((plus_plus @ num) @ N) @ one2)))))).
% 9.43/3.44 thf(fact_1140_not__neg__one__le__neg__numeral__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: (((~ ((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))))) = (M2 != one2)))))))).
% 9.43/3.44 thf(fact_1141_one__add__one, axiom, (! [A:$tType]: (((numeral @ A) => ((((plus_plus @ A) @ (one_one @ A)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1142_neg__numeral__less__neg__one__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [M2:num]: (((((ord_less @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (M2 != one2)))))))).
% 9.43/3.44 thf(fact_1143_bits__one__mod__two__eq__one, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((((modulo_modulo @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1144_one__mod__two__eq__one, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => ((((modulo_modulo @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1145_add__2__eq__Suc, axiom, (! [N:nat]: (((((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) = (suc @ (suc @ N)))))).
% 9.43/3.44 thf(fact_1146_add__2__eq__Suc_H, axiom, (! [N:nat]: (((((plus_plus @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (suc @ (suc @ N)))))).
% 9.43/3.44 thf(fact_1147_Suc__1, axiom, ((suc @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))).
% 9.43/3.44 thf(fact_1148_numeral__plus__one, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ (one_one @ A)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ N) @ one2))))))))).
% 9.43/3.44 thf(fact_1149_one__plus__numeral, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ N))))))))).
% 9.43/3.44 thf(fact_1150_add__self__div__2, axiom, (! [M2:nat]: (((((divide_divide @ nat) @ (((plus_plus @ nat) @ M2) @ M2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = M2)))).
% 9.43/3.44 thf(fact_1151_bits__1__div__2, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_1152_one__div__two__eq__zero, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => ((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_1153_add__neg__numeral__special_I9_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 9.43/3.44 thf(fact_1154_diff__numeral__special_I11_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1155_diff__numeral__special_I10_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 9.43/3.44 thf(fact_1156_not__mod__2__eq__0__eq__1, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: ((((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (zero_zero @ A)) = ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1157_not__mod__2__eq__1__eq__0, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: ((((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (one_one @ A)) = ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1158_minus__1__div__2__eq, axiom, (! [A:$tType]: (((euclid8789492081693882211th_nat @ A) => ((((divide_divide @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_1159_bits__minus__1__mod__2__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1160_minus__1__mod__2__eq, axiom, (! [A:$tType]: (((euclid8789492081693882211th_nat @ A) => ((((modulo_modulo @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1161_diff__numeral__special_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((minus_minus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ N))))))))).
% 9.43/3.44 thf(fact_1162_diff__numeral__special_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ M2) @ one2)))))))))).
% 9.43/3.44 thf(fact_1163_add__self__mod__2, axiom, (! [M2:nat]: (((((modulo_modulo @ nat) @ (((plus_plus @ nat) @ M2) @ M2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (zero_zero @ nat))))).
% 9.43/3.44 thf(fact_1164_power__minus1__even, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: (((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1165_mod2__gr__0, axiom, (! [M2:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ (((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = ((((modulo_modulo @ nat) @ M2) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ nat)))))).
% 9.43/3.44 thf(fact_1166_lex__prod__def, axiom, (! [B:$tType,A:$tType]: ((((lex_prod @ A) @ B) = (^ [Ra:(set @ ((product_prod @ A) @ A)),Rb:(set @ ((product_prod @ B) @ B))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [A7:A,B5:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [A8:A,B7:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ A8)) @ Ra) | ((A7 = A8) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ B5) @ B7)) @ Rb))))))))))))))))).
% 9.43/3.44 thf(fact_1167_same__fst__def, axiom, (! [B:$tType,A:$tType]: ((((same_fst @ A) @ B) = (^ [P4:(A > $o),R6:(A > (set @ ((product_prod @ B) @ B)))]: (((collect @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ $o) @ ((((product_case_prod @ A) @ B) @ (((product_prod @ A) @ B) > $o)) @ (^ [X9:A,Y8:B]: (((((product_case_prod @ A) @ B) @ $o) @ (^ [X3:A,Y3:B]: (((X9 = X3) & ((P4 @ X3) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y8) @ Y3)) @ (R6 @ X3)))))))))))))))))).
% 9.43/3.44 thf(fact_1168_mem__case__prodE, axiom, (! [B:$tType,A:$tType,C:$tType,Z2:A,C2:(B > (C > (set @ A))),P5:((product_prod @ B) @ C)]: (((((member @ A) @ Z2) @ (((((product_case_prod @ B) @ C) @ (set @ A)) @ C2) @ P5)) => (~ ((! [X2:B,Y4:C]: (((P5 = ((((product_Pair @ B) @ C) @ X2) @ Y4)) => (~ ((((member @ A) @ Z2) @ ((C2 @ X2) @ Y4))))))))))))).
% 9.43/3.44 thf(fact_1169_case__prodD, axiom, (! [A:$tType,B:$tType,F:(A > (B > $o)),A3:A,B2:B]: (((((((product_case_prod @ A) @ B) @ $o) @ F) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) => ((F @ A3) @ B2))))).
% 9.43/3.44 thf(fact_1170_case__prodE, axiom, (! [A:$tType,B:$tType,C2:(A > (B > $o)),P5:((product_prod @ A) @ B)]: (((((((product_case_prod @ A) @ B) @ $o) @ C2) @ P5) => (~ ((! [X2:A,Y4:B]: (((P5 = ((((product_Pair @ A) @ B) @ X2) @ Y4)) => (~ (((C2 @ X2) @ Y4)))))))))))).
% 9.43/3.44 thf(fact_1171_add__One__commute, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ N) = (((plus_plus @ num) @ N) @ one2))))).
% 9.43/3.44 thf(fact_1172_case__prodD_H, axiom, (! [B:$tType,A:$tType,C:$tType,R5:(A > (B > (C > $o))),A3:A,B2:B,C2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ R5) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ C2) => (((R5 @ A3) @ B2) @ C2))))).
% 9.43/3.44 thf(fact_1173_case__prodE_H, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(A > (B > (C > $o))),P5:((product_prod @ A) @ B),Z2:C]: ((((((((product_case_prod @ A) @ B) @ (C > $o)) @ C2) @ P5) @ Z2) => (~ ((! [X2:A,Y4:B]: (((P5 = ((((product_Pair @ A) @ B) @ X2) @ Y4)) => (~ ((((C2 @ X2) @ Y4) @ Z2)))))))))))).
% 9.43/3.44 thf(fact_1174_sum_OatLeast__int__atMost__int__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(int > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ int) @ A) @ G) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1175_less__int_Oabs__eq, axiom, (! [Xa:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((ord_less @ int) @ (abs_Integ @ Xa)) @ (abs_Integ @ X)) = ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3)))))))) @ Xa) @ X))))).
% 9.43/3.44 thf(fact_1176_less__eq__int_Oabs__eq, axiom, (! [Xa:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((ord_less_eq @ int) @ (abs_Integ @ Xa)) @ (abs_Integ @ X)) = ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3)))))))) @ Xa) @ X))))).
% 9.43/3.44 thf(fact_1177_Collect__case__prod__mono, axiom, (! [B:$tType,A:$tType,A4:(A > (B > $o)),B6:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ A4) @ B6) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A4))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ B6))))))).
% 9.43/3.44 thf(fact_1178_zadd__int__left, axiom, (! [M2:nat,N:nat,Z2:int]: (((((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ Z2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ M2) @ N))) @ Z2))))).
% 9.43/3.44 thf(fact_1179_int__plus, axiom, (! [N:nat,M2:nat]: ((((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ N) @ M2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ ((semiring_1_of_nat @ int) @ M2)))))).
% 9.43/3.44 thf(fact_1180_int__ops_I5_J, axiom, (! [A3:nat,B2:nat]: ((((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ A3) @ B2)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 9.43/3.44 thf(fact_1181_int__ops_I2_J, axiom, (((semiring_1_of_nat @ int) @ (one_one @ nat)) = (one_one @ int))).
% 9.43/3.44 thf(fact_1182_zle__iff__zadd, axiom, ((ord_less_eq @ int) = (^ [W2:int,Z5:int]: ((? [N5:nat]: ((Z5 = (((plus_plus @ int) @ W2) @ ((semiring_1_of_nat @ int) @ N5))))))))).
% 9.43/3.44 thf(fact_1183_numeral__One, axiom, (! [A:$tType]: (((numeral @ A) => (((numeral_numeral @ A) @ one2) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1184_numerals_I1_J, axiom, (((numeral_numeral @ nat) @ one2) = (one_one @ nat))).
% 9.43/3.44 thf(fact_1185_execute__bind__case, axiom, (! [A:$tType,B:$tType,F:(heap_Time_Heap @ B),G:(B > (heap_Time_Heap @ A)),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ ((((heap_Time_bind @ B) @ A) @ F) @ G)) @ H3) = (((((case_option @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ ((((product_case_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [X3:B]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H7:(heap_ext @ product_unit),N5:nat]: ((((heap_Time_timeFrame @ A) @ N5) @ (((heap_Time_execute @ A) @ (G @ X3)) @ H7))))))))) @ (((heap_Time_execute @ B) @ F) @ H3)))))).
% 9.43/3.44 thf(fact_1186_Sum__Icc__int, axiom, (! [M2:int,N:int]: (((((ord_less_eq @ int) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ int) @ int) @ (^ [X3:int]: (X3))) @ (((set_or1337092689740270186AtMost @ int) @ M2) @ N)) = (((divide_divide @ int) @ (((minus_minus @ int) @ (((times_times @ int) @ N) @ (((plus_plus @ int) @ N) @ (one_one @ int)))) @ (((times_times @ int) @ M2) @ (((minus_minus @ int) @ M2) @ (one_one @ int))))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))).
% 9.43/3.44 thf(fact_1187_Heap__Time__Monad_Obind__def, axiom, (! [B:$tType,A:$tType]: ((((heap_Time_bind @ A) @ B) = (^ [F2:(heap_Time_Heap @ A),G2:(A > (heap_Time_Heap @ B))]: (((heap_Time_Heap2 @ B) @ (^ [H:(heap_ext @ product_unit)]: ((((((case_option @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (none @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ ((((product_case_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [R3:A]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [H7:(heap_ext @ product_unit),N5:nat]: ((((heap_Time_timeFrame @ B) @ N5) @ (((heap_Time_execute @ B) @ (G2 @ R3)) @ H7))))))))) @ (((heap_Time_execute @ A) @ F2) @ H))))))))))).
% 9.43/3.44 thf(fact_1188_zless__iff__Suc__zadd, axiom, ((ord_less @ int) = (^ [W2:int,Z5:int]: ((? [N5:nat]: ((Z5 = (((plus_plus @ int) @ W2) @ ((semiring_1_of_nat @ int) @ (suc @ N5)))))))))).
% 9.43/3.44 thf(fact_1189_int__ops_I4_J, axiom, (! [A3:nat]: ((((semiring_1_of_nat @ int) @ (suc @ A3)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1190_int__Suc, axiom, (! [N:nat]: ((((semiring_1_of_nat @ int) @ (suc @ N)) = (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ N)) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1191_int__bit__induct, axiom, (! [P:(int > $o),K:int]: (((P @ (zero_zero @ int)) => ((P @ ((uminus_uminus @ int) @ (one_one @ int))) => ((! [K5:int]: (((P @ K5) => ((K5 != (zero_zero @ int)) => (P @ (((times_times @ int) @ K5) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))) => ((! [K5:int]: (((P @ K5) => ((K5 != ((uminus_uminus @ int) @ (one_one @ int))) => (P @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ K5) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))) => (P @ K)))))))).
% 9.43/3.44 thf(fact_1192_uminus__numeral__One, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((uminus_uminus @ A) @ ((numeral_numeral @ A) @ one2)) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_1193_int__def, axiom, ((semiring_1_of_nat @ int) = (^ [N5:nat]: ((abs_Integ @ ((((product_Pair @ nat) @ nat) @ N5) @ (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_1194_num__of__nat__One, axiom, (! [N:nat]: (((((ord_less_eq @ nat) @ N) @ (one_one @ nat)) => ((num_of_nat @ N) = one2))))).
% 9.43/3.44 thf(fact_1195_neg__zdiv__mult__2, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ A3) @ (zero_zero @ int)) => ((((divide_divide @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3)) = (((divide_divide @ int) @ (((plus_plus @ int) @ B2) @ (one_one @ int))) @ A3)))))).
% 9.43/3.44 thf(fact_1196_pos__zdiv__mult__2, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A3) => ((((divide_divide @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3)) = (((divide_divide @ int) @ B2) @ A3)))))).
% 9.43/3.44 thf(fact_1197_pos__zmod__mult__2, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ A3) => ((((modulo_modulo @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3)) = (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((modulo_modulo @ int) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_1198_mod__mult2__eq_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,M2:nat,N:nat]: (((((modulo_modulo @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ ((semiring_1_of_nat @ A) @ N))) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ (((modulo_modulo @ A) @ (((divide_divide @ A) @ A3) @ ((semiring_1_of_nat @ A) @ M2))) @ ((semiring_1_of_nat @ A) @ N)))) @ (((modulo_modulo @ A) @ A3) @ ((semiring_1_of_nat @ A) @ M2)))))))))).
% 9.43/3.44 thf(fact_1199_numeral__num__of__nat__unfold, axiom, (! [A:$tType]: (((semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => (((numeral_numeral @ A) @ (num_of_nat @ N)) = (one_one @ A))) & ((N != (zero_zero @ nat)) => (((numeral_numeral @ A) @ (num_of_nat @ N)) = ((semiring_1_of_nat @ A) @ N)))))))))).
% 9.43/3.44 thf(fact_1200_double__arith__series, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,D3:A,N:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ D3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) = (((times_times @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ D3)))))))))).
% 9.43/3.44 thf(fact_1201_double__gauss__sum, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1202_neg__zmod__mult__2, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ int) @ A3) @ (zero_zero @ int)) => ((((modulo_modulo @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3)) = (((minus_minus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((modulo_modulo @ int) @ (((plus_plus @ int) @ B2) @ (one_one @ int))) @ A3))) @ (one_one @ int))))))).
% 9.43/3.44 thf(fact_1203_mult__2, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [Z2:A]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Z2) = (((plus_plus @ A) @ Z2) @ Z2)))))))).
% 9.43/3.44 thf(fact_1204_mult__2__right, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [Z2:A]: (((((times_times @ A) @ Z2) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((plus_plus @ A) @ Z2) @ Z2)))))))).
% 9.43/3.44 thf(fact_1205_left__add__twice, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [A3:A,B2:A]: (((((plus_plus @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ B2)))))))).
% 9.43/3.44 thf(fact_1206_pos__eucl__rel__int__mult__2, axiom, (! [B2:int,A3:int,Q3:int,R:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ B2) => ((((eucl_rel_int @ A3) @ B2) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => (((eucl_rel_int @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2)) @ ((((product_Pair @ int) @ int) @ Q3) @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ R))))))))).
% 9.43/3.44 thf(fact_1207_one__power2, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((((power_power @ A) @ (one_one @ A)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1208_nat__1__add__1, axiom, ((((plus_plus @ nat) @ (one_one @ nat)) @ (one_one @ nat)) = ((numeral_numeral @ nat) @ (bit0 @ one2)))).
% 9.43/3.44 thf(fact_1209_Suc__nat__number__of__add, axiom, (! [V:num,N:nat]: (((suc @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V)) @ N)) = (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (((plus_plus @ num) @ V) @ one2))) @ N))))).
% 9.43/3.44 thf(fact_1210_double__gauss__sum__from__Suc__0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: (((((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N))) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1211_arith__series, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A,D3:A,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A3) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ I3)) @ D3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((divide_divide @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A))) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ D3)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1212_gauss__sum, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((divide_divide @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1213_nat__approx__posE, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [E2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ E2) => (~ ((! [N3:nat]: ((~ ((((ord_less @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((semiring_1_of_nat @ A) @ (suc @ N3)))) @ E2)))))))))))))).
% 9.43/3.44 thf(fact_1214_neg__eucl__rel__int__mult__2, axiom, (! [B2:int,A3:int,Q3:int,R:int]: (((((ord_less_eq @ int) @ B2) @ (zero_zero @ int)) => ((((eucl_rel_int @ (((plus_plus @ int) @ A3) @ (one_one @ int))) @ B2) @ ((((product_Pair @ int) @ int) @ Q3) @ R)) => (((eucl_rel_int @ (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ A3))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ B2)) @ ((((product_Pair @ int) @ int) @ Q3) @ (((minus_minus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ R)) @ (one_one @ int))))))))).
% 9.43/3.44 thf(fact_1215_gauss__sum__from__Suc__0, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (semiring_1_of_nat @ A)) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = (((divide_divide @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A)))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1216_exp__add__not__zero__imp__right, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M2:nat,N:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M2) @ N)) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1217_exp__add__not__zero__imp__left, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M2:nat,N:nat]: ((((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M2) @ N)) != (zero_zero @ A)) => ((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2) != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1218_sum__power2__eq__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: ((((((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1219_div__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,M2:nat,N:nat]: (((((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((divide_divide @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M2) @ N)))))))))).
% 9.43/3.44 thf(fact_1220_power2__eq__1__iff, axiom, (! [A:$tType]: (((ring_15535105094025558882visors @ A) => (! [A3:A]: ((((((power_power @ A) @ A3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ A)) = ((A3 = (one_one @ A)) | (A3 = ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1221_plus__int_Oabs__eq, axiom, (! [Xa:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((plus_plus @ int) @ (abs_Integ @ Xa)) @ (abs_Integ @ X)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ U2)) @ (((plus_plus @ nat) @ Y3) @ V2)))))))) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1222_abs__square__eq__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: ((((((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (one_one @ A)) = (((abs_abs @ A) @ X) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1223_minus__int_Oabs__eq, axiom, (! [Xa:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((minus_minus @ int) @ (abs_Integ @ Xa)) @ (abs_Integ @ X)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ Y3) @ U2)))))))) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1224_nat__induct2, axiom, (! [P:(nat > $o),N:nat]: (((P @ (zero_zero @ nat)) => ((P @ (one_one @ nat)) => ((! [N3:nat]: (((P @ N3) => (P @ (((plus_plus @ nat) @ N3) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))) => (P @ N))))))).
% 9.43/3.44 thf(fact_1225_times__int_Oabs__eq, axiom, (! [Xa:((product_prod @ nat) @ nat),X:((product_prod @ nat) @ nat)]: (((((times_times @ int) @ (abs_Integ @ Xa)) @ (abs_Integ @ X)) = (abs_Integ @ ((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ U2)) @ (((times_times @ nat) @ Y3) @ V2))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ V2)) @ (((times_times @ nat) @ Y3) @ U2))))))))) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1226_sum__power2__le__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (zero_zero @ A)) = ((X = (zero_zero @ A)) & (Y = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1227_sum__power2__ge__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))).
% 9.43/3.44 thf(fact_1228_bits__stable__imp__add__self, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: ((((((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A3) => ((((plus_plus @ A) @ A3) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1229_not__sum__power2__lt__zero, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: ((~ ((((ord_less @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1230_sum__power2__gt__zero__iff, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) = ((X != (zero_zero @ A)) | (Y != (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1231_power2__sum, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [X:A,Y:A]: (((((power_power @ A) @ (((plus_plus @ A) @ X) @ Y)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X)) @ Y))))))))).
% 9.43/3.44 thf(fact_1232_square__le__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X) => ((((ord_less_eq @ A) @ X) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1233_abs__square__le__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((ord_less_eq @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)) = (((ord_less_eq @ A) @ ((abs_abs @ A) @ X)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1234_abs__square__less__1, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [X:A]: (((((ord_less @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (one_one @ A)) = (((ord_less @ A) @ ((abs_abs @ A) @ X)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1235_div__exp__mod__exp__eq, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat,M2:nat]: (((((modulo_modulo @ A) @ (((divide_divide @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) = (((divide_divide @ A) @ (((modulo_modulo @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ N) @ M2)))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))))))))).
% 9.43/3.44 thf(fact_1236_ex__power__ivl2, axiom, (! [B2:nat,K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K) => (? [N3:nat]: (((((ord_less @ nat) @ (((power_power @ nat) @ B2) @ N3)) @ K) & (((ord_less_eq @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N3) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1237_ex__power__ivl1, axiom, (! [B2:nat,K:nat]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ B2) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ K) => (? [N3:nat]: (((((ord_less_eq @ nat) @ (((power_power @ nat) @ B2) @ N3)) @ K) & (((ord_less @ nat) @ K) @ (((power_power @ nat) @ B2) @ (((plus_plus @ nat) @ N3) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1238_power2__diff, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [X:A,Y:A]: (((((power_power @ A) @ (((minus_minus @ A) @ X) @ Y)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((minus_minus @ A) @ (((plus_plus @ A) @ (((power_power @ A) @ X) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((power_power @ A) @ Y) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ X)) @ Y))))))))).
% 9.43/3.44 thf(fact_1239_power__minus1__odd, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: (((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1240_mask__eq__sum__exp, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [N:nat]: (((((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ (one_one @ A)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((collect @ nat) @ (^ [Q5:nat]: ((((ord_less @ nat) @ Q5) @ N)))))))))))).
% 9.43/3.44 thf(fact_1241_sum_Oin__pairs, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1242_gauss__sum__nat, axiom, (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((divide_divide @ nat) @ (((times_times @ nat) @ N) @ (suc @ N))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1243_divmod__step__eq, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [L:num,R:A,Q3:A]: ((((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L)) @ R) => ((((unique1321980374590559556d_step @ A) @ L) @ ((((product_Pair @ A) @ A) @ Q3) @ R)) = ((((product_Pair @ A) @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q3)) @ (one_one @ A))) @ (((minus_minus @ A) @ R) @ ((numeral_numeral @ A) @ L))))) & ((~ ((((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L)) @ R))) => ((((unique1321980374590559556d_step @ A) @ L) @ ((((product_Pair @ A) @ A) @ Q3) @ R)) = ((((product_Pair @ A) @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q3)) @ R)))))))))).
% 9.43/3.44 thf(fact_1244_divmod__step__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique1321980374590559556d_step @ A) = (^ [L3:num]: (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q5:A,R3:A]: (((((if @ ((product_prod @ A) @ A)) @ (((ord_less_eq @ A) @ ((numeral_numeral @ A) @ L3)) @ R3)) @ ((((product_Pair @ A) @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q5)) @ (one_one @ A))) @ (((minus_minus @ A) @ R3) @ ((numeral_numeral @ A) @ L3)))) @ ((((product_Pair @ A) @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Q5)) @ R3)))))))))))).
% 9.43/3.44 thf(fact_1245_divmod__step__nat__def, axiom, ((unique1321980374590559556d_step @ nat) = (^ [L3:num]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [Q5:nat,R3:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ (((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ L3)) @ R3)) @ ((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Q5)) @ (one_one @ nat))) @ (((minus_minus @ nat) @ R3) @ ((numeral_numeral @ nat) @ L3)))) @ ((((product_Pair @ nat) @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ Q5)) @ R3))))))))).
% 9.43/3.44 thf(fact_1246_divmod__step__int__def, axiom, ((unique1321980374590559556d_step @ int) = (^ [L3:num]: (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [Q5:int,R3:int]: (((((if @ ((product_prod @ int) @ int)) @ (((ord_less_eq @ int) @ ((numeral_numeral @ int) @ L3)) @ R3)) @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Q5)) @ (one_one @ int))) @ (((minus_minus @ int) @ R3) @ ((numeral_numeral @ int) @ L3)))) @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Q5)) @ R3))))))))).
% 9.43/3.44 thf(fact_1247_set__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5668285175392031749et_bit @ A) @ (zero_zero @ nat)) @ A3) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1248_mlex__eq, axiom, (! [A:$tType]: (((mlex_prod @ A) = (^ [F2:(A > nat),R6:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X3:A,Y3:A]: (((((ord_less @ nat) @ (F2 @ X3)) @ (F2 @ Y3)) | ((((ord_less_eq @ nat) @ (F2 @ X3)) @ (F2 @ Y3)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R6))))))))))))).
% 9.43/3.44 thf(fact_1249_unset__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se2638667681897837118et_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se2638667681897837118et_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))).
% 9.43/3.44 thf(fact_1250_flip__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se8732182000553998342ip_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se8732182000553998342ip_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))).
% 9.43/3.44 thf(fact_1251_set__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se5668285175392031749et_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5668285175392031749et_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))).
% 9.43/3.44 thf(fact_1252_split__part, axiom, (! [B:$tType,A:$tType,P:$o,Q:(A > (B > $o))]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [A7:A,B5:B]: ((P & ((Q @ A7) @ B5))))) = (^ [Ab:((product_prod @ A) @ B)]: ((P & (((((product_case_prod @ A) @ B) @ $o) @ Q) @ Ab)))))))).
% 9.43/3.44 thf(fact_1253_prod_Odisc__eq__case, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_case_prod @ A) @ B) @ $o) @ (^ [Uu:A,Uv:B]: ($true))) @ Prod)))).
% 9.43/3.44 thf(fact_1254_divmod__step__integer__def, axiom, ((unique1321980374590559556d_step @ code_integer) = (^ [L3:num]: (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [Q5:code_integer,R3:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less_eq @ code_integer) @ ((numeral_numeral @ code_integer) @ L3)) @ R3)) @ ((((product_Pair @ code_integer) @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ Q5)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ R3) @ ((numeral_numeral @ code_integer) @ L3)))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ Q5)) @ R3))))))))).
% 9.43/3.44 thf(fact_1255_signed__take__bit__rec, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N5:nat,A7:A]: (((((if @ A) @ (N5 = (zero_zero @ nat))) @ ((uminus_uminus @ A) @ (((modulo_modulo @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (((plus_plus @ A) @ (((modulo_modulo @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_ri4674362597316999326ke_bit @ A) @ (((minus_minus @ nat) @ N5) @ (one_one @ nat))) @ (((divide_divide @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))))).
% 9.43/3.44 thf(fact_1256_round__unique, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,Y:int]: (((((ord_less @ A) @ (((minus_minus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ Y)) => ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ Y)) @ (((plus_plus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) => (((archimedean_round @ A) @ X) = Y))))))))).
% 9.43/3.44 thf(fact_1257_rel__of__def, axiom, (! [B:$tType,A:$tType]: ((((rel_of @ A) @ B) = (^ [M:(A > (option @ B)),P4:(((product_prod @ A) @ B) > $o)]: (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [K4:A,V2:B]: ((((M @ K4) = ((some @ B) @ V2)) & (P4 @ ((((product_Pair @ A) @ B) @ K4) @ V2))))))))))))).
% 9.43/3.44 thf(fact_1258_dbl__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 9.43/3.44 thf(fact_1259_round__altdef, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_round @ A) = (^ [X3:A]: (((((if @ int) @ (((ord_less_eq @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((archimedean_frac @ A) @ X3))) @ ((archimedean_ceiling @ A) @ X3)) @ ((archim6421214686448440834_floor @ A) @ X3))))))))).
% 9.43/3.44 thf(fact_1260_round__unique_H, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A,N:int]: (((((ord_less @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ X) @ ((ring_1_of_int @ A) @ N)))) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) => (((archimedean_round @ A) @ X) = N)))))))).
% 9.43/3.44 thf(fact_1261_of__int__round__abs__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ ((abs_abs @ A) @ (((minus_minus @ A) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X))) @ X))) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1262_divmod__algorithm__code_I5_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M2:num,N:num]: (((((unique8689654367752047608divmod @ A) @ (bit0 @ M2)) @ (bit0 @ N)) = (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q5:A,R3:A]: (((((product_Pair @ A) @ A) @ Q5) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ R3))))) @ (((unique8689654367752047608divmod @ A) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1263_signed__take__bit__of__minus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ N) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1264_signed__take__bit__Suc__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_ri4674362597316999326ke_bit @ A) @ (suc @ N)) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1265_signed__take__bit__numeral__of__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [K:num]: (((((bit_ri4674362597316999326ke_bit @ A) @ ((numeral_numeral @ nat) @ K)) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1266_round__1, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (((archimedean_round @ A) @ (one_one @ A)) = (one_one @ int)))))).
% 9.43/3.44 thf(fact_1267_divmod__algorithm__code_I2_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M2:num]: (((((unique8689654367752047608divmod @ A) @ M2) @ one2) = ((((product_Pair @ A) @ A) @ ((numeral_numeral @ A) @ M2)) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1268_dbl__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl @ A) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1269_divmod__algorithm__code_I3_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((unique8689654367752047608divmod @ A) @ one2) @ (bit0 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ one2))))))))).
% 9.43/3.44 thf(fact_1270_less__eq__integer__code_I1_J, axiom, (((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))).
% 9.43/3.44 thf(fact_1271_sgn__integer__code, axiom, ((sgn_sgn @ code_integer) = (^ [K4:code_integer]: (((((if @ code_integer) @ (K4 = (zero_zero @ code_integer))) @ (zero_zero @ code_integer)) @ ((((if @ code_integer) @ (((ord_less @ code_integer) @ K4) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ code_integer) @ (one_one @ code_integer))) @ (one_one @ code_integer))))))).
% 9.43/3.44 thf(fact_1272_exhaustive__integer_H_Ocases, axiom, (! [X:((product_prod @ (code_integer > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer))]: ((~ ((! [F4:(code_integer > (option @ ((product_prod @ $o) @ (list @ code_term)))),D2:code_integer,I2:code_integer]: ((X != ((((product_Pair @ (code_integer > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer)) @ F4) @ ((((product_Pair @ code_integer) @ code_integer) @ D2) @ I2)))))))))).
% 9.43/3.44 thf(fact_1273_full__exhaustive__integer_H_Ocases, axiom, (! [X:((product_prod @ (((product_prod @ code_integer) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer))]: ((~ ((! [F4:(((product_prod @ code_integer) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term)))),D2:code_integer,I2:code_integer]: ((X != ((((product_Pair @ (((product_prod @ code_integer) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_integer) @ code_integer)) @ F4) @ ((((product_Pair @ code_integer) @ code_integer) @ D2) @ I2)))))))))).
% 9.43/3.44 thf(fact_1274_times__integer__code_I1_J, axiom, (! [K:code_integer]: (((((times_times @ code_integer) @ K) @ (zero_zero @ code_integer)) = (zero_zero @ code_integer))))).
% 9.43/3.44 thf(fact_1275_times__integer__code_I2_J, axiom, (! [L:code_integer]: (((((times_times @ code_integer) @ (zero_zero @ code_integer)) @ L) = (zero_zero @ code_integer))))).
% 9.43/3.44 thf(fact_1276_minus__integer__code_I1_J, axiom, (! [K:code_integer]: (((((minus_minus @ code_integer) @ K) @ (zero_zero @ code_integer)) = K)))).
% 9.43/3.44 thf(fact_1277_plus__integer__code_I2_J, axiom, (! [L:code_integer]: (((((plus_plus @ code_integer) @ (zero_zero @ code_integer)) @ L) = L)))).
% 9.43/3.44 thf(fact_1278_plus__integer__code_I1_J, axiom, (! [K:code_integer]: (((((plus_plus @ code_integer) @ K) @ (zero_zero @ code_integer)) = K)))).
% 9.43/3.44 thf(fact_1279_minus__integer__code_I2_J, axiom, (! [L:code_integer]: (((((minus_minus @ code_integer) @ (zero_zero @ code_integer)) @ L) = ((uminus_uminus @ code_integer) @ L))))).
% 9.43/3.44 thf(fact_1280_divmod__integer_H__def, axiom, ((unique8689654367752047608divmod @ code_integer) = (^ [M:num,N5:num]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ ((numeral_numeral @ code_integer) @ M)) @ ((numeral_numeral @ code_integer) @ N5))) @ (((modulo_modulo @ code_integer) @ ((numeral_numeral @ code_integer) @ M)) @ ((numeral_numeral @ code_integer) @ N5))))))).
% 9.43/3.44 thf(fact_1281_signed__take__bit__add, axiom, (! [N:nat,K:int,L:int]: (((((bit_ri4674362597316999326ke_bit @ int) @ N) @ (((plus_plus @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ L))) = (((bit_ri4674362597316999326ke_bit @ int) @ N) @ (((plus_plus @ int) @ K) @ L)))))).
% 9.43/3.44 thf(fact_1282_dbl__def, axiom, (! [A:$tType]: (((neg_numeral @ A) => ((neg_numeral_dbl @ A) = (^ [X3:A]: ((((plus_plus @ A) @ X3) @ X3)))))))).
% 9.43/3.44 thf(fact_1283_divmod__int__def, axiom, ((unique8689654367752047608divmod @ int) = (^ [M:num,N5:num]: (((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N5))) @ (((modulo_modulo @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N5))))))).
% 9.43/3.44 thf(fact_1284_divmod__def, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique8689654367752047608divmod @ A) = (^ [M:num,N5:num]: (((((product_Pair @ A) @ A) @ (((divide_divide @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N5))) @ (((modulo_modulo @ A) @ ((numeral_numeral @ A) @ M)) @ ((numeral_numeral @ A) @ N5)))))))))).
% 9.43/3.44 thf(fact_1285_divmod_H__nat__def, axiom, ((unique8689654367752047608divmod @ nat) = (^ [M:num,N5:num]: (((((product_Pair @ nat) @ nat) @ (((divide_divide @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N5))) @ (((modulo_modulo @ nat) @ ((numeral_numeral @ nat) @ M)) @ ((numeral_numeral @ nat) @ N5))))))).
% 9.43/3.44 thf(fact_1286_signed__take__bit__int__greater__eq, axiom, (! [K:int,N:nat]: (((((ord_less @ int) @ K) @ ((uminus_uminus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (suc @ N)))) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ K)))))).
% 9.43/3.44 thf(fact_1287_divmod__divmod__step, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => ((unique8689654367752047608divmod @ A) = (^ [M:num,N5:num]: (((((if @ ((product_prod @ A) @ A)) @ (((ord_less @ num) @ M) @ N5)) @ ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ M))) @ (((unique1321980374590559556d_step @ A) @ N5) @ (((unique8689654367752047608divmod @ A) @ M) @ (bit0 @ N5))))))))))).
% 9.43/3.44 thf(fact_1288_round__def, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => ((archimedean_round @ A) = (^ [X3:A]: (((archim6421214686448440834_floor @ A) @ (((plus_plus @ A) @ X3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1289_signed__take__bit__Suc, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat,A3:A]: (((((bit_ri4674362597316999326ke_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_ri4674362597316999326ke_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))))).
% 9.43/3.44 thf(fact_1290_of__int__round__le, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X))) @ (((plus_plus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 9.43/3.44 thf(fact_1291_of__int__round__ge, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less_eq @ A) @ (((minus_minus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X))))))))).
% 9.43/3.44 thf(fact_1292_of__int__round__gt, axiom, (! [A:$tType]: (((archim2362893244070406136eiling @ A) => (! [X:A]: ((((ord_less @ A) @ (((minus_minus @ A) @ X) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((ring_1_of_int @ A) @ ((archimedean_round @ A) @ X))))))))).
% 9.43/3.44 thf(fact_1293_integer__of__int__code, axiom, (code_integer_of_int = (^ [K4:int]: (((((if @ code_integer) @ (((ord_less @ int) @ K4) @ (zero_zero @ int))) @ ((uminus_uminus @ code_integer) @ (code_integer_of_int @ ((uminus_uminus @ int) @ K4)))) @ ((((if @ code_integer) @ (K4 = (zero_zero @ int))) @ (zero_zero @ code_integer)) @ ((((if @ code_integer) @ ((((modulo_modulo @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2))) = (zero_zero @ int))) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ (code_integer_of_int @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ (code_integer_of_int @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (one_one @ code_integer))))))))).
% 9.43/3.44 thf(fact_1294_divmod__algorithm__code_I6_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M2:num,N:num]: (((((unique8689654367752047608divmod @ A) @ (bit1 @ M2)) @ (bit0 @ N)) = (((((product_case_prod @ A) @ A) @ ((product_prod @ A) @ A)) @ (^ [Q5:A,R3:A]: (((((product_Pair @ A) @ A) @ Q5) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ R3)) @ (one_one @ A)))))) @ (((unique8689654367752047608divmod @ A) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1295_of__nat__code, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N5:nat]: (((((semiri8178284476397505188at_aux @ A) @ (^ [I3:A]: ((((plus_plus @ A) @ I3) @ (one_one @ A))))) @ N5) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1296_signed__take__bit__Suc__minus__bit1, axiom, (! [N:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) @ (one_one @ int)))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1297_concat__bit__Suc, axiom, (! [N:nat,K:int,L:int]: (((((bit_concat_bit @ (suc @ N)) @ K) @ L) = (((plus_plus @ int) @ (((modulo_modulo @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_concat_bit @ N) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ L))))))).
% 9.43/3.44 thf(fact_1298_even__succ__mod__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((modulo_modulo @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((plus_plus @ A) @ (one_one @ A)) @ (((modulo_modulo @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)))))))))))).
% 9.43/3.44 thf(fact_1299_even__succ__div__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = (((divide_divide @ A) @ A3) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))))))))))).
% 9.43/3.44 thf(fact_1300_take__bit__rec, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se2584673776208193580ke_bit @ A) = (^ [N5:nat,A7:A]: (((((if @ A) @ (N5 = (zero_zero @ nat))) @ (zero_zero @ A)) @ (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (((minus_minus @ nat) @ N5) @ (one_one @ nat))) @ (((divide_divide @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((modulo_modulo @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1301_nat__dvd__1__iff__1, axiom, (! [M2:nat]: (((((dvd_dvd @ nat) @ M2) @ (one_one @ nat)) = (M2 = (one_one @ nat)))))).
% 9.43/3.44 thf(fact_1302_integer__of__int__eq__of__int, axiom, (code_integer_of_int = (ring_1_of_int @ code_integer))).
% 9.43/3.44 thf(fact_1303_dvd__add__triv__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ A3)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_1304_dvd__add__triv__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ A3) @ B2)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_1305_dvd__add__times__triv__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ (((times_times @ A) @ C2) @ A3))) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_1306_dvd__add__times__triv__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ (((times_times @ A) @ C2) @ A3)) @ B2)) = (((dvd_dvd @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_1307_unit__prod, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1308_div__add, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2))))))))))).
% 9.43/3.44 thf(fact_1309_unit__div__1__div__1, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((divide_divide @ A) @ (one_one @ A)) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) = A3)))))))).
% 9.43/3.44 thf(fact_1310_unit__div__1__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1311_unit__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1312_take__bit__Suc__1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1313_take__bit__numeral__1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [L:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ L)) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1314_semiring__norm_I9_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ num) @ (bit1 @ M2)) @ (bit0 @ N)) = (bit1 @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_1315_semiring__norm_I7_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ num) @ (bit0 @ M2)) @ (bit1 @ N)) = (bit1 @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_1316_semiring__norm_I16_J, axiom, (! [M2:num,N:num]: (((((times_times @ num) @ (bit1 @ M2)) @ (bit1 @ N)) = (bit1 @ (((plus_plus @ num) @ (((plus_plus @ num) @ M2) @ N)) @ (bit0 @ (((times_times @ num) @ M2) @ N)))))))).
% 9.43/3.44 thf(fact_1317_unit__mult__div__div, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((times_times @ A) @ B2) @ (((divide_divide @ A) @ (one_one @ A)) @ A3)) = (((divide_divide @ A) @ B2) @ A3))))))))).
% 9.43/3.44 thf(fact_1318_unit__div__mult__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((times_times @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ A3) = B2)))))))).
% 9.43/3.44 thf(fact_1319_take__bit__of__1__eq__0__iff, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat]: ((((((bit_se2584673776208193580ke_bit @ A) @ N) @ (one_one @ A)) = (zero_zero @ A)) = (N = (zero_zero @ nat))))))))).
% 9.43/3.44 thf(fact_1320_semiring__norm_I10_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ num) @ (bit1 @ M2)) @ (bit1 @ N)) = (bit0 @ (((plus_plus @ num) @ (((plus_plus @ num) @ M2) @ N)) @ one2)))))).
% 9.43/3.44 thf(fact_1321_semiring__norm_I8_J, axiom, (! [M2:num]: (((((plus_plus @ num) @ (bit1 @ M2)) @ one2) = (bit0 @ (((plus_plus @ num) @ M2) @ one2)))))).
% 9.43/3.44 thf(fact_1322_semiring__norm_I5_J, axiom, (! [M2:num]: (((((plus_plus @ num) @ (bit0 @ M2)) @ one2) = (bit1 @ M2))))).
% 9.43/3.44 thf(fact_1323_semiring__norm_I4_J, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ (bit1 @ N)) = (bit0 @ (((plus_plus @ num) @ N) @ one2)))))).
% 9.43/3.44 thf(fact_1324_semiring__norm_I3_J, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ (bit0 @ N)) = (bit1 @ N))))).
% 9.43/3.44 thf(fact_1325_odd__add, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A,B2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A3) @ B2)))) = ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) != (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))))))))))).
% 9.43/3.44 thf(fact_1326_even__add, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A3) @ B2)) = ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) = (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))))))))).
% 9.43/3.44 thf(fact_1327_dbl__inc__simps_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_inc @ A) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ one2))))))).
% 9.43/3.44 thf(fact_1328_even__plus__one__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) = (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))).
% 9.43/3.44 thf(fact_1329_even__diff, axiom, (! [A:$tType]: (((ring_parity @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ A) @ A3) @ B2)) = (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_1330_neg__one__even__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1331_neg__one__odd__power, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = ((uminus_uminus @ A) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1332_even__diff__nat, axiom, (! [M2:nat,N:nat]: (((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ M2) @ N)) = ((((ord_less @ nat) @ M2) @ N) | (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((plus_plus @ nat) @ M2) @ N))))))).
% 9.43/3.44 thf(fact_1333_divmod__algorithm__code_I4_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((unique8689654367752047608divmod @ A) @ one2) @ (bit1 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ one2))))))))).
% 9.43/3.44 thf(fact_1334_mod__Suc__eq__mod__add3, axiom, (! [M2:nat,N:nat]: (((((modulo_modulo @ nat) @ M2) @ (suc @ (suc @ (suc @ N)))) = (((modulo_modulo @ nat) @ M2) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N)))))).
% 9.43/3.44 thf(fact_1335_Suc__mod__eq__add3__mod__numeral, axiom, (! [M2:nat,V:num]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (suc @ M2)))) @ ((numeral_numeral @ nat) @ V)) = (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M2)) @ ((numeral_numeral @ nat) @ V)))))).
% 9.43/3.44 thf(fact_1336_div__Suc__eq__div__add3, axiom, (! [M2:nat,N:nat]: (((((divide_divide @ nat) @ M2) @ (suc @ (suc @ (suc @ N)))) = (((divide_divide @ nat) @ M2) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N)))))).
% 9.43/3.44 thf(fact_1337_Suc__div__eq__add3__div__numeral, axiom, (! [M2:nat,V:num]: (((((divide_divide @ nat) @ (suc @ (suc @ (suc @ M2)))) @ ((numeral_numeral @ nat) @ V)) = (((divide_divide @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M2)) @ ((numeral_numeral @ nat) @ V)))))).
% 9.43/3.44 thf(fact_1338_dbl__dec__simps_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (((neg_numeral_dbl_dec @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit1 @ one2)))))))).
% 9.43/3.44 thf(fact_1339_even__succ__div__2, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 9.43/3.44 thf(fact_1340_odd__succ__div__two, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1341_even__succ__div__two, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 9.43/3.44 thf(fact_1342_odd__two__times__div__two__nat, axiom, (! [N:nat]: (((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((divide_divide @ nat) @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) = (((minus_minus @ nat) @ N) @ (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1343_odd__two__times__div__two__succ, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (one_one @ A)) = A3)))))))).
% 9.43/3.44 thf(fact_1344_semiring__parity__class_Oeven__mask__iff, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ (one_one @ A))) = (N = (zero_zero @ nat))))))))).
% 9.43/3.44 thf(fact_1345_zmod__numeral__Bit1, axiom, (! [V:num,W:num]: (((((modulo_modulo @ int) @ ((numeral_numeral @ int) @ (bit1 @ V))) @ ((numeral_numeral @ int) @ (bit0 @ W))) = (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((modulo_modulo @ int) @ ((numeral_numeral @ int) @ V)) @ ((numeral_numeral @ int) @ W)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1346_divmod__algorithm__code_I8_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M2:num,N:num]: ((((((ord_less @ num) @ M2) @ N) => ((((unique8689654367752047608divmod @ A) @ (bit1 @ M2)) @ (bit1 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ (bit1 @ M2))))) & ((~ ((((ord_less @ num) @ M2) @ N))) => ((((unique8689654367752047608divmod @ A) @ (bit1 @ M2)) @ (bit1 @ N)) = (((unique1321980374590559556d_step @ A) @ (bit1 @ N)) @ (((unique8689654367752047608divmod @ A) @ (bit1 @ M2)) @ (bit0 @ (bit1 @ N))))))))))))).
% 9.43/3.44 thf(fact_1347_divmod__algorithm__code_I7_J, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [M2:num,N:num]: ((((((ord_less_eq @ num) @ M2) @ N) => ((((unique8689654367752047608divmod @ A) @ (bit0 @ M2)) @ (bit1 @ N)) = ((((product_Pair @ A) @ A) @ (zero_zero @ A)) @ ((numeral_numeral @ A) @ (bit0 @ M2))))) & ((~ ((((ord_less_eq @ num) @ M2) @ N))) => ((((unique8689654367752047608divmod @ A) @ (bit0 @ M2)) @ (bit1 @ N)) = (((unique1321980374590559556d_step @ A) @ (bit1 @ N)) @ (((unique8689654367752047608divmod @ A) @ (bit0 @ M2)) @ (bit0 @ (bit1 @ N))))))))))))).
% 9.43/3.44 thf(fact_1348_signed__take__bit__Suc__bit1, axiom, (! [N:nat,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ (suc @ N)) @ ((numeral_numeral @ int) @ (bit1 @ K))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ N) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1349_uminus__integer__code_I1_J, axiom, (((uminus_uminus @ code_integer) @ (zero_zero @ code_integer)) = (zero_zero @ code_integer))).
% 9.43/3.44 thf(fact_1350_take__bit__integer_Oabs__eq, axiom, (! [Xa:nat,X:int]: (((((bit_se2584673776208193580ke_bit @ code_integer) @ Xa) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se2584673776208193580ke_bit @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1351_abs__integer__code, axiom, ((abs_abs @ code_integer) = (^ [K4:code_integer]: (((((if @ code_integer) @ (((ord_less @ code_integer) @ K4) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ code_integer) @ K4)) @ K4))))).
% 9.43/3.44 thf(fact_1352_less__integer__code_I1_J, axiom, (~ ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))))).
% 9.43/3.44 thf(fact_1353_unset__bit__integer_Oabs__eq, axiom, (! [Xa:nat,X:int]: (((((bit_se2638667681897837118et_bit @ code_integer) @ Xa) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se2638667681897837118et_bit @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1354_flip__bit__integer_Oabs__eq, axiom, (! [Xa:nat,X:int]: (((((bit_se8732182000553998342ip_bit @ code_integer) @ Xa) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se8732182000553998342ip_bit @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1355_set__bit__integer_Oabs__eq, axiom, (! [Xa:nat,X:int]: (((((bit_se5668285175392031749et_bit @ code_integer) @ Xa) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se5668285175392031749et_bit @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1356_take__bit__add, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,A3:A,B2:A]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ (((plus_plus @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3)) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ B2))) = (((bit_se2584673776208193580ke_bit @ A) @ N) @ (((plus_plus @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_1357_zero__integer__def, axiom, ((zero_zero @ code_integer) = (code_integer_of_int @ (zero_zero @ int)))).
% 9.43/3.44 thf(fact_1358_less__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((ord_less @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (((ord_less @ int) @ Xa) @ X))))).
% 9.43/3.44 thf(fact_1359_uminus__integer_Oabs__eq, axiom, (! [X:int]: ((((uminus_uminus @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((uminus_uminus @ int) @ X)))))).
% 9.43/3.44 thf(fact_1360_abs__integer_Oabs__eq, axiom, (! [X:int]: ((((abs_abs @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((abs_abs @ int) @ X)))))).
% 9.43/3.44 thf(fact_1361_dvd__add__right__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_1362_dvd__add__left__iff, axiom, (! [A:$tType]: (((comm_s4317794764714335236cancel @ A) => (! [A3:A,C2:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ C2) => ((((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_1363_dvd__add, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ A3) @ C2) => (((dvd_dvd @ A) @ A3) @ (((plus_plus @ A) @ B2) @ C2)))))))))).
% 9.43/3.44 thf(fact_1364_divide__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((divide_divide @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((divide_divide @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1365_dvd__unit__imp__unit, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ A3) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1366_unit__imp__dvd, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((dvd_dvd @ A) @ B2) @ A3)))))))).
% 9.43/3.44 thf(fact_1367_one__dvd, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ (one_one @ A)) @ A3))))))).
% 9.43/3.44 thf(fact_1368_modulo__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((modulo_modulo @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((modulo_modulo @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1369_sgn__integer_Oabs__eq, axiom, (! [X:int]: ((((sgn_sgn @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((sgn_sgn @ int) @ X)))))).
% 9.43/3.44 thf(fact_1370_concat__bit__assoc, axiom, (! [N:nat,K:int,M2:nat,L:int,R:int]: (((((bit_concat_bit @ N) @ K) @ (((bit_concat_bit @ M2) @ L) @ R)) = (((bit_concat_bit @ (((plus_plus @ nat) @ M2) @ N)) @ (((bit_concat_bit @ N) @ K) @ L)) @ R))))).
% 9.43/3.44 thf(fact_1371_not__is__unit__0, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (~ ((((dvd_dvd @ A) @ (zero_zero @ A)) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1372_pinf_I9_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D3:B,S:B]: ((? [Z3:B]: ((! [X7:B]: (((((ord_less @ B) @ Z3) @ X7) => ((((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)) = (((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)))))))))))))).
% 9.43/3.44 thf(fact_1373_pinf_I10_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D3:B,S:B]: ((? [Z3:B]: ((! [X7:B]: (((((ord_less @ B) @ Z3) @ X7) => ((~ ((((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)))) = (~ ((((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)))))))))))))))).
% 9.43/3.44 thf(fact_1374_minf_I9_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D3:B,S:B]: ((? [Z3:B]: ((! [X7:B]: (((((ord_less @ B) @ X7) @ Z3) => ((((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)) = (((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)))))))))))))).
% 9.43/3.44 thf(fact_1375_minf_I10_J, axiom, (! [B:$tType]: ((((plus @ B) & ((linorder @ B) & (dvd @ B))) => (! [D3:B,S:B]: ((? [Z3:B]: ((! [X7:B]: (((((ord_less @ B) @ X7) @ Z3) => ((~ ((((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)))) = (~ ((((dvd_dvd @ B) @ D3) @ (((plus_plus @ B) @ X7) @ S)))))))))))))))).
% 9.43/3.44 thf(fact_1376_unit__mult__right__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((((times_times @ A) @ B2) @ A3) = (((times_times @ A) @ C2) @ A3)) = (B2 = C2))))))))).
% 9.43/3.44 thf(fact_1377_unit__mult__left__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((((times_times @ A) @ A3) @ B2) = (((times_times @ A) @ A3) @ C2)) = (B2 = C2))))))))).
% 9.43/3.44 thf(fact_1378_mult__unit__dvd__iff_H, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (((dvd_dvd @ A) @ B2) @ C2))))))))).
% 9.43/3.44 thf(fact_1379_dvd__mult__unit__iff_H, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_1380_mult__unit__dvd__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_1381_dvd__mult__unit__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A3) @ (((times_times @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_1382_is__unit__mult__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ (((times_times @ A) @ A3) @ B2)) @ (one_one @ A)) = ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) & (((dvd_dvd @ A) @ B2) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1383_div__plus__div__distrib__dvd__left, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A3) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))).
% 9.43/3.44 thf(fact_1384_div__plus__div__distrib__dvd__right, axiom, (! [A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ B2) => ((((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ C2) = (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ C2)) @ (((divide_divide @ A) @ B2) @ C2)))))))))).
% 9.43/3.44 thf(fact_1385_unit__div__cancel, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((((divide_divide @ A) @ B2) @ A3) = (((divide_divide @ A) @ C2) @ A3)) = (B2 = C2))))))))).
% 9.43/3.44 thf(fact_1386_div__unit__dvd__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_1387_dvd__div__unit__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ A3) @ (((divide_divide @ A) @ C2) @ B2)) = (((dvd_dvd @ A) @ A3) @ C2))))))))).
% 9.43/3.44 thf(fact_1388_xor__num_Ocases, axiom, (! [X:((product_prod @ num) @ num)]: (((X != ((((product_Pair @ num) @ num) @ one2) @ one2)) => ((! [N3:num]: ((X != ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ N3))))) => ((! [N3:num]: ((X != ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ N3))))) => ((! [M5:num]: ((X != ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ one2)))) => ((! [M5:num,N3:num]: ((X != ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit0 @ N3))))) => ((! [M5:num,N3:num]: ((X != ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit1 @ N3))))) => ((! [M5:num]: ((X != ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ one2)))) => ((! [M5:num,N3:num]: ((X != ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit0 @ N3))))) => (~ ((! [M5:num,N3:num]: ((X != ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit1 @ N3)))))))))))))))))).
% 9.43/3.44 thf(fact_1389_plus__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((plus_plus @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((plus_plus @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1390_bezout__lemma__nat, axiom, (! [D3:nat,A3:nat,B2:nat,X:nat,Y:nat]: (((((dvd_dvd @ nat) @ D3) @ A3) => ((((dvd_dvd @ nat) @ D3) @ B2) => ((((((times_times @ nat) @ A3) @ X) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y)) @ D3)) | ((((times_times @ nat) @ B2) @ X) = (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ Y)) @ D3))) => (? [X2:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D3) @ A3) & ((((dvd_dvd @ nat) @ D3) @ (((plus_plus @ nat) @ A3) @ B2)) & (((((times_times @ nat) @ A3) @ X2) = (((plus_plus @ nat) @ (((times_times @ nat) @ (((plus_plus @ nat) @ A3) @ B2)) @ Y4)) @ D3)) | ((((times_times @ nat) @ (((plus_plus @ nat) @ A3) @ B2)) @ X2) = (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ Y4)) @ D3))))))))))))).
% 9.43/3.44 thf(fact_1391_bezout__add__nat, axiom, (! [A3:nat,B2:nat]: ((? [D2:nat,X2:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D2) @ A3) & ((((dvd_dvd @ nat) @ D2) @ B2) & (((((times_times @ nat) @ A3) @ X2) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ D2)) | ((((times_times @ nat) @ B2) @ X2) = (((plus_plus @ nat) @ (((times_times @ nat) @ A3) @ Y4)) @ D2)))))))))).
% 9.43/3.44 thf(fact_1392_one__integer__def, axiom, ((one_one @ code_integer) = (code_integer_of_int @ (one_one @ int)))).
% 9.43/3.44 thf(fact_1393_less__eq__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((ord_less_eq @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (((ord_less_eq @ int) @ Xa) @ X))))).
% 9.43/3.44 thf(fact_1394_times__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((times_times @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((times_times @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1395_zdvd__period, axiom, (! [A3:int,D3:int,X:int,T2:int,C2:int]: (((((dvd_dvd @ int) @ A3) @ D3) => ((((dvd_dvd @ int) @ A3) @ (((plus_plus @ int) @ X) @ T2)) = (((dvd_dvd @ int) @ A3) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X) @ (((times_times @ int) @ C2) @ D3))) @ T2))))))).
% 9.43/3.44 thf(fact_1396_zdvd__reduce, axiom, (! [K:int,N:int,M2:int]: (((((dvd_dvd @ int) @ K) @ (((plus_plus @ int) @ N) @ (((times_times @ int) @ K) @ M2))) = (((dvd_dvd @ int) @ K) @ N))))).
% 9.43/3.44 thf(fact_1397_minus__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((minus_minus @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((minus_minus @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1398_unity__coeff__ex, axiom, (! [A:$tType]: ((((dvd @ A) & (semiring_0 @ A)) => (! [P:(A > $o),L:A]: (((? [X3:A]: ((P @ (((times_times @ A) @ L) @ X3)))) = (? [X3:A]: (((((dvd_dvd @ A) @ L) @ (((plus_plus @ A) @ X3) @ (zero_zero @ A))) & (P @ X3))))))))))).
% 9.43/3.44 thf(fact_1399_unit__dvdE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (~ (((A3 != (zero_zero @ A)) => (! [C4:A]: ((B2 != (((times_times @ A) @ A3) @ C4)))))))))))))).
% 9.43/3.44 thf(fact_1400_inf__period_I4_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [D3:A,D4:A,T2:A]: (((((dvd_dvd @ A) @ D3) @ D4) => (! [X7:A,K8:A]: (((~ ((((dvd_dvd @ A) @ D3) @ (((plus_plus @ A) @ X7) @ T2)))) = (~ ((((dvd_dvd @ A) @ D3) @ (((plus_plus @ A) @ (((minus_minus @ A) @ X7) @ (((times_times @ A) @ K8) @ D4))) @ T2)))))))))))))).
% 9.43/3.44 thf(fact_1401_inf__period_I3_J, axiom, (! [A:$tType]: ((((comm_ring @ A) & (dvd @ A)) => (! [D3:A,D4:A,T2:A]: (((((dvd_dvd @ A) @ D3) @ D4) => (! [X7:A,K8:A]: (((((dvd_dvd @ A) @ D3) @ (((plus_plus @ A) @ X7) @ T2)) = (((dvd_dvd @ A) @ D3) @ (((plus_plus @ A) @ (((minus_minus @ A) @ X7) @ (((times_times @ A) @ K8) @ D4))) @ T2)))))))))))).
% 9.43/3.44 thf(fact_1402_unit__div__eq__0__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((((divide_divide @ A) @ A3) @ B2) = (zero_zero @ A)) = (A3 = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1403_numeral__Bit1, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bit1 @ N)) = (((plus_plus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ N))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1404_unit__eq__div1, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((((divide_divide @ A) @ A3) @ B2) = C2) = (A3 = (((times_times @ A) @ C2) @ B2)))))))))).
% 9.43/3.44 thf(fact_1405_unit__eq__div2, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((A3 = (((divide_divide @ A) @ C2) @ B2)) = ((((times_times @ A) @ A3) @ B2) = C2))))))))).
% 9.43/3.44 thf(fact_1406_div__mult__unit2, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,B2:A,A3:A]: (((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ B2) @ A3) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2)))))))))).
% 9.43/3.44 thf(fact_1407_unit__div__commute, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A,C2:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ C2)) @ B2))))))))).
% 9.43/3.44 thf(fact_1408_unit__div__mult__swap, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((times_times @ A) @ A3) @ (((divide_divide @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2))))))))).
% 9.43/3.44 thf(fact_1409_is__unit__div__mult2__eq, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,C2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((dvd_dvd @ A) @ C2) @ (one_one @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ C2)) = (((divide_divide @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ C2)))))))))).
% 9.43/3.44 thf(fact_1410_unit__imp__mod__eq__0, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((modulo_modulo @ A) @ A3) @ B2) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1411_is__unit__power__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ (((power_power @ A) @ A3) @ N)) @ (one_one @ A)) = ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) | (N = (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_1412_bezout__add__strong__nat, axiom, (! [A3:nat,B2:nat]: (((A3 != (zero_zero @ nat)) => (? [D2:nat,X2:nat,Y4:nat]: (((((dvd_dvd @ nat) @ D2) @ A3) & ((((dvd_dvd @ nat) @ D2) @ B2) & ((((times_times @ nat) @ A3) @ X2) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ D2)))))))))).
% 9.43/3.44 thf(fact_1413_take__bit__Suc__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1414_numeral__code_I3_J, axiom, (! [A:$tType]: (((numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bit1 @ N)) = (((plus_plus @ A) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ N)) @ ((numeral_numeral @ A) @ N))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1415_stable__imp__take__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,N:nat]: ((((((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A3) => (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ A3) = (zero_zero @ A))) & ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((bit_se2584673776208193580ke_bit @ A) @ N) @ A3) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ (one_one @ A)))))))))))).
% 9.43/3.44 thf(fact_1416_is__unitE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (~ (((A3 != (zero_zero @ A)) => (! [B3:A]: (((B3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B3) @ (one_one @ A)) => (((((divide_divide @ A) @ (one_one @ A)) @ A3) = B3) => (((((divide_divide @ A) @ (one_one @ A)) @ B3) = A3) => (((((times_times @ A) @ A3) @ B3) = (one_one @ A)) => ((((divide_divide @ A) @ C2) @ A3) != (((times_times @ A) @ C2) @ B3))))))))))))))))))).
% 9.43/3.44 thf(fact_1417_is__unit__div__mult__cancel__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ A3) @ B2)) = (((divide_divide @ A) @ (one_one @ A)) @ B2)))))))))).
% 9.43/3.44 thf(fact_1418_is__unit__div__mult__cancel__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => ((((divide_divide @ A) @ A3) @ (((times_times @ A) @ B2) @ A3)) = (((divide_divide @ A) @ (one_one @ A)) @ B2)))))))))).
% 9.43/3.44 thf(fact_1419_odd__even__add, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A,B2:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) => (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((plus_plus @ A) @ A3) @ B2)))))))))).
% 9.43/3.44 thf(fact_1420_odd__one, axiom, (! [A:$tType]: (((semiring_parity @ A) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1421_dvd__power__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [X:A,M2:nat,N:nat]: (((X != (zero_zero @ A)) => ((((dvd_dvd @ A) @ (((power_power @ A) @ X) @ M2)) @ (((power_power @ A) @ X) @ N)) = ((((dvd_dvd @ A) @ X) @ (one_one @ A)) | (((ord_less_eq @ nat) @ M2) @ N)))))))))).
% 9.43/3.44 thf(fact_1422_dvd__power, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat,X:A]: ((((((ord_less @ nat) @ (zero_zero @ nat)) @ N) | (X = (one_one @ A))) => (((dvd_dvd @ A) @ X) @ (((power_power @ A) @ X) @ N))))))))).
% 9.43/3.44 thf(fact_1423_Suc3__eq__add__3, axiom, (! [N:nat]: (((suc @ (suc @ (suc @ N))) = (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ N))))).
% 9.43/3.44 thf(fact_1424_dvd__mult__cancel2, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ N) @ M2)) @ M2) = (N = (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1425_dvd__mult__cancel1, axiom, (! [M2:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M2) => ((((dvd_dvd @ nat) @ (((times_times @ nat) @ M2) @ N)) @ M2) = (N = (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1426_dvd__minus__add, axiom, (! [Q3:nat,N:nat,R:nat,M2:nat]: (((((ord_less_eq @ nat) @ Q3) @ N) => ((((ord_less_eq @ nat) @ Q3) @ (((times_times @ nat) @ R) @ M2)) => ((((dvd_dvd @ nat) @ M2) @ (((minus_minus @ nat) @ N) @ Q3)) = (((dvd_dvd @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ (((minus_minus @ nat) @ (((times_times @ nat) @ R) @ M2)) @ Q3))))))))).
% 9.43/3.44 thf(fact_1427_power__dvd__imp__le, axiom, (! [I:nat,M2:nat,N:nat]: (((((dvd_dvd @ nat) @ (((power_power @ nat) @ I) @ M2)) @ (((power_power @ nat) @ I) @ N)) => ((((ord_less @ nat) @ (one_one @ nat)) @ I) => (((ord_less_eq @ nat) @ M2) @ N)))))).
% 9.43/3.44 thf(fact_1428_aset_I10_J, axiom, (! [D3:int,D4:int,A4:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D3) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((~ ((((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ X7) @ T2)))) => (~ ((((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X7) @ D4)) @ T2)))))))))))).
% 9.43/3.44 thf(fact_1429_aset_I9_J, axiom, (! [D3:int,D4:int,A4:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D3) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ A4) => (X7 != (((minus_minus @ int) @ Xb3) @ Xa3)))))))) => ((((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ X7) @ T2)) => (((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ (((plus_plus @ int) @ X7) @ D4)) @ T2)))))))))).
% 9.43/3.44 thf(fact_1430_bset_I10_J, axiom, (! [D3:int,D4:int,B6:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D3) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((~ ((((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ X7) @ T2)))) => (~ ((((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ (((minus_minus @ int) @ X7) @ D4)) @ T2)))))))))))).
% 9.43/3.44 thf(fact_1431_bset_I9_J, axiom, (! [D3:int,D4:int,B6:(set @ int),T2:int]: (((((dvd_dvd @ int) @ D3) @ D4) => (! [X7:int]: (((! [Xa3:int]: (((((member @ int) @ Xa3) @ (((set_or1337092689740270186AtMost @ int) @ (one_one @ int)) @ D4)) => (! [Xb3:int]: (((((member @ int) @ Xb3) @ B6) => (X7 != (((plus_plus @ int) @ Xb3) @ Xa3)))))))) => ((((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ X7) @ T2)) => (((dvd_dvd @ int) @ D3) @ (((plus_plus @ int) @ (((minus_minus @ int) @ X7) @ D4)) @ T2)))))))))).
% 9.43/3.44 thf(fact_1432_num_Osize__gen_I3_J, axiom, (! [X32:num]: (((size_num @ (bit1 @ X32)) = (((plus_plus @ nat) @ (size_num @ X32)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_1433_num_Osize_I6_J, axiom, (! [X32:num]: ((((size_size @ num) @ (bit1 @ X32)) = (((plus_plus @ nat) @ ((size_size @ num) @ X32)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.44 thf(fact_1434_odd__iff__mod__2__eq__one, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) = ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1435_Suc__mod__eq__add3__mod, axiom, (! [M2:nat,N:nat]: (((((modulo_modulo @ nat) @ (suc @ (suc @ (suc @ M2)))) @ N) = (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M2)) @ N))))).
% 9.43/3.44 thf(fact_1436_Suc__div__eq__add3__div, axiom, (! [M2:nat,N:nat]: (((((divide_divide @ nat) @ (suc @ (suc @ (suc @ M2)))) @ N) = (((divide_divide @ nat) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ one2))) @ M2)) @ N))))).
% 9.43/3.44 thf(fact_1437_even__diff__iff, axiom, (! [K:int,L:int]: (((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((minus_minus @ int) @ K) @ L)) = (((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((plus_plus @ int) @ K) @ L)))))).
% 9.43/3.44 thf(fact_1438_even__abs__add__iff, axiom, (! [K:int,L:int]: (((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((plus_plus @ int) @ ((abs_abs @ int) @ K)) @ L)) = (((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((plus_plus @ int) @ K) @ L)))))).
% 9.43/3.44 thf(fact_1439_even__add__abs__iff, axiom, (! [K:int,L:int]: (((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((plus_plus @ int) @ K) @ ((abs_abs @ int) @ L))) = (((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((plus_plus @ int) @ K) @ L)))))).
% 9.43/3.44 thf(fact_1440_signed__take__bit__eq__take__bit__shift, axiom, ((bit_ri4674362597316999326ke_bit @ int) = (^ [N5:nat,K4:int]: ((((minus_minus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (suc @ N5)) @ (((plus_plus @ int) @ K4) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N5)))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N5)))))).
% 9.43/3.44 thf(fact_1441_take__bit__incr__eq, axiom, (! [N:nat,K:int]: ((((((bit_se2584673776208193580ke_bit @ int) @ N) @ K) != (((minus_minus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ (one_one @ int))) => ((((bit_se2584673776208193580ke_bit @ int) @ N) @ (((plus_plus @ int) @ K) @ (one_one @ int))) = (((plus_plus @ int) @ (one_one @ int)) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K))))))).
% 9.43/3.44 thf(fact_1442_oddE, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => (~ ((! [B3:A]: ((A3 != (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B3)) @ (one_one @ A)))))))))))))).
% 9.43/3.44 thf(fact_1443_parity__cases, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: ((((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (zero_zero @ A))) => (~ (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) != (one_one @ A)))))))))))).
% 9.43/3.44 thf(fact_1444_mod2__eq__if, axiom, (! [A:$tType]: (((semiring_parity @ A) => (! [A3:A]: ((((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (zero_zero @ A))) & ((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1445_minus__one__power__iff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [N:nat]: ((((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = (one_one @ A))) & ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) => ((((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1446_take__bit__int__greater__eq, axiom, (! [K:int,N:nat]: (((((ord_less @ int) @ K) @ (zero_zero @ int)) => (((ord_less_eq @ int) @ (((plus_plus @ int) @ K) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N))) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)))))).
% 9.43/3.44 thf(fact_1447_take__bit__Suc__minus__1__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (suc @ N))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1448_take__bit__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((bit_se2584673776208193580ke_bit @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))))))).
% 9.43/3.44 thf(fact_1449_take__bit__numeral__minus__1__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ K)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ K))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1450_even__mask__div__iff_H, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [M2:nat,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) @ (one_one @ A))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))) = (((ord_less_eq @ nat) @ M2) @ N)))))))).
% 9.43/3.44 thf(fact_1451_even__mask__div__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M2:nat,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) @ (one_one @ A))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N))) = (((((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) = (zero_zero @ A)) | (((ord_less_eq @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1452_Code__Numeral_Onegative__def, axiom, (code_negative = (((((comp @ code_integer) @ code_integer) @ num) @ (uminus_uminus @ code_integer)) @ (numeral_numeral @ code_integer)))).
% 9.43/3.44 thf(fact_1453_take__bit__numeral__minus__numeral__int, axiom, (! [M2:num,N:num]: (((((bit_se2584673776208193580ke_bit @ int) @ ((numeral_numeral @ nat) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (^ [Q5:num]: ((((bit_se2584673776208193580ke_bit @ int) @ ((numeral_numeral @ nat) @ M2)) @ (((minus_minus @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ M2))) @ ((numeral_numeral @ int) @ Q5)))))) @ ((bit_take_bit_num @ ((numeral_numeral @ nat) @ M2)) @ N)))))).
% 9.43/3.44 thf(fact_1454_signed__take__bit__numeral__minus__bit1, axiom, (! [L:num,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ (pred_numeral @ L)) @ (((minus_minus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) @ (one_one @ int)))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1455_execute__change, axiom, (! [A:$tType]: (((heap @ A) => (! [F:(A > A),R:(ref @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ A) @ (((ref_change @ A) @ F) @ R)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ (((ref_get @ A) @ H3) @ R))) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ ((((ref_set @ A) @ R) @ (F @ (((ref_get @ A) @ H3) @ R))) @ H3)) @ ((numeral_numeral @ nat) @ (bit1 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1456_signed__take__bit__numeral__bit1, axiom, (! [L:num,K:num]: (((((bit_ri4674362597316999326ke_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((numeral_numeral @ int) @ (bit1 @ K))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_ri4674362597316999326ke_bit @ int) @ (pred_numeral @ L)) @ ((numeral_numeral @ int) @ K))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1457_take__bit__Suc__minus__bit1, axiom, (! [N:nat,K:num]: (((((bit_se2584673776208193580ke_bit @ int) @ (suc @ N)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (inc @ K))))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1458_divmod__BitM__2__eq, axiom, (! [M2:num]: (((((unique8689654367752047608divmod @ int) @ (bitM @ M2)) @ (bit0 @ one2)) = ((((product_Pair @ int) @ int) @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ M2)) @ (one_one @ int))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1459_flip__bit__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se8732182000553998342ip_bit @ A) @ (zero_zero @ nat)) @ A3) = (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1460_of__bool__eq_I2_J, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (((zero_neq_one_of_bool @ A) @ $true) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1461_of__bool__eq__1__iff, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:$o]: (((((zero_neq_one_of_bool @ A) @ P) = (one_one @ A)) = P))))))).
% 9.43/3.44 thf(fact_1462_of__bool__less__one__iff, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: (((((ord_less @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ (one_one @ A)) = (~ (P))))))))).
% 9.43/3.44 thf(fact_1463_of__bool__not__iff, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [P:$o]: ((((zero_neq_one_of_bool @ A) @ (~ (P))) = (((minus_minus @ A) @ (one_one @ A)) @ ((zero_neq_one_of_bool @ A) @ P))))))))).
% 9.43/3.44 thf(fact_1464_take__bit__num__simps_I3_J, axiom, (! [N:nat,M2:num]: ((((bit_take_bit_num @ (suc @ N)) @ (bit0 @ M2)) = (((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ N) @ M2)))))).
% 9.43/3.44 thf(fact_1465_take__bit__num__simps_I4_J, axiom, (! [N:nat,M2:num]: ((((bit_take_bit_num @ (suc @ N)) @ (bit1 @ M2)) = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ N) @ M2))))))).
% 9.43/3.44 thf(fact_1466_take__bit__num__simps_I6_J, axiom, (! [R:num,M2:num]: ((((bit_take_bit_num @ ((numeral_numeral @ nat) @ R)) @ (bit0 @ M2)) = (((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ (pred_numeral @ R)) @ M2)))))).
% 9.43/3.44 thf(fact_1467_take__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ (one_one @ A)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 9.43/3.44 thf(fact_1468_add__neg__numeral__special_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ M2)))))))))).
% 9.43/3.44 thf(fact_1469_add__neg__numeral__special_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ N)))))))))).
% 9.43/3.44 thf(fact_1470_diff__numeral__special_I5_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ N)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (inc @ N)))))))))).
% 9.43/3.44 thf(fact_1471_diff__numeral__special_I6_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((numeral_numeral @ A) @ (inc @ M2))))))))).
% 9.43/3.44 thf(fact_1472_take__bit__num__simps_I7_J, axiom, (! [R:num,M2:num]: ((((bit_take_bit_num @ ((numeral_numeral @ nat) @ R)) @ (bit1 @ M2)) = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ (pred_numeral @ R)) @ M2))))))).
% 9.43/3.44 thf(fact_1473_take__bit__numeral__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M2:num,N:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ M2)) @ ((numeral_numeral @ A) @ N)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_take_bit_num @ ((numeral_numeral @ nat) @ M2)) @ N))))))))).
% 9.43/3.44 thf(fact_1474_bits__1__div__exp, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat]: (((((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = ((zero_neq_one_of_bool @ A) @ (N = (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_1475_one__div__2__pow__eq, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: (((((divide_divide @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = ((zero_neq_one_of_bool @ A) @ (N = (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_1476_one__mod__2__pow__eq, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: (((((modulo_modulo @ A) @ (one_one @ A)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) = ((zero_neq_one_of_bool @ A) @ (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 9.43/3.44 thf(fact_1477_mult__inc, axiom, (! [X:num,Y:num]: (((((times_times @ num) @ X) @ (inc @ Y)) = (((plus_plus @ num) @ (((times_times @ num) @ X) @ Y)) @ X))))).
% 9.43/3.44 thf(fact_1478_add__inc, axiom, (! [X:num,Y:num]: (((((plus_plus @ num) @ X) @ (inc @ Y)) = (inc @ (((plus_plus @ num) @ X) @ Y)))))).
% 9.43/3.44 thf(fact_1479_of__bool__less__eq__one, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [P:$o]: ((((ord_less_eq @ A) @ ((zero_neq_one_of_bool @ A) @ P)) @ (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1480_of__bool__def, axiom, (! [A:$tType]: (((zero_neq_one @ A) => ((zero_neq_one_of_bool @ A) = (^ [P6:$o]: (((((if @ A) @ P6) @ (one_one @ A)) @ (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1481_split__of__bool, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:(A > $o),P5:$o]: (((P @ ((zero_neq_one_of_bool @ A) @ P5)) = ((P5 => (P @ (one_one @ A))) & ((~ (P5)) => (P @ (zero_zero @ A))))))))))).
% 9.43/3.44 thf(fact_1482_split__of__bool__asm, axiom, (! [A:$tType]: (((zero_neq_one @ A) => (! [P:(A > $o),P5:$o]: (((P @ ((zero_neq_one_of_bool @ A) @ P5)) = (~ (((P5 & (~ ((P @ (one_one @ A))))) | ((~ (P5)) & (~ ((P @ (zero_zero @ A))))))))))))))).
% 9.43/3.44 thf(fact_1483_add__One, axiom, (! [X:num]: (((((plus_plus @ num) @ X) @ one2) = (inc @ X))))).
% 9.43/3.44 thf(fact_1484_effect__updateI, axiom, (! [A:$tType]: (((heap @ A) => (! [H4:(heap_ext @ product_unit),R:(ref @ A),V:A,H3:(heap_ext @ product_unit),N:nat,X:product_unit]: (((H4 = ((((ref_set @ A) @ R) @ V) @ H3)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ product_unit) @ (((ref_update @ A) @ R) @ V)) @ H3) @ H4) @ X) @ N))))))))).
% 9.43/3.44 thf(fact_1485_effect__updateE, axiom, (! [A:$tType]: (((heap @ A) => (! [R4:(ref @ A),V:A,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:product_unit,N:nat]: ((((((((heap_Time_effect @ product_unit) @ (((ref_update @ A) @ R4) @ V)) @ H3) @ H4) @ R) @ N) => (~ (((H4 = ((((ref_set @ A) @ R4) @ V) @ H3)) => (N != (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1486_pred__numeral__def, axiom, (pred_numeral = (^ [K4:num]: ((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ K4)) @ (one_one @ nat)))))).
% 9.43/3.44 thf(fact_1487_BitM__plus__one, axiom, (! [N:num]: (((((plus_plus @ num) @ (bitM @ N)) @ one2) = (bit0 @ N))))).
% 9.43/3.44 thf(fact_1488_one__plus__BitM, axiom, (! [N:num]: (((((plus_plus @ num) @ one2) @ (bitM @ N)) = (bit0 @ N))))).
% 9.43/3.44 thf(fact_1489_numeral__inc, axiom, (! [A:$tType]: (((numeral @ A) => (! [X:num]: ((((numeral_numeral @ A) @ (inc @ X)) = (((plus_plus @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1490_numeral__BitM, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: ((((numeral_numeral @ A) @ (bitM @ N)) = (((minus_minus @ A) @ ((numeral_numeral @ A) @ (bit0 @ N))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1491_take__bit__numeral__minus__bit1, axiom, (! [L:num,K:num]: (((((bit_se2584673776208193580ke_bit @ int) @ ((numeral_numeral @ nat) @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ K)))) = (((plus_plus @ int) @ (((times_times @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ (pred_numeral @ L)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (inc @ K))))) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1492_bits__induct, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [P:(A > $o),A3:A]: (((! [A5:A]: ((((((divide_divide @ A) @ A5) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A5) => (P @ A5)))) => ((! [A5:A,B3:$o]: (((P @ A5) => (((((divide_divide @ A) @ (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ B3)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A5))) @ ((numeral_numeral @ A) @ (bit0 @ one2))) = A5) => (P @ (((plus_plus @ A) @ ((zero_neq_one_of_bool @ A) @ B3)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A5))))))) => (P @ A3))))))))).
% 9.43/3.44 thf(fact_1493_effect__changeE, axiom, (! [A:$tType]: (((heap @ A) => (! [F:(A > A),R4:(ref @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ (((ref_change @ A) @ F) @ R4)) @ H3) @ H4) @ R) @ N) => (~ (((H4 = ((((ref_set @ A) @ R4) @ (F @ (((ref_get @ A) @ H3) @ R4))) @ H3)) => ((R = (F @ (((ref_get @ A) @ H3) @ R4))) => (N != ((numeral_numeral @ nat) @ (bit1 @ one2)))))))))))))).
% 9.43/3.44 thf(fact_1494_effect__changeI, axiom, (! [A:$tType]: (((heap @ A) => (! [H4:(heap_ext @ product_unit),R:(ref @ A),F:(A > A),H3:(heap_ext @ product_unit),X:A,N:nat]: (((H4 = ((((ref_set @ A) @ R) @ (F @ (((ref_get @ A) @ H3) @ R))) @ H3)) => ((X = (F @ (((ref_get @ A) @ H3) @ R))) => ((N = ((numeral_numeral @ nat) @ (bit1 @ one2))) => ((((((heap_Time_effect @ A) @ (((ref_change @ A) @ F) @ R)) @ H3) @ H4) @ X) @ N)))))))))).
% 9.43/3.44 thf(fact_1495_divide__int__unfold, axiom, (! [L:int,K:int,N:nat,M2:nat]: (((((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N = (zero_zero @ nat)))) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M2))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = (zero_zero @ int))) & ((~ (((((sgn_sgn @ int) @ L) = (zero_zero @ int)) | ((((sgn_sgn @ int) @ K) = (zero_zero @ int)) | (N = (zero_zero @ nat)))))) => (((((sgn_sgn @ int) @ K) = ((sgn_sgn @ int) @ L)) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M2))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ M2) @ N)))) & ((((sgn_sgn @ int) @ K) != ((sgn_sgn @ int) @ L)) => ((((divide_divide @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ K)) @ ((semiring_1_of_nat @ int) @ M2))) @ (((times_times @ int) @ ((sgn_sgn @ int) @ L)) @ ((semiring_1_of_nat @ int) @ N))) = ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ M2) @ N)) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ N) @ M2))))))))))))))).
% 9.43/3.44 thf(fact_1496_take__bit__numeral__bit1, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [L:num,K:num]: (((((bit_se2584673776208193580ke_bit @ A) @ ((numeral_numeral @ nat) @ L)) @ ((numeral_numeral @ A) @ (bit1 @ K))) = (((plus_plus @ A) @ (((times_times @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (pred_numeral @ L)) @ ((numeral_numeral @ A) @ K))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1497_Divides_Oadjust__div__eq, axiom, (! [Q3:int,R:int]: (((adjust_div @ ((((product_Pair @ int) @ int) @ Q3) @ R)) = (((plus_plus @ int) @ Q3) @ ((zero_neq_one_of_bool @ int) @ (R != (zero_zero @ int)))))))).
% 9.43/3.44 thf(fact_1498_mask__numeral, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:num]: ((((bit_se2239418461657761734s_mask @ A) @ ((numeral_numeral @ nat) @ N)) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((bit_se2239418461657761734s_mask @ A) @ (pred_numeral @ N))))))))))).
% 9.43/3.44 thf(fact_1499_and__int__unfold, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K4:int,L3:int]: (((((if @ int) @ ((K4 = (zero_zero @ int)) | (L3 = (zero_zero @ int)))) @ (zero_zero @ int)) @ ((((if @ int) @ (K4 = ((uminus_uminus @ int) @ (one_one @ int)))) @ L3) @ ((((if @ int) @ (L3 = ((uminus_uminus @ int) @ (one_one @ int)))) @ K4) @ (((plus_plus @ int) @ (((times_times @ int) @ (((modulo_modulo @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((modulo_modulo @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 9.43/3.44 thf(fact_1500_the__res__def, axiom, (! [A:$tType]: (((time_the_res @ A) = (^ [M:(heap_Time_Heap @ A),H:(heap_ext @ product_unit)]: ((((((case_option @ A) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (undefined @ A)) @ ((((product_case_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A) @ (^ [R3:A]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ A) @ (^ [X3:(heap_ext @ product_unit),Xa4:nat]: (R3))))))) @ (((heap_Time_execute @ A) @ M) @ H)))))))).
% 9.43/3.44 thf(fact_1501_take__bit__eq__mask__iff__exp__dvd, axiom, (! [N:nat,K:int]: ((((((bit_se2584673776208193580ke_bit @ int) @ N) @ K) = ((bit_se2239418461657761734s_mask @ int) @ N)) = (((dvd_dvd @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ (((plus_plus @ int) @ K) @ (one_one @ int))))))).
% 9.43/3.44 thf(fact_1502_Code__Numeral_Opositive__def, axiom, (code_positive = (numeral_numeral @ code_integer))).
% 9.43/3.44 thf(fact_1503_divide__int__def, axiom, ((divide_divide @ int) = (^ [K4:int,L3:int]: (((((if @ int) @ (L3 = (zero_zero @ int))) @ (zero_zero @ int)) @ ((((if @ int) @ (((sgn_sgn @ int) @ K4) = ((sgn_sgn @ int) @ L3))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K4))) @ (nat2 @ ((abs_abs @ int) @ L3))))) @ ((uminus_uminus @ int) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ (((divide_divide @ nat) @ (nat2 @ ((abs_abs @ int) @ K4))) @ (nat2 @ ((abs_abs @ int) @ L3)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ int) @ L3) @ K4))))))))))))).
% 9.43/3.44 thf(fact_1504_sum__gp0, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: ((((X = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_atMost @ nat) @ N)) = ((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))) & ((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_atMost @ nat) @ N)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ (suc @ N)))) @ (((minus_minus @ A) @ (one_one @ A)) @ X))))))))))).
% 9.43/3.44 thf(fact_1505_atMost__eq__iff, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((set_ord_atMost @ A) @ X) = ((set_ord_atMost @ A) @ Y)) = (X = Y)))))))).
% 9.43/3.44 thf(fact_1506_atMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atMost @ A) @ K)) = (((ord_less_eq @ A) @ I) @ K)))))))).
% 9.43/3.44 thf(fact_1507_atMost__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ X)) @ ((set_ord_atMost @ A) @ Y)) = (((ord_less_eq @ A) @ X) @ Y)))))))).
% 9.43/3.44 thf(fact_1508_bit_Oconj__one__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344872417868541ns_and @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = X))))))).
% 9.43/3.44 thf(fact_1509_and_Oright__neutral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ A3) @ ((uminus_uminus @ A) @ (one_one @ A))) = A3))))))).
% 9.43/3.44 thf(fact_1510_and_Oleft__neutral, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ A3) = A3))))))).
% 9.43/3.44 thf(fact_1511_and__numerals_I2_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1512_and__numerals_I8_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1513_Icc__subset__Iic__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H3:A,H4:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H3)) @ ((set_ord_atMost @ A) @ H4)) = ((~ ((((ord_less_eq @ A) @ L) @ H3))) | (((ord_less_eq @ A) @ H3) @ H4))))))))).
% 9.43/3.44 thf(fact_1514_mask__Suc__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (((bit_se2239418461657761734s_mask @ A) @ (suc @ (zero_zero @ nat))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1515_sum_OatMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N))) @ (G @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_1516_take__bit__minus__one__eq__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se2584673776208193580ke_bit @ A) @ N) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((bit_se2239418461657761734s_mask @ A) @ N)))))))).
% 9.43/3.44 thf(fact_1517_and__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_1518_and__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ (one_one @ A)) = (zero_zero @ A)))))))).
% 9.43/3.44 thf(fact_1519_nat__numeral__diff__1, axiom, (! [V:num]: (((((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ (one_one @ nat)) = (nat2 @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ V)) @ (one_one @ int))))))).
% 9.43/3.44 thf(fact_1520_and__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y))))))))))).
% 9.43/3.44 thf(fact_1521_mask__integer_Oabs__eq, axiom, ((bit_se2239418461657761734s_mask @ code_integer) = (^ [X3:nat]: ((code_integer_of_int @ ((bit_se2239418461657761734s_mask @ int) @ X3)))))).
% 9.43/3.44 thf(fact_1522_not__Iic__eq__Icc, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H4:A,L:A,H3:A]: ((((set_ord_atMost @ A) @ H4) != (((set_or1337092689740270186AtMost @ A) @ L) @ H3)))))))).
% 9.43/3.44 thf(fact_1523_atMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atMost @ A) = (^ [U2:A]: (((collect @ A) @ (^ [X3:A]: ((((ord_less_eq @ A) @ X3) @ U2))))))))))).
% 9.43/3.44 thf(fact_1524_and__eq__minus__1__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,B2:A]: ((((((bit_se5824344872417868541ns_and @ A) @ A3) @ B2) = ((uminus_uminus @ A) @ (one_one @ A))) = ((A3 = ((uminus_uminus @ A) @ (one_one @ A))) & (B2 = ((uminus_uminus @ A) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1525_atMost__atLeast0, axiom, ((set_ord_atMost @ nat) = ((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)))).
% 9.43/3.44 thf(fact_1526_nat__one__as__int, axiom, ((one_one @ nat) = (nat2 @ (one_one @ int)))).
% 9.43/3.44 thf(fact_1527_not__Iic__le__Icc, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H3:A,L4:A,H4:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H3)) @ (((set_or1337092689740270186AtMost @ A) @ L4) @ H4)))))))))).
% 9.43/3.44 thf(fact_1528_nat__int__add, axiom, (! [A3:nat,B2:nat]: (((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2))) = (((plus_plus @ nat) @ A3) @ B2))))).
% 9.43/3.44 thf(fact_1529_option_Othe__def, axiom, (! [A:$tType]: (((the2 @ A) = ((((case_option @ A) @ A) @ (undefined @ A)) @ (^ [X24:A]: (X24))))))).
% 9.43/3.44 thf(fact_1530_nat__plus__as__int, axiom, ((plus_plus @ nat) = (^ [A7:nat,B5:nat]: ((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A7)) @ ((semiring_1_of_nat @ int) @ B5))))))).
% 9.43/3.44 thf(fact_1531_nat__add__distrib, axiom, (! [Z2:int,Z7:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z7) => ((nat2 @ (((plus_plus @ int) @ Z2) @ Z7)) = (((plus_plus @ nat) @ (nat2 @ Z2)) @ (nat2 @ Z7)))))))).
% 9.43/3.44 thf(fact_1532_Suc__as__int, axiom, (suc = (^ [A7:nat]: ((nat2 @ (((plus_plus @ int) @ ((semiring_1_of_nat @ int) @ A7)) @ (one_one @ int))))))).
% 9.43/3.44 thf(fact_1533_nat__abs__triangle__ineq, axiom, (! [K:int,L:int]: ((((ord_less_eq @ nat) @ (nat2 @ ((abs_abs @ int) @ (((plus_plus @ int) @ K) @ L)))) @ (((plus_plus @ nat) @ (nat2 @ ((abs_abs @ int) @ K))) @ (nat2 @ ((abs_abs @ int) @ L))))))).
% 9.43/3.44 thf(fact_1534_sum_OatMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_atMost @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1535_sum__telescope, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F:(nat > A),I:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ (F @ I3)) @ (F @ (suc @ I3)))))) @ ((set_ord_atMost @ nat) @ I)) = (((minus_minus @ A) @ (F @ (zero_zero @ nat))) @ (F @ (suc @ I)))))))))).
% 9.43/3.44 thf(fact_1536_the__heap__def, axiom, (! [A:$tType]: (((time_the_heap @ A) = (^ [M:(heap_Time_Heap @ A),H:(heap_ext @ product_unit)]: ((((((case_option @ (heap_ext @ product_unit)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (undefined @ (heap_ext @ product_unit))) @ ((((product_case_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (heap_ext @ product_unit)) @ (^ [X3:A]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ (heap_ext @ product_unit)) @ (^ [H7:(heap_ext @ product_unit),Xa4:nat]: (H7))))))) @ (((heap_Time_execute @ A) @ M) @ H)))))))).
% 9.43/3.44 thf(fact_1537_and__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ A3) @ (one_one @ A)) = (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1538_one__and__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344872417868541ns_and @ A) @ (one_one @ A)) @ A3) = (((modulo_modulo @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1539_take__bit__eq__mask__iff, axiom, (! [N:nat,K:int]: ((((((bit_se2584673776208193580ke_bit @ int) @ N) @ K) = ((bit_se2239418461657761734s_mask @ int) @ N)) = ((((bit_se2584673776208193580ke_bit @ int) @ N) @ (((plus_plus @ int) @ K) @ (one_one @ int))) = (zero_zero @ int)))))).
% 9.43/3.44 thf(fact_1540_Suc__nat__eq__nat__zadd1, axiom, (! [Z2:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ Z2) => ((suc @ (nat2 @ Z2)) = (nat2 @ (((plus_plus @ int) @ (one_one @ int)) @ Z2))))))).
% 9.43/3.44 thf(fact_1541_Divides_Oadjust__div__def, axiom, (adjust_div = ((((product_case_prod @ int) @ int) @ int) @ (^ [Q5:int,R3:int]: ((((plus_plus @ int) @ Q5) @ ((zero_neq_one_of_bool @ int) @ (R3 != (zero_zero @ int))))))))).
% 9.43/3.44 thf(fact_1542_sum__gp__basic, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: (((((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_atMost @ nat) @ N))) = (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_1543_sum__power__shift, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [M2:nat,N:nat,X:A]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((times_times @ A) @ (((power_power @ A) @ X) @ M2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ N) @ M2)))))))))))).
% 9.43/3.44 thf(fact_1544_sum_Otriangle__reindex__eq, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > (nat > A)),N:nat]: ((((((groups7311177749621191930dd_sum @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I3) @ J2)) @ N)))))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K4) @ I3))))) @ ((set_ord_atMost @ nat) @ K4))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1545_and__int__rec, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K4:int,L3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K4))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1546_mask__half__int, axiom, (! [N:nat]: (((((divide_divide @ int) @ ((bit_se2239418461657761734s_mask @ int) @ N)) @ ((numeral_numeral @ int) @ (bit0 @ one2))) = ((bit_se2239418461657761734s_mask @ int) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1547_sum_Oin__pairs__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1548_mask__nat__def, axiom, ((bit_se2239418461657761734s_mask @ nat) = (^ [N5:nat]: ((((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N5)) @ (one_one @ nat)))))).
% 9.43/3.44 thf(fact_1549_mask__eq__exp__minus__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se2239418461657761734s_mask @ A) = (^ [N5:nat]: ((((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N5)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1550_sum_Ozero__middle, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [P5:nat,K:nat,G:(nat > A),H3:(nat > A)]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ P5) => ((((ord_less_eq @ nat) @ K) @ P5) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J2:nat]: (((((if @ A) @ (((ord_less @ nat) @ J2) @ K)) @ (G @ J2)) @ ((((if @ A) @ (J2 = K)) @ (zero_zero @ A)) @ (H3 @ (((minus_minus @ nat) @ J2) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P5)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J2:nat]: (((((if @ A) @ (((ord_less @ nat) @ J2) @ K)) @ (G @ J2)) @ (H3 @ J2))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P5) @ (suc @ (zero_zero @ nat)))))))))))))).
% 9.43/3.44 thf(fact_1551_and__minus__numerals_I3_J, axiom, (! [M2:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M2) @ (bitM @ N))))))).
% 9.43/3.44 thf(fact_1552_and__minus__numerals_I7_J, axiom, (! [N:num,M2:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) @ ((numeral_numeral @ int) @ M2)) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M2) @ (bitM @ N))))))).
% 9.43/3.44 thf(fact_1553_and__minus__numerals_I4_J, axiom, (! [M2:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M2) @ (bit0 @ N))))))).
% 9.43/3.44 thf(fact_1554_and__minus__numerals_I8_J, axiom, (! [N:num,M2:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) @ ((numeral_numeral @ int) @ M2)) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M2) @ (bit0 @ N))))))).
% 9.43/3.44 thf(fact_1555_gbinomial__partial__row__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ A3) @ K4)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ ((semiring_1_of_nat @ A) @ K4)))))) @ ((set_ord_atMost @ nat) @ M2)) = (((times_times @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ (one_one @ A))) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((gbinomial @ A) @ A3) @ (((plus_plus @ nat) @ M2) @ (one_one @ nat))))))))))).
% 9.43/3.44 thf(fact_1556_gbinomial__r__part__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ ((gbinomial @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((semiring_1_of_nat @ A) @ M2))) @ (one_one @ A)))) @ ((set_ord_atMost @ nat) @ M2)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2))))))))).
% 9.43/3.44 thf(fact_1557_and__num_Oelims, axiom, (! [X:num,Xa:num,Y:(option @ num)]: (((((bit_un7362597486090784418nd_num @ X) @ Xa) = Y) => (((X = one2) => ((Xa = one2) => (Y != ((some @ num) @ one2)))) => (((X = one2) => ((? [N3:num]: ((Xa = (bit0 @ N3)))) => (Y != (none @ num)))) => (((X = one2) => ((? [N3:num]: ((Xa = (bit1 @ N3)))) => (Y != ((some @ num) @ one2)))) => (((? [M5:num]: ((X = (bit0 @ M5)))) => ((Xa = one2) => (Y != (none @ num)))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))))))))) => (((? [M5:num]: ((X = (bit1 @ M5)))) => ((Xa = one2) => (Y != ((some @ num) @ one2)))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N8:num]: (((some @ num) @ (bit1 @ N8))))) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))))))))))))))))))))))).
% 9.43/3.44 thf(fact_1558_take__bit__Suc__from__most, axiom, (! [N:nat,K:int]: (((((bit_se2584673776208193580ke_bit @ int) @ (suc @ N)) @ K) = (((plus_plus @ int) @ (((times_times @ int) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N)) @ ((zero_neq_one_of_bool @ int) @ (((bit_se5641148757651400278ts_bit @ int) @ K) @ N)))) @ (((bit_se2584673776208193580ke_bit @ int) @ N) @ K)))))).
% 9.43/3.44 thf(fact_1559_gbinomial__1, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A3:A]: (((((gbinomial @ A) @ A3) @ (one_one @ nat)) = A3))))))).
% 9.43/3.44 thf(fact_1560_gbinomial__0_I1_J, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (semidom_divide @ A)) => (! [A3:A]: (((((gbinomial @ A) @ A3) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1561_and__nat__numerals_I2_J, axiom, (! [Y:num]: (((((bit_se5824344872417868541ns_and @ nat) @ (suc @ (zero_zero @ nat))) @ ((numeral_numeral @ nat) @ (bit1 @ Y))) = (one_one @ nat))))).
% 9.43/3.44 thf(fact_1562_and__nat__numerals_I4_J, axiom, (! [X:num]: (((((bit_se5824344872417868541ns_and @ nat) @ ((numeral_numeral @ nat) @ (bit1 @ X))) @ (suc @ (zero_zero @ nat))) = (one_one @ nat))))).
% 9.43/3.44 thf(fact_1563_bit__disjunctive__add__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,B2:A,N:nat]: (((! [N3:nat]: (((~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N3))) | (~ ((((bit_se5641148757651400278ts_bit @ A) @ B2) @ N3)))))) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ N) = ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N) | (((bit_se5641148757651400278ts_bit @ A) @ B2) @ N)))))))))).
% 9.43/3.44 thf(fact_1564_bit__1__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ N) = (N = (zero_zero @ nat))))))))).
% 9.43/3.44 thf(fact_1565_not__bit__1__Suc, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_1566_bit__numeral__simps_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [N:num]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ (one_one @ A)) @ ((numeral_numeral @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1567_gbinomial__Suc__Suc, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (suc @ K)) = (((plus_plus @ A) @ (((gbinomial @ A) @ A3) @ K)) @ (((gbinomial @ A) @ A3) @ (suc @ K)))))))))).
% 9.43/3.44 thf(fact_1568_and__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((bit_se5824344872417868541ns_and @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se5824344872417868541ns_and @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1569_gbinomial__addition__formula, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ A3) @ (suc @ K)) = (((plus_plus @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ (suc @ K))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K))))))))).
% 9.43/3.44 thf(fact_1570_gbinomial__absorb__comp, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((times_times @ A) @ (((minus_minus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ (((gbinomial @ A) @ A3) @ K)) = (((times_times @ A) @ A3) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K))))))))).
% 9.43/3.44 thf(fact_1571_gbinomial__mult__1_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((times_times @ A) @ (((gbinomial @ A) @ A3) @ K)) @ A3) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ K)) @ (((gbinomial @ A) @ A3) @ K))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A3) @ (suc @ K))))))))))).
% 9.43/3.44 thf(fact_1572_gbinomial__mult__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((times_times @ A) @ A3) @ (((gbinomial @ A) @ A3) @ K)) = (((plus_plus @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ K)) @ (((gbinomial @ A) @ A3) @ K))) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A3) @ (suc @ K))))))))))).
% 9.43/3.44 thf(fact_1573_Suc__times__gbinomial, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A3:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (suc @ K))) = (((times_times @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (((gbinomial @ A) @ A3) @ K))))))))).
% 9.43/3.44 thf(fact_1574_gbinomial__absorption, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A3:A]: (((((times_times @ A) @ ((semiring_1_of_nat @ A) @ (suc @ K))) @ (((gbinomial @ A) @ A3) @ (suc @ K))) = (((times_times @ A) @ A3) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K))))))))).
% 9.43/3.44 thf(fact_1575_gbinomial__parallel__sum, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K4))) @ K4)))) @ ((set_ord_atMost @ nat) @ N)) = (((gbinomial @ A) @ (((plus_plus @ A) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ N))) @ (one_one @ A))) @ N)))))))).
% 9.43/3.44 thf(fact_1576_gbinomial__rec, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (suc @ K)) = (((times_times @ A) @ (((gbinomial @ A) @ A3) @ K)) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ ((semiring_1_of_nat @ A) @ (suc @ K))))))))))).
% 9.43/3.44 thf(fact_1577_gbinomial__factors, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ (suc @ K)) = (((times_times @ A) @ (((divide_divide @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ ((semiring_1_of_nat @ A) @ (suc @ K)))) @ (((gbinomial @ A) @ A3) @ K))))))))).
% 9.43/3.44 thf(fact_1578_gbinomial__negated__upper, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A7:A,K4:nat]: ((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K4)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((minus_minus @ A) @ ((semiring_1_of_nat @ A) @ K4)) @ A7)) @ (one_one @ A))) @ K4))))))))).
% 9.43/3.44 thf(fact_1579_gbinomial__index__swap, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: (((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N))) @ (one_one @ A))) @ K)) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ N))))))))).
% 9.43/3.44 thf(fact_1580_int__bit__bound, axiom, (! [K:int]: ((~ ((! [N3:nat]: (((! [M6:nat]: (((((ord_less_eq @ nat) @ N3) @ M6) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ M6) = (((bit_se5641148757651400278ts_bit @ int) @ K) @ N3))))) => (~ (((((ord_less @ nat) @ (zero_zero @ nat)) @ N3) => ((((bit_se5641148757651400278ts_bit @ int) @ K) @ (((minus_minus @ nat) @ N3) @ (one_one @ nat))) = (~ ((((bit_se5641148757651400278ts_bit @ int) @ K) @ N3))))))))))))))).
% 9.43/3.44 thf(fact_1581_gbinomial__minus, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,K:nat]: (((((gbinomial @ A) @ ((uminus_uminus @ A) @ A3)) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K))))))))).
% 9.43/3.44 thf(fact_1582_gbinomial__reduce__nat, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((gbinomial @ A) @ A3) @ K) = (((plus_plus @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat)))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ K)))))))))).
% 9.43/3.44 thf(fact_1583_gbinomial__sum__lower__neg, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:A,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((gbinomial @ A) @ A3) @ K4)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K4))))) @ ((set_ord_atMost @ nat) @ M2)) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M2)) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ M2))))))))).
% 9.43/3.44 thf(fact_1584_numeral__and__num, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M2:num,N:num]: (((((bit_se5824344872417868541ns_and @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_un7362597486090784418nd_num @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1585_gbinomial__sum__up__index, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J2:nat]: ((((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ J2)) @ K)))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ N)) @ (one_one @ A))) @ (((plus_plus @ nat) @ K) @ (one_one @ nat)))))))))).
% 9.43/3.44 thf(fact_1586_gbinomial__partial__sum__poly, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M2:nat,A3:A,X:A,Y:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ A3)) @ K4)) @ (((power_power @ A) @ X) @ K4))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ M2) @ K4)))))) @ ((set_ord_atMost @ nat) @ M2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ ((uminus_uminus @ A) @ A3)) @ K4)) @ (((power_power @ A) @ ((uminus_uminus @ A) @ X)) @ K4))) @ (((power_power @ A) @ (((plus_plus @ A) @ X) @ Y)) @ (((minus_minus @ nat) @ M2) @ K4)))))) @ ((set_ord_atMost @ nat) @ M2))))))))).
% 9.43/3.44 thf(fact_1587_even__bit__succ__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ (one_one @ A)) @ A3)) @ N) = ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N) | (N = (zero_zero @ nat))))))))))).
% 9.43/3.44 thf(fact_1588_odd__bit__iff__bit__pred, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3))) => ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N) = ((((bit_se5641148757651400278ts_bit @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ N) | (N = (zero_zero @ nat))))))))))).
% 9.43/3.44 thf(fact_1589_and__num_Osimps_I9_J, axiom, (! [M2:num,N:num]: ((((bit_un7362597486090784418nd_num @ (bit1 @ M2)) @ (bit1 @ N)) = (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N8:num]: (((some @ num) @ (bit1 @ N8))))) @ ((bit_un7362597486090784418nd_num @ M2) @ N)))))).
% 9.43/3.44 thf(fact_1590_gbinomial__absorption_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [K:nat,A3:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((gbinomial @ A) @ A3) @ K) = (((times_times @ A) @ (((divide_divide @ A) @ A3) @ ((semiring_1_of_nat @ A) @ K))) @ (((gbinomial @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1591_gbinomial__partial__sum__poly__xpos, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M2:nat,A3:A,X:A,Y:A]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ M2)) @ A3)) @ K4)) @ (((power_power @ A) @ X) @ K4))) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ M2) @ K4)))))) @ ((set_ord_atMost @ nat) @ M2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((gbinomial @ A) @ (((minus_minus @ A) @ (((plus_plus @ A) @ ((semiring_1_of_nat @ A) @ K4)) @ A3)) @ (one_one @ A))) @ K4)) @ (((power_power @ A) @ X) @ K4))) @ (((power_power @ A) @ (((plus_plus @ A) @ X) @ Y)) @ (((minus_minus @ nat) @ M2) @ K4)))))) @ ((set_ord_atMost @ nat) @ M2))))))))).
% 9.43/3.44 thf(fact_1592_gbinomial__sum__nat__pow2, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((divide_divide @ A) @ (((gbinomial @ A) @ ((semiring_1_of_nat @ A) @ (((plus_plus @ nat) @ M2) @ K4))) @ K4)) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ K4))))) @ ((set_ord_atMost @ nat) @ M2)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)))))))).
% 9.43/3.44 thf(fact_1593_bit__sum__mult__2__cases, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,B2:A,N:nat]: (((! [J3:nat]: ((~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ (suc @ J3)))))) => ((((bit_se5641148757651400278ts_bit @ A) @ (((plus_plus @ A) @ A3) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2))) @ N) = (((N = (zero_zero @ nat)) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))) & ((N != (zero_zero @ nat)) => (((bit_se5641148757651400278ts_bit @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ B2)) @ N))))))))))).
% 9.43/3.44 thf(fact_1594_and__not__num_Osimps_I8_J, axiom, (! [M2:num,N:num]: ((((bit_and_not_num @ (bit1 @ M2)) @ (bit0 @ N)) = (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N8:num]: (((some @ num) @ (bit1 @ N8))))) @ ((bit_and_not_num @ M2) @ N)))))).
% 9.43/3.44 thf(fact_1595_bit__rec, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A7:A,N5:nat]: ((((N5 = (zero_zero @ nat)) => (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A7)))) & ((N5 != (zero_zero @ nat)) => (((bit_se5641148757651400278ts_bit @ A) @ (((divide_divide @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ (((minus_minus @ nat) @ N5) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1596_and__nat__unfold, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M:nat,N5:nat]: (((((if @ nat) @ ((M = (zero_zero @ nat)) | (N5 = (zero_zero @ nat)))) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ nat) @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))).
% 9.43/3.44 thf(fact_1597_and__nat__rec, axiom, ((bit_se5824344872417868541ns_and @ nat) = (^ [M:nat,N5:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M))) & (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N5)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ nat) @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1598_set__bit__eq, axiom, ((bit_se5668285175392031749et_bit @ int) = (^ [N5:nat,K4:int]: ((((plus_plus @ int) @ K4) @ (((times_times @ int) @ ((zero_neq_one_of_bool @ int) @ (~ ((((bit_se5641148757651400278ts_bit @ int) @ K4) @ N5))))) @ (((power_power @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ N5))))))).
% 9.43/3.44 thf(fact_1599_and__not__num_Oelims, axiom, (! [X:num,Xa:num,Y:(option @ num)]: (((((bit_and_not_num @ X) @ Xa) = Y) => (((X = one2) => ((Xa = one2) => (Y != (none @ num)))) => (((X = one2) => ((? [N3:num]: ((Xa = (bit0 @ N3)))) => (Y != ((some @ num) @ one2)))) => (((X = one2) => ((? [N3:num]: ((Xa = (bit1 @ N3)))) => (Y != (none @ num)))) => ((! [M5:num]: (((X = (bit0 @ M5)) => ((Xa = one2) => (Y != ((some @ num) @ (bit0 @ M5))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M5) @ N3))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M5) @ N3))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => ((Xa = one2) => (Y != ((some @ num) @ (bit0 @ M5))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N8:num]: (((some @ num) @ (bit1 @ N8))))) @ ((bit_and_not_num @ M5) @ N3))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M5) @ N3))))))))))))))))))))))).
% 9.43/3.44 thf(fact_1600_and__not__num_Opelims, axiom, (! [X:num,Xa:num,Y:(option @ num)]: (((((bit_and_not_num @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ X) @ Xa)) => (((X = one2) => ((Xa = one2) => ((Y = (none @ num)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ one2) @ one2))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((some @ num) @ one2)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ N3)))))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = (none @ num)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ N3)))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => ((Xa = one2) => ((Y = ((some @ num) @ (bit0 @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit0 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit1 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => ((Xa = one2) => ((Y = ((some @ num) @ (bit0 @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N8:num]: (((some @ num) @ (bit1 @ N8))))) @ ((bit_and_not_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit0 @ N3)))))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_and_not_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_and_not_num_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit1 @ N3))))))))))))))))))))))))))).
% 9.43/3.44 thf(fact_1601_and__num_Opelims, axiom, (! [X:num,Xa:num,Y:(option @ num)]: (((((bit_un7362597486090784418nd_num @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ X) @ Xa)) => (((X = one2) => ((Xa = one2) => ((Y = ((some @ num) @ one2)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ one2))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = (none @ num)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ N3)))))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((some @ num) @ one2)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ N3)))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => ((Xa = one2) => ((Y = (none @ num)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit0 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit1 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => ((Xa = one2) => ((Y = ((some @ num) @ one2)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit0 @ N3)))))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = (((((case_option @ (option @ num)) @ num) @ ((some @ num) @ one2)) @ (^ [N8:num]: (((some @ num) @ (bit1 @ N8))))) @ ((bit_un7362597486090784418nd_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4731106466462545111um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit1 @ N3))))))))))))))))))))))))))).
% 9.43/3.44 thf(fact_1602_gbinomial__code, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A7:A,K4:nat]: (((((if @ A) @ (K4 = (zero_zero @ nat))) @ (one_one @ A)) @ (((divide_divide @ A) @ (((((set_fo6178422350223883121st_nat @ A) @ (^ [L3:nat]: (((times_times @ A) @ (((minus_minus @ A) @ A7) @ ((semiring_1_of_nat @ A) @ L3)))))) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ K4) @ (one_one @ nat))) @ (one_one @ A))) @ ((semiring_char_0_fact @ A) @ K4)))))))))).
% 9.43/3.44 thf(fact_1603_xor__num_Oelims, axiom, (! [X:num,Xa:num,Y:(option @ num)]: (((((bit_un2480387367778600638or_num @ X) @ Xa) = Y) => (((X = one2) => ((Xa = one2) => (Y != (none @ num)))) => (((X = one2) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != ((some @ num) @ (bit1 @ N3))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != ((some @ num) @ (bit0 @ N3))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => ((Xa = one2) => (Y != ((some @ num) @ (bit1 @ M5))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un2480387367778600638or_num @ M5) @ N3))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M5) @ N3)))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => ((Xa = one2) => (Y != ((some @ num) @ (bit0 @ M5))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => (Y != ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M5) @ N3)))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => (Y != ((((map_option @ num) @ num) @ bit0) @ ((bit_un2480387367778600638or_num @ M5) @ N3))))))))))))))))))))))).
% 9.43/3.44 thf(fact_1604_choose__alternating__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ I3)) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ I3)))))) @ ((set_ord_atMost @ nat) @ N)) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1605_choose__linear__sum, axiom, (! [N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [I3:nat]: ((((times_times @ nat) @ I3) @ ((binomial @ N) @ I3))))) @ ((set_ord_atMost @ nat) @ N)) = (((times_times @ nat) @ N) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 9.43/3.44 thf(fact_1606_binomial__r__part__sum, axiom, (! [M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (binomial @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2)) @ (one_one @ nat)))) @ ((set_ord_atMost @ nat) @ M2)) = (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2)))))).
% 9.43/3.44 thf(fact_1607_sum_Otriangle__reindex, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > (nat > A)),N:nat]: ((((((groups7311177749621191930dd_sum @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ J2)) @ N)))))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K4) @ I3))))) @ ((set_ord_atMost @ nat) @ K4))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1608_lessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((set_ord_lessThan @ A) @ X) = ((set_ord_lessThan @ A) @ Y)) = (X = Y)))))))).
% 9.43/3.44 thf(fact_1609_lessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_lessThan @ A) @ K)) = (((ord_less @ A) @ I) @ K)))))))).
% 9.43/3.44 thf(fact_1610_binomial__n__n, axiom, (! [N:nat]: ((((binomial @ N) @ N) = (one_one @ nat))))).
% 9.43/3.44 thf(fact_1611_lessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_lessThan @ A) @ X)) @ ((set_ord_lessThan @ A) @ Y)) = (((ord_less_eq @ A) @ X) @ Y)))))))).
% 9.43/3.44 thf(fact_1612_binomial__Suc__Suc, axiom, (! [N:nat,K:nat]: ((((binomial @ (suc @ N)) @ (suc @ K)) = (((plus_plus @ nat) @ ((binomial @ N) @ K)) @ ((binomial @ N) @ (suc @ K))))))).
% 9.43/3.44 thf(fact_1613_binomial__n__0, axiom, (! [N:nat]: ((((binomial @ N) @ (zero_zero @ nat)) = (one_one @ nat))))).
% 9.43/3.44 thf(fact_1614_sum_OlessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))) @ (G @ N))))))))).
% 9.43/3.44 thf(fact_1615_choose__one, axiom, (! [N:nat]: ((((binomial @ N) @ (one_one @ nat)) = N)))).
% 9.43/3.44 thf(fact_1616_lessThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_lessThan @ A) = (^ [U2:A]: (((collect @ A) @ (^ [X3:A]: ((((ord_less @ A) @ X3) @ U2))))))))))).
% 9.43/3.44 thf(fact_1617_choose__mult__lemma, axiom, (! [M2:nat,R:nat,K:nat]: (((((times_times @ nat) @ ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ M2) @ R)) @ K)) @ (((plus_plus @ nat) @ M2) @ K))) @ ((binomial @ (((plus_plus @ nat) @ M2) @ K)) @ K)) = (((times_times @ nat) @ ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ M2) @ R)) @ K)) @ K)) @ ((binomial @ (((plus_plus @ nat) @ M2) @ R)) @ M2)))))).
% 9.43/3.44 thf(fact_1618_lessThan__Suc__atMost, axiom, (! [K:nat]: ((((set_ord_lessThan @ nat) @ (suc @ K)) = ((set_ord_atMost @ nat) @ K))))).
% 9.43/3.44 thf(fact_1619_bit__integer_Oabs__eq, axiom, (! [X:int]: ((((bit_se5641148757651400278ts_bit @ code_integer) @ (code_integer_of_int @ X)) = ((bit_se5641148757651400278ts_bit @ int) @ X))))).
% 9.43/3.44 thf(fact_1620_lessThan__strict__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:A,N:A]: (((((ord_less @ (set @ A)) @ ((set_ord_lessThan @ A) @ M2)) @ ((set_ord_lessThan @ A) @ N)) = (((ord_less @ A) @ M2) @ N)))))))).
% 9.43/3.44 thf(fact_1621_Suc__times__binomial__add, axiom, (! [A3:nat,B2:nat]: (((((times_times @ nat) @ (suc @ A3)) @ ((binomial @ (suc @ (((plus_plus @ nat) @ A3) @ B2))) @ (suc @ A3))) = (((times_times @ nat) @ (suc @ B2)) @ ((binomial @ (suc @ (((plus_plus @ nat) @ A3) @ B2))) @ A3)))))).
% 9.43/3.44 thf(fact_1622_binomial__absorb__comp, axiom, (! [N:nat,K:nat]: (((((times_times @ nat) @ (((minus_minus @ nat) @ N) @ K)) @ ((binomial @ N) @ K)) = (((times_times @ nat) @ N) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ K)))))).
% 9.43/3.44 thf(fact_1623_fact__fact__dvd__fact, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:nat,N:nat]: ((((dvd_dvd @ A) @ (((times_times @ A) @ ((semiring_char_0_fact @ A) @ K)) @ ((semiring_char_0_fact @ A) @ N))) @ ((semiring_char_0_fact @ A) @ (((plus_plus @ nat) @ K) @ N))))))))).
% 9.43/3.44 thf(fact_1624_sum_Onat__diff__reindex, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ N) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1625_Iic__subset__Iio__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ A3)) @ ((set_ord_lessThan @ A) @ B2)) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.44 thf(fact_1626_binomial__absorption, axiom, (! [K:nat,N:nat]: (((((times_times @ nat) @ (suc @ K)) @ ((binomial @ N) @ (suc @ K))) = (((times_times @ nat) @ N) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ K)))))).
% 9.43/3.44 thf(fact_1627_sum__diff__distrib, axiom, (! [A:$tType]: (((ord @ A) => (! [Q:(A > nat),P:(A > nat),N:A]: (((! [X2:A]: ((((ord_less_eq @ nat) @ (Q @ X2)) @ (P @ X2)))) => ((((minus_minus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ P) @ ((set_ord_lessThan @ A) @ N))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ Q) @ ((set_ord_lessThan @ A) @ N))) = ((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X3:A]: ((((minus_minus @ nat) @ (P @ X3)) @ (Q @ X3))))) @ ((set_ord_lessThan @ A) @ N)))))))))).
% 9.43/3.44 thf(fact_1628_sum_OlessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1629_sum__lessThan__telescope, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F:(nat > A),M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N5:nat]: ((((minus_minus @ A) @ (F @ (suc @ N5))) @ (F @ N5))))) @ ((set_ord_lessThan @ nat) @ M2)) = (((minus_minus @ A) @ (F @ M2)) @ (F @ (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_1630_sum__lessThan__telescope_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [F:(nat > A),M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [N5:nat]: ((((minus_minus @ A) @ (F @ N5)) @ (F @ (suc @ N5)))))) @ ((set_ord_lessThan @ nat) @ M2)) = (((minus_minus @ A) @ (F @ (zero_zero @ nat))) @ (F @ M2))))))))).
% 9.43/3.44 thf(fact_1631_sum_OatLeast1__atMost__eq, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((G @ (suc @ K4))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1632_sum__bounds__lt__plus1, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),Mm:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((F @ (suc @ K4))))) @ ((set_ord_lessThan @ nat) @ Mm)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ Mm))))))))).
% 9.43/3.44 thf(fact_1633_sum_Onested__swap_H, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (A3 @ I3)) @ ((set_ord_lessThan @ nat) @ I3))))) @ ((set_ord_atMost @ nat) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J2:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((A3 @ I3) @ J2)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J2)) @ N))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1634_sum__choose__lower, axiom, (! [R:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K4:nat]: (((binomial @ (((plus_plus @ nat) @ R) @ K4)) @ K4)))) @ ((set_ord_atMost @ nat) @ N)) = ((binomial @ (suc @ (((plus_plus @ nat) @ R) @ N))) @ N))))).
% 9.43/3.44 thf(fact_1635_choose__rising__sum_I2_J, axiom, (! [N:nat,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J2:nat]: (((binomial @ (((plus_plus @ nat) @ N) @ J2)) @ N)))) @ ((set_ord_atMost @ nat) @ M2)) = ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ N) @ M2)) @ (one_one @ nat))) @ M2))))).
% 9.43/3.44 thf(fact_1636_choose__rising__sum_I1_J, axiom, (! [N:nat,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [J2:nat]: (((binomial @ (((plus_plus @ nat) @ N) @ J2)) @ N)))) @ ((set_ord_atMost @ nat) @ M2)) = ((binomial @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ N) @ M2)) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1637_binomial__code, axiom, (binomial = (^ [N5:nat,K4:nat]: (((((if @ nat) @ (((ord_less @ nat) @ N5) @ K4)) @ (zero_zero @ nat)) @ ((((if @ nat) @ (((ord_less @ nat) @ N5) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K4))) @ ((binomial @ N5) @ (((minus_minus @ nat) @ N5) @ K4))) @ (((divide_divide @ nat) @ (((((set_fo6178422350223883121st_nat @ nat) @ (times_times @ nat)) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ N5) @ K4)) @ (one_one @ nat))) @ N5) @ (one_one @ nat))) @ ((semiring_char_0_fact @ nat) @ K4)))))))).
% 9.43/3.44 thf(fact_1638_choose__reduce__nat, axiom, (! [N:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => (((binomial @ N) @ K) = (((plus_plus @ nat) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat)))) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ K)))))))).
% 9.43/3.44 thf(fact_1639_times__binomial__minus1__eq, axiom, (! [K:nat,N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ K) => ((((times_times @ nat) @ K) @ ((binomial @ N) @ K)) = (((times_times @ nat) @ N) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (((minus_minus @ nat) @ K) @ (one_one @ nat))))))))).
% 9.43/3.44 thf(fact_1640_one__diff__power__eq, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: (((((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ N)) = (((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1641_power__diff__1__eq, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: (((((minus_minus @ A) @ (((power_power @ A) @ X) @ N)) @ (one_one @ A)) = (((times_times @ A) @ (((minus_minus @ A) @ X) @ (one_one @ A))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1642_geometric__sum, axiom, (! [A:$tType]: (((field @ A) => (! [X:A,N:nat]: (((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_lessThan @ nat) @ N)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ X) @ N)) @ (one_one @ A))) @ (((minus_minus @ A) @ X) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1643_sum_OatMost__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N)) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1644_vandermonde, axiom, (! [M2:nat,N:nat,R:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K4:nat]: ((((times_times @ nat) @ ((binomial @ M2) @ K4)) @ ((binomial @ N) @ (((minus_minus @ nat) @ R) @ K4)))))) @ ((set_ord_atMost @ nat) @ R)) = ((binomial @ (((plus_plus @ nat) @ M2) @ N)) @ R))))).
% 9.43/3.44 thf(fact_1645_binomial__addition__formula, axiom, (! [N:nat,K:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((binomial @ N) @ (suc @ K)) = (((plus_plus @ nat) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ (suc @ K))) @ ((binomial @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ K))))))).
% 9.43/3.44 thf(fact_1646_power__diff__sumr2, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat,Y:A]: (((((minus_minus @ A) @ (((power_power @ A) @ X) @ N)) @ (((power_power @ A) @ Y) @ N)) = (((times_times @ A) @ (((minus_minus @ A) @ X) @ Y)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ N) @ (suc @ I3)))) @ (((power_power @ A) @ X) @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1647_diff__power__eq__sum, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat,Y:A]: (((((minus_minus @ A) @ (((power_power @ A) @ X) @ (suc @ N))) @ (((power_power @ A) @ Y) @ (suc @ N))) = (((times_times @ A) @ (((minus_minus @ A) @ X) @ Y)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [P6:nat]: ((((times_times @ A) @ (((power_power @ A) @ X) @ P6)) @ (((power_power @ A) @ Y) @ (((minus_minus @ nat) @ N) @ P6)))))) @ ((set_ord_lessThan @ nat) @ (suc @ N))))))))))).
% 9.43/3.44 thf(fact_1648_sum__gp__strict, axiom, (! [A:$tType]: ((((division_ring @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: ((((X = (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_lessThan @ nat) @ N)) = ((semiring_1_of_nat @ A) @ N))) & ((X != (one_one @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ ((power_power @ A) @ X)) @ ((set_ord_lessThan @ nat) @ N)) = (((divide_divide @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ N))) @ (((minus_minus @ A) @ (one_one @ A)) @ X))))))))))).
% 9.43/3.44 thf(fact_1649_binomial, axiom, (! [A3:nat,B2:nat,N:nat]: (((((power_power @ nat) @ (((plus_plus @ nat) @ A3) @ B2)) @ N) = ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [K4:nat]: ((((times_times @ nat) @ (((times_times @ nat) @ ((semiring_1_of_nat @ nat) @ ((binomial @ N) @ K4))) @ (((power_power @ nat) @ A3) @ K4))) @ (((power_power @ nat) @ B2) @ (((minus_minus @ nat) @ N) @ K4)))))) @ ((set_ord_atMost @ nat) @ N)))))).
% 9.43/3.44 thf(fact_1650_choose__two, axiom, (! [N:nat]: ((((binomial @ N) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) = (((divide_divide @ nat) @ (((times_times @ nat) @ N) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1651_one__diff__power__eq_H, axiom, (! [A:$tType]: ((((monoid_mult @ A) & (comm_ring @ A)) => (! [X:A,N:nat]: (((((minus_minus @ A) @ (one_one @ A)) @ (((power_power @ A) @ X) @ N)) = (((times_times @ A) @ (((minus_minus @ A) @ (one_one @ A)) @ X)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((power_power @ A) @ X) @ (((minus_minus @ nat) @ N) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1652_binomial__ring, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,B2:A,N:nat]: (((((power_power @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ N) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ K4))) @ (((power_power @ A) @ A3) @ K4))) @ (((power_power @ A) @ B2) @ (((minus_minus @ nat) @ N) @ K4)))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1653_xor__num_Osimps_I6_J, axiom, (! [M2:num,N:num]: ((((bit_un2480387367778600638or_num @ (bit0 @ M2)) @ (bit1 @ N)) = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M2) @ N))))))).
% 9.43/3.44 thf(fact_1654_xor__num_Osimps_I8_J, axiom, (! [M2:num,N:num]: ((((bit_un2480387367778600638or_num @ (bit1 @ M2)) @ (bit0 @ N)) = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M2) @ N))))))).
% 9.43/3.44 thf(fact_1655_choose__alternating__linear__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [N:nat]: (((N != (one_one @ nat)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ I3)) @ ((semiring_1_of_nat @ A) @ I3))) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ I3)))))) @ ((set_ord_atMost @ nat) @ N)) = (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1656_fact__Suc__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (suc @ (zero_zero @ nat))) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1657_fact__1, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (one_one @ nat)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1658_fact__0, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (((semiring_char_0_fact @ A) @ (zero_zero @ nat)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1659_fact__code, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N5:nat]: (((semiring_1_of_nat @ A) @ (((((set_fo6178422350223883121st_nat @ nat) @ (times_times @ nat)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N5) @ (one_one @ nat)))))))))).
% 9.43/3.44 thf(fact_1660_fact__reduce, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((semiring_char_0_fact @ A) @ N) = (((times_times @ A) @ ((semiring_1_of_nat @ A) @ N)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1661_fact__num__eq__if, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [M:nat]: (((((if @ A) @ (M = (zero_zero @ nat))) @ (one_one @ A)) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ M)) @ ((semiring_char_0_fact @ A) @ (((minus_minus @ nat) @ M) @ (one_one @ nat)))))))))))).
% 9.43/3.44 thf(fact_1662_xor__num_Opelims, axiom, (! [X:num,Xa:num,Y:(option @ num)]: (((((bit_un2480387367778600638or_num @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ X) @ Xa)) => (((X = one2) => ((Xa = one2) => ((Y = (none @ num)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ one2))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((some @ num) @ (bit1 @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ N3)))))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((some @ num) @ (bit0 @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ N3)))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => ((Xa = one2) => ((Y = ((some @ num) @ (bit1 @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_un2480387367778600638or_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit0 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M5) @ N3)))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit1 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => ((Xa = one2) => ((Y = ((some @ num) @ (bit0 @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = ((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_un2480387367778600638or_num @ M5) @ N3)))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit0 @ N3)))))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = ((((map_option @ num) @ num) @ bit0) @ ((bit_un2480387367778600638or_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un2901131394128224187um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit1 @ N3))))))))))))))))))))))))))).
% 9.43/3.44 thf(fact_1663_dvd__fact, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ M2) => ((((ord_less_eq @ nat) @ M2) @ N) => (((dvd_dvd @ nat) @ M2) @ ((semiring_char_0_fact @ nat) @ N))))))).
% 9.43/3.44 thf(fact_1664_fact__ge__1, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [N:nat]: ((((ord_less_eq @ A) @ (one_one @ A)) @ ((semiring_char_0_fact @ A) @ N)))))))).
% 9.43/3.44 thf(fact_1665_pochhammer__double, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [Z2:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ Z2)) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ N))) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ N))))))))).
% 9.43/3.44 thf(fact_1666_fact__double, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [N:nat]: ((((semiring_char_0_fact @ A) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) = (((times_times @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))) @ (((comm_s3205402744901411588hammer @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2)))) @ N))) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1667_pochhammer__code, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A7:A,N5:nat]: (((((if @ A) @ (N5 = (zero_zero @ nat))) @ (one_one @ A)) @ (((((set_fo6178422350223883121st_nat @ A) @ (^ [O:nat]: (((times_times @ A) @ (((plus_plus @ A) @ A7) @ ((semiring_1_of_nat @ A) @ O)))))) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N5) @ (one_one @ nat))) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1668_xor__Suc__0__eq, axiom, (! [N:nat]: (((((bit_se5824344971392196577ns_xor @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))))).
% 9.43/3.44 thf(fact_1669_Suc__0__xor__eq, axiom, (! [N:nat]: (((((bit_se5824344971392196577ns_xor @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) @ ((zero_neq_one_of_bool @ nat) @ (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))))).
% 9.43/3.44 thf(fact_1670_push__bit__numeral__minus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:num]: (((((bit_se4730199178511100633sh_bit @ A) @ ((numeral_numeral @ nat) @ N)) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((numeral_numeral @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1671_num__of__integer__code, axiom, (code_num_of_integer = (^ [K4:code_integer]: (((((if @ num) @ (((ord_less_eq @ code_integer) @ K4) @ (one_one @ code_integer))) @ one2) @ (((((product_case_prod @ code_integer) @ code_integer) @ num) @ (^ [L3:code_integer,J2:code_integer]: (((((if @ num) @ (J2 = (zero_zero @ code_integer))) @ (((plus_plus @ num) @ (code_num_of_integer @ L3)) @ (code_num_of_integer @ L3))) @ (((plus_plus @ num) @ (((plus_plus @ num) @ (code_num_of_integer @ L3)) @ (code_num_of_integer @ L3))) @ one2))))) @ ((code_divmod_integer @ K4) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))).
% 9.43/3.44 thf(fact_1672_Sum__Ico__nat, axiom, (! [M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = (((divide_divide @ nat) @ (((minus_minus @ nat) @ (((times_times @ nat) @ N) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))) @ (((times_times @ nat) @ M2) @ (((minus_minus @ nat) @ M2) @ (one_one @ nat))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))).
% 9.43/3.44 thf(fact_1673_ivl__diff, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,N:A,M2:A]: (((((ord_less_eq @ A) @ I) @ N) => ((((minus_minus @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ N)) = (((set_or7035219750837199246ssThan @ A) @ N) @ M2))))))))).
% 9.43/3.44 thf(fact_1674_push__bit__push__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M2:nat,N:nat,A3:A]: (((((bit_se4730199178511100633sh_bit @ A) @ M2) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ A3)) = (((bit_se4730199178511100633sh_bit @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ A3)))))))).
% 9.43/3.44 thf(fact_1675_lessThan__minus__lessThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [N:A,M2:A]: (((((minus_minus @ (set @ A)) @ ((set_ord_lessThan @ A) @ N)) @ ((set_ord_lessThan @ A) @ M2)) = (((set_or7035219750837199246ssThan @ A) @ M2) @ N)))))))).
% 9.43/3.44 thf(fact_1676_pochhammer__1, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (one_one @ nat)) = A3))))))).
% 9.43/3.44 thf(fact_1677_atLeastLessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) = ((((ord_less_eq @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))).
% 9.43/3.44 thf(fact_1678_ivl__subset, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:A,J:A,M2:A,N:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ N)) = ((((ord_less_eq @ A) @ J) @ I) | ((((ord_less_eq @ A) @ M2) @ I) & (((ord_less_eq @ A) @ J) @ N)))))))))).
% 9.43/3.44 thf(fact_1679_pochhammer__0, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (zero_zero @ nat)) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1680_xor__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = ((numeral_numeral @ A) @ (bit1 @ Y))))))))).
% 9.43/3.44 thf(fact_1681_xor__numerals_I2_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = ((numeral_numeral @ A) @ (bit0 @ Y))))))))).
% 9.43/3.44 thf(fact_1682_xor__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ X))))))))).
% 9.43/3.44 thf(fact_1683_xor__numerals_I8_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit0 @ X))))))))).
% 9.43/3.44 thf(fact_1684_push__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ (one_one @ A)) = (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)))))))).
% 9.43/3.44 thf(fact_1685_sum_Oop__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N:nat,M2:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N))) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ (G @ N))))))))))).
% 9.43/3.44 thf(fact_1686_xor__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y))))))))))).
% 9.43/3.44 thf(fact_1687_xor__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y))))))))))).
% 9.43/3.44 thf(fact_1688_flip__bit__nat__def, axiom, ((bit_se8732182000553998342ip_bit @ nat) = (^ [M:nat,N5:nat]: ((((bit_se5824344971392196577ns_xor @ nat) @ N5) @ (((bit_se4730199178511100633sh_bit @ nat) @ M) @ (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1689_push__bit__integer_Oabs__eq, axiom, (! [Xa:nat,X:int]: (((((bit_se4730199178511100633sh_bit @ code_integer) @ Xa) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se4730199178511100633sh_bit @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1690_atLeastLessThan__inj_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) => ((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D3) => (B2 = D3)))))))))).
% 9.43/3.44 thf(fact_1691_atLeastLessThan__inj_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) => ((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D3) => (A3 = C2)))))))))).
% 9.43/3.44 thf(fact_1692_atLeastLessThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_less @ A) @ C2) @ D3) => (((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) = ((A3 = C2) & (B2 = D3))))))))))).
% 9.43/3.44 thf(fact_1693_flip__bit__eq__xor, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se8732182000553998342ip_bit @ A) = (^ [N5:nat,A7:A]: ((((bit_se5824344971392196577ns_xor @ A) @ A7) @ (((bit_se4730199178511100633sh_bit @ A) @ N5) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1694_push__bit__add, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A,B2:A]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ (((plus_plus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ A3)) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ B2))))))))).
% 9.43/3.44 thf(fact_1695_atLeastLessThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) => ((((ord_less_eq @ A) @ B2) @ A3) | ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_1696_atLeastLessThanSuc__atLeastAtMost, axiom, (! [L:nat,U:nat]: (((((set_or7035219750837199246ssThan @ nat) @ L) @ (suc @ U)) = (((set_or1337092689740270186AtMost @ nat) @ L) @ U))))).
% 9.43/3.44 thf(fact_1697_lessThan__atLeast0, axiom, ((set_ord_lessThan @ nat) = ((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)))).
% 9.43/3.44 thf(fact_1698_push__bit__take__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M2:nat,N:nat,A3:A]: (((((bit_se4730199178511100633sh_bit @ A) @ M2) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3)) = (((bit_se2584673776208193580ke_bit @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ (((bit_se4730199178511100633sh_bit @ A) @ M2) @ A3))))))))).
% 9.43/3.44 thf(fact_1699_pochhammer__fact, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & (comm_semiring_1 @ A)) => ((semiring_char_0_fact @ A) = ((comm_s3205402744901411588hammer @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_1700_sum_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1701_sum_Oshift__bounds__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1702_sum_Oivl__cong, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (comm_monoid_add @ A)) => (! [A3:B,C2:B,B2:B,D3:B,G:(B > A),H3:(B > A)]: (((A3 = C2) => ((B2 = D3) => ((! [X2:B]: (((((ord_less_eq @ B) @ C2) @ X2) => ((((ord_less @ B) @ X2) @ D3) => ((G @ X2) = (H3 @ X2)))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A3) @ B2)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ H3) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D3)))))))))))).
% 9.43/3.44 thf(fact_1703_sum_OatLeastLessThan__concat, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,P5:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ P5))))))))))).
% 9.43/3.44 thf(fact_1704_sum__diff__nat__ivl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M2:nat,N:nat,P5:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => ((((minus_minus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ P5))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))))))))))).
% 9.43/3.44 thf(fact_1705_pochhammer__0__left, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [N:nat]: ((((N = (zero_zero @ nat)) => ((((comm_s3205402744901411588hammer @ A) @ (zero_zero @ A)) @ N) = (one_one @ A))) & ((N != (zero_zero @ nat)) => ((((comm_s3205402744901411588hammer @ A) @ (zero_zero @ A)) @ N) = (zero_zero @ A)))))))))).
% 9.43/3.44 thf(fact_1706_sum_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1707_sum_OatLeastLessThan__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1708_concat__bit__eq, axiom, (bit_concat_bit = (^ [N5:nat,K4:int,L3:int]: ((((plus_plus @ int) @ (((bit_se2584673776208193580ke_bit @ int) @ N5) @ K4)) @ (((bit_se4730199178511100633sh_bit @ int) @ N5) @ L3)))))).
% 9.43/3.44 thf(fact_1709_atLeastAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) = ((((ord_less_eq @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_1710_atLeastLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_1711_sum__shift__lb__Suc0__0__upt, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [F:(nat > A),K:nat]: ((((F @ (zero_zero @ nat)) = (zero_zero @ A)) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ K)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ F) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K)))))))))).
% 9.43/3.44 thf(fact_1712_sum_OatLeast0__lessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ N))))))))).
% 9.43/3.44 thf(fact_1713_sum_OatLeast__Suc__lessThan, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = (((plus_plus @ A) @ (G @ M2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ N))))))))))).
% 9.43/3.44 thf(fact_1714_sum_OatLeastLessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:nat,B2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ A3) @ B2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ (suc @ B2))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2))) @ (G @ B2)))))))))).
% 9.43/3.44 thf(fact_1715_sum_Olast__plus, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((plus_plus @ A) @ (G @ N)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))))).
% 9.43/3.44 thf(fact_1716_pochhammer__rec, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = (((times_times @ A) @ A3) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ N))))))))).
% 9.43/3.44 thf(fact_1717_pochhammer__Suc, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ A3) @ N)) @ (((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ N)))))))))).
% 9.43/3.44 thf(fact_1718_pochhammer__rec_H, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [Z2:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ Z2) @ (suc @ N)) = (((times_times @ A) @ (((plus_plus @ A) @ Z2) @ ((semiring_1_of_nat @ A) @ N))) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ N))))))))).
% 9.43/3.44 thf(fact_1719_pochhammer__product_H, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [Z2:A,N:nat,M2:nat]: (((((comm_s3205402744901411588hammer @ A) @ Z2) @ (((plus_plus @ nat) @ N) @ M2)) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ N)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ ((semiring_1_of_nat @ A) @ N))) @ M2))))))))).
% 9.43/3.44 thf(fact_1720_bit__iff__and__push__bit__not__eq__0, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A7:A,N5:nat]: (((((bit_se5824344872417868541ns_and @ A) @ A7) @ (((bit_se4730199178511100633sh_bit @ A) @ N5) @ (one_one @ A))) != (zero_zero @ A))))))))).
% 9.43/3.44 thf(fact_1721_sum__Suc__diff_H, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [M2:nat,N:nat,F:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((((minus_minus @ A) @ (F @ (suc @ I3))) @ (F @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = (((minus_minus @ A) @ (F @ N)) @ (F @ M2)))))))))).
% 9.43/3.44 thf(fact_1722_sum_OatLeastLessThan__rev, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ (suc @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2))))))))).
% 9.43/3.44 thf(fact_1723_sum_Onested__swap, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (A3 @ I3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [J2:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: (((A3 @ I3) @ J2)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J2)) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 9.43/3.44 thf(fact_1724_sum_Onat__group, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),K:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [M:nat]: (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N) @ K)))))))))).
% 9.43/3.44 thf(fact_1725_numeral__xor__num, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [M2:num,N:num]: (((((bit_se5824344971392196577ns_xor @ A) @ ((numeral_numeral @ A) @ M2)) @ ((numeral_numeral @ A) @ N)) = (((((case_option @ A) @ num) @ (zero_zero @ A)) @ (numeral_numeral @ A)) @ ((bit_un2480387367778600638or_num @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1726_sum_Ohead__if, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [N:nat,M2:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M2) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (zero_zero @ A))) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ (G @ N))))))))))).
% 9.43/3.44 thf(fact_1727_sum_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((plus_plus @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))))))).
% 9.43/3.44 thf(fact_1728_sum_OatLeastLessThan__shift__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M2))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M2)))))))))).
% 9.43/3.44 thf(fact_1729_pochhammer__product, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [M2:nat,N:nat,Z2:A]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((comm_s3205402744901411588hammer @ A) @ Z2) @ N) = (((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ M2)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ ((semiring_1_of_nat @ A) @ M2))) @ (((minus_minus @ nat) @ N) @ M2))))))))))).
% 9.43/3.44 thf(fact_1730_sum_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),N:nat,M2:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2)) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M2))))))))).
% 9.43/3.44 thf(fact_1731_sum_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N5:nat]: ((((minus_minus @ nat) @ N5) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1732_pochhammer__absorb__comp, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [R:A,K:nat]: (((((times_times @ A) @ (((minus_minus @ A) @ R) @ ((semiring_1_of_nat @ A) @ K))) @ (((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ R)) @ K)) = (((times_times @ A) @ R) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ R)) @ (one_one @ A))) @ K))))))))).
% 9.43/3.44 thf(fact_1733_pochhammer__same, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) & ((comm_ring_1 @ A) & (semiri3467727345109120633visors @ A))) => (! [N:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ ((semiring_1_of_nat @ A) @ N))) @ N) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ ((semiring_char_0_fact @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1734_pochhammer__minus, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [B2:A,K:nat]: (((((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ B2)) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K))))))))).
% 9.43/3.44 thf(fact_1735_pochhammer__minus_H, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [B2:A,K:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ B2) @ ((semiring_1_of_nat @ A) @ K))) @ (one_one @ A))) @ K) = (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K)) @ (((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ B2)) @ K))))))))).
% 9.43/3.44 thf(fact_1736_gbinomial__pochhammer_H, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A7:A,K4:nat]: ((((divide_divide @ A) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ (((minus_minus @ A) @ A7) @ ((semiring_1_of_nat @ A) @ K4))) @ (one_one @ A))) @ K4)) @ ((semiring_char_0_fact @ A) @ K4))))))))).
% 9.43/3.44 thf(fact_1737_gbinomial__pochhammer, axiom, (! [A:$tType]: (((field_char_0 @ A) => ((gbinomial @ A) = (^ [A7:A,K4:nat]: ((((divide_divide @ A) @ (((times_times @ A) @ (((power_power @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ K4)) @ (((comm_s3205402744901411588hammer @ A) @ ((uminus_uminus @ A) @ A7)) @ K4))) @ ((semiring_char_0_fact @ A) @ K4))))))))).
% 9.43/3.44 thf(fact_1738_xor__nat__unfold, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M:nat,N5:nat]: (((((if @ nat) @ (M = (zero_zero @ nat))) @ N5) @ ((((if @ nat) @ (N5 = (zero_zero @ nat))) @ M) @ (((plus_plus @ nat) @ (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ nat) @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1739_num__of__integer_Oabs__eq, axiom, (! [X:int]: (((code_num_of_integer @ (code_integer_of_int @ X)) = (num_of_nat @ (nat2 @ X)))))).
% 9.43/3.44 thf(fact_1740_signed__take__bit__code, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4674362597316999326ke_bit @ A) = (^ [N5:nat,A7:A]: (((((if @ A) @ (((bit_se5641148757651400278ts_bit @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N5)) @ A7)) @ N5)) @ (((plus_plus @ A) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N5)) @ A7)) @ (((bit_se4730199178511100633sh_bit @ A) @ (suc @ N5)) @ ((uminus_uminus @ A) @ (one_one @ A))))) @ (((bit_se2584673776208193580ke_bit @ A) @ (suc @ N5)) @ A7))))))))).
% 9.43/3.44 thf(fact_1741_xor__nat__rec, axiom, ((bit_se5824344971392196577ns_xor @ nat) = (^ [M:nat,N5:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M))) != (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N5)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ nat) @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1742_xor__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ A3) @ (one_one @ A)) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))) @ ((zero_neq_one_of_bool @ A) @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))))).
% 9.43/3.44 thf(fact_1743_one__xor__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se5824344971392196577ns_xor @ A) @ (one_one @ A)) @ A3) = (((minus_minus @ A) @ (((plus_plus @ A) @ A3) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))) @ ((zero_neq_one_of_bool @ A) @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))))).
% 9.43/3.44 thf(fact_1744_pochhammer__binomial__sum, axiom, (! [A:$tType]: (((comm_ring_1 @ A) => (! [A3:A,B2:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ N) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (^ [K4:nat]: ((((times_times @ A) @ (((times_times @ A) @ ((semiring_1_of_nat @ A) @ ((binomial @ N) @ K4))) @ (((comm_s3205402744901411588hammer @ A) @ A3) @ K4))) @ (((comm_s3205402744901411588hammer @ A) @ B2) @ (((minus_minus @ nat) @ N) @ K4)))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1745_divmod__integer__def, axiom, (code_divmod_integer = (^ [K4:code_integer,L3:code_integer]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ K4) @ L3)) @ (((modulo_modulo @ code_integer) @ K4) @ L3)))))).
% 9.43/3.44 thf(fact_1746_sum__power2, axiom, (! [K:nat]: ((((((groups7311177749621191930dd_sum @ nat) @ nat) @ ((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ K)) = (((minus_minus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ K)) @ (one_one @ nat)))))).
% 9.43/3.44 thf(fact_1747_nat__of__integer__code, axiom, (code_nat_of_integer = (^ [K4:code_integer]: (((((if @ nat) @ (((ord_less_eq @ code_integer) @ K4) @ (zero_zero @ code_integer))) @ (zero_zero @ nat)) @ (((((product_case_prod @ code_integer) @ code_integer) @ nat) @ (^ [L3:code_integer,J2:code_integer]: (((((if @ nat) @ (J2 = (zero_zero @ code_integer))) @ (((plus_plus @ nat) @ (code_nat_of_integer @ L3)) @ (code_nat_of_integer @ L3))) @ (((plus_plus @ nat) @ (((plus_plus @ nat) @ (code_nat_of_integer @ L3)) @ (code_nat_of_integer @ L3))) @ (one_one @ nat)))))) @ ((code_divmod_integer @ K4) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))).
% 9.43/3.44 thf(fact_1748_pochhammer__times__pochhammer__half, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [Z2:A,N:nat]: (((((times_times @ A) @ (((comm_s3205402744901411588hammer @ A) @ Z2) @ (suc @ N))) @ (((comm_s3205402744901411588hammer @ A) @ (((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K4:nat]: ((((plus_plus @ A) @ Z2) @ (((divide_divide @ A) @ ((semiring_1_of_nat @ A) @ K4)) @ ((numeral_numeral @ A) @ (bit0 @ one2))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (one_one @ nat))))))))))).
% 9.43/3.44 thf(fact_1749_int__of__integer__code, axiom, (code_int_of_integer = (^ [K4:code_integer]: (((((if @ int) @ (((ord_less @ code_integer) @ K4) @ (zero_zero @ code_integer))) @ ((uminus_uminus @ int) @ (code_int_of_integer @ ((uminus_uminus @ code_integer) @ K4)))) @ ((((if @ int) @ (K4 = (zero_zero @ code_integer))) @ (zero_zero @ int)) @ (((((product_case_prod @ code_integer) @ code_integer) @ int) @ (^ [L3:code_integer,J2:code_integer]: (((((if @ int) @ (J2 = (zero_zero @ code_integer))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (code_int_of_integer @ L3))) @ (((plus_plus @ int) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (code_int_of_integer @ L3))) @ (one_one @ int)))))) @ ((code_divmod_integer @ K4) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1750_divmod__integer__eq__cases, axiom, (code_divmod_integer = (^ [K4:code_integer,L3:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (K4 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (L3 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ K4)) @ (((((((comp @ code_integer) @ (((product_prod @ code_integer) @ code_integer) > ((product_prod @ code_integer) @ code_integer))) @ code_integer) @ (((((comp @ (code_integer > code_integer)) @ (((product_prod @ code_integer) @ code_integer) > ((product_prod @ code_integer) @ code_integer))) @ code_integer) @ (((product_apsnd @ code_integer) @ code_integer) @ code_integer)) @ (times_times @ code_integer))) @ (sgn_sgn @ code_integer)) @ L3) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((sgn_sgn @ code_integer) @ K4) = ((sgn_sgn @ code_integer) @ L3))) @ ((code_divmod_abs @ K4) @ L3)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R3:code_integer,S4:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S4 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ ((abs_abs @ code_integer) @ L3)) @ S4)))))) @ ((code_divmod_abs @ K4) @ L3)))))))))).
% 9.43/3.44 thf(fact_1751_time__def, axiom, (! [A:$tType]: (((time_time @ A) = (^ [C3:(heap_Time_Heap @ A),H:(heap_ext @ product_unit)]: ((((((case_option @ nat) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (zero_zero @ nat)) @ ((((product_case_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ nat) @ (^ [X3:A]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ nat) @ (^ [Xa4:(heap_ext @ product_unit),T3:nat]: (T3))))))) @ (((heap_Time_execute @ A) @ C3) @ H)))))))).
% 9.43/3.44 thf(fact_1752_bit__cut__integer__def, axiom, (code_bit_cut_integer = (^ [K4:code_integer]: (((((product_Pair @ code_integer) @ $o) @ (((divide_divide @ code_integer) @ K4) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ (~ ((((dvd_dvd @ code_integer) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))) @ K4)))))))).
% 9.43/3.44 thf(fact_1753_integer__of__num_I3_J, axiom, (! [N:num]: (((code_integer_of_num @ (bit1 @ N)) = (((plus_plus @ code_integer) @ (((plus_plus @ code_integer) @ (code_integer_of_num @ N)) @ (code_integer_of_num @ N))) @ (one_one @ code_integer)))))).
% 9.43/3.44 thf(fact_1754_less__eq__int_Orep__eq, axiom, ((ord_less_eq @ int) = (^ [X3:int,Xa4:int]: (((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [Y3:nat,Z5:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ Y3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Z5)))))))) @ (rep_Integ @ X3)) @ (rep_Integ @ Xa4)))))).
% 9.43/3.44 thf(fact_1755_nat__of__integer__of__nat, axiom, (! [N:nat]: (((code_nat_of_integer @ ((semiring_1_of_nat @ code_integer) @ N)) = N)))).
% 9.43/3.44 thf(fact_1756_int__of__integer__of__nat, axiom, (! [N:nat]: (((code_int_of_integer @ ((semiring_1_of_nat @ code_integer) @ N)) = ((semiring_1_of_nat @ int) @ N))))).
% 9.43/3.44 thf(fact_1757_integer__of__int__int__of__integer, axiom, (! [K:code_integer]: (((code_integer_of_int @ (code_int_of_integer @ K)) = K)))).
% 9.43/3.44 thf(fact_1758_int__of__integer__integer__of__int, axiom, (! [K:int]: (((code_int_of_integer @ (code_integer_of_int @ K)) = K)))).
% 9.43/3.44 thf(fact_1759_int__of__integer__inverse, axiom, (! [X:code_integer]: (((code_integer_of_int @ (code_int_of_integer @ X)) = X)))).
% 9.43/3.44 thf(fact_1760_apsnd__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),X:A,Y:C]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ ((((product_Pair @ A) @ C) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ X) @ (F @ Y)))))).
% 9.43/3.44 thf(fact_1761_prod_Oneutral__const, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ (^ [Uu:B]: ((one_one @ A)))) @ A4) = (one_one @ A)))))))).
% 9.43/3.44 thf(fact_1762_of__int__integer__of, axiom, (! [K:code_integer]: ((((ring_1_of_int @ code_integer) @ (code_int_of_integer @ K)) = K)))).
% 9.43/3.44 thf(fact_1763_int__of__integer__of__int, axiom, (! [K:int]: (((code_int_of_integer @ ((ring_1_of_int @ code_integer) @ K)) = K)))).
% 9.43/3.44 thf(fact_1764_zero__integer_Orep__eq, axiom, ((code_int_of_integer @ (zero_zero @ code_integer)) = (zero_zero @ int))).
% 9.43/3.44 thf(fact_1765_int__of__integer__numeral, axiom, (! [K:num]: (((code_int_of_integer @ ((numeral_numeral @ code_integer) @ K)) = ((numeral_numeral @ int) @ K))))).
% 9.43/3.44 thf(fact_1766_plus__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((plus_plus @ code_integer) @ X) @ Xa)) = (((plus_plus @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1767_one__integer_Orep__eq, axiom, ((code_int_of_integer @ (one_one @ code_integer)) = (one_one @ int))).
% 9.43/3.44 thf(fact_1768_uminus__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ ((uminus_uminus @ code_integer) @ X)) = ((uminus_uminus @ int) @ (code_int_of_integer @ X)))))).
% 9.43/3.44 thf(fact_1769_times__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((times_times @ code_integer) @ X) @ Xa)) = (((times_times @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1770_minus__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((minus_minus @ code_integer) @ X) @ Xa)) = (((minus_minus @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1771_abs__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ ((abs_abs @ code_integer) @ X)) = ((abs_abs @ int) @ (code_int_of_integer @ X)))))).
% 9.43/3.44 thf(fact_1772_divide__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((divide_divide @ code_integer) @ X) @ Xa)) = (((divide_divide @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1773_modulo__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((modulo_modulo @ code_integer) @ X) @ Xa)) = (((modulo_modulo @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1774_sgn__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ ((sgn_sgn @ code_integer) @ X)) = ((sgn_sgn @ int) @ (code_int_of_integer @ X)))))).
% 9.43/3.44 thf(fact_1775_prod_OatMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N))) @ (G @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_1776_prod_OlessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))) @ (G @ N))))))))).
% 9.43/3.44 thf(fact_1777_nat__of__integer__non__positive, axiom, (! [K:code_integer]: (((((ord_less_eq @ code_integer) @ K) @ (zero_zero @ code_integer)) => ((code_nat_of_integer @ K) = (zero_zero @ nat)))))).
% 9.43/3.44 thf(fact_1778_prod_Ocl__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N:nat,M2:nat,G:(nat > A)]: ((((((ord_less @ nat) @ (suc @ N)) @ M2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ (suc @ N)) @ M2))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ (G @ (suc @ N)))))))))))).
% 9.43/3.44 thf(fact_1779_prod_Oop__ivl__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N:nat,M2:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N))) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ (G @ N))))))))))).
% 9.43/3.44 thf(fact_1780_xor__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((bit_se5824344971392196577ns_xor @ code_integer) @ X) @ Xa)) = (((bit_se5824344971392196577ns_xor @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1781_push__bit__integer_Orep__eq, axiom, (! [X:nat,Xa:code_integer]: (((code_int_of_integer @ (((bit_se4730199178511100633sh_bit @ code_integer) @ X) @ Xa)) = (((bit_se4730199178511100633sh_bit @ int) @ X) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1782_xor__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((bit_se5824344971392196577ns_xor @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se5824344971392196577ns_xor @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1783_nat__of__integer_Orep__eq, axiom, (code_nat_of_integer = (^ [X3:code_integer]: ((nat2 @ (code_int_of_integer @ X3)))))).
% 9.43/3.44 thf(fact_1784_integer__eq__iff, axiom, ((^ [Y5:code_integer,Z4:code_integer]: ((Y5 = Z4))) = (^ [K4:code_integer,L3:code_integer]: (((code_int_of_integer @ K4) = (code_int_of_integer @ L3)))))).
% 9.43/3.44 thf(fact_1785_integer__eqI, axiom, (! [K:code_integer,L:code_integer]: ((((code_int_of_integer @ K) = (code_int_of_integer @ L)) => (K = L))))).
% 9.43/3.44 thf(fact_1786_int__of__integer__inject, axiom, (! [X:code_integer,Y:code_integer]: ((((code_int_of_integer @ X) = (code_int_of_integer @ Y)) = (X = Y))))).
% 9.43/3.44 thf(fact_1787_prod_Onot__neutral__contains__not__neutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A),A4:(set @ B)]: (((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4) != (one_one @ A)) => (~ ((! [A5:B]: (((((member @ B) @ A5) @ A4) => ((G @ A5) = (one_one @ A)))))))))))))).
% 9.43/3.44 thf(fact_1788_prod_Oneutral, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),G:(B > A)]: (((! [X2:B]: (((((member @ B) @ X2) @ A4) => ((G @ X2) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1789_apsnd__compose, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,F:(C > B),G:(D > C),X:((product_prod @ A) @ D)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apsnd @ D) @ C) @ A) @ G) @ X)) = (((((product_apsnd @ D) @ B) @ A) @ (((((comp @ C) @ B) @ D) @ F) @ G)) @ X))))).
% 9.43/3.44 thf(fact_1790_prod__ge__1, axiom, (! [A:$tType,B:$tType]: (((linord181362715937106298miring @ A) => (! [A4:(set @ B),F:(B > A)]: (((! [X2:B]: (((((member @ B) @ X2) @ A4) => (((ord_less_eq @ A) @ (one_one @ A)) @ (F @ X2))))) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A4))))))))).
% 9.43/3.44 thf(fact_1791_int__of__integer__less__iff, axiom, (! [X:code_integer,Y:code_integer]: (((((ord_less @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Y)) = (((ord_less @ code_integer) @ X) @ Y))))).
% 9.43/3.44 thf(fact_1792_integer__less__iff, axiom, ((ord_less @ code_integer) = (^ [K4:code_integer,L3:code_integer]: ((((ord_less @ int) @ (code_int_of_integer @ K4)) @ (code_int_of_integer @ L3)))))).
% 9.43/3.44 thf(fact_1793_less__integer_Orep__eq, axiom, ((ord_less @ code_integer) = (^ [X3:code_integer,Xa4:code_integer]: ((((ord_less @ int) @ (code_int_of_integer @ X3)) @ (code_int_of_integer @ Xa4)))))).
% 9.43/3.44 thf(fact_1794_less__eq__integer_Orep__eq, axiom, ((ord_less_eq @ code_integer) = (^ [X3:code_integer,Xa4:code_integer]: ((((ord_less_eq @ int) @ (code_int_of_integer @ X3)) @ (code_int_of_integer @ Xa4)))))).
% 9.43/3.44 thf(fact_1795_integer__less__eq__iff, axiom, ((ord_less_eq @ code_integer) = (^ [K4:code_integer,L3:code_integer]: ((((ord_less_eq @ int) @ (code_int_of_integer @ K4)) @ (code_int_of_integer @ L3)))))).
% 9.43/3.44 thf(fact_1796_prod_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1797_prod_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1798_prod_Oshift__bounds__cl__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1799_prod_Oshift__bounds__nat__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1800_atLeastLessThanPlusOne__atLeastAtMost__integer, axiom, (! [L:code_integer,U:code_integer]: (((((set_or7035219750837199246ssThan @ code_integer) @ L) @ (((plus_plus @ code_integer) @ U) @ (one_one @ code_integer))) = (((set_or1337092689740270186AtMost @ code_integer) @ L) @ U))))).
% 9.43/3.44 thf(fact_1801_prod__le__1, axiom, (! [B:$tType,A:$tType]: (((linord181362715937106298miring @ A) => (! [A4:(set @ B),F:(B > A)]: (((! [X2:B]: (((((member @ B) @ X2) @ A4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ X2)) & (((ord_less_eq @ A) @ (F @ X2)) @ (one_one @ A)))))) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A4)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1802_prod_Oivl__cong, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (comm_monoid_mult @ A)) => (! [A3:B,C2:B,B2:B,D3:B,G:(B > A),H3:(B > A)]: (((A3 = C2) => ((B2 = D3) => ((! [X2:B]: (((((ord_less_eq @ B) @ C2) @ X2) => ((((ord_less @ B) @ X2) @ D3) => ((G @ X2) = (H3 @ X2)))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A3) @ B2)) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D3)))))))))))).
% 9.43/3.44 thf(fact_1803_take__bit__integer_Orep__eq, axiom, (! [X:nat,Xa:code_integer]: (((code_int_of_integer @ (((bit_se2584673776208193580ke_bit @ code_integer) @ X) @ Xa)) = (((bit_se2584673776208193580ke_bit @ int) @ X) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1804_and__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((bit_se5824344872417868541ns_and @ code_integer) @ X) @ Xa)) = (((bit_se5824344872417868541ns_and @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1805_bit__integer_Orep__eq, axiom, ((bit_se5641148757651400278ts_bit @ code_integer) = (^ [X3:code_integer]: (((bit_se5641148757651400278ts_bit @ int) @ (code_int_of_integer @ X3)))))).
% 9.43/3.44 thf(fact_1806_prod_OatLeastLessThan__concat, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,P5:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ P5))))))))))).
% 9.43/3.44 thf(fact_1807_mask__integer_Orep__eq, axiom, (! [X:nat]: (((code_int_of_integer @ ((bit_se2239418461657761734s_mask @ code_integer) @ X)) = ((bit_se2239418461657761734s_mask @ int) @ X))))).
% 9.43/3.44 thf(fact_1808_integer__of__num__def, axiom, (code_integer_of_num = (numeral_numeral @ code_integer))).
% 9.43/3.44 thf(fact_1809_unset__bit__integer_Orep__eq, axiom, (! [X:nat,Xa:code_integer]: (((code_int_of_integer @ (((bit_se2638667681897837118et_bit @ code_integer) @ X) @ Xa)) = (((bit_se2638667681897837118et_bit @ int) @ X) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1810_set__bit__integer_Orep__eq, axiom, (! [X:nat,Xa:code_integer]: (((code_int_of_integer @ (((bit_se5668285175392031749et_bit @ code_integer) @ X) @ Xa)) = (((bit_se5668285175392031749et_bit @ int) @ X) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1811_flip__bit__integer_Orep__eq, axiom, (! [X:nat,Xa:code_integer]: (((code_int_of_integer @ (((bit_se8732182000553998342ip_bit @ code_integer) @ X) @ Xa)) = (((bit_se8732182000553998342ip_bit @ int) @ X) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1812_prod_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1813_prod_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1814_atLeastLessThanPlusOne__atLeastAtMost__int, axiom, (! [L:int,U:int]: (((((set_or7035219750837199246ssThan @ int) @ L) @ (((plus_plus @ int) @ U) @ (one_one @ int))) = (((set_or1337092689740270186AtMost @ int) @ L) @ U))))).
% 9.43/3.44 thf(fact_1815_prod_OatLeastAtMost__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1816_prod_OatLeastLessThan__shift__bounds, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,K:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1817_prod_OatLeastAtMost__rev, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat,M2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ N) @ M2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ N) @ M2))))))))).
% 9.43/3.44 thf(fact_1818_prod_Onat__diff__reindex, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ N) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1819_nat__of__integer_Oabs__eq, axiom, (! [X:int]: (((code_nat_of_integer @ (code_integer_of_int @ X)) = (nat2 @ X))))).
% 9.43/3.44 thf(fact_1820_num__of__integer_Orep__eq, axiom, (code_num_of_integer = (^ [X3:code_integer]: ((num_of_nat @ (nat2 @ (code_int_of_integer @ X3))))))).
% 9.43/3.44 thf(fact_1821_prod_OatLeast0__atMost__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ (suc @ N)))))))))).
% 9.43/3.44 thf(fact_1822_prod_OatLeast0__lessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ N))))))))).
% 9.43/3.44 thf(fact_1823_prod_OatLeast__Suc__lessThan, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less @ nat) @ M2) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = (((times_times @ A) @ (G @ M2)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ N))))))))))).
% 9.43/3.44 thf(fact_1824_prod_Onat__ivl__Suc_H, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ (suc @ N)) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = (((times_times @ A) @ (G @ (suc @ N))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.44 thf(fact_1825_prod_OatLeast__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((times_times @ A) @ (G @ M2)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ N))))))))))).
% 9.43/3.44 thf(fact_1826_prod_OatLeastLessThan__Suc, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:nat,B2:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ A3) @ B2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ (suc @ B2))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2))) @ (G @ B2)))))))))).
% 9.43/3.44 thf(fact_1827_divmod__abs__code_I6_J, axiom, (! [J:code_integer]: ((((code_divmod_abs @ (zero_zero @ code_integer)) @ J) = ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer)))))).
% 9.43/3.44 thf(fact_1828_prod_Olast__plus, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((times_times @ A) @ (G @ N)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))))).
% 9.43/3.44 thf(fact_1829_prod_OSuc__reindex__ivl, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ (G @ (suc @ N))) = (((times_times @ A) @ (G @ M2)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.44 thf(fact_1830_prod_OatMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_atMost @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1831_prod_OlessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1832_prod_OatLeastLessThan__rev, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat,M2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ (suc @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2))))))))).
% 9.43/3.44 thf(fact_1833_prod_Onested__swap, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (A3 @ I3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J2:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((A3 @ I3) @ J2)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J2)) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 9.43/3.44 thf(fact_1834_prod_OatLeast1__atMost__eq, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K4:nat]: ((G @ (suc @ K4))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1835_prod_Onat__group, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),K:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [M:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N) @ K)))))))))).
% 9.43/3.44 thf(fact_1836_fact__prod, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => ((semiring_char_0_fact @ A) = (^ [N5:nat]: (((semiring_1_of_nat @ A) @ ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N5)))))))))).
% 9.43/3.44 thf(fact_1837_prod_Onested__swap_H, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A3:(nat > (nat > A)),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (A3 @ I3)) @ ((set_ord_lessThan @ nat) @ I3))))) @ ((set_ord_atMost @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J2:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((A3 @ I3) @ J2)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J2)) @ N))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1838_prod__atLeastAtMost__code, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [F:(nat > A),A3:nat,B2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ F) @ (((set_or1337092689740270186AtMost @ nat) @ A3) @ B2)) = (((((set_fo6178422350223883121st_nat @ A) @ (^ [A7:nat]: (((times_times @ A) @ (F @ A7))))) @ A3) @ B2) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1839_prod_Ohead__if, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [N:nat,M2:nat,G:(nat > A)]: ((((((ord_less @ nat) @ N) @ M2) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (one_one @ A))) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ (G @ N))))))))))).
% 9.43/3.44 thf(fact_1840_prod_Oub__add__nat, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A),P5:nat]: (((((ord_less_eq @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ P5))) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ P5)))))))))))).
% 9.43/3.44 thf(fact_1841_prod_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))))))).
% 9.43/3.44 thf(fact_1842_prod_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))))))).
% 9.43/3.44 thf(fact_1843_prod_OatLeastLessThan__shift__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M2))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M2)))))))))).
% 9.43/3.44 thf(fact_1844_integer__of__num__triv_I1_J, axiom, ((code_integer_of_num @ one2) = (one_one @ code_integer))).
% 9.43/3.44 thf(fact_1845_prod_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat,M2:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M2))))))))).
% 9.43/3.44 thf(fact_1846_prod_OatMost__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ N)) = (((times_times @ A) @ (G @ (zero_zero @ nat))) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))))).
% 9.43/3.44 thf(fact_1847_pochhammer__prod, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A7:A,N5:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A7) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N5))))))))).
% 9.43/3.44 thf(fact_1848_prod_OatLeast__atMost__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N5:nat]: ((((minus_minus @ nat) @ N5) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1849_prod_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N5:nat]: ((((minus_minus @ nat) @ N5) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1850_integer__of__num_I2_J, axiom, (! [N:num]: (((code_integer_of_num @ (bit0 @ N)) = (((plus_plus @ code_integer) @ (code_integer_of_num @ N)) @ (code_integer_of_num @ N)))))).
% 9.43/3.44 thf(fact_1851_divmod__abs__code_I5_J, axiom, (! [J:code_integer]: ((((code_divmod_abs @ J) @ (zero_zero @ code_integer)) = ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ ((abs_abs @ code_integer) @ J)))))).
% 9.43/3.44 thf(fact_1852_prod_OatLeastAtMost__shift__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M2))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M2))))))))))).
% 9.43/3.44 thf(fact_1853_sum_OatLeast__int__lessThan__int__shift, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [G:(int > A),M2:nat,N:nat]: ((((((groups7311177749621191930dd_sum @ int) @ A) @ G) @ (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1854_pochhammer__Suc__prod, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 9.43/3.44 thf(fact_1855_pochhammer__prod__rev, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => ((comm_s3205402744901411588hammer @ A) = (^ [A7:A,N5:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A7) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N5) @ I3)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N5))))))))).
% 9.43/3.44 thf(fact_1856_fact__div__fact, axiom, (! [N:nat,M2:nat]: (((((ord_less_eq @ nat) @ N) @ M2) => ((((divide_divide @ nat) @ ((semiring_char_0_fact @ nat) @ M2)) @ ((semiring_char_0_fact @ nat) @ N)) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ M2))))))).
% 9.43/3.44 thf(fact_1857_integer__of__num__triv_I2_J, axiom, ((code_integer_of_num @ (bit0 @ one2)) = ((numeral_numeral @ code_integer) @ (bit0 @ one2)))).
% 9.43/3.44 thf(fact_1858_prod_Oin__pairs, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1859_prod_Oin__pairs__0, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > A),N:nat]: ((((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((times_times @ A) @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1860_nat__of__integer__less__iff, axiom, (! [X:code_integer,Y:code_integer]: (((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ X) => ((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ Y) => ((((ord_less @ nat) @ (code_nat_of_integer @ X)) @ (code_nat_of_integer @ Y)) = (((ord_less @ code_integer) @ X) @ Y))))))).
% 9.43/3.44 thf(fact_1861_pochhammer__Suc__prod__rev, axiom, (! [A:$tType]: (((comm_semiring_1 @ A) => (! [A3:A,N:nat]: (((((comm_s3205402744901411588hammer @ A) @ A3) @ (suc @ N)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: ((((plus_plus @ A) @ A3) @ ((semiring_1_of_nat @ A) @ (((minus_minus @ nat) @ N) @ I3)))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))))))).
% 9.43/3.44 thf(fact_1862_divmod__integer__code, axiom, (code_divmod_integer = (^ [K4:code_integer,L3:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (K4 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ (zero_zero @ code_integer))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ L3)) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ K4)) @ ((code_divmod_abs @ K4) @ L3)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R3:code_integer,S4:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S4 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ L3) @ S4)))))) @ ((code_divmod_abs @ K4) @ L3)))) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (L3 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (zero_zero @ code_integer)) @ K4)) @ (((((product_apsnd @ code_integer) @ code_integer) @ code_integer) @ (uminus_uminus @ code_integer)) @ ((((if @ ((product_prod @ code_integer) @ code_integer)) @ (((ord_less @ code_integer) @ K4) @ (zero_zero @ code_integer))) @ ((code_divmod_abs @ K4) @ L3)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ code_integer)) @ (^ [R3:code_integer,S4:code_integer]: (((((if @ ((product_prod @ code_integer) @ code_integer)) @ (S4 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ code_integer) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ (one_one @ code_integer))) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ L3)) @ S4)))))) @ ((code_divmod_abs @ K4) @ L3))))))))))).
% 9.43/3.44 thf(fact_1863_bit__cut__integer__code, axiom, (code_bit_cut_integer = (^ [K4:code_integer]: (((((if @ ((product_prod @ code_integer) @ $o)) @ (K4 = (zero_zero @ code_integer))) @ ((((product_Pair @ code_integer) @ $o) @ (zero_zero @ code_integer)) @ $false)) @ (((((product_case_prod @ code_integer) @ code_integer) @ ((product_prod @ code_integer) @ $o)) @ (^ [R3:code_integer,S4:code_integer]: (((((product_Pair @ code_integer) @ $o) @ ((((if @ code_integer) @ (((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ K4)) @ R3) @ (((minus_minus @ code_integer) @ ((uminus_uminus @ code_integer) @ R3)) @ S4))) @ (S4 = (one_one @ code_integer)))))) @ ((code_divmod_abs @ K4) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2))))))))).
% 9.43/3.44 thf(fact_1864_prod_Ozero__middle, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [P5:nat,K:nat,G:(nat > A),H3:(nat > A)]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ P5) => ((((ord_less_eq @ nat) @ K) @ P5) => (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J2:nat]: (((((if @ A) @ (((ord_less @ nat) @ J2) @ K)) @ (G @ J2)) @ ((((if @ A) @ (J2 = K)) @ (one_one @ A)) @ (H3 @ (((minus_minus @ nat) @ J2) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P5)) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [J2:nat]: (((((if @ A) @ (((ord_less @ nat) @ J2) @ K)) @ (G @ J2)) @ (H3 @ J2))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P5) @ (suc @ (zero_zero @ nat)))))))))))))).
% 9.43/3.44 thf(fact_1865_xor__int__rec, axiom, ((bit_se5824344971392196577ns_xor @ int) = (^ [K4:int,L3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K4))) != (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1866_divmod__abs__def, axiom, (code_divmod_abs = (^ [K4:code_integer,L3:code_integer]: (((((product_Pair @ code_integer) @ code_integer) @ (((divide_divide @ code_integer) @ ((abs_abs @ code_integer) @ K4)) @ ((abs_abs @ code_integer) @ L3))) @ (((modulo_modulo @ code_integer) @ ((abs_abs @ code_integer) @ K4)) @ ((abs_abs @ code_integer) @ L3))))))).
% 9.43/3.44 thf(fact_1867_less__int_Orep__eq, axiom, ((ord_less @ int) = (^ [X3:int,Xa4:int]: (((((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [Y3:nat,Z5:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ Y3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Z5)))))))) @ (rep_Integ @ X3)) @ (rep_Integ @ Xa4)))))).
% 9.43/3.44 thf(fact_1868_make_H__def, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (semiring_1 @ B)) => (((array_make2 @ B) @ A) = (^ [I3:code_integer,F2:(B > A)]: ((((array_make @ A) @ (code_nat_of_integer @ I3)) @ (((((comp @ B) @ A) @ nat) @ F2) @ (semiring_1_of_nat @ B)))))))))).
% 9.43/3.44 thf(fact_1869_new_H__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_new2 @ A) = (((((comp @ nat) @ (A > (heap_Time_Heap @ (array @ A)))) @ code_integer) @ (array_new @ A)) @ code_nat_of_integer)))))).
% 9.43/3.44 thf(fact_1870_xor__int__unfold, axiom, ((bit_se5824344971392196577ns_xor @ int) = (^ [K4:int,L3:int]: (((((if @ int) @ (K4 = ((uminus_uminus @ int) @ (one_one @ int)))) @ ((bit_ri4277139882892585799ns_not @ int) @ L3)) @ ((((if @ int) @ (L3 = ((uminus_uminus @ int) @ (one_one @ int)))) @ ((bit_ri4277139882892585799ns_not @ int) @ K4)) @ ((((if @ int) @ (K4 = (zero_zero @ int))) @ L3) @ ((((if @ int) @ (L3 = (zero_zero @ int))) @ K4) @ (((plus_plus @ int) @ ((abs_abs @ int) @ (((minus_minus @ int) @ (((modulo_modulo @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((modulo_modulo @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344971392196577ns_xor @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))))))).
% 9.43/3.44 thf(fact_1871_take__bit__num__code, axiom, (bit_take_bit_num = (^ [N5:nat,M:num]: ((((((product_case_prod @ nat) @ num) @ (option @ num)) @ (^ [A7:nat,X3:num]: (((((case_nat @ (option @ num)) @ (none @ num)) @ (^ [O:nat]: ((((((case_num @ (option @ num)) @ ((some @ num) @ one2)) @ (^ [P6:num]: ((((((case_option @ (option @ num)) @ num) @ (none @ num)) @ (^ [Q5:num]: (((some @ num) @ (bit0 @ Q5))))) @ ((bit_take_bit_num @ O) @ P6))))) @ (^ [P6:num]: (((some @ num) @ (((((case_option @ num) @ num) @ one2) @ bit1) @ ((bit_take_bit_num @ O) @ P6)))))) @ X3)))) @ A7)))) @ ((((product_Pair @ nat) @ num) @ N5) @ M)))))).
% 9.43/3.44 thf(fact_1872_prod_Otriangle__reindex, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > (nat > A)),N:nat]: ((((((groups7121269368397514597t_prod @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ J2)) @ N)))))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K4:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K4) @ I3))))) @ ((set_ord_atMost @ nat) @ K4))))) @ ((set_ord_lessThan @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1873_case__nat__add__eq__if, axiom, (! [A:$tType,A3:A,F:(nat > A),V:num,N:nat]: ((((((case_nat @ A) @ A3) @ F) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V)) @ N)) = (F @ (((plus_plus @ nat) @ (pred_numeral @ V)) @ N)))))).
% 9.43/3.44 thf(fact_1874_rec__nat__add__eq__if, axiom, (! [A:$tType,A3:A,F:(nat > (A > A)),V:num,N:nat]: ((((((rec_nat @ A) @ A3) @ F) @ (((plus_plus @ nat) @ ((numeral_numeral @ nat) @ V)) @ N)) = ((F @ (((plus_plus @ nat) @ (pred_numeral @ V)) @ N)) @ ((((rec_nat @ A) @ A3) @ F) @ (((plus_plus @ nat) @ (pred_numeral @ V)) @ N))))))).
% 9.43/3.44 thf(fact_1875_bit_Ocompl__zero, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ (zero_zero @ A)) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_1876_bit_Ocompl__one, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) = (zero_zero @ A)))))).
% 9.43/3.44 thf(fact_1877_bit_Oxor__cancel__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ X) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1878_bit_Oxor__cancel__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) @ X) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1879_bit_Oxor__one__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((bit_ri4277139882892585799ns_not @ A) @ X)))))))).
% 9.43/3.44 thf(fact_1880_bit_Oxor__one__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se5824344971392196577ns_xor @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X) = ((bit_ri4277139882892585799ns_not @ A) @ X)))))))).
% 9.43/3.44 thf(fact_1881_push__bit__minus__one__eq__not__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N))))))))).
% 9.43/3.44 thf(fact_1882_not__one__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((bit_ri4277139882892585799ns_not @ A) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))).
% 9.43/3.44 thf(fact_1883_not__add__distrib, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,B2:A]: ((((bit_ri4277139882892585799ns_not @ A) @ (((plus_plus @ A) @ A3) @ B2)) = (((minus_minus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A3)) @ B2)))))))).
% 9.43/3.44 thf(fact_1884_not__diff__distrib, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,B2:A]: ((((bit_ri4277139882892585799ns_not @ A) @ (((minus_minus @ A) @ A3) @ B2)) = (((plus_plus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A3)) @ B2)))))))).
% 9.43/3.44 thf(fact_1885_minus__eq__not__plus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((uminus_uminus @ A) = (^ [A7:A]: ((((plus_plus @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ A7)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1886_minus__eq__not__minus__1, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((uminus_uminus @ A) = (^ [A7:A]: (((bit_ri4277139882892585799ns_not @ A) @ (((minus_minus @ A) @ A7) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1887_not__eq__complement, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_ri4277139882892585799ns_not @ A) = (^ [A7:A]: ((((minus_minus @ A) @ ((uminus_uminus @ A) @ A7)) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1888_prod__int__eq, axiom, (! [I:nat,J:nat]: ((((((groups7121269368397514597t_prod @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ J)) = ((((groups7121269368397514597t_prod @ int) @ int) @ (^ [X3:int]: (X3))) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ I)) @ ((semiring_1_of_nat @ int) @ J))))))).
% 9.43/3.44 thf(fact_1889_prod__int__plus__eq, axiom, (! [I:nat,J:nat]: ((((((groups7121269368397514597t_prod @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ (((plus_plus @ nat) @ I) @ J))) = ((((groups7121269368397514597t_prod @ int) @ int) @ (^ [X3:int]: (X3))) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ I)) @ ((semiring_1_of_nat @ int) @ (((plus_plus @ nat) @ I) @ J)))))))).
% 9.43/3.44 thf(fact_1890_push__bit__mask__eq, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [M2:nat,N:nat]: (((((bit_se4730199178511100633sh_bit @ A) @ M2) @ ((bit_se2239418461657761734s_mask @ A) @ N)) = (((bit_se5824344872417868541ns_and @ A) @ ((bit_se2239418461657761734s_mask @ A) @ (((plus_plus @ nat) @ N) @ M2))) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ M2)))))))))).
% 9.43/3.44 thf(fact_1891_prod_OatLeast__int__lessThan__int__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(int > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ int) @ A) @ G) @ (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1892_unset__bit__eq__and__not, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((bit_se2638667681897837118et_bit @ A) = (^ [N5:nat,A7:A]: ((((bit_se5824344872417868541ns_and @ A) @ A7) @ ((bit_ri4277139882892585799ns_not @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N5) @ (one_one @ A))))))))))).
% 9.43/3.44 thf(fact_1893_prod_OatLeast__int__atMost__int__shift, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(int > A),M2:nat,N:nat]: ((((((groups7121269368397514597t_prod @ int) @ A) @ G) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.44 thf(fact_1894_int__numeral__not__and__num, axiom, (! [M2:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ M2))) @ ((numeral_numeral @ int) @ N)) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ N) @ M2)))))).
% 9.43/3.44 thf(fact_1895_int__numeral__and__not__num, axiom, (! [M2:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M2)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))) = (((((case_option @ int) @ num) @ (zero_zero @ int)) @ (numeral_numeral @ int)) @ ((bit_and_not_num @ M2) @ N)))))).
% 9.43/3.44 thf(fact_1896_prod_Otriangle__reindex__eq, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [G:(nat > (nat > A)),N:nat]: ((((((groups7121269368397514597t_prod @ ((product_prod @ nat) @ nat)) @ A) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I3) @ J2)) @ N)))))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [K4:nat]: (((((groups7121269368397514597t_prod @ nat) @ A) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K4) @ I3))))) @ ((set_ord_atMost @ nat) @ K4))))) @ ((set_ord_atMost @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1897_and__not__numerals_I8_J, axiom, (! [M2:num,N:num]: (((((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ (bit1 @ M2))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ ((numeral_numeral @ int) @ M2)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 9.43/3.44 thf(fact_1898_not__int__rec, axiom, ((bit_ri4277139882892585799ns_not @ int) = (^ [K4:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ (((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K4))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ ((bit_ri4277139882892585799ns_not @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1899_Nitpick_Ocase__nat__unfold, axiom, (! [A:$tType]: (((case_nat @ A) = (^ [X3:A,F2:(nat > A),N5:nat]: (((((if @ A) @ (N5 = (zero_zero @ nat))) @ X3) @ (F2 @ (((minus_minus @ nat) @ N5) @ (one_one @ nat)))))))))).
% 9.43/3.44 thf(fact_1900_or__int__rec, axiom, ((bit_se1065995026697491101ons_or @ int) = (^ [K4:int,L3:int]: ((((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K4))) | (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1901_or__numerals_I4_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y))))))))))).
% 9.43/3.44 thf(fact_1902_or__numerals_I6_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y))))))))))).
% 9.43/3.44 thf(fact_1903_or__numerals_I7_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num,Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = (((plus_plus @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ X)) @ ((numeral_numeral @ A) @ Y))))))))))).
% 9.43/3.44 thf(fact_1904_or__nat__rec, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M:nat,N5:nat]: ((((plus_plus @ nat) @ ((zero_neq_one_of_bool @ nat) @ ((~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M))) | (~ ((((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N5)))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ nat) @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))).
% 9.43/3.44 thf(fact_1905_or__not__numerals_I9_J, axiom, (! [M2:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit1 @ M2))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit1 @ N)))) = (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M2)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 9.43/3.44 thf(fact_1906_bit_Odisj__one__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ X) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1907_bit_Odisj__one__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ X) @ ((uminus_uminus @ A) @ (one_one @ A))) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1908_or__numerals_I8_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit1 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ X))))))))).
% 9.43/3.44 thf(fact_1909_or__numerals_I2_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit1 @ Y))) = ((numeral_numeral @ A) @ (bit1 @ Y))))))))).
% 9.43/3.44 thf(fact_1910_bit_Odisj__cancel__left, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) @ X) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1911_bit_Odisj__cancel__right, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A]: (((((bit_se1065995026697491101ons_or @ A) @ X) @ ((bit_ri4277139882892585799ns_not @ A) @ X)) = ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1912_or__numerals_I5_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [X:num]: (((((bit_se1065995026697491101ons_or @ A) @ ((numeral_numeral @ A) @ (bit0 @ X))) @ (one_one @ A)) = ((numeral_numeral @ A) @ (bit1 @ X))))))))).
% 9.43/3.44 thf(fact_1913_or__numerals_I1_J, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => (! [Y:num]: (((((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ (bit0 @ Y))) = ((numeral_numeral @ A) @ (bit1 @ Y))))))))).
% 9.43/3.44 thf(fact_1914_or__integer_Oabs__eq, axiom, (! [Xa:int,X:int]: (((((bit_se1065995026697491101ons_or @ code_integer) @ (code_integer_of_int @ Xa)) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se1065995026697491101ons_or @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1915_or__integer_Orep__eq, axiom, (! [X:code_integer,Xa:code_integer]: (((code_int_of_integer @ (((bit_se1065995026697491101ons_or @ code_integer) @ X) @ Xa)) = (((bit_se1065995026697491101ons_or @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1916_disjunctive__add, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,B2:A]: (((! [N3:nat]: (((~ ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ N3))) | (~ ((((bit_se5641148757651400278ts_bit @ A) @ B2) @ N3)))))) => ((((plus_plus @ A) @ A3) @ B2) = (((bit_se1065995026697491101ons_or @ A) @ A3) @ B2))))))))).
% 9.43/3.44 thf(fact_1917_plus__and__or, axiom, (! [X:int,Y:int]: (((((plus_plus @ int) @ (((bit_se5824344872417868541ns_and @ int) @ X) @ Y)) @ (((bit_se1065995026697491101ons_or @ int) @ X) @ Y)) = (((plus_plus @ int) @ X) @ Y))))).
% 9.43/3.44 thf(fact_1918_not__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ ((bit_ri4277139882892585799ns_not @ code_integer) @ X)) = ((bit_ri4277139882892585799ns_not @ int) @ (code_int_of_integer @ X)))))).
% 9.43/3.44 thf(fact_1919_set__bit__nat__def, axiom, ((bit_se5668285175392031749et_bit @ nat) = (^ [M:nat,N5:nat]: ((((bit_se1065995026697491101ons_or @ nat) @ N5) @ (((bit_se4730199178511100633sh_bit @ nat) @ M) @ (one_one @ nat))))))).
% 9.43/3.44 thf(fact_1920_not__integer_Oabs__eq, axiom, (! [X:int]: ((((bit_ri4277139882892585799ns_not @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((bit_ri4277139882892585799ns_not @ int) @ X)))))).
% 9.43/3.44 thf(fact_1921_set__bit__eq__or, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5668285175392031749et_bit @ A) = (^ [N5:nat,A7:A]: ((((bit_se1065995026697491101ons_or @ A) @ A7) @ (((bit_se4730199178511100633sh_bit @ A) @ N5) @ (one_one @ A)))))))))).
% 9.43/3.44 thf(fact_1922_bit_Ocomplement__unique, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,X:A,Y:A]: ((((((bit_se5824344872417868541ns_and @ A) @ A3) @ X) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ A3) @ X) = ((uminus_uminus @ A) @ (one_one @ A))) => (((((bit_se5824344872417868541ns_and @ A) @ A3) @ Y) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ A3) @ Y) = ((uminus_uminus @ A) @ (one_one @ A))) => (X = Y))))))))))).
% 9.43/3.44 thf(fact_1923_bit_Ocompl__unique, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [X:A,Y:A]: ((((((bit_se5824344872417868541ns_and @ A) @ X) @ Y) = (zero_zero @ A)) => (((((bit_se1065995026697491101ons_or @ A) @ X) @ Y) = ((uminus_uminus @ A) @ (one_one @ A))) => (((bit_ri4277139882892585799ns_not @ A) @ X) = Y))))))))).
% 9.43/3.44 thf(fact_1924_or__one__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se1065995026697491101ons_or @ A) @ A3) @ (one_one @ A)) = (((plus_plus @ A) @ A3) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))).
% 9.43/3.44 thf(fact_1925_one__or__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A]: (((((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ A3) = (((plus_plus @ A) @ A3) @ ((zero_neq_one_of_bool @ A) @ (((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)))))))))).
% 9.43/3.44 thf(fact_1926_mask__Suc__double, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: ((((bit_se2239418461657761734s_mask @ A) @ (suc @ N)) = (((bit_se1065995026697491101ons_or @ A) @ (one_one @ A)) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ ((bit_se2239418461657761734s_mask @ A) @ N)))))))))).
% 9.43/3.44 thf(fact_1927_or__not__numerals_I5_J, axiom, (! [M2:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit0 @ M2))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M2)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 9.43/3.44 thf(fact_1928_Suc__0__or__eq, axiom, (! [N:nat]: (((((bit_se1065995026697491101ons_or @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((plus_plus @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))).
% 9.43/3.44 thf(fact_1929_or__Suc__0__eq, axiom, (! [N:nat]: (((((bit_se1065995026697491101ons_or @ nat) @ N) @ (suc @ (zero_zero @ nat))) = (((plus_plus @ nat) @ N) @ ((zero_neq_one_of_bool @ nat) @ (((dvd_dvd @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N))))))).
% 9.43/3.44 thf(fact_1930_or__not__numerals_I8_J, axiom, (! [M2:num,N:num]: (((((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ (bit1 @ M2))) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ (bit0 @ N)))) = (((plus_plus @ int) @ (one_one @ int)) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ ((numeral_numeral @ int) @ M2)) @ ((bit_ri4277139882892585799ns_not @ int) @ ((numeral_numeral @ int) @ N))))))))).
% 9.43/3.44 thf(fact_1931_or__int__unfold, axiom, ((bit_se1065995026697491101ons_or @ int) = (^ [K4:int,L3:int]: (((((if @ int) @ ((K4 = ((uminus_uminus @ int) @ (one_one @ int))) | (L3 = ((uminus_uminus @ int) @ (one_one @ int))))) @ ((uminus_uminus @ int) @ (one_one @ int))) @ ((((if @ int) @ (K4 = (zero_zero @ int))) @ L3) @ ((((if @ int) @ (L3 = (zero_zero @ int))) @ K4) @ (((plus_plus @ int) @ (((ord_max @ int) @ (((modulo_modulo @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((modulo_modulo @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 9.43/3.44 thf(fact_1932_bit_Oabstract__boolean__algebra__sym__diff__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((((((boolea3799213064322606851m_diff @ A) @ (bit_se5824344872417868541ns_and @ A)) @ (bit_se1065995026697491101ons_or @ A)) @ (bit_ri4277139882892585799ns_not @ A)) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (bit_se5824344971392196577ns_xor @ A)))))).
% 9.43/3.44 thf(fact_1933_or__nat__unfold, axiom, ((bit_se1065995026697491101ons_or @ nat) = (^ [M:nat,N5:nat]: (((((if @ nat) @ (M = (zero_zero @ nat))) @ N5) @ ((((if @ nat) @ (N5 = (zero_zero @ nat))) @ M) @ (((plus_plus @ nat) @ (((ord_max @ nat) @ (((modulo_modulo @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((modulo_modulo @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ (((bit_se1065995026697491101ons_or @ nat) @ (((divide_divide @ nat) @ M) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((divide_divide @ nat) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1934_drop__bit__rec, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se4197421643247451524op_bit @ A) = (^ [N5:nat,A7:A]: (((((if @ A) @ (N5 = (zero_zero @ nat))) @ A7) @ (((bit_se4197421643247451524op_bit @ A) @ (((minus_minus @ nat) @ N5) @ (one_one @ nat))) @ (((divide_divide @ A) @ A7) @ ((numeral_numeral @ A) @ (bit0 @ one2)))))))))))).
% 9.43/3.44 thf(fact_1935_image__atLeastZeroLessThan__integer, axiom, (! [U:code_integer]: (((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ U) => ((((set_or7035219750837199246ssThan @ code_integer) @ (zero_zero @ code_integer)) @ U) = ((((image2 @ nat) @ code_integer) @ (semiring_1_of_nat @ code_integer)) @ ((set_ord_lessThan @ nat) @ (code_nat_of_integer @ U)))))))).
% 9.43/3.44 thf(fact_1936_uminus__int__def, axiom, ((uminus_uminus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X3:nat,Y3:nat]: (((((product_Pair @ nat) @ nat) @ Y3) @ X3))))))).
% 9.43/3.44 thf(fact_1937_bit_Oabstract__boolean__algebra__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => ((((((boolea2506097494486148201lgebra @ A) @ (bit_se5824344872417868541ns_and @ A)) @ (bit_se1065995026697491101ons_or @ A)) @ (bit_ri4277139882892585799ns_not @ A)) @ (zero_zero @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.44 thf(fact_1938_map__fun__apply, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType]: ((((((map_fun @ B) @ C) @ D) @ A) = (^ [F2:(B > C),G2:(D > A),H:(C > D),X3:B]: ((G2 @ (H @ (F2 @ X3))))))))).
% 9.43/3.44 thf(fact_1939_drop__bit__drop__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M2:nat,N:nat,A3:A]: (((((bit_se4197421643247451524op_bit @ A) @ M2) @ (((bit_se4197421643247451524op_bit @ A) @ N) @ A3)) = (((bit_se4197421643247451524op_bit @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ A3)))))))).
% 9.43/3.44 thf(fact_1940_image__add__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ K)) @ (((set_or1337092689740270186AtMost @ A) @ I) @ J)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))).
% 9.43/3.44 thf(fact_1941_image__diff__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [D3:A,A3:A,B2:A]: ((((((image2 @ A) @ A) @ ((minus_minus @ A) @ D3)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ D3) @ B2)) @ (((minus_minus @ A) @ D3) @ A3))))))))).
% 9.43/3.44 thf(fact_1942_image__add__atLeastLessThan, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ K)) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) = (((set_or7035219750837199246ssThan @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))).
% 9.43/3.44 thf(fact_1943_max__0__1_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_max @ A) @ (zero_zero @ A)) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1944_max__0__1_I2_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_max @ A) @ (one_one @ A)) @ (zero_zero @ A)) = (one_one @ A)))))).
% 9.43/3.44 thf(fact_1945_image__uminus__atLeastAtMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.44 thf(fact_1946_max__0__1_I6_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_max @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A)) = ((numeral_numeral @ A) @ X)))))))).
% 9.43/3.44 thf(fact_1947_max__0__1_I5_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_max @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ X)) = ((numeral_numeral @ A) @ X)))))))).
% 9.43/3.44 thf(fact_1948_image__add__atMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [C2:A,A3:A]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ C2)) @ ((set_ord_atMost @ A) @ A3)) = ((set_ord_atMost @ A) @ (((plus_plus @ A) @ C2) @ A3))))))))).
% 9.43/3.44 thf(fact_1949_image__add__atLeastAtMost_H, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image2 @ A) @ A) @ (^ [N5:A]: ((((plus_plus @ A) @ N5) @ K)))) @ (((set_or1337092689740270186AtMost @ A) @ I) @ J)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))).
% 9.43/3.44 thf(fact_1950_image__minus__const__atLeastAtMost_H, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [D3:A,A3:A,B2:A]: ((((((image2 @ A) @ A) @ (^ [T3:A]: ((((minus_minus @ A) @ T3) @ D3)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ A3) @ D3)) @ (((minus_minus @ A) @ B2) @ D3))))))))).
% 9.43/3.44 thf(fact_1951_image__add__atLeastLessThan_H, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A,J:A]: ((((((image2 @ A) @ A) @ (^ [N5:A]: ((((plus_plus @ A) @ N5) @ K)))) @ (((set_or7035219750837199246ssThan @ A) @ I) @ J)) = (((set_or7035219750837199246ssThan @ A) @ (((plus_plus @ A) @ I) @ K)) @ (((plus_plus @ A) @ J) @ K))))))))).
% 9.43/3.44 thf(fact_1952_drop__bit__of__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat]: (((((bit_se4197421643247451524op_bit @ A) @ N) @ (one_one @ A)) = ((zero_neq_one_of_bool @ A) @ (N = (zero_zero @ nat)))))))))).
% 9.43/3.44 thf(fact_1953_image__mult__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D3:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D3) => (((((image2 @ A) @ A) @ ((times_times @ A) @ D3)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ D3) @ A3)) @ (((times_times @ A) @ D3) @ B2)))))))))).
% 9.43/3.44 thf(fact_1954_image__divide__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [D3:A,A3:A,B2:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ D3) => (((((image2 @ A) @ A) @ (^ [C3:A]: ((((divide_divide @ A) @ C3) @ D3)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((divide_divide @ A) @ A3) @ D3)) @ (((divide_divide @ A) @ B2) @ D3)))))))))).
% 9.43/3.44 thf(fact_1955_drop__bit__integer_Orep__eq, axiom, (! [X:nat,Xa:code_integer]: (((code_int_of_integer @ (((bit_se4197421643247451524op_bit @ code_integer) @ X) @ Xa)) = (((bit_se4197421643247451524op_bit @ int) @ X) @ (code_int_of_integer @ Xa)))))).
% 9.43/3.44 thf(fact_1956_drop__bit__integer_Oabs__eq, axiom, (! [Xa:nat,X:int]: (((((bit_se4197421643247451524op_bit @ code_integer) @ Xa) @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((bit_se4197421643247451524op_bit @ int) @ Xa) @ X)))))).
% 9.43/3.44 thf(fact_1957_max__add__distrib__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y:A,Z2:A]: (((((plus_plus @ A) @ (((ord_max @ A) @ X) @ Y)) @ Z2) = (((ord_max @ A) @ (((plus_plus @ A) @ X) @ Z2)) @ (((plus_plus @ A) @ Y) @ Z2))))))))).
% 9.43/3.44 thf(fact_1958_max__add__distrib__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y:A,Z2:A]: (((((plus_plus @ A) @ X) @ (((ord_max @ A) @ Y) @ Z2)) = (((ord_max @ A) @ (((plus_plus @ A) @ X) @ Y)) @ (((plus_plus @ A) @ X) @ Z2))))))))).
% 9.43/3.44 thf(fact_1959_max__diff__distrib__left, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A,Z2:A]: (((((minus_minus @ A) @ (((ord_max @ A) @ X) @ Y)) @ Z2) = (((ord_max @ A) @ (((minus_minus @ A) @ X) @ Z2)) @ (((minus_minus @ A) @ Y) @ Z2))))))))).
% 9.43/3.44 thf(fact_1960_translation__Compl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,T2:(set @ A)]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ ((uminus_uminus @ (set @ A)) @ T2)) = ((uminus_uminus @ (set @ A)) @ ((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ T2))))))))).
% 9.43/3.44 thf(fact_1961_nat__add__max__right, axiom, (! [M2:nat,N:nat,Q3:nat]: (((((plus_plus @ nat) @ M2) @ (((ord_max @ nat) @ N) @ Q3)) = (((ord_max @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ (((plus_plus @ nat) @ M2) @ Q3)))))).
% 9.43/3.44 thf(fact_1962_nat__add__max__left, axiom, (! [M2:nat,N:nat,Q3:nat]: (((((plus_plus @ nat) @ (((ord_max @ nat) @ M2) @ N)) @ Q3) = (((ord_max @ nat) @ (((plus_plus @ nat) @ M2) @ Q3)) @ (((plus_plus @ nat) @ N) @ Q3)))))).
% 9.43/3.44 thf(fact_1963_image__eq__imp__comp, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(B > A),A4:(set @ B),G:(C > A),B6:(set @ C),H3:(A > D)]: (((((((image2 @ B) @ A) @ F) @ A4) = ((((image2 @ C) @ A) @ G) @ B6)) => (((((image2 @ B) @ D) @ (((((comp @ A) @ D) @ B) @ H3) @ F)) @ A4) = ((((image2 @ C) @ D) @ (((((comp @ A) @ D) @ C) @ H3) @ G)) @ B6)))))).
% 9.43/3.44 thf(fact_1964_image__comp, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),R:(set @ C)]: ((((((image2 @ B) @ A) @ F) @ ((((image2 @ C) @ B) @ G) @ R)) = ((((image2 @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F) @ G)) @ R))))).
% 9.43/3.44 thf(fact_1965_map__fun__def, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType]: ((((((map_fun @ C) @ A) @ B) @ D) = (^ [F2:(C > A),G2:(B > D),H:(A > B)]: ((((((comp @ A) @ D) @ C) @ (((((comp @ B) @ D) @ A) @ G2) @ H)) @ F2))))))).
% 9.43/3.44 thf(fact_1966_map__fun_Ocompositionality, axiom, (! [D:$tType,F5:$tType,C:$tType,E:$tType,B:$tType,A:$tType,F:(E > C),G:(D > F5),H3:(C > A),I:(B > D),Fun:(A > B)]: (((((((((map_fun @ E) @ C) @ D) @ F5) @ F) @ G) @ (((((((map_fun @ C) @ A) @ B) @ D) @ H3) @ I) @ Fun)) = (((((((map_fun @ E) @ A) @ B) @ F5) @ (((((comp @ C) @ A) @ E) @ H3) @ F)) @ (((((comp @ D) @ F5) @ B) @ G) @ I)) @ Fun))))).
% 9.43/3.44 thf(fact_1967_map__fun_Ocomp, axiom, (! [E:$tType,C:$tType,A:$tType,F5:$tType,D:$tType,B:$tType,F:(E > C),G:(D > F5),H3:(C > A),I:(B > D)]: (((((((comp @ (C > D)) @ (E > F5)) @ (A > B)) @ ((((((map_fun @ E) @ C) @ D) @ F5) @ F) @ G)) @ ((((((map_fun @ C) @ A) @ B) @ D) @ H3) @ I)) = ((((((map_fun @ E) @ A) @ B) @ F5) @ (((((comp @ C) @ A) @ E) @ H3) @ F)) @ (((((comp @ D) @ F5) @ B) @ G) @ I)))))).
% 9.43/3.44 thf(fact_1968_translation__subtract__Compl, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,T2:(set @ A)]: ((((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ ((uminus_uminus @ (set @ A)) @ T2)) = ((uminus_uminus @ (set @ A)) @ ((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ T2))))))))).
% 9.43/3.44 thf(fact_1969_nat__minus__add__max, axiom, (! [N:nat,M2:nat]: (((((plus_plus @ nat) @ (((minus_minus @ nat) @ N) @ M2)) @ M2) = (((ord_max @ nat) @ N) @ M2))))).
% 9.43/3.44 thf(fact_1970_translation__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,S:(set @ A),T2:(set @ A)]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ (((minus_minus @ (set @ A)) @ S) @ T2)) = (((minus_minus @ (set @ A)) @ ((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ S)) @ ((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ T2))))))))).
% 9.43/3.44 thf(fact_1971_translation__subtract__diff, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,S:(set @ A),T2:(set @ A)]: ((((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ (((minus_minus @ (set @ A)) @ S) @ T2)) = (((minus_minus @ (set @ A)) @ ((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ S)) @ ((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ T2))))))))).
% 9.43/3.44 thf(fact_1972_take__bit__drop__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [M2:nat,N:nat,A3:A]: (((((bit_se2584673776208193580ke_bit @ A) @ M2) @ (((bit_se4197421643247451524op_bit @ A) @ N) @ A3)) = (((bit_se4197421643247451524op_bit @ A) @ N) @ (((bit_se2584673776208193580ke_bit @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ A3))))))))).
% 9.43/3.44 thf(fact_1973_bit__drop__bit__eq, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: ((((bit_se5641148757651400278ts_bit @ A) @ (((bit_se4197421643247451524op_bit @ A) @ N) @ A3)) = (((((comp @ nat) @ $o) @ nat) @ ((bit_se5641148757651400278ts_bit @ A) @ A3)) @ ((plus_plus @ nat) @ N))))))))).
% 9.43/3.44 thf(fact_1974_div__push__bit__of__1__eq__drop__bit, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [A3:A,N:nat]: (((((divide_divide @ A) @ A3) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ (one_one @ A))) = (((bit_se4197421643247451524op_bit @ A) @ N) @ A3)))))))).
% 9.43/3.44 thf(fact_1975_bit__iff__and__drop__bit__eq__1, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => ((bit_se5641148757651400278ts_bit @ A) = (^ [A7:A,N5:nat]: (((((bit_se5824344872417868541ns_and @ A) @ (((bit_se4197421643247451524op_bit @ A) @ N5) @ A7)) @ (one_one @ A)) = (one_one @ A))))))))).
% 9.43/3.44 thf(fact_1976_bits__ident, axiom, (! [A:$tType]: (((bit_se359711467146920520ations @ A) => (! [N:nat,A3:A]: (((((plus_plus @ A) @ (((bit_se4730199178511100633sh_bit @ A) @ N) @ (((bit_se4197421643247451524op_bit @ A) @ N) @ A3))) @ (((bit_se2584673776208193580ke_bit @ A) @ N) @ A3)) = A3))))))).
% 9.43/3.44 thf(fact_1977_slice__eq__mask, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat,M2:nat,A3:A]: (((((bit_se4730199178511100633sh_bit @ A) @ N) @ (((bit_se2584673776208193580ke_bit @ A) @ M2) @ (((bit_se4197421643247451524op_bit @ A) @ N) @ A3))) = (((bit_se5824344872417868541ns_and @ A) @ A3) @ (((bit_se5824344872417868541ns_and @ A) @ ((bit_se2239418461657761734s_mask @ A) @ (((plus_plus @ nat) @ M2) @ N))) @ ((bit_ri4277139882892585799ns_not @ A) @ ((bit_se2239418461657761734s_mask @ A) @ N))))))))))).
% 9.43/3.44 thf(fact_1978_image__add__0, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [S2:(set @ A)]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ (zero_zero @ A))) @ S2) = S2))))))).
% 9.43/3.44 thf(fact_1979_times__int__def, axiom, ((times_times @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ U2)) @ (((times_times @ nat) @ Y3) @ V2))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ V2)) @ (((times_times @ nat) @ Y3) @ U2)))))))))))).
% 9.43/3.44 thf(fact_1980_minus__int__def, axiom, ((minus_minus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ Y3) @ U2))))))))))).
% 9.43/3.44 thf(fact_1981_plus__int__def, axiom, ((plus_plus @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ int) @ rep_Integ) @ abs_Integ)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ U2)) @ (((plus_plus @ nat) @ Y3) @ V2))))))))))).
% 9.43/3.44 thf(fact_1982_mlex__prod__def, axiom, (! [A:$tType]: (((mlex_prod @ A) = (^ [F2:(A > nat),R6:(set @ ((product_prod @ A) @ A))]: (((((inv_image @ ((product_prod @ nat) @ A)) @ A) @ ((((lex_prod @ nat) @ A) @ less_than) @ R6)) @ (^ [X3:A]: (((((product_Pair @ nat) @ A) @ (F2 @ X3)) @ X3)))))))))).
% 9.43/3.44 thf(fact_1983_rat__minus__code, axiom, (! [P5:rat,Q3:rat]: (((quotient_of @ (((minus_minus @ rat) @ P5) @ Q3)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int,C3:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B5:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((minus_minus @ int) @ (((times_times @ int) @ A7) @ D5)) @ (((times_times @ int) @ B5) @ C3))) @ (((times_times @ int) @ C3) @ D5)))))) @ (quotient_of @ Q3))))) @ (quotient_of @ P5)))))).
% 9.43/3.44 thf(fact_1984_rat__plus__code, axiom, (! [P5:rat,Q3:rat]: (((quotient_of @ (((plus_plus @ rat) @ P5) @ Q3)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int,C3:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B5:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ A7) @ D5)) @ (((times_times @ int) @ B5) @ C3))) @ (((times_times @ int) @ C3) @ D5)))))) @ (quotient_of @ Q3))))) @ (quotient_of @ P5)))))).
% 9.43/3.44 thf(fact_1985_image__Suc__atLeastAtMost, axiom, (! [I:nat,J:nat]: ((((((image2 @ nat) @ nat) @ suc) @ (((set_or1337092689740270186AtMost @ nat) @ I) @ J)) = (((set_or1337092689740270186AtMost @ nat) @ (suc @ I)) @ (suc @ J)))))).
% 9.43/3.44 thf(fact_1986_image__Suc__atLeastLessThan, axiom, (! [I:nat,J:nat]: ((((((image2 @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ I) @ J)) = (((set_or7035219750837199246ssThan @ nat) @ (suc @ I)) @ (suc @ J)))))).
% 9.43/3.44 thf(fact_1987_in__inv__image, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((((inv_image @ B) @ A) @ R) @ F)) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y))) @ R))))).
% 9.43/3.44 thf(fact_1988_pair__imageI, axiom, (! [C:$tType,B:$tType,A:$tType,A3:A,B2:B,A4:(set @ ((product_prod @ A) @ B)),F:(A > (B > C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ A4) => (((member @ C) @ ((F @ A3) @ B2)) @ ((((image2 @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) @ A4)))))).
% 9.43/3.44 thf(fact_1989_int__of__integer__max, axiom, (! [K:code_integer,L:code_integer]: (((code_int_of_integer @ (((ord_max @ code_integer) @ K) @ L)) = (((ord_max @ int) @ (code_int_of_integer @ K)) @ (code_int_of_integer @ L)))))).
% 9.43/3.44 thf(fact_1990_rat__one__code, axiom, ((quotient_of @ (one_one @ rat)) = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int)))).
% 9.43/3.44 thf(fact_1991_of__nat__of__integer, axiom, (! [K:code_integer]: ((((semiring_1_of_nat @ code_integer) @ (code_nat_of_integer @ K)) = (((ord_max @ code_integer) @ (zero_zero @ code_integer)) @ K))))).
% 9.43/3.44 thf(fact_1992_rat__zero__code, axiom, ((quotient_of @ (zero_zero @ rat)) = ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int)))).
% 9.43/3.44 thf(fact_1993_quotient__of__number_I3_J, axiom, (! [K:num]: (((quotient_of @ ((numeral_numeral @ rat) @ K)) = ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1994_quotient__of__number_I4_J, axiom, ((quotient_of @ ((uminus_uminus @ rat) @ (one_one @ rat))) = ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (one_one @ int)))).
% 9.43/3.44 thf(fact_1995_quotient__of__number_I5_J, axiom, (! [K:num]: (((quotient_of @ ((uminus_uminus @ rat) @ ((numeral_numeral @ rat) @ K))) = ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ K))) @ (one_one @ int)))))).
% 9.43/3.44 thf(fact_1996_diff__rat__def, axiom, ((minus_minus @ rat) = (^ [Q5:rat,R3:rat]: ((((plus_plus @ rat) @ Q5) @ ((uminus_uminus @ rat) @ R3)))))).
% 9.43/3.44 thf(fact_1997_quotient__of__div, axiom, (! [R:rat,N:int,D3:int]: ((((quotient_of @ R) = ((((product_Pair @ int) @ int) @ N) @ D3)) => (R = (((divide_divide @ rat) @ ((ring_1_of_int @ rat) @ N)) @ ((ring_1_of_int @ rat) @ D3))))))).
% 9.43/3.44 thf(fact_1998_zero__notin__Suc__image, axiom, (! [A4:(set @ nat)]: ((~ ((((member @ nat) @ (zero_zero @ nat)) @ ((((image2 @ nat) @ nat) @ suc) @ A4))))))).
% 9.43/3.44 thf(fact_1999_None__notin__image__Some, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((((member @ (option @ A)) @ (none @ A)) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ A4))))))).
% 9.43/3.44 thf(fact_2000_plus__integer__def, axiom, ((plus_plus @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (plus_plus @ int)))).
% 9.43/3.44 thf(fact_2001_times__integer__def, axiom, ((times_times @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (times_times @ int)))).
% 9.43/3.45 thf(fact_2002_minus__integer__def, axiom, ((minus_minus @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (minus_minus @ int)))).
% 9.43/3.45 thf(fact_2003_divide__integer__def, axiom, ((divide_divide @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (divide_divide @ int)))).
% 9.43/3.45 thf(fact_2004_modulo__integer__def, axiom, ((modulo_modulo @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (modulo_modulo @ int)))).
% 9.43/3.45 thf(fact_2005_and__integer__def, axiom, ((bit_se5824344872417868541ns_and @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se5824344872417868541ns_and @ int)))).
% 9.43/3.45 thf(fact_2006_or__integer__def, axiom, ((bit_se1065995026697491101ons_or @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se1065995026697491101ons_or @ int)))).
% 9.43/3.45 thf(fact_2007_xor__integer__def, axiom, ((bit_se5824344971392196577ns_xor @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > int)) @ (code_integer > code_integer)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se5824344971392196577ns_xor @ int)))).
% 9.43/3.45 thf(fact_2008_image__int__atLeastAtMost, axiom, (! [A3:nat,B2:nat]: ((((((image2 @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or1337092689740270186AtMost @ nat) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 9.43/3.45 thf(fact_2009_image__int__atLeastLessThan, axiom, (! [A3:nat,B2:nat]: ((((((image2 @ nat) @ int) @ (semiring_1_of_nat @ int)) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2)) = (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ A3)) @ ((semiring_1_of_nat @ int) @ B2)))))).
% 9.43/3.45 thf(fact_2010_quotient__of__denom__pos, axiom, (! [R:rat,P5:int,Q3:int]: ((((quotient_of @ R) = ((((product_Pair @ int) @ int) @ P5) @ Q3)) => (((ord_less @ int) @ (zero_zero @ int)) @ Q3))))).
% 9.43/3.45 thf(fact_2011_image__Suc__atMost, axiom, (! [N:nat]: ((((((image2 @ nat) @ nat) @ suc) @ ((set_ord_atMost @ nat) @ N)) = (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ (suc @ N)))))).
% 9.43/3.45 thf(fact_2012_image__Suc__lessThan, axiom, (! [N:nat]: ((((((image2 @ nat) @ nat) @ suc) @ ((set_ord_lessThan @ nat) @ N)) = (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N))))).
% 9.43/3.45 thf(fact_2013_inv__image__def, axiom, (! [A:$tType,B:$tType]: ((((inv_image @ B) @ A) = (^ [R3:(set @ ((product_prod @ B) @ B)),F2:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F2 @ X3)) @ (F2 @ Y3))) @ R3))))))))))).
% 9.43/3.45 thf(fact_2014_rat__uminus__code, axiom, (! [P5:rat]: (((quotient_of @ ((uminus_uminus @ rat) @ P5)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int]: ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ A7))))) @ (quotient_of @ P5)))))).
% 9.43/3.45 thf(fact_2015_rat__abs__code, axiom, (! [P5:rat]: (((quotient_of @ ((abs_abs @ rat) @ P5)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int]: ((((product_Pair @ int) @ int) @ ((abs_abs @ int) @ A7))))) @ (quotient_of @ P5)))))).
% 9.43/3.45 thf(fact_2016_uminus__integer__def, axiom, ((uminus_uminus @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (uminus_uminus @ int)))).
% 9.43/3.45 thf(fact_2017_abs__integer__def, axiom, ((abs_abs @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (abs_abs @ int)))).
% 9.43/3.45 thf(fact_2018_image__add__int__atLeastLessThan, axiom, (! [L:int,U:int]: ((((((image2 @ int) @ int) @ (^ [X3:int]: ((((plus_plus @ int) @ X3) @ L)))) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ (((minus_minus @ int) @ U) @ L))) = (((set_or7035219750837199246ssThan @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2019_sgn__integer__def, axiom, ((sgn_sgn @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (sgn_sgn @ int)))).
% 9.43/3.45 thf(fact_2020_not__integer__def, axiom, ((bit_ri4277139882892585799ns_not @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (bit_ri4277139882892585799ns_not @ int)))).
% 9.43/3.45 thf(fact_2021_image__add__integer__atLeastLessThan, axiom, (! [L:code_integer,U:code_integer]: ((((((image2 @ code_integer) @ code_integer) @ (^ [X3:code_integer]: ((((plus_plus @ code_integer) @ X3) @ L)))) @ (((set_or7035219750837199246ssThan @ code_integer) @ (zero_zero @ code_integer)) @ (((minus_minus @ code_integer) @ U) @ L))) = (((set_or7035219750837199246ssThan @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2022_rat__times__code, axiom, (! [P5:rat,Q3:rat]: (((quotient_of @ (((times_times @ rat) @ P5) @ Q3)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int,C3:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B5:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ A7) @ B5)) @ (((times_times @ int) @ C3) @ D5)))))) @ (quotient_of @ Q3))))) @ (quotient_of @ P5)))))).
% 9.43/3.45 thf(fact_2023_rat__divide__code, axiom, (! [P5:rat,Q3:rat]: (((quotient_of @ (((divide_divide @ rat) @ P5) @ Q3)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int,C3:int]: ((((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [B5:int,D5:int]: ((normalize @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ A7) @ D5)) @ (((times_times @ int) @ C3) @ B5)))))) @ (quotient_of @ Q3))))) @ (quotient_of @ P5)))))).
% 9.43/3.45 thf(fact_2024_image__atLeastZeroLessThan__int, axiom, (! [U:int]: (((((ord_less_eq @ int) @ (zero_zero @ int)) @ U) => ((((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U) = ((((image2 @ nat) @ int) @ (semiring_1_of_nat @ int)) @ ((set_ord_lessThan @ nat) @ (nat2 @ U)))))))).
% 9.43/3.45 thf(fact_2025_quotient__of__int, axiom, (! [A3:int]: (((quotient_of @ (of_int @ A3)) = ((((product_Pair @ int) @ int) @ A3) @ (one_one @ int)))))).
% 9.43/3.45 thf(fact_2026_rp__inv__image__def, axiom, (! [B:$tType,A:$tType]: ((((fun_rp_inv_image @ A) @ B) = ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((B > A) > ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))))) @ (^ [R6:(set @ ((product_prod @ A) @ A)),S5:(set @ ((product_prod @ A) @ A)),F2:(B > A)]: (((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((((inv_image @ A) @ B) @ R6) @ F2)) @ ((((inv_image @ A) @ B) @ S5) @ F2))))))))).
% 9.43/3.45 thf(fact_2027_rat__inverse__code, axiom, (! [P5:rat]: (((quotient_of @ ((inverse_inverse @ rat) @ P5)) = (((((product_case_prod @ int) @ int) @ ((product_prod @ int) @ int)) @ (^ [A7:int,B5:int]: (((((if @ ((product_prod @ int) @ int)) @ (A7 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ ((sgn_sgn @ int) @ A7)) @ B5)) @ ((abs_abs @ int) @ A7)))))) @ (quotient_of @ P5)))))).
% 9.43/3.45 thf(fact_2028_diff__numeral__special_I8_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((neg_numeral_sub @ A) @ one2) @ M2)))))))).
% 9.43/3.45 thf(fact_2029_diff__numeral__special_I7_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((neg_numeral_sub @ A) @ N) @ one2)))))))).
% 9.43/3.45 thf(fact_2030_minus__sub__one__diff__one, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [M2:num]: (((((minus_minus @ A) @ ((uminus_uminus @ A) @ (((neg_numeral_sub @ A) @ M2) @ one2))) @ (one_one @ A)) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))))))))).
% 9.43/3.45 thf(fact_2031_add__neg__numeral__special_I4_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((numeral_numeral @ A) @ N)) = (((neg_numeral_sub @ A) @ N) @ one2)))))))).
% 9.43/3.45 thf(fact_2032_int__of__integer__sub, axiom, (! [K:num,L:num]: (((code_int_of_integer @ (((neg_numeral_sub @ code_integer) @ K) @ L)) = (((neg_numeral_sub @ int) @ K) @ L))))).
% 9.43/3.45 thf(fact_2033_inverse__1, axiom, (! [A:$tType]: (((division_ring @ A) => (((inverse_inverse @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.45 thf(fact_2034_inverse__eq__1__iff, axiom, (! [A:$tType]: (((field @ A) => (! [X:A]: (((((inverse_inverse @ A) @ X) = (one_one @ A)) = (X = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2035_right__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((times_times @ A) @ A3) @ ((inverse_inverse @ A) @ A3)) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2036_left__inverse, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ A3) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2037_inverse__eq__divide__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W:num]: ((((inverse_inverse @ A) @ ((numeral_numeral @ A) @ W)) = (((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ W))))))))).
% 9.43/3.45 thf(fact_2038_add__neg__numeral__simps_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num,N:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ ((numeral_numeral @ A) @ N)) = (((neg_numeral_sub @ A) @ N) @ M2)))))))).
% 9.43/3.45 thf(fact_2039_add__neg__numeral__simps_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num,N:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ N))) = (((neg_numeral_sub @ A) @ M2) @ N)))))))).
% 9.43/3.45 thf(fact_2040_semiring__norm_I166_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V:num,W:num,Y:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ V)) @ (((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W))) @ Y)) = (((plus_plus @ A) @ (((neg_numeral_sub @ A) @ V) @ W)) @ Y)))))))).
% 9.43/3.45 thf(fact_2041_semiring__norm_I167_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [V:num,W:num,Y:A]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ V))) @ (((plus_plus @ A) @ ((numeral_numeral @ A) @ W)) @ Y)) = (((plus_plus @ A) @ (((neg_numeral_sub @ A) @ W) @ V)) @ Y)))))))).
% 9.43/3.45 thf(fact_2042_inverse__eq__divide__neg__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [W:num]: ((((inverse_inverse @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W))) = (((divide_divide @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ W)))))))))).
% 9.43/3.45 thf(fact_2043_diff__numeral__special_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((minus_minus @ A) @ ((numeral_numeral @ A) @ M2)) @ (one_one @ A)) = (((neg_numeral_sub @ A) @ M2) @ one2)))))))).
% 9.43/3.45 thf(fact_2044_diff__numeral__special_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [N:num]: (((((minus_minus @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((neg_numeral_sub @ A) @ one2) @ N)))))))).
% 9.43/3.45 thf(fact_2045_add__neg__numeral__special_I1_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((plus_plus @ A) @ (one_one @ A)) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) = (((neg_numeral_sub @ A) @ one2) @ M2)))))))).
% 9.43/3.45 thf(fact_2046_add__neg__numeral__special_I2_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((plus_plus @ A) @ ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ M2))) @ (one_one @ A)) = (((neg_numeral_sub @ A) @ one2) @ M2)))))))).
% 9.43/3.45 thf(fact_2047_add__neg__numeral__special_I3_J, axiom, (! [A:$tType]: (((neg_numeral @ A) => (! [M2:num]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ M2)) @ ((uminus_uminus @ A) @ (one_one @ A))) = (((neg_numeral_sub @ A) @ M2) @ one2)))))))).
% 9.43/3.45 thf(fact_2048_inverse__unique, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: ((((((times_times @ A) @ A3) @ B2) = (one_one @ A)) => (((inverse_inverse @ A) @ A3) = B2)))))))).
% 9.43/3.45 thf(fact_2049_inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => ((inverse_inverse @ A) = ((divide_divide @ A) @ (one_one @ A))))))).
% 9.43/3.45 thf(fact_2050_prod__inversef, axiom, (! [A:$tType,B:$tType]: (((field @ A) => (! [F:(B > A),A4:(set @ B)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ (((((comp @ A) @ A) @ B) @ (inverse_inverse @ A)) @ F)) @ A4) = ((inverse_inverse @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A4))))))))).
% 9.43/3.45 thf(fact_2051_inverse__le__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less_eq @ A) @ ((inverse_inverse @ A) @ X)) @ (one_one @ A)) = ((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) | (((ord_less_eq @ A) @ (one_one @ A)) @ X))))))))).
% 9.43/3.45 thf(fact_2052_one__less__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X) & (((ord_less @ A) @ X) @ (one_one @ A)))))))))).
% 9.43/3.45 thf(fact_2053_one__less__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ (one_one @ A)) => (((ord_less @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 9.43/3.45 thf(fact_2054_division__ring__inverse__add, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ (((plus_plus @ A) @ A3) @ B2))) @ ((inverse_inverse @ A) @ B2))))))))))).
% 9.43/3.45 thf(fact_2055_inverse__add, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((((plus_plus @ A) @ ((inverse_inverse @ A) @ A3)) @ ((inverse_inverse @ A) @ B2)) = (((times_times @ A) @ (((times_times @ A) @ (((plus_plus @ A) @ A3) @ B2)) @ ((inverse_inverse @ A) @ A3))) @ ((inverse_inverse @ A) @ B2))))))))))).
% 9.43/3.45 thf(fact_2056_field__class_Ofield__inverse, axiom, (! [A:$tType]: (((field @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => ((((times_times @ A) @ ((inverse_inverse @ A) @ A3)) @ A3) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2057_nonzero__inverse__eq__divide, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((inverse_inverse @ A) @ A3) = (((divide_divide @ A) @ (one_one @ A)) @ A3))))))))).
% 9.43/3.45 thf(fact_2058_one__le__inverse__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ X)) = ((((ord_less @ A) @ (zero_zero @ A)) @ X) & (((ord_less_eq @ A) @ X) @ (one_one @ A)))))))))).
% 9.43/3.45 thf(fact_2059_inverse__less__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A]: (((((ord_less @ A) @ ((inverse_inverse @ A) @ X)) @ (one_one @ A)) = ((((ord_less_eq @ A) @ X) @ (zero_zero @ A)) | (((ord_less @ A) @ (one_one @ A)) @ X))))))))).
% 9.43/3.45 thf(fact_2060_one__le__inverse, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A]: (((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((ord_less_eq @ A) @ (one_one @ A)) @ ((inverse_inverse @ A) @ A3)))))))))).
% 9.43/3.45 thf(fact_2061_Frct__code__post_I5_J, axiom, (! [K:num]: (((frct @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ ((numeral_numeral @ int) @ K))) = (((divide_divide @ rat) @ (one_one @ rat)) @ ((numeral_numeral @ rat) @ K)))))).
% 9.43/3.45 thf(fact_2062_Code__Numeral_Odup__def, axiom, (code_dup = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (^ [K4:int]: ((((plus_plus @ int) @ K4) @ K4)))))).
% 9.43/3.45 thf(fact_2063_Frct__code__post_I6_J, axiom, (! [K:num,L:num]: (((frct @ ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ ((numeral_numeral @ int) @ L))) = (((divide_divide @ rat) @ ((numeral_numeral @ rat) @ K)) @ ((numeral_numeral @ rat) @ L)))))).
% 9.43/3.45 thf(fact_2064_Frct__code__post_I4_J, axiom, (! [K:num]: (((frct @ ((((product_Pair @ int) @ int) @ ((numeral_numeral @ int) @ K)) @ (one_one @ int))) = ((numeral_numeral @ rat) @ K))))).
% 9.43/3.45 thf(fact_2065_eq__numeral__iff__iszero_I8_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Y:num]: ((((one_one @ A) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ Y))) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ one2) @ Y)))))))))).
% 9.43/3.45 thf(fact_2066_eq__numeral__iff__iszero_I7_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X)) = (one_one @ A)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ X) @ one2)))))))))).
% 9.43/3.45 thf(fact_2067_Inf_OINF__image, axiom, (! [B:$tType,A:$tType,C:$tType,Inf:((set @ A) > A),G:(B > A),F:(C > B),A4:(set @ C)]: (((Inf @ ((((image2 @ B) @ A) @ G) @ ((((image2 @ C) @ B) @ F) @ A4))) = (Inf @ ((((image2 @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A4)))))).
% 9.43/3.45 thf(fact_2068_not__iszero__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (~ (((ring_1_iszero @ A) @ (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2069_Code__Numeral_Odup__code_I1_J, axiom, ((code_dup @ (zero_zero @ code_integer)) = (zero_zero @ code_integer))).
% 9.43/3.45 thf(fact_2070_not__iszero__neg__1, axiom, (! [A:$tType]: (((ring_1 @ A) => (~ (((ring_1_iszero @ A) @ ((uminus_uminus @ A) @ (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2071_dup_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ (code_dup @ X)) = (((plus_plus @ int) @ (code_int_of_integer @ X)) @ (code_int_of_integer @ X)))))).
% 9.43/3.45 thf(fact_2072_dup_Oabs__eq, axiom, (! [X:int]: (((code_dup @ (code_integer_of_int @ X)) = (code_integer_of_int @ (((plus_plus @ int) @ X) @ X)))))).
% 9.43/3.45 thf(fact_2073_eq__numeral__iff__iszero_I2_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num,Y:num]: (((((numeral_numeral @ A) @ X) = ((uminus_uminus @ A) @ ((numeral_numeral @ A) @ Y))) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ X) @ Y)))))))))).
% 9.43/3.45 thf(fact_2074_eq__numeral__iff__iszero_I3_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num,Y:num]: (((((uminus_uminus @ A) @ ((numeral_numeral @ A) @ X)) = ((numeral_numeral @ A) @ Y)) = ((ring_1_iszero @ A) @ ((numeral_numeral @ A) @ (((plus_plus @ num) @ X) @ Y)))))))))).
% 9.43/3.45 thf(fact_2075_Frct__code__post_I2_J, axiom, (! [A3:int]: (((frct @ ((((product_Pair @ int) @ int) @ A3) @ (zero_zero @ int))) = (zero_zero @ rat))))).
% 9.43/3.45 thf(fact_2076_Frct__code__post_I1_J, axiom, (! [A3:int]: (((frct @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ A3)) = (zero_zero @ rat))))).
% 9.43/3.45 thf(fact_2077_Frct__code__post_I3_J, axiom, ((frct @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))) = (one_one @ rat))).
% 9.43/3.45 thf(fact_2078_eq__numeral__iff__iszero_I6_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [Y:num]: ((((one_one @ A) = ((numeral_numeral @ A) @ Y)) = ((ring_1_iszero @ A) @ (((neg_numeral_sub @ A) @ one2) @ Y))))))))).
% 9.43/3.45 thf(fact_2079_eq__numeral__iff__iszero_I5_J, axiom, (! [A:$tType]: (((ring_1 @ A) => (! [X:num]: (((((numeral_numeral @ A) @ X) = (one_one @ A)) = ((ring_1_iszero @ A) @ (((neg_numeral_sub @ A) @ X) @ one2))))))))).
% 9.43/3.45 thf(fact_2080_Frct__code__post_I7_J, axiom, (! [A3:int,B2:int]: (((frct @ ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ A3)) @ B2)) = ((uminus_uminus @ rat) @ (frct @ ((((product_Pair @ int) @ int) @ A3) @ B2))))))).
% 9.43/3.45 thf(fact_2081_Frct__code__post_I8_J, axiom, (! [A3:int,B2:int]: (((frct @ ((((product_Pair @ int) @ int) @ A3) @ ((uminus_uminus @ int) @ B2))) = ((uminus_uminus @ rat) @ (frct @ ((((product_Pair @ int) @ int) @ A3) @ B2))))))).
% 9.43/3.45 thf(fact_2082_Sup_OSUP__image, axiom, (! [B:$tType,A:$tType,C:$tType,Sup:((set @ A) > A),G:(B > A),F:(C > B),A4:(set @ C)]: (((Sup @ ((((image2 @ B) @ A) @ G) @ ((((image2 @ C) @ B) @ F) @ A4))) = (Sup @ ((((image2 @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A4)))))).
% 9.43/3.45 thf(fact_2083_Code__Numeral_Osub__code_I9_J, axiom, (! [M2:num,N:num]: ((((code_sub @ (bit0 @ M2)) @ (bit1 @ N)) = (((minus_minus @ code_integer) @ (code_dup @ ((code_sub @ M2) @ N))) @ (one_one @ code_integer)))))).
% 9.43/3.45 thf(fact_2084_Code__Numeral_Osub__code_I8_J, axiom, (! [M2:num,N:num]: ((((code_sub @ (bit1 @ M2)) @ (bit0 @ N)) = (((plus_plus @ code_integer) @ (code_dup @ ((code_sub @ M2) @ N))) @ (one_one @ code_integer)))))).
% 9.43/3.45 thf(fact_2085_less__eq__int__def, axiom, ((ord_less_eq @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > $o)) @ (int > $o)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ $o) @ $o) @ rep_Integ) @ (id @ $o))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3))))))))))).
% 9.43/3.45 thf(fact_2086_less__int__def, axiom, ((ord_less @ int) = (((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > $o)) @ (int > $o)) @ rep_Integ) @ ((((((map_fun @ int) @ ((product_prod @ nat) @ nat)) @ $o) @ $o) @ rep_Integ) @ (id @ $o))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3))))))))))).
% 9.43/3.45 thf(fact_2087_image__affinity__atLeastAtMost__div__diff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M2:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X3) @ M2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X3) @ M2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ M2)) @ C2)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ B2) @ M2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ (((divide_divide @ A) @ X3) @ M2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((divide_divide @ A) @ B2) @ M2)) @ C2)) @ (((minus_minus @ A) @ (((divide_divide @ A) @ A3) @ M2)) @ C2))))))))))))).
% 9.43/3.45 thf(fact_2088_image__affinity__atLeastAtMost__div, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M2:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ M2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ M2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ M2)) @ C2)) @ (((plus_plus @ A) @ (((divide_divide @ A) @ B2) @ M2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((plus_plus @ A) @ (((divide_divide @ A) @ X3) @ M2)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((divide_divide @ A) @ B2) @ M2)) @ C2)) @ (((plus_plus @ A) @ (((divide_divide @ A) @ A3) @ M2)) @ C2))))))))))))).
% 9.43/3.45 thf(fact_2089_image__affinity__atLeastAtMost__diff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M2:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M2) @ X3)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M2) @ X3)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ M2) @ A3)) @ C2)) @ (((minus_minus @ A) @ (((times_times @ A) @ M2) @ B2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ (((times_times @ A) @ M2) @ X3)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((minus_minus @ A) @ (((times_times @ A) @ M2) @ B2)) @ C2)) @ (((minus_minus @ A) @ (((times_times @ A) @ M2) @ A3)) @ C2))))))))))))).
% 9.43/3.45 thf(fact_2090_id__apply, axiom, (! [A:$tType]: (((id @ A) = (^ [X3:A]: (X3)))))).
% 9.43/3.45 thf(fact_2091_bijective__Empty, axiom, (! [B:$tType,A:$tType]: ((((bijective @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2092_image__id, axiom, (! [A:$tType]: (((((image2 @ A) @ A) @ (id @ A)) = (id @ (set @ A)))))).
% 9.43/3.45 thf(fact_2093_fun_Omap__id, axiom, (! [A:$tType,D:$tType,T2:(D > A)]: (((((((comp @ A) @ A) @ D) @ (id @ A)) @ T2) = T2)))).
% 9.43/3.45 thf(fact_2094_id__comp, axiom, (! [B:$tType,A:$tType,G:(A > B)]: (((((((comp @ B) @ B) @ A) @ (id @ B)) @ G) = G)))).
% 9.43/3.45 thf(fact_2095_comp__id, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((comp @ A) @ B) @ A) @ F) @ (id @ A)) = F)))).
% 9.43/3.45 thf(fact_2096_lessThan__0, axiom, (((set_ord_lessThan @ nat) @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))).
% 9.43/3.45 thf(fact_2097_apsnd__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apsnd @ B) @ B) @ A) @ (id @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 9.43/3.45 thf(fact_2098_map__mmupd__empty, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),V:B]: (((((((map_mmupd @ A) @ B) @ M2) @ (bot_bot @ (set @ A))) @ V) = M2)))).
% 9.43/3.45 thf(fact_2099_set__to__map__empty, axiom, (! [B:$tType,A:$tType]: (((((set_to_map @ A) @ B) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_2100_atLeastatMost__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (~ ((((ord_less_eq @ A) @ A3) @ B2)))))))))).
% 9.43/3.45 thf(fact_2101_atLeastatMost__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) = (~ ((((ord_less_eq @ A) @ A3) @ B2)))))))))).
% 9.43/3.45 thf(fact_2102_atLeastatMost__empty, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2103_atLeastLessThan__empty, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less_eq @ A) @ B2) @ A3) => ((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2104_atLeastLessThan__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((((set_or7035219750837199246ssThan @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) = (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 9.43/3.45 thf(fact_2105_atLeastLessThan__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) = (~ ((((ord_less @ A) @ A3) @ B2)))))))))).
% 9.43/3.45 thf(fact_2106_prod_Oempty, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A)]: ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (bot_bot @ (set @ B))) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2107_group__add__class_Ominus__comp__minus, axiom, (! [A:$tType]: (((group_add @ A) => ((((((comp @ A) @ A) @ A) @ (uminus_uminus @ A)) @ (uminus_uminus @ A)) = (id @ A)))))).
% 9.43/3.45 thf(fact_2108_boolean__algebra__class_Ominus__comp__minus, axiom, (! [A:$tType]: (((boolea8198339166811842893lgebra @ A) => ((((((comp @ A) @ A) @ A) @ (uminus_uminus @ A)) @ (uminus_uminus @ A)) = (id @ A)))))).
% 9.43/3.45 thf(fact_2109_rel__of__empty, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o)]: ((((((rel_of @ A) @ B) @ (^ [X3:A]: ((none @ B)))) @ P) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2110_comp__the__Some, axiom, (! [A:$tType]: (((((((comp @ (option @ A)) @ A) @ A) @ (the2 @ A)) @ (some @ A)) = (id @ A))))).
% 9.43/3.45 thf(fact_2111_fun_Omap__id0, axiom, (! [A:$tType,D:$tType]: ((((((comp @ A) @ A) @ D) @ (id @ A)) = (id @ (D > A)))))).
% 9.43/3.45 thf(fact_2112_Iio__eq__empty__iff, axiom, (! [A:$tType]: ((((linorder @ A) & (order_bot @ A)) => (! [N:A]: (((((set_ord_lessThan @ A) @ N) = (bot_bot @ (set @ A))) = (N = (bot_bot @ A))))))))).
% 9.43/3.45 thf(fact_2113_option_Omap__id0, axiom, (! [A:$tType]: (((((map_option @ A) @ A) @ (id @ A)) = (id @ (option @ A)))))).
% 9.43/3.45 thf(fact_2114_set__notEmptyE, axiom, (! [A:$tType,S2:(set @ A)]: (((S2 != (bot_bot @ (set @ A))) => (~ ((! [X2:A]: ((~ ((((member @ A) @ X2) @ S2))))))))))).
% 9.43/3.45 thf(fact_2115_memb__imp__not__empty, axiom, (! [A:$tType,X:A,S2:(set @ A)]: (((((member @ A) @ X) @ S2) => (S2 != (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_2116_id__def, axiom, (! [A:$tType]: (((id @ A) = (^ [X3:A]: (X3)))))).
% 9.43/3.45 thf(fact_2117_eq__id__iff, axiom, (! [A:$tType,F:(A > A)]: (((! [X3:A]: (((F @ X3) = X3))) = (F = (id @ A)))))).
% 9.43/3.45 thf(fact_2118_comp__eq__id__dest, axiom, (! [C:$tType,B:$tType,A:$tType,A3:(C > B),B2:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B2) = (((((comp @ B) @ B) @ A) @ (id @ B)) @ C2)) => ((A3 @ (B2 @ V)) = (C2 @ V)))))).
% 9.43/3.45 thf(fact_2119_pointfree__idE, axiom, (! [B:$tType,A:$tType,F:(B > A),G:(A > B),X:A]: ((((((((comp @ B) @ A) @ A) @ F) @ G) = (id @ A)) => ((F @ (G @ X)) = X))))).
% 9.43/3.45 thf(fact_2120_option_Omap__id, axiom, (! [A:$tType,T2:(option @ A)]: ((((((map_option @ A) @ A) @ (id @ A)) @ T2) = T2)))).
% 9.43/3.45 thf(fact_2121_not__empty__eq__Iic__eq__empty, axiom, (! [A:$tType]: (((preorder @ A) => (! [H3:A]: (((bot_bot @ (set @ A)) != ((set_ord_atMost @ A) @ H3)))))))).
% 9.43/3.45 thf(fact_2122_lessThan__non__empty, axiom, (! [A:$tType]: (((no_bot @ A) => (! [X:A]: ((((set_ord_lessThan @ A) @ X) != (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_2123_subset__minus__empty, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B6) => ((((minus_minus @ (set @ A)) @ A4) @ B6) = (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_2124_atLeastLessThan0, axiom, (! [M2:nat]: (((((set_or7035219750837199246ssThan @ nat) @ M2) @ (zero_zero @ nat)) = (bot_bot @ (set @ nat)))))).
% 9.43/3.45 thf(fact_2125_lessThan__empty__iff, axiom, (! [N:nat]: (((((set_ord_lessThan @ nat) @ N) = (bot_bot @ (set @ nat))) = (N = (zero_zero @ nat)))))).
% 9.43/3.45 thf(fact_2126_set__to__map__empty__iff_I1_J, axiom, (! [B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B))]: ((((((set_to_map @ A) @ B) @ S2) = (^ [X3:A]: ((none @ B)))) = (S2 = (bot_bot @ (set @ ((product_prod @ A) @ B)))))))).
% 9.43/3.45 thf(fact_2127_Code__Numeral_Osub__code_I1_J, axiom, (((code_sub @ one2) @ one2) = (zero_zero @ code_integer))).
% 9.43/3.45 thf(fact_2128_set__to__map__empty__iff_I2_J, axiom, (! [B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B))]: ((((^ [X3:A]: ((none @ B))) = (((set_to_map @ A) @ B) @ S2)) = (S2 = (bot_bot @ (set @ ((product_prod @ A) @ B)))))))).
% 9.43/3.45 thf(fact_2129_less__integer__def, axiom, ((ord_less @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > $o)) @ (code_integer > $o)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ $o) @ $o) @ code_int_of_integer) @ (id @ $o))) @ (ord_less @ int)))).
% 9.43/3.45 thf(fact_2130_less__eq__integer__def, axiom, ((ord_less_eq @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (int > $o)) @ (code_integer > $o)) @ code_int_of_integer) @ ((((((map_fun @ code_integer) @ int) @ $o) @ $o) @ code_int_of_integer) @ (id @ $o))) @ (ord_less_eq @ int)))).
% 9.43/3.45 thf(fact_2131_Code__Numeral_Osub__code_I6_J, axiom, (! [M2:num,N:num]: ((((code_sub @ (bit0 @ M2)) @ (bit0 @ N)) = (code_dup @ ((code_sub @ M2) @ N)))))).
% 9.43/3.45 thf(fact_2132_Code__Numeral_Osub__code_I7_J, axiom, (! [M2:num,N:num]: ((((code_sub @ (bit1 @ M2)) @ (bit1 @ N)) = (code_dup @ ((code_sub @ M2) @ N)))))).
% 9.43/3.45 thf(fact_2133_sub_Orep__eq, axiom, (! [X:num,Xa:num]: (((code_int_of_integer @ ((code_sub @ X) @ Xa)) = (((minus_minus @ int) @ ((numeral_numeral @ int) @ X)) @ ((numeral_numeral @ int) @ Xa)))))).
% 9.43/3.45 thf(fact_2134_sub_Oabs__eq, axiom, (code_sub = (^ [Xa4:num,X3:num]: ((code_integer_of_int @ (((minus_minus @ int) @ ((numeral_numeral @ int) @ Xa4)) @ ((numeral_numeral @ int) @ X3))))))).
% 9.43/3.45 thf(fact_2135_image__mult__atLeastAtMost__if, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [C2:A,X:A,Y:A]: ((((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((image2 @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ C2) @ X)) @ (((times_times @ A) @ C2) @ Y)))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((ord_less_eq @ A) @ X) @ Y) => (((((image2 @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ C2) @ Y)) @ (((times_times @ A) @ C2) @ X)))) & ((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((((image2 @ A) @ A) @ ((times_times @ A) @ C2)) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (bot_bot @ (set @ A))))))))))))).
% 9.43/3.45 thf(fact_2136_image__mult__atLeastAtMost__if_H, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,Y:A,C2:A]: ((((((ord_less_eq @ A) @ X) @ Y) => (((((ord_less @ A) @ (zero_zero @ A)) @ C2) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((times_times @ A) @ X3) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ X) @ C2)) @ (((times_times @ A) @ Y) @ C2)))) & ((~ ((((ord_less @ A) @ (zero_zero @ A)) @ C2))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((times_times @ A) @ X3) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (((set_or1337092689740270186AtMost @ A) @ (((times_times @ A) @ Y) @ C2)) @ (((times_times @ A) @ X) @ C2)))))) & ((~ ((((ord_less_eq @ A) @ X) @ Y))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((times_times @ A) @ X3) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_2137_image__affinity__atLeastAtMost, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,B2:A,M2:A,C2:A]: (((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M2) @ X3)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (bot_bot @ (set @ A)))) & (((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) != (bot_bot @ (set @ A))) => (((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M2) @ X3)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ M2) @ A3)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ M2) @ B2)) @ C2)))) & ((~ ((((ord_less_eq @ A) @ (zero_zero @ A)) @ M2))) => (((((image2 @ A) @ A) @ (^ [X3:A]: ((((plus_plus @ A) @ (((times_times @ A) @ M2) @ X3)) @ C2)))) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) = (((set_or1337092689740270186AtMost @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ M2) @ B2)) @ C2)) @ (((plus_plus @ A) @ (((times_times @ A) @ M2) @ A3)) @ C2))))))))))))).
% 9.43/3.45 thf(fact_2138_and__int_Osimps, axiom, ((bit_se5824344872417868541ns_and @ int) = (^ [K4:int,L3:int]: (((((if @ int) @ ((((member @ int) @ K4) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L3) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))))) @ ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K4))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3))))))) @ (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K4))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L3)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K4) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L3) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))).
% 9.43/3.45 thf(fact_2139_and__int_Oelims, axiom, (! [X:int,Xa:int,Y:int]: ((((((bit_se5824344872417868541ns_and @ int) @ X) @ Xa) = Y) => ((((((member @ int) @ X) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => (Y = ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa)))))))) & ((~ (((((member @ int) @ X) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => (Y = (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ X) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ Xa) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 9.43/3.45 thf(fact_2140_and__int_Opelims, axiom, (! [X:int,Xa:int,Y:int]: ((((((bit_se5824344872417868541ns_and @ int) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa)) => (~ ((((((((member @ int) @ X) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => (Y = ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa)))))))) & ((~ (((((member @ int) @ X) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ Xa) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => (Y = (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ X))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ Xa)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ X) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ Xa) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))) => (~ ((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_2141_and__int_Opsimps, axiom, (! [K:int,L:int]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ K) @ L)) => ((((((member @ int) @ K) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))) => ((((bit_se5824344872417868541ns_and @ int) @ K) @ L) = ((uminus_uminus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L)))))))) & ((~ (((((member @ int) @ K) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => ((((bit_se5824344872417868541ns_and @ int) @ K) @ L) = (((plus_plus @ int) @ ((zero_neq_one_of_bool @ int) @ ((~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ K))) & (~ ((((dvd_dvd @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ L)))))) @ (((times_times @ int) @ ((numeral_numeral @ int) @ (bit0 @ one2))) @ (((bit_se5824344872417868541ns_and @ int) @ (((divide_divide @ int) @ K) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L) @ ((numeral_numeral @ int) @ (bit0 @ one2))))))))))))).
% 9.43/3.45 thf(fact_2142_and__int_Opinduct, axiom, (! [A0:int,A1:int,P:(int > (int > $o))]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ A0) @ A1)) => ((! [K5:int,L5:int]: (((((accp @ ((product_prod @ int) @ int)) @ bit_and_int_rel) @ ((((product_Pair @ int) @ int) @ K5) @ L5)) => (((~ (((((member @ int) @ K5) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int))))) & (((member @ int) @ L5) @ (((insert2 @ int) @ (zero_zero @ int)) @ (((insert2 @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (bot_bot @ (set @ int)))))))) => ((P @ (((divide_divide @ int) @ K5) @ ((numeral_numeral @ int) @ (bit0 @ one2)))) @ (((divide_divide @ int) @ L5) @ ((numeral_numeral @ int) @ (bit0 @ one2))))) => ((P @ K5) @ L5))))) => ((P @ A0) @ A1)))))).
% 9.43/3.45 thf(fact_2143_mask__mod__exp, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat,M2:nat]: (((((modulo_modulo @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N)) @ (one_one @ A))) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) = (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ (((ord_min @ nat) @ M2) @ N))) @ (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2144_pair__leqI2, axiom, (! [A3:nat,B2:nat,S:nat,T2:nat]: (((((ord_less_eq @ nat) @ A3) @ B2) => ((((ord_less_eq @ nat) @ S) @ T2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A3) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_leq)))))).
% 9.43/3.45 thf(fact_2145_min__eq__arg_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:A,N:A]: ((((((ord_min @ A) @ M2) @ N) = M2) = (((ord_less_eq @ A) @ M2) @ N)))))))).
% 9.43/3.45 thf(fact_2146_min__eq__arg_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:A,N:A]: ((((((ord_min @ A) @ M2) @ N) = N) = (((ord_less_eq @ A) @ N) @ M2)))))))).
% 9.43/3.45 thf(fact_2147_min__arg__le_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [N:A,M2:A]: (((((ord_less_eq @ A) @ N) @ (((ord_min @ A) @ M2) @ N)) = ((((ord_min @ A) @ M2) @ N) = N)))))))).
% 9.43/3.45 thf(fact_2148_min__arg__le_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [M2:A,N:A]: (((((ord_less_eq @ A) @ M2) @ (((ord_min @ A) @ M2) @ N)) = ((((ord_min @ A) @ M2) @ N) = M2)))))))).
% 9.43/3.45 thf(fact_2149_min__arg__not__ge_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:A,N:A]: (((~ ((((ord_less @ A) @ (((ord_min @ A) @ M2) @ N)) @ M2))) = ((((ord_min @ A) @ M2) @ N) = M2)))))))).
% 9.43/3.45 thf(fact_2150_min__arg__not__ge_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [M2:A,N:A]: (((~ ((((ord_less @ A) @ (((ord_min @ A) @ M2) @ N)) @ N))) = ((((ord_min @ A) @ M2) @ N) = N)))))))).
% 9.43/3.45 thf(fact_2151_min__less__self__conv_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (((ord_min @ A) @ A3) @ B2)) @ A3) = (((ord_less @ A) @ B2) @ A3)))))))).
% 9.43/3.45 thf(fact_2152_min__less__self__conv_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ (((ord_min @ A) @ A3) @ B2)) @ B2) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2153_min__simps_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => ((((ord_min @ A) @ A3) @ B2) = A3)))))))).
% 9.43/3.45 thf(fact_2154_min__simps_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [B2:A,A3:A]: (((((ord_less @ A) @ B2) @ A3) => ((((ord_min @ A) @ A3) @ B2) = B2)))))))).
% 9.43/3.45 thf(fact_2155_min__0__1_I2_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_min @ A) @ (one_one @ A)) @ (zero_zero @ A)) = (zero_zero @ A)))))).
% 9.43/3.45 thf(fact_2156_min__0__1_I1_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => ((((ord_min @ A) @ (zero_zero @ A)) @ (one_one @ A)) = (zero_zero @ A)))))).
% 9.43/3.45 thf(fact_2157_min__0__1_I5_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_min @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ X)) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2158_min__0__1_I6_J, axiom, (! [A:$tType]: (((linord181362715937106298miring @ A) => (! [X:num]: (((((ord_min @ A) @ ((numeral_numeral @ A) @ X)) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2159_atLeastAtMost__singleton__iff, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A,C2:A]: ((((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (((insert2 @ A) @ C2) @ (bot_bot @ (set @ A)))) = ((A3 = B2) & (B2 = C2))))))))).
% 9.43/3.45 thf(fact_2160_atLeastAtMost__singleton, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: (((((set_or1337092689740270186AtMost @ A) @ A3) @ A3) = (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2161_min__Suc__gt_I2_J, axiom, (! [A3:nat,B2:nat]: (((((ord_less @ nat) @ A3) @ B2) => ((((ord_min @ nat) @ B2) @ (suc @ A3)) = (suc @ A3)))))).
% 9.43/3.45 thf(fact_2162_min__Suc__gt_I1_J, axiom, (! [A3:nat,B2:nat]: (((((ord_less @ nat) @ A3) @ B2) => ((((ord_min @ nat) @ (suc @ A3)) @ B2) = (suc @ A3)))))).
% 9.43/3.45 thf(fact_2163_case__prod__Pair, axiom, (! [B:$tType,A:$tType]: ((((((product_case_prod @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 9.43/3.45 thf(fact_2164_single__Diff__lessThan, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A]: (((((minus_minus @ (set @ A)) @ (((insert2 @ A) @ K) @ (bot_bot @ (set @ A)))) @ ((set_ord_lessThan @ A) @ K)) = (((insert2 @ A) @ K) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2165_min__add__distrib__right, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y:A,Z2:A]: (((((plus_plus @ A) @ X) @ (((ord_min @ A) @ Y) @ Z2)) = (((ord_min @ A) @ (((plus_plus @ A) @ X) @ Y)) @ (((plus_plus @ A) @ X) @ Z2))))))))).
% 9.43/3.45 thf(fact_2166_min__add__distrib__left, axiom, (! [A:$tType]: (((ordere2412721322843649153imp_le @ A) => (! [X:A,Y:A,Z2:A]: (((((plus_plus @ A) @ (((ord_min @ A) @ X) @ Y)) @ Z2) = (((ord_min @ A) @ (((plus_plus @ A) @ X) @ Z2)) @ (((plus_plus @ A) @ Y) @ Z2))))))))).
% 9.43/3.45 thf(fact_2167_min__diff__distrib__left, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A,Z2:A]: (((((minus_minus @ A) @ (((ord_min @ A) @ X) @ Y)) @ Z2) = (((ord_min @ A) @ (((minus_minus @ A) @ X) @ Z2)) @ (((minus_minus @ A) @ Y) @ Z2))))))))).
% 9.43/3.45 thf(fact_2168_bot__option__def, axiom, (! [A:$tType]: (((order @ A) => ((bot_bot @ (option @ A)) = (none @ A)))))).
% 9.43/3.45 thf(fact_2169_bot__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((bot_bot @ (A > (B > $o))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))))))))).
% 9.43/3.45 thf(fact_2170_map__fun_Oidentity, axiom, (! [B:$tType,A:$tType]: ((((((((map_fun @ A) @ A) @ B) @ B) @ (^ [X3:A]: (X3))) @ (^ [X3:B]: (X3))) = (id @ (A > B)))))).
% 9.43/3.45 thf(fact_2171_map__option_Oidentity, axiom, (! [A:$tType]: (((((map_option @ A) @ A) @ (^ [X3:A]: (X3))) = (id @ (option @ A)))))).
% 9.43/3.45 thf(fact_2172_minus__max__eq__min, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [X:A,Y:A]: ((((uminus_uminus @ A) @ (((ord_max @ A) @ X) @ Y)) = (((ord_min @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y))))))))).
% 9.43/3.45 thf(fact_2173_minus__min__eq__max, axiom, (! [A:$tType]: (((linord5086331880401160121up_add @ A) => (! [X:A,Y:A]: ((((uminus_uminus @ A) @ (((ord_min @ A) @ X) @ Y)) = (((ord_max @ A) @ ((uminus_uminus @ A) @ X)) @ ((uminus_uminus @ A) @ Y))))))))).
% 9.43/3.45 thf(fact_2174_insert__minus__eq, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ A)]: (((X != Y) => ((((minus_minus @ (set @ A)) @ (((insert2 @ A) @ X) @ A4)) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A)))) = (((insert2 @ A) @ X) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2175_set__minus__singleton__eq, axiom, (! [A:$tType,X:A,X10:(set @ A)]: (((~ ((((member @ A) @ X) @ X10))) => ((((minus_minus @ (set @ A)) @ X10) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) = X10))))).
% 9.43/3.45 thf(fact_2176_atLeastAtMost__singleton_H, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B2:A]: (((A3 = B2) => ((((set_or1337092689740270186AtMost @ A) @ A3) @ B2) = (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_2177_remove__subset, axiom, (! [A:$tType,X:A,S2:(set @ A)]: (((((member @ A) @ X) @ S2) => (((ord_less @ (set @ A)) @ (((minus_minus @ (set @ A)) @ S2) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) @ S2))))).
% 9.43/3.45 thf(fact_2178_nat__of__integer__def, axiom, (code_nat_of_integer = (((((((map_fun @ code_integer) @ int) @ nat) @ nat) @ code_int_of_integer) @ (id @ nat)) @ nat2))).
% 9.43/3.45 thf(fact_2179_bit__integer__def, axiom, ((bit_se5641148757651400278ts_bit @ code_integer) = (((((((map_fun @ code_integer) @ int) @ (nat > $o)) @ (nat > $o)) @ code_int_of_integer) @ (id @ (nat > $o))) @ (bit_se5641148757651400278ts_bit @ int)))).
% 9.43/3.45 thf(fact_2180_mask__integer__def, axiom, ((bit_se2239418461657761734s_mask @ code_integer) = (((((((map_fun @ nat) @ nat) @ int) @ code_integer) @ (id @ nat)) @ code_integer_of_int) @ (bit_se2239418461657761734s_mask @ int)))).
% 9.43/3.45 thf(fact_2181_atLeastLessThan__eq__atLeastAtMost__diff, axiom, (! [A:$tType]: (((order @ A) => ((set_or7035219750837199246ssThan @ A) = (^ [A7:A,B5:A]: ((((minus_minus @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A7) @ B5)) @ (((insert2 @ A) @ B5) @ (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_2182_atLeastAtMostPlus1__int__conv, axiom, (! [M2:int,N:int]: (((((ord_less_eq @ int) @ M2) @ (((plus_plus @ int) @ (one_one @ int)) @ N)) => ((((set_or1337092689740270186AtMost @ int) @ M2) @ (((plus_plus @ int) @ (one_one @ int)) @ N)) = (((insert2 @ int) @ (((plus_plus @ int) @ (one_one @ int)) @ N)) @ (((set_or1337092689740270186AtMost @ int) @ M2) @ N))))))).
% 9.43/3.45 thf(fact_2183_simp__from__to, axiom, ((set_or1337092689740270186AtMost @ int) = (^ [I3:int,J2:int]: (((((if @ (set @ int)) @ (((ord_less @ int) @ J2) @ I3)) @ (bot_bot @ (set @ int))) @ (((insert2 @ int) @ I3) @ (((set_or1337092689740270186AtMost @ int) @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ J2))))))).
% 9.43/3.45 thf(fact_2184_Code__Numeral_Osub__def, axiom, (code_sub = (((((((map_fun @ num) @ num) @ (num > int)) @ (num > code_integer)) @ (id @ num)) @ ((((((map_fun @ num) @ num) @ int) @ code_integer) @ (id @ num)) @ code_integer_of_int)) @ (^ [M:num,N5:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N5))))))).
% 9.43/3.45 thf(fact_2185_num__of__integer__def, axiom, (code_num_of_integer = (((((((map_fun @ code_integer) @ int) @ num) @ num) @ code_int_of_integer) @ (id @ num)) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2)))).
% 9.43/3.45 thf(fact_2186_take__bit__integer__def, axiom, ((bit_se2584673776208193580ke_bit @ code_integer) = (((((((map_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (id @ nat)) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se2584673776208193580ke_bit @ int)))).
% 9.43/3.45 thf(fact_2187_drop__bit__integer__def, axiom, ((bit_se4197421643247451524op_bit @ code_integer) = (((((((map_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (id @ nat)) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se4197421643247451524op_bit @ int)))).
% 9.43/3.45 thf(fact_2188_push__bit__integer__def, axiom, ((bit_se4730199178511100633sh_bit @ code_integer) = (((((((map_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (id @ nat)) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se4730199178511100633sh_bit @ int)))).
% 9.43/3.45 thf(fact_2189_unset__bit__integer__def, axiom, ((bit_se2638667681897837118et_bit @ code_integer) = (((((((map_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (id @ nat)) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se2638667681897837118et_bit @ int)))).
% 9.43/3.45 thf(fact_2190_set__bit__integer__def, axiom, ((bit_se5668285175392031749et_bit @ code_integer) = (((((((map_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (id @ nat)) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se5668285175392031749et_bit @ int)))).
% 9.43/3.45 thf(fact_2191_flip__bit__integer__def, axiom, ((bit_se8732182000553998342ip_bit @ code_integer) = (((((((map_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (id @ nat)) @ ((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int)) @ (bit_se8732182000553998342ip_bit @ int)))).
% 9.43/3.45 thf(fact_2192_pair__leqI1, axiom, (! [A3:nat,B2:nat,S:nat,T2:nat]: (((((ord_less @ nat) @ A3) @ B2) => (((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ A3) @ S)) @ ((((product_Pair @ nat) @ nat) @ B2) @ T2))) @ fun_pair_leq))))).
% 9.43/3.45 thf(fact_2193_image__minus__const__atLeastLessThan__nat, axiom, (! [C2:nat,Y:nat,X:nat]: ((((((ord_less @ nat) @ C2) @ Y) => (((((image2 @ nat) @ nat) @ (^ [I3:nat]: ((((minus_minus @ nat) @ I3) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X) @ Y)) = (((set_or7035219750837199246ssThan @ nat) @ (((minus_minus @ nat) @ X) @ C2)) @ (((minus_minus @ nat) @ Y) @ C2)))) & ((~ ((((ord_less @ nat) @ C2) @ Y))) => (((((ord_less @ nat) @ X) @ Y) => (((((image2 @ nat) @ nat) @ (^ [I3:nat]: ((((minus_minus @ nat) @ I3) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X) @ Y)) = (((insert2 @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat))))) & ((~ ((((ord_less @ nat) @ X) @ Y))) => (((((image2 @ nat) @ nat) @ (^ [I3:nat]: ((((minus_minus @ nat) @ I3) @ C2)))) @ (((set_or7035219750837199246ssThan @ nat) @ X) @ Y)) = (bot_bot @ (set @ nat)))))))))).
% 9.43/3.45 thf(fact_2194_card__atLeastAtMost__int, axiom, (! [L:int,U:int]: ((((finite_card @ int) @ (((set_or1337092689740270186AtMost @ int) @ L) @ U)) = (nat2 @ (((plus_plus @ int) @ (((minus_minus @ int) @ U) @ L)) @ (one_one @ int))))))).
% 9.43/3.45 thf(fact_2195_range__mod, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((((image2 @ nat) @ nat) @ (^ [M:nat]: ((((modulo_modulo @ nat) @ M) @ N)))) @ (top_top @ (set @ nat))) = (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))).
% 9.43/3.45 thf(fact_2196_Some__image__these__eq, axiom, (! [A:$tType,A4:(set @ (option @ A))]: ((((((image2 @ A) @ (option @ A)) @ (some @ A)) @ ((these @ A) @ A4)) = ((collect @ (option @ A)) @ (^ [X3:(option @ A)]: (((((member @ (option @ A)) @ X3) @ A4) & (X3 != (none @ A)))))))))).
% 9.43/3.45 thf(fact_2197_prod__mono2, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [B6:(set @ A),A4:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ B6) => ((((ord_less_eq @ (set @ A)) @ A4) @ B6) => ((! [B3:A]: (((((member @ A) @ B3) @ (((minus_minus @ (set @ A)) @ B6) @ A4)) => (((ord_less_eq @ B) @ (one_one @ B)) @ (F @ B3))))) => ((! [A5:A]: (((((member @ A) @ A5) @ A4) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (F @ A5))))) => (((ord_less_eq @ B) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ A4)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ B6)))))))))))).
% 9.43/3.45 thf(fact_2198_the__dflt__None__nonempty, axiom, (! [A:$tType,S2:(set @ A)]: (((S2 != (bot_bot @ (set @ A))) => (((dflt_None_set @ A) @ S2) = ((some @ (set @ A)) @ S2)))))).
% 9.43/3.45 thf(fact_2199_the__dflt__None__empty, axiom, (! [A:$tType]: ((((dflt_None_set @ A) @ (bot_bot @ (set @ A))) = (none @ (set @ A)))))).
% 9.43/3.45 thf(fact_2200_atMost__UNIV__triv, axiom, (! [A:$tType]: ((((set_ord_atMost @ (set @ A)) @ (top_top @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 9.43/3.45 thf(fact_2201_finite__atLeastAtMost, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or1337092689740270186AtMost @ nat) @ L) @ U))))).
% 9.43/3.45 thf(fact_2202_finite__atLeastLessThan, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or7035219750837199246ssThan @ nat) @ L) @ U))))).
% 9.43/3.45 thf(fact_2203_int__of__integer__min, axiom, (! [K:code_integer,L:code_integer]: (((code_int_of_integer @ (((ord_min @ code_integer) @ K) @ L)) = (((ord_min @ int) @ (code_int_of_integer @ K)) @ (code_int_of_integer @ L)))))).
% 9.43/3.45 thf(fact_2204_finite__atMost, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((set_ord_atMost @ nat) @ K))))).
% 9.43/3.45 thf(fact_2205_finite__lessThan, axiom, (! [K:nat]: (((finite_finite2 @ nat) @ ((set_ord_lessThan @ nat) @ K))))).
% 9.43/3.45 thf(fact_2206_finite__atLeastAtMost__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or1337092689740270186AtMost @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2207_finite__atLeastLessThan__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or7035219750837199246ssThan @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2208_finite__atLeastLessThan__integer, axiom, (! [L:code_integer,U:code_integer]: (((finite_finite2 @ code_integer) @ (((set_or7035219750837199246ssThan @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2209_finite__atLeastAtMost__integer, axiom, (! [L:code_integer,U:code_integer]: (((finite_finite2 @ code_integer) @ (((set_or1337092689740270186AtMost @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2210_finite__option__UNIV, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ (top_top @ (set @ (option @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 9.43/3.45 thf(fact_2211_card__eq__UNIV, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [S2:(set @ A)]: (((((finite_card @ A) @ S2) = ((finite_card @ A) @ (top_top @ (set @ A)))) = (S2 = (top_top @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2212_card__eq__UNIV2, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [S2:(set @ A)]: (((((finite_card @ A) @ (top_top @ (set @ A))) = ((finite_card @ A) @ S2)) = (S2 = (top_top @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2213_these__empty, axiom, (! [A:$tType]: ((((these @ A) @ (bot_bot @ (set @ (option @ A)))) = (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_2214_these__insert__None, axiom, (! [A:$tType,A4:(set @ (option @ A))]: ((((these @ A) @ (((insert2 @ (option @ A)) @ (none @ A)) @ A4)) = ((these @ A) @ A4))))).
% 9.43/3.45 thf(fact_2215_the__dflt__None__set, axiom, (! [A:$tType,X:(set @ A)]: (((((the_default @ (set @ A)) @ (bot_bot @ (set @ A))) @ ((dflt_None_set @ A) @ X)) = X)))).
% 9.43/3.45 thf(fact_2216_surj__plus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 9.43/3.45 thf(fact_2217_infinite__Icc__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2218_infinite__Ico__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2219_prod_Oinfinite, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),G:(B > A)]: (((~ (((finite_finite2 @ B) @ A4))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2220_card__ge__UNIV, axiom, (! [A:$tType]: (((finite_finite @ A) => (! [S2:(set @ A)]: (((((ord_less_eq @ nat) @ ((finite_card @ A) @ (top_top @ (set @ A)))) @ ((finite_card @ A) @ S2)) = (S2 = (top_top @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2221_surj__diff__right, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A]: ((((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ (top_top @ (set @ A))) = (top_top @ (set @ A))))))))).
% 9.43/3.45 thf(fact_2222_prod__eq__1__iff, axiom, (! [A:$tType,A4:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A4) => ((((((groups7121269368397514597t_prod @ A) @ nat) @ F) @ A4) = (one_one @ nat)) = (! [X3:A]: (((((member @ A) @ X3) @ A4) => ((F @ X3) = (one_one @ nat)))))))))).
% 9.43/3.45 thf(fact_2223_prod_Odelta, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),A3:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S2) => (((((member @ B) @ A3) @ S2) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (K4 = A3)) @ (B2 @ K4)) @ (one_one @ A))))) @ S2) = (B2 @ A3))) & ((~ ((((member @ B) @ A3) @ S2))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (K4 = A3)) @ (B2 @ K4)) @ (one_one @ A))))) @ S2) = (one_one @ A))))))))))).
% 9.43/3.45 thf(fact_2224_prod_Odelta_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),A3:B,B2:(B > A)]: ((((finite_finite2 @ B) @ S2) => (((((member @ B) @ A3) @ S2) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (A3 = K4)) @ (B2 @ K4)) @ (one_one @ A))))) @ S2) = (B2 @ A3))) & ((~ ((((member @ B) @ A3) @ S2))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (A3 = K4)) @ (B2 @ K4)) @ (one_one @ A))))) @ S2) = (one_one @ A))))))))))).
% 9.43/3.45 thf(fact_2225_these__insert__Some, axiom, (! [A:$tType,X:A,A4:(set @ (option @ A))]: ((((these @ A) @ (((insert2 @ (option @ A)) @ ((some @ A) @ X)) @ A4)) = (((insert2 @ A) @ X) @ ((these @ A) @ A4)))))).
% 9.43/3.45 thf(fact_2226_these__image__Some__eq, axiom, (! [A:$tType,A4:(set @ A)]: ((((these @ A) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ A4)) = A4)))).
% 9.43/3.45 thf(fact_2227_sum_Oinsert, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),X:B,G:(B > A)]: ((((finite_finite2 @ B) @ A4) => ((~ ((((member @ B) @ X) @ A4))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert2 @ B) @ X) @ A4)) = (((plus_plus @ A) @ (G @ X)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4))))))))))).
% 9.43/3.45 thf(fact_2228_atLeastLessThan__singleton, axiom, (! [M2:nat]: (((((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ M2)) = (((insert2 @ nat) @ M2) @ (bot_bot @ (set @ nat))))))).
% 9.43/3.45 thf(fact_2229_atMost__0, axiom, (((set_ord_atMost @ nat) @ (zero_zero @ nat)) = (((insert2 @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat))))).
% 9.43/3.45 thf(fact_2230_card__atLeastLessThan__int, axiom, (! [L:int,U:int]: ((((finite_card @ int) @ (((set_or7035219750837199246ssThan @ int) @ L) @ U)) = (nat2 @ (((minus_minus @ int) @ U) @ L)))))).
% 9.43/3.45 thf(fact_2231_card__doubleton__eq__2__iff, axiom, (! [A:$tType,A3:A,B2:A]: (((((finite_card @ A) @ (((insert2 @ A) @ A3) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) = ((numeral_numeral @ nat) @ (bit0 @ one2))) = (A3 != B2))))).
% 9.43/3.45 thf(fact_2232_atMost__eq__UNIV__iff, axiom, (! [A:$tType]: (((order_top @ A) => (! [X:A]: (((((set_ord_atMost @ A) @ X) = (top_top @ (set @ A))) = (X = (top_top @ A))))))))).
% 9.43/3.45 thf(fact_2233_card__le__if__inj__on__rel, axiom, (! [B:$tType,A:$tType,B6:(set @ A),A4:(set @ B),R:(B > (A > $o))]: ((((finite_finite2 @ A) @ B6) => ((! [A5:B]: (((((member @ B) @ A5) @ A4) => (? [B8:A]: (((((member @ A) @ B8) @ B6) & ((R @ A5) @ B8))))))) => ((! [A13:B,A24:B,B3:A]: (((((member @ B) @ A13) @ A4) => ((((member @ B) @ A24) @ A4) => ((((member @ A) @ B3) @ B6) => (((R @ A13) @ B3) => (((R @ A24) @ B3) => (A13 = A24)))))))) => (((ord_less_eq @ nat) @ ((finite_card @ B) @ A4)) @ ((finite_card @ A) @ B6)))))))).
% 9.43/3.45 thf(fact_2234_top__option__def, axiom, (! [A:$tType]: (((order_top @ A) => ((top_top @ (option @ A)) = ((some @ A) @ (top_top @ A))))))).
% 9.43/3.45 thf(fact_2235_finite__if__eq__beyond__finite, axiom, (! [A:$tType,S2:(set @ A),S3:(set @ A)]: ((((finite_finite2 @ A) @ S2) => ((finite_finite2 @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [S4:(set @ A)]: (((((minus_minus @ (set @ A)) @ S4) @ S2) = (((minus_minus @ (set @ A)) @ S3) @ S2)))))))))).
% 9.43/3.45 thf(fact_2236_finite__range__Some, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ (top_top @ (set @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 9.43/3.45 thf(fact_2237_obtain__subset__with__card__n, axiom, (! [A:$tType,N:nat,S2:(set @ A)]: (((((ord_less_eq @ nat) @ N) @ ((finite_card @ A) @ S2)) => (~ ((! [T4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T4) @ S2) => ((((finite_card @ A) @ T4) = N) => (~ (((finite_finite2 @ A) @ T4))))))))))))).
% 9.43/3.45 thf(fact_2238_atLeastAtMost__eq__UNIV__iff, axiom, (! [A:$tType]: (((bounded_lattice @ A) => (! [X:A,Y:A]: ((((((set_or1337092689740270186AtMost @ A) @ X) @ Y) = (top_top @ (set @ A))) = ((X = (bot_bot @ A)) & (Y = (top_top @ A)))))))))).
% 9.43/3.45 thf(fact_2239_UNIV__option__conv, axiom, (! [A:$tType]: (((top_top @ (set @ (option @ A))) = (((insert2 @ (option @ A)) @ (none @ A)) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ (top_top @ (set @ A)))))))).
% 9.43/3.45 thf(fact_2240_bounded__nat__set__is__finite, axiom, (! [N6:(set @ nat),N:nat]: (((! [X2:nat]: (((((member @ nat) @ X2) @ N6) => (((ord_less @ nat) @ X2) @ N)))) => ((finite_finite2 @ nat) @ N6))))).
% 9.43/3.45 thf(fact_2241_finite__nat__set__iff__bounded, axiom, ((finite_finite2 @ nat) = (^ [N9:(set @ nat)]: ((? [M:nat]: ((! [X3:nat]: (((((member @ nat) @ X3) @ N9) => (((ord_less @ nat) @ X3) @ M)))))))))).
% 9.43/3.45 thf(fact_2242_finite__nat__set__iff__bounded__le, axiom, ((finite_finite2 @ nat) = (^ [N9:(set @ nat)]: ((? [M:nat]: ((! [X3:nat]: (((((member @ nat) @ X3) @ N9) => (((ord_less_eq @ nat) @ X3) @ M)))))))))).
% 9.43/3.45 thf(fact_2243_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X2:B]: ((Y = (F @ X2)))))))).
% 9.43/3.45 thf(fact_2244_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y:A]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X2:B]: ((Y != (F @ X2)))))))))).
% 9.43/3.45 thf(fact_2245_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X2:A]: (((G @ (F @ X2)) = X2))) => (((((image2 @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 9.43/3.45 thf(fact_2246_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y3:A]: ((? [X3:B]: ((Y3 = (F @ X3)))))))))).
% 9.43/3.45 thf(fact_2247_these__empty__eq, axiom, (! [A:$tType,B6:(set @ (option @ A))]: (((((these @ A) @ B6) = (bot_bot @ (set @ A))) = ((B6 = (bot_bot @ (set @ (option @ A)))) | (B6 = (((insert2 @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A)))))))))).
% 9.43/3.45 thf(fact_2248_these__not__empty__eq, axiom, (! [A:$tType,B6:(set @ (option @ A))]: (((((these @ A) @ B6) != (bot_bot @ (set @ A))) = ((B6 != (bot_bot @ (set @ (option @ A)))) & (B6 != (((insert2 @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A)))))))))).
% 9.43/3.45 thf(fact_2249_not__UNIV__eq__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L4:A,H4:A]: (((top_top @ (set @ A)) != (((set_or1337092689740270186AtMost @ A) @ L4) @ H4)))))))).
% 9.43/3.45 thf(fact_2250_infinite__Iic, axiom, (! [A:$tType]: ((((linorder @ A) & (no_bot @ A)) => (! [A3:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_atMost @ A) @ A3)))))))))).
% 9.43/3.45 thf(fact_2251_infinite__Iio, axiom, (! [A:$tType]: ((((linorder @ A) & (no_bot @ A)) => (! [A3:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_lessThan @ A) @ A3)))))))))).
% 9.43/3.45 thf(fact_2252_not__UNIV__eq__Iic, axiom, (! [A:$tType]: (((no_top @ A) => (! [H4:A]: (((top_top @ (set @ A)) != ((set_ord_atMost @ A) @ H4)))))))).
% 9.43/3.45 thf(fact_2253_card__1__singletonI, axiom, (! [A:$tType,S2:(set @ A),X:A]: ((((finite_finite2 @ A) @ S2) => ((((finite_card @ A) @ S2) = (one_one @ nat)) => ((((member @ A) @ X) @ S2) => (S2 = (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2254_finite__M__bounded__by__nat, axiom, (! [P:(nat > $o),I:nat]: (((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [K4:nat]: (((P @ K4) & (((ord_less @ nat) @ K4) @ I))))))))).
% 9.43/3.45 thf(fact_2255_finite__less__ub, axiom, (! [F:(nat > nat),U:nat]: (((! [N3:nat]: ((((ord_less_eq @ nat) @ N3) @ (F @ N3)))) => ((finite_finite2 @ nat) @ ((collect @ nat) @ (^ [N5:nat]: ((((ord_less_eq @ nat) @ (F @ N5)) @ U))))))))).
% 9.43/3.45 thf(fact_2256_card__insert__disjoint_H, axiom, (! [A:$tType,A4:(set @ A),X:A]: ((((finite_finite2 @ A) @ A4) => ((~ ((((member @ A) @ X) @ A4))) => ((((minus_minus @ nat) @ ((finite_card @ A) @ (((insert2 @ A) @ X) @ A4))) @ (suc @ (zero_zero @ nat))) = ((finite_card @ A) @ A4))))))).
% 9.43/3.45 thf(fact_2257_in__these__eq, axiom, (! [A:$tType,X:A,A4:(set @ (option @ A))]: (((((member @ A) @ X) @ ((these @ A) @ A4)) = (((member @ (option @ A)) @ ((some @ A) @ X)) @ A4))))).
% 9.43/3.45 thf(fact_2258_infinite__Icc, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2))))))))))).
% 9.43/3.45 thf(fact_2259_infinite__Ico, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2))))))))))).
% 9.43/3.45 thf(fact_2260_subset__eq__atLeast0__atMost__finite, axiom, (! [N6:(set @ nat),N:nat]: (((((ord_less_eq @ (set @ nat)) @ N6) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) => ((finite_finite2 @ nat) @ N6))))).
% 9.43/3.45 thf(fact_2261_subset__eq__atLeast0__lessThan__finite, axiom, (! [N6:(set @ nat),N:nat]: (((((ord_less_eq @ (set @ nat)) @ N6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) => ((finite_finite2 @ nat) @ N6))))).
% 9.43/3.45 thf(fact_2262_atMost__Suc, axiom, (! [K:nat]: ((((set_ord_atMost @ nat) @ (suc @ K)) = (((insert2 @ nat) @ (suc @ K)) @ ((set_ord_atMost @ nat) @ K)))))).
% 9.43/3.45 thf(fact_2263_surj__fun__eq, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > A),X10:(set @ B),G1:(A > C),G22:(A > C)]: (((((((image2 @ B) @ A) @ F) @ X10) = (top_top @ (set @ A))) => ((! [X2:B]: (((((member @ B) @ X2) @ X10) => (((((((comp @ A) @ C) @ B) @ G1) @ F) @ X2) = ((((((comp @ A) @ C) @ B) @ G22) @ F) @ X2))))) => (G1 = G22)))))).
% 9.43/3.45 thf(fact_2264_comp__surj, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(A > C)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((image2 @ A) @ C) @ G) @ (top_top @ (set @ A))) = (top_top @ (set @ C))) => (((((image2 @ B) @ C) @ (((((comp @ A) @ C) @ B) @ G) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ C)))))))).
% 9.43/3.45 thf(fact_2265_fun_Oset__map, axiom, (! [B:$tType,A:$tType,D:$tType,F:(A > B),V:(D > A)]: ((((((image2 @ D) @ B) @ (((((comp @ A) @ B) @ D) @ F) @ V)) @ (top_top @ (set @ D))) = ((((image2 @ A) @ B) @ F) @ ((((image2 @ D) @ A) @ V) @ (top_top @ (set @ D)))))))).
% 9.43/3.45 thf(fact_2266_fun_Omap__cong, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Ya:(D > A),F:(A > B),G:(A > B)]: (((X = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((((image2 @ D) @ A) @ Ya) @ (top_top @ (set @ D)))) => ((F @ Z3) = (G @ Z3))))) => ((((((comp @ A) @ B) @ D) @ F) @ X) = (((((comp @ A) @ B) @ D) @ G) @ Ya))))))).
% 9.43/3.45 thf(fact_2267_fun_Omap__cong0, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),F:(A > B),G:(A > B)]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((((image2 @ D) @ A) @ X) @ (top_top @ (set @ D)))) => ((F @ Z3) = (G @ Z3))))) => ((((((comp @ A) @ B) @ D) @ F) @ X) = (((((comp @ A) @ B) @ D) @ G) @ X)))))).
% 9.43/3.45 thf(fact_2268_fun_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Xa:(D > A),F:(A > B),Fa:(A > B)]: (((! [Z3:A,Za:A]: (((((member @ A) @ Z3) @ ((((image2 @ D) @ A) @ X) @ (top_top @ (set @ D)))) => ((((member @ A) @ Za) @ ((((image2 @ D) @ A) @ Xa) @ (top_top @ (set @ D)))) => (((F @ Z3) = (Fa @ Za)) => (Z3 = Za)))))) => (((((((comp @ A) @ B) @ D) @ F) @ X) = (((((comp @ A) @ B) @ D) @ Fa) @ Xa)) => (X = Xa)))))).
% 9.43/3.45 thf(fact_2269_lessThan__Suc, axiom, (! [K:nat]: ((((set_ord_lessThan @ nat) @ (suc @ K)) = (((insert2 @ nat) @ K) @ ((set_ord_lessThan @ nat) @ K)))))).
% 9.43/3.45 thf(fact_2270_not__UNIV__le__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,H3:A]: ((~ ((((ord_less_eq @ (set @ A)) @ (top_top @ (set @ A))) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H3)))))))))).
% 9.43/3.45 thf(fact_2271_surj__id, axiom, (! [A:$tType]: ((((((image2 @ A) @ A) @ (id @ A)) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 9.43/3.45 thf(fact_2272_not__UNIV__le__Iic, axiom, (! [A:$tType]: (((no_top @ A) => (! [H3:A]: ((~ ((((ord_less_eq @ (set @ A)) @ (top_top @ (set @ A))) @ ((set_ord_atMost @ A) @ H3)))))))))).
% 9.43/3.45 thf(fact_2273_sum_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I4:(set @ B),X:(B > A),Y:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((X @ I3) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((Y @ I3) != (zero_zero @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((((plus_plus @ A) @ (X @ I3)) @ (Y @ I3)) != (zero_zero @ A))))))))))))))).
% 9.43/3.45 thf(fact_2274_prod_Ofinite__Collect__op, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I4:(set @ B),X:(B > A),Y:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((X @ I3) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((Y @ I3) != (one_one @ A))))))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [I3:B]: (((((member @ B) @ I3) @ I4) & ((((times_times @ A) @ (X @ I3)) @ (Y @ I3)) != (one_one @ A))))))))))))))).
% 9.43/3.45 thf(fact_2275_finite__atLeastZeroLessThan__int, axiom, (! [U:int]: (((finite_finite2 @ int) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U))))).
% 9.43/3.45 thf(fact_2276_card__eq__sum, axiom, (! [A:$tType]: (((finite_card @ A) = (((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X3:A]: ((one_one @ nat)))))))).
% 9.43/3.45 thf(fact_2277_prod_Ointer__filter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),G:(B > A),P:(B > $o)]: ((((finite_finite2 @ B) @ A4) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ A4) & (P @ X3)))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X3:B]: (((((if @ A) @ (P @ X3)) @ (G @ X3)) @ (one_one @ A))))) @ A4))))))))).
% 9.43/3.45 thf(fact_2278_prod__gen__delta, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),A3:B,B2:(B > A),C2:A]: ((((finite_finite2 @ B) @ S2) => (((((member @ B) @ A3) @ S2) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (K4 = A3)) @ (B2 @ K4)) @ C2)))) @ S2) = (((times_times @ A) @ (B2 @ A3)) @ (((power_power @ A) @ C2) @ (((minus_minus @ nat) @ ((finite_card @ B) @ S2)) @ (one_one @ nat)))))) & ((~ ((((member @ B) @ A3) @ S2))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (K4 = A3)) @ (B2 @ K4)) @ C2)))) @ S2) = (((power_power @ A) @ C2) @ ((finite_card @ B) @ S2)))))))))))).
% 9.43/3.45 thf(fact_2279_finite__atLeastZeroLessThan__integer, axiom, (! [U:code_integer]: (((finite_finite2 @ code_integer) @ (((set_or7035219750837199246ssThan @ code_integer) @ (zero_zero @ code_integer)) @ U))))).
% 9.43/3.45 thf(fact_2280_card__2__iff_H, axiom, (! [A:$tType,S2:(set @ A)]: (((((finite_card @ A) @ S2) = ((numeral_numeral @ nat) @ (bit0 @ one2))) = (? [X3:A]: (((((member @ A) @ X3) @ S2) & (? [Y3:A]: (((((member @ A) @ Y3) @ S2) & ((X3 != Y3) & (! [Z5:A]: (((((member @ A) @ Z5) @ S2) => ((Z5 = X3) | (Z5 = Y3)))))))))))))))).
% 9.43/3.45 thf(fact_2281_sum_Orelated, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [R5:(A > (A > $o)),S2:(set @ B),H3:(B > A),G:(B > A)]: ((((R5 @ (zero_zero @ A)) @ (zero_zero @ A)) => ((! [X13:A,Y12:A,X23:A,Y22:A]: (((((R5 @ X13) @ X23) & ((R5 @ Y12) @ Y22)) => ((R5 @ (((plus_plus @ A) @ X13) @ Y12)) @ (((plus_plus @ A) @ X23) @ Y22))))) => (((finite_finite2 @ B) @ S2) => ((! [X2:B]: (((((member @ B) @ X2) @ S2) => ((R5 @ (H3 @ X2)) @ (G @ X2))))) => ((R5 @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H3) @ S2)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ S2)))))))))))).
% 9.43/3.45 thf(fact_2282_prod_Orelated, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [R5:(A > (A > $o)),S2:(set @ B),H3:(B > A),G:(B > A)]: ((((R5 @ (one_one @ A)) @ (one_one @ A)) => ((! [X13:A,Y12:A,X23:A,Y22:A]: (((((R5 @ X13) @ X23) & ((R5 @ Y12) @ Y22)) => ((R5 @ (((times_times @ A) @ X13) @ Y12)) @ (((times_times @ A) @ X23) @ Y22))))) => (((finite_finite2 @ B) @ S2) => ((! [X2:B]: (((((member @ B) @ X2) @ S2) => ((R5 @ (H3 @ X2)) @ (G @ X2))))) => ((R5 @ ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ S2)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S2)))))))))))).
% 9.43/3.45 thf(fact_2283_sum_Oinsert__if, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),X:B,G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((((member @ B) @ X) @ A4) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert2 @ B) @ X) @ A4)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4))) & ((~ ((((member @ B) @ X) @ A4))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert2 @ B) @ X) @ A4)) = (((plus_plus @ A) @ (G @ X)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4)))))))))))).
% 9.43/3.45 thf(fact_2284_atLeast0__atMost__Suc, axiom, (! [N:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert2 @ nat) @ (suc @ N)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))).
% 9.43/3.45 thf(fact_2285_atLeast0__lessThan__Suc, axiom, (! [N:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert2 @ nat) @ N) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))).
% 9.43/3.45 thf(fact_2286_Icc__eq__insert__lb__nat, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((set_or1337092689740270186AtMost @ nat) @ M2) @ N) = (((insert2 @ nat) @ M2) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ N))))))).
% 9.43/3.45 thf(fact_2287_atLeastAtMostSuc__conv, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ (suc @ N)) => ((((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N)) = (((insert2 @ nat) @ (suc @ N)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.45 thf(fact_2288_atLeastAtMost__insertL, axiom, (! [M2:nat,N:nat]: (((((ord_less_eq @ nat) @ M2) @ N) => ((((insert2 @ nat) @ M2) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ N)) = (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)))))).
% 9.43/3.45 thf(fact_2289_prod_Oreindex__bij__witness__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [S7:(set @ B),T5:(set @ C),S2:(set @ B),I:(C > B),J:(B > C),T6:(set @ C),G:(B > A),H3:(C > A)]: ((((finite_finite2 @ B) @ S7) => (((finite_finite2 @ C) @ T5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ S2) @ S7)) => ((I @ (J @ A5)) = A5)))) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ S2) @ S7)) => (((member @ C) @ (J @ A5)) @ (((minus_minus @ (set @ C)) @ T6) @ T5))))) => ((! [B3:C]: (((((member @ C) @ B3) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => ((J @ (I @ B3)) = B3)))) => ((! [B3:C]: (((((member @ C) @ B3) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => (((member @ B) @ (I @ B3)) @ (((minus_minus @ (set @ B)) @ S2) @ S7))))) => ((! [A5:B]: (((((member @ B) @ A5) @ S7) => ((G @ A5) = (one_one @ A))))) => ((! [B3:C]: (((((member @ C) @ B3) @ T5) => ((H3 @ B3) = (one_one @ A))))) => ((! [A5:B]: (((((member @ B) @ A5) @ S2) => ((H3 @ (J @ A5)) = (G @ A5))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S2) = ((((groups7121269368397514597t_prod @ C) @ A) @ H3) @ T6))))))))))))))))).
% 9.43/3.45 thf(fact_2290_dflt__None__set__def, axiom, (! [A:$tType]: (((dflt_None_set @ A) = (^ [S5:(set @ A)]: (((((if @ (option @ (set @ A))) @ (S5 = (bot_bot @ (set @ A)))) @ (none @ (set @ A))) @ ((some @ (set @ A)) @ S5)))))))).
% 9.43/3.45 thf(fact_2291_sum__eq__1__iff, axiom, (! [A:$tType,A4:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A4) => ((((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A4) = (one_one @ nat)) = (? [X3:A]: (((((member @ A) @ X3) @ A4) & (((F @ X3) = (one_one @ nat)) & (! [Y3:A]: (((((member @ A) @ Y3) @ A4) => ((X3 != Y3) => ((F @ Y3) = (zero_zero @ nat))))))))))))))).
% 9.43/3.45 thf(fact_2292_surj__Compl__image__subset, axiom, (! [A:$tType,B:$tType,F:(B > A),A4:(set @ B)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((((image2 @ B) @ A) @ F) @ A4))) @ ((((image2 @ B) @ A) @ F) @ ((uminus_uminus @ (set @ B)) @ A4))))))).
% 9.43/3.45 thf(fact_2293_card__atLeastZeroLessThan__int, axiom, (! [U:int]: ((((finite_card @ int) @ (((set_or7035219750837199246ssThan @ int) @ (zero_zero @ int)) @ U)) = (nat2 @ U))))).
% 9.43/3.45 thf(fact_2294_atMost__nat__numeral, axiom, (! [K:num]: ((((set_ord_atMost @ nat) @ ((numeral_numeral @ nat) @ K)) = (((insert2 @ nat) @ ((numeral_numeral @ nat) @ K)) @ ((set_ord_atMost @ nat) @ (pred_numeral @ K))))))).
% 9.43/3.45 thf(fact_2295_lessThan__nat__numeral, axiom, (! [K:num]: ((((set_ord_lessThan @ nat) @ ((numeral_numeral @ nat) @ K)) = (((insert2 @ nat) @ (pred_numeral @ K)) @ ((set_ord_lessThan @ nat) @ (pred_numeral @ K))))))).
% 9.43/3.45 thf(fact_2296_sum__Suc, axiom, (! [A:$tType,F:(A > nat),A4:(set @ A)]: ((((((groups7311177749621191930dd_sum @ A) @ nat) @ (^ [X3:A]: ((suc @ (F @ X3))))) @ A4) = (((plus_plus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A4)) @ ((finite_card @ A) @ A4)))))).
% 9.43/3.45 thf(fact_2297_prod_Osetdiff__irrelevant, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A4) @ ((collect @ B) @ (^ [X3:B]: (((G @ X3) = (one_one @ A))))))) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4))))))))).
% 9.43/3.45 thf(fact_2298_notin__range__Some, axiom, (! [A:$tType,X:(option @ A)]: (((~ ((((member @ (option @ A)) @ X) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ (top_top @ (set @ A)))))) = (X = (none @ A)))))).
% 9.43/3.45 thf(fact_2299_less__1__prod2, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [I4:(set @ A),I:A,F:(A > B)]: ((((finite_finite2 @ A) @ I4) => ((((member @ A) @ I) @ I4) => ((((ord_less @ B) @ (one_one @ B)) @ (F @ I)) => ((! [I2:A]: (((((member @ A) @ I2) @ I4) => (((ord_less_eq @ B) @ (one_one @ B)) @ (F @ I2))))) => (((ord_less @ B) @ (one_one @ B)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ I4)))))))))))).
% 9.43/3.45 thf(fact_2300_less__1__prod, axiom, (! [B:$tType,A:$tType]: (((linordered_idom @ B) => (! [I4:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ I4) => ((I4 != (bot_bot @ (set @ A))) => ((! [I2:A]: (((((member @ A) @ I2) @ I4) => (((ord_less @ B) @ (one_one @ B)) @ (F @ I2))))) => (((ord_less @ B) @ (one_one @ B)) @ ((((groups7121269368397514597t_prod @ A) @ B) @ F) @ I4))))))))))).
% 9.43/3.45 thf(fact_2301_sum_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),H3:(B > C),G:(C > A)]: ((((finite_finite2 @ B) @ A4) => ((! [X2:B,Y4:B]: (((((member @ B) @ X2) @ A4) => ((((member @ B) @ Y4) @ A4) => ((X2 != Y4) => (((H3 @ X2) = (H3 @ Y4)) => ((G @ (H3 @ X2)) = (zero_zero @ A)))))))) => (((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ ((((image2 @ B) @ C) @ H3) @ A4)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H3)) @ A4)))))))))).
% 9.43/3.45 thf(fact_2302_prod_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),H3:(B > C),G:(C > A)]: ((((finite_finite2 @ B) @ A4) => ((! [X2:B,Y4:B]: (((((member @ B) @ X2) @ A4) => ((((member @ B) @ Y4) @ A4) => ((X2 != Y4) => (((H3 @ X2) = (H3 @ Y4)) => ((G @ (H3 @ X2)) = (one_one @ A)))))))) => (((((groups7121269368397514597t_prod @ C) @ A) @ G) @ ((((image2 @ B) @ C) @ H3) @ A4)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H3)) @ A4)))))))))).
% 9.43/3.45 thf(fact_2303_sum_Osubset__diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [B6:(set @ B),A4:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ B6) @ A4) => (((finite_finite2 @ B) @ A4) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A4) @ B6))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B6))))))))))).
% 9.43/3.45 thf(fact_2304_prod_Osame__carrier, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C5:(set @ B),A4:(set @ B),B6:(set @ B),G:(B > A),H3:(B > A)]: ((((finite_finite2 @ B) @ C5) => ((((ord_less_eq @ (set @ B)) @ A4) @ C5) => ((((ord_less_eq @ (set @ B)) @ B6) @ C5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ C5) @ A4)) => ((G @ A5) = (one_one @ A))))) => ((! [B3:B]: (((((member @ B) @ B3) @ (((minus_minus @ (set @ B)) @ C5) @ B6)) => ((H3 @ B3) = (one_one @ A))))) => ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ B6)) = (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ C5) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ C5)))))))))))))).
% 9.43/3.45 thf(fact_2305_prod_Osame__carrierI, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [C5:(set @ B),A4:(set @ B),B6:(set @ B),G:(B > A),H3:(B > A)]: ((((finite_finite2 @ B) @ C5) => ((((ord_less_eq @ (set @ B)) @ A4) @ C5) => ((((ord_less_eq @ (set @ B)) @ B6) @ C5) => ((! [A5:B]: (((((member @ B) @ A5) @ (((minus_minus @ (set @ B)) @ C5) @ A4)) => ((G @ A5) = (one_one @ A))))) => ((! [B3:B]: (((((member @ B) @ B3) @ (((minus_minus @ (set @ B)) @ C5) @ B6)) => ((H3 @ B3) = (one_one @ A))))) => ((((((groups7121269368397514597t_prod @ B) @ A) @ G) @ C5) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ C5)) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ B6)))))))))))))).
% 9.43/3.45 thf(fact_2306_prod_Omono__neutral__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S2:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((G @ X2) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S2) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T6))))))))))).
% 9.43/3.45 thf(fact_2307_prod_Omono__neutral__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S2:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((G @ X2) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T6) = ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S2))))))))))).
% 9.43/3.45 thf(fact_2308_prod_Omono__neutral__cong__left, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S2:(set @ B),H3:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((H3 @ X2) = (one_one @ A))))) => ((! [X2:B]: (((((member @ B) @ X2) @ S2) => ((G @ X2) = (H3 @ X2))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S2) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ T6)))))))))))).
% 9.43/3.45 thf(fact_2309_prod_Omono__neutral__cong__right, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S2:(set @ B),G:(B > A),H3:(B > A)]: ((((finite_finite2 @ B) @ T6) => ((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((G @ X2) = (one_one @ A))))) => ((! [X2:B]: (((((member @ B) @ X2) @ S2) => ((G @ X2) = (H3 @ X2))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ T6) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ S2)))))))))))).
% 9.43/3.45 thf(fact_2310_atLeastLessThanSuc, axiom, (! [M2:nat,N:nat]: ((((((ord_less_eq @ nat) @ M2) @ N) => ((((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N)) = (((insert2 @ nat) @ N) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)))) & ((~ ((((ord_less_eq @ nat) @ M2) @ N))) => ((((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N)) = (bot_bot @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_2311_Option_Othese__def, axiom, (! [A:$tType]: (((these @ A) = (^ [A9:(set @ (option @ A))]: (((((image2 @ (option @ A)) @ A) @ (the2 @ A)) @ ((collect @ (option @ A)) @ (^ [X3:(option @ A)]: (((((member @ (option @ A)) @ X3) @ A9) & (X3 != (none @ A))))))))))))).
% 9.43/3.45 thf(fact_2312_atLeast0__atMost__Suc__eq__insert__0, axiom, (! [N:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert2 @ nat) @ (zero_zero @ nat)) @ ((((image2 @ nat) @ nat) @ suc) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))))))).
% 9.43/3.45 thf(fact_2313_atLeast0__lessThan__Suc__eq__insert__0, axiom, (! [N:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N)) = (((insert2 @ nat) @ (zero_zero @ nat)) @ ((((image2 @ nat) @ nat) @ suc) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))).
% 9.43/3.45 thf(fact_2314_atMost__Suc__eq__insert__0, axiom, (! [N:nat]: ((((set_ord_atMost @ nat) @ (suc @ N)) = (((insert2 @ nat) @ (zero_zero @ nat)) @ ((((image2 @ nat) @ nat) @ suc) @ ((set_ord_atMost @ nat) @ N))))))).
% 9.43/3.45 thf(fact_2315_lessThan__Suc__eq__insert__0, axiom, (! [N:nat]: ((((set_ord_lessThan @ nat) @ (suc @ N)) = (((insert2 @ nat) @ (zero_zero @ nat)) @ ((((image2 @ nat) @ nat) @ suc) @ ((set_ord_lessThan @ nat) @ N))))))).
% 9.43/3.45 thf(fact_2316_card__2__iff, axiom, (! [A:$tType,S2:(set @ A)]: (((((finite_card @ A) @ S2) = ((numeral_numeral @ nat) @ (bit0 @ one2))) = (? [X3:A,Y3:A]: (((S2 = (((insert2 @ A) @ X3) @ (((insert2 @ A) @ Y3) @ (bot_bot @ (set @ A))))) & (X3 != Y3)))))))).
% 9.43/3.45 thf(fact_2317_card__3__iff, axiom, (! [A:$tType,S2:(set @ A)]: (((((finite_card @ A) @ S2) = ((numeral_numeral @ nat) @ (bit1 @ one2))) = (? [X3:A,Y3:A,Z5:A]: (((S2 = (((insert2 @ A) @ X3) @ (((insert2 @ A) @ Y3) @ (((insert2 @ A) @ Z5) @ (bot_bot @ (set @ A)))))) & ((X3 != Y3) & ((Y3 != Z5) & (X3 != Z5)))))))))).
% 9.43/3.45 thf(fact_2318_sum__image__le, axiom, (! [A:$tType,B:$tType,C:$tType]: (((ordere6911136660526730532id_add @ B) => (! [I4:(set @ C),G:(A > B),F:(C > A)]: ((((finite_finite2 @ C) @ I4) => ((! [I2:C]: (((((member @ C) @ I2) @ I4) => (((ord_less_eq @ B) @ (zero_zero @ B)) @ (G @ (F @ I2)))))) => (((ord_less_eq @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ G) @ ((((image2 @ C) @ A) @ F) @ I4))) @ ((((groups7311177749621191930dd_sum @ C) @ B) @ (((((comp @ A) @ B) @ C) @ G) @ F)) @ I4)))))))))).
% 9.43/3.45 thf(fact_2319_sum_Oremove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),X:B,G:(B > A)]: ((((finite_finite2 @ B) @ A4) => ((((member @ B) @ X) @ A4) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4) = (((plus_plus @ A) @ (G @ X)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A4) @ (((insert2 @ B) @ X) @ (bot_bot @ (set @ B))))))))))))))).
% 9.43/3.45 thf(fact_2320_sum_Oinsert__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),G:(B > A),X:B]: ((((finite_finite2 @ B) @ A4) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((insert2 @ B) @ X) @ A4)) = (((plus_plus @ A) @ (G @ X)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A4) @ (((insert2 @ B) @ X) @ (bot_bot @ (set @ B)))))))))))))).
% 9.43/3.45 thf(fact_2321_atLeast1__atMost__eq__remove0, axiom, (! [N:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((minus_minus @ (set @ nat)) @ ((set_ord_atMost @ nat) @ N)) @ (((insert2 @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_2322_atLeast1__lessThan__eq__remove0, axiom, (! [N:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (suc @ (zero_zero @ nat))) @ N) = (((minus_minus @ (set @ nat)) @ ((set_ord_lessThan @ nat) @ N)) @ (((insert2 @ nat) @ (zero_zero @ nat)) @ (bot_bot @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_2323_atLeastLessThan__nat__numeral, axiom, (! [M2:nat,K:num]: ((((((ord_less_eq @ nat) @ M2) @ (pred_numeral @ K)) => ((((set_or7035219750837199246ssThan @ nat) @ M2) @ ((numeral_numeral @ nat) @ K)) = (((insert2 @ nat) @ (pred_numeral @ K)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (pred_numeral @ K))))) & ((~ ((((ord_less_eq @ nat) @ M2) @ (pred_numeral @ K)))) => ((((set_or7035219750837199246ssThan @ nat) @ M2) @ ((numeral_numeral @ nat) @ K)) = (bot_bot @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_2324_sum_Odelta__remove, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [S2:(set @ B),A3:B,B2:(B > A),C2:(B > A)]: ((((finite_finite2 @ B) @ S2) => (((((member @ B) @ A3) @ S2) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (K4 = A3)) @ (B2 @ K4)) @ (C2 @ K4))))) @ S2) = (((plus_plus @ A) @ (B2 @ A3)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S2) @ (((insert2 @ B) @ A3) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ A3) @ S2))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [K4:B]: (((((if @ A) @ (K4 = A3)) @ (B2 @ K4)) @ (C2 @ K4))))) @ S2) = ((((groups7311177749621191930dd_sum @ B) @ A) @ C2) @ (((minus_minus @ (set @ B)) @ S2) @ (((insert2 @ B) @ A3) @ (bot_bot @ (set @ B))))))))))))))).
% 9.43/3.45 thf(fact_2325_prod__le__power, axiom, (! [B:$tType,A:$tType]: (((linordered_semidom @ A) => (! [A4:(set @ B),F:(B > A),N:A,K:nat]: (((! [I2:B]: (((((member @ B) @ I2) @ A4) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ (F @ I2)) & (((ord_less_eq @ A) @ (F @ I2)) @ N))))) => ((((ord_less_eq @ nat) @ ((finite_card @ B) @ A4)) @ K) => ((((ord_less_eq @ A) @ (one_one @ A)) @ N) => (((ord_less_eq @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ F) @ A4)) @ (((power_power @ A) @ N) @ K))))))))))).
% 9.43/3.45 thf(fact_2326_card__Diff__insert, axiom, (! [A:$tType,A3:A,A4:(set @ A),B6:(set @ A)]: (((((member @ A) @ A3) @ A4) => ((~ ((((member @ A) @ A3) @ B6))) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert2 @ A) @ A3) @ B6))) = (((minus_minus @ nat) @ ((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A4) @ B6))) @ (one_one @ nat)))))))).
% 9.43/3.45 thf(fact_2327_card__insert__le__m1, axiom, (! [A:$tType,N:nat,Y:(set @ A),X:A]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((ord_less_eq @ nat) @ ((finite_card @ A) @ Y)) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) => (((ord_less_eq @ nat) @ ((finite_card @ A) @ (((insert2 @ A) @ X) @ Y))) @ N)))))).
% 9.43/3.45 thf(fact_2328_card__Diff__singleton__if, axiom, (! [A:$tType,X:A,A4:(set @ A)]: ((((((member @ A) @ X) @ A4) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((finite_card @ A) @ A4)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X) @ A4))) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) = ((finite_card @ A) @ A4))))))).
% 9.43/3.45 thf(fact_2329_card__Diff__singleton, axiom, (! [A:$tType,X:A,A4:(set @ A)]: (((((member @ A) @ X) @ A4) => (((finite_card @ A) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) = (((minus_minus @ nat) @ ((finite_card @ A) @ A4)) @ (one_one @ nat))))))).
% 9.43/3.45 thf(fact_2330_in__finite__psubset, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A4) @ B6)) @ (finite_psubset @ A)) = ((((ord_less @ (set @ A)) @ A4) @ B6) & ((finite_finite2 @ A) @ B6)))))).
% 9.43/3.45 thf(fact_2331_card__1__singletonE, axiom, (! [A:$tType,A4:(set @ A)]: (((((finite_card @ A) @ A4) = (one_one @ nat)) => (~ ((! [X2:A]: ((A4 != (((insert2 @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 9.43/3.45 thf(fact_2332_integer__of__nat__def, axiom, (code_integer_of_nat = (((((((map_fun @ nat) @ nat) @ int) @ code_integer) @ (id @ nat)) @ code_integer_of_int) @ (semiring_1_of_nat @ int)))).
% 9.43/3.45 thf(fact_2333_card__UNIV__unit, axiom, (((finite_card @ product_unit) @ (top_top @ (set @ product_unit))) = (one_one @ nat))).
% 9.43/3.45 thf(fact_2334_card__lessThan, axiom, (! [U:nat]: ((((finite_card @ nat) @ ((set_ord_lessThan @ nat) @ U)) = U)))).
% 9.43/3.45 thf(fact_2335_card__atLeastLessThan, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or7035219750837199246ssThan @ nat) @ L) @ U)) = (((minus_minus @ nat) @ U) @ L))))).
% 9.43/3.45 thf(fact_2336_card__atMost, axiom, (! [U:nat]: ((((finite_card @ nat) @ ((set_ord_atMost @ nat) @ U)) = (suc @ U))))).
% 9.43/3.45 thf(fact_2337_Collect__const__case__prod, axiom, (! [B:$tType,A:$tType,P:$o]: (((P => (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A7:A,B5:B]: (P)))) = (top_top @ (set @ ((product_prod @ A) @ B))))) & ((~ (P)) => (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A7:A,B5:B]: (P)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))))).
% 9.43/3.45 thf(fact_2338_nat__of__integer__integer__of__nat, axiom, (! [N:nat]: (((code_nat_of_integer @ (code_integer_of_nat @ N)) = N)))).
% 9.43/3.45 thf(fact_2339_card__atLeastAtMost, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or1337092689740270186AtMost @ nat) @ L) @ U)) = (((minus_minus @ nat) @ (suc @ U)) @ L))))).
% 9.43/3.45 thf(fact_2340_integer__of__nat_Orep__eq, axiom, (! [X:nat]: (((code_int_of_integer @ (code_integer_of_nat @ X)) = ((semiring_1_of_nat @ int) @ X))))).
% 9.43/3.45 thf(fact_2341_int__of__integer__integer__of__nat, axiom, (! [N:nat]: (((code_int_of_integer @ (code_integer_of_nat @ N)) = ((semiring_1_of_nat @ int) @ N))))).
% 9.43/3.45 thf(fact_2342_top__empty__eq2, axiom, (! [B:$tType,A:$tType]: (((top_top @ (A > (B > $o))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (top_top @ (set @ ((product_prod @ A) @ B)))))))))).
% 9.43/3.45 thf(fact_2343_int__of__integer__induct, axiom, (! [Y:int,P:(int > $o)]: (((((member @ int) @ Y) @ (top_top @ (set @ int))) => ((! [X2:code_integer]: ((P @ (code_int_of_integer @ X2)))) => (P @ Y)))))).
% 9.43/3.45 thf(fact_2344_int__of__integer__cases, axiom, (! [Y:int]: (((((member @ int) @ Y) @ (top_top @ (set @ int))) => (~ ((! [X2:code_integer]: ((Y != (code_int_of_integer @ X2)))))))))).
% 9.43/3.45 thf(fact_2345_int__of__integer, axiom, (! [X:code_integer]: ((((member @ int) @ (code_int_of_integer @ X)) @ (top_top @ (set @ int)))))).
% 9.43/3.45 thf(fact_2346_integer__of__int__inject, axiom, (! [X:int,Y:int]: (((((member @ int) @ X) @ (top_top @ (set @ int))) => ((((member @ int) @ Y) @ (top_top @ (set @ int))) => (((code_integer_of_int @ X) = (code_integer_of_int @ Y)) = (X = Y))))))).
% 9.43/3.45 thf(fact_2347_integer__of__int__induct, axiom, (! [P:(code_integer > $o),X:code_integer]: (((! [Y4:int]: (((((member @ int) @ Y4) @ (top_top @ (set @ int))) => (P @ (code_integer_of_int @ Y4))))) => (P @ X))))).
% 9.43/3.45 thf(fact_2348_integer__of__int__cases, axiom, (! [X:code_integer]: ((~ ((! [Y4:int]: (((X = (code_integer_of_int @ Y4)) => (~ ((((member @ int) @ Y4) @ (top_top @ (set @ int))))))))))))).
% 9.43/3.45 thf(fact_2349_subset__card__intvl__is__intvl, axiom, (! [A4:(set @ nat),K:nat]: (((((ord_less_eq @ (set @ nat)) @ A4) @ (((set_or7035219750837199246ssThan @ nat) @ K) @ (((plus_plus @ nat) @ K) @ ((finite_card @ nat) @ A4)))) => (A4 = (((set_or7035219750837199246ssThan @ nat) @ K) @ (((plus_plus @ nat) @ K) @ ((finite_card @ nat) @ A4)))))))).
% 9.43/3.45 thf(fact_2350_integer__of__int__inverse, axiom, (! [Y:int]: (((((member @ int) @ Y) @ (top_top @ (set @ int))) => ((code_int_of_integer @ (code_integer_of_int @ Y)) = Y))))).
% 9.43/3.45 thf(fact_2351_subset__eq__atLeast0__lessThan__card, axiom, (! [N6:(set @ nat),N:nat]: (((((ord_less_eq @ (set @ nat)) @ N6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)) => (((ord_less_eq @ nat) @ ((finite_card @ nat) @ N6)) @ N))))).
% 9.43/3.45 thf(fact_2352_integer__of__nat__eq__of__nat, axiom, (code_integer_of_nat = (semiring_1_of_nat @ code_integer))).
% 9.43/3.45 thf(fact_2353_card__less__Suc2, axiom, (! [M3:(set @ nat),I:nat]: (((~ ((((member @ nat) @ (zero_zero @ nat)) @ M3))) => (((finite_card @ nat) @ ((collect @ nat) @ (^ [K4:nat]: (((((member @ nat) @ (suc @ K4)) @ M3) & (((ord_less @ nat) @ K4) @ I)))))) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [K4:nat]: (((((member @ nat) @ K4) @ M3) & (((ord_less @ nat) @ K4) @ (suc @ I)))))))))))).
% 9.43/3.45 thf(fact_2354_card__less__Suc, axiom, (! [M3:(set @ nat),I:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ M3) => ((suc @ ((finite_card @ nat) @ ((collect @ nat) @ (^ [K4:nat]: (((((member @ nat) @ (suc @ K4)) @ M3) & (((ord_less @ nat) @ K4) @ I))))))) = ((finite_card @ nat) @ ((collect @ nat) @ (^ [K4:nat]: (((((member @ nat) @ K4) @ M3) & (((ord_less @ nat) @ K4) @ (suc @ I)))))))))))).
% 9.43/3.45 thf(fact_2355_card__less, axiom, (! [M3:(set @ nat),I:nat]: (((((member @ nat) @ (zero_zero @ nat)) @ M3) => (((finite_card @ nat) @ ((collect @ nat) @ (^ [K4:nat]: (((((member @ nat) @ K4) @ M3) & (((ord_less @ nat) @ K4) @ (suc @ I))))))) != (zero_zero @ nat)))))).
% 9.43/3.45 thf(fact_2356_integer__of__nat__0, axiom, ((code_integer_of_nat @ (zero_zero @ nat)) = (zero_zero @ code_integer))).
% 9.43/3.45 thf(fact_2357_integer__of__nat_Oabs__eq, axiom, (code_integer_of_nat = (^ [X3:nat]: ((code_integer_of_int @ ((semiring_1_of_nat @ int) @ X3)))))).
% 9.43/3.45 thf(fact_2358_integer__of__nat__numeral, axiom, (! [N:num]: (((code_integer_of_nat @ ((numeral_numeral @ nat) @ N)) = ((numeral_numeral @ code_integer) @ N))))).
% 9.43/3.45 thf(fact_2359_integer__of__nat__1, axiom, ((code_integer_of_nat @ (one_one @ nat)) = (one_one @ code_integer))).
% 9.43/3.45 thf(fact_2360_card__sum__le__nat__sum, axiom, (! [S2:(set @ nat)]: ((((ord_less_eq @ nat) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ nat) @ S2)))) @ ((((groups7311177749621191930dd_sum @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ S2))))).
% 9.43/3.45 thf(fact_2361_set__encode__insert, axiom, (! [A4:(set @ nat),N:nat]: ((((finite_finite2 @ nat) @ A4) => ((~ ((((member @ nat) @ N) @ A4))) => ((nat_set_encode @ (((insert2 @ nat) @ N) @ A4)) = (((plus_plus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ (nat_set_encode @ A4)))))))).
% 9.43/3.45 thf(fact_2362_card__greaterThanLessThan__int, axiom, (! [L:int,U:int]: ((((finite_card @ int) @ (((set_or5935395276787703475ssThan @ int) @ L) @ U)) = (nat2 @ (((minus_minus @ int) @ U) @ (((plus_plus @ int) @ L) @ (one_one @ int)))))))).
% 9.43/3.45 thf(fact_2363_wmax__insertI, axiom, (! [Y:((product_prod @ nat) @ nat),YS:(set @ ((product_prod @ nat) @ nat)),X:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ Y) @ YS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y)) @ fun_pair_leq) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ YS)) @ fun_max_weak) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (((insert2 @ ((product_prod @ nat) @ nat)) @ X) @ XS)) @ YS)) @ fun_max_weak))))))).
% 9.43/3.45 thf(fact_2364_wmin__insertI, axiom, (! [X:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat)),Y:((product_prod @ nat) @ nat),YS:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ X) @ XS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y)) @ fun_pair_leq) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ YS)) @ fun_min_weak) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ (((insert2 @ ((product_prod @ nat) @ nat)) @ Y) @ YS))) @ fun_min_weak))))))).
% 9.43/3.45 thf(fact_2365_smin__insertI, axiom, (! [X:((product_prod @ nat) @ nat),XS:(set @ ((product_prod @ nat) @ nat)),Y:((product_prod @ nat) @ nat),YS:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ X) @ XS) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y)) @ fun_pair_less) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ YS)) @ fun_min_strict) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ XS) @ (((insert2 @ ((product_prod @ nat) @ nat)) @ Y) @ YS))) @ fun_min_strict))))))).
% 9.43/3.45 thf(fact_2366_prod_Oinsert_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I4:(set @ B),P5:(B > A),I:B]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((P5 @ X3) != (one_one @ A))))))) => (((((member @ B) @ I) @ I4) => (((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ (((insert2 @ B) @ I) @ I4)) = ((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ I4))) & ((~ ((((member @ B) @ I) @ I4))) => (((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ (((insert2 @ B) @ I) @ I4)) = (((times_times @ A) @ (P5 @ I)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ I4)))))))))))).
% 9.43/3.45 thf(fact_2367_greaterThanLessThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = ((((ord_less @ A) @ L) @ I) & (((ord_less @ A) @ I) @ U))))))))).
% 9.43/3.45 thf(fact_2368_finite__greaterThanLessThan__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or5935395276787703475ssThan @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2369_greaterThanLessThan__empty__iff2, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: ((((bot_bot @ (set @ A)) = (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) = (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 9.43/3.45 thf(fact_2370_greaterThanLessThan__empty__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: ((((((set_or5935395276787703475ssThan @ A) @ A3) @ B2) = (bot_bot @ (set @ A))) = (((ord_less_eq @ A) @ B2) @ A3)))))))).
% 9.43/3.45 thf(fact_2371_greaterThanLessThan__empty, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,K:A]: (((((ord_less_eq @ A) @ L) @ K) => ((((set_or5935395276787703475ssThan @ A) @ K) @ L) = (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2372_infinite__Ioo__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2373_image__uminus__greaterThanLessThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or5935395276787703475ssThan @ A) @ X) @ Y)) = (((set_or5935395276787703475ssThan @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_2374_prod_Oempty_H, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [P5:(B > A)]: ((((((groups1962203154675924110t_prod @ B) @ A) @ P5) @ (bot_bot @ (set @ B))) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2375_smin__emptyI, axiom, (! [X10:(set @ ((product_prod @ nat) @ nat))]: (((X10 != (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ X10) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_strict))))).
% 9.43/3.45 thf(fact_2376_wmax__emptyI, axiom, (! [X10:(set @ ((product_prod @ nat) @ nat))]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ X10) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ X10)) @ fun_max_weak))))).
% 9.43/3.45 thf(fact_2377_wmin__emptyI, axiom, (! [X10:(set @ ((product_prod @ nat) @ nat))]: ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ X10) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_weak)))).
% 9.43/3.45 thf(fact_2378_prod_Onon__neutral_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [G:(B > A),I4:(set @ B)]: ((((((groups1962203154675924110t_prod @ B) @ A) @ G) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((G @ X3) != (one_one @ A))))))) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I4)))))))).
% 9.43/3.45 thf(fact_2379_infinite__Ioo, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2))))))))))).
% 9.43/3.45 thf(fact_2380_greaterThanLessThan__subseteq__greaterThanLessThan, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or5935395276787703475ssThan @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2381_prod__decode__aux_Ocases, axiom, (! [X:((product_prod @ nat) @ nat)]: ((~ ((! [K5:nat,M5:nat]: ((X != ((((product_Pair @ nat) @ nat) @ K5) @ M5))))))))).
% 9.43/3.45 thf(fact_2382_atLeastPlusOneLessThan__greaterThanLessThan__int, axiom, (! [L:int,U:int]: (((((set_or7035219750837199246ssThan @ int) @ (((plus_plus @ int) @ L) @ (one_one @ int))) @ U) = (((set_or5935395276787703475ssThan @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2383_prod_Omono__neutral__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),T6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((G @ X2) = (one_one @ A))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S2) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T6)))))))))).
% 9.43/3.45 thf(fact_2384_prod_Omono__neutral__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),T6:(set @ B),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((G @ X2) = (one_one @ A))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T6) = ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S2)))))))))).
% 9.43/3.45 thf(fact_2385_prod_Omono__neutral__cong__left_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),T6:(set @ B),H3:(B > A),G:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [I2:B]: (((((member @ B) @ I2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((H3 @ I2) = (one_one @ A))))) => ((! [X2:B]: (((((member @ B) @ X2) @ S2) => ((G @ X2) = (H3 @ X2))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ S2) = ((((groups1962203154675924110t_prod @ B) @ A) @ H3) @ T6))))))))))).
% 9.43/3.45 thf(fact_2386_prod_Omono__neutral__cong__right_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [S2:(set @ B),T6:(set @ B),G:(B > A),H3:(B > A)]: (((((ord_less_eq @ (set @ B)) @ S2) @ T6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((G @ X2) = (one_one @ A))))) => ((! [X2:B]: (((((member @ B) @ X2) @ S2) => ((G @ X2) = (H3 @ X2))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ G) @ T6) = ((((groups1962203154675924110t_prod @ B) @ A) @ H3) @ S2))))))))))).
% 9.43/3.45 thf(fact_2387_prod_Odistrib_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [I4:(set @ B),G:(B > A),H3:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((G @ X3) != (one_one @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((H3 @ X3) != (one_one @ A))))))) => (((((groups1962203154675924110t_prod @ B) @ A) @ (^ [I3:B]: ((((times_times @ A) @ (G @ I3)) @ (H3 @ I3))))) @ I4) = (((times_times @ A) @ ((((groups1962203154675924110t_prod @ B) @ A) @ G) @ I4)) @ ((((groups1962203154675924110t_prod @ B) @ A) @ H3) @ I4))))))))))).
% 9.43/3.45 thf(fact_2388_prod_OG__def, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (((groups1962203154675924110t_prod @ B) @ A) = (^ [P6:(B > A),I5:(set @ B)]: (((((if @ A) @ ((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I5) & ((P6 @ X3) != (one_one @ A)))))))) @ ((((groups7121269368397514597t_prod @ B) @ A) @ P6) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I5) & ((P6 @ X3) != (one_one @ A)))))))) @ (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2389_greaterThanLessThan__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2390_greaterThanLessThan__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2391_atLeastAtMost__diff__ends, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((minus_minus @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((insert2 @ A) @ A3) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) = (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2392_smax__insertI, axiom, (! [Y:((product_prod @ nat) @ nat),Y9:(set @ ((product_prod @ nat) @ nat)),X:((product_prod @ nat) @ nat),X10:(set @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ nat) @ nat)) @ Y) @ Y9) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y)) @ fun_pair_less) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ X10) @ Y9)) @ fun_max_strict) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (((insert2 @ ((product_prod @ nat) @ nat)) @ X) @ X10)) @ Y9)) @ fun_max_strict))))))).
% 9.43/3.45 thf(fact_2393_set__decode__plus__power__2, axiom, (! [N:nat,Z2:nat]: (((~ ((((member @ nat) @ N) @ (nat_set_decode @ Z2)))) => ((nat_set_decode @ (((plus_plus @ nat) @ (((power_power @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)) @ Z2)) = (((insert2 @ nat) @ N) @ (nat_set_decode @ Z2))))))).
% 9.43/3.45 thf(fact_2394_prod__decode__aux_Osimps, axiom, (nat_prod_decode_aux = (^ [K4:nat,M:nat]: (((((if @ ((product_prod @ nat) @ nat)) @ (((ord_less_eq @ nat) @ M) @ K4)) @ ((((product_Pair @ nat) @ nat) @ M) @ (((minus_minus @ nat) @ K4) @ M))) @ ((nat_prod_decode_aux @ (suc @ K4)) @ (((minus_minus @ nat) @ M) @ (suc @ K4)))))))).
% 9.43/3.45 thf(fact_2395_prod__decode__aux_Oelims, axiom, (! [X:nat,Xa:nat,Y:((product_prod @ nat) @ nat)]: (((((nat_prod_decode_aux @ X) @ Xa) = Y) => (((((ord_less_eq @ nat) @ Xa) @ X) => (Y = ((((product_Pair @ nat) @ nat) @ Xa) @ (((minus_minus @ nat) @ X) @ Xa)))) & ((~ ((((ord_less_eq @ nat) @ Xa) @ X))) => (Y = ((nat_prod_decode_aux @ (suc @ X)) @ (((minus_minus @ nat) @ Xa) @ (suc @ X)))))))))).
% 9.43/3.45 thf(fact_2396_min__rpair__set, axiom, ((fun_reduction_pair @ (set @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ fun_min_strict) @ fun_min_weak))).
% 9.43/3.45 thf(fact_2397_triangle__Suc, axiom, (! [N:nat]: (((nat_triangle @ (suc @ N)) = (((plus_plus @ nat) @ (nat_triangle @ N)) @ (suc @ N)))))).
% 9.43/3.45 thf(fact_2398_finite__greaterThanLessThan, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ L) @ U))))).
% 9.43/3.45 thf(fact_2399_finite__greaterThanLessThan__integer, axiom, (! [L:code_integer,U:code_integer]: (((finite_finite2 @ code_integer) @ (((set_or5935395276787703475ssThan @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2400_card__greaterThanLessThan, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ L) @ U)) = (((minus_minus @ nat) @ U) @ (suc @ L)))))).
% 9.43/3.45 thf(fact_2401_max__rpair__set, axiom, ((fun_reduction_pair @ (set @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ fun_max_strict) @ fun_max_weak))).
% 9.43/3.45 thf(fact_2402_atLeastSucLessThan__greaterThanLessThan, axiom, (! [L:nat,U:nat]: (((((set_or7035219750837199246ssThan @ nat) @ (suc @ L)) @ U) = (((set_or5935395276787703475ssThan @ nat) @ L) @ U))))).
% 9.43/3.45 thf(fact_2403_smax__emptyI, axiom, (! [Y9:(set @ ((product_prod @ nat) @ nat))]: ((((finite_finite2 @ ((product_prod @ nat) @ nat)) @ Y9) => ((Y9 != (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ Y9)) @ fun_max_strict)))))).
% 9.43/3.45 thf(fact_2404_atLeastPlusOneLessThan__greaterThanLessThan__integer, axiom, (! [L:code_integer,U:code_integer]: (((((set_or7035219750837199246ssThan @ code_integer) @ (((plus_plus @ code_integer) @ L) @ (one_one @ code_integer))) @ U) = (((set_or5935395276787703475ssThan @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2405_prod__decode__aux_Opelims, axiom, (! [X:nat,Xa:nat,Y:((product_prod @ nat) @ nat)]: (((((nat_prod_decode_aux @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ nat) @ nat)) @ nat_pr5047031295181774490ux_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa)) => (~ (((((((ord_less_eq @ nat) @ Xa) @ X) => (Y = ((((product_Pair @ nat) @ nat) @ Xa) @ (((minus_minus @ nat) @ X) @ Xa)))) & ((~ ((((ord_less_eq @ nat) @ Xa) @ X))) => (Y = ((nat_prod_decode_aux @ (suc @ X)) @ (((minus_minus @ nat) @ Xa) @ (suc @ X)))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ nat_pr5047031295181774490ux_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_2406_total__on__singleton, axiom, (! [A:$tType,X:A]: ((((total_on @ A) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.45 thf(fact_2407_prod__encode__prod__decode__aux, axiom, (! [K:nat,M2:nat]: (((nat_prod_encode @ ((nat_prod_decode_aux @ K) @ M2)) = (((plus_plus @ nat) @ (nat_triangle @ K)) @ M2))))).
% 9.43/3.45 thf(fact_2408_sum_Oinsert_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I4:(set @ B),P5:(B > A),I:B]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((P5 @ X3) != (zero_zero @ A))))))) => (((((member @ B) @ I) @ I4) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ (((insert2 @ B) @ I) @ I4)) = ((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ I4))) & ((~ ((((member @ B) @ I) @ I4))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ (((insert2 @ B) @ I) @ I4)) = (((plus_plus @ A) @ (P5 @ I)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ P5) @ I4)))))))))))).
% 9.43/3.45 thf(fact_2409_max__ext_Omax__extI, axiom, (! [A:$tType,X10:(set @ A),Y9:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ A) @ X10) => (((finite_finite2 @ A) @ Y9) => ((Y9 != (bot_bot @ (set @ A))) => ((! [X2:A]: (((((member @ A) @ X2) @ X10) => (? [Xa2:A]: (((((member @ A) @ Xa2) @ Y9) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Xa2)) @ R5))))))) => (((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ X10) @ Y9)) @ ((max_ext @ A) @ R5))))))))).
% 9.43/3.45 thf(fact_2410_total__on__def, axiom, (! [A:$tType]: (((total_on @ A) = (^ [A9:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X3:A]: (((((member @ A) @ X3) @ A9) => (! [Y3:A]: (((((member @ A) @ Y3) @ A9) => ((X3 != Y3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3))))))))))))))).
% 9.43/3.45 thf(fact_2411_total__onI, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y4) @ A4) => ((X2 != Y4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ R))))))) => (((total_on @ A) @ A4) @ R))))).
% 9.43/3.45 thf(fact_2412_sum_Odistrib__triv_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I4:(set @ B),G:(B > A),H3:(B > A)]: ((((finite_finite2 @ B) @ I4) => (((((groups1027152243600224163dd_sum @ B) @ A) @ (^ [I3:B]: ((((plus_plus @ A) @ (G @ I3)) @ (H3 @ I3))))) @ I4) = (((plus_plus @ A) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I4)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ H3) @ I4)))))))))).
% 9.43/3.45 thf(fact_2413_le__prod__encode__1, axiom, (! [A3:nat,B2:nat]: ((((ord_less_eq @ nat) @ A3) @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_2414_le__prod__encode__2, axiom, (! [B2:nat,A3:nat]: ((((ord_less_eq @ nat) @ B2) @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_2415_sum_Odistrib_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [I4:(set @ B),G:(B > A),H3:(B > A)]: ((((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((G @ X3) != (zero_zero @ A))))))) => (((finite_finite2 @ B) @ ((collect @ B) @ (^ [X3:B]: (((((member @ B) @ X3) @ I4) & ((H3 @ X3) != (zero_zero @ A))))))) => (((((groups1027152243600224163dd_sum @ B) @ A) @ (^ [I3:B]: ((((plus_plus @ A) @ (G @ I3)) @ (H3 @ I3))))) @ I4) = (((plus_plus @ A) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ G) @ I4)) @ ((((groups1027152243600224163dd_sum @ B) @ A) @ H3) @ I4))))))))))).
% 9.43/3.45 thf(fact_2416_prod__encode__def, axiom, (nat_prod_encode = ((((product_case_prod @ nat) @ nat) @ nat) @ (^ [M:nat,N5:nat]: ((((plus_plus @ nat) @ (nat_triangle @ (((plus_plus @ nat) @ M) @ N5))) @ M)))))).
% 9.43/3.45 thf(fact_2417_max__ext_Ocases, axiom, (! [A:$tType,A1:(set @ A),A22:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A1) @ A22)) @ ((max_ext @ A) @ R5)) => (~ ((((finite_finite2 @ A) @ A1) => (((finite_finite2 @ A) @ A22) => ((A22 != (bot_bot @ (set @ A))) => (~ ((! [X7:A]: (((((member @ A) @ X7) @ A1) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ A22) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X7) @ Xa3)) @ R5)))))))))))))))))).
% 9.43/3.45 thf(fact_2418_max__ext_Osimps, axiom, (! [A:$tType,A1:(set @ A),A22:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A1) @ A22)) @ ((max_ext @ A) @ R5)) = (((finite_finite2 @ A) @ A1) & (((finite_finite2 @ A) @ A22) & ((A22 != (bot_bot @ (set @ A))) & (! [X3:A]: (((((member @ A) @ X3) @ A1) => (? [Y3:A]: (((((member @ A) @ Y3) @ A22) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R5)))))))))))))).
% 9.43/3.45 thf(fact_2419_max__extp__eq, axiom, (! [A:$tType]: (((max_extp @ A) = (^ [R3:(A > (A > $o)),X3:(set @ A),Y3:(set @ A)]: ((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ X3) @ Y3)) @ ((max_ext @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.45 thf(fact_2420_max__extp__max__ext__eq, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((max_extp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R5)))) = (^ [X3:(set @ A),Y3:(set @ A)]: ((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ X3) @ Y3)) @ ((max_ext @ A) @ R5)))))))).
% 9.43/3.45 thf(fact_2421_one__div__numeral, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((divide_divide @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((product_fst @ A) @ A) @ (((unique8689654367752047608divmod @ A) @ one2) @ N))))))))).
% 9.43/3.45 thf(fact_2422_sqr_Osimps_I3_J, axiom, (! [N:num]: (((sqr @ (bit1 @ N)) = (bit1 @ (bit0 @ (((plus_plus @ num) @ (sqr @ N)) @ N))))))).
% 9.43/3.45 thf(fact_2423_one__mod__numeral, axiom, (! [A:$tType]: (((unique1627219031080169319umeral @ A) => (! [N:num]: (((((modulo_modulo @ A) @ (one_one @ A)) @ ((numeral_numeral @ A) @ N)) = (((product_snd @ A) @ A) @ (((unique8689654367752047608divmod @ A) @ one2) @ N))))))))).
% 9.43/3.45 thf(fact_2424_card__UNION, axiom, (! [A:$tType,A4:(set @ (set @ A))]: ((((finite_finite2 @ (set @ A)) @ A4) => ((! [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ A4) => ((finite_finite2 @ A) @ X2)))) => (((finite_card @ A) @ ((complete_Sup_Sup @ (set @ A)) @ A4)) = (nat2 @ ((((groups7311177749621191930dd_sum @ (set @ (set @ A))) @ int) @ (^ [I5:(set @ (set @ A))]: ((((times_times @ int) @ (((power_power @ int) @ ((uminus_uminus @ int) @ (one_one @ int))) @ (((plus_plus @ nat) @ ((finite_card @ (set @ A)) @ I5)) @ (one_one @ nat)))) @ ((semiring_1_of_nat @ int) @ ((finite_card @ A) @ ((complete_Inf_Inf @ (set @ A)) @ I5))))))) @ ((collect @ (set @ (set @ A))) @ (^ [I5:(set @ (set @ A))]: (((((ord_less_eq @ (set @ (set @ A))) @ I5) @ A4) & (I5 != (bot_bot @ (set @ (set @ A)))))))))))))))).
% 9.43/3.45 thf(fact_2425_Sup__atMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [Y:A]: ((((complete_Sup_Sup @ A) @ ((set_ord_atMost @ A) @ Y)) = Y))))))).
% 9.43/3.45 thf(fact_2426_Sup__lessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [Y:A]: ((((complete_Sup_Sup @ A) @ ((set_ord_lessThan @ A) @ Y)) = Y))))))).
% 9.43/3.45 thf(fact_2427_fst__apsnd, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C)]: (((((product_fst @ A) @ B) @ (((((product_apsnd @ C) @ B) @ A) @ F) @ X)) = (((product_fst @ A) @ C) @ X))))).
% 9.43/3.45 thf(fact_2428_fst__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) = ((product_fst @ A) @ B))))).
% 9.43/3.45 thf(fact_2429_apsnd__eq__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),X:((product_prod @ A) @ C),G:(C > B)]: ((((((((product_apsnd @ C) @ B) @ A) @ F) @ X) = (((((product_apsnd @ C) @ B) @ A) @ G) @ X)) = ((F @ (((product_snd @ A) @ C) @ X)) = (G @ (((product_snd @ A) @ C) @ X))))))).
% 9.43/3.45 thf(fact_2430_snd__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),X:((product_prod @ B) @ C)]: (((((product_snd @ B) @ A) @ (((((product_apsnd @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_snd @ B) @ C) @ X)))))).
% 9.43/3.45 thf(fact_2431_snd__comp__apsnd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C)]: (((((((comp @ ((product_prod @ A) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_snd @ A) @ B)))))).
% 9.43/3.45 thf(fact_2432_fst__divmod__integer, axiom, (! [K:code_integer,L:code_integer]: (((((product_fst @ code_integer) @ code_integer) @ ((code_divmod_integer @ K) @ L)) = (((divide_divide @ code_integer) @ K) @ L))))).
% 9.43/3.45 thf(fact_2433_snd__divmod__integer, axiom, (! [K:code_integer,L:code_integer]: (((((product_snd @ code_integer) @ code_integer) @ ((code_divmod_integer @ K) @ L)) = (((modulo_modulo @ code_integer) @ K) @ L))))).
% 9.43/3.45 thf(fact_2434_Sup__atLeastAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = Y)))))))).
% 9.43/3.45 thf(fact_2435_Inf__atLeastAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ A) @ X) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or1337092689740270186AtMost @ A) @ X) @ Y)) = X)))))))).
% 9.43/3.45 thf(fact_2436_Sup__atLeastLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or7035219750837199246ssThan @ A) @ X) @ Y)) = Y)))))))).
% 9.43/3.45 thf(fact_2437_Inf__atLeastLessThan, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or7035219750837199246ssThan @ A) @ X) @ Y)) = X)))))))).
% 9.43/3.45 thf(fact_2438_prod_Ocollapse, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)) = Prod)))).
% 9.43/3.45 thf(fact_2439_Inf__atMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: ((((complete_Inf_Inf @ A) @ ((set_ord_atMost @ A) @ X)) = (bot_bot @ A)))))))).
% 9.43/3.45 thf(fact_2440_Sup__greaterThanLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or5935395276787703475ssThan @ A) @ X) @ Y)) = Y)))))))).
% 9.43/3.45 thf(fact_2441_img__fst, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,S2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ S2) => (((member @ A) @ A3) @ ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S2)))))).
% 9.43/3.45 thf(fact_2442_Inf__greaterThanLessThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or5935395276787703475ssThan @ A) @ X) @ Y)) = X)))))))).
% 9.43/3.45 thf(fact_2443_img__snd, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,S2:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ S2) => (((member @ B) @ B2) @ ((((image2 @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ S2)))))).
% 9.43/3.45 thf(fact_2444_fst__divmod__abs, axiom, (! [K:code_integer,L:code_integer]: (((((product_fst @ code_integer) @ code_integer) @ ((code_divmod_abs @ K) @ L)) = (((divide_divide @ code_integer) @ ((abs_abs @ code_integer) @ K)) @ ((abs_abs @ code_integer) @ L)))))).
% 9.43/3.45 thf(fact_2445_snd__divmod__abs, axiom, (! [K:code_integer,L:code_integer]: (((((product_snd @ code_integer) @ code_integer) @ ((code_divmod_abs @ K) @ L)) = (((modulo_modulo @ code_integer) @ ((abs_abs @ code_integer) @ K)) @ ((abs_abs @ code_integer) @ L)))))).
% 9.43/3.45 thf(fact_2446_Inf__atMostLessThan, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: (((((ord_less @ A) @ (top_top @ A)) @ X) => (((complete_Inf_Inf @ A) @ ((set_ord_lessThan @ A) @ X)) = (bot_bot @ A))))))))).
% 9.43/3.45 thf(fact_2447_split__comp__eq, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,F:(A > (B > C)),G:(D > A)]: (((^ [U2:((product_prod @ D) @ B)]: (((F @ (G @ (((product_fst @ D) @ B) @ U2))) @ (((product_snd @ D) @ B) @ U2)))) = ((((product_case_prod @ D) @ B) @ C) @ (^ [X3:D]: ((F @ (G @ X3))))))))).
% 9.43/3.45 thf(fact_2448_case__prod__beta_H, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F2:(A > (B > C)),X3:((product_prod @ A) @ B)]: (((F2 @ (((product_fst @ A) @ B) @ X3)) @ (((product_snd @ A) @ B) @ X3)))))))).
% 9.43/3.45 thf(fact_2449_case__prod__unfold, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [C3:(A > (B > C)),P6:((product_prod @ A) @ B)]: (((C3 @ (((product_fst @ A) @ B) @ P6)) @ (((product_snd @ A) @ B) @ P6)))))))).
% 9.43/3.45 thf(fact_2450_Product__Type_OCollect__case__prodD, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A4:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ A4))) => ((A4 @ (((product_fst @ A) @ B) @ X)) @ (((product_snd @ A) @ B) @ X)))))).
% 9.43/3.45 thf(fact_2451_case__prod__beta, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((product_case_prod @ B) @ C) @ A) = (^ [F2:(B > (C > A)),P6:((product_prod @ B) @ C)]: (((F2 @ (((product_fst @ B) @ C) @ P6)) @ (((product_snd @ B) @ C) @ P6)))))))).
% 9.43/3.45 thf(fact_2452_split__beta, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((product_case_prod @ A) @ B) @ C) = (^ [F2:(A > (B > C)),Prod2:((product_prod @ A) @ B)]: (((F2 @ (((product_fst @ A) @ B) @ Prod2)) @ (((product_snd @ A) @ B) @ Prod2)))))))).
% 9.43/3.45 thf(fact_2453_effect__heapI, axiom, (! [A:$tType,N:nat,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A]: (((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((H4 = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3))) => ((((((heap_Time_effect @ A) @ ((heap_Time_heap @ A) @ F)) @ H3) @ H4) @ R) @ N))))))).
% 9.43/3.45 thf(fact_2454_effect__heapE, axiom, (! [A:$tType,F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((heap_Time_heap @ A) @ F)) @ H3) @ H4) @ R) @ N) => (~ (((H4 = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => (R != (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3))))))))))).
% 9.43/3.45 thf(fact_2455_effect__guardI, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),N:nat,R:A]: (((P @ H3) => ((H4 = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3))) => ((((((heap_Time_effect @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H3) @ H4) @ R) @ N)))))))).
% 9.43/3.45 thf(fact_2456_effect__guardE, axiom, (! [A:$tType,P:((heap_ext @ product_unit) > $o),F:((heap_ext @ product_unit) > ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ (((heap_Time_guard @ A) @ P) @ F)) @ H3) @ H4) @ R) @ N) => (~ (((H4 = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3)))) => ((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (F @ H3))) => (~ ((P @ H3)))))))))))).
% 9.43/3.45 thf(fact_2457_effectE, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ C2) @ H3) @ H4) @ R) @ N) => (~ (((R = (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ C2) @ H3)))) => ((H4 = (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ C2) @ H3))))) => ((N = (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ C2) @ H3))))) => (~ ((((heap_Time_success @ A) @ C2) @ H3)))))))))))).
% 9.43/3.45 thf(fact_2458_prod__eq__iff, axiom, (! [B:$tType,A:$tType]: (((^ [Y5:((product_prod @ A) @ B),Z4:((product_prod @ A) @ B)]: ((Y5 = Z4))) = (^ [S4:((product_prod @ A) @ B),T3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ S4) = (((product_fst @ A) @ B) @ T3)) & ((((product_snd @ A) @ B) @ S4) = (((product_snd @ A) @ B) @ T3))))))))).
% 9.43/3.45 thf(fact_2459_prod__eqI, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B),Q3:((product_prod @ A) @ B)]: ((((((product_fst @ A) @ B) @ P5) = (((product_fst @ A) @ B) @ Q3)) => (((((product_snd @ A) @ B) @ P5) = (((product_snd @ A) @ B) @ Q3)) => (P5 = Q3)))))).
% 9.43/3.45 thf(fact_2460_prod_Oexpand, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B),Prod3:((product_prod @ A) @ B)]: (((((((product_fst @ A) @ B) @ Prod) = (((product_fst @ A) @ B) @ Prod3)) & ((((product_snd @ A) @ B) @ Prod) = (((product_snd @ A) @ B) @ Prod3))) => (Prod = Prod3))))).
% 9.43/3.45 thf(fact_2461_All__prod__contract, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: (((! [A7:A,X8:B]: (((P @ A7) @ X8))) = (! [Z5:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Z5)) @ (((product_snd @ A) @ B) @ Z5)))))))).
% 9.43/3.45 thf(fact_2462_Ex__prod__contract, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: (((? [A7:A,X8:B]: (((P @ A7) @ X8))) = (? [Z5:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Z5)) @ (((product_snd @ A) @ B) @ Z5)))))))).
% 9.43/3.45 thf(fact_2463_prod_Osplit__sel, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) => (P @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))).
% 9.43/3.45 thf(fact_2464_prod_Osplit__sel__asm, axiom, (! [C:$tType,B:$tType,A:$tType,P:(C > $o),F:(A > (B > C)),Prod:((product_prod @ A) @ B)]: (((P @ (((((product_case_prod @ A) @ B) @ C) @ F) @ Prod)) = (~ (((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod))) & (~ ((P @ ((F @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))))))))).
% 9.43/3.45 thf(fact_2465_surjective__pairing, axiom, (! [B:$tType,A:$tType,T2:((product_prod @ A) @ B)]: ((T2 = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ T2)) @ (((product_snd @ A) @ B) @ T2)))))).
% 9.43/3.45 thf(fact_2466_prod_Oexhaust__sel, axiom, (! [B:$tType,A:$tType,Prod:((product_prod @ A) @ B)]: ((Prod = ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ B) @ Prod)) @ (((product_snd @ A) @ B) @ Prod)))))).
% 9.43/3.45 thf(fact_2467_exI__realizer, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),Y:A,X:B]: ((((P @ Y) @ X) => ((P @ (((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))) @ (((product_fst @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y))))))).
% 9.43/3.45 thf(fact_2468_conjI__realizer, axiom, (! [A:$tType,B:$tType,P:(A > $o),P5:A,Q:(B > $o),Q3:B]: (((P @ P5) => ((Q @ Q3) => ((P @ (((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ P5) @ Q3))) & (Q @ (((product_snd @ A) @ B) @ ((((product_Pair @ A) @ B) @ P5) @ Q3))))))))).
% 9.43/3.45 thf(fact_2469_execute__bind__success, axiom, (! [B:$tType,A:$tType,F:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit),G:(A > (heap_Time_Heap @ B))]: (((((heap_Time_success @ A) @ F) @ H3) => ((((heap_Time_execute @ B) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ H3) = (((heap_Time_timeFrame @ B) @ (((product_snd @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ F) @ H3))))) @ (((heap_Time_execute @ B) @ (G @ (((product_fst @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ F) @ H3))))) @ (((product_fst @ (heap_ext @ product_unit)) @ nat) @ (((product_snd @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((the2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((heap_Time_execute @ A) @ F) @ H3))))))))))).
% 9.43/3.45 thf(fact_2470_sndI, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),Y:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_snd @ A) @ B) @ X) = Z2))))).
% 9.43/3.45 thf(fact_2471_sndE, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),A3:A,B2:B,P:(B > $o)]: (((X = ((((product_Pair @ A) @ B) @ A3) @ B2)) => ((P @ (((product_snd @ A) @ B) @ X)) => (P @ B2)))))).
% 9.43/3.45 thf(fact_2472_snd__eqD, axiom, (! [B:$tType,A:$tType,X:B,Y:A,A3:A]: ((((((product_snd @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = A3) => (Y = A3))))).
% 9.43/3.45 thf(fact_2473_snd__conv, axiom, (! [Aa:$tType,A:$tType,X1:Aa,X22:A]: (((((product_snd @ Aa) @ A) @ ((((product_Pair @ Aa) @ A) @ X1) @ X22)) = X22)))).
% 9.43/3.45 thf(fact_2474_Some__Inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A4:(set @ A)]: ((((some @ A) @ ((complete_Inf_Inf @ A) @ A4)) = ((complete_Inf_Inf @ (option @ A)) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ A4))))))))).
% 9.43/3.45 thf(fact_2475_fstI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),Y:A,Z2:B]: (((X = ((((product_Pair @ A) @ B) @ Y) @ Z2)) => ((((product_fst @ A) @ B) @ X) = Y))))).
% 9.43/3.45 thf(fact_2476_fstE, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A3:A,B2:B,P:(A > $o)]: (((X = ((((product_Pair @ A) @ B) @ A3) @ B2)) => ((P @ (((product_fst @ A) @ B) @ X)) => (P @ A3)))))).
% 9.43/3.45 thf(fact_2477_fst__eqD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,A3:A]: ((((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = A3) => (X = A3))))).
% 9.43/3.45 thf(fact_2478_fst__conv, axiom, (! [B:$tType,A:$tType,X1:A,X22:B]: (((((product_fst @ A) @ B) @ ((((product_Pair @ A) @ B) @ X1) @ X22)) = X1)))).
% 9.43/3.45 thf(fact_2479_snd__fst__flip, axiom, (! [A:$tType,B:$tType]: ((((product_snd @ B) @ A) = (((((comp @ ((product_prod @ A) @ B)) @ A) @ ((product_prod @ B) @ A)) @ ((product_fst @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:B,Y3:A]: (((((product_Pair @ A) @ B) @ Y3) @ X3))))))))).
% 9.43/3.45 thf(fact_2480_fst__snd__flip, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = (((((comp @ ((product_prod @ B) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_snd @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [X3:A,Y3:B]: (((((product_Pair @ B) @ A) @ Y3) @ X3))))))))).
% 9.43/3.45 thf(fact_2481_finite__UNION__then__finite, axiom, (! [A:$tType,B:$tType,B6:(B > (set @ A)),A4:(set @ B),A3:B]: ((((finite_finite2 @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ B) @ (set @ A)) @ B6) @ A4))) => ((((member @ B) @ A3) @ A4) => ((finite_finite2 @ A) @ (B6 @ A3))))))).
% 9.43/3.45 thf(fact_2482_case__prod__comp, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType,F:(D > (C > A)),G:(B > D),X:((product_prod @ B) @ C)]: (((((((product_case_prod @ B) @ C) @ A) @ (((((comp @ D) @ (C > A)) @ B) @ F) @ G)) @ X) = ((F @ (G @ (((product_fst @ B) @ C) @ X))) @ (((product_snd @ B) @ C) @ X)))))).
% 9.43/3.45 thf(fact_2483_fn__snd__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > C)]: (((^ [X3:((product_prod @ A) @ B)]: ((F @ (((product_snd @ A) @ B) @ X3)))) = ((((product_case_prod @ A) @ B) @ C) @ (^ [Uu:A]: (F))))))).
% 9.43/3.45 thf(fact_2484_snd__def, axiom, (! [B:$tType,A:$tType]: ((((product_snd @ A) @ B) = ((((product_case_prod @ A) @ B) @ B) @ (^ [X12:A,X24:B]: (X24))))))).
% 9.43/3.45 thf(fact_2485_fn__fst__conv, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C)]: (((^ [X3:((product_prod @ A) @ B)]: ((F @ (((product_fst @ A) @ B) @ X3)))) = ((((product_case_prod @ A) @ B) @ C) @ (^ [A7:A,Uu:B]: ((F @ A7)))))))).
% 9.43/3.45 thf(fact_2486_fst__def, axiom, (! [B:$tType,A:$tType]: ((((product_fst @ A) @ B) = ((((product_case_prod @ A) @ B) @ A) @ (^ [X12:A,X24:B]: (X12))))))).
% 9.43/3.45 thf(fact_2487_Collect__split__mono__strong, axiom, (! [B:$tType,A:$tType,X10:(set @ A),A4:(set @ ((product_prod @ A) @ B)),Y9:(set @ B),P:(A > (B > $o)),Q:(A > (B > $o))]: (((X10 = ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A4)) => ((Y9 = ((((image2 @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A4)) => ((! [X2:A]: (((((member @ A) @ X2) @ X10) => (! [Xa3:B]: (((((member @ B) @ Xa3) @ Y9) => (((P @ X2) @ Xa3) => ((Q @ X2) @ Xa3)))))))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A4) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A4) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ Q)))))))))).
% 9.43/3.45 thf(fact_2488_Some__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A4:(set @ A)]: (((A4 != (bot_bot @ (set @ A))) => (((some @ A) @ ((complete_Sup_Sup @ A) @ A4)) = ((complete_Sup_Sup @ (option @ A)) @ ((((image2 @ A) @ (option @ A)) @ (some @ A)) @ A4)))))))))).
% 9.43/3.45 thf(fact_2489_Inf__option__def, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_Inf_Inf @ (option @ A)) = (^ [A9:(set @ (option @ A))]: (((((if @ (option @ A)) @ (((member @ (option @ A)) @ (none @ A)) @ A9)) @ (none @ A)) @ ((some @ A) @ ((complete_Inf_Inf @ A) @ ((these @ A) @ A9))))))))))).
% 9.43/3.45 thf(fact_2490_Some__INF, axiom, (! [A:$tType,B:$tType]: (((comple6319245703460814977attice @ A) => (! [F:(B > A),A4:(set @ B)]: ((((some @ A) @ ((complete_Inf_Inf @ A) @ ((((image2 @ B) @ A) @ F) @ A4))) = ((complete_Inf_Inf @ (option @ A)) @ ((((image2 @ B) @ (option @ A)) @ (^ [X3:B]: (((some @ A) @ (F @ X3))))) @ A4))))))))).
% 9.43/3.45 thf(fact_2491_INF__image, axiom, (! [B:$tType,A:$tType,C:$tType]: (((complete_Inf @ A) => (! [G:(B > A),F:(C > B),A4:(set @ C)]: ((((complete_Inf_Inf @ A) @ ((((image2 @ B) @ A) @ G) @ ((((image2 @ C) @ B) @ F) @ A4))) = ((complete_Inf_Inf @ A) @ ((((image2 @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A4))))))))).
% 9.43/3.45 thf(fact_2492_SUP__image, axiom, (! [B:$tType,A:$tType,C:$tType]: (((complete_Sup @ A) => (! [G:(B > A),F:(C > B),A4:(set @ C)]: ((((complete_Sup_Sup @ A) @ ((((image2 @ B) @ A) @ G) @ ((((image2 @ C) @ B) @ F) @ A4))) = ((complete_Sup_Sup @ A) @ ((((image2 @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A4))))))))).
% 9.43/3.45 thf(fact_2493_in__snd__imageE, axiom, (! [A:$tType,B:$tType,Y:A,S2:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ Y) @ ((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ S2)) => (~ ((! [X2:B]: ((~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X2) @ Y)) @ S2))))))))))).
% 9.43/3.45 thf(fact_2494_in__fst__imageE, axiom, (! [B:$tType,A:$tType,X:A,S2:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ X) @ ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S2)) => (~ ((! [Y4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ S2))))))))))).
% 9.43/3.45 thf(fact_2495_Some__SUP, axiom, (! [B:$tType,A:$tType]: (((comple6319245703460814977attice @ B) => (! [A4:(set @ A),F:(A > B)]: (((A4 != (bot_bot @ (set @ A))) => (((some @ B) @ ((complete_Sup_Sup @ B) @ ((((image2 @ A) @ B) @ F) @ A4))) = ((complete_Sup_Sup @ (option @ B)) @ ((((image2 @ A) @ (option @ B)) @ (^ [X3:A]: (((some @ B) @ (F @ X3))))) @ A4)))))))))).
% 9.43/3.45 thf(fact_2496_finite__range__prod, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > ((product_prod @ A) @ C))]: ((((finite_finite2 @ A) @ ((((image2 @ B) @ A) @ (((((comp @ ((product_prod @ A) @ C)) @ A) @ B) @ ((product_fst @ A) @ C)) @ F)) @ (top_top @ (set @ B)))) => (((finite_finite2 @ C) @ ((((image2 @ B) @ C) @ (((((comp @ ((product_prod @ A) @ C)) @ C) @ B) @ ((product_snd @ A) @ C)) @ F)) @ (top_top @ (set @ B)))) => ((finite_finite2 @ ((product_prod @ A) @ C)) @ ((((image2 @ B) @ ((product_prod @ A) @ C)) @ F) @ (top_top @ (set @ B))))))))).
% 9.43/3.45 thf(fact_2497_UN__atMost__UNIV, axiom, (((complete_Sup_Sup @ (set @ nat)) @ ((((image2 @ nat) @ (set @ nat)) @ (set_ord_atMost @ nat)) @ (top_top @ (set @ nat)))) = (top_top @ (set @ nat)))).
% 9.43/3.45 thf(fact_2498_UN__lessThan__UNIV, axiom, (((complete_Sup_Sup @ (set @ nat)) @ ((((image2 @ nat) @ (set @ nat)) @ (set_ord_lessThan @ nat)) @ (top_top @ (set @ nat)))) = (top_top @ (set @ nat)))).
% 9.43/3.45 thf(fact_2499_Sup__option__def, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => ((complete_Sup_Sup @ (option @ A)) = (^ [A9:(set @ (option @ A))]: (((((if @ (option @ A)) @ ((A9 = (bot_bot @ (set @ (option @ A)))) | (A9 = (((insert2 @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A))))))) @ (none @ A)) @ ((some @ A) @ ((complete_Sup_Sup @ A) @ ((these @ A) @ A9))))))))))).
% 9.43/3.45 thf(fact_2500_snd__image__mp, axiom, (! [B:$tType,A:$tType,A4:(set @ ((product_prod @ B) @ A)),B6:(set @ A),X:B,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ A4)) @ B6) => ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A4) => (((member @ A) @ Y) @ B6)))))).
% 9.43/3.45 thf(fact_2501_fst__image__mp, axiom, (! [B:$tType,A:$tType,A4:(set @ ((product_prod @ A) @ B)),B6:(set @ A),X:A,Y:B]: (((((ord_less_eq @ (set @ A)) @ ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A4)) @ B6) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ A4) => (((member @ A) @ X) @ B6)))))).
% 9.43/3.45 thf(fact_2502_prod_OUnion__comp, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [B6:(set @ (set @ B)),G:(B > A)]: (((! [X2:(set @ B)]: (((((member @ (set @ B)) @ X2) @ B6) => ((finite_finite2 @ B) @ X2)))) => ((! [A14:(set @ B)]: (((((member @ (set @ B)) @ A14) @ B6) => (! [A25:(set @ B)]: (((((member @ (set @ B)) @ A25) @ B6) => ((A14 != A25) => (! [X2:B]: (((((member @ B) @ X2) @ A14) => ((((member @ B) @ X2) @ A25) => ((G @ X2) = (one_one @ A))))))))))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ ((complete_Sup_Sup @ (set @ B)) @ B6)) = (((((((comp @ ((set @ B) > A)) @ ((set @ (set @ B)) > A)) @ (B > A)) @ ((groups7121269368397514597t_prod @ (set @ B)) @ A)) @ ((groups7121269368397514597t_prod @ B) @ A)) @ G) @ B6)))))))))).
% 9.43/3.45 thf(fact_2503_UN__UN__finite__eq, axiom, (! [A:$tType,A4:(nat > (set @ A))]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ (^ [N5:nat]: (((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N5)))))) @ (top_top @ (set @ nat)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (top_top @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_2504_UN__le__add__shift, axiom, (! [A:$tType,M3:(nat > (set @ A)),K:nat,N:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ (^ [I3:nat]: ((M3 @ (((plus_plus @ nat) @ I3) @ K))))) @ ((set_ord_atMost @ nat) @ N))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ M3) @ (((set_or1337092689740270186AtMost @ nat) @ K) @ (((plus_plus @ nat) @ N) @ K)))))))).
% 9.43/3.45 thf(fact_2505_UN__le__add__shift__strict, axiom, (! [A:$tType,M3:(nat > (set @ A)),K:nat,N:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ (^ [I3:nat]: ((M3 @ (((plus_plus @ nat) @ I3) @ K))))) @ ((set_ord_lessThan @ nat) @ N))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ M3) @ (((set_or7035219750837199246ssThan @ nat) @ K) @ (((plus_plus @ nat) @ N) @ K)))))))).
% 9.43/3.45 thf(fact_2506_rat__sgn__code, axiom, (! [P5:rat]: (((quotient_of @ ((sgn_sgn @ rat) @ P5)) = ((((product_Pair @ int) @ int) @ ((sgn_sgn @ int) @ (((product_fst @ int) @ int) @ (quotient_of @ P5)))) @ (one_one @ int)))))).
% 9.43/3.45 thf(fact_2507_bezw__non__0, axiom, (! [Y:nat,X:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ Y) => (((bezw @ X) @ Y) = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y) @ (((modulo_modulo @ nat) @ X) @ Y)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Y) @ (((modulo_modulo @ nat) @ X) @ Y)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y) @ (((modulo_modulo @ nat) @ X) @ Y)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Y)))))))))).
% 9.43/3.45 thf(fact_2508_UN__finite__subset, axiom, (! [A:$tType,A4:(nat > (set @ A)),C5:(set @ A)]: (((! [N3:nat]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))) @ C5))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (top_top @ (set @ nat))))) @ C5))))).
% 9.43/3.45 thf(fact_2509_UN__finite2__eq, axiom, (! [A:$tType,A4:(nat > (set @ A)),B6:(nat > (set @ A)),K:nat]: (((! [N3:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ B6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N3) @ K))))))) => (((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (top_top @ (set @ nat)))) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ B6) @ (top_top @ (set @ nat))))))))).
% 9.43/3.45 thf(fact_2510_bezw_Oelims, axiom, (! [X:nat,Xa:nat,Y:((product_prod @ int) @ int)]: (((((bezw @ X) @ Xa) = Y) => (((Xa = (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))) & ((Xa != (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_2511_bezw_Osimps, axiom, (bezw = (^ [X3:nat,Y3:nat]: (((((if @ ((product_prod @ int) @ int)) @ (Y3 = (zero_zero @ nat))) @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y3) @ (((modulo_modulo @ nat) @ X3) @ Y3)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Y3) @ (((modulo_modulo @ nat) @ X3) @ Y3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Y3) @ (((modulo_modulo @ nat) @ X3) @ Y3)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X3) @ Y3)))))))))).
% 9.43/3.45 thf(fact_2512_max__ext__def, axiom, (! [A:$tType]: (((max_ext @ A) = (^ [R6:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ ((max_extp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R6)))))))))))).
% 9.43/3.45 thf(fact_2513_UN__finite2__subset, axiom, (! [A:$tType,A4:(nat > (set @ A)),B6:(nat > (set @ A)),K:nat]: (((! [N3:nat]: ((((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N3)))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ B6) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((plus_plus @ nat) @ N3) @ K))))))) => (((ord_less_eq @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ A4) @ (top_top @ (set @ nat))))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ B6) @ (top_top @ (set @ nat))))))))).
% 9.43/3.45 thf(fact_2514_bezw_Opelims, axiom, (! [X:nat,Xa:nat,Y:((product_prod @ int) @ int)]: (((((bezw @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ nat) @ nat)) @ bezw_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa)) => (~ (((((Xa = (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (zero_zero @ int)))) & ((Xa != (zero_zero @ nat)) => (Y = ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa)))) @ (((minus_minus @ int) @ (((product_fst @ int) @ int) @ ((bezw @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa)))) @ ((semiring_1_of_nat @ int) @ (((divide_divide @ nat) @ X) @ Xa)))))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ bezw_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_2515_snd__diag__id, axiom, (! [A:$tType,Z2:A]: ((((((((comp @ ((product_prod @ A) @ A)) @ A) @ A) @ ((product_snd @ A) @ A)) @ (^ [X3:A]: (((((product_Pair @ A) @ A) @ X3) @ X3)))) @ Z2) = ((id @ A) @ Z2))))).
% 9.43/3.45 thf(fact_2516_fst__diag__id, axiom, (! [A:$tType,Z2:A]: ((((((((comp @ ((product_prod @ A) @ A)) @ A) @ A) @ ((product_fst @ A) @ A)) @ (^ [X3:A]: (((((product_Pair @ A) @ A) @ X3) @ X3)))) @ Z2) = ((id @ A) @ Z2))))).
% 9.43/3.45 thf(fact_2517_size__prod__simp, axiom, (! [B:$tType,A:$tType]: ((((basic_BNF_size_prod @ A) @ B) = (^ [F2:(A > nat),G2:(B > nat),P6:((product_prod @ A) @ B)]: ((((plus_plus @ nat) @ (((plus_plus @ nat) @ (F2 @ (((product_fst @ A) @ B) @ P6))) @ (G2 @ (((product_snd @ A) @ B) @ P6)))) @ (suc @ (zero_zero @ nat))))))))).
% 9.43/3.45 thf(fact_2518_fst__diag__snd, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_fst @ B) @ B)) @ (((((comp @ B) @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ B)) @ (^ [X3:B]: (((((product_Pair @ B) @ B) @ X3) @ X3)))) @ ((product_snd @ A) @ B))) = ((product_snd @ A) @ B))))).
% 9.43/3.45 thf(fact_2519_snd__diag__fst, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ A) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_snd @ A) @ A)) @ (((((comp @ A) @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ A) @ X3) @ X3)))) @ ((product_fst @ A) @ B))) = ((product_fst @ A) @ B))))).
% 9.43/3.45 thf(fact_2520_None__in__Inf, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [A4:(set @ (option @ A))]: (((((member @ (option @ A)) @ (none @ A)) @ A4) => (((complete_Inf_Inf @ (option @ A)) @ A4) = (none @ A))))))))).
% 9.43/3.45 thf(fact_2521_empty__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (((complete_Sup_Sup @ (option @ A)) @ (bot_bot @ (set @ (option @ A)))) = (none @ A)))))).
% 9.43/3.45 thf(fact_2522_singleton__None__Sup, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (((complete_Sup_Sup @ (option @ A)) @ (((insert2 @ (option @ A)) @ (none @ A)) @ (bot_bot @ (set @ (option @ A))))) = (none @ A)))))).
% 9.43/3.45 thf(fact_2523_INF__INT__eq2, axiom, (! [B:$tType,C:$tType,A:$tType,R:(C > (set @ ((product_prod @ A) @ B))),S2:(set @ C)]: ((((complete_Inf_Inf @ (A > (B > $o))) @ ((((image2 @ C) @ (A > (B > $o))) @ (^ [I3:C,X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (R @ I3))))) @ S2)) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((complete_Inf_Inf @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ C) @ (set @ ((product_prod @ A) @ B))) @ R) @ S2))))))))).
% 9.43/3.45 thf(fact_2524_INF__Int__eq2, axiom, (! [B:$tType,A:$tType,S2:(set @ (set @ ((product_prod @ A) @ B)))]: ((((complete_Inf_Inf @ (A > (B > $o))) @ ((((image2 @ (set @ ((product_prod @ A) @ B))) @ (A > (B > $o))) @ (^ [I3:(set @ ((product_prod @ A) @ B)),X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ I3)))) @ S2)) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((complete_Inf_Inf @ (set @ ((product_prod @ A) @ B))) @ S2)))))))).
% 9.43/3.45 thf(fact_2525_SUP__UN__eq2, axiom, (! [B:$tType,C:$tType,A:$tType,R:(C > (set @ ((product_prod @ A) @ B))),S2:(set @ C)]: ((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image2 @ C) @ (A > (B > $o))) @ (^ [I3:C,X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (R @ I3))))) @ S2)) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ C) @ (set @ ((product_prod @ A) @ B))) @ R) @ S2))))))))).
% 9.43/3.45 thf(fact_2526_SUP__Sup__eq2, axiom, (! [B:$tType,A:$tType,S2:(set @ (set @ ((product_prod @ A) @ B)))]: ((((complete_Sup_Sup @ (A > (B > $o))) @ ((((image2 @ (set @ ((product_prod @ A) @ B))) @ (A > (B > $o))) @ (^ [I3:(set @ ((product_prod @ A) @ B)),X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ I3)))) @ S2)) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ S2)))))))).
% 9.43/3.45 thf(fact_2527_Inf__INT__eq2, axiom, (! [B:$tType,A:$tType]: (((complete_Inf_Inf @ (A > (B > $o))) = (^ [S5:(set @ (A > (B > $o))),X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((complete_Inf_Inf @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ (((product_prod @ A) @ B) > $o)) @ (set @ ((product_prod @ A) @ B))) @ (collect @ ((product_prod @ A) @ B))) @ ((((image2 @ (A > (B > $o))) @ (((product_prod @ A) @ B) > $o)) @ (((product_case_prod @ A) @ B) @ $o)) @ S5)))))))))).
% 9.43/3.45 thf(fact_2528_Sup__SUP__eq2, axiom, (! [B:$tType,A:$tType]: (((complete_Sup_Sup @ (A > (B > $o))) = (^ [S5:(set @ (A > (B > $o))),X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ (((product_prod @ A) @ B) > $o)) @ (set @ ((product_prod @ A) @ B))) @ (collect @ ((product_prod @ A) @ B))) @ ((((image2 @ (A > (B > $o))) @ (((product_prod @ A) @ B) > $o)) @ (((product_case_prod @ A) @ B) @ $o)) @ S5)))))))))).
% 9.43/3.45 thf(fact_2529_fst__diag__fst, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ A) @ A)) @ A) @ ((product_prod @ A) @ B)) @ ((product_fst @ A) @ A)) @ (((((comp @ A) @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ A) @ X3) @ X3)))) @ ((product_fst @ A) @ B))) = ((product_fst @ A) @ B))))).
% 9.43/3.45 thf(fact_2530_snd__diag__snd, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ B) @ B)) @ (((((comp @ B) @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ B)) @ (^ [X3:B]: (((((product_Pair @ B) @ B) @ X3) @ X3)))) @ ((product_snd @ A) @ B))) = ((product_snd @ A) @ B))))).
% 9.43/3.45 thf(fact_2531_BNF__Greatest__Fixpoint_Osubst__Pair, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X:A,Y:B,A3:((product_prod @ A) @ B)]: ((((P @ X) @ Y) => ((A3 = ((((product_Pair @ A) @ B) @ X) @ Y)) => ((P @ (((product_fst @ A) @ B) @ A3)) @ (((product_snd @ A) @ B) @ A3))))))).
% 9.43/3.45 thf(fact_2532_normalize__def, axiom, (normalize = (^ [P6:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ (((ord_less @ int) @ (zero_zero @ int)) @ (((product_snd @ int) @ int) @ P6))) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6)))) @ (((divide_divide @ int) @ (((product_snd @ int) @ int) @ P6)) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))) @ ((((if @ ((product_prod @ int) @ int)) @ ((((product_snd @ int) @ int) @ P6) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((divide_divide @ int) @ (((product_fst @ int) @ int) @ P6)) @ ((uminus_uminus @ int) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))) @ (((divide_divide @ int) @ (((product_snd @ int) @ int) @ P6)) @ ((uminus_uminus @ int) @ (((gcd_gcd @ int) @ (((product_fst @ int) @ int) @ P6)) @ (((product_snd @ int) @ int) @ P6))))))))))).
% 9.43/3.45 thf(fact_2533_range__prod, axiom, (! [C:$tType,B:$tType,A:$tType,F:(C > ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ C) @ ((product_prod @ A) @ B)) @ F) @ (top_top @ (set @ C)))) @ ((((product_Sigma @ A) @ B) @ ((((image2 @ C) @ A) @ (((((comp @ ((product_prod @ A) @ B)) @ A) @ C) @ ((product_fst @ A) @ B)) @ F)) @ (top_top @ (set @ C)))) @ (^ [Uu:A]: (((((image2 @ C) @ B) @ (((((comp @ ((product_prod @ A) @ B)) @ B) @ C) @ ((product_snd @ A) @ B)) @ F)) @ (top_top @ (set @ C)))))))))).
% 9.43/3.45 thf(fact_2534_fun_Oin__rel, axiom, (! [B:$tType,A:$tType,D:$tType,R5:(A > (B > $o)),A3:(D > A),B2:(D > B)]: ((((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ R5) @ A3) @ B2) = (? [Z5:(D > ((product_prod @ A) @ B))]: (((((member @ (D > ((product_prod @ A) @ B))) @ Z5) @ ((collect @ (D > ((product_prod @ A) @ B))) @ (^ [X3:(D > ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ D) @ ((product_prod @ A) @ B)) @ X3) @ (top_top @ (set @ D)))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R5))))))) & (((((((comp @ ((product_prod @ A) @ B)) @ A) @ D) @ ((product_fst @ A) @ B)) @ Z5) = A3) & ((((((comp @ ((product_prod @ A) @ B)) @ B) @ D) @ ((product_snd @ A) @ B)) @ Z5) = B2))))))))).
% 9.43/3.45 thf(fact_2535_apfst__apsnd, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apsnd @ D) @ B) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (F @ (((product_fst @ C) @ D) @ X))) @ (G @ (((product_snd @ C) @ D) @ X))))))).
% 9.43/3.45 thf(fact_2536_apsnd__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),X:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ X)) = ((((product_Pair @ A) @ B) @ (G @ (((product_fst @ D) @ C) @ X))) @ (F @ (((product_snd @ D) @ C) @ X))))))).
% 9.43/3.45 thf(fact_2537_eq__snd__iff, axiom, (! [A:$tType,B:$tType,B2:A,P5:((product_prod @ B) @ A)]: (((B2 = (((product_snd @ B) @ A) @ P5)) = (? [A7:B]: ((P5 = ((((product_Pair @ B) @ A) @ A7) @ B2)))))))).
% 9.43/3.45 thf(fact_2538_SigmaI, axiom, (! [B:$tType,A:$tType,A3:A,A4:(set @ A),B2:B,B6:(A > (set @ B))]: (((((member @ A) @ A3) @ A4) => ((((member @ B) @ B2) @ (B6 @ A3)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Sigma @ A) @ B) @ A4) @ B6))))))).
% 9.43/3.45 thf(fact_2539_mem__Sigma__iff, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,A4:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) = ((((member @ A) @ A3) @ A4) & (((member @ B) @ B2) @ (B6 @ A3))))))).
% 9.43/3.45 thf(fact_2540_gcd__add1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M2:A,N:A]: (((((gcd_gcd @ A) @ (((plus_plus @ A) @ M2) @ N)) @ N) = (((gcd_gcd @ A) @ M2) @ N)))))))).
% 9.43/3.45 thf(fact_2541_gcd__add2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M2:A,N:A]: (((((gcd_gcd @ A) @ M2) @ (((plus_plus @ A) @ M2) @ N)) = (((gcd_gcd @ A) @ M2) @ N)))))))).
% 9.43/3.45 thf(fact_2542_gcd_Obottom__left__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_gcd @ A) @ (one_one @ A)) @ A3) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2543_gcd_Obottom__right__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_gcd @ A) @ A3) @ (one_one @ A)) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2544_apfst__conv, axiom, (! [C:$tType,A:$tType,B:$tType,F:(C > A),X:C,Y:B]: (((((((product_apfst @ C) @ A) @ B) @ F) @ ((((product_Pair @ C) @ B) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ (F @ X)) @ Y))))).
% 9.43/3.45 thf(fact_2545_apfst__eq__conv, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B),G:(C > A)]: ((((((((product_apfst @ C) @ A) @ B) @ F) @ X) = (((((product_apfst @ C) @ A) @ B) @ G) @ X)) = ((F @ (((product_fst @ C) @ B) @ X)) = (G @ (((product_fst @ C) @ B) @ X))))))).
% 9.43/3.45 thf(fact_2546_fst__apfst, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > A),X:((product_prod @ C) @ B)]: (((((product_fst @ A) @ B) @ (((((product_apfst @ C) @ A) @ B) @ F) @ X)) = (F @ (((product_fst @ C) @ B) @ X)))))).
% 9.43/3.45 thf(fact_2547_snd__apfst, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > B),X:((product_prod @ C) @ A)]: (((((product_snd @ B) @ A) @ (((((product_apfst @ C) @ B) @ A) @ F) @ X)) = (((product_snd @ C) @ A) @ X))))).
% 9.43/3.45 thf(fact_2548_Collect__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(B > $o)]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A7:A,B5:B]: (((P @ A7) & (Q @ B5)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [Uu:A]: (((collect @ B) @ Q)))))))).
% 9.43/3.45 thf(fact_2549_is__unit__gcd__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ (one_one @ A)) = ((((gcd_gcd @ A) @ A3) @ B2) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2550_Sigma__empty1, axiom, (! [B:$tType,A:$tType,B6:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (bot_bot @ (set @ A))) @ B6) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2551_Sigma__empty2, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: ((((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: ((bot_bot @ (set @ B))))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2552_Times__empty, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = ((A4 = (bot_bot @ (set @ A))) | (B6 = (bot_bot @ (set @ B)))))))).
% 9.43/3.45 thf(fact_2553_Sigma__UNIV__cancel, axiom, (! [B:$tType,A:$tType,A4:(set @ A),X10:(set @ B)]: (((((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (X10)))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: ((top_top @ (set @ B)))))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2554_Compl__Times__UNIV2, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: ((((uminus_uminus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: ((top_top @ (set @ B)))))) = ((((product_Sigma @ A) @ B) @ ((uminus_uminus @ (set @ A)) @ A4)) @ (^ [Uu:A]: ((top_top @ (set @ B))))))))).
% 9.43/3.45 thf(fact_2555_Compl__Times__UNIV1, axiom, (! [B:$tType,A:$tType,A4:(set @ B)]: ((((uminus_uminus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (A4)))) = ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (((uminus_uminus @ (set @ B)) @ A4)))))))).
% 9.43/3.45 thf(fact_2556_snd__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ B) @ ((product_prod @ A) @ B)) @ ((product_snd @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) = ((product_snd @ A) @ B))))).
% 9.43/3.45 thf(fact_2557_UNIV__Times__UNIV, axiom, (! [B:$tType,A:$tType]: ((((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: ((top_top @ (set @ B))))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2558_apfst__id, axiom, (! [B:$tType,A:$tType]: ((((((product_apfst @ A) @ A) @ B) @ (id @ A)) = (id @ ((product_prod @ A) @ B)))))).
% 9.43/3.45 thf(fact_2559_pairself__image__cart, axiom, (! [B:$tType,A:$tType,F:(B > A),A4:(set @ B),B6:(set @ B)]: ((((((image2 @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ A)) @ (((pairself @ B) @ A) @ F)) @ ((((product_Sigma @ B) @ B) @ A4) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ A) @ ((((image2 @ B) @ A) @ F) @ A4)) @ (^ [Uu:A]: (((((image2 @ B) @ A) @ F) @ B6)))))))).
% 9.43/3.45 thf(fact_2560_fst__image__times, axiom, (! [B:$tType,A:$tType,B6:(set @ B),A4:(set @ A)]: ((((B6 = (bot_bot @ (set @ B))) => (((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))) = (bot_bot @ (set @ A)))) & ((B6 != (bot_bot @ (set @ B))) => (((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))) = A4)))))).
% 9.43/3.45 thf(fact_2561_snd__image__times, axiom, (! [B:$tType,A:$tType,A4:(set @ B),B6:(set @ A)]: ((((A4 = (bot_bot @ (set @ B))) => (((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (B6)))) = (bot_bot @ (set @ A)))) & ((A4 != (bot_bot @ (set @ B))) => (((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (B6)))) = B6)))))).
% 9.43/3.45 thf(fact_2562_fst__comp__apfst, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C)]: (((((((comp @ ((product_prod @ C) @ B)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)))))).
% 9.43/3.45 thf(fact_2563_fun_Omap__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,G3:$tType,F5:$tType,Rb2:(A > (F5 > $o)),Sd:(B > (G3 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (F5 > G3)) @ ((D > A) > (D > B))) @ ((D > F5) > (D > G3))) @ ((((((bNF_rel_fun @ A) @ F5) @ B) @ G3) @ Rb2) @ Sd)) @ ((((((bNF_rel_fun @ (D > A)) @ (D > F5)) @ (D > B)) @ (D > G3)) @ ((((((bNF_rel_fun @ D) @ D) @ A) @ F5) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ Rb2)) @ ((((((bNF_rel_fun @ D) @ D) @ B) @ G3) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ Sd))) @ (((comp @ A) @ B) @ D)) @ (((comp @ F5) @ G3) @ D))))).
% 9.43/3.45 thf(fact_2564_comp__transfer, axiom, (! [A:$tType,B:$tType,E:$tType,F5:$tType,D:$tType,C:$tType,B6:(A > (C > $o)),C5:(B > (D > $o)),A4:(E > (F5 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > A) > (E > B))) @ ((F5 > C) > (F5 > D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ B6) @ C5)) @ ((((((bNF_rel_fun @ (E > A)) @ (F5 > C)) @ (E > B)) @ (F5 > D)) @ ((((((bNF_rel_fun @ E) @ F5) @ A) @ C) @ A4) @ B6)) @ ((((((bNF_rel_fun @ E) @ F5) @ B) @ D) @ A4) @ C5))) @ (((comp @ A) @ B) @ E)) @ (((comp @ C) @ D) @ F5))))).
% 9.43/3.45 thf(fact_2565_power__transfer, axiom, (! [A:$tType,B:$tType]: ((((power @ B) & (power @ A)) => (! [R5:(A > (B > $o))]: ((((R5 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R5) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R5) @ R5)) @ (times_times @ A)) @ (times_times @ B)) => ((((((((bNF_rel_fun @ A) @ B) @ (nat > A)) @ (nat > B)) @ R5) @ ((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ R5)) @ (power_power @ A)) @ (power_power @ B)))))))))).
% 9.43/3.45 thf(fact_2566_times__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (times_times @ int)) @ (times_times @ int))).
% 9.43/3.45 thf(fact_2567_times__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (times_times @ nat)) @ (times_times @ nat))).
% 9.43/3.45 thf(fact_2568_Times__eq__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A4:(set @ B),B6:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (C5))) = ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (C5)))) = (A4 = B6)))))).
% 9.43/3.45 thf(fact_2569_Sigma__cong, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ A),C5:(A > (set @ B)),D4:(A > (set @ B))]: (((A4 = B6) => ((! [X2:A]: (((((member @ A) @ X2) @ B6) => ((C5 @ X2) = (D4 @ X2))))) => (((((product_Sigma @ A) @ B) @ A4) @ C5) = ((((product_Sigma @ A) @ B) @ B6) @ D4))))))).
% 9.43/3.45 thf(fact_2570_divide__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (divide_divide @ nat)) @ (divide_divide @ nat))).
% 9.43/3.45 thf(fact_2571_divide__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (divide_divide @ int)) @ (divide_divide @ int))).
% 9.43/3.45 thf(fact_2572_or__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se1065995026697491101ons_or @ nat)) @ (bit_se1065995026697491101ons_or @ nat))).
% 9.43/3.45 thf(fact_2573_or__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se1065995026697491101ons_or @ int)) @ (bit_se1065995026697491101ons_or @ int))).
% 9.43/3.45 thf(fact_2574_set__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se5668285175392031749et_bit @ int)) @ (bit_se5668285175392031749et_bit @ int))).
% 9.43/3.45 thf(fact_2575_set__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se5668285175392031749et_bit @ nat)) @ (bit_se5668285175392031749et_bit @ nat))).
% 9.43/3.45 thf(fact_2576_xor__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se5824344971392196577ns_xor @ int)) @ (bit_se5824344971392196577ns_xor @ int))).
% 9.43/3.45 thf(fact_2577_xor__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se5824344971392196577ns_xor @ nat)) @ (bit_se5824344971392196577ns_xor @ nat))).
% 9.43/3.45 thf(fact_2578_and__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se5824344872417868541ns_and @ int)) @ (bit_se5824344872417868541ns_and @ int))).
% 9.43/3.45 thf(fact_2579_and__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se5824344872417868541ns_and @ nat)) @ (bit_se5824344872417868541ns_and @ nat))).
% 9.43/3.45 thf(fact_2580_modulo__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (modulo_modulo @ int)) @ (modulo_modulo @ int))).
% 9.43/3.45 thf(fact_2581_modulo__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (modulo_modulo @ nat)) @ (modulo_modulo @ nat))).
% 9.43/3.45 thf(fact_2582_less__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > $o)) @ (int > $o)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ $o) @ $o) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less @ int)) @ (ord_less @ int))).
% 9.43/3.45 thf(fact_2583_less__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > $o)) @ (nat > $o)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ $o) @ $o) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less @ nat)) @ (ord_less @ nat))).
% 9.43/3.45 thf(fact_2584_unset__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se2638667681897837118et_bit @ int)) @ (bit_se2638667681897837118et_bit @ int))).
% 9.43/3.45 thf(fact_2585_take__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se2584673776208193580ke_bit @ int)) @ (bit_se2584673776208193580ke_bit @ int))).
% 9.43/3.45 thf(fact_2586_flip__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se8732182000553998342ip_bit @ int)) @ (bit_se8732182000553998342ip_bit @ int))).
% 9.43/3.45 thf(fact_2587_flip__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se8732182000553998342ip_bit @ nat)) @ (bit_se8732182000553998342ip_bit @ nat))).
% 9.43/3.45 thf(fact_2588_push__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se4730199178511100633sh_bit @ nat)) @ (bit_se4730199178511100633sh_bit @ nat))).
% 9.43/3.45 thf(fact_2589_push__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se4730199178511100633sh_bit @ int)) @ (bit_se4730199178511100633sh_bit @ int))).
% 9.43/3.45 thf(fact_2590_plus__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (plus_plus @ int)) @ (plus_plus @ int))).
% 9.43/3.45 thf(fact_2591_plus__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (plus_plus @ nat)) @ (plus_plus @ nat))).
% 9.43/3.45 thf(fact_2592_drop__bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (int > int)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (bit_se4197421643247451524op_bit @ int)) @ (bit_se4197421643247451524op_bit @ int))).
% 9.43/3.45 thf(fact_2593_minus__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (minus_minus @ nat)) @ (minus_minus @ nat))).
% 9.43/3.45 thf(fact_2594_minus__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > int)) @ (int > int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (minus_minus @ int)) @ (minus_minus @ int))).
% 9.43/3.45 thf(fact_2595_less__eq__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > $o)) @ (int > $o)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ $o) @ $o) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less_eq @ int)) @ (ord_less_eq @ int))).
% 9.43/3.45 thf(fact_2596_less__eq__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > $o)) @ (nat > $o)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ $o) @ $o) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less_eq @ nat)) @ (ord_less_eq @ nat))).
% 9.43/3.45 thf(fact_2597_SigmaE, axiom, (! [A:$tType,B:$tType,C2:((product_prod @ A) @ B),A4:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (! [Y4:B]: (((((member @ B) @ Y4) @ (B6 @ X2)) => (C2 != ((((product_Pair @ A) @ B) @ X2) @ Y4)))))))))))))).
% 9.43/3.45 thf(fact_2598_SigmaD1, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,A4:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) => (((member @ A) @ A3) @ A4))))).
% 9.43/3.45 thf(fact_2599_SigmaD2, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,A4:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) => (((member @ B) @ B2) @ (B6 @ A3)))))).
% 9.43/3.45 thf(fact_2600_SigmaE2, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,A4:(set @ A),B6:(A > (set @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) => (~ (((((member @ A) @ A3) @ A4) => (~ ((((member @ B) @ B2) @ (B6 @ A3))))))))))).
% 9.43/3.45 thf(fact_2601_Suc_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ suc) @ suc)).
% 9.43/3.45 thf(fact_2602_sub_Orsp, axiom, ((((((((bNF_rel_fun @ num) @ num) @ (num > int)) @ (num > int)) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ num) @ num) @ int) @ int) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4))))) @ (^ [M:num,N5:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N5))))) @ (^ [M:num,N5:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N5)))))).
% 9.43/3.45 thf(fact_2603_transfer__rule__numeral, axiom, (! [A:$tType,B:$tType]: ((((monoid_add @ B) & ((semiring_numeral @ B) & ((monoid_add @ A) & (semiring_numeral @ A)))) => (! [R5:(A > (B > $o))]: ((((R5 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R5 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R5) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R5) @ R5)) @ (plus_plus @ A)) @ (plus_plus @ B)) => ((((((((bNF_rel_fun @ num) @ num) @ A) @ B) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ R5) @ (numeral_numeral @ A)) @ (numeral_numeral @ B))))))))))).
% 9.43/3.45 thf(fact_2604_uminus__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (uminus_uminus @ int)) @ (uminus_uminus @ int))).
% 9.43/3.45 thf(fact_2605_integer__of__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ int) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (semiring_1_of_nat @ int)) @ (semiring_1_of_nat @ int))).
% 9.43/3.45 thf(fact_2606_Sigma__Diff__distrib1, axiom, (! [B:$tType,A:$tType,I4:(set @ A),J4:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((minus_minus @ (set @ A)) @ I4) @ J4)) @ C5) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I4) @ C5)) @ ((((product_Sigma @ A) @ B) @ J4) @ C5)))))).
% 9.43/3.45 thf(fact_2607_natural__of__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ nat) @ nat) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ nat2) @ nat2)).
% 9.43/3.45 thf(fact_2608_abs__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (abs_abs @ int)) @ (abs_abs @ int))).
% 9.43/3.45 thf(fact_2609_bit__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (nat > $o)) @ (nat > $o)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:(nat > $o),Z4:(nat > $o)]: ((Y5 = Z4)))) @ (bit_se5641148757651400278ts_bit @ int)) @ (bit_se5641148757651400278ts_bit @ int))).
% 9.43/3.45 thf(fact_2610_not__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (bit_ri4277139882892585799ns_not @ int)) @ (bit_ri4277139882892585799ns_not @ int))).
% 9.43/3.45 thf(fact_2611_sgn__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (sgn_sgn @ int)) @ (sgn_sgn @ int))).
% 9.43/3.45 thf(fact_2612_mask__integer_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ int) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (bit_se2239418461657761734s_mask @ int)) @ (bit_se2239418461657761734s_mask @ int))).
% 9.43/3.45 thf(fact_2613_mask__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (bit_se2239418461657761734s_mask @ nat)) @ (bit_se2239418461657761734s_mask @ nat))).
% 9.43/3.45 thf(fact_2614_transfer__rule__of__int, axiom, (! [A:$tType,B:$tType]: ((((ring_1 @ B) & (ring_1 @ A)) => (! [R5:(A > (B > $o))]: ((((R5 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R5 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R5) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R5) @ R5)) @ (plus_plus @ A)) @ (plus_plus @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R5) @ R5) @ (uminus_uminus @ A)) @ (uminus_uminus @ B)) => ((((((((bNF_rel_fun @ int) @ int) @ A) @ B) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ R5) @ (ring_1_of_int @ A)) @ (ring_1_of_int @ B)))))))))))).
% 9.43/3.45 thf(fact_2615_fun_Orel__map_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,Sb:(C > (B > $o)),I:(A > C),X:(D > A),Y:(D > B)]: ((((((((((bNF_rel_fun @ D) @ D) @ C) @ B) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ Sb) @ (((((comp @ A) @ C) @ D) @ I) @ X)) @ Y) = ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ (^ [X3:A]: ((Sb @ (I @ X3))))) @ X) @ Y))))).
% 9.43/3.45 thf(fact_2616_fun_Orel__map_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,Sa:(A > (C > $o)),X:(D > A),G:(B > C),Y:(D > B)]: ((((((((((bNF_rel_fun @ D) @ D) @ A) @ C) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ Sa) @ X) @ (((((comp @ B) @ C) @ D) @ G) @ Y)) = ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ (^ [X3:A,Y3:B]: (((Sa @ X3) @ (G @ Y3))))) @ X) @ Y))))).
% 9.43/3.45 thf(fact_2617_times__eq__iff, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ A),D4:(set @ B)]: (((((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))) = ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu:A]: (D4)))) = (((A4 = C5) & (B6 = D4)) | (((A4 = (bot_bot @ (set @ A))) | (B6 = (bot_bot @ (set @ B)))) & ((C5 = (bot_bot @ (set @ A))) | (D4 = (bot_bot @ (set @ B)))))))))).
% 9.43/3.45 thf(fact_2618_dup_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [K4:int]: ((((plus_plus @ int) @ K4) @ K4)))) @ (^ [K4:int]: ((((plus_plus @ int) @ K4) @ K4))))).
% 9.43/3.45 thf(fact_2619_Sigma__Diff__distrib2, axiom, (! [B:$tType,A:$tType,I4:(set @ A),A4:(A > (set @ B)),B6:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I4) @ (^ [I3:A]: ((((minus_minus @ (set @ B)) @ (A4 @ I3)) @ (B6 @ I3))))) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I4) @ A4)) @ ((((product_Sigma @ A) @ B) @ I4) @ B6)))))).
% 9.43/3.45 thf(fact_2620_Times__Diff__distrib1, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((minus_minus @ (set @ A)) @ A4) @ B6)) @ (^ [Uu:A]: (C5))) = (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (C5)))))))).
% 9.43/3.45 thf(fact_2621_Sigma__Union, axiom, (! [B:$tType,A:$tType,X10:(set @ (set @ A)),B6:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ ((complete_Sup_Sup @ (set @ A)) @ X10)) @ B6) = ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ (set @ A)) @ (set @ ((product_prod @ A) @ B))) @ (^ [A9:(set @ A)]: (((((product_Sigma @ A) @ B) @ A9) @ B6)))) @ X10)))))).
% 9.43/3.45 thf(fact_2622_Collect__case__prod__Sigma, axiom, (! [B:$tType,A:$tType,P:(A > $o),Q:(A > (B > $o))]: ((((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X3:A,Y3:B]: (((P @ X3) & ((Q @ X3) @ Y3)))))) = ((((product_Sigma @ A) @ B) @ ((collect @ A) @ P)) @ (^ [X3:A]: (((collect @ B) @ (Q @ X3))))))))).
% 9.43/3.45 thf(fact_2623_apfst__compose, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > C),X:((product_prod @ D) @ B)]: (((((((product_apfst @ C) @ A) @ B) @ F) @ (((((product_apfst @ D) @ C) @ B) @ G) @ X)) = (((((product_apfst @ D) @ A) @ B) @ (((((comp @ C) @ A) @ D) @ F) @ G)) @ X))))).
% 9.43/3.45 thf(fact_2624_apsnd__apfst__commute, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,F:(C > B),G:(D > A),P5:((product_prod @ D) @ C)]: (((((((product_apsnd @ C) @ B) @ A) @ F) @ (((((product_apfst @ D) @ A) @ C) @ G) @ P5)) = (((((product_apfst @ D) @ A) @ B) @ G) @ (((((product_apsnd @ C) @ B) @ D) @ F) @ P5)))))).
% 9.43/3.45 thf(fact_2625_gcd__add__mult, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [M2:A,K:A,N:A]: (((((gcd_gcd @ A) @ M2) @ (((plus_plus @ A) @ (((times_times @ A) @ K) @ M2)) @ N)) = (((gcd_gcd @ A) @ M2) @ N)))))))).
% 9.43/3.45 thf(fact_2626_Sigma__mono, axiom, (! [B:$tType,A:$tType,A4:(set @ A),C5:(set @ A),B6:(A > (set @ B)),D4:(A > (set @ B))]: (((((ord_less_eq @ (set @ A)) @ A4) @ C5) => ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (((ord_less_eq @ (set @ B)) @ (B6 @ X2)) @ (D4 @ X2))))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) @ ((((product_Sigma @ A) @ B) @ C5) @ D4))))))).
% 9.43/3.45 thf(fact_2627_bezout__int, axiom, (! [X:int,Y:int]: ((? [U3:int,V4:int]: (((((plus_plus @ int) @ (((times_times @ int) @ U3) @ X)) @ (((times_times @ int) @ V4) @ Y)) = (((gcd_gcd @ int) @ X) @ Y))))))).
% 9.43/3.45 thf(fact_2628_Times__subset__cancel2, axiom, (! [A:$tType,B:$tType,X:A,C5:(set @ A),A4:(set @ B),B6:(set @ B)]: (((((member @ A) @ X) @ C5) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (C5)))) @ ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (C5)))) = (((ord_less_eq @ (set @ B)) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_2629_UN__Times__distrib, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,E4:(C > (set @ A)),F6:(D > (set @ B)),A4:(set @ C),B6:(set @ D)]: ((((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ ((product_prod @ C) @ D)) @ (set @ ((product_prod @ A) @ B))) @ ((((product_case_prod @ C) @ D) @ (set @ ((product_prod @ A) @ B))) @ (^ [A7:C,B5:D]: (((((product_Sigma @ A) @ B) @ (E4 @ A7)) @ (^ [Uu:A]: ((F6 @ B5)))))))) @ ((((product_Sigma @ C) @ D) @ A4) @ (^ [Uu:C]: (B6))))) = ((((product_Sigma @ A) @ B) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ C) @ (set @ A)) @ E4) @ A4))) @ (^ [Uu:A]: (((complete_Sup_Sup @ (set @ B)) @ ((((image2 @ D) @ (set @ B)) @ F6) @ B6))))))))).
% 9.43/3.45 thf(fact_2630_mem__Times__iff, axiom, (! [A:$tType,B:$tType,X:((product_prod @ A) @ B),A4:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))) = ((((member @ A) @ (((product_fst @ A) @ B) @ X)) @ A4) & (((member @ B) @ (((product_snd @ A) @ B) @ X)) @ B6)))))).
% 9.43/3.45 thf(fact_2631_in__prod__fst__sndI, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A4:(set @ A),B6:(set @ B)]: (((((member @ A) @ (((product_fst @ A) @ B) @ X)) @ A4) => ((((member @ B) @ (((product_snd @ A) @ B) @ X)) @ B6) => (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))))))))).
% 9.43/3.45 thf(fact_2632_swap__product, axiom, (! [B:$tType,A:$tType,A4:(set @ B),B6:(set @ A)]: ((((((image2 @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [I3:B,J2:A]: (((((product_Pair @ A) @ B) @ J2) @ I3))))) @ ((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (A4))))))).
% 9.43/3.45 thf(fact_2633_Sigma__empty__iff, axiom, (! [B:$tType,A:$tType,I4:(set @ A),X10:(A > (set @ B))]: (((((((product_Sigma @ A) @ B) @ I4) @ X10) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (! [X3:A]: (((((member @ A) @ X3) @ I4) => ((X10 @ X3) = (bot_bot @ (set @ B)))))))))).
% 9.43/3.45 thf(fact_2634_gcd__mult__unit2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_gcd @ A) @ B2) @ (((times_times @ A) @ C2) @ A3)) = (((gcd_gcd @ A) @ B2) @ C2))))))))).
% 9.43/3.45 thf(fact_2635_gcd__mult__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_gcd @ A) @ (((times_times @ A) @ B2) @ A3)) @ C2) = (((gcd_gcd @ A) @ B2) @ C2))))))))).
% 9.43/3.45 thf(fact_2636_gcd__div__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_gcd @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ C2) = (((gcd_gcd @ A) @ B2) @ C2))))))))).
% 9.43/3.45 thf(fact_2637_gcd__div__unit2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_gcd @ A) @ B2) @ (((divide_divide @ A) @ C2) @ A3)) = (((gcd_gcd @ A) @ B2) @ C2))))))))).
% 9.43/3.45 thf(fact_2638_less__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > $o)) @ (int > $o)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ $o) @ $o) @ pcr_int) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3))))))))) @ (ord_less @ int))).
% 9.43/3.45 thf(fact_2639_less__eq__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > $o)) @ (int > $o)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ $o) @ $o) @ pcr_int) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3))))))))) @ (ord_less_eq @ int))).
% 9.43/3.45 thf(fact_2640_plus__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) @ pcr_int)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ U2)) @ (((plus_plus @ nat) @ Y3) @ V2))))))))) @ (plus_plus @ int))).
% 9.43/3.45 thf(fact_2641_minus__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) @ pcr_int)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ Y3) @ U2))))))))) @ (minus_minus @ int))).
% 9.43/3.45 thf(fact_2642_times__subset__iff, axiom, (! [A:$tType,B:$tType,A4:(set @ A),C5:(set @ B),B6:(set @ A),D4:(set @ B)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (D4)))) = ((A4 = (bot_bot @ (set @ A))) | ((C5 = (bot_bot @ (set @ B))) | ((((ord_less_eq @ (set @ A)) @ A4) @ B6) & (((ord_less_eq @ (set @ B)) @ C5) @ D4)))))))).
% 9.43/3.45 thf(fact_2643_image__paired__Times, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F:(C > A),G:(D > B),A4:(set @ C),B6:(set @ D)]: ((((((image2 @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ D) @ ((product_prod @ A) @ B)) @ (^ [X3:C,Y3:D]: (((((product_Pair @ A) @ B) @ (F @ X3)) @ (G @ Y3)))))) @ ((((product_Sigma @ C) @ D) @ A4) @ (^ [Uu:C]: (B6)))) = ((((product_Sigma @ A) @ B) @ ((((image2 @ C) @ A) @ F) @ A4)) @ (^ [Uu:A]: (((((image2 @ D) @ B) @ G) @ B6)))))))).
% 9.43/3.45 thf(fact_2644_fst__image__Sigma, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(A > (set @ B))]: ((((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ B6)) = ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A4) & ((B6 @ X3) != (bot_bot @ (set @ B))))))))))).
% 9.43/3.45 thf(fact_2645_Sigma__def, axiom, (! [B:$tType,A:$tType]: ((((product_Sigma @ A) @ B) = (^ [A9:(set @ A),B9:(A > (set @ B))]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ A) @ (set @ ((product_prod @ A) @ B))) @ (^ [X3:A]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ B) @ (set @ ((product_prod @ A) @ B))) @ (^ [Y3:B]: ((((insert2 @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (bot_bot @ (set @ ((product_prod @ A) @ B))))))) @ (B9 @ X3)))))) @ A9)))))))).
% 9.43/3.45 thf(fact_2646_times__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (int > int)) @ pcr_int) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) @ pcr_int)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ U2)) @ (((times_times @ nat) @ Y3) @ V2))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ V2)) @ (((times_times @ nat) @ Y3) @ U2)))))))))) @ (times_times @ int))).
% 9.43/3.45 thf(fact_2647_num__of__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ num) @ num) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2)) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2))).
% 9.43/3.45 thf(fact_2648_snd__image__Sigma, axiom, (! [A:$tType,B:$tType,A4:(set @ B),B6:(B > (set @ A))]: ((((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A4) @ B6)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ B) @ (set @ A)) @ B6) @ A4)))))).
% 9.43/3.45 thf(fact_2649_subset__fst__snd, axiom, (! [B:$tType,A:$tType,A4:(set @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A4) @ ((((product_Sigma @ A) @ B) @ ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A4)) @ (^ [Uu:A]: (((((image2 @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A4)))))))).
% 9.43/3.45 thf(fact_2650_int__transfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ int) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ pcr_int) @ (^ [N5:nat]: (((((product_Pair @ nat) @ nat) @ N5) @ (zero_zero @ nat))))) @ (semiring_1_of_nat @ int))).
% 9.43/3.45 thf(fact_2651_uminus__int_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ int) @ ((product_prod @ nat) @ nat)) @ int) @ pcr_int) @ pcr_int) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X3:nat,Y3:nat]: (((((product_Pair @ nat) @ nat) @ Y3) @ X3))))) @ (uminus_uminus @ int))).
% 9.43/3.45 thf(fact_2652_eq__fst__iff, axiom, (! [A:$tType,B:$tType,A3:A,P5:((product_prod @ A) @ B)]: (((A3 = (((product_fst @ A) @ B) @ P5)) = (? [B5:B]: ((P5 = ((((product_Pair @ A) @ B) @ A3) @ B5)))))))).
% 9.43/3.45 thf(fact_2653_transfer__rule__of__nat, axiom, (! [A:$tType,B:$tType]: ((((semiring_1 @ B) & (semiring_1 @ A)) => (! [R5:(A > (B > $o))]: ((((R5 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R5 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ R5) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ R5) @ R5)) @ (plus_plus @ A)) @ (plus_plus @ B)) => ((((((((bNF_rel_fun @ nat) @ nat) @ A) @ B) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ R5) @ (semiring_1_of_nat @ A)) @ (semiring_1_of_nat @ B))))))))))).
% 9.43/3.45 thf(fact_2654_transfer__rule__of__bool, axiom, (! [A:$tType,B:$tType]: ((((zero_neq_one @ B) & (zero_neq_one @ A)) => (! [R5:(A > (B > $o))]: ((((R5 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((R5 @ (one_one @ A)) @ (one_one @ B)) => ((((((((bNF_rel_fun @ $o) @ $o) @ A) @ B) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))) @ R5) @ (zero_neq_one_of_bool @ A)) @ (zero_neq_one_of_bool @ B)))))))))).
% 9.43/3.45 thf(fact_2655_plus__rat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ (rat > rat)) @ pcr_rat) @ ((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) @ pcr_rat)) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y3)) @ (((product_snd @ int) @ int) @ X3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3)))))) @ (plus_plus @ rat))).
% 9.43/3.45 thf(fact_2656_inverse__rat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) @ pcr_rat) @ (^ [X3:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X3) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ X3)))))) @ (inverse_inverse @ rat))).
% 9.43/3.45 thf(fact_2657_Id__on__def, axiom, (! [A:$tType]: (((id_on @ A) = (^ [A9:(set @ A)]: (((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ ((((image2 @ A) @ (set @ ((product_prod @ A) @ A))) @ (^ [X3:A]: ((((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))) @ A9)))))))).
% 9.43/3.45 thf(fact_2658_times__rat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ (rat > rat)) @ pcr_rat) @ ((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) @ pcr_rat)) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3)))))) @ (times_times @ rat))).
% 9.43/3.45 thf(fact_2659_gcd__1__nat, axiom, (! [M2:nat]: (((((gcd_gcd @ nat) @ M2) @ (one_one @ nat)) = (one_one @ nat))))).
% 9.43/3.45 thf(fact_2660_Id__onI, axiom, (! [A:$tType,A3:A,A4:(set @ A)]: (((((member @ A) @ A3) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ ((id_on @ A) @ A4)))))).
% 9.43/3.45 thf(fact_2661_unset__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se2638667681897837118et_bit @ nat)) @ (bit_se2638667681897837118et_bit @ nat))).
% 9.43/3.45 thf(fact_2662_take__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se2584673776208193580ke_bit @ nat)) @ (bit_se2584673776208193580ke_bit @ nat))).
% 9.43/3.45 thf(fact_2663_bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > $o)) @ (nat > $o)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:(nat > $o),Z4:(nat > $o)]: ((Y5 = Z4)))) @ (bit_se5641148757651400278ts_bit @ nat)) @ (bit_se5641148757651400278ts_bit @ nat))).
% 9.43/3.45 thf(fact_2664_drop__bit__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (nat > nat)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4))))) @ (bit_se4197421643247451524op_bit @ nat)) @ (bit_se4197421643247451524op_bit @ nat))).
% 9.43/3.45 thf(fact_2665_Id__on__iff, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((id_on @ A) @ A4)) = ((X = Y) & (((member @ A) @ X) @ A4)))))).
% 9.43/3.45 thf(fact_2666_Id__on__eqI, axiom, (! [A:$tType,A3:A,B2:A,A4:(set @ A)]: (((A3 = B2) => ((((member @ A) @ A3) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((id_on @ A) @ A4))))))).
% 9.43/3.45 thf(fact_2667_Id__onE, axiom, (! [A:$tType,C2:((product_prod @ A) @ A),A4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ C2) @ ((id_on @ A) @ A4)) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (C2 != ((((product_Pair @ A) @ A) @ X2) @ X2))))))))))).
% 9.43/3.45 thf(fact_2668_bezout__nat, axiom, (! [A3:nat,B2:nat]: (((A3 != (zero_zero @ nat)) => (? [X2:nat,Y4:nat]: (((((times_times @ nat) @ A3) @ X2) = (((plus_plus @ nat) @ (((times_times @ nat) @ B2) @ Y4)) @ (((gcd_gcd @ nat) @ A3) @ B2))))))))).
% 9.43/3.45 thf(fact_2669_one__rat_Otransfer, axiom, ((pcr_rat @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))) @ (one_one @ rat))).
% 9.43/3.45 thf(fact_2670_zero__rat_Otransfer, axiom, ((pcr_rat @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ (zero_zero @ rat))).
% 9.43/3.45 thf(fact_2671_bezw__aux, axiom, (! [X:nat,Y:nat]: ((((semiring_1_of_nat @ int) @ (((gcd_gcd @ nat) @ X) @ Y)) = (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ ((bezw @ X) @ Y))) @ ((semiring_1_of_nat @ int) @ X))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ ((bezw @ X) @ Y))) @ ((semiring_1_of_nat @ int) @ Y))))))).
% 9.43/3.45 thf(fact_2672_uminus__rat_Otransfer, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ rat) @ ((product_prod @ int) @ int)) @ rat) @ pcr_rat) @ pcr_rat) @ (^ [X3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X3))) @ (((product_snd @ int) @ int) @ X3))))) @ (uminus_uminus @ rat))).
% 9.43/3.45 thf(fact_2673_gcd__nat_Opelims, axiom, (! [X:nat,Xa:nat,Y:nat]: ((((((gcd_gcd @ nat) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ nat) @ nat)) @ gcd_nat_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa)) => (~ (((((Xa = (zero_zero @ nat)) => (Y = X)) & ((Xa != (zero_zero @ nat)) => (Y = (((gcd_gcd @ nat) @ Xa) @ (((modulo_modulo @ nat) @ X) @ Xa))))) => (~ ((((accp @ ((product_prod @ nat) @ nat)) @ gcd_nat_rel) @ ((((product_Pair @ nat) @ nat) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_2674_o__rsp_I1_J, axiom, (! [A:$tType,B:$tType,E:$tType,F5:$tType,D:$tType,C:$tType,R23:(A > (C > $o)),R32:(B > (D > $o)),R12:(E > (F5 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (C > D)) @ ((E > A) > (E > B))) @ ((F5 > C) > (F5 > D))) @ ((((((bNF_rel_fun @ A) @ C) @ B) @ D) @ R23) @ R32)) @ ((((((bNF_rel_fun @ (E > A)) @ (F5 > C)) @ (E > B)) @ (F5 > D)) @ ((((((bNF_rel_fun @ E) @ F5) @ A) @ C) @ R12) @ R23)) @ ((((((bNF_rel_fun @ E) @ F5) @ B) @ D) @ R12) @ R32))) @ (((comp @ A) @ B) @ E)) @ (((comp @ C) @ D) @ F5))))).
% 9.43/3.45 thf(fact_2675_o__rsp_I2_J, axiom, (! [E:$tType,F5:$tType,H8:$tType,G3:$tType,R12:(E > (F5 > $o))]: (((((((((bNF_rel_fun @ (G3 > H8)) @ (G3 > H8)) @ ((E > G3) > (E > H8))) @ ((F5 > G3) > (F5 > H8))) @ (^ [Y5:(G3 > H8),Z4:(G3 > H8)]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ (E > G3)) @ (F5 > G3)) @ (E > H8)) @ (F5 > H8)) @ ((((((bNF_rel_fun @ E) @ F5) @ G3) @ G3) @ R12) @ (^ [Y5:G3,Z4:G3]: ((Y5 = Z4))))) @ ((((((bNF_rel_fun @ E) @ F5) @ H8) @ H8) @ R12) @ (^ [Y5:H8,Z4:H8]: ((Y5 = Z4)))))) @ (((comp @ G3) @ H8) @ E)) @ (((comp @ G3) @ H8) @ F5))))).
% 9.43/3.45 thf(fact_2676_apfst__convE, axiom, (! [C:$tType,A:$tType,B:$tType,Q3:((product_prod @ A) @ B),F:(C > A),P5:((product_prod @ C) @ B)]: (((Q3 = (((((product_apfst @ C) @ A) @ B) @ F) @ P5)) => (~ ((! [X2:C,Y4:B]: (((P5 = ((((product_Pair @ C) @ B) @ X2) @ Y4)) => (Q3 != ((((product_Pair @ A) @ B) @ (F @ X2)) @ Y4))))))))))).
% 9.43/3.45 thf(fact_2677_UN__le__eq__Un0, axiom, (! [A:$tType,M3:(nat > (set @ A)),N:nat]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ M3) @ ((set_ord_atMost @ nat) @ N))) = (((sup_sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ nat) @ (set @ A)) @ M3) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ N)))) @ (M3 @ (zero_zero @ nat))))))).
% 9.43/3.45 thf(fact_2678_times__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ intrel) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ intrel) @ intrel)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ U2)) @ (((times_times @ nat) @ Y3) @ V2))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ V2)) @ (((times_times @ nat) @ Y3) @ U2)))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ U2)) @ (((times_times @ nat) @ Y3) @ V2))) @ (((plus_plus @ nat) @ (((times_times @ nat) @ X3) @ V2)) @ (((times_times @ nat) @ Y3) @ U2))))))))))).
% 9.43/3.45 thf(fact_2679_sup__Some, axiom, (! [A:$tType]: (((sup @ A) => (! [X:A,Y:A]: (((((sup_sup @ (option @ A)) @ ((some @ A) @ X)) @ ((some @ A) @ Y)) = ((some @ A) @ (((sup_sup @ A) @ X) @ Y))))))))).
% 9.43/3.45 thf(fact_2680_intrel__iff, axiom, (! [X:nat,Y:nat,U:nat,V:nat]: ((((intrel @ ((((product_Pair @ nat) @ nat) @ X) @ Y)) @ ((((product_Pair @ nat) @ nat) @ U) @ V)) = ((((plus_plus @ nat) @ X) @ V) = (((plus_plus @ nat) @ U) @ Y)))))).
% 9.43/3.45 thf(fact_2681_insert__Times__insert, axiom, (! [B:$tType,A:$tType,A3:A,A4:(set @ A),B2:B,B6:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((insert2 @ A) @ A3) @ A4)) @ (^ [Uu:A]: ((((insert2 @ B) @ B2) @ B6)))) = (((insert2 @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: ((((insert2 @ B) @ B2) @ B6))))) @ ((((product_Sigma @ A) @ B) @ (((insert2 @ A) @ A3) @ A4)) @ (^ [Uu:A]: (B6))))))))).
% 9.43/3.45 thf(fact_2682_Sigma__Un__distrib1, axiom, (! [B:$tType,A:$tType,I4:(set @ A),J4:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((sup_sup @ (set @ A)) @ I4) @ J4)) @ C5) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I4) @ C5)) @ ((((product_Sigma @ A) @ B) @ J4) @ C5)))))).
% 9.43/3.45 thf(fact_2683_Sigma__Un__distrib2, axiom, (! [B:$tType,A:$tType,I4:(set @ A),A4:(A > (set @ B)),B6:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I4) @ (^ [I3:A]: ((((sup_sup @ (set @ B)) @ (A4 @ I3)) @ (B6 @ I3))))) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I4) @ A4)) @ ((((product_Sigma @ A) @ B) @ I4) @ B6)))))).
% 9.43/3.45 thf(fact_2684_Times__Un__distrib1, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) @ (^ [Uu:A]: (C5))) = (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (C5)))))))).
% 9.43/3.45 thf(fact_2685_set__diff__diff__left, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),C5:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B6)) @ C5) = (((minus_minus @ (set @ A)) @ A4) @ (((sup_sup @ (set @ A)) @ B6) @ C5)))))).
% 9.43/3.45 thf(fact_2686_sup__option__def, axiom, (! [A:$tType]: (((sup @ A) => ((sup_sup @ (option @ A)) = (^ [X3:(option @ A),Y3:(option @ A)]: ((((((case_option @ (option @ A)) @ A) @ Y3) @ (^ [X9:A]: ((((((case_option @ (option @ A)) @ A) @ X3) @ (^ [Z5:A]: (((some @ A) @ (((sup_sup @ A) @ X9) @ Z5))))) @ Y3)))) @ X3)))))))).
% 9.43/3.45 thf(fact_2687_ivl__disj__un__two__touch_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M2)) @ (((set_or1337092689740270186AtMost @ A) @ M2) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2688_ivl__disj__un__two_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2689_max__ext__additive, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),R5:(set @ ((product_prod @ A) @ A)),C5:(set @ A),D4:(set @ A)]: (((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ A4) @ B6)) @ ((max_ext @ A) @ R5)) => ((((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ C5) @ D4)) @ ((max_ext @ A) @ R5)) => (((member @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A4) @ C5)) @ (((sup_sup @ (set @ A)) @ B6) @ D4))) @ ((max_ext @ A) @ R5))))))).
% 9.43/3.45 thf(fact_2690_zero__int_Orsp, axiom, ((intrel @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat))) @ ((((product_Pair @ nat) @ nat) @ (zero_zero @ nat)) @ (zero_zero @ nat)))).
% 9.43/3.45 thf(fact_2691_ivl__disj__un__two_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M2)) @ (((set_or1337092689740270186AtMost @ A) @ M2) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2692_card__Un__le, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((ord_less_eq @ nat) @ ((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6))) @ (((plus_plus @ nat) @ ((finite_card @ A) @ A4)) @ ((finite_card @ A) @ B6)))))).
% 9.43/3.45 thf(fact_2693_ivl__disj__un__one_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) = ((set_ord_lessThan @ A) @ U))))))))).
% 9.43/3.45 thf(fact_2694_atLeastLessThan__add__Un, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => ((((set_or7035219750837199246ssThan @ nat) @ I) @ (((plus_plus @ nat) @ J) @ K)) = (((sup_sup @ (set @ nat)) @ (((set_or7035219750837199246ssThan @ nat) @ I) @ J)) @ (((set_or7035219750837199246ssThan @ nat) @ J) @ (((plus_plus @ nat) @ J) @ K)))))))).
% 9.43/3.45 thf(fact_2695_uminus__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ intrel) @ intrel) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X3:nat,Y3:nat]: (((((product_Pair @ nat) @ nat) @ Y3) @ X3))))) @ ((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [X3:nat,Y3:nat]: (((((product_Pair @ nat) @ nat) @ Y3) @ X3)))))).
% 9.43/3.45 thf(fact_2696_one__int_Orsp, axiom, ((intrel @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat))) @ ((((product_Pair @ nat) @ nat) @ (one_one @ nat)) @ (zero_zero @ nat)))).
% 9.43/3.45 thf(fact_2697_ivl__disj__un__two__touch_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2698_ivl__disj__un__two_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2699_ivl__disj__un__singleton_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [U:A]: (((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ U)) @ (((insert2 @ A) @ U) @ (bot_bot @ (set @ A)))) = ((set_ord_atMost @ A) @ U)))))))).
% 9.43/3.45 thf(fact_2700_ivl__disj__un__one_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = ((set_ord_atMost @ A) @ U))))))))).
% 9.43/3.45 thf(fact_2701_ivl__disj__un__one_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = ((set_ord_lessThan @ A) @ U))))))))).
% 9.43/3.45 thf(fact_2702_intrel__def, axiom, (intrel = ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: (((((plus_plus @ nat) @ X3) @ V2) = (((plus_plus @ nat) @ U2) @ Y3)))))))))).
% 9.43/3.45 thf(fact_2703_ivl__disj__un__singleton_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ (((insert2 @ A) @ U) @ (bot_bot @ (set @ A)))) = (((set_or1337092689740270186AtMost @ A) @ L) @ U))))))))).
% 9.43/3.45 thf(fact_2704_ivl__disj__un__singleton_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((insert2 @ A) @ L) @ (bot_bot @ (set @ A)))) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U))))))))).
% 9.43/3.45 thf(fact_2705_ivl__disj__un__two_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M2)) @ (((set_or5935395276787703475ssThan @ A) @ M2) @ U)) = (((set_or7035219750837199246ssThan @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2706_less__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > $o)) @ (((product_prod @ nat) @ nat) > $o)) @ intrel) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ $o) @ $o) @ intrel) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3)))))))))).
% 9.43/3.45 thf(fact_2707_less__eq__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > $o)) @ (((product_prod @ nat) @ nat) > $o)) @ intrel) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ $o) @ $o) @ intrel) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > $o)) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ $o) @ (^ [U2:nat,V2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ U2) @ Y3)))))))))).
% 9.43/3.45 thf(fact_2708_plus__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ intrel) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ intrel) @ intrel)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ U2)) @ (((plus_plus @ nat) @ Y3) @ V2))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ U2)) @ (((plus_plus @ nat) @ Y3) @ V2)))))))))).
% 9.43/3.45 thf(fact_2709_minus__int_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ intrel) @ ((((((bNF_rel_fun @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ intrel) @ intrel)) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ Y3) @ U2))))))))) @ ((((product_case_prod @ nat) @ nat) @ (((product_prod @ nat) @ nat) > ((product_prod @ nat) @ nat))) @ (^ [X3:nat,Y3:nat]: (((((product_case_prod @ nat) @ nat) @ ((product_prod @ nat) @ nat)) @ (^ [U2:nat,V2:nat]: (((((product_Pair @ nat) @ nat) @ (((plus_plus @ nat) @ X3) @ V2)) @ (((plus_plus @ nat) @ Y3) @ U2)))))))))).
% 9.43/3.45 thf(fact_2710_max__weak__def, axiom, (fun_max_weak = (((sup_sup @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ ((max_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_leq)) @ (((insert2 @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))))))).
% 9.43/3.45 thf(fact_2711_min__weak__def, axiom, (fun_min_weak = (((sup_sup @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))) @ ((min_ext @ ((product_prod @ nat) @ nat)) @ fun_pair_leq)) @ (((insert2 @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat)))) @ (bot_bot @ (set @ ((product_prod @ nat) @ nat))))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))))))))).
% 9.43/3.45 thf(fact_2712_fold__union__pair, axiom, (! [B:$tType,A:$tType,B6:(set @ A),X:B,A4:(set @ ((product_prod @ B) @ A))]: ((((finite_finite2 @ A) @ B6) => ((((sup_sup @ (set @ ((product_prod @ B) @ A))) @ ((complete_Sup_Sup @ (set @ ((product_prod @ B) @ A))) @ ((((image2 @ A) @ (set @ ((product_prod @ B) @ A))) @ (^ [Y3:A]: ((((insert2 @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y3)) @ (bot_bot @ (set @ ((product_prod @ B) @ A))))))) @ B6))) @ A4) = (((((finite_fold @ A) @ (set @ ((product_prod @ B) @ A))) @ (^ [Y3:A]: (((insert2 @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y3))))) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_2713_Field__insert, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: ((((field2 @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)) = (((sup_sup @ (set @ A)) @ (((insert2 @ A) @ A3) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((field2 @ A) @ R)))))).
% 9.43/3.45 thf(fact_2714_set__to__map__insert, axiom, (! [B:$tType,A:$tType,Kv:((product_prod @ A) @ B),S2:(set @ ((product_prod @ A) @ B))]: (((~ ((((member @ A) @ (((product_fst @ A) @ B) @ Kv)) @ ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S2)))) => ((((set_to_map @ A) @ B) @ (((insert2 @ ((product_prod @ A) @ B)) @ Kv) @ S2)) = (((((fun_upd @ A) @ (option @ B)) @ (((set_to_map @ A) @ B) @ S2)) @ (((product_fst @ A) @ B) @ Kv)) @ ((some @ B) @ (((product_snd @ A) @ B) @ Kv)))))))).
% 9.43/3.45 thf(fact_2715_plus__rat_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ ratrel) @ ((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ratrel) @ ratrel)) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y3)) @ (((product_snd @ int) @ int) @ X3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3)))))) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y3)) @ (((product_snd @ int) @ int) @ X3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3))))))).
% 9.43/3.45 thf(fact_2716_inverse__rat_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ratrel) @ ratrel) @ (^ [X3:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X3) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ X3)))))) @ (^ [X3:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X3) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ X3))))))).
% 9.43/3.45 thf(fact_2717_fun__upd__upd, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,Z2:B]: (((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ Y)) @ X) @ Z2) = (((((fun_upd @ A) @ B) @ F) @ X) @ Z2))))).
% 9.43/3.45 thf(fact_2718_fun__upd__triv, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: (((((((fun_upd @ A) @ B) @ F) @ X) @ (F @ X)) = F)))).
% 9.43/3.45 thf(fact_2719_fun__upd__apply, axiom, (! [A:$tType,B:$tType]: ((((fun_upd @ B) @ A) = (^ [F2:(B > A),X3:B,Y3:A,Z5:B]: (((((if @ A) @ (Z5 = X3)) @ Y3) @ (F2 @ Z5)))))))).
% 9.43/3.45 thf(fact_2720_image__update, axiom, (! [B:$tType,A:$tType,X:A,A4:(set @ A),F:(A > B),N:B]: (((~ ((((member @ A) @ X) @ A4))) => (((((image2 @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ N)) @ A4) = ((((image2 @ A) @ B) @ F) @ A4)))))).
% 9.43/3.45 thf(fact_2721_empty__upd__none, axiom, (! [B:$tType,A:$tType,X:A]: (((((((fun_upd @ A) @ (option @ B)) @ (^ [X3:A]: ((none @ B)))) @ X) @ (none @ B)) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_2722_sup__None__1, axiom, (! [A:$tType]: (((sup @ A) => (! [Y:(option @ A)]: (((((sup_sup @ (option @ A)) @ (none @ A)) @ Y) = Y))))))).
% 9.43/3.45 thf(fact_2723_sup__None__2, axiom, (! [A:$tType]: (((sup @ A) => (! [X:(option @ A)]: (((((sup_sup @ (option @ A)) @ X) @ (none @ A)) = X))))))).
% 9.43/3.45 thf(fact_2724_finite__Field__eq__finite, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ A) @ ((field2 @ A) @ R5)) = ((finite_finite2 @ ((product_prod @ A) @ A)) @ R5))))).
% 9.43/3.45 thf(fact_2725_image__map__upd, axiom, (! [B:$tType,A:$tType,X:A,A4:(set @ A),M2:(A > (option @ B)),Y:B]: (((~ ((((member @ A) @ X) @ A4))) => (((((image2 @ A) @ (option @ B)) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ X) @ ((some @ B) @ Y))) @ A4) = ((((image2 @ A) @ (option @ B)) @ M2) @ A4)))))).
% 9.43/3.45 thf(fact_2726_map__option__o__map__upd, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),M2:(A > (option @ C)),A3:A,B2:C]: (((((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F)) @ (((((fun_upd @ A) @ (option @ C)) @ M2) @ A3) @ ((some @ C) @ B2))) = (((((fun_upd @ A) @ (option @ B)) @ (((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F)) @ M2)) @ A3) @ ((some @ B) @ (F @ B2))))))).
% 9.43/3.45 thf(fact_2727_fun__upd__def, axiom, (! [B:$tType,A:$tType]: ((((fun_upd @ A) @ B) = (^ [F2:(A > B),A7:A,B5:B,X3:A]: (((((if @ B) @ (X3 = A7)) @ B5) @ (F2 @ X3)))))))).
% 9.43/3.45 thf(fact_2728_fun__upd__eqD, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B,G:(A > B),Z2:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = (((((fun_upd @ A) @ B) @ G) @ X) @ Z2)) => (Y = Z2))))).
% 9.43/3.45 thf(fact_2729_fun__upd__idem, axiom, (! [A:$tType,B:$tType,F:(B > A),X:B,Y:A]: ((((F @ X) = Y) => ((((((fun_upd @ B) @ A) @ F) @ X) @ Y) = F))))).
% 9.43/3.45 thf(fact_2730_fun__upd__same, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,Y:A]: ((((((((fun_upd @ B) @ A) @ F) @ X) @ Y) @ X) = Y)))).
% 9.43/3.45 thf(fact_2731_fun__upd__other, axiom, (! [B:$tType,A:$tType,Z2:A,X:A,F:(A > B),Y:B]: (((Z2 != X) => (((((((fun_upd @ A) @ B) @ F) @ X) @ Y) @ Z2) = (F @ Z2)))))).
% 9.43/3.45 thf(fact_2732_fun__upd__twist, axiom, (! [A:$tType,B:$tType,A3:A,C2:A,M2:(A > B),B2:B,D3:B]: (((A3 != C2) => ((((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M2) @ A3) @ B2)) @ C2) @ D3) = (((((fun_upd @ A) @ B) @ (((((fun_upd @ A) @ B) @ M2) @ C2) @ D3)) @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_2733_fun__upd__idem__iff, axiom, (! [A:$tType,B:$tType,F:(A > B),X:A,Y:B]: ((((((((fun_upd @ A) @ B) @ F) @ X) @ Y) = F) = ((F @ X) = Y))))).
% 9.43/3.45 thf(fact_2734_map__upd__Some__unfold, axiom, (! [B:$tType,A:$tType,M2:(B > (option @ A)),A3:B,B2:A,X:B,Y:A]: (((((((((fun_upd @ B) @ (option @ A)) @ M2) @ A3) @ ((some @ A) @ B2)) @ X) = ((some @ A) @ Y)) = (((X = A3) & (B2 = Y)) | ((X != A3) & ((M2 @ X) = ((some @ A) @ Y)))))))).
% 9.43/3.45 thf(fact_2735_map__upd__triv, axiom, (! [A:$tType,B:$tType,T2:(B > (option @ A)),K:B,X:A]: ((((T2 @ K) = ((some @ A) @ X)) => ((((((fun_upd @ B) @ (option @ A)) @ T2) @ K) @ ((some @ A) @ X)) = T2))))).
% 9.43/3.45 thf(fact_2736_map__upd__eqD1, axiom, (! [A:$tType,B:$tType,M2:(A > (option @ B)),A3:A,X:B,N:(A > (option @ B)),Y:B]: ((((((((fun_upd @ A) @ (option @ B)) @ M2) @ A3) @ ((some @ B) @ X)) = (((((fun_upd @ A) @ (option @ B)) @ N) @ A3) @ ((some @ B) @ Y))) => (X = Y))))).
% 9.43/3.45 thf(fact_2737_FieldI2, axiom, (! [A:$tType,I:A,J:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R5) => (((member @ A) @ J) @ ((field2 @ A) @ R5)))))).
% 9.43/3.45 thf(fact_2738_FieldI1, axiom, (! [A:$tType,I:A,J:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R5) => (((member @ A) @ I) @ ((field2 @ A) @ R5)))))).
% 9.43/3.45 thf(fact_2739_sup__Un__eq2, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((((sup_sup @ (A > (B > $o))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R5)))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S2)))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (((sup_sup @ (set @ ((product_prod @ A) @ B))) @ R5) @ S2)))))))).
% 9.43/3.45 thf(fact_2740_fun__upd__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),G:(A > C),X:A,Y:C]: (((((((comp @ C) @ B) @ A) @ F) @ (((((fun_upd @ A) @ C) @ G) @ X) @ Y)) = (((((fun_upd @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F) @ G)) @ X) @ (F @ Y)))))).
% 9.43/3.45 thf(fact_2741_map__upd__nonempty, axiom, (! [B:$tType,A:$tType,T2:(A > (option @ B)),K:A,X:B]: (((((((fun_upd @ A) @ (option @ B)) @ T2) @ K) @ ((some @ B) @ X)) != (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_2742_one__rat_Orsp, axiom, ((ratrel @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int)))).
% 9.43/3.45 thf(fact_2743_R__subset__Field, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ ((field2 @ A) @ R5)) @ (^ [Uu:A]: (((field2 @ A) @ R5)))))))).
% 9.43/3.45 thf(fact_2744_fst__in__Field, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ A)) @ ((((image2 @ ((product_prod @ A) @ A)) @ A) @ ((product_fst @ A) @ A)) @ R5)) @ ((field2 @ A) @ R5))))).
% 9.43/3.45 thf(fact_2745_snd__in__Field, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ A)) @ ((((image2 @ ((product_prod @ A) @ A)) @ A) @ ((product_snd @ A) @ A)) @ R5)) @ ((field2 @ A) @ R5))))).
% 9.43/3.45 thf(fact_2746_zero__rat_Orsp, axiom, ((ratrel @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int)))).
% 9.43/3.45 thf(fact_2747_Fract_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > ((product_prod @ int) @ int))) @ (int > ((product_prod @ int) @ int))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ratrel)) @ (^ [A7:int,B5:int]: (((((if @ ((product_prod @ int) @ int)) @ (B5 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A7) @ B5))))) @ (^ [A7:int,B5:int]: (((((if @ ((product_prod @ int) @ int)) @ (B5 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A7) @ B5)))))).
% 9.43/3.45 thf(fact_2748_sum_Oeq__fold, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (((groups7311177749621191930dd_sum @ B) @ A) = (^ [G2:(B > A)]: (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (plus_plus @ A)) @ G2)) @ (zero_zero @ A))))))))).
% 9.43/3.45 thf(fact_2749_prod_Oeq__fold, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (((groups7121269368397514597t_prod @ B) @ A) = (^ [G2:(B > A)]: (((((finite_fold @ B) @ A) @ (((((comp @ A) @ (A > A)) @ B) @ (times_times @ A)) @ G2)) @ (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2750_finite__range__updI, axiom, (! [A:$tType,B:$tType,F:(B > (option @ A)),A3:B,B2:A]: ((((finite_finite2 @ (option @ A)) @ ((((image2 @ B) @ (option @ A)) @ F) @ (top_top @ (set @ B)))) => ((finite_finite2 @ (option @ A)) @ ((((image2 @ B) @ (option @ A)) @ (((((fun_upd @ B) @ (option @ A)) @ F) @ A3) @ ((some @ A) @ B2))) @ (top_top @ (set @ B)))))))).
% 9.43/3.45 thf(fact_2751_fun__upd__image, axiom, (! [A:$tType,B:$tType,X:B,A4:(set @ B),F:(B > A),Y:A]: ((((((member @ B) @ X) @ A4) => (((((image2 @ B) @ A) @ (((((fun_upd @ B) @ A) @ F) @ X) @ Y)) @ A4) = (((insert2 @ A) @ Y) @ ((((image2 @ B) @ A) @ F) @ (((minus_minus @ (set @ B)) @ A4) @ (((insert2 @ B) @ X) @ (bot_bot @ (set @ B)))))))) & ((~ ((((member @ B) @ X) @ A4))) => (((((image2 @ B) @ A) @ (((((fun_upd @ B) @ A) @ F) @ X) @ Y)) @ A4) = ((((image2 @ B) @ A) @ F) @ A4))))))).
% 9.43/3.45 thf(fact_2752_times__rat_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ ratrel) @ ((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ratrel) @ ratrel)) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3)))))) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3))))))).
% 9.43/3.45 thf(fact_2753_uminus__rat_Orsp, axiom, ((((((((bNF_rel_fun @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ ratrel) @ ratrel) @ (^ [X3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X3))) @ (((product_snd @ int) @ int) @ X3))))) @ (^ [X3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X3))) @ (((product_snd @ int) @ int) @ X3)))))).
% 9.43/3.45 thf(fact_2754_product__fold, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ B) @ B6) => (((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))) = (((((finite_fold @ A) @ (set @ ((product_prod @ A) @ B))) @ (^ [X3:A,Z5:(set @ ((product_prod @ A) @ B))]: ((((((finite_fold @ B) @ (set @ ((product_prod @ A) @ B))) @ (^ [Y3:B]: (((insert2 @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3))))) @ Z5) @ B6)))) @ (bot_bot @ (set @ ((product_prod @ A) @ B)))) @ A4))))))).
% 9.43/3.45 thf(fact_2755_Id__on__fold, axiom, (! [A:$tType,A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (((id_on @ A) @ A4) = (((((finite_fold @ A) @ (set @ ((product_prod @ A) @ A))) @ (^ [X3:A]: (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3))))) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))) @ A4)))))).
% 9.43/3.45 thf(fact_2756_plus__rat_Oabs__eq, axiom, (! [Xa:((product_prod @ int) @ int),X:((product_prod @ int) @ int)]: ((((ratrel @ Xa) @ Xa) => (((ratrel @ X) @ X) => ((((plus_plus @ rat) @ (abs_Rat @ Xa)) @ (abs_Rat @ X)) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Xa)) @ (((product_snd @ int) @ int) @ X))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X)) @ (((product_snd @ int) @ int) @ Xa)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ Xa)) @ (((product_snd @ int) @ int) @ X)))))))))).
% 9.43/3.45 thf(fact_2757_inverse__rat_Oabs__eq, axiom, (! [X:((product_prod @ int) @ int)]: ((((ratrel @ X) @ X) => (((inverse_inverse @ rat) @ (abs_Rat @ X)) = (abs_Rat @ ((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X)) @ (((product_fst @ int) @ int) @ X))))))))).
% 9.43/3.45 thf(fact_2758_times__rat_Oabs__eq, axiom, (! [Xa:((product_prod @ int) @ int),X:((product_prod @ int) @ int)]: ((((ratrel @ Xa) @ Xa) => (((ratrel @ X) @ X) => ((((times_times @ rat) @ (abs_Rat @ Xa)) @ (abs_Rat @ X)) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Xa)) @ (((product_fst @ int) @ int) @ X))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ Xa)) @ (((product_snd @ int) @ int) @ X)))))))))).
% 9.43/3.45 thf(fact_2759_uminus__rat_Oabs__eq, axiom, (! [X:((product_prod @ int) @ int)]: ((((ratrel @ X) @ X) => (((uminus_uminus @ rat) @ (abs_Rat @ X)) = (abs_Rat @ ((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X))) @ (((product_snd @ int) @ int) @ X)))))))).
% 9.43/3.45 thf(fact_2760_graph__map__upd, axiom, (! [A:$tType,B:$tType,M2:(A > (option @ B)),K:A,V:B]: (((((graph @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ K) @ ((some @ B) @ V))) = (((insert2 @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V)) @ (((graph @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ K) @ (none @ B)))))))).
% 9.43/3.45 thf(fact_2761_insert__relcomp__union__fold, axiom, (! [C:$tType,B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B)),X:((product_prod @ C) @ A),X10:(set @ ((product_prod @ C) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ S2) => ((((sup_sup @ (set @ ((product_prod @ C) @ B))) @ (((((relcomp @ C) @ A) @ B) @ (((insert2 @ ((product_prod @ C) @ A)) @ X) @ (bot_bot @ (set @ ((product_prod @ C) @ A))))) @ S2)) @ X10) = (((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [W2:A,Z5:B,A10:(set @ ((product_prod @ C) @ B))]: (((((if @ (set @ ((product_prod @ C) @ B))) @ ((((product_snd @ C) @ A) @ X) = W2)) @ (((insert2 @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ (((product_fst @ C) @ A) @ X)) @ Z5)) @ A10)) @ A10))))) @ X10) @ S2)))))).
% 9.43/3.45 thf(fact_2762_graph__empty, axiom, (! [B:$tType,A:$tType]: (((((graph @ A) @ B) @ (^ [X3:A]: ((none @ B)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2763_relcomp_Ocases, axiom, (! [A:$tType,C:$tType,B:$tType,A1:A,A22:C,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A1) @ A22)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S)) => (~ ((! [B3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A1) @ B3)) @ R) => (~ ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B3) @ A22)) @ S)))))))))))).
% 9.43/3.45 thf(fact_2764_relcomp_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,A1:A,A22:C,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A1) @ A22)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S)) = (? [A7:A,B5:B,C3:C]: (((A1 = A7) & ((A22 = C3) & ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B5)) @ R) & (((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B5) @ C3)) @ S)))))))))).
% 9.43/3.45 thf(fact_2765_relcomp_OrelcompI, axiom, (! [A:$tType,C:$tType,B:$tType,A3:A,B2:B,R:(set @ ((product_prod @ A) @ B)),C2:C,S:(set @ ((product_prod @ B) @ C))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R) => ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ B2) @ C2)) @ S) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A3) @ C2)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S))))))).
% 9.43/3.45 thf(fact_2766_relcompE, axiom, (! [A:$tType,B:$tType,C:$tType,Xz:((product_prod @ A) @ B),R:(set @ ((product_prod @ A) @ C)),S:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ Xz) @ (((((relcomp @ A) @ C) @ B) @ R) @ S)) => (~ ((! [X2:A,Y4:C,Z3:B]: (((Xz = ((((product_Pair @ A) @ B) @ X2) @ Z3)) => ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X2) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ Y4) @ Z3)) @ S))))))))))))).
% 9.43/3.45 thf(fact_2767_relcompEpair, axiom, (! [A:$tType,B:$tType,C:$tType,A3:A,C2:B,R:(set @ ((product_prod @ A) @ C)),S:(set @ ((product_prod @ C) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ C2)) @ (((((relcomp @ A) @ C) @ B) @ R) @ S)) => (~ ((! [B3:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ A3) @ B3)) @ R) => (~ ((((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ B3) @ C2)) @ S)))))))))))).
% 9.43/3.45 thf(fact_2768_in__graphI, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),K:B,V:A]: ((((M2 @ K) = ((some @ A) @ V)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ K) @ V)) @ (((graph @ B) @ A) @ M2)))))).
% 9.43/3.45 thf(fact_2769_in__graphD, axiom, (! [A:$tType,B:$tType,K:A,V:B,M2:(A > (option @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V)) @ (((graph @ A) @ B) @ M2)) => ((M2 @ K) = ((some @ B) @ V)))))).
% 9.43/3.45 thf(fact_2770_max__ext__compat, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R5) @ S2)) @ R5) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (((((relcomp @ (set @ A)) @ (set @ A)) @ (set @ A)) @ ((max_ext @ A) @ R5)) @ (((sup_sup @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((max_ext @ A) @ S2)) @ (((insert2 @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ A)))) @ (bot_bot @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))))))) @ ((max_ext @ A) @ R5)))))).
% 9.43/3.45 thf(fact_2771_min__ext__compat, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R5) @ S2)) @ R5) => (((ord_less_eq @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (((((relcomp @ (set @ A)) @ (set @ A)) @ (set @ A)) @ ((min_ext @ A) @ R5)) @ (((sup_sup @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((min_ext @ A) @ S2)) @ (((insert2 @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_Pair @ (set @ A)) @ (set @ A)) @ (bot_bot @ (set @ A))) @ (bot_bot @ (set @ A)))) @ (bot_bot @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))))))) @ ((min_ext @ A) @ R5)))))).
% 9.43/3.45 thf(fact_2772_relcomp__fold, axiom, (! [C:$tType,B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ B) @ C))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ R5) => (((finite_finite2 @ ((product_prod @ B) @ C)) @ S2) => ((((((relcomp @ A) @ B) @ C) @ R5) @ S2) = (((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ A) @ C))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ A) @ C)) > (set @ ((product_prod @ A) @ C)))) @ (^ [X3:A,Y3:B,A9:(set @ ((product_prod @ A) @ C))]: ((((((finite_fold @ ((product_prod @ B) @ C)) @ (set @ ((product_prod @ A) @ C))) @ ((((product_case_prod @ B) @ C) @ ((set @ ((product_prod @ A) @ C)) > (set @ ((product_prod @ A) @ C)))) @ (^ [W2:B,Z5:C,A10:(set @ ((product_prod @ A) @ C))]: (((((if @ (set @ ((product_prod @ A) @ C))) @ (Y3 = W2)) @ (((insert2 @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X3) @ Z5)) @ A10)) @ A10))))) @ A9) @ S2))))) @ (bot_bot @ (set @ ((product_prod @ A) @ C)))) @ R5))))))).
% 9.43/3.45 thf(fact_2773_one__rat__def, axiom, ((one_one @ rat) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (one_one @ int)) @ (one_one @ int))))).
% 9.43/3.45 thf(fact_2774_graph__fun__upd__None, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),K:A]: (((((graph @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ K) @ (none @ B))) = ((collect @ ((product_prod @ A) @ B)) @ (^ [E5:((product_prod @ A) @ B)]: (((((member @ ((product_prod @ A) @ B)) @ E5) @ (((graph @ A) @ B) @ M2)) & ((((product_fst @ A) @ B) @ E5) != K))))))))).
% 9.43/3.45 thf(fact_2775_insert__relcomp__fold, axiom, (! [C:$tType,B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B)),X:((product_prod @ C) @ A),R5:(set @ ((product_prod @ C) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ S2) => ((((((relcomp @ C) @ A) @ B) @ (((insert2 @ ((product_prod @ C) @ A)) @ X) @ R5)) @ S2) = (((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [W2:A,Z5:B,A10:(set @ ((product_prod @ C) @ B))]: (((((if @ (set @ ((product_prod @ C) @ B))) @ ((((product_snd @ C) @ A) @ X) = W2)) @ (((insert2 @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ (((product_fst @ C) @ A) @ X)) @ Z5)) @ A10)) @ A10))))) @ (((((relcomp @ C) @ A) @ B) @ R5) @ S2)) @ S2)))))).
% 9.43/3.45 thf(fact_2776_zero__rat__def, axiom, ((zero_zero @ rat) = (abs_Rat @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))))).
% 9.43/3.45 thf(fact_2777_plus__rat__def, axiom, ((plus_plus @ rat) = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ (rat > rat)) @ rep_Rat) @ ((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ rat) @ rep_Rat) @ abs_Rat)) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((plus_plus @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ Y3)) @ (((product_snd @ int) @ int) @ X3)))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3)))))))).
% 9.43/3.45 thf(fact_2778_inverse__rat__def, axiom, ((inverse_inverse @ rat) = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ rat) @ rep_Rat) @ abs_Rat) @ (^ [X3:((product_prod @ int) @ int)]: (((((if @ ((product_prod @ int) @ int)) @ ((((product_fst @ int) @ int) @ X3) = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ X3)))))))).
% 9.43/3.45 thf(fact_2779_times__rat__def, axiom, ((times_times @ rat) = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ (((product_prod @ int) @ int) > ((product_prod @ int) @ int))) @ (rat > rat)) @ rep_Rat) @ ((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ rat) @ rep_Rat) @ abs_Rat)) @ (^ [X3:((product_prod @ int) @ int),Y3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ (((times_times @ int) @ (((product_fst @ int) @ int) @ X3)) @ (((product_fst @ int) @ int) @ Y3))) @ (((times_times @ int) @ (((product_snd @ int) @ int) @ X3)) @ (((product_snd @ int) @ int) @ Y3)))))))).
% 9.43/3.45 thf(fact_2780_uminus__rat__def, axiom, ((uminus_uminus @ rat) = (((((((map_fun @ rat) @ ((product_prod @ int) @ int)) @ ((product_prod @ int) @ int)) @ rat) @ rep_Rat) @ abs_Rat) @ (^ [X3:((product_prod @ int) @ int)]: (((((product_Pair @ int) @ int) @ ((uminus_uminus @ int) @ (((product_fst @ int) @ int) @ X3))) @ (((product_snd @ int) @ int) @ X3))))))).
% 9.43/3.45 thf(fact_2781_comp__fun__commute__relcomp__fold, axiom, (! [A:$tType,B:$tType,C:$tType,S2:(set @ ((product_prod @ A) @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ S2) => (((finite6289374366891150609ommute @ ((product_prod @ C) @ A)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ C) @ A) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [X3:C,Y3:A,A9:(set @ ((product_prod @ C) @ B))]: ((((((finite_fold @ ((product_prod @ A) @ B)) @ (set @ ((product_prod @ C) @ B))) @ ((((product_case_prod @ A) @ B) @ ((set @ ((product_prod @ C) @ B)) > (set @ ((product_prod @ C) @ B)))) @ (^ [W2:A,Z5:B,A10:(set @ ((product_prod @ C) @ B))]: (((((if @ (set @ ((product_prod @ C) @ B))) @ (Y3 = W2)) @ (((insert2 @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ X3) @ Z5)) @ A10)) @ A10))))) @ A9) @ S2))))))))).
% 9.43/3.45 thf(fact_2782_restrict__upd__same, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),X:A,Y:B]: ((((((restrict_map @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ X) @ ((some @ B) @ Y))) @ ((uminus_uminus @ (set @ A)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) = ((((restrict_map @ A) @ B) @ M2) @ ((uminus_uminus @ (set @ A)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_2783_restrict__map__UNIV, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B))]: ((((((restrict_map @ A) @ B) @ F) @ (top_top @ (set @ A))) = F)))).
% 9.43/3.45 thf(fact_2784_restrict__out, axiom, (! [A:$tType,B:$tType,X:A,A4:(set @ A),M2:(A > (option @ B))]: (((~ ((((member @ A) @ X) @ A4))) => ((((((restrict_map @ A) @ B) @ M2) @ A4) @ X) = (none @ B)))))).
% 9.43/3.45 thf(fact_2785_restrict__map__empty, axiom, (! [B:$tType,A:$tType,D4:(set @ A)]: ((((((restrict_map @ A) @ B) @ (^ [X3:A]: ((none @ B)))) @ D4) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_2786_restrict__map__to__empty, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((((restrict_map @ A) @ B) @ M2) @ (bot_bot @ (set @ A))) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_2787_fun__upd__None__restrict, axiom, (! [B:$tType,A:$tType,X:A,D4:(set @ A),M2:(A > (option @ B))]: ((((((member @ A) @ X) @ D4) => ((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M2) @ D4)) @ X) @ (none @ B)) = ((((restrict_map @ A) @ B) @ M2) @ (((minus_minus @ (set @ A)) @ D4) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))))) & ((~ ((((member @ A) @ X) @ D4))) => ((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ M2) @ D4)) @ X) @ (none @ B)) = ((((restrict_map @ A) @ B) @ M2) @ D4))))))).
% 9.43/3.45 thf(fact_2788_le__map__restrict, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [M2:(A > (option @ B)),X10:(set @ A)]: ((((ord_less_eq @ (A > (option @ B))) @ ((((restrict_map @ A) @ B) @ M2) @ X10)) @ M2))))))).
% 9.43/3.45 thf(fact_2789_restrict__map__subset__eq, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),R5:(set @ A),M7:(A > (option @ B)),R7:(set @ A)]: (((((((restrict_map @ A) @ B) @ M2) @ R5) = M7) => ((((ord_less_eq @ (set @ A)) @ R7) @ R5) => (((((restrict_map @ A) @ B) @ M2) @ R7) = ((((restrict_map @ A) @ B) @ M7) @ R7))))))).
% 9.43/3.45 thf(fact_2790_restrict__map__eq_I2_J, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),A4:(set @ B),K:B,V:A]: ((((((((restrict_map @ B) @ A) @ M2) @ A4) @ K) = ((some @ A) @ V)) = (((M2 @ K) = ((some @ A) @ V)) & (((member @ B) @ K) @ A4)))))).
% 9.43/3.45 thf(fact_2791_restrict__map__def, axiom, (! [B:$tType,A:$tType]: ((((restrict_map @ A) @ B) = (^ [M:(A > (option @ B)),A9:(set @ A),X3:A]: (((((if @ (option @ B)) @ (((member @ A) @ X3) @ A9)) @ (M @ X3)) @ (none @ B)))))))).
% 9.43/3.45 thf(fact_2792_comp__fun__commute_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [Y4:A,X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ Y4)) @ (F @ X2)) = (((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y4))))) => (((finite6289374366891150609ommute @ A) @ B) @ F))))).
% 9.43/3.45 thf(fact_2793_comp__fun__commute_Ocomp__fun__commute, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),Y:A,X:A]: (((((finite6289374366891150609ommute @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ Y)) @ (F @ X)) = (((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ Y))))))).
% 9.43/3.45 thf(fact_2794_comp__fun__commute__def, axiom, (! [B:$tType,A:$tType]: ((((finite6289374366891150609ommute @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [Y3:A,X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ Y3)) @ (F2 @ X3)) = (((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ Y3))))))))))).
% 9.43/3.45 thf(fact_2795_graph__restrictD_I1_J, axiom, (! [B:$tType,A:$tType,K:A,V:B,M2:(A > (option @ B)),A4:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V)) @ (((graph @ A) @ B) @ ((((restrict_map @ A) @ B) @ M2) @ A4))) => (((member @ A) @ K) @ A4))))).
% 9.43/3.45 thf(fact_2796_map__restrict__insert__none__simp, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),X:B,S:(set @ B)]: ((((M2 @ X) = (none @ A)) => (((((restrict_map @ B) @ A) @ M2) @ ((uminus_uminus @ (set @ B)) @ (((insert2 @ B) @ X) @ S))) = ((((restrict_map @ B) @ A) @ M2) @ ((uminus_uminus @ (set @ B)) @ S))))))).
% 9.43/3.45 thf(fact_2797_restrict__map__upd, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B)),S2:(set @ A),K:A,V:B]: (((((((fun_upd @ A) @ (option @ B)) @ ((((restrict_map @ A) @ B) @ F) @ S2)) @ K) @ ((some @ B) @ V)) = ((((restrict_map @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F) @ K) @ ((some @ B) @ V))) @ (((insert2 @ A) @ K) @ S2)))))).
% 9.43/3.45 thf(fact_2798_graph__restrictD_I2_J, axiom, (! [A:$tType,B:$tType,K:A,V:B,M2:(A > (option @ B)),A4:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V)) @ (((graph @ A) @ B) @ ((((restrict_map @ A) @ B) @ M2) @ A4))) => ((M2 @ K) = ((some @ B) @ V)))))).
% 9.43/3.45 thf(fact_2799_restrict__complement__singleton__eq, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B)),X:A]: ((((((restrict_map @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) = (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ (none @ B)))))).
% 9.43/3.45 thf(fact_2800_map__upd__eq__restrict, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),X:A]: (((((((fun_upd @ A) @ (option @ B)) @ M2) @ X) @ (none @ B)) = ((((restrict_map @ A) @ B) @ M2) @ ((uminus_uminus @ (set @ A)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_2801_comp__fun__commute__product__fold, axiom, (! [A:$tType,B:$tType,B6:(set @ A)]: ((((finite_finite2 @ A) @ B6) => (((finite6289374366891150609ommute @ B) @ (set @ ((product_prod @ B) @ A))) @ (^ [X3:B,Z5:(set @ ((product_prod @ B) @ A))]: ((((((finite_fold @ A) @ (set @ ((product_prod @ B) @ A))) @ (^ [Y3:A]: (((insert2 @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y3))))) @ Z5) @ B6)))))))).
% 9.43/3.45 thf(fact_2802_Sup__fin_Oeq__fold_H, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((lattic5882676163264333800up_fin @ A) = (^ [A9:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X3:A,Y3:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X3) @ ((sup_sup @ A) @ X3)) @ Y3))))) @ (none @ A)) @ A9))))))))).
% 9.43/3.45 thf(fact_2803_ran__map__upd, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),A3:B,B2:A]: ((((M2 @ A3) = (none @ A)) => ((((ran @ B) @ A) @ (((((fun_upd @ B) @ (option @ A)) @ M2) @ A3) @ ((some @ A) @ B2))) = (((insert2 @ A) @ B2) @ (((ran @ B) @ A) @ M2))))))).
% 9.43/3.45 thf(fact_2804_Gcd__fin_Oeq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Gcd_fin @ A) = (^ [A9:(set @ A)]: (((((if @ A) @ ((finite_finite2 @ A) @ A9)) @ (((((finite_fold @ A) @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ A9)) @ (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2805_Max_Oeq__fold_H, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798349783984er_Max @ A) = (^ [A9:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X3:A,Y3:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X3) @ ((ord_max @ A) @ X3)) @ Y3))))) @ (none @ A)) @ A9))))))))).
% 9.43/3.45 thf(fact_2806_Min_Oeq__fold_H, axiom, (! [A:$tType]: (((linorder @ A) => ((lattic643756798350308766er_Min @ A) = (^ [A9:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X3:A,Y3:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X3) @ ((ord_min @ A) @ X3)) @ Y3))))) @ (none @ A)) @ A9))))))))).
% 9.43/3.45 thf(fact_2807_pairself__image__eq, axiom, (! [B:$tType,A:$tType,F:(B > A),P:(B > (B > $o))]: ((((((image2 @ ((product_prod @ B) @ B)) @ ((product_prod @ A) @ A)) @ (((pairself @ B) @ A) @ F)) @ ((collect @ ((product_prod @ B) @ B)) @ ((((product_case_prod @ B) @ B) @ $o) @ P))) = ((collect @ ((product_prod @ A) @ A)) @ (^ [Uu:((product_prod @ A) @ A)]: ((? [A7:B,B5:B]: (((Uu = ((((product_Pair @ A) @ A) @ (F @ A7)) @ (F @ B5))) & ((P @ A7) @ B5))))))))))).
% 9.43/3.45 thf(fact_2808_eq__or__mem__image__simp, axiom, (! [B:$tType,A:$tType,F:(B > A),A3:B,B6:(set @ B)]: ((((collect @ A) @ (^ [Uu:A]: ((? [L3:B]: (((Uu = (F @ L3)) & ((L3 = A3) | (((member @ B) @ L3) @ B6)))))))) = (((insert2 @ A) @ (F @ A3)) @ ((collect @ A) @ (^ [Uu:A]: ((? [L3:B]: (((Uu = (F @ L3)) & (((member @ B) @ L3) @ B6)))))))))))).
% 9.43/3.45 thf(fact_2809_Gcd__fin_Oinfinite, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (((semiring_gcd_Gcd_fin @ A) @ A4) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2810_is__unit__Gcd__fin__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A4:(set @ A)]: (((((dvd_dvd @ A) @ ((semiring_gcd_Gcd_fin @ A) @ A4)) @ (one_one @ A)) = (((semiring_gcd_Gcd_fin @ A) @ A4) = (one_one @ A))))))))).
% 9.43/3.45 thf(fact_2811_Eps__Opt__eq__None, axiom, (! [A:$tType,P:(A > $o)]: (((((eps_Opt @ A) @ P) = (none @ A)) = (~ ((? [X8:A]: ((P @ X8))))))))).
% 9.43/3.45 thf(fact_2812_ran__empty, axiom, (! [B:$tType,A:$tType]: (((((ran @ B) @ A) @ (^ [X3:B]: ((none @ A)))) = (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_2813_map__update__eta__repair_I2_J, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),K:A,V:B]: ((((M2 @ K) = (none @ B)) => ((((ran @ A) @ B) @ (^ [X3:A]: (((((if @ (option @ B)) @ (X3 = K)) @ ((some @ B) @ V)) @ (M2 @ X3))))) = (((insert2 @ B) @ V) @ (((ran @ A) @ B) @ M2))))))).
% 9.43/3.45 thf(fact_2814_ran__def, axiom, (! [B:$tType,A:$tType]: ((((ran @ A) @ B) = (^ [M:(A > (option @ B))]: (((collect @ B) @ (^ [B5:B]: ((? [A7:A]: (((M @ A7) = ((some @ B) @ B5))))))))))))).
% 9.43/3.45 thf(fact_2815_ranI, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),A3:B,B2:A]: ((((M2 @ A3) = ((some @ A) @ B2)) => (((member @ A) @ B2) @ (((ran @ B) @ A) @ M2)))))).
% 9.43/3.45 thf(fact_2816_fs__contract, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > (B > C)),S2:(set @ C)]: ((((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [P6:((product_prod @ A) @ B)]: (((Uu = P6) & (((member @ C) @ ((F @ (((product_fst @ A) @ B) @ P6)) @ (((product_snd @ A) @ B) @ P6))) @ S2)))))))) = ((collect @ A) @ (^ [A7:A]: ((? [B5:B]: ((((member @ C) @ ((F @ A7) @ B5)) @ S2)))))))))).
% 9.43/3.45 thf(fact_2817_ran__restrictD, axiom, (! [B:$tType,A:$tType,Y:A,M2:(B > (option @ A)),A4:(set @ B)]: (((((member @ A) @ Y) @ (((ran @ B) @ A) @ ((((restrict_map @ B) @ A) @ M2) @ A4))) => (? [X2:B]: (((((member @ B) @ X2) @ A4) & ((M2 @ X2) = ((some @ A) @ Y))))))))).
% 9.43/3.45 thf(fact_2818_Un__interval, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [B1:A,B22:A,B32:A,F:(A > B)]: (((((ord_less_eq @ A) @ B1) @ B22) => ((((ord_less_eq @ A) @ B22) @ B32) => ((((sup_sup @ (set @ B)) @ ((collect @ B) @ (^ [Uu:B]: ((? [I3:A]: (((Uu = (F @ I3)) & ((((ord_less_eq @ A) @ B1) @ I3) & (((ord_less @ A) @ I3) @ B22))))))))) @ ((collect @ B) @ (^ [Uu:B]: ((? [I3:A]: (((Uu = (F @ I3)) & ((((ord_less_eq @ A) @ B22) @ I3) & (((ord_less @ A) @ I3) @ B32))))))))) = ((collect @ B) @ (^ [Uu:B]: ((? [I3:A]: (((Uu = (F @ I3)) & ((((ord_less_eq @ A) @ B1) @ I3) & (((ord_less @ A) @ I3) @ B32))))))))))))))))).
% 9.43/3.45 thf(fact_2819_fold__atLeastAtMost__nat, axiom, (! [A:$tType,F:(nat > (A > A)),A3:nat,B2:nat,Acc2:A]: (((((finite6289374366891150609ommute @ nat) @ A) @ F) => ((((((set_fo6178422350223883121st_nat @ A) @ F) @ A3) @ B2) @ Acc2) = (((((finite_fold @ nat) @ A) @ F) @ Acc2) @ (((set_or1337092689740270186AtMost @ nat) @ A3) @ B2))))))).
% 9.43/3.45 thf(fact_2820_Min_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (((lattic643756798350308766er_Min @ A) @ A4) = ((the2 @ A) @ (none @ A)))))))))).
% 9.43/3.45 thf(fact_2821_relcomp__unfold, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((relcomp @ A) @ C) @ B) = (^ [R3:(set @ ((product_prod @ A) @ C)),S4:(set @ ((product_prod @ C) @ B))]: (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X3:A,Z5:B]: ((? [Y3:C]: (((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X3) @ Y3)) @ R3) & (((member @ ((product_prod @ C) @ B)) @ ((((product_Pair @ C) @ B) @ Y3) @ Z5)) @ S4)))))))))))))).
% 9.43/3.45 thf(fact_2822_Max_Oinfinite, axiom, (! [A:$tType]: (((linorder @ A) => (! [A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (((lattic643756798349783984er_Max @ A) @ A4) = ((the2 @ A) @ (none @ A)))))))))).
% 9.43/3.45 thf(fact_2823_graph__def, axiom, (! [B:$tType,A:$tType]: ((((graph @ A) @ B) = (^ [M:(A > (option @ B))]: (((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [A7:A,B5:B]: (((Uu = ((((product_Pair @ A) @ B) @ A7) @ B5)) & ((M @ A7) = ((some @ B) @ B5)))))))))))))).
% 9.43/3.45 thf(fact_2824_Sup__fin_Oinfinite, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (((lattic5882676163264333800up_fin @ A) @ A4) = ((the2 @ A) @ (none @ A)))))))))).
% 9.43/3.45 thf(fact_2825_Min__add__commute, axiom, (! [B:$tType,A:$tType]: (((linord4140545234300271783up_add @ A) => (! [S2:(set @ B),F:(B > A),K:A]: ((((finite_finite2 @ B) @ S2) => ((S2 != (bot_bot @ (set @ B))) => (((lattic643756798350308766er_Min @ A) @ ((((image2 @ B) @ A) @ (^ [X3:B]: ((((plus_plus @ A) @ (F @ X3)) @ K)))) @ S2)) = (((plus_plus @ A) @ ((lattic643756798350308766er_Min @ A) @ ((((image2 @ B) @ A) @ F) @ S2))) @ K)))))))))).
% 9.43/3.45 thf(fact_2826_card__le__Suc__Max, axiom, (! [S2:(set @ nat)]: ((((finite_finite2 @ nat) @ S2) => (((ord_less_eq @ nat) @ ((finite_card @ nat) @ S2)) @ (suc @ ((lattic643756798349783984er_Max @ nat) @ S2))))))).
% 9.43/3.45 thf(fact_2827_Max__add__commute, axiom, (! [B:$tType,A:$tType]: (((linord4140545234300271783up_add @ A) => (! [S2:(set @ B),F:(B > A),K:A]: ((((finite_finite2 @ B) @ S2) => ((S2 != (bot_bot @ (set @ B))) => (((lattic643756798349783984er_Max @ A) @ ((((image2 @ B) @ A) @ (^ [X3:B]: ((((plus_plus @ A) @ (F @ X3)) @ K)))) @ S2)) = (((plus_plus @ A) @ ((lattic643756798349783984er_Max @ A) @ ((((image2 @ B) @ A) @ F) @ S2))) @ K)))))))))).
% 9.43/3.45 thf(fact_2828_set__to__map__ran, axiom, (! [A:$tType,B:$tType,S2:(set @ ((product_prod @ B) @ A))]: ((((ord_less_eq @ (set @ A)) @ (((ran @ B) @ A) @ (((set_to_map @ B) @ A) @ S2))) @ ((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ S2))))).
% 9.43/3.45 thf(fact_2829_map__to__set__upd, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),K:A,V:B]: (((((map_to_set @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ K) @ ((some @ B) @ V))) = (((insert2 @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V)) @ (((minus_minus @ (set @ ((product_prod @ A) @ B))) @ (((map_to_set @ A) @ B) @ M2)) @ ((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [V5:B]: ((Uu = ((((product_Pair @ A) @ B) @ K) @ V5))))))))))))).
% 9.43/3.45 thf(fact_2830_brk__rel__def, axiom, (! [B:$tType,A:$tType]: ((((brk_rel @ A) @ B) = (^ [R6:(set @ ((product_prod @ A) @ B))]: ((((sup_sup @ (set @ ((product_prod @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B)))) @ ((collect @ ((product_prod @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B))) @ (^ [Uu:((product_prod @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B))]: ((? [X3:A,Y3:B]: (((Uu = ((((product_Pair @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B)) @ ((((product_Pair @ $o) @ A) @ $false) @ X3)) @ ((((product_Pair @ $o) @ B) @ $false) @ Y3))) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R6)))))))) @ ((collect @ ((product_prod @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B))) @ (^ [Uu:((product_prod @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B))]: ((? [X3:A,Y3:B]: ((Uu = ((((product_Pair @ ((product_prod @ $o) @ A)) @ ((product_prod @ $o) @ B)) @ ((((product_Pair @ $o) @ A) @ $true) @ X3)) @ ((((product_Pair @ $o) @ B) @ $false) @ Y3))))))))))))))).
% 9.43/3.45 thf(fact_2831_image2__def, axiom, (! [A:$tType,B:$tType,C:$tType]: (((((bNF_Greatest_image2 @ C) @ A) @ B) = (^ [A9:(set @ C),F2:(C > A),G2:(C > B)]: (((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [A7:C]: (((Uu = ((((product_Pair @ A) @ B) @ (F2 @ A7)) @ (G2 @ A7))) & (((member @ C) @ A7) @ A9))))))))))))).
% 9.43/3.45 thf(fact_2832_greaterThan__Suc, axiom, (! [K:nat]: ((((set_ord_greaterThan @ nat) @ (suc @ K)) = (((minus_minus @ (set @ nat)) @ ((set_ord_greaterThan @ nat) @ K)) @ (((insert2 @ nat) @ (suc @ K)) @ (bot_bot @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_2833_refl__on__singleton, axiom, (! [A:$tType,X:A]: ((((refl_on @ A) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.45 thf(fact_2834_time__refines, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),C6:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: (((((refine_Imp_refines @ A) @ C2) @ C6) => ((~ ((((time_fails @ A) @ C6) @ H3))) => (((ord_less_eq @ nat) @ (((time_time @ A) @ C2) @ H3)) @ (((time_time @ A) @ C6) @ H3))))))).
% 9.43/3.45 thf(fact_2835_greaterThan__eq__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((set_ord_greaterThan @ A) @ X) = ((set_ord_greaterThan @ A) @ Y)) = (X = Y)))))))).
% 9.43/3.45 thf(fact_2836_greaterThan__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_greaterThan @ A) @ K)) = (((ord_less @ A) @ K) @ I)))))))).
% 9.43/3.45 thf(fact_2837_Inf__greaterThan, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A]: ((((complete_Inf_Inf @ A) @ ((set_ord_greaterThan @ A) @ X)) = X))))))).
% 9.43/3.45 thf(fact_2838_greaterThan__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ X)) @ ((set_ord_greaterThan @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 9.43/3.45 thf(fact_2839_Compl__greaterThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_greaterThan @ A) @ K)) = ((set_ord_atMost @ A) @ K)))))))).
% 9.43/3.45 thf(fact_2840_Compl__atMost, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_atMost @ A) @ K)) = ((set_ord_greaterThan @ A) @ K)))))))).
% 9.43/3.45 thf(fact_2841_map__to__set__empty, axiom, (! [B:$tType,A:$tType]: (((((map_to_set @ A) @ B) @ (^ [X3:A]: ((none @ B)))) = (bot_bot @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2842_Sup__greaterThanAtLeast, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: (((((ord_less @ A) @ X) @ (top_top @ A)) => (((complete_Sup_Sup @ A) @ ((set_ord_greaterThan @ A) @ X)) = (top_top @ A))))))))).
% 9.43/3.45 thf(fact_2843_image__uminus__greaterThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_greaterThan @ A) @ X)) = ((set_ord_lessThan @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_2844_image__uminus__lessThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_lessThan @ A) @ X)) = ((set_ord_greaterThan @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_2845_refl__onD2, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (((member @ A) @ Y) @ A4)))))).
% 9.43/3.45 thf(fact_2846_refl__onD1, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (((member @ A) @ X) @ A4)))))).
% 9.43/3.45 thf(fact_2847_refl__onD, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ A) @ A3) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ R)))))).
% 9.43/3.45 thf(fact_2848_greaterThan__non__empty, axiom, (! [A:$tType]: (((no_top @ A) => (! [X:A]: ((((set_ord_greaterThan @ A) @ X) != (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_2849_infinite__Ioi, axiom, (! [A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [A3:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_greaterThan @ A) @ A3)))))))))).
% 9.43/3.45 thf(fact_2850_greaterThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_greaterThan @ A) = (^ [L3:A]: (((collect @ A) @ ((ord_less @ A) @ L3))))))))).
% 9.43/3.45 thf(fact_2851_fails__refines, axiom, (! [A:$tType,C2:(heap_Time_Heap @ A),C6:(heap_Time_Heap @ A),H3:(heap_ext @ product_unit)]: (((((refine_Imp_refines @ A) @ C2) @ C6) => ((((time_fails @ A) @ C2) @ H3) => (((time_fails @ A) @ C6) @ H3)))))).
% 9.43/3.45 thf(fact_2852_map__to__set__inverse, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: (((((set_to_map @ A) @ B) @ (((map_to_set @ A) @ B) @ M2)) = M2)))).
% 9.43/3.45 thf(fact_2853_map__to__set__empty__iff_I1_J, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((((map_to_set @ A) @ B) @ M2) = (bot_bot @ (set @ ((product_prod @ A) @ B)))) = (M2 = (^ [X3:A]: ((none @ B)))))))).
% 9.43/3.45 thf(fact_2854_map__to__set__empty__iff_I2_J, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((bot_bot @ (set @ ((product_prod @ A) @ B))) = (((map_to_set @ A) @ B) @ M2)) = (M2 = (^ [X3:A]: ((none @ B)))))))).
% 9.43/3.45 thf(fact_2855_image2__eqI, axiom, (! [A:$tType,C:$tType,B:$tType,B2:A,F:(B > A),X:B,C2:C,G:(B > C),A4:(set @ B)]: (((B2 = (F @ X)) => ((C2 = (G @ X)) => ((((member @ B) @ X) @ A4) => (((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ B2) @ C2)) @ ((((((bNF_Greatest_image2 @ B) @ A) @ C) @ A4) @ F) @ G)))))))).
% 9.43/3.45 thf(fact_2856_refl__on__def, axiom, (! [A:$tType]: (((refl_on @ A) = (^ [A9:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R3) @ ((((product_Sigma @ A) @ A) @ A9) @ (^ [Uu:A]: (A9)))) & (! [X3:A]: (((((member @ A) @ X3) @ A9) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ R3))))))))))).
% 9.43/3.45 thf(fact_2857_refl__onI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4)))) => ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ R)))) => (((refl_on @ A) @ A4) @ R)))))).
% 9.43/3.45 thf(fact_2858_map__to__set__def, axiom, (! [B:$tType,A:$tType]: ((((map_to_set @ A) @ B) = (^ [M:(A > (option @ B))]: (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [K4:A,V2:B]: (((M @ K4) = ((some @ B) @ V2)))))))))))).
% 9.43/3.45 thf(fact_2859_INT__greaterThan__UNIV, axiom, (((complete_Inf_Inf @ (set @ nat)) @ ((((image2 @ nat) @ (set @ nat)) @ (set_ord_greaterThan @ nat)) @ (top_top @ (set @ nat)))) = (bot_bot @ (set @ nat)))).
% 9.43/3.45 thf(fact_2860_map__to__set__ran, axiom, (! [A:$tType,B:$tType]: ((((ran @ B) @ A) = (^ [M:(B > (option @ A))]: (((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ (((map_to_set @ B) @ A) @ M)))))))).
% 9.43/3.45 thf(fact_2861_greaterThan__0, axiom, (((set_ord_greaterThan @ nat) @ (zero_zero @ nat)) = ((((image2 @ nat) @ nat) @ suc) @ (top_top @ (set @ nat))))).
% 9.43/3.45 thf(fact_2862_refines__def, axiom, (! [A:$tType]: (((refine_Imp_refines @ A) = (^ [P6:(heap_Time_Heap @ A),Q5:(heap_Time_Heap @ A)]: ((! [H:(heap_ext @ product_unit)]: ((((((case_option @ $o) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ $true) @ ((((product_case_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ $o) @ (^ [R3:A]: (((((product_case_prod @ (heap_ext @ product_unit)) @ nat) @ $o) @ (^ [H7:(heap_ext @ product_unit),T3:nat]: (((((heap_Time_execute @ A) @ P6) @ H) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ T3))))))))))) @ (((heap_Time_execute @ A) @ Q5) @ H)))))))))).
% 9.43/3.45 thf(fact_2863_refines__empty, axiom, (! [A:$tType,M2:(heap_Time_Heap @ A)]: ((((refine_Imp_refines @ A) @ M2) @ ((heap_Time_Heap2 @ A) @ (^ [X3:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 9.43/3.45 thf(fact_2864_refines__bind, axiom, (! [B:$tType,A:$tType,M2:(heap_Time_Heap @ A),M7:(heap_Time_Heap @ A),F:(A > (heap_Time_Heap @ B)),F3:(A > (heap_Time_Heap @ B))]: (((((refine_Imp_refines @ A) @ M2) @ M7) => ((! [X2:A]: ((((refine_Imp_refines @ B) @ (F @ X2)) @ (F3 @ X2)))) => (((refine_Imp_refines @ B) @ ((((heap_Time_bind @ A) @ B) @ M2) @ F)) @ ((((heap_Time_bind @ A) @ B) @ M7) @ F3))))))).
% 9.43/3.45 thf(fact_2865_refl__on__domain, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((refl_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((((member @ A) @ A3) @ A4) & (((member @ A) @ B2) @ A4))))))).
% 9.43/3.45 thf(fact_2866_refines__option, axiom, (! [B:$tType,A:$tType,A3:(option @ A),A6:(option @ A),M1:(heap_Time_Heap @ B),M12:(heap_Time_Heap @ B),M22:(A > (heap_Time_Heap @ B)),M23:(A > (heap_Time_Heap @ B))]: (((A3 = A6) => ((((refine_Imp_refines @ B) @ M1) @ M12) => ((! [X2:A]: ((((refine_Imp_refines @ B) @ (M22 @ X2)) @ (M23 @ X2)))) => (((refine_Imp_refines @ B) @ (((((case_option @ (heap_Time_Heap @ B)) @ A) @ M1) @ M22) @ A3)) @ (((((case_option @ (heap_Time_Heap @ B)) @ A) @ M12) @ M23) @ A6)))))))).
% 9.43/3.45 thf(fact_2867_linear__order__on__singleton, axiom, (! [A:$tType,X:A]: ((((order_679001287576687338der_on @ A) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.45 thf(fact_2868_Total__subset__Id, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((total_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ (id2 @ A)) => ((R = (bot_bot @ (set @ ((product_prod @ A) @ A)))) | (? [A5:A]: ((R = (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))))))))).
% 9.43/3.45 thf(fact_2869_relation__of__def, axiom, (! [A:$tType]: (((order_relation_of @ A) = (^ [P4:(A > (A > $o)),A9:(set @ A)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [A7:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ ((((product_Sigma @ A) @ A) @ A9) @ (^ [Uu:A]: (A9)))) & ((P4 @ A7) @ B5)))))))))))).
% 9.43/3.45 thf(fact_2870_euclidean__size__times__nonunit, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((A3 != (zero_zero @ A)) => ((B2 != (zero_zero @ A)) => ((~ ((((dvd_dvd @ A) @ A3) @ (one_one @ A)))) => (((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ B2)) @ ((euclid6346220572633701492n_size @ A) @ (((times_times @ A) @ A3) @ B2)))))))))))).
% 9.43/3.45 thf(fact_2871_surj__swap, axiom, (! [B:$tType,A:$tType]: ((((((image2 @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ (top_top @ (set @ ((product_prod @ B) @ A)))) = (top_top @ (set @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_2872_euclidean__size__integer_Orep__eq, axiom, ((euclid6346220572633701492n_size @ code_integer) = (^ [X3:code_integer]: (((euclid6346220572633701492n_size @ int) @ (code_int_of_integer @ X3)))))).
% 9.43/3.45 thf(fact_2873_swap__swap, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B)]: (((((product_swap @ B) @ A) @ (((product_swap @ A) @ B) @ P5)) = P5)))).
% 9.43/3.45 thf(fact_2874_pair__in__Id__conv, axiom, (! [A:$tType,A3:A,B2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (id2 @ A)) = (A3 = B2))))).
% 9.43/3.45 thf(fact_2875_IdI, axiom, (! [A:$tType,A3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ (id2 @ A))))).
% 9.43/3.45 thf(fact_2876_swap__simp, axiom, (! [A:$tType,B:$tType,X:B,Y:A]: (((((product_swap @ B) @ A) @ ((((product_Pair @ B) @ A) @ X) @ Y)) = ((((product_Pair @ A) @ B) @ Y) @ X))))).
% 9.43/3.45 thf(fact_2877_case__swap, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > (B > A)),P5:((product_prod @ C) @ B)]: (((((((product_case_prod @ B) @ C) @ A) @ (^ [Y3:B,X3:C]: (((F @ X3) @ Y3)))) @ (((product_swap @ C) @ B) @ P5)) = (((((product_case_prod @ C) @ B) @ A) @ F) @ P5))))).
% 9.43/3.45 thf(fact_2878_swap__comp__swap, axiom, (! [B:$tType,A:$tType]: (((((((comp @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((product_swap @ A) @ B)) = (id @ ((product_prod @ A) @ B)))))).
% 9.43/3.45 thf(fact_2879_bijective__Id, axiom, (! [A:$tType]: ((((bijective @ A) @ A) @ (id2 @ A))))).
% 9.43/3.45 thf(fact_2880_euclidean__size__1, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (((euclid6346220572633701492n_size @ A) @ (one_one @ A)) = (one_one @ nat)))))).
% 9.43/3.45 thf(fact_2881_fst__swap, axiom, (! [A:$tType,B:$tType,X:((product_prod @ B) @ A)]: (((((product_fst @ A) @ B) @ (((product_swap @ B) @ A) @ X)) = (((product_snd @ B) @ A) @ X))))).
% 9.43/3.45 thf(fact_2882_snd__swap, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B)]: (((((product_snd @ B) @ A) @ (((product_swap @ A) @ B) @ X)) = (((product_fst @ A) @ B) @ X))))).
% 9.43/3.45 thf(fact_2883_pair__in__swap__image, axiom, (! [A:$tType,B:$tType,Y:A,X:B,A4:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ X)) @ ((((image2 @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ A4)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X) @ Y)) @ A4))))).
% 9.43/3.45 thf(fact_2884_euclidean__size__integer_Oabs__eq, axiom, (! [X:int]: ((((euclid6346220572633701492n_size @ code_integer) @ (code_integer_of_int @ X)) = ((euclid6346220572633701492n_size @ int) @ X))))).
% 9.43/3.45 thf(fact_2885_BNF__Greatest__Fixpoint_OIdD, axiom, (! [A:$tType,A3:A,B2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (id2 @ A)) => (A3 = B2))))).
% 9.43/3.45 thf(fact_2886_IdE, axiom, (! [A:$tType,P5:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ P5) @ (id2 @ A)) => (~ ((! [X2:A]: ((P5 != ((((product_Pair @ A) @ A) @ X2) @ X2)))))))))).
% 9.43/3.45 thf(fact_2887_euclidean__size__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (euclid6346220572633701492n_size @ nat)) @ (euclid6346220572633701492n_size @ nat))).
% 9.43/3.45 thf(fact_2888_Linear__order__in__diff__Id, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) = (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R) @ (id2 @ A)))))))))))).
% 9.43/3.45 thf(fact_2889_euclidean__size__unit, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((euclid6346220572633701492n_size @ A) @ A3) = ((euclid6346220572633701492n_size @ A) @ (one_one @ A)))))))))).
% 9.43/3.45 thf(fact_2890_euclidean__size__integer__def, axiom, ((euclid6346220572633701492n_size @ code_integer) = (((((((map_fun @ code_integer) @ int) @ nat) @ nat) @ code_int_of_integer) @ (id @ nat)) @ (euclid6346220572633701492n_size @ int)))).
% 9.43/3.45 thf(fact_2891_euclidean__size__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ nat) @ nat) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (euclid6346220572633701492n_size @ int)) @ (euclid6346220572633701492n_size @ int))).
% 9.43/3.45 thf(fact_2892_Id__def, axiom, (! [A:$tType]: (((id2 @ A) = ((collect @ ((product_prod @ A) @ A)) @ (^ [P6:((product_prod @ A) @ A)]: ((? [X3:A]: ((P6 = ((((product_Pair @ A) @ A) @ X3) @ X3))))))))))).
% 9.43/3.45 thf(fact_2893_unit__iff__euclidean__size, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) = ((((euclid6346220572633701492n_size @ A) @ A3) = ((euclid6346220572633701492n_size @ A) @ (one_one @ A))) & (A3 != (zero_zero @ A)))))))))).
% 9.43/3.45 thf(fact_2894_euclidean__size__times__unit, axiom, (! [A:$tType]: (((euclid3725896446679973847miring @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((euclid6346220572633701492n_size @ A) @ (((times_times @ A) @ A3) @ B2)) = ((euclid6346220572633701492n_size @ A) @ B2))))))))).
% 9.43/3.45 thf(fact_2895_product__swap, axiom, (! [B:$tType,A:$tType,A4:(set @ B),B6:(set @ A)]: ((((((image2 @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((product_swap @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (A4))))))).
% 9.43/3.45 thf(fact_2896_prod_Oswap__def, axiom, (! [B:$tType,A:$tType]: ((((product_swap @ A) @ B) = (^ [P6:((product_prod @ A) @ B)]: (((((product_Pair @ B) @ A) @ (((product_snd @ A) @ B) @ P6)) @ (((product_fst @ A) @ B) @ P6)))))))).
% 9.43/3.45 thf(fact_2897_reflcl__set__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((sup_sup @ (A > (A > $o))) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) = (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R) @ (id2 @ A))))))))).
% 9.43/3.45 thf(fact_2898_divmod__cases, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,A3:A]: ((((B2 != (zero_zero @ A)) => (((((modulo_modulo @ A) @ A3) @ B2) = (zero_zero @ A)) => (A3 != (((times_times @ A) @ (((divide_divide @ A) @ A3) @ B2)) @ B2)))) => (((B2 != (zero_zero @ A)) => (! [Q2:A,R2:A]: (((((euclid7384307370059645450egment @ A) @ R2) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R2)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => ((R2 != (zero_zero @ A)) => (((((divide_divide @ A) @ A3) @ B2) = Q2) => (((((modulo_modulo @ A) @ A3) @ B2) = R2) => (A3 != (((plus_plus @ A) @ (((times_times @ A) @ Q2) @ B2)) @ R2)))))))))) => (B2 = (zero_zero @ A)))))))))).
% 9.43/3.45 thf(fact_2899_mod__eqI, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,R:A,Q3:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((euclid7384307370059645450egment @ A) @ R) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => (((((plus_plus @ A) @ (((times_times @ A) @ Q3) @ B2)) @ R) = A3) => ((((modulo_modulo @ A) @ A3) @ B2) = R))))))))))).
% 9.43/3.45 thf(fact_2900_div__bounded, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,R:A,Q3:A]: (((B2 != (zero_zero @ A)) => ((((euclid7384307370059645450egment @ A) @ R) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => ((((divide_divide @ A) @ (((plus_plus @ A) @ (((times_times @ A) @ Q3) @ B2)) @ R)) @ B2) = Q3)))))))))).
% 9.43/3.45 thf(fact_2901_div__eqI, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [B2:A,R:A,Q3:A,A3:A]: (((B2 != (zero_zero @ A)) => ((((euclid7384307370059645450egment @ A) @ R) = ((euclid7384307370059645450egment @ A) @ B2)) => ((((ord_less @ nat) @ ((euclid6346220572633701492n_size @ A) @ R)) @ ((euclid6346220572633701492n_size @ A) @ B2)) => (((((plus_plus @ A) @ (((times_times @ A) @ Q3) @ B2)) @ R) = A3) => ((((divide_divide @ A) @ A3) @ B2) = Q3))))))))))).
% 9.43/3.45 thf(fact_2902_division__segment__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_int_of_integer @ ((euclid7384307370059645450egment @ code_integer) @ X)) = ((euclid7384307370059645450egment @ int) @ (code_int_of_integer @ X)))))).
% 9.43/3.45 thf(fact_2903_division__segment__1, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (((euclid7384307370059645450egment @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.45 thf(fact_2904_division__segment__numeral, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [K:num]: ((((euclid7384307370059645450egment @ A) @ ((numeral_numeral @ A) @ K)) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2905_division__segment__of__nat, axiom, (! [A:$tType]: (((euclid5411537665997757685th_nat @ A) => (! [N:nat]: ((((euclid7384307370059645450egment @ A) @ ((semiring_1_of_nat @ A) @ N)) = (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2906_division__segment__integer_Oabs__eq, axiom, (! [X:int]: ((((euclid7384307370059645450egment @ code_integer) @ (code_integer_of_int @ X)) = (code_integer_of_int @ ((euclid7384307370059645450egment @ int) @ X)))))).
% 9.43/3.45 thf(fact_2907_division__segment__nat__def, axiom, ((euclid7384307370059645450egment @ nat) = (^ [N5:nat]: ((one_one @ nat))))).
% 9.43/3.45 thf(fact_2908_division__segment__integer_Orsp, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (euclid7384307370059645450egment @ int)) @ (euclid7384307370059645450egment @ int))).
% 9.43/3.45 thf(fact_2909_division__segment__natural_Orsp, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (euclid7384307370059645450egment @ nat)) @ (euclid7384307370059645450egment @ nat))).
% 9.43/3.45 thf(fact_2910_division__segment__integer__def, axiom, ((euclid7384307370059645450egment @ code_integer) = (((((((map_fun @ code_integer) @ int) @ int) @ code_integer) @ code_int_of_integer) @ code_integer_of_int) @ (euclid7384307370059645450egment @ int)))).
% 9.43/3.45 thf(fact_2911_is__unit__division__segment, axiom, (! [A:$tType]: (((euclid3128863361964157862miring @ A) => (! [A3:A]: ((((dvd_dvd @ A) @ ((euclid7384307370059645450egment @ A) @ A3)) @ (one_one @ A)))))))).
% 9.43/3.45 thf(fact_2912_Linear__order__wf__diff__Id, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((wf @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R) @ (id2 @ A))) = (! [A9:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A9) @ ((field2 @ A) @ R)) => ((A9 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ A9) & (! [Y3:A]: (((((member @ A) @ Y3) @ A9) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))))))))))))))).
% 9.43/3.45 thf(fact_2913_bsqr__def, axiom, (! [A:$tType]: (((bNF_Wellorder_bsqr @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))) @ ((((product_case_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)) @ $o) @ ((((product_case_prod @ A) @ A) @ (((product_prod @ A) @ A) > $o)) @ (^ [A12:A,A23:A]: (((((product_case_prod @ A) @ A) @ $o) @ (^ [B12:A,B23:A]: (((((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ A12) @ (((insert2 @ A) @ A23) @ (((insert2 @ A) @ B12) @ (((insert2 @ A) @ B23) @ (bot_bot @ (set @ A))))))) @ ((field2 @ A) @ R3)) & (((A12 = B12) & (A23 = B23)) | ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R3) @ A12) @ A23)) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R3) @ B12) @ B23))) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R3) @ (id2 @ A))) | (((((((bNF_We1388413361240627857o_max2 @ A) @ R3) @ A12) @ A23) = ((((bNF_We1388413361240627857o_max2 @ A) @ R3) @ B12) @ B23)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A12) @ B12)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R3) @ (id2 @ A)))) | ((((((bNF_We1388413361240627857o_max2 @ A) @ R3) @ A12) @ A23) = ((((bNF_We1388413361240627857o_max2 @ A) @ R3) @ B12) @ B23)) & ((A12 = B12) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A23) @ B23)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R3) @ (id2 @ A))))))))))))))))))))))).
% 9.43/3.45 thf(fact_2914_wo__rel_Ocases__Total3, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A,Phi:(A > (A > $o))]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ A3) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((field2 @ A) @ R)) => ((((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R) @ (id2 @ A))) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R) @ (id2 @ A)))) => ((Phi @ A3) @ B2)) => (((A3 = B2) => ((Phi @ A3) @ B2)) => ((Phi @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2915_ivl__disj__un__two_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M2)) @ (((set_or1337092689740270186AtMost @ A) @ M2) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2916_ivl__disj__un__singleton_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ (((insert2 @ A) @ U) @ (bot_bot @ (set @ A)))) = (((set_or3652927894154168847AtMost @ A) @ L) @ U))))))))).
% 9.43/3.45 thf(fact_2917_finite__greaterThanAtMost, axiom, (! [L:nat,U:nat]: (((finite_finite2 @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ L) @ U))))).
% 9.43/3.45 thf(fact_2918_finite__greaterThanAtMost__int, axiom, (! [L:int,U:int]: (((finite_finite2 @ int) @ (((set_or3652927894154168847AtMost @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2919_finite__greaterThanAtMost__integer, axiom, (! [L:code_integer,U:code_integer]: (((finite_finite2 @ code_integer) @ (((set_or3652927894154168847AtMost @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2920_card__greaterThanAtMost, axiom, (! [L:nat,U:nat]: ((((finite_card @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ L) @ U)) = (((minus_minus @ nat) @ U) @ L))))).
% 9.43/3.45 thf(fact_2921_greaterThanAtMost__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,L:A,U:A]: (((((member @ A) @ I) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = ((((ord_less @ A) @ L) @ I) & (((ord_less_eq @ A) @ I) @ U))))))))).
% 9.43/3.45 thf(fact_2922_greaterThanAtMost__empty, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,K:A]: (((((ord_less_eq @ A) @ L) @ K) => ((((set_or3652927894154168847AtMost @ A) @ K) @ L) = (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_2923_greaterThanAtMost__empty__iff2, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A,L:A]: ((((bot_bot @ (set @ A)) = (((set_or3652927894154168847AtMost @ A) @ K) @ L)) = (~ ((((ord_less @ A) @ K) @ L)))))))))).
% 9.43/3.45 thf(fact_2924_greaterThanAtMost__empty__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [K:A,L:A]: ((((((set_or3652927894154168847AtMost @ A) @ K) @ L) = (bot_bot @ (set @ A))) = (~ ((((ord_less @ A) @ K) @ L)))))))))).
% 9.43/3.45 thf(fact_2925_infinite__Ioc__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((~ (((finite_finite2 @ A) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)))) = (((ord_less @ A) @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2926_image__add__greaterThanAtMost, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [C2:A,A3:A,B2:A]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ C2)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) = (((set_or3652927894154168847AtMost @ A) @ (((plus_plus @ A) @ C2) @ A3)) @ (((plus_plus @ A) @ C2) @ B2))))))))).
% 9.43/3.45 thf(fact_2927_Sup__greaterThanAtMost, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((complete_Sup_Sup @ A) @ (((set_or3652927894154168847AtMost @ A) @ X) @ Y)) = Y)))))))).
% 9.43/3.45 thf(fact_2928_Inf__greaterThanAtMost, axiom, (! [A:$tType]: ((((comple6319245703460814977attice @ A) & (dense_linorder @ A)) => (! [X:A,Y:A]: (((((ord_less @ A) @ X) @ Y) => (((complete_Inf_Inf @ A) @ (((set_or3652927894154168847AtMost @ A) @ X) @ Y)) = X)))))))).
% 9.43/3.45 thf(fact_2929_brk__rel__wf, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R5) => ((wf @ ((product_prod @ $o) @ A)) @ (((brk_rel @ A) @ A) @ R5)))))).
% 9.43/3.45 thf(fact_2930_card__greaterThanAtMost__int, axiom, (! [L:int,U:int]: ((((finite_card @ int) @ (((set_or3652927894154168847AtMost @ int) @ L) @ U)) = (nat2 @ (((minus_minus @ int) @ U) @ L)))))).
% 9.43/3.45 thf(fact_2931_image__minus__const__greaterThanAtMost, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A,B2:A]: ((((((image2 @ A) @ A) @ ((minus_minus @ A) @ C2)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) = (((set_or7035219750837199246ssThan @ A) @ (((minus_minus @ A) @ C2) @ B2)) @ (((minus_minus @ A) @ C2) @ A3))))))))).
% 9.43/3.45 thf(fact_2932_image__diff__atLeastLessThan, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A,B2:A]: ((((((image2 @ A) @ A) @ ((minus_minus @ A) @ C2)) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) = (((set_or3652927894154168847AtMost @ A) @ (((minus_minus @ A) @ C2) @ B2)) @ (((minus_minus @ A) @ C2) @ A3))))))))).
% 9.43/3.45 thf(fact_2933_image__uminus__greaterThanAtMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or3652927894154168847AtMost @ A) @ X) @ Y)) = (((set_or7035219750837199246ssThan @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_2934_image__uminus__atLeastLessThan, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A,Y:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ (((set_or7035219750837199246ssThan @ A) @ X) @ Y)) = (((set_or3652927894154168847AtMost @ A) @ ((uminus_uminus @ A) @ Y)) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_2935_reduction__pairI, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R5) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((((relcomp @ A) @ A) @ A) @ R5) @ S2)) @ R5) => ((fun_reduction_pair @ A) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R5) @ S2))))))).
% 9.43/3.45 thf(fact_2936_wo__rel_Omax2__equals1, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2) = A3) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ R)))))))).
% 9.43/3.45 thf(fact_2937_wo__rel_Omax2__equals2, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2) = B2) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)))))))).
% 9.43/3.45 thf(fact_2938_wo__rel_Omax2__greater, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2))) @ R) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2))) @ R)))))))).
% 9.43/3.45 thf(fact_2939_wf__induct__rule, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(A > $o),A3:A]: ((((wf @ A) @ R) => ((! [X2:A]: (((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R) => (P @ Y6)))) => (P @ X2)))) => (P @ A3)))))).
% 9.43/3.45 thf(fact_2940_wf__eq__minimal, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [Q6:(set @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ Q6))) => (? [X3:A]: (((((member @ A) @ X3) @ Q6) & (! [Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3) => (~ ((((member @ A) @ Y3) @ Q6)))))))))))))))))).
% 9.43/3.45 thf(fact_2941_wf__not__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ R))))))).
% 9.43/3.45 thf(fact_2942_wf__not__sym, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,X:A]: ((((wf @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ X)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A3)) @ R)))))))).
% 9.43/3.45 thf(fact_2943_wf__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A]: ((((wf @ A) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ R))))))).
% 9.43/3.45 thf(fact_2944_wf__induct, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(A > $o),A3:A]: ((((wf @ A) @ R) => ((! [X2:A]: (((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R) => (P @ Y6)))) => (P @ X2)))) => (P @ A3)))))).
% 9.43/3.45 thf(fact_2945_wf__asym, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,X:A]: ((((wf @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ X)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A3)) @ R)))))))).
% 9.43/3.45 thf(fact_2946_wfUNIVI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [P7:(A > $o),X2:A]: (((! [Xa2:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Xa2)) @ R) => (P7 @ Y4)))) => (P7 @ Xa2)))) => (P7 @ X2)))) => ((wf @ A) @ R))))).
% 9.43/3.45 thf(fact_2947_wfI__min, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((! [X2:A,Q7:(set @ A)]: (((((member @ A) @ X2) @ Q7) => (? [Xa2:A]: (((((member @ A) @ Xa2) @ Q7) & (! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Xa2)) @ R5) => (~ ((((member @ A) @ Y4) @ Q7)))))))))))) => ((wf @ A) @ R5))))).
% 9.43/3.45 thf(fact_2948_wfE__min, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),X:A,Q:(set @ A)]: ((((wf @ A) @ R5) => ((((member @ A) @ X) @ Q) => (~ ((! [Z3:A]: (((((member @ A) @ Z3) @ Q) => (~ ((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ Z3)) @ R5) => (~ ((((member @ A) @ Y6) @ Q)))))))))))))))))).
% 9.43/3.45 thf(fact_2949_wf__def, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [P4:(A > $o)]: (((! [X3:A]: (((! [Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3) => (P4 @ Y3)))) => (P4 @ X3)))) => (! [X8:A]: ((P4 @ X8)))))))))))).
% 9.43/3.45 thf(fact_2950_wo__rel_Omax2__def, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2) = B2)) & ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R))) => (((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2) = A3))))))).
% 9.43/3.45 thf(fact_2951_wo__rel_Owell__order__induct, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(A > $o),A3:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((! [X2:A]: (((! [Y6:A]: ((((Y6 != X2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R)) => (P @ Y6)))) => (P @ X2)))) => (P @ A3)))))).
% 9.43/3.45 thf(fact_2952_Ioc__inj, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((set_or3652927894154168847AtMost @ A) @ A3) @ B2) = (((set_or3652927894154168847AtMost @ A) @ C2) @ D3)) = (((((ord_less_eq @ A) @ B2) @ A3) & (((ord_less_eq @ A) @ D3) @ C2)) | ((A3 = C2) & (B2 = D3)))))))))).
% 9.43/3.45 thf(fact_2953_wo__rel_OTOTALS, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((bNF_Wellorder_wo_rel @ A) @ R) => (! [X7:A]: (((((member @ A) @ X7) @ ((field2 @ A) @ R)) => (! [Xa2:A]: (((((member @ A) @ Xa2) @ ((field2 @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X7) @ Xa2)) @ R) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Xa2) @ X7)) @ R)))))))))))).
% 9.43/3.45 thf(fact_2954_wo__rel_Omax2__greater__among, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2))) @ R) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2))) @ R) & (((member @ A) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A3) @ B2)) @ (((insert2 @ A) @ A3) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))))))))))).
% 9.43/3.45 thf(fact_2955_wfE__min_H, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),Q:(set @ A)]: ((((wf @ A) @ R5) => ((Q != (bot_bot @ (set @ A))) => (~ ((! [Z3:A]: (((((member @ A) @ Z3) @ Q) => (~ ((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ Z3)) @ R5) => (~ ((((member @ A) @ Y6) @ Q)))))))))))))))))).
% 9.43/3.45 thf(fact_2956_infinite__Ioc, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A]: (((((ord_less @ A) @ A3) @ B2) => (~ (((finite_finite2 @ A) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2))))))))))).
% 9.43/3.45 thf(fact_2957_Ioc__subset__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D3)) = ((((ord_less_eq @ A) @ B2) @ A3) | ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2958_atLeastSucAtMost__greaterThanAtMost, axiom, (! [L:nat,U:nat]: (((((set_or1337092689740270186AtMost @ nat) @ (suc @ L)) @ U) = (((set_or3652927894154168847AtMost @ nat) @ L) @ U))))).
% 9.43/3.45 thf(fact_2959_ivl__disj__un__two_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M2)) @ (((set_or3652927894154168847AtMost @ A) @ M2) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2960_well__order__induct__imp, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(A > $o),A3:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((! [X2:A]: (((! [Y6:A]: ((((Y6 != X2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R)) => ((((member @ A) @ Y6) @ ((field2 @ A) @ R)) => (P @ Y6))))) => ((((member @ A) @ X2) @ ((field2 @ A) @ R)) => (P @ X2))))) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => (P @ A3))))))).
% 9.43/3.45 thf(fact_2961_wf__bounded__measure, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Ub:(A > nat),F:(A > nat)]: (((! [A5:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A5)) @ R) => ((((ord_less_eq @ nat) @ (Ub @ B3)) @ (Ub @ A5)) & ((((ord_less_eq @ nat) @ (F @ B3)) @ (Ub @ A5)) & (((ord_less @ nat) @ (F @ A5)) @ (F @ B3))))))) => ((wf @ A) @ R))))).
% 9.43/3.45 thf(fact_2962_wfI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (B6)))) => ((! [X2:A,P7:(A > $o)]: (((! [Xa2:A]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Xa2)) @ R) => (P7 @ Y4)))) => (P7 @ Xa2)))) => ((((member @ A) @ X2) @ A4) => ((((member @ A) @ X2) @ B6) => (P7 @ X2)))))) => ((wf @ A) @ R)))))).
% 9.43/3.45 thf(fact_2963_ivl__disj__un__two_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M2)) @ (((set_or3652927894154168847AtMost @ A) @ M2) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2964_ivl__disj__un__one_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = ((set_ord_atMost @ A) @ U))))))))).
% 9.43/3.45 thf(fact_2965_atLeastPlusOneAtMost__greaterThanAtMost__int, axiom, (! [L:int,U:int]: (((((set_or1337092689740270186AtMost @ int) @ (((plus_plus @ int) @ L) @ (one_one @ int))) @ U) = (((set_or3652927894154168847AtMost @ int) @ L) @ U))))).
% 9.43/3.45 thf(fact_2966_ivl__disj__un__one_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = ((set_ord_greaterThan @ A) @ L))))))))).
% 9.43/3.45 thf(fact_2967_wf__eq__minimal2, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [A9:(set @ A)]: ((((((ord_less_eq @ (set @ A)) @ A9) @ ((field2 @ A) @ R3)) & (A9 != (bot_bot @ (set @ A)))) => (? [X3:A]: (((((member @ A) @ X3) @ A9) & (! [Y3:A]: (((((member @ A) @ Y3) @ A9) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3)))))))))))))))))).
% 9.43/3.45 thf(fact_2968_wf__bounded__set, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),Ub:(A > (set @ B)),F:(A > (set @ B))]: (((! [A5:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A5)) @ R) => (((finite_finite2 @ B) @ (Ub @ A5)) & ((((ord_less_eq @ (set @ B)) @ (Ub @ B3)) @ (Ub @ A5)) & ((((ord_less_eq @ (set @ B)) @ (F @ B3)) @ (Ub @ A5)) & (((ord_less @ (set @ B)) @ (F @ A5)) @ (F @ B3)))))))) => ((wf @ A) @ R))))).
% 9.43/3.45 thf(fact_2969_wo__rel_Ocases__Total, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A,Phi:(A > (A > $o))]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ A3) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((field2 @ A) @ R)) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((Phi @ A3) @ B2)) => (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ R) => ((Phi @ A3) @ B2)) => ((Phi @ A3) @ B2)))))))).
% 9.43/3.45 thf(fact_2970_sum_Ohead, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((plus_plus @ A) @ (G @ M2)) @ ((((groups7311177749621191930dd_sum @ nat) @ A) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.45 thf(fact_2971_prod_Ohead, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M2:nat,N:nat,G:(nat > A)]: (((((ord_less_eq @ nat) @ M2) @ N) => (((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = (((times_times @ A) @ (G @ M2)) @ ((((groups7121269368397514597t_prod @ nat) @ A) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.45 thf(fact_2972_greaterThanAtMost__subseteq__atLeastAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2973_greaterThanAtMost__subseteq__atLeastLessThan__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2974_ivl__disj__un__two__touch_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less @ A) @ L) @ M2) => ((((ord_less_eq @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M2)) @ (((set_or1337092689740270186AtMost @ A) @ M2) @ U)) = (((set_or3652927894154168847AtMost @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2975_greaterThanAtMost__eq__atLeastAtMost__diff, axiom, (! [A:$tType]: (((order @ A) => ((set_or3652927894154168847AtMost @ A) = (^ [A7:A,B5:A]: ((((minus_minus @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A7) @ B5)) @ (((insert2 @ A) @ A7) @ (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_2976_greaterThanLessThan__subseteq__greaterThanAtMost__iff, axiom, (! [A:$tType]: (((dense_linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D3)) = ((((ord_less @ A) @ A3) @ B2) => ((((ord_less_eq @ A) @ C2) @ A3) & (((ord_less_eq @ A) @ B2) @ D3)))))))))).
% 9.43/3.45 thf(fact_2977_ivl__disj__un__two_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less_eq @ A) @ L) @ M2) => ((((ord_less @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M2)) @ (((set_or5935395276787703475ssThan @ A) @ M2) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2978_ivl__disj__un__two__touch_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,M2:A,U:A]: (((((ord_less @ A) @ L) @ M2) => ((((ord_less @ A) @ M2) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ U)) = (((set_or5935395276787703475ssThan @ A) @ L) @ U)))))))))).
% 9.43/3.45 thf(fact_2979_wf__bounded__supset, axiom, (! [A:$tType,S2:(set @ A)]: ((((finite_finite2 @ A) @ S2) => ((wf @ (set @ A)) @ ((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [Q8:(set @ A),Q6:(set @ A)]: (((((ord_less @ (set @ A)) @ Q6) @ Q8) & (((ord_less_eq @ (set @ A)) @ Q8) @ S2))))))))))).
% 9.43/3.45 thf(fact_2980_atLeastPlusOneAtMost__greaterThanAtMost__integer, axiom, (! [L:code_integer,U:code_integer]: (((((set_or1337092689740270186AtMost @ code_integer) @ (((plus_plus @ code_integer) @ L) @ (one_one @ code_integer))) @ U) = (((set_or3652927894154168847AtMost @ code_integer) @ L) @ U))))).
% 9.43/3.45 thf(fact_2981_ivl__disj__un__singleton_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((insert2 @ A) @ L) @ (bot_bot @ (set @ A)))) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = (((set_or1337092689740270186AtMost @ A) @ L) @ U))))))))).
% 9.43/3.45 thf(fact_2982_wf__iff__no__infinite__down__chain, axiom, (! [A:$tType]: (((wf @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((~ ((? [F2:(nat > A)]: ((! [I3:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F2 @ (suc @ I3))) @ (F2 @ I3))) @ R3))))))))))))).
% 9.43/3.45 thf(fact_2983_wf__no__infinite__down__chainE, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(nat > A)]: ((((wf @ A) @ R) => (~ ((! [K5:nat]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F @ (suc @ K5))) @ (F @ K5))) @ R))))))))).
% 9.43/3.45 thf(fact_2984_wo__rel_Ominim__least, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),B6:(set @ A),B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ B6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((bNF_We6954850376910717587_minim @ A) @ R) @ B6)) @ B2)) @ R))))))).
% 9.43/3.45 thf(fact_2985_wo__rel_Oequals__minim, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),B6:(set @ A),A3:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((field2 @ A) @ R)) => ((((member @ A) @ A3) @ B6) => ((! [B3:A]: (((((member @ A) @ B3) @ B6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B3)) @ R)))) => (A3 = (((bNF_We6954850376910717587_minim @ A) @ R) @ B6))))))))).
% 9.43/3.45 thf(fact_2986_dependent__wf__choice, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ A)),P:((A > B) > (A > (B > $o)))]: ((((wf @ A) @ R5) => ((! [F4:(A > B),G4:(A > B),X2:A,R2:B]: (((! [Z6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z6) @ X2)) @ R5) => ((F4 @ Z6) = (G4 @ Z6))))) => ((((P @ F4) @ X2) @ R2) = (((P @ G4) @ X2) @ R2))))) => ((! [X2:A,F4:(A > B)]: (((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R5) => (((P @ F4) @ Y6) @ (F4 @ Y6))))) => (? [X_1:B]: ((((P @ F4) @ X2) @ X_1)))))) => (? [F4:(A > B)]: ((! [X7:A]: ((((P @ F4) @ X7) @ (F4 @ X7)))))))))))).
% 9.43/3.45 thf(fact_2987_of__rat__neg__one, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((field_char_0_of_rat @ A) @ ((uminus_uminus @ rat) @ (one_one @ rat))) = ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.45 thf(fact_2988_Ref__Time_Oupdate__def, axiom, (! [A:$tType]: (((heap @ A) => ((ref_update @ A) = (^ [R3:(ref @ A),V2:A]: (((heap_Time_heap @ product_unit) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ product_Unity) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ ((((ref_set @ A) @ R3) @ V2) @ H)) @ (one_one @ nat))))))))))))).
% 9.43/3.45 thf(fact_2989_sum__div__partition, axiom, (! [B:$tType,A:$tType]: (((euclid4440199948858584721cancel @ A) => (! [A4:(set @ B),F:(B > A),B2:A]: ((((finite_finite2 @ B) @ A4) => ((((divide_divide @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A4)) @ B2) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [A7:B]: ((((divide_divide @ A) @ (F @ A7)) @ B2)))) @ (((inf_inf @ (set @ B)) @ A4) @ ((collect @ B) @ (^ [A7:B]: ((((dvd_dvd @ A) @ B2) @ (F @ A7)))))))) @ (((divide_divide @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ (((inf_inf @ (set @ B)) @ A4) @ ((collect @ B) @ (^ [A7:B]: ((~ ((((dvd_dvd @ A) @ B2) @ (F @ A7)))))))))) @ B2)))))))))).
% 9.43/3.45 thf(fact_2990_bsqr__max2, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A1:A,A22:A,B1:A,B22:A]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))) @ ((((product_Pair @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((((product_Pair @ A) @ A) @ B1) @ B22))) @ ((bNF_Wellorder_bsqr @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ A1) @ A22)) @ ((((bNF_We1388413361240627857o_max2 @ A) @ R) @ B1) @ B22))) @ R)))))).
% 9.43/3.45 thf(fact_2991_UN__atLeast__UNIV, axiom, (((complete_Sup_Sup @ (set @ nat)) @ ((((image2 @ nat) @ (set @ nat)) @ (set_ord_atLeast @ nat)) @ (top_top @ (set @ nat)))) = (top_top @ (set @ nat)))).
% 9.43/3.45 thf(fact_2992_atLeast__eq__iff, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,Y:A]: (((((set_ord_atLeast @ A) @ X) = ((set_ord_atLeast @ A) @ Y)) = (X = Y)))))))).
% 9.43/3.45 thf(fact_2993_unit__abs__eta__conv, axiom, (! [A:$tType,F:(product_unit > A)]: (((^ [U2:product_unit]: ((F @ product_Unity))) = F)))).
% 9.43/3.45 thf(fact_2994_inf__Some, axiom, (! [A:$tType]: (((inf @ A) => (! [X:A,Y:A]: (((((inf_inf @ (option @ A)) @ ((some @ A) @ X)) @ ((some @ A) @ Y)) = ((some @ A) @ (((inf_inf @ A) @ X) @ Y))))))))).
% 9.43/3.45 thf(fact_2995_atLeast__iff, axiom, (! [A:$tType]: (((ord @ A) => (! [I:A,K:A]: (((((member @ A) @ I) @ ((set_ord_atLeast @ A) @ K)) = (((ord_less_eq @ A) @ K) @ I)))))))).
% 9.43/3.45 thf(fact_2996_Inf__atLeast, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: ((((complete_Inf_Inf @ A) @ ((set_ord_atLeast @ A) @ X)) = X))))))).
% 9.43/3.45 thf(fact_2997_atLeast__subset__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [X:A,Y:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ X)) @ ((set_ord_atLeast @ A) @ Y)) = (((ord_less_eq @ A) @ Y) @ X)))))))).
% 9.43/3.45 thf(fact_2998_image__add__atLeast, axiom, (! [A:$tType]: (((linordered_semidom @ A) => (! [K:A,I:A]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ K)) @ ((set_ord_atLeast @ A) @ I)) = ((set_ord_atLeast @ A) @ (((plus_plus @ A) @ K) @ I))))))))).
% 9.43/3.45 thf(fact_2999_Sup__atLeast, axiom, (! [A:$tType]: (((comple6319245703460814977attice @ A) => (! [X:A]: ((((complete_Sup_Sup @ A) @ ((set_ord_atLeast @ A) @ X)) = (top_top @ A)))))))).
% 9.43/3.45 thf(fact_3000_inter__compl__diff__conv, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((inf_inf @ (set @ A)) @ A4) @ ((uminus_uminus @ (set @ A)) @ B6)) = (((minus_minus @ (set @ A)) @ A4) @ B6))))).
% 9.43/3.45 thf(fact_3001_Int__atMost, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ A3)) @ ((set_ord_atMost @ A) @ B2)) = ((set_ord_atMost @ A) @ (((ord_min @ A) @ A3) @ B2))))))))).
% 9.43/3.45 thf(fact_3002_Compl__lessThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) = ((set_ord_atLeast @ A) @ K)))))))).
% 9.43/3.45 thf(fact_3003_Compl__atLeast, axiom, (! [A:$tType]: (((linorder @ A) => (! [K:A]: ((((uminus_uminus @ (set @ A)) @ ((set_ord_atLeast @ A) @ K)) = ((set_ord_lessThan @ A) @ K)))))))).
% 9.43/3.45 thf(fact_3004_one__eq__of__rat__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: ((((one_one @ A) = ((field_char_0_of_rat @ A) @ A3)) = ((one_one @ rat) = A3)))))))).
% 9.43/3.45 thf(fact_3005_of__rat__eq__1__iff, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat]: (((((field_char_0_of_rat @ A) @ A3) = (one_one @ A)) = (A3 = (one_one @ rat))))))))).
% 9.43/3.45 thf(fact_3006_of__rat__1, axiom, (! [A:$tType]: (((field_char_0 @ A) => (((field_char_0_of_rat @ A) @ (one_one @ rat)) = (one_one @ A)))))).
% 9.43/3.45 thf(fact_3007_atLeast__0, axiom, (((set_ord_atLeast @ nat) @ (zero_zero @ nat)) = (top_top @ (set @ nat)))).
% 9.43/3.45 thf(fact_3008_Icc__subset__Ici__iff, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A,H3:A,L4:A]: (((((ord_less_eq @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ H3)) @ ((set_ord_atLeast @ A) @ L4)) = ((~ ((((ord_less_eq @ A) @ L) @ H3))) | (((ord_less_eq @ A) @ L4) @ L))))))))).
% 9.43/3.45 thf(fact_3009_Int__atLeastAtMost, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = (((set_or1337092689740270186AtMost @ A) @ (((ord_max @ A) @ A3) @ C2)) @ (((ord_min @ A) @ B2) @ D3))))))))).
% 9.43/3.45 thf(fact_3010_Int__atLeastLessThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ A3) @ B2)) @ (((set_or7035219750837199246ssThan @ A) @ C2) @ D3)) = (((set_or7035219750837199246ssThan @ A) @ (((ord_max @ A) @ A3) @ C2)) @ (((ord_min @ A) @ B2) @ D3))))))))).
% 9.43/3.45 thf(fact_3011_Int__atLeastAtMostR1, axiom, (! [A:$tType]: (((linorder @ A) => (! [B2:A,C2:A,D3:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ B2)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = (((set_or1337092689740270186AtMost @ A) @ C2) @ (((ord_min @ A) @ B2) @ D3))))))))).
% 9.43/3.45 thf(fact_3012_Int__atLeastAtMostL1, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,D3:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ ((set_ord_atMost @ A) @ D3)) = (((set_or1337092689740270186AtMost @ A) @ A3) @ (((ord_min @ A) @ B2) @ D3))))))))).
% 9.43/3.45 thf(fact_3013_image__minus__const__atLeast, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,A3:A]: ((((((image2 @ A) @ A) @ ((minus_minus @ A) @ C2)) @ ((set_ord_atLeast @ A) @ A3)) = ((set_ord_atMost @ A) @ (((minus_minus @ A) @ C2) @ A3))))))))).
% 9.43/3.45 thf(fact_3014_image__minus__const__AtMost, axiom, (! [A:$tType]: (((linordered_idom @ A) => (! [C2:A,B2:A]: ((((((image2 @ A) @ A) @ ((minus_minus @ A) @ C2)) @ ((set_ord_atMost @ A) @ B2)) = ((set_ord_atLeast @ A) @ (((minus_minus @ A) @ C2) @ B2))))))))).
% 9.43/3.45 thf(fact_3015_image__uminus__atLeast, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_atLeast @ A) @ X)) = ((set_ord_atMost @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_3016_image__uminus__atMost, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [X:A]: ((((((image2 @ A) @ A) @ (uminus_uminus @ A)) @ ((set_ord_atMost @ A) @ X)) = ((set_ord_atLeast @ A) @ ((uminus_uminus @ A) @ X))))))))).
% 9.43/3.45 thf(fact_3017_Int__atLeastAtMostL2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ A3) @ B2)) @ ((set_ord_atLeast @ A) @ C2)) = (((set_or1337092689740270186AtMost @ A) @ (((ord_max @ A) @ A3) @ C2)) @ B2)))))))).
% 9.43/3.45 thf(fact_3018_Int__atLeastAtMostR2, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,C2:A,D3:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atLeast @ A) @ A3)) @ (((set_or1337092689740270186AtMost @ A) @ C2) @ D3)) = (((set_or1337092689740270186AtMost @ A) @ (((ord_max @ A) @ A3) @ C2)) @ D3)))))))).
% 9.43/3.45 thf(fact_3019_Int__greaterThanAtMost, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D3)) = (((set_or3652927894154168847AtMost @ A) @ (((ord_max @ A) @ A3) @ C2)) @ (((ord_min @ A) @ B2) @ D3))))))))).
% 9.43/3.45 thf(fact_3020_Int__greaterThanLessThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ A3) @ B2)) @ (((set_or5935395276787703475ssThan @ A) @ C2) @ D3)) = (((set_or5935395276787703475ssThan @ A) @ (((ord_max @ A) @ A3) @ C2)) @ (((ord_min @ A) @ B2) @ D3))))))))).
% 9.43/3.45 thf(fact_3021_one__le__of__rat__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less_eq @ A) @ (one_one @ A)) @ ((field_char_0_of_rat @ A) @ R)) = (((ord_less_eq @ rat) @ (one_one @ rat)) @ R)))))))).
% 9.43/3.45 thf(fact_3022_of__rat__le__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less_eq @ A) @ ((field_char_0_of_rat @ A) @ R)) @ (one_one @ A)) = (((ord_less_eq @ rat) @ R) @ (one_one @ rat))))))))).
% 9.43/3.45 thf(fact_3023_one__less__of__rat__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less @ A) @ (one_one @ A)) @ ((field_char_0_of_rat @ A) @ R)) = (((ord_less @ rat) @ (one_one @ rat)) @ R)))))))).
% 9.43/3.45 thf(fact_3024_of__rat__less__1__iff, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [R:rat]: (((((ord_less @ A) @ ((field_char_0_of_rat @ A) @ R)) @ (one_one @ A)) = (((ord_less @ rat) @ R) @ (one_one @ rat))))))))).
% 9.43/3.45 thf(fact_3025_atLeastLessThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_or7035219750837199246ssThan @ A) = (^ [L3:A,U2:A]: ((((inf_inf @ (set @ A)) @ ((set_ord_atLeast @ A) @ L3)) @ ((set_ord_lessThan @ A) @ U2))))))))).
% 9.43/3.45 thf(fact_3026_well__order__on__domain, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((order_well_order_on @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((((member @ A) @ A3) @ A4) & (((member @ A) @ B2) @ A4))))))).
% 9.43/3.45 thf(fact_3027_Inf__unit__def, axiom, ((complete_Inf_Inf @ product_unit) = (^ [Uu3:(set @ product_unit)]: (product_Unity)))).
% 9.43/3.45 thf(fact_3028_Sup__unit__def, axiom, ((complete_Sup_Sup @ product_unit) = (^ [Uu3:(set @ product_unit)]: (product_Unity)))).
% 9.43/3.45 thf(fact_3029_Times__Int__distrib1, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ A),C5:(set @ B)]: ((((((product_Sigma @ A) @ B) @ (((inf_inf @ (set @ A)) @ A4) @ B6)) @ (^ [Uu:A]: (C5))) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (C5)))))))).
% 9.43/3.45 thf(fact_3030_Sigma__Int__distrib2, axiom, (! [B:$tType,A:$tType,I4:(set @ A),A4:(A > (set @ B)),B6:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ I4) @ (^ [I3:A]: ((((inf_inf @ (set @ B)) @ (A4 @ I3)) @ (B6 @ I3))))) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I4) @ A4)) @ ((((product_Sigma @ A) @ B) @ I4) @ B6)))))).
% 9.43/3.45 thf(fact_3031_Times__Int__Times, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ A),D4:(set @ B)]: (((((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))) @ ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu:A]: (D4)))) = ((((product_Sigma @ A) @ B) @ (((inf_inf @ (set @ A)) @ A4) @ C5)) @ (^ [Uu:A]: ((((inf_inf @ (set @ B)) @ B6) @ D4)))))))).
% 9.43/3.45 thf(fact_3032_Sigma__Int__distrib1, axiom, (! [B:$tType,A:$tType,I4:(set @ A),J4:(set @ A),C5:(A > (set @ B))]: ((((((product_Sigma @ A) @ B) @ (((inf_inf @ (set @ A)) @ I4) @ J4)) @ C5) = (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((((product_Sigma @ A) @ B) @ I4) @ C5)) @ ((((product_Sigma @ A) @ B) @ J4) @ C5)))))).
% 9.43/3.45 thf(fact_3033_old_Ounit_Oexhaust, axiom, (! [Y:product_unit]: ((Y = product_Unity)))).
% 9.43/3.45 thf(fact_3034_top__unit__def, axiom, ((top_top @ product_unit) = product_Unity)).
% 9.43/3.45 thf(fact_3035_uminus__unit__def, axiom, ((uminus_uminus @ product_unit) = (^ [Uu3:product_unit]: (product_Unity)))).
% 9.43/3.45 thf(fact_3036_bot__unit__def, axiom, ((bot_bot @ product_unit) = product_Unity)).
% 9.43/3.45 thf(fact_3037_ivl__disj__int__one_I8_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3038_sup__unit__def, axiom, ((sup_sup @ product_unit) = (^ [Uu3:product_unit,Uv2:product_unit]: (product_Unity)))).
% 9.43/3.45 thf(fact_3039_atLeastAtMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_or1337092689740270186AtMost @ A) = (^ [L3:A,U2:A]: ((((inf_inf @ (set @ A)) @ ((set_ord_atLeast @ A) @ L3)) @ ((set_ord_atMost @ A) @ U2))))))))).
% 9.43/3.45 thf(fact_3040_ivl__disj__int__one_I6_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3041_disjointI, axiom, (! [A:$tType,A3:(set @ A),B2:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (~ ((((member @ A) @ X2) @ B2)))))) => ((((inf_inf @ (set @ A)) @ A3) @ B2) = (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_3042_inter__eq__subsetI, axiom, (! [A:$tType,S2:(set @ A),S7:(set @ A),A4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ S7) => (((((inf_inf @ (set @ A)) @ A4) @ S7) = (((inf_inf @ (set @ A)) @ B6) @ S7)) => ((((inf_inf @ (set @ A)) @ A4) @ S2) = (((inf_inf @ (set @ A)) @ B6) @ S2))))))).
% 9.43/3.45 thf(fact_3043_not__empty__eq__Ici__eq__empty, axiom, (! [A:$tType]: (((preorder @ A) => (! [L:A]: (((bot_bot @ (set @ A)) != ((set_ord_atLeast @ A) @ L)))))))).
% 9.43/3.45 thf(fact_3044_infinite__Ici, axiom, (! [A:$tType]: ((((linorder @ A) & (no_top @ A)) => (! [A3:A]: ((~ (((finite_finite2 @ A) @ ((set_ord_atLeast @ A) @ A3)))))))))).
% 9.43/3.45 thf(fact_3045_not__UNIV__eq__Ici, axiom, (! [A:$tType]: (((no_bot @ A) => (! [L4:A]: (((top_top @ (set @ A)) != ((set_ord_atLeast @ A) @ L4)))))))).
% 9.43/3.45 thf(fact_3046_not__Ici__eq__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L4:A,L:A,H3:A]: ((((set_ord_atLeast @ A) @ L4) != (((set_or1337092689740270186AtMost @ A) @ L) @ H3)))))))).
% 9.43/3.45 thf(fact_3047_not__Iic__eq__Ici, axiom, (! [A:$tType]: (((no_top @ A) => (! [H3:A,L4:A]: ((((set_ord_atMost @ A) @ H3) != ((set_ord_atLeast @ A) @ L4)))))))).
% 9.43/3.45 thf(fact_3048_atMost__Int__atLeast, axiom, (! [A:$tType]: (((order @ A) => (! [N:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ N)) @ ((set_ord_atLeast @ A) @ N)) = (((insert2 @ A) @ N) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_3049_atLeast__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_ord_atLeast @ A) = (^ [L3:A]: (((collect @ A) @ ((ord_less_eq @ A) @ L3))))))))).
% 9.43/3.45 thf(fact_3050_inf__option__def, axiom, (! [A:$tType]: (((inf @ A) => ((inf_inf @ (option @ A)) = (^ [X3:(option @ A),Y3:(option @ A)]: ((((((case_option @ (option @ A)) @ A) @ (none @ A)) @ (^ [Z5:A]: ((((((case_option @ (option @ A)) @ A) @ (none @ A)) @ (^ [Aa2:A]: (((some @ A) @ (((inf_inf @ A) @ Z5) @ Aa2))))) @ Y3)))) @ X3)))))))).
% 9.43/3.45 thf(fact_3051_translation__Int, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,S:(set @ A),T2:(set @ A)]: ((((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ (((inf_inf @ (set @ A)) @ S) @ T2)) = (((inf_inf @ (set @ A)) @ ((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ S)) @ ((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ T2))))))))).
% 9.43/3.45 thf(fact_3052_disjoint__mono, axiom, (! [A:$tType,A3:(set @ A),A6:(set @ A),B2:(set @ A),B4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ A6) => ((((ord_less_eq @ (set @ A)) @ B2) @ B4) => (((((inf_inf @ (set @ A)) @ A6) @ B4) = (bot_bot @ (set @ A))) => ((((inf_inf @ (set @ A)) @ A3) @ B2) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3053_disjoint__alt__simp2, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A4) @ B6) != A4) = ((((inf_inf @ (set @ A)) @ A4) @ B6) != (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_3054_disjoint__alt__simp1, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A4) @ B6) = A4) = ((((inf_inf @ (set @ A)) @ A4) @ B6) = (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_3055_ivl__disj__int__two_I3_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3056_ivl__disj__int__two_I6_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M2)) @ (((set_or3652927894154168847AtMost @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3057_greaterThan__Int__greaterThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ A3)) @ ((set_ord_lessThan @ A) @ B2)) = ((set_ord_lessThan @ A) @ (((ord_min @ A) @ A3) @ B2))))))))).
% 9.43/3.45 thf(fact_3058_atLeast__eq__UNIV__iff, axiom, (! [A:$tType]: (((order_bot @ A) => (! [X:A]: (((((set_ord_atLeast @ A) @ X) = (top_top @ (set @ A))) = (X = (bot_bot @ A))))))))).
% 9.43/3.45 thf(fact_3059_not__UNIV__le__Ici, axiom, (! [A:$tType]: (((no_bot @ A) => (! [L:A]: ((~ ((((ord_less_eq @ (set @ A)) @ (top_top @ (set @ A))) @ ((set_ord_atLeast @ A) @ L)))))))))).
% 9.43/3.45 thf(fact_3060_UNIV__unit, axiom, ((top_top @ (set @ product_unit)) = (((insert2 @ product_unit) @ product_Unity) @ (bot_bot @ (set @ product_unit))))).
% 9.43/3.45 thf(fact_3061_not__Ici__le__Icc, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,L4:A,H4:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L4) @ H4)))))))))).
% 9.43/3.45 thf(fact_3062_lessThan__Int__lessThan, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_greaterThan @ A) @ A3)) @ ((set_ord_greaterThan @ A) @ B2)) = ((set_ord_greaterThan @ A) @ (((ord_max @ A) @ A3) @ B2))))))))).
% 9.43/3.45 thf(fact_3063_not__Iic__le__Ici, axiom, (! [A:$tType]: (((no_bot @ A) => (! [H3:A,L4:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atMost @ A) @ H3)) @ ((set_ord_atLeast @ A) @ L4)))))))))).
% 9.43/3.45 thf(fact_3064_not__Ici__le__Iic, axiom, (! [A:$tType]: (((no_top @ A) => (! [L:A,H4:A]: ((~ ((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ L)) @ ((set_ord_atMost @ A) @ H4)))))))))).
% 9.43/3.45 thf(fact_3065_translation__subtract__Int, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A,S:(set @ A),T2:(set @ A)]: ((((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ (((inf_inf @ (set @ A)) @ S) @ T2)) = (((inf_inf @ (set @ A)) @ ((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ S)) @ ((((image2 @ A) @ A) @ (^ [X3:A]: ((((minus_minus @ A) @ X3) @ A3)))) @ T2))))))))).
% 9.43/3.45 thf(fact_3066_of__rat__add, axiom, (! [A:$tType]: (((field_char_0 @ A) => (! [A3:rat,B2:rat]: ((((field_char_0_of_rat @ A) @ (((plus_plus @ rat) @ A3) @ B2)) = (((plus_plus @ A) @ ((field_char_0_of_rat @ A) @ A3)) @ ((field_char_0_of_rat @ A) @ B2))))))))).
% 9.43/3.45 thf(fact_3067_Ioi__le__Ico, axiom, (! [A:$tType]: (((preorder @ A) => (! [A3:A]: ((((ord_less_eq @ (set @ A)) @ ((set_ord_greaterThan @ A) @ A3)) @ ((set_ord_atLeast @ A) @ A3)))))))).
% 9.43/3.45 thf(fact_3068_atLeast__Suc__greaterThan, axiom, (! [K:nat]: ((((set_ord_atLeast @ nat) @ (suc @ K)) = ((set_ord_greaterThan @ nat) @ K))))).
% 9.43/3.45 thf(fact_3069_ivl__disj__int__two_I7_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ M2)) @ (((set_or1337092689740270186AtMost @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3070_disjoint__alt__simp3, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((ord_less @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B6)) @ A4) = ((((inf_inf @ (set @ A)) @ A4) @ B6) != (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_3071_Ioc__disjoint, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A,C2:A,D3:A]: ((((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ A3) @ B2)) @ (((set_or3652927894154168847AtMost @ A) @ C2) @ D3)) = (bot_bot @ (set @ A))) = ((((ord_less_eq @ A) @ B2) @ A3) | ((((ord_less_eq @ A) @ D3) @ C2) | ((((ord_less_eq @ A) @ B2) @ C2) | (((ord_less_eq @ A) @ D3) @ A3))))))))))).
% 9.43/3.45 thf(fact_3072_ivl__disj__int__one_I4_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3073_ivl__disj__int__one_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ L)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3074_wait__def, axiom, (heap_Time_wait = (^ [N5:nat]: (((heap_Time_Heap2 @ product_unit) @ (^ [H:(heap_ext @ product_unit)]: (((some @ ((product_prod @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ product_Unity) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ N5)))))))))).
% 9.43/3.45 thf(fact_3075_ivl__disj__int__two_I8_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M2)) @ (((set_or3652927894154168847AtMost @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3076_ivl__disj__int__two_I5_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M2)) @ (((set_or1337092689740270186AtMost @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3077_ivl__disj__int__two_I4_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ M2)) @ (((set_or5935395276787703475ssThan @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3078_ivl__disj__int__two_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ M2)) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3079_ivl__disj__int__one_I3_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3080_ivl__disj__int__one_I7_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3081_ivl__disj__int__one_I1_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ ((set_ord_atMost @ A) @ L)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3082_ivl__disj__int__two_I2_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,M2:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ M2)) @ (((set_or5935395276787703475ssThan @ A) @ M2) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3083_ivl__disj__int__one_I5_J, axiom, (! [A:$tType]: (((order @ A) => (! [L:A,U:A]: (((((inf_inf @ (set @ A)) @ (((set_or3652927894154168847AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3084_ivl__disj__un__one_I8_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or7035219750837199246ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = ((set_ord_atLeast @ A) @ L))))))))).
% 9.43/3.45 thf(fact_3085_prod_Ointer__restrict, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),G:(B > A),B6:(set @ B)]: ((((finite_finite2 @ B) @ A4) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A4) @ B6)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X3:B]: (((((if @ A) @ (((member @ B) @ X3) @ B6)) @ (G @ X3)) @ (one_one @ A))))) @ A4))))))))).
% 9.43/3.45 thf(fact_3086_greaterThanAtMost__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_or3652927894154168847AtMost @ A) = (^ [L3:A,U2:A]: ((((inf_inf @ (set @ A)) @ ((set_ord_greaterThan @ A) @ L3)) @ ((set_ord_atMost @ A) @ U2))))))))).
% 9.43/3.45 thf(fact_3087_greaterThanLessThan__eq, axiom, (! [A:$tType]: (((ord @ A) => ((set_or5935395276787703475ssThan @ A) = (^ [A7:A,B5:A]: ((((inf_inf @ (set @ A)) @ ((set_ord_greaterThan @ A) @ A7)) @ ((set_ord_lessThan @ A) @ B5))))))))).
% 9.43/3.45 thf(fact_3088_greaterThanLessThan__def, axiom, (! [A:$tType]: (((ord @ A) => ((set_or5935395276787703475ssThan @ A) = (^ [L3:A,U2:A]: ((((inf_inf @ (set @ A)) @ ((set_ord_greaterThan @ A) @ L3)) @ ((set_ord_lessThan @ A) @ U2))))))))).
% 9.43/3.45 thf(fact_3089_Ici__subset__Ioi__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [A3:A,B2:A]: (((((ord_less_eq @ (set @ A)) @ ((set_ord_atLeast @ A) @ A3)) @ ((set_ord_greaterThan @ A) @ B2)) = (((ord_less @ A) @ B2) @ A3)))))))).
% 9.43/3.45 thf(fact_3090_execute__assert_H, axiom, (! [P:$o,H3:(heap_ext @ product_unit)]: (((P => ((((heap_Time_execute @ product_unit) @ (refine_Imp_assert @ P)) @ H3) = ((some @ ((product_prod @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ product_Unity) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (zero_zero @ nat)))))) & ((~ (P)) => ((((heap_Time_execute @ product_unit) @ (refine_Imp_assert @ P)) @ H3) = (none @ ((product_prod @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))).
% 9.43/3.45 thf(fact_3091_sum_Ounion__inter, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),B6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ B) @ B6) => ((((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A4) @ B6))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A4) @ B6))) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B6))))))))))).
% 9.43/3.45 thf(fact_3092_sum_OInt__Diff, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),G:(B > A),B6:(set @ B)]: ((((finite_finite2 @ B) @ A4) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A4) @ B6))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A4) @ B6))))))))))).
% 9.43/3.45 thf(fact_3093_prod_Omono__neutral__cong, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [T6:(set @ B),S2:(set @ B),H3:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ T6) => (((finite_finite2 @ B) @ S2) => ((! [I2:B]: (((((member @ B) @ I2) @ (((minus_minus @ (set @ B)) @ T6) @ S2)) => ((H3 @ I2) = (one_one @ A))))) => ((! [I2:B]: (((((member @ B) @ I2) @ (((minus_minus @ (set @ B)) @ S2) @ T6)) => ((G @ I2) = (one_one @ A))))) => ((! [X2:B]: (((((member @ B) @ X2) @ (((inf_inf @ (set @ B)) @ S2) @ T6)) => ((G @ X2) = (H3 @ X2))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ S2) = ((((groups7121269368397514597t_prod @ B) @ A) @ H3) @ T6))))))))))))).
% 9.43/3.45 thf(fact_3094_Iio__Int__singleton, axiom, (! [A:$tType]: (((order @ A) => (! [X:A,K:A]: ((((((ord_less @ A) @ X) @ K) => ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) = (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))) & ((~ ((((ord_less @ A) @ X) @ K))) => ((((inf_inf @ (set @ A)) @ ((set_ord_lessThan @ A) @ K)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_3095_card__Un__Int, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ A) @ B6) => ((((plus_plus @ nat) @ ((finite_card @ A) @ A4)) @ ((finite_card @ A) @ B6)) = (((plus_plus @ nat) @ ((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6))) @ ((finite_card @ A) @ (((inf_inf @ (set @ A)) @ A4) @ B6))))))))).
% 9.43/3.45 thf(fact_3096_sum_OIf__cases, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),P:(B > $o),H3:(B > A),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((((groups7311177749621191930dd_sum @ B) @ A) @ (^ [X3:B]: (((((if @ A) @ (P @ X3)) @ (H3 @ X3)) @ (G @ X3))))) @ A4) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ H3) @ (((inf_inf @ (set @ B)) @ A4) @ ((collect @ B) @ P)))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A4) @ ((uminus_uminus @ (set @ B)) @ ((collect @ B) @ P))))))))))))).
% 9.43/3.45 thf(fact_3097_ivl__disj__un__singleton_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A]: (((((sup_sup @ (set @ A)) @ (((insert2 @ A) @ L) @ (bot_bot @ (set @ A)))) @ ((set_ord_greaterThan @ A) @ L)) = ((set_ord_atLeast @ A) @ L)))))))).
% 9.43/3.45 thf(fact_3098_ivl__disj__un__one_I7_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less_eq @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or1337092689740270186AtMost @ A) @ L) @ U)) @ ((set_ord_greaterThan @ A) @ U)) = ((set_ord_atLeast @ A) @ L))))))))).
% 9.43/3.45 thf(fact_3099_ivl__disj__un__one_I6_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:A,U:A]: (((((ord_less @ A) @ L) @ U) => ((((sup_sup @ (set @ A)) @ (((set_or5935395276787703475ssThan @ A) @ L) @ U)) @ ((set_ord_atLeast @ A) @ U)) = ((set_ord_greaterThan @ A) @ L))))))))).
% 9.43/3.45 thf(fact_3100_Linear__order__Well__order__iff, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) = (! [A9:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A9) @ ((field2 @ A) @ R)) => ((A9 != (bot_bot @ (set @ A))) => (? [X3:A]: (((((member @ A) @ X3) @ A9) & (! [Y3:A]: (((((member @ A) @ Y3) @ A9) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))))))))))))))).
% 9.43/3.45 thf(fact_3101_sum_Ounion__inter__neutral, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),B6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ B) @ B6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((inf_inf @ (set @ B)) @ A4) @ B6)) => ((G @ X2) = (zero_zero @ A))))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A4) @ B6)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B6)))))))))))).
% 9.43/3.45 thf(fact_3102_sum__Un, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [A4:(set @ B),B6:(set @ B),F:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ B) @ B6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ A4) @ B6)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ A4)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ B6))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ F) @ (((inf_inf @ (set @ B)) @ A4) @ B6)))))))))))).
% 9.43/3.45 thf(fact_3103_sum_Ounion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),B6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ B) @ B6) => (((((inf_inf @ (set @ B)) @ A4) @ B6) = (bot_bot @ (set @ B))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A4) @ B6)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ A4)) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ B6)))))))))))).
% 9.43/3.45 thf(fact_3104_prod_Ounion__inter__neutral, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [A4:(set @ B),B6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ B) @ B6) => ((! [X2:B]: (((((member @ B) @ X2) @ (((inf_inf @ (set @ B)) @ A4) @ B6)) => ((G @ X2) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A4) @ B6)) = (((times_times @ A) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ A4)) @ ((((groups7121269368397514597t_prod @ B) @ A) @ G) @ B6)))))))))))).
% 9.43/3.45 thf(fact_3105_execute__update, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),V:A,H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ product_unit) @ (((ref_update @ A) @ R) @ V)) @ H3) = ((some @ ((product_prod @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ product_unit) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ product_Unity) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ ((((ref_set @ A) @ R) @ V) @ H3)) @ (one_one @ nat))))))))))).
% 9.43/3.45 thf(fact_3106_sum_Ounion__diff2, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),B6:(set @ B),G:(B > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ B) @ B6) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((sup_sup @ (set @ B)) @ A4) @ B6)) = (((plus_plus @ A) @ (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ A4) @ B6))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((minus_minus @ (set @ B)) @ B6) @ A4)))) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((inf_inf @ (set @ B)) @ A4) @ B6)))))))))))).
% 9.43/3.45 thf(fact_3107_sum__Un2, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ B) => (! [A4:(set @ A),B6:(set @ A),F:(A > B)]: ((((finite_finite2 @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) => (((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((plus_plus @ B) @ (((plus_plus @ B) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ A4) @ B6))) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ B6) @ A4)))) @ ((((groups7311177749621191930dd_sum @ A) @ B) @ F) @ (((inf_inf @ (set @ A)) @ A4) @ B6))))))))))).
% 9.43/3.45 thf(fact_3108_card__Un__disjoint, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ A) @ B6) => (((((inf_inf @ (set @ A)) @ A4) @ B6) = (bot_bot @ (set @ A))) => (((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((plus_plus @ nat) @ ((finite_card @ A) @ A4)) @ ((finite_card @ A) @ B6))))))))).
% 9.43/3.45 thf(fact_3109_sum__Un__nat, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A),F:(A > nat)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ A) @ B6) => (((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((minus_minus @ nat) @ (((plus_plus @ nat) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ A4)) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ B6))) @ ((((groups7311177749621191930dd_sum @ A) @ nat) @ F) @ (((inf_inf @ (set @ A)) @ A4) @ B6))))))))).
% 9.43/3.45 thf(fact_3110_atLeast__Suc, axiom, (! [K:nat]: ((((set_ord_atLeast @ nat) @ (suc @ K)) = (((minus_minus @ (set @ nat)) @ ((set_ord_atLeast @ nat) @ K)) @ (((insert2 @ nat) @ K) @ (bot_bot @ (set @ nat)))))))).
% 9.43/3.45 thf(fact_3111_old_Ounit_Orec, axiom, (! [T:$tType,F1:T]: (((((product_rec_unit @ T) @ F1) @ product_Unity) = F1)))).
% 9.43/3.45 thf(fact_3112_default__unit__def, axiom, ((default_default @ product_unit) = product_Unity)).
% 9.43/3.45 thf(fact_3113_CODE__ABORT__def, axiom, (! [A:$tType]: (((cODE_ABORT @ A) = (^ [F2:(product_unit > A)]: ((F2 @ product_Unity))))))).
% 9.43/3.45 thf(fact_3114_Inf__fin_Oeq__fold_H, axiom, (! [A:$tType]: (((semilattice_inf @ A) => ((lattic7752659483105999362nf_fin @ A) = (^ [A9:(set @ A)]: (((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X3:A,Y3:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X3) @ ((inf_inf @ A) @ X3)) @ Y3))))) @ (none @ A)) @ A9))))))))).
% 9.43/3.45 thf(fact_3115_image__split__eq__Sigma, axiom, (! [C:$tType,B:$tType,A:$tType,F:(C > A),G:(C > B),A4:(set @ C)]: ((((((image2 @ C) @ ((product_prod @ A) @ B)) @ (^ [X3:C]: (((((product_Pair @ A) @ B) @ (F @ X3)) @ (G @ X3))))) @ A4) = ((((product_Sigma @ A) @ B) @ ((((image2 @ C) @ A) @ F) @ A4)) @ (^ [X3:A]: (((((image2 @ C) @ B) @ G) @ (((inf_inf @ (set @ C)) @ ((((vimage @ C) @ A) @ F) @ (((insert2 @ A) @ X3) @ (bot_bot @ (set @ A))))) @ A4))))))))).
% 9.43/3.45 thf(fact_3116_atLeast__empty__triv, axiom, (! [A:$tType]: ((((set_ord_atLeast @ (set @ A)) @ (bot_bot @ (set @ A))) = (top_top @ (set @ (set @ A))))))).
% 9.43/3.45 thf(fact_3117_inf__None__2, axiom, (! [A:$tType]: (((inf @ A) => (! [X:(option @ A)]: (((((inf_inf @ (option @ A)) @ X) @ (none @ A)) = (none @ A)))))))).
% 9.43/3.45 thf(fact_3118_inf__None__1, axiom, (! [A:$tType]: (((inf @ A) => (! [Y:(option @ A)]: (((((inf_inf @ (option @ A)) @ (none @ A)) @ Y) = (none @ A)))))))).
% 9.43/3.45 thf(fact_3119_vimage__id, axiom, (! [A:$tType]: (((((vimage @ A) @ A) @ (id @ A)) = (id @ (set @ A)))))).
% 9.43/3.45 thf(fact_3120_set_Ocomp, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),G:(B > A)]: (((((((comp @ (set @ B)) @ (set @ C)) @ (set @ A)) @ (((vimage @ C) @ B) @ F)) @ (((vimage @ B) @ A) @ G)) = (((vimage @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)))))).
% 9.43/3.45 thf(fact_3121_set_Ocompositionality, axiom, (! [C:$tType,B:$tType,A:$tType,F:(C > B),G:(B > A),Set:(set @ A)]: ((((((vimage @ C) @ B) @ F) @ ((((vimage @ B) @ A) @ G) @ Set)) = ((((vimage @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ Set))))).
% 9.43/3.45 thf(fact_3122_vimage__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),G:(B > C),X:(set @ C)]: ((((((vimage @ A) @ B) @ F) @ ((((vimage @ B) @ C) @ G) @ X)) = ((((vimage @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ X))))).
% 9.43/3.45 thf(fact_3123_inf__unit__def, axiom, ((inf_inf @ product_unit) = (^ [Uu3:product_unit,Uv2:product_unit]: (product_Unity)))).
% 9.43/3.45 thf(fact_3124_inf__Int__eq2, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((((inf_inf @ (A > (B > $o))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R5)))) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ S2)))) = (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ R5) @ S2)))))))).
% 9.43/3.45 thf(fact_3125_surj__image__vimage__eq, axiom, (! [B:$tType,A:$tType,F:(B > A),A4:(set @ A)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((((image2 @ B) @ A) @ F) @ ((((vimage @ B) @ A) @ F) @ A4)) = A4))))).
% 9.43/3.45 thf(fact_3126_set_Oidentity, axiom, (! [A:$tType]: (((((vimage @ A) @ A) @ (^ [X3:A]: (X3))) = (id @ (set @ A)))))).
% 9.43/3.45 thf(fact_3127_vimage__Times, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > ((product_prod @ B) @ C)),A4:(set @ B),B6:(set @ C)]: ((((((vimage @ A) @ ((product_prod @ B) @ C)) @ F) @ ((((product_Sigma @ B) @ C) @ A4) @ (^ [Uu:B]: (B6)))) = (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ (((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ F)) @ A4)) @ ((((vimage @ A) @ C) @ (((((comp @ ((product_prod @ B) @ C)) @ C) @ A) @ ((product_snd @ B) @ C)) @ F)) @ B6)))))).
% 9.43/3.45 thf(fact_3128_Pair__vimage__Sigma, axiom, (! [B:$tType,A:$tType,X:B,A4:(set @ B),F:(B > (set @ A))]: ((((((member @ B) @ X) @ A4) => (((((vimage @ A) @ ((product_prod @ B) @ A)) @ (((product_Pair @ B) @ A) @ X)) @ ((((product_Sigma @ B) @ A) @ A4) @ F)) = (F @ X))) & ((~ ((((member @ B) @ X) @ A4))) => (((((vimage @ A) @ ((product_prod @ B) @ A)) @ (((product_Pair @ B) @ A) @ X)) @ ((((product_Sigma @ B) @ A) @ A4) @ F)) = (bot_bot @ (set @ A)))))))).
% 9.43/3.45 thf(fact_3129_surj__vimage__empty, axiom, (! [B:$tType,A:$tType,F:(B > A),A4:(set @ A)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((vimage @ B) @ A) @ F) @ A4) = (bot_bot @ (set @ B))) = (A4 = (bot_bot @ (set @ A)))))))).
% 9.43/3.45 thf(fact_3130_vimage__subsetD, axiom, (! [A:$tType,B:$tType,F:(B > A),B6:(set @ A),A4:(set @ B)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((vimage @ B) @ A) @ F) @ B6)) @ A4) => (((ord_less_eq @ (set @ A)) @ B6) @ ((((image2 @ B) @ A) @ F) @ A4))))))).
% 9.43/3.45 thf(fact_3131_vimage__fst, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: ((((((vimage @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ A4) = ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: ((top_top @ (set @ B))))))))).
% 9.43/3.45 thf(fact_3132_vimage__snd, axiom, (! [B:$tType,A:$tType,A4:(set @ B)]: ((((((vimage @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ A4) = ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (A4))))))).
% 9.43/3.45 thf(fact_3133_Inf__fin_Oinfinite, axiom, (! [A:$tType]: (((semilattice_inf @ A) => (! [A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (((lattic7752659483105999362nf_fin @ A) @ A4) = ((the2 @ A) @ (none @ A)))))))))).
% 9.43/3.45 thf(fact_3134_rtrancl__last__visit__node, axiom, (! [A:$tType,S:A,S3:A,R5:(set @ ((product_prod @ A) @ A)),Sh:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ S) @ S3)) @ ((transitive_rtrancl @ A) @ R5)) => (((S != Sh) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ S) @ S3)) @ ((transitive_rtrancl @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (((uminus_uminus @ (set @ A)) @ (((insert2 @ A) @ Sh) @ (bot_bot @ (set @ A))))))))))) | ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ S) @ Sh)) @ ((transitive_rtrancl @ A) @ R5)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Sh) @ S3)) @ ((transitive_rtrancl @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (((uminus_uminus @ (set @ A)) @ (((insert2 @ A) @ Sh) @ (bot_bot @ (set @ A)))))))))))))))).
% 9.43/3.45 thf(fact_3135_floor__rat__def, axiom, ((archim6421214686448440834_floor @ rat) = (^ [X3:rat]: (((the @ int) @ (^ [Z5:int]: (((((ord_less_eq @ rat) @ ((ring_1_of_int @ rat) @ Z5)) @ X3) & (((ord_less @ rat) @ X3) @ ((ring_1_of_int @ rat) @ (((plus_plus @ int) @ Z5) @ (one_one @ int)))))))))))).
% 9.43/3.45 thf(fact_3136_rat__floor__lemma, axiom, (! [A3:int,B2:int]: (((((ord_less_eq @ rat) @ ((ring_1_of_int @ rat) @ (((divide_divide @ int) @ A3) @ B2))) @ ((fract @ A3) @ B2)) & (((ord_less @ rat) @ ((fract @ A3) @ B2)) @ ((ring_1_of_int @ rat) @ (((plus_plus @ int) @ (((divide_divide @ int) @ A3) @ B2)) @ (one_one @ int)))))))).
% 9.43/3.45 thf(fact_3137_num__of__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ num) @ num) @ code_pcr_integer) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ (((((comp @ nat) @ num) @ int) @ num_of_nat) @ nat2)) @ code_num_of_integer)).
% 9.43/3.45 thf(fact_3138_prod__decode__triangle__add, axiom, (! [K:nat,M2:nat]: (((nat_prod_decode @ (((plus_plus @ nat) @ (nat_triangle @ K)) @ M2)) = ((nat_prod_decode_aux @ K) @ M2))))).
% 9.43/3.45 thf(fact_3139_wf__insert, axiom, (! [A:$tType,Y:A,X:A,R:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ R)) = (((wf @ A) @ R) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R))))))))).
% 9.43/3.45 thf(fact_3140_add__rat, axiom, (! [B2:int,D3:int,A3:int,C2:int]: (((B2 != (zero_zero @ int)) => ((D3 != (zero_zero @ int)) => ((((plus_plus @ rat) @ ((fract @ A3) @ B2)) @ ((fract @ C2) @ D3)) = ((fract @ (((plus_plus @ int) @ (((times_times @ int) @ A3) @ D3)) @ (((times_times @ int) @ C2) @ B2))) @ (((times_times @ int) @ B2) @ D3)))))))).
% 9.43/3.45 thf(fact_3141_r__le__rtrancl, axiom, (! [A:$tType,S2:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S2) @ ((transitive_rtrancl @ A) @ S2))))).
% 9.43/3.45 thf(fact_3142_rtrancl__mono__mp, axiom, (! [A:$tType,U4:(set @ ((product_prod @ A) @ A)),V6:(set @ ((product_prod @ A) @ A)),X:((product_prod @ A) @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ U4) @ V6) => ((((member @ ((product_prod @ A) @ A)) @ X) @ ((transitive_rtrancl @ A) @ U4)) => (((member @ ((product_prod @ A) @ A)) @ X) @ ((transitive_rtrancl @ A) @ V6))))))).
% 9.43/3.45 thf(fact_3143_rtrancl__mono__rightI, axiom, (! [A:$tType,S2:(set @ ((product_prod @ A) @ A)),S7:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S2) @ S7) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ S2) @ ((transitive_rtrancl @ A) @ S7)))))).
% 9.43/3.45 thf(fact_3144_converse__rtranclE_H, axiom, (! [A:$tType,U:A,V:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ R5)) => ((U != V) => (~ ((! [Vh:A]: (((U != Vh) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ Vh)) @ R5) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Vh) @ V)) @ ((transitive_rtrancl @ A) @ R5))))))))))))))).
% 9.43/3.45 thf(fact_3145_rtrancl_Ocases, axiom, (! [A:$tType,A1:A,A22:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((transitive_rtrancl @ A) @ R)) => ((A22 != A1) => (~ ((! [B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ B3)) @ ((transitive_rtrancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A22)) @ R))))))))))))).
% 9.43/3.45 thf(fact_3146_rtrancl_Osimps, axiom, (! [A:$tType,A1:A,A22:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((transitive_rtrancl @ A) @ R)) = ((? [A7:A]: (((A1 = A7) & (A22 = A7)))) | (? [A7:A,B5:A,C3:A]: (((A1 = A7) & ((A22 = C3) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ ((transitive_rtrancl @ A) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ C3)) @ R))))))))))).
% 9.43/3.45 thf(fact_3147_rtrancl_Ortrancl__refl, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ ((transitive_rtrancl @ A) @ R))))).
% 9.43/3.45 thf(fact_3148_rtrancl_Ortrancl__into__rtrancl, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_rtrancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3149_rtranclE, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((A3 != B2) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ Y4)) @ ((transitive_rtrancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ B2)) @ R))))))))))))).
% 9.43/3.45 thf(fact_3150_rtrancl__trans, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ ((transitive_rtrancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3151_rtrancl__induct, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((P @ A3) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ Y4)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => ((P @ Y4) => (P @ Z3)))))) => (P @ B2))))))).
% 9.43/3.45 thf(fact_3152_converse__rtranclE, axiom, (! [A:$tType,X:A,Z2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ ((transitive_rtrancl @ A) @ R)) => ((X != Z2) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ ((transitive_rtrancl @ A) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3153_converse__rtrancl__induct, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((P @ B2) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((P @ Z3) => (P @ Y4)))))) => (P @ A3))))))).
% 9.43/3.45 thf(fact_3154_converse__rtrancl__into__rtrancl, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_rtrancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3155_in__rtrancl__insert, axiom, (! [A:$tType,X:((product_prod @ A) @ A),R5:(set @ ((product_prod @ A) @ A)),R:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X) @ ((transitive_rtrancl @ A) @ R5)) => (((member @ ((product_prod @ A) @ A)) @ X) @ ((transitive_rtrancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ R) @ R5))))))).
% 9.43/3.45 thf(fact_3156_integer_Orep__transfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ int) @ code_pcr_integer) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ (^ [X3:int]: (X3))) @ code_int_of_integer)).
% 9.43/3.45 thf(fact_3157_integer_Oid__abs__transfer, axiom, ((((((((bNF_rel_fun @ int) @ int) @ int) @ code_integer) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ code_pcr_integer) @ (^ [X3:int]: (X3))) @ code_integer_of_int)).
% 9.43/3.45 thf(fact_3158_rtrancl__Un__separator__converseE, axiom, (! [A:$tType,A3:A,B2:A,P:(set @ ((product_prod @ A) @ A)),Q:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ P) @ Q))) => ((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ B2)) @ ((transitive_rtrancl @ A) @ P)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ Q) => (Y4 = X2))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ P))))))).
% 9.43/3.45 thf(fact_3159_rtrancl__Un__separatorE, axiom, (! [A:$tType,A3:A,B2:A,P:(set @ ((product_prod @ A) @ A)),Q:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ P) @ Q))) => ((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ X2)) @ ((transitive_rtrancl @ A) @ P)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ Q) => (X2 = Y4))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ P))))))).
% 9.43/3.45 thf(fact_3160_zero__integer_Otransfer, axiom, ((code_pcr_integer @ (zero_zero @ int)) @ (zero_zero @ code_integer))).
% 9.43/3.45 thf(fact_3161_converse__rtrancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => (((P @ Bx) @ By) => ((! [A5:A,B3:B,Aa3:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba))) @ R) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => (((P @ Aa3) @ Ba) => ((P @ A5) @ B3)))))) => ((P @ Ax) @ Ay))))))).
% 9.43/3.45 thf(fact_3162_converse__rtranclE2, axiom, (! [B:$tType,A:$tType,Xa:A,Xb:B,Za2:A,Zb:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Xa) @ Xb)) @ ((((product_Pair @ A) @ B) @ Za2) @ Zb))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => ((((((product_Pair @ A) @ B) @ Xa) @ Xb) != ((((product_Pair @ A) @ B) @ Za2) @ Zb)) => (~ ((! [A5:A,B3:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Xa) @ Xb)) @ ((((product_Pair @ A) @ B) @ A5) @ B3))) @ R) => (~ ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Za2) @ Zb))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3163_rtrancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => (((P @ Ax) @ Ay) => ((! [A5:A,B3:B,Aa3:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B3))) @ ((transitive_rtrancl @ ((product_prod @ A) @ B)) @ R)) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba))) @ R) => (((P @ A5) @ B3) => ((P @ Aa3) @ Ba)))))) => ((P @ Bx) @ By))))))).
% 9.43/3.45 thf(fact_3164_rtrancl__sub__insert__rtrancl, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),X:((product_prod @ A) @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R5)) @ ((transitive_rtrancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ X) @ R5)))))).
% 9.43/3.45 thf(fact_3165_plus__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (plus_plus @ int)) @ (plus_plus @ code_integer))).
% 9.43/3.45 thf(fact_3166_quotient__of__eq, axiom, (! [A3:int,B2:int,P5:int,Q3:int]: ((((quotient_of @ ((fract @ A3) @ B2)) = ((((product_Pair @ int) @ int) @ P5) @ Q3)) => (((fract @ P5) @ Q3) = ((fract @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_3167_less__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > $o)) @ (code_integer > $o)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ $o) @ $o) @ code_pcr_integer) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less @ int)) @ (ord_less @ code_integer))).
% 9.43/3.45 thf(fact_3168_one__integer_Otransfer, axiom, ((code_pcr_integer @ (one_one @ int)) @ (one_one @ code_integer))).
% 9.43/3.45 thf(fact_3169_less__eq__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > $o)) @ (code_integer > $o)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ $o) @ $o) @ code_pcr_integer) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less_eq @ int)) @ (ord_less_eq @ code_integer))).
% 9.43/3.45 thf(fact_3170_uminus__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (uminus_uminus @ int)) @ (uminus_uminus @ code_integer))).
% 9.43/3.45 thf(fact_3171_times__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (times_times @ int)) @ (times_times @ code_integer))).
% 9.43/3.45 thf(fact_3172_minus__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (minus_minus @ int)) @ (minus_minus @ code_integer))).
% 9.43/3.45 thf(fact_3173_abs__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (abs_abs @ int)) @ (abs_abs @ code_integer))).
% 9.43/3.45 thf(fact_3174_nat__of__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ nat) @ code_pcr_integer) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ nat2) @ code_nat_of_integer)).
% 9.43/3.45 thf(fact_3175_normalize__eq, axiom, (! [A3:int,B2:int,P5:int,Q3:int]: ((((normalize @ ((((product_Pair @ int) @ int) @ A3) @ B2)) = ((((product_Pair @ int) @ int) @ P5) @ Q3)) => (((fract @ P5) @ Q3) = ((fract @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_3176_integer__of__nat_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ code_integer) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ code_pcr_integer) @ (semiring_1_of_nat @ int)) @ code_integer_of_nat)).
% 9.43/3.45 thf(fact_3177_divide__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (divide_divide @ int)) @ (divide_divide @ code_integer))).
% 9.43/3.45 thf(fact_3178_modulo__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (modulo_modulo @ int)) @ (modulo_modulo @ code_integer))).
% 9.43/3.45 thf(fact_3179_sgn__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (sgn_sgn @ int)) @ (sgn_sgn @ code_integer))).
% 9.43/3.45 thf(fact_3180_dup_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (^ [K4:int]: ((((plus_plus @ int) @ K4) @ K4)))) @ code_dup)).
% 9.43/3.45 thf(fact_3181_take__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se2584673776208193580ke_bit @ int)) @ (bit_se2584673776208193580ke_bit @ code_integer))).
% 9.43/3.45 thf(fact_3182_not__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (bit_ri4277139882892585799ns_not @ int)) @ (bit_ri4277139882892585799ns_not @ code_integer))).
% 9.43/3.45 thf(fact_3183_and__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se5824344872417868541ns_and @ int)) @ (bit_se5824344872417868541ns_and @ code_integer))).
% 9.43/3.45 thf(fact_3184_bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (nat > $o)) @ (nat > $o)) @ code_pcr_integer) @ (^ [Y5:(nat > $o),Z4:(nat > $o)]: ((Y5 = Z4)))) @ (bit_se5641148757651400278ts_bit @ int)) @ (bit_se5641148757651400278ts_bit @ code_integer))).
% 9.43/3.45 thf(fact_3185_or__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se1065995026697491101ons_or @ int)) @ (bit_se1065995026697491101ons_or @ code_integer))).
% 9.43/3.45 thf(fact_3186_xor__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ (int > int)) @ (code_integer > code_integer)) @ code_pcr_integer) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se5824344971392196577ns_xor @ int)) @ (bit_se5824344971392196577ns_xor @ code_integer))).
% 9.43/3.45 thf(fact_3187_drop__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se4197421643247451524op_bit @ int)) @ (bit_se4197421643247451524op_bit @ code_integer))).
% 9.43/3.45 thf(fact_3188_push__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se4730199178511100633sh_bit @ int)) @ (bit_se4730199178511100633sh_bit @ code_integer))).
% 9.43/3.45 thf(fact_3189_mask__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ int) @ code_integer) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ code_pcr_integer) @ (bit_se2239418461657761734s_mask @ int)) @ (bit_se2239418461657761734s_mask @ code_integer))).
% 9.43/3.45 thf(fact_3190_unset__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se2638667681897837118et_bit @ int)) @ (bit_se2638667681897837118et_bit @ code_integer))).
% 9.43/3.45 thf(fact_3191_set__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se5668285175392031749et_bit @ int)) @ (bit_se5668285175392031749et_bit @ code_integer))).
% 9.43/3.45 thf(fact_3192_division__segment__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer) @ (euclid7384307370059645450egment @ int)) @ (euclid7384307370059645450egment @ code_integer))).
% 9.43/3.45 thf(fact_3193_flip__bit__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (int > int)) @ (code_integer > code_integer)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ code_integer) @ int) @ code_integer) @ code_pcr_integer) @ code_pcr_integer)) @ (bit_se8732182000553998342ip_bit @ int)) @ (bit_se8732182000553998342ip_bit @ code_integer))).
% 9.43/3.45 thf(fact_3194_euclidean__size__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ nat) @ code_pcr_integer) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (euclid6346220572633701492n_size @ int)) @ (euclid6346220572633701492n_size @ code_integer))).
% 9.43/3.45 thf(fact_3195_trancl__subset__Sigma__aux, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4)))) => ((A3 = B2) | (((member @ A) @ A3) @ A4))))))).
% 9.43/3.45 thf(fact_3196_Restr__rtrancl__mono, axiom, (! [A:$tType,V:A,W:A,E4:(set @ ((product_prod @ A) @ A)),U4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_rtrancl @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ E4) @ ((((product_Sigma @ A) @ A) @ U4) @ (^ [Uu:A]: (U4)))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_rtrancl @ A) @ E4)))))).
% 9.43/3.45 thf(fact_3197_quotient__of__Fract, axiom, (! [A3:int,B2:int]: (((quotient_of @ ((fract @ A3) @ B2)) = (normalize @ ((((product_Pair @ int) @ int) @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_3198_sub_Otransfer, axiom, ((((((((bNF_rel_fun @ num) @ num) @ (num > int)) @ (num > code_integer)) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ num) @ num) @ int) @ code_integer) @ (^ [Y5:num,Z4:num]: ((Y5 = Z4)))) @ code_pcr_integer)) @ (^ [M:num,N5:num]: ((((minus_minus @ int) @ ((numeral_numeral @ int) @ M)) @ ((numeral_numeral @ int) @ N5))))) @ code_sub)).
% 9.43/3.45 thf(fact_3199_rtrancl__mapI, axiom, (! [B:$tType,A:$tType,A3:A,B2:A,E4:(set @ ((product_prod @ A) @ A)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ E4)) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ A3)) @ (F @ B2))) @ ((transitive_rtrancl @ B) @ ((((image2 @ ((product_prod @ A) @ A)) @ ((product_prod @ B) @ B)) @ (((pairself @ A) @ B) @ F)) @ E4))))))).
% 9.43/3.45 thf(fact_3200_Fract_Oabs__eq, axiom, (fract = (^ [Xa4:int,X3:int]: ((abs_Rat @ ((((if @ ((product_prod @ int) @ int)) @ (X3 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ Xa4) @ X3))))))).
% 9.43/3.45 thf(fact_3201_rtrancl__last__touch, axiom, (! [A:$tType,Q3:A,Q4:A,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Q4)) @ ((transitive_rtrancl @ A) @ R5)) => ((((member @ A) @ Q3) @ S2) => (~ ((! [Qt:A]: (((((member @ A) @ Qt) @ S2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Qt)) @ ((transitive_rtrancl @ A) @ R5)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Qt) @ Q4)) @ ((transitive_rtrancl @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (S2))))))))))))))))))).
% 9.43/3.45 thf(fact_3202_rtrancl__last__visit_H, axiom, (! [A:$tType,Q3:A,Q4:A,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Q4)) @ ((transitive_rtrancl @ A) @ R5)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Q4)) @ ((transitive_rtrancl @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (S2)))))))) => (~ ((! [Qt:A]: (((((member @ A) @ Qt) @ S2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Qt)) @ ((transitive_rtrancl @ A) @ R5)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Qt) @ Q4)) @ ((transitive_rtrancl @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (S2))))))))))))))))))).
% 9.43/3.45 thf(fact_3203_rtrancl__insert, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_rtrancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_rtrancl @ A) @ R)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ A3)) @ ((transitive_rtrancl @ A) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ Y3)) @ ((transitive_rtrancl @ A) @ R)))))))))))).
% 9.43/3.45 thf(fact_3204_Fract_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ int) @ (int > ((product_prod @ int) @ int))) @ (int > rat)) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ int) @ int) @ ((product_prod @ int) @ int)) @ rat) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ pcr_rat)) @ (^ [A7:int,B5:int]: (((((if @ ((product_prod @ int) @ int)) @ (B5 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A7) @ B5))))) @ fract)).
% 9.43/3.45 thf(fact_3205_Fract__add__one, axiom, (! [N:int,M2:int]: (((N != (zero_zero @ int)) => (((fract @ (((plus_plus @ int) @ M2) @ N)) @ N) = (((plus_plus @ rat) @ ((fract @ M2) @ N)) @ (one_one @ rat))))))).
% 9.43/3.45 thf(fact_3206_Fract__def, axiom, (fract = (((((((map_fun @ int) @ int) @ (int > ((product_prod @ int) @ int))) @ (int > rat)) @ (id @ int)) @ ((((((map_fun @ int) @ int) @ ((product_prod @ int) @ int)) @ rat) @ (id @ int)) @ abs_Rat)) @ (^ [A7:int,B5:int]: (((((if @ ((product_prod @ int) @ int)) @ (B5 = (zero_zero @ int))) @ ((((product_Pair @ int) @ int) @ (zero_zero @ int)) @ (one_one @ int))) @ ((((product_Pair @ int) @ int) @ A7) @ B5))))))).
% 9.43/3.45 thf(fact_3207_old_Orec__unit__def, axiom, (! [T:$tType]: (((product_rec_unit @ T) = (^ [F12:T,X3:product_unit]: (((the @ T) @ (((product_rec_set_unit @ T) @ F12) @ X3)))))))).
% 9.43/3.45 thf(fact_3208_old_Orec__prod__def, axiom, (! [T:$tType,B:$tType,A:$tType]: (((((product_rec_prod @ A) @ B) @ T) = (^ [F12:(A > (B > T)),X3:((product_prod @ A) @ B)]: (((the @ T) @ (((((product_rec_set_prod @ A) @ B) @ T) @ F12) @ X3)))))))).
% 9.43/3.45 thf(fact_3209_list__decode_Opinduct, axiom, (! [A0:nat,P:(nat > $o)]: (((((accp @ nat) @ nat_list_decode_rel) @ A0) => (((((accp @ nat) @ nat_list_decode_rel) @ (zero_zero @ nat)) => (P @ (zero_zero @ nat))) => ((! [N3:nat]: (((((accp @ nat) @ nat_list_decode_rel) @ (suc @ N3)) => ((! [X7:nat,Y6:nat]: (((((((product_Pair @ nat) @ nat) @ X7) @ Y6) = (nat_prod_decode @ N3)) => (P @ Y6)))) => (P @ (suc @ N3)))))) => (P @ A0))))))).
% 9.43/3.45 thf(fact_3210_rtrancl__restrictI, axiom, (! [A:$tType,U:A,V:A,E4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ E4) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (R5)))))) => ((~ ((((member @ A) @ U) @ R5))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ (((rel_restrict @ A) @ E4) @ R5)))))))).
% 9.43/3.45 thf(fact_3211_inj__on__vimage__singleton, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),A3:B]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((ord_less_eq @ (set @ A)) @ (((inf_inf @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ (((insert2 @ B) @ A3) @ (bot_bot @ (set @ B))))) @ A4)) @ (((insert2 @ A) @ ((the @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ A4) & ((F @ X3) = A3)))))) @ (bot_bot @ (set @ A)))))))).
% 9.43/3.45 thf(fact_3212_inj__on__empty, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((inj_on @ A) @ B) @ F) @ (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_3213_inj__uminus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A4:(set @ A)]: (((((inj_on @ A) @ A) @ (uminus_uminus @ A)) @ A4))))))).
% 9.43/3.45 thf(fact_3214_rel__restrict__empty, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((((rel_restrict @ A) @ R5) @ (bot_bot @ (set @ A))) = R5)))).
% 9.43/3.45 thf(fact_3215_The__split__eq, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X9:A,Y8:B]: (((X = X9) & (Y = Y8)))))) = ((((product_Pair @ A) @ B) @ X) @ Y))))).
% 9.43/3.45 thf(fact_3216_inj__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ A) @ C) @ F) @ (top_top @ (set @ A))))))).
% 9.43/3.45 thf(fact_3217_inj__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = ((((inj_on @ B) @ C) @ F) @ (top_top @ (set @ B))))))).
% 9.43/3.45 thf(fact_3218_inj__on__apfst, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),A4:(set @ A)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ B)) @ ((((product_apfst @ A) @ C) @ B) @ F)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: ((top_top @ (set @ B)))))) = ((((inj_on @ A) @ C) @ F) @ A4))))).
% 9.43/3.45 thf(fact_3219_inj__on__apsnd, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),A4:(set @ B)]: ((((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_apsnd @ B) @ C) @ A) @ F)) @ ((((product_Sigma @ A) @ B) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (A4)))) = ((((inj_on @ B) @ C) @ F) @ A4))))).
% 9.43/3.45 thf(fact_3220_inj__on__insert, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:A,A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (((insert2 @ A) @ A3) @ A4)) = (((((inj_on @ A) @ B) @ F) @ A4) & (~ ((((member @ B) @ (F @ A3)) @ ((((image2 @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ A4) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))))))))))).
% 9.43/3.45 thf(fact_3221_inj__on__Int, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: (((((((inj_on @ A) @ B) @ F) @ A4) | ((((inj_on @ A) @ B) @ F) @ B6)) => ((((inj_on @ A) @ B) @ F) @ (((inf_inf @ (set @ A)) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_3222_fun_Oinj__map, axiom, (! [B:$tType,A:$tType,D:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ (D > A)) @ (D > B)) @ ((((comp @ A) @ B) @ D) @ F)) @ (top_top @ (set @ (D > A)))))))).
% 9.43/3.45 thf(fact_3223_inj__on__imageI2, axiom, (! [B:$tType,C:$tType,A:$tType,F3:(C > B),F:(A > C),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F3) @ F)) @ A4) => ((((inj_on @ A) @ C) @ F) @ A4))))).
% 9.43/3.45 thf(fact_3224_inj__on__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N6:(set @ nat)]: (((((inj_on @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N6))))))).
% 9.43/3.45 thf(fact_3225_inj__on__add, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,A4:(set @ A)]: (((((inj_on @ A) @ A) @ ((plus_plus @ A) @ A3)) @ A4))))))).
% 9.43/3.45 thf(fact_3226_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A4:(set @ A),G:(A > B),F:(A > A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A4) => (! [Xa3:A]: (((((member @ A) @ Xa3) @ A4) => (((G @ (F @ X2)) = (G @ (F @ Xa3))) = ((G @ X2) = (G @ Xa3))))))))) => (((((inj_on @ A) @ A) @ F) @ A4) => (((((inj_on @ A) @ B) @ G) @ ((((image2 @ A) @ A) @ F) @ A4)) = ((((inj_on @ A) @ B) @ G) @ A4))))))).
% 9.43/3.45 thf(fact_3227_inj__on__id2, axiom, (! [A:$tType,A4:(set @ A)]: (((((inj_on @ A) @ A) @ (^ [X3:A]: (X3))) @ A4)))).
% 9.43/3.45 thf(fact_3228_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A4:(set @ A),G:(B > A),F:(A > B)]: (((! [X2:A]: (((((member @ A) @ X2) @ A4) => ((G @ (F @ X2)) = X2)))) => ((((inj_on @ A) @ B) @ F) @ A4))))).
% 9.43/3.45 thf(fact_3229_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((X != Y) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => ((F @ X) != (F @ Y))))))))).
% 9.43/3.45 thf(fact_3230_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => (((F @ X) = (F @ Y)) = (X = Y)))))))).
% 9.43/3.45 thf(fact_3231_inj__on__cong, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F:(A > B),G:(A > B)]: (((! [A5:A]: (((((member @ A) @ A5) @ A4) => ((F @ A5) = (G @ A5))))) => (((((inj_on @ A) @ B) @ F) @ A4) = ((((inj_on @ A) @ B) @ G) @ A4)))))).
% 9.43/3.45 thf(fact_3232_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F2:(A > B),A9:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A9) => (! [Y3:A]: (((((member @ A) @ Y3) @ A9) => (((F2 @ X3) = (F2 @ Y3)) => (X3 = Y3)))))))))))))).
% 9.43/3.45 thf(fact_3233_inj__onI, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F:(A > B)]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y4) @ A4) => (((F @ X2) = (F @ Y4)) => (X2 = Y4)))))) => ((((inj_on @ A) @ B) @ F) @ A4))))).
% 9.43/3.45 thf(fact_3234_inj__onD, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((F @ X) = (F @ Y)) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => (X = Y)))))))).
% 9.43/3.45 thf(fact_3235_inj__on__diff, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((inj_on @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_3236_inj__on__id, axiom, (! [A:$tType,A4:(set @ A)]: (((((inj_on @ A) @ A) @ (id @ A)) @ A4)))).
% 9.43/3.45 thf(fact_3237_inj__Some, axiom, (! [A:$tType,A4:(set @ A)]: (((((inj_on @ A) @ (option @ A)) @ (some @ A)) @ A4)))).
% 9.43/3.45 thf(fact_3238_inj__on__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((ord_less_eq @ (set @ A)) @ B6) @ A4) => ((((inj_on @ A) @ B) @ F) @ B6)))))).
% 9.43/3.45 thf(fact_3239_subset__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ A),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B6) => ((((ord_less_eq @ (set @ A)) @ A4) @ B6) => ((((inj_on @ A) @ B) @ F) @ A4)))))).
% 9.43/3.45 thf(fact_3240_inj__fun, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ A) @ (C > B)) @ (^ [X3:A,Y3:C]: ((F @ X3)))) @ (top_top @ (set @ A))))))).
% 9.43/3.45 thf(fact_3241_inj__def, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) = (! [X3:A,Y3:A]: ((((F @ X3) = (F @ Y3)) => (X3 = Y3)))))))).
% 9.43/3.45 thf(fact_3242_inj__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = (F @ Y)) = (X = Y)))))).
% 9.43/3.45 thf(fact_3243_injI, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((! [X2:A,Y4:A]: ((((F @ X2) = (F @ Y4)) => (X2 = Y4)))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))))))).
% 9.43/3.45 thf(fact_3244_injD, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A,Y:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X) = (F @ Y)) => (X = Y)))))).
% 9.43/3.45 thf(fact_3245_inj__swap, axiom, (! [B:$tType,A:$tType,A4:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((product_swap @ A) @ B)) @ A4)))).
% 9.43/3.45 thf(fact_3246_inj__on__convol__ident, axiom, (! [B:$tType,A:$tType,F:(A > B),X10:(set @ A)]: (((((inj_on @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ B) @ X3) @ (F @ X3))))) @ X10)))).
% 9.43/3.45 thf(fact_3247_inj__Pair_I1_J, axiom, (! [B:$tType,A:$tType,C2:(A > B),S2:(set @ A)]: (((((inj_on @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ B) @ X3) @ (C2 @ X3))))) @ S2)))).
% 9.43/3.45 thf(fact_3248_inj__Pair_I2_J, axiom, (! [B:$tType,A:$tType,C2:(A > B),S2:(set @ A)]: (((((inj_on @ A) @ ((product_prod @ B) @ A)) @ (^ [X3:A]: (((((product_Pair @ B) @ A) @ (C2 @ X3)) @ X3)))) @ S2)))).
% 9.43/3.45 thf(fact_3249_inj__on__add_H, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,A4:(set @ A)]: (((((inj_on @ A) @ A) @ (^ [B5:A]: ((((plus_plus @ A) @ B5) @ A3)))) @ A4))))))).
% 9.43/3.45 thf(fact_3250_rel__restrict__lift, axiom, (! [A:$tType,X:A,Y:A,E4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((rel_restrict @ A) @ E4) @ R5)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ E4))))).
% 9.43/3.45 thf(fact_3251_rel__restrictI, axiom, (! [A:$tType,X:A,R5:(set @ A),Y:A,E4:(set @ ((product_prod @ A) @ A))]: (((~ ((((member @ A) @ X) @ R5))) => ((~ ((((member @ A) @ Y) @ R5))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ E4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((rel_restrict @ A) @ E4) @ R5)))))))).
% 9.43/3.45 thf(fact_3252_rel__restrict__notR_I1_J, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((rel_restrict @ A) @ A4) @ R5)) => (~ ((((member @ A) @ X) @ R5))))))).
% 9.43/3.45 thf(fact_3253_rel__restrict__notR_I2_J, axiom, (! [A:$tType,X:A,Y:A,A4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((rel_restrict @ A) @ A4) @ R5)) => (~ ((((member @ A) @ Y) @ R5))))))).
% 9.43/3.45 thf(fact_3254_rel__restrict__union, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B6:(set @ A)]: (((((rel_restrict @ A) @ R5) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((rel_restrict @ A) @ (((rel_restrict @ A) @ R5) @ A4)) @ B6))))).
% 9.43/3.45 thf(fact_3255_rel__restrict__sub, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ R5) @ A4)) @ R5)))).
% 9.43/3.45 thf(fact_3256_rel__restrict__mono, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),B6:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ A4) @ B6) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ A4) @ R5)) @ (((rel_restrict @ A) @ B6) @ R5)))))).
% 9.43/3.45 thf(fact_3257_finite__rel__restrict, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R5) => ((finite_finite2 @ ((product_prod @ A) @ A)) @ (((rel_restrict @ A) @ R5) @ A4)))))).
% 9.43/3.45 thf(fact_3258_option_Oinj__map, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ (option @ A)) @ (option @ B)) @ (((map_option @ A) @ B) @ F)) @ (top_top @ (set @ (option @ A)))))))).
% 9.43/3.45 thf(fact_3259_linorder__inj__onI, axiom, (! [B:$tType,A:$tType]: (((order @ A) => (! [A4:(set @ A),F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => ((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y4) @ A4) => ((F @ X2) != (F @ Y4))))))) => ((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y4) @ A4) => ((((ord_less_eq @ A) @ X2) @ Y4) | (((ord_less_eq @ A) @ Y4) @ X2)))))) => ((((inj_on @ A) @ B) @ F) @ A4))))))))).
% 9.43/3.45 thf(fact_3260_linorder__injI, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [F:(A > B)]: (((! [X2:A,Y4:A]: (((((ord_less @ A) @ X2) @ Y4) => ((F @ X2) != (F @ Y4))))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_3261_inj__add__left, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A]: (((((inj_on @ A) @ A) @ ((plus_plus @ A) @ A3)) @ (top_top @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3262_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),C5:(set @ A),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C5) => ((((ord_less_eq @ (set @ A)) @ A4) @ C5) => ((((ord_less_eq @ (set @ A)) @ B6) @ C5) => ((((((image2 @ A) @ B) @ F) @ A4) = ((((image2 @ A) @ B) @ F) @ B6)) = (A4 = B6)))))))).
% 9.43/3.45 thf(fact_3263_inj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ A),A3:A,A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B6) => ((((member @ A) @ A3) @ B6) => ((((ord_less_eq @ (set @ A)) @ A4) @ B6) => ((((member @ B) @ (F @ A3)) @ ((((image2 @ A) @ B) @ F) @ A4)) = (((member @ A) @ A3) @ A4)))))))).
% 9.43/3.45 thf(fact_3264_inj__img__insertE, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:B,B6:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((~ ((((member @ B) @ X) @ B6))) => (((((insert2 @ B) @ X) @ B6) = ((((image2 @ A) @ B) @ F) @ A4)) => (~ ((! [X5:A,A11:(set @ A)]: (((~ ((((member @ A) @ X5) @ A11))) => ((A4 = (((insert2 @ A) @ X5) @ A11)) => ((X = (F @ X5)) => (B6 != ((((image2 @ A) @ B) @ F) @ A11))))))))))))))).
% 9.43/3.45 thf(fact_3265_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:B]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ B2) @ ((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A)))) = (? [X3:A]: (((B2 = (F @ X3)) & (! [Y3:A]: (((B2 = (F @ Y3)) => (Y3 = X3)))))))))))).
% 9.43/3.45 thf(fact_3266_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((image2 @ A) @ B) @ F) @ A4) = ((((image2 @ A) @ B) @ F) @ B6)) = (A4 = B6)))))).
% 9.43/3.45 thf(fact_3267_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:A,A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ (F @ A3)) @ ((((image2 @ A) @ B) @ F) @ A4)) = (((member @ A) @ A3) @ A4)))))).
% 9.43/3.45 thf(fact_3268_comp__inj__on, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),A4:(set @ A),G:(B > C)]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((((inj_on @ B) @ C) @ G) @ ((((image2 @ A) @ B) @ F) @ A4)) => ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ A4)))))).
% 9.43/3.45 thf(fact_3269_inj__on__imageI, axiom, (! [B:$tType,C:$tType,A:$tType,G:(C > B),F:(A > C),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ G) @ F)) @ A4) => ((((inj_on @ C) @ B) @ G) @ ((((image2 @ A) @ C) @ F) @ A4)))))).
% 9.43/3.45 thf(fact_3270_comp__inj__on__iff, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),A4:(set @ A),F3:(B > C)]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((((inj_on @ B) @ C) @ F3) @ ((((image2 @ A) @ B) @ F) @ A4)) = ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F3) @ F)) @ A4)))))).
% 9.43/3.45 thf(fact_3271_inj__on__Un__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) => ((((((image2 @ A) @ B) @ F) @ A4) = ((((image2 @ A) @ B) @ F) @ B6)) = (A4 = B6)))))).
% 9.43/3.45 thf(fact_3272_inj__compose, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),G:(C > A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((inj_on @ C) @ A) @ G) @ (top_top @ (set @ C))) => ((((inj_on @ C) @ B) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ (top_top @ (set @ C)))))))).
% 9.43/3.45 thf(fact_3273_inj__on__strict__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ A),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B6) => ((((ord_less @ (set @ A)) @ A4) @ B6) => (((ord_less @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ F) @ B6))))))).
% 9.43/3.45 thf(fact_3274_inj__on__fun__updI, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),Y:B,X:A]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((~ ((((member @ B) @ Y) @ ((((image2 @ A) @ B) @ F) @ A4)))) => ((((inj_on @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ Y)) @ A4)))))).
% 9.43/3.45 thf(fact_3275_inj__on__fst__map__to__set, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (((map_to_set @ A) @ B) @ M2))))).
% 9.43/3.45 thf(fact_3276_inj__diff__right, axiom, (! [A:$tType]: (((ab_group_add @ A) => (! [A3:A]: (((((inj_on @ A) @ A) @ (^ [B5:A]: ((((minus_minus @ A) @ B5) @ A3)))) @ (top_top @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3277_inj__singleton, axiom, (! [A:$tType,A4:(set @ A)]: (((((inj_on @ A) @ (set @ A)) @ (^ [X3:A]: ((((insert2 @ A) @ X3) @ (bot_bot @ (set @ A)))))) @ A4)))).
% 9.43/3.45 thf(fact_3278_finite__Collect, axiom, (! [A:$tType,B:$tType,S2:(set @ A),F:(B > A)]: ((((finite_finite2 @ A) @ S2) => (((((inj_on @ B) @ A) @ F) @ (top_top @ (set @ B))) => ((finite_finite2 @ B) @ ((collect @ B) @ (^ [A7:B]: ((((member @ A) @ (F @ A7)) @ S2)))))))))).
% 9.43/3.45 thf(fact_3279_swap__inj__on, axiom, (! [B:$tType,A:$tType,A4:(set @ ((product_prod @ A) @ B))]: (((((inj_on @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [I3:A,J2:B]: (((((product_Pair @ B) @ A) @ J2) @ I3))))) @ A4)))).
% 9.43/3.45 thf(fact_3280_rel__restrict__mono2, axiom, (! [A:$tType,R5:(set @ A),S2:(set @ A),A4:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ A)) @ R5) @ S2) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ A4) @ S2)) @ (((rel_restrict @ A) @ A4) @ R5)))))).
% 9.43/3.45 thf(fact_3281_inj__image__subset__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ F) @ B6)) = (((ord_less_eq @ (set @ A)) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_3282_inj__on__image__Int, axiom, (! [B:$tType,A:$tType,F:(A > B),C5:(set @ A),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C5) => ((((ord_less_eq @ (set @ A)) @ A4) @ C5) => ((((ord_less_eq @ (set @ A)) @ B6) @ C5) => (((((image2 @ A) @ B) @ F) @ (((inf_inf @ (set @ A)) @ A4) @ B6)) = (((inf_inf @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ F) @ B6))))))))).
% 9.43/3.45 thf(fact_3283_image__Int, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((image2 @ A) @ B) @ F) @ (((inf_inf @ (set @ A)) @ A4) @ B6)) = (((inf_inf @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ F) @ B6))))))).
% 9.43/3.45 thf(fact_3284_inj__on__image__set__diff, axiom, (! [B:$tType,A:$tType,F:(A > B),C5:(set @ A),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C5) => ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A4) @ B6)) @ C5) => ((((ord_less_eq @ (set @ A)) @ B6) @ C5) => (((((image2 @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ A4) @ B6)) = (((minus_minus @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ F) @ B6))))))))).
% 9.43/3.45 thf(fact_3285_sum_Oreindex, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [H3:(B > C),A4:(set @ B),G:(C > A)]: ((((((inj_on @ B) @ C) @ H3) @ A4) => (((((groups7311177749621191930dd_sum @ C) @ A) @ G) @ ((((image2 @ B) @ C) @ H3) @ A4)) = ((((groups7311177749621191930dd_sum @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H3)) @ A4))))))))).
% 9.43/3.45 thf(fact_3286_image__set__diff, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((image2 @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ A4) @ B6)) = (((minus_minus @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ F) @ B6))))))).
% 9.43/3.45 thf(fact_3287_prod_Oreindex, axiom, (! [C:$tType,A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [H3:(B > C),A4:(set @ B),G:(C > A)]: ((((((inj_on @ B) @ C) @ H3) @ A4) => (((((groups7121269368397514597t_prod @ C) @ A) @ G) @ ((((image2 @ B) @ C) @ H3) @ A4)) = ((((groups7121269368397514597t_prod @ B) @ A) @ (((((comp @ C) @ A) @ B) @ G) @ H3)) @ A4))))))))).
% 9.43/3.45 thf(fact_3288_inj__vimage__image__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((vimage @ A) @ B) @ F) @ ((((image2 @ A) @ B) @ F) @ A4)) = A4))))).
% 9.43/3.45 thf(fact_3289_rel__restrict__def, axiom, (! [A:$tType]: (((rel_restrict @ A) = (^ [R6:(set @ ((product_prod @ A) @ A)),A9:(set @ A)]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [V2:A,W2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V2) @ W2)) @ R6) & ((~ ((((member @ A) @ V2) @ A9))) & (~ ((((member @ A) @ W2) @ A9))))))))))))))).
% 9.43/3.45 thf(fact_3290_finite__imp__inj__to__nat__seg_H, axiom, (! [A:$tType,A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (~ ((! [F4:(A > nat)]: (((? [N3:nat]: ((((((image2 @ A) @ nat) @ F4) @ A4) = ((collect @ nat) @ (^ [I3:nat]: ((((ord_less @ nat) @ I3) @ N3))))))) => (~ (((((inj_on @ A) @ nat) @ F4) @ A4)))))))))))).
% 9.43/3.45 thf(fact_3291_set__to__map__inverse, axiom, (! [B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B))]: ((((((inj_on @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S2) => ((((map_to_set @ A) @ B) @ (((set_to_map @ A) @ B) @ S2)) = S2))))).
% 9.43/3.45 thf(fact_3292_rel__restrict__Int__empty, axiom, (! [A:$tType,A4:(set @ A),R5:(set @ ((product_prod @ A) @ A))]: ((((((inf_inf @ (set @ A)) @ A4) @ ((field2 @ A) @ R5)) = (bot_bot @ (set @ A))) => ((((rel_restrict @ A) @ R5) @ A4) = R5))))).
% 9.43/3.45 thf(fact_3293_Field__rel__restrict, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((field2 @ A) @ (((rel_restrict @ A) @ R5) @ A4))) @ (((minus_minus @ (set @ A)) @ ((field2 @ A) @ R5)) @ A4))))).
% 9.43/3.45 thf(fact_3294_vimage__subsetI, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((((image2 @ A) @ B) @ F) @ A4)) => (((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ B6)) @ A4)))))).
% 9.43/3.45 thf(fact_3295_inj__image__Compl__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ A4))) @ ((uminus_uminus @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4))))))).
% 9.43/3.45 thf(fact_3296_The__case__prod, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: ((((the @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P)) = ((the @ ((product_prod @ A) @ B)) @ (^ [Xy:((product_prod @ A) @ B)]: (((P @ (((product_fst @ A) @ B) @ Xy)) @ (((product_snd @ A) @ B) @ Xy))))))))).
% 9.43/3.45 thf(fact_3297_inj__on__disjoint__Un, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),G:(A > B),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((((inj_on @ A) @ B) @ G) @ B6) => (((((inf_inf @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ ((((image2 @ A) @ B) @ G) @ B6)) = (bot_bot @ (set @ B))) => ((((inj_on @ A) @ B) @ (^ [X3:A]: (((((if @ B) @ (((member @ A) @ X3) @ A4)) @ (F @ X3)) @ (G @ X3))))) @ (((sup_sup @ (set @ A)) @ A4) @ B6)))))))).
% 9.43/3.45 thf(fact_3298_set__to__map__simp, axiom, (! [B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B)),K:A,V:B]: ((((((inj_on @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S2) => ((((((set_to_map @ A) @ B) @ S2) @ K) = ((some @ B) @ V)) = (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V)) @ S2)))))).
% 9.43/3.45 thf(fact_3299_rel__restrict__compl, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: (((((inf_inf @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ R5) @ A4)) @ (((rel_restrict @ A) @ R5) @ ((uminus_uminus @ (set @ A)) @ A4))) = (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.45 thf(fact_3300_homo__rel__restrict__mono, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),B6:(set @ A),A4:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ B6) @ (^ [Uu:A]: (B6)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ R5) @ A4)) @ ((((product_Sigma @ A) @ A) @ (((minus_minus @ (set @ A)) @ B6) @ A4)) @ (^ [Uu:A]: ((((minus_minus @ (set @ A)) @ B6) @ A4))))))))).
% 9.43/3.45 thf(fact_3301_rel__restrict__alt__def, axiom, (! [A:$tType]: (((rel_restrict @ A) = (^ [R6:(set @ ((product_prod @ A) @ A)),A9:(set @ A)]: ((((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R6) @ ((((product_Sigma @ A) @ A) @ ((uminus_uminus @ (set @ A)) @ A9)) @ (^ [Uu:A]: (((uminus_uminus @ (set @ A)) @ A9))))))))))).
% 9.43/3.45 thf(fact_3302_inj__on__Un, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((((inj_on @ A) @ B) @ F) @ A4) & (((((inj_on @ A) @ B) @ F) @ B6) & ((((inf_inf @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ A4) @ B6))) @ ((((image2 @ A) @ B) @ F) @ (((minus_minus @ (set @ A)) @ B6) @ A4))) = (bot_bot @ (set @ B))))))))).
% 9.43/3.45 thf(fact_3303_inj__vimage__singleton, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:B]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ (((insert2 @ B) @ A3) @ (bot_bot @ (set @ B))))) @ (((insert2 @ A) @ ((the @ A) @ (^ [X3:A]: (((F @ X3) = A3))))) @ (bot_bot @ (set @ A)))))))).
% 9.43/3.45 thf(fact_3304_If__the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,I:A,C5:(set @ A),G:(A > B),X:A]: (((((member @ A) @ I) @ C5) => (((((inj_on @ A) @ B) @ G) @ C5) => (((((((comp @ B) @ A) @ A) @ (^ [I3:B]: (((((if @ A) @ (((member @ B) @ I3) @ ((((image2 @ A) @ B) @ G) @ C5))) @ (((((the_inv_into @ A) @ B) @ C5) @ G) @ I3)) @ X)))) @ G) @ I) = ((id @ A) @ I))))))).
% 9.43/3.45 thf(fact_3305_ran__map__upd__Some, axiom, (! [B:$tType,A:$tType,M2:(B > (option @ A)),X:B,Y:A,Z2:A]: ((((M2 @ X) = ((some @ A) @ Y)) => (((((inj_on @ B) @ (option @ A)) @ M2) @ (((dom @ B) @ A) @ M2)) => ((~ ((((member @ A) @ Z2) @ (((ran @ B) @ A) @ M2)))) => ((((ran @ B) @ A) @ (((((fun_upd @ B) @ (option @ A)) @ M2) @ X) @ ((some @ A) @ Z2))) = (((sup_sup @ (set @ A)) @ (((minus_minus @ (set @ A)) @ (((ran @ B) @ A) @ M2)) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A))))) @ (((insert2 @ A) @ Z2) @ (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_3306_trancl__multi__insert, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),X10:(set @ A),M2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ X10) @ (^ [Uu:A]: ((((insert2 @ A) @ M2) @ (bot_bot @ (set @ A))))))))) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)))) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ X10) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ X2)) @ ((transitive_rtrancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ M2) @ B2)) @ ((transitive_rtrancl @ A) @ R))))))))))))))).
% 9.43/3.45 thf(fact_3307_trancl__multi__insert2, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),M2:A,X10:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ (((insert2 @ A) @ M2) @ (bot_bot @ (set @ A)))) @ (^ [Uu:A]: (X10)))))) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)))) => (~ ((! [X2:A]: (((((member @ A) @ X2) @ X10) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ M2)) @ ((transitive_rtrancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ B2)) @ ((transitive_rtrancl @ A) @ R))))))))))))))).
% 9.43/3.45 thf(fact_3308_the__inv__f__o__f__id, axiom, (! [B:$tType,A:$tType,F:(A > B),Z2:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((((comp @ B) @ A) @ A) @ ((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ F) @ Z2) = ((id @ A) @ Z2)))))).
% 9.43/3.45 thf(fact_3309_restrict__map__self, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((((restrict_map @ A) @ B) @ M2) @ (((dom @ A) @ B) @ M2)) = M2)))).
% 9.43/3.45 thf(fact_3310_same__fst__trancl, axiom, (! [B:$tType,A:$tType,P:(A > $o),R5:(A > (set @ ((product_prod @ B) @ B)))]: ((((transitive_trancl @ ((product_prod @ A) @ B)) @ ((((same_fst @ A) @ B) @ P) @ R5)) = ((((same_fst @ A) @ B) @ P) @ (^ [X3:A]: (((transitive_trancl @ B) @ (R5 @ X3))))))))).
% 9.43/3.45 thf(fact_3311_dom__eq__empty__conv, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B))]: ((((((dom @ A) @ B) @ F) = (bot_bot @ (set @ A))) = (F = (^ [X3:A]: ((none @ B)))))))).
% 9.43/3.45 thf(fact_3312_fun__upd__None__if__notin__dom, axiom, (! [B:$tType,A:$tType,K:A,M2:(A > (option @ B))]: (((~ ((((member @ A) @ K) @ (((dom @ A) @ B) @ M2)))) => ((((((fun_upd @ A) @ (option @ B)) @ M2) @ K) @ (none @ B)) = M2))))).
% 9.43/3.45 thf(fact_3313_the__inv__into__onto, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((((image2 @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A4) @ F)) @ ((((image2 @ A) @ B) @ F) @ A4)) = A4))))).
% 9.43/3.45 thf(fact_3314_dom__empty, axiom, (! [B:$tType,A:$tType]: (((((dom @ A) @ B) @ (^ [X3:A]: ((none @ B)))) = (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_3315_map__update__eta__repair_I1_J, axiom, (! [B:$tType,A:$tType,K:A,V:B,M2:(A > (option @ B))]: (((((dom @ A) @ B) @ (^ [X3:A]: (((((if @ (option @ B)) @ (X3 = K)) @ ((some @ B) @ V)) @ (M2 @ X3))))) = (((insert2 @ A) @ K) @ (((dom @ A) @ B) @ M2)))))).
% 9.43/3.45 thf(fact_3316_dom__const_H, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((dom @ A) @ B) @ (^ [X3:A]: (((some @ B) @ (F @ X3))))) = (top_top @ (set @ A)))))).
% 9.43/3.45 thf(fact_3317_dom__mmupd, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),K2:(set @ A),V:B]: (((((dom @ A) @ B) @ (((((map_mmupd @ A) @ B) @ M2) @ K2) @ V)) = (((sup_sup @ (set @ A)) @ (((dom @ A) @ B) @ M2)) @ K2))))).
% 9.43/3.45 thf(fact_3318_trancl__single, axiom, (! [A:$tType,A3:A,B2:A]: ((((transitive_trancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))) = (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.45 thf(fact_3319_restrict__map__inv, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B))]: ((((((restrict_map @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ (((dom @ A) @ B) @ F))) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_3320_dom__fun__upd, axiom, (! [B:$tType,A:$tType,Y:(option @ B),F:(A > (option @ B)),X:A]: ((((Y = (none @ B)) => ((((dom @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ Y)) = (((minus_minus @ (set @ A)) @ (((dom @ A) @ B) @ F)) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))))) & ((Y != (none @ B)) => ((((dom @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ Y)) = (((insert2 @ A) @ X) @ (((dom @ A) @ B) @ F)))))))).
% 9.43/3.45 thf(fact_3321_domI, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),A3:B,B2:A]: ((((M2 @ A3) = ((some @ A) @ B2)) => (((member @ B) @ A3) @ (((dom @ B) @ A) @ M2)))))).
% 9.43/3.45 thf(fact_3322_domD, axiom, (! [A:$tType,B:$tType,A3:A,M2:(A > (option @ B))]: (((((member @ A) @ A3) @ (((dom @ A) @ B) @ M2)) => (? [B3:B]: (((M2 @ A3) = ((some @ B) @ B3)))))))).
% 9.43/3.45 thf(fact_3323_domIff, axiom, (! [A:$tType,B:$tType,A3:A,M2:(A > (option @ B))]: (((((member @ A) @ A3) @ (((dom @ A) @ B) @ M2)) = ((M2 @ A3) != (none @ B)))))).
% 9.43/3.45 thf(fact_3324_nempty__dom, axiom, (! [B:$tType,A:$tType,E2:(A > (option @ B))]: (((E2 != (^ [X3:A]: ((none @ B)))) => (~ ((! [M5:A]: ((~ ((((member @ A) @ M5) @ (((dom @ A) @ B) @ E2)))))))))))).
% 9.43/3.45 thf(fact_3325_trancl__sub, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((transitive_trancl @ A) @ R5))))).
% 9.43/3.45 thf(fact_3326_trancl__mono__mp, axiom, (! [A:$tType,U4:(set @ ((product_prod @ A) @ A)),V6:(set @ ((product_prod @ A) @ A)),X:((product_prod @ A) @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ U4) @ V6) => ((((member @ ((product_prod @ A) @ A)) @ X) @ ((transitive_trancl @ A) @ U4)) => (((member @ ((product_prod @ A) @ A)) @ X) @ ((transitive_trancl @ A) @ V6))))))).
% 9.43/3.45 thf(fact_3327_trancl_Ocases, axiom, (! [A:$tType,A1:A,A22:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((transitive_trancl @ A) @ R)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ R))) => (~ ((! [B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ B3)) @ ((transitive_trancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A22)) @ R))))))))))))).
% 9.43/3.45 thf(fact_3328_trancl_Osimps, axiom, (! [A:$tType,A1:A,A22:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A1) @ A22)) @ ((transitive_trancl @ A) @ R)) = ((? [A7:A,B5:A]: (((A1 = A7) & ((A22 = B5) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R))))) | (? [A7:A,B5:A,C3:A]: (((A1 = A7) & ((A22 = C3) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ ((transitive_trancl @ A) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ C3)) @ R))))))))))).
% 9.43/3.45 thf(fact_3329_trancl_Or__into__trancl, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)))))).
% 9.43/3.45 thf(fact_3330_tranclE, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R))) => (~ ((! [C4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C4)) @ ((transitive_trancl @ A) @ R)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ C4) @ B2)) @ R))))))))))))).
% 9.43/3.45 thf(fact_3331_trancl__trans, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_trancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3332_trancl__induct, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)) => ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ Y4)) @ R) => (P @ Y4)))) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ Y4)) @ ((transitive_trancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => ((P @ Y4) => (P @ Z3)))))) => (P @ B2))))))).
% 9.43/3.45 thf(fact_3333_r__r__into__trancl, axiom, (! [A:$tType,A3:A,B2:A,R5:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_trancl @ A) @ R5))))))).
% 9.43/3.45 thf(fact_3334_converse__tranclE, axiom, (! [A:$tType,X:A,Z2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ ((transitive_trancl @ A) @ R)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ R))) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ ((transitive_trancl @ A) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3335_irrefl__trancl__rD, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ ((transitive_trancl @ A) @ R)))))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (X != Y)))))).
% 9.43/3.45 thf(fact_3336_Transitive__Closure_Otrancl__into__trancl, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3337_trancl__into__trancl2, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_trancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3338_trancl__trans__induct, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),P:(A > (A > $o))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ R)) => ((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R) => ((P @ X2) @ Y4)))) => ((! [X2:A,Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ ((transitive_trancl @ A) @ R)) => (((P @ X2) @ Y4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ ((transitive_trancl @ A) @ R)) => (((P @ Y4) @ Z3) => ((P @ X2) @ Z3))))))) => ((P @ X) @ Y))))))).
% 9.43/3.45 thf(fact_3339_converse__trancl__induct, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)) => ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ B2)) @ R) => (P @ Y4)))) => ((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ B2)) @ ((transitive_trancl @ A) @ R)) => ((P @ Z3) => (P @ Y4)))))) => (P @ A3))))))).
% 9.43/3.45 thf(fact_3340_dom__def, axiom, (! [B:$tType,A:$tType]: ((((dom @ A) @ B) = (^ [M:(A > (option @ B))]: (((collect @ A) @ (^ [A7:A]: (((M @ A7) != (none @ B))))))))))).
% 9.43/3.45 thf(fact_3341_the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:A]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((member @ A) @ X) @ A4) => ((((((the_inv_into @ A) @ B) @ A4) @ F) @ (F @ X)) = X)))))).
% 9.43/3.45 thf(fact_3342_the__inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:A,Y:B]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((F @ X) = Y) => ((((member @ A) @ X) @ A4) => ((((((the_inv_into @ A) @ B) @ A4) @ F) @ Y) = X))))))).
% 9.43/3.45 thf(fact_3343_insert__dom, axiom, (! [A:$tType,B:$tType,F:(B > (option @ A)),X:B,Y:A]: ((((F @ X) = ((some @ A) @ Y)) => ((((insert2 @ B) @ X) @ (((dom @ B) @ A) @ F)) = (((dom @ B) @ A) @ F)))))).
% 9.43/3.45 thf(fact_3344_trancl__rtrancl__trancl, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3345_rtrancl__trancl__trancl, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_trancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3346_rtrancl__into__trancl2, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3347_rtrancl__into__trancl1, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),C2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ C2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ C2)) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_3348_rtrancl__eq__or__trancl, axiom, (! [A:$tType,X:A,Y:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R5)) = ((X = Y) | ((X != Y) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ R5)))))))).
% 9.43/3.45 thf(fact_3349_trancl__into__rtrancl, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R)))))).
% 9.43/3.45 thf(fact_3350_tranclD2, axiom, (! [A:$tType,X:A,Y:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ R5)) => (? [Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ ((transitive_rtrancl @ A) @ R5)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ Y)) @ R5)))))))).
% 9.43/3.45 thf(fact_3351_rtranclD, axiom, (! [A:$tType,A3:A,B2:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_rtrancl @ A) @ R5)) => ((A3 = B2) | ((A3 != B2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ R5)))))))).
% 9.43/3.45 thf(fact_3352_tranclD, axiom, (! [A:$tType,X:A,Y:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ R5)) => (? [Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z3)) @ R5) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ Y)) @ ((transitive_rtrancl @ A) @ R5))))))))).
% 9.43/3.45 thf(fact_3353_map__dom__ran__finite, axiom, (! [B:$tType,A:$tType,M3:(A > (option @ B))]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ M3)) => ((finite_finite2 @ B) @ (((ran @ A) @ B) @ M3)))))).
% 9.43/3.45 thf(fact_3354_rel__restrict__trancl__notR_I2_J, axiom, (! [A:$tType,V:A,W:A,E4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ E4) @ R5))) => (~ ((((member @ A) @ W) @ R5))))))).
% 9.43/3.45 thf(fact_3355_rel__restrict__trancl__notR_I1_J, axiom, (! [A:$tType,V:A,W:A,E4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ E4) @ R5))) => (~ ((((member @ A) @ V) @ R5))))))).
% 9.43/3.45 thf(fact_3356_rel__restrict__trancl__mem, axiom, (! [A:$tType,A3:A,B2:A,A4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ A4) @ R5))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (((rel_restrict @ A) @ ((transitive_trancl @ A) @ A4)) @ R5)))))).
% 9.43/3.45 thf(fact_3357_trancl__induct2, axiom, (! [A:$tType,B:$tType,Ax:A,Ay:B,Bx:A,By:B,R:(set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By))) @ ((transitive_trancl @ ((product_prod @ A) @ B)) @ R)) => ((! [A5:A,B3:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B3))) @ R) => ((P @ A5) @ B3)))) => ((! [A5:A,B3:B,Aa3:A,Ba:B]: (((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B3))) @ ((transitive_trancl @ ((product_prod @ A) @ B)) @ R)) => ((((member @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))) @ ((((product_Pair @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba))) @ R) => (((P @ A5) @ B3) => ((P @ Aa3) @ Ba)))))) => ((P @ Bx) @ By))))))).
% 9.43/3.45 thf(fact_3358_card__map__to__set, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((finite_card @ ((product_prod @ A) @ B)) @ (((map_to_set @ A) @ B) @ M2)) = ((finite_card @ A) @ (((dom @ A) @ B) @ M2)))))).
% 9.43/3.45 thf(fact_3359_trancl__sub__insert__trancl, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),X:((product_prod @ A) @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ R5)) @ ((transitive_trancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ X) @ R5)))))).
% 9.43/3.45 thf(fact_3360_rel__restrict__trancl__sub, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ A4) @ R5))) @ (((rel_restrict @ A) @ ((transitive_trancl @ A) @ A4)) @ R5))))).
% 9.43/3.45 thf(fact_3361_le__map__dom__mono, axiom, (! [B:$tType,A:$tType]: (((preorder @ B) => (! [M2:(A > (option @ B)),M7:(A > (option @ B))]: (((((ord_less_eq @ (A > (option @ B))) @ M2) @ M7) => (((ord_less_eq @ (set @ A)) @ (((dom @ A) @ B) @ M2)) @ (((dom @ A) @ B) @ M7))))))))).
% 9.43/3.45 thf(fact_3362_the__inv__into__def, axiom, (! [B:$tType,A:$tType]: ((((the_inv_into @ A) @ B) = (^ [A9:(set @ A),F2:(A > B),X3:B]: (((the @ A) @ (^ [Y3:A]: (((((member @ A) @ Y3) @ A9) & ((F2 @ Y3) = X3))))))))))).
% 9.43/3.45 thf(fact_3363_finite__map__freshness, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B))]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ F)) => ((~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))) => (? [X2:A]: (((F @ X2) = (none @ B))))))))).
% 9.43/3.45 thf(fact_3364_dom__minus, axiom, (! [A:$tType,B:$tType,F:(B > (option @ A)),X:B,A4:(set @ B)]: ((((F @ X) = (none @ A)) => ((((minus_minus @ (set @ B)) @ (((dom @ B) @ A) @ F)) @ (((insert2 @ B) @ X) @ A4)) = (((minus_minus @ (set @ B)) @ (((dom @ B) @ A) @ F)) @ A4)))))).
% 9.43/3.45 thf(fact_3365_restrict__map__eq_I1_J, axiom, (! [A:$tType,B:$tType,M2:(B > (option @ A)),A4:(set @ B),K:B]: ((((((((restrict_map @ B) @ A) @ M2) @ A4) @ K) = (none @ A)) = (~ ((((member @ B) @ K) @ (((inf_inf @ (set @ B)) @ (((dom @ B) @ A) @ M2)) @ A4)))))))).
% 9.43/3.45 thf(fact_3366_map__card__eq__iff, axiom, (! [B:$tType,A:$tType,M3:(A > (option @ B)),X:A,Y:A]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ M3)) => ((((finite_card @ A) @ (((dom @ A) @ B) @ M3)) = ((finite_card @ B) @ (((ran @ A) @ B) @ M3))) => ((((member @ A) @ X) @ (((dom @ A) @ B) @ M3)) => (((M3 @ X) = (M3 @ Y)) = (X = Y)))))))).
% 9.43/3.45 thf(fact_3367_wf__linord__ex__has__least, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),P:(B > $o),K:B,M2:(B > A)]: ((((wf @ A) @ R) => ((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ ((transitive_trancl @ A) @ R)) = (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ ((transitive_rtrancl @ A) @ R))))))) => ((P @ K) => (? [X2:B]: (((P @ X2) & (! [Y6:B]: (((P @ Y6) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (M2 @ X2)) @ (M2 @ Y6))) @ ((transitive_rtrancl @ A) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3368_trancl__union__outside, axiom, (! [A:$tType,V:A,W:A,E4:(set @ ((product_prod @ A) @ A)),U4:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_trancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ E4) @ U4))) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_trancl @ A) @ E4)))) => (? [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ X2)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ E4) @ U4))) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ U4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ W)) @ ((transitive_rtrancl @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ E4) @ U4)))))))))))).
% 9.43/3.45 thf(fact_3369_trancl__over__edgeE, axiom, (! [A:$tType,U:A,W:A,V1:A,V22:A,E4:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ W)) @ ((transitive_trancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V1) @ V22)) @ E4))) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ W)) @ ((transitive_trancl @ A) @ E4)))) => (~ (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V1)) @ ((transitive_rtrancl @ A) @ E4)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V22) @ W)) @ ((transitive_rtrancl @ A) @ E4)))))))))))).
% 9.43/3.45 thf(fact_3370_finite__map__to__set, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ (((map_to_set @ A) @ B) @ M2)) = ((finite_finite2 @ A) @ (((dom @ A) @ B) @ M2)))))).
% 9.43/3.45 thf(fact_3371_map__to__set__dom, axiom, (! [B:$tType,A:$tType]: ((((dom @ A) @ B) = (^ [M:(A > (option @ B))]: (((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (((map_to_set @ A) @ B) @ M)))))))).
% 9.43/3.45 thf(fact_3372_inj__on__the__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((inj_on @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A4) @ F)) @ ((((image2 @ A) @ B) @ F) @ A4)))))).
% 9.43/3.45 thf(fact_3373_f__the__inv__into__f, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),Y:B]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((member @ B) @ Y) @ ((((image2 @ A) @ B) @ F) @ A4)) => ((F @ (((((the_inv_into @ A) @ B) @ A4) @ F) @ Y)) = Y)))))).
% 9.43/3.45 thf(fact_3374_the__inv__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),X:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ (F @ X)) = X))))).
% 9.43/3.45 thf(fact_3375_set__to__map__dom, axiom, (! [B:$tType,A:$tType,S2:(set @ ((product_prod @ A) @ B))]: (((((dom @ A) @ B) @ (((set_to_map @ A) @ B) @ S2)) = ((((image2 @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ S2))))).
% 9.43/3.45 thf(fact_3376_Restr__trancl__mono, axiom, (! [A:$tType,V:A,W:A,E4:(set @ ((product_prod @ A) @ A)),U4:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_trancl @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ E4) @ ((((product_Sigma @ A) @ A) @ U4) @ (^ [Uu:A]: (U4)))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V) @ W)) @ ((transitive_trancl @ A) @ E4)))))).
% 9.43/3.45 thf(fact_3377_inj__on__map__the, axiom, (! [B:$tType,A:$tType,D4:(set @ A),M2:(A > (option @ B))]: (((((ord_less_eq @ (set @ A)) @ D4) @ (((dom @ A) @ B) @ M2)) => (((((inj_on @ A) @ (option @ B)) @ M2) @ D4) => ((((inj_on @ A) @ B) @ (((((comp @ (option @ B)) @ B) @ A) @ (the2 @ B)) @ M2)) @ D4)))))).
% 9.43/3.45 thf(fact_3378_finite__Map__induct, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),P:((A > (option @ B)) > $o)]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ M2)) => ((P @ (^ [X3:A]: ((none @ B)))) => ((! [K5:A,V4:B,M5:(A > (option @ B))]: ((((finite_finite2 @ A) @ (((dom @ A) @ B) @ M5)) => ((~ ((((member @ A) @ K5) @ (((dom @ A) @ B) @ M5)))) => ((P @ M5) => (P @ (((((fun_upd @ A) @ (option @ B)) @ M5) @ K5) @ ((some @ B) @ V4)))))))) => (P @ M2))))))).
% 9.43/3.45 thf(fact_3379_ran__is__image, axiom, (! [A:$tType,B:$tType]: ((((ran @ B) @ A) = (^ [M8:(B > (option @ A))]: (((((image2 @ B) @ A) @ (((((comp @ (option @ A)) @ A) @ B) @ (the2 @ A)) @ M8)) @ (((dom @ B) @ A) @ M8)))))))).
% 9.43/3.45 thf(fact_3380_graph__eq__to__snd__dom, axiom, (! [B:$tType,A:$tType]: ((((graph @ A) @ B) = (^ [M:(A > (option @ B))]: (((((image2 @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ B) @ X3) @ ((the2 @ B) @ (M @ X3)))))) @ (((dom @ A) @ B) @ M)))))))).
% 9.43/3.45 thf(fact_3381_trancl__insert2, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ R)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [X3:A,Y3:A]: ((((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ A3)) @ ((transitive_trancl @ A) @ R)) | (X3 = A3)) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ Y3)) @ ((transitive_trancl @ A) @ R)) | (Y3 = B2)))))))))))).
% 9.43/3.45 thf(fact_3382_rel__restrict__Sigma__sub, axiom, (! [A:$tType,A4:(set @ A),R5:(set @ A)]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((rel_restrict @ A) @ ((transitive_trancl @ A) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4))))) @ R5)) @ ((transitive_trancl @ A) @ ((((product_Sigma @ A) @ A) @ (((minus_minus @ (set @ A)) @ A4) @ R5)) @ (^ [Uu:A]: ((((minus_minus @ (set @ A)) @ A4) @ R5))))))))).
% 9.43/3.45 thf(fact_3383_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:B,B6:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((member @ B) @ X) @ ((((image2 @ A) @ B) @ F) @ A4)) => ((((ord_less_eq @ (set @ A)) @ A4) @ B6) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A4) @ F) @ X)) @ B6))))))).
% 9.43/3.45 thf(fact_3384_the__inv__into__comp, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > B),G:(C > A),A4:(set @ C),X:B]: ((((((inj_on @ A) @ B) @ F) @ ((((image2 @ C) @ A) @ G) @ A4)) => (((((inj_on @ C) @ A) @ G) @ A4) => ((((member @ B) @ X) @ ((((image2 @ A) @ B) @ F) @ ((((image2 @ C) @ A) @ G) @ A4))) => ((((((the_inv_into @ C) @ B) @ A4) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ X) = ((((((comp @ A) @ C) @ B) @ ((((the_inv_into @ C) @ A) @ A4) @ G)) @ ((((the_inv_into @ A) @ B) @ ((((image2 @ C) @ A) @ G) @ A4)) @ F)) @ X)))))))).
% 9.43/3.45 thf(fact_3385_dom__eq__singleton__conv, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B)),X:A]: ((((((dom @ A) @ B) @ F) = (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) = (? [V2:B]: ((F = (((((fun_upd @ A) @ (option @ B)) @ (^ [X3:A]: ((none @ B)))) @ X) @ ((some @ B) @ V2))))))))).
% 9.43/3.45 thf(fact_3386_rtrancl__last__visit, axiom, (! [A:$tType,Q3:A,Q4:A,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Q4)) @ ((transitive_rtrancl @ A) @ R5)) => ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Q4)) @ ((transitive_rtrancl @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (S2)))))))) => (~ ((! [Qt:A]: (((((member @ A) @ Qt) @ S2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ Qt)) @ ((transitive_trancl @ A) @ R5)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Qt) @ Q4)) @ ((transitive_rtrancl @ A) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ R5) @ ((((product_Sigma @ A) @ A) @ (top_top @ (set @ A))) @ (^ [Uu:A]: (S2))))))))))))))))))).
% 9.43/3.45 thf(fact_3387_trancl__insert, axiom, (! [A:$tType,Y:A,X:A,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ R)) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ ((transitive_trancl @ A) @ R)) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (^ [A7:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ Y)) @ ((transitive_rtrancl @ A) @ R)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ B5)) @ ((transitive_rtrancl @ A) @ R)))))))))))).
% 9.43/3.45 thf(fact_3388_finite__trancl__ntranl, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R5) => (((transitive_trancl @ A) @ R5) = (((transitive_ntrancl @ A) @ (((minus_minus @ nat) @ ((finite_card @ ((product_prod @ A) @ A)) @ R5)) @ (one_one @ nat))) @ R5)))))).
% 9.43/3.45 thf(fact_3389_ran__add, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B)),G:(A > (option @ B))]: ((((((inf_inf @ (set @ A)) @ (((dom @ A) @ B) @ F)) @ (((dom @ A) @ B) @ G)) = (bot_bot @ (set @ A))) => ((((ran @ A) @ B) @ ((((map_add @ A) @ B) @ F) @ G)) = (((sup_sup @ (set @ B)) @ (((ran @ A) @ B) @ F)) @ (((ran @ A) @ B) @ G))))))).
% 9.43/3.45 thf(fact_3390_eq__f__restr__ss__eq, axiom, (! [B:$tType,A:$tType,S:(set @ A),F:((A > (option @ B)) > (A > (option @ B))),A4:(A > (option @ B))]: (((((ord_less_eq @ (set @ A)) @ S) @ (((dom @ A) @ B) @ (F @ A4))) => ((A4 = ((((restrict_map @ A) @ B) @ (F @ A4)) @ ((uminus_uminus @ (set @ A)) @ S))) = (((((map_le @ A) @ B) @ A4) @ (F @ A4)) & (S = (((minus_minus @ (set @ A)) @ (((dom @ A) @ B) @ (F @ A4))) @ (((dom @ A) @ B) @ A4))))))))).
% 9.43/3.45 thf(fact_3391_eq__f__restr__conv, axiom, (! [B:$tType,A:$tType,S:(set @ A),F:((A > (option @ B)) > (A > (option @ B))),A4:(A > (option @ B))]: ((((((ord_less_eq @ (set @ A)) @ S) @ (((dom @ A) @ B) @ (F @ A4))) & (A4 = ((((restrict_map @ A) @ B) @ (F @ A4)) @ ((uminus_uminus @ (set @ A)) @ S)))) = (((((map_le @ A) @ B) @ A4) @ (F @ A4)) & (S = (((minus_minus @ (set @ A)) @ (((dom @ A) @ B) @ (F @ A4))) @ (((dom @ A) @ B) @ A4)))))))).
% 9.43/3.45 thf(fact_3392_type__copy__set__map0, axiom, (! [A:$tType,B:$tType,D:$tType,E:$tType,C:$tType,F5:$tType,Rep:(A > B),Abs:(B > A),S2:(B > (set @ D)),M3:(C > B),F:(E > D),S7:(C > (set @ E)),G:(F5 > C)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => (((((((comp @ B) @ (set @ D)) @ C) @ S2) @ M3) = (((((comp @ (set @ E)) @ (set @ D)) @ C) @ (((image2 @ E) @ D) @ F)) @ S7)) => ((((((comp @ A) @ (set @ D)) @ F5) @ (((((comp @ B) @ (set @ D)) @ A) @ S2) @ Rep)) @ (((((comp @ C) @ A) @ F5) @ (((((comp @ B) @ A) @ C) @ Abs) @ M3)) @ G)) = (((((comp @ (set @ E)) @ (set @ D)) @ F5) @ (((image2 @ E) @ D) @ F)) @ (((((comp @ C) @ (set @ E)) @ F5) @ S7) @ G)))))))).
% 9.43/3.45 thf(fact_3393_map__add__find__right, axiom, (! [B:$tType,A:$tType,N:(B > (option @ A)),K:B,Xx:A,M2:(B > (option @ A))]: ((((N @ K) = ((some @ A) @ Xx)) => ((((((map_add @ B) @ A) @ M2) @ N) @ K) = ((some @ A) @ Xx)))))).
% 9.43/3.45 thf(fact_3394_map__add__eq__empty__iff, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B)),G:(A > (option @ B))]: (((((((map_add @ A) @ B) @ F) @ G) = (^ [X3:A]: ((none @ B)))) = ((F = (^ [X3:A]: ((none @ B)))) & (G = (^ [X3:A]: ((none @ B))))))))).
% 9.43/3.45 thf(fact_3395_map__add__None, axiom, (! [B:$tType,A:$tType,M2:(B > (option @ A)),N:(B > (option @ A)),K:B]: ((((((((map_add @ B) @ A) @ M2) @ N) @ K) = (none @ A)) = (((N @ K) = (none @ A)) & ((M2 @ K) = (none @ A))))))).
% 9.43/3.45 thf(fact_3396_empty__map__add, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((((map_add @ A) @ B) @ (^ [X3:A]: ((none @ B)))) @ M2) = M2)))).
% 9.43/3.45 thf(fact_3397_map__add__empty, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B))]: ((((((map_add @ A) @ B) @ M2) @ (^ [X3:A]: ((none @ B)))) = M2)))).
% 9.43/3.45 thf(fact_3398_empty__eq__map__add__iff, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B)),G:(A > (option @ B))]: ((((^ [X3:A]: ((none @ B))) = ((((map_add @ A) @ B) @ F) @ G)) = ((F = (^ [X3:A]: ((none @ B)))) & (G = (^ [X3:A]: ((none @ B))))))))).
% 9.43/3.45 thf(fact_3399_map__add__upd, axiom, (! [A:$tType,B:$tType,F:(A > (option @ B)),G:(A > (option @ B)),X:A,Y:B]: ((((((map_add @ A) @ B) @ F) @ (((((fun_upd @ A) @ (option @ B)) @ G) @ X) @ ((some @ B) @ Y))) = (((((fun_upd @ A) @ (option @ B)) @ ((((map_add @ A) @ B) @ F) @ G)) @ X) @ ((some @ B) @ Y)))))).
% 9.43/3.45 thf(fact_3400_map__add__upd__left, axiom, (! [A:$tType,B:$tType,M2:A,E22:(A > (option @ B)),E1:(A > (option @ B)),U1:B]: (((~ ((((member @ A) @ M2) @ (((dom @ A) @ B) @ E22)))) => (((((map_add @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ E1) @ M2) @ ((some @ B) @ U1))) @ E22) = (((((fun_upd @ A) @ (option @ B)) @ ((((map_add @ A) @ B) @ E1) @ E22)) @ M2) @ ((some @ B) @ U1))))))).
% 9.43/3.45 thf(fact_3401_le__map__mmupd__not__dom, axiom, (! [A:$tType,B:$tType,M2:(A > (option @ B)),K2:(set @ A),V:B]: (((((map_le @ A) @ B) @ M2) @ (((((map_mmupd @ A) @ B) @ M2) @ (((minus_minus @ (set @ A)) @ K2) @ (((dom @ A) @ B) @ M2))) @ V))))).
% 9.43/3.45 thf(fact_3402_map__add__find__left, axiom, (! [A:$tType,B:$tType,G:(B > (option @ A)),K:B,F:(B > (option @ A))]: ((((G @ K) = (none @ A)) => ((((((map_add @ B) @ A) @ F) @ G) @ K) = (F @ K)))))).
% 9.43/3.45 thf(fact_3403_map__add__left__None, axiom, (! [A:$tType,B:$tType,F:(B > (option @ A)),K:B,G:(B > (option @ A))]: ((((F @ K) = (none @ A)) => ((((((map_add @ B) @ A) @ F) @ G) @ K) = (G @ K)))))).
% 9.43/3.45 thf(fact_3404_map__leI, axiom, (! [B:$tType,A:$tType,M1:(A > (option @ B)),M22:(A > (option @ B))]: (((! [X2:A,V4:B]: ((((M1 @ X2) = ((some @ B) @ V4)) => ((M22 @ X2) = ((some @ B) @ V4))))) => ((((map_le @ A) @ B) @ M1) @ M22))))).
% 9.43/3.45 thf(fact_3405_map__leD, axiom, (! [A:$tType,B:$tType,M1:(A > (option @ B)),M22:(A > (option @ B)),K:A,V:B]: ((((((map_le @ A) @ B) @ M1) @ M22) => (((M1 @ K) = ((some @ B) @ V)) => ((M22 @ K) = ((some @ B) @ V))))))).
% 9.43/3.45 thf(fact_3406_map__add__first__le, axiom, (! [B:$tType,A:$tType]: (((order @ B) => (! [M2:(A > (option @ B)),M7:(A > (option @ B)),N:(A > (option @ B))]: (((((ord_less_eq @ (A > (option @ B))) @ M2) @ M7) => (((ord_less_eq @ (A > (option @ B))) @ ((((map_add @ A) @ B) @ M2) @ N)) @ ((((map_add @ A) @ B) @ M7) @ N))))))))).
% 9.43/3.45 thf(fact_3407_map__le__empty, axiom, (! [B:$tType,A:$tType,G:(A > (option @ B))]: (((((map_le @ A) @ B) @ (^ [X3:A]: ((none @ B)))) @ G)))).
% 9.43/3.45 thf(fact_3408_map__add__Some__iff, axiom, (! [B:$tType,A:$tType,M2:(B > (option @ A)),N:(B > (option @ A)),K:B,X:A]: ((((((((map_add @ B) @ A) @ M2) @ N) @ K) = ((some @ A) @ X)) = (((N @ K) = ((some @ A) @ X)) | (((N @ K) = (none @ A)) & ((M2 @ K) = ((some @ A) @ X)))))))).
% 9.43/3.45 thf(fact_3409_map__add__SomeD, axiom, (! [B:$tType,A:$tType,M2:(B > (option @ A)),N:(B > (option @ A)),K:B,X:A]: ((((((((map_add @ B) @ A) @ M2) @ N) @ K) = ((some @ A) @ X)) => (((N @ K) = ((some @ A) @ X)) | (((N @ K) = (none @ A)) & ((M2 @ K) = ((some @ A) @ X)))))))).
% 9.43/3.45 thf(fact_3410_type__copy__map__comp0__undo, axiom, (! [E:$tType,A:$tType,C:$tType,B:$tType,D:$tType,F5:$tType,Rep:(A > B),Abs:(B > A),Rep2:(C > D),Abs2:(D > C),Rep3:(E > F5),Abs3:(F5 > E),M3:(F5 > D),M13:(B > D),M24:(F5 > B)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((((((type_definition @ C) @ D) @ Rep2) @ Abs2) @ (top_top @ (set @ D))) => ((((((type_definition @ E) @ F5) @ Rep3) @ Abs3) @ (top_top @ (set @ F5))) => (((((((comp @ F5) @ C) @ E) @ (((((comp @ D) @ C) @ F5) @ Abs2) @ M3)) @ Rep3) = (((((comp @ A) @ C) @ E) @ (((((comp @ B) @ C) @ A) @ (((((comp @ D) @ C) @ B) @ Abs2) @ M13)) @ Rep)) @ (((((comp @ F5) @ A) @ E) @ (((((comp @ B) @ A) @ F5) @ Abs) @ M24)) @ Rep3))) => ((((((comp @ B) @ D) @ F5) @ M13) @ M24) = M3)))))))).
% 9.43/3.45 thf(fact_3411_type__copy__map__comp0, axiom, (! [F5:$tType,D:$tType,B:$tType,A:$tType,C:$tType,E:$tType,Rep:(A > B),Abs:(B > A),M3:(C > D),M13:(B > D),M24:(C > B),F:(D > F5),G:(E > C)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((M3 = (((((comp @ B) @ D) @ C) @ M13) @ M24)) => ((((((comp @ C) @ F5) @ E) @ (((((comp @ D) @ F5) @ C) @ F) @ M3)) @ G) = (((((comp @ A) @ F5) @ E) @ (((((comp @ B) @ F5) @ A) @ (((((comp @ D) @ F5) @ B) @ F) @ M13)) @ Rep)) @ (((((comp @ C) @ A) @ E) @ (((((comp @ B) @ A) @ C) @ Abs) @ M24)) @ G)))))))).
% 9.43/3.45 thf(fact_3412_upd__None__map__le, axiom, (! [B:$tType,A:$tType,F:(A > (option @ B)),X:A]: (((((map_le @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ (none @ B))) @ F)))).
% 9.43/3.45 thf(fact_3413_map__add__def, axiom, (! [B:$tType,A:$tType]: ((((map_add @ A) @ B) = (^ [M14:(A > (option @ B)),M25:(A > (option @ B)),X3:A]: ((((((case_option @ (option @ B)) @ B) @ (M14 @ X3)) @ (some @ B)) @ (M25 @ X3)))))))).
% 9.43/3.45 thf(fact_3414_map__add__left__comm, axiom, (! [B:$tType,A:$tType,A4:(A > (option @ B)),B6:(A > (option @ B)),C5:(A > (option @ B))]: ((((((inf_inf @ (set @ A)) @ (((dom @ A) @ B) @ A4)) @ (((dom @ A) @ B) @ B6)) = (bot_bot @ (set @ A))) => (((((map_add @ A) @ B) @ A4) @ ((((map_add @ A) @ B) @ B6) @ C5)) = ((((map_add @ A) @ B) @ B6) @ ((((map_add @ A) @ B) @ A4) @ C5))))))).
% 9.43/3.45 thf(fact_3415_type__copy__map__id0, axiom, (! [B:$tType,A:$tType,Rep:(A > B),Abs:(B > A),M3:(B > B)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((M3 = (id @ B)) => ((((((comp @ B) @ A) @ A) @ (((((comp @ B) @ A) @ B) @ Abs) @ M3)) @ Rep) = (id @ A))))))).
% 9.43/3.45 thf(fact_3416_type__copy__Abs__o__Rep, axiom, (! [B:$tType,A:$tType,Rep:(A > B),Abs:(B > A)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((((((comp @ B) @ A) @ A) @ Abs) @ Rep) = (id @ A)))))).
% 9.43/3.45 thf(fact_3417_type__copy__Rep__o__Abs, axiom, (! [A:$tType,B:$tType,Rep:(A > B),Abs:(B > A)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => ((((((comp @ A) @ B) @ B) @ Rep) @ Abs) = (id @ B)))))).
% 9.43/3.45 thf(fact_3418_type__definition__integer, axiom, (((((type_definition @ code_integer) @ int) @ code_int_of_integer) @ code_integer_of_int) @ (top_top @ (set @ int)))).
% 9.43/3.45 thf(fact_3419_map__le__imp__upd__le, axiom, (! [A:$tType,B:$tType,M1:(A > (option @ B)),M22:(A > (option @ B)),X:A,Y:B]: ((((((map_le @ A) @ B) @ M1) @ M22) => ((((map_le @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M1) @ X) @ (none @ B))) @ (((((fun_upd @ A) @ (option @ B)) @ M22) @ X) @ ((some @ B) @ Y))))))).
% 9.43/3.45 thf(fact_3420_map__add__distinct__le, axiom, (! [B:$tType,A:$tType]: (((preorder @ B) => (! [M2:(A > (option @ B)),M7:(A > (option @ B)),N:(A > (option @ B)),N2:(A > (option @ B))]: (((((ord_less_eq @ (A > (option @ B))) @ M2) @ M7) => ((((ord_less_eq @ (A > (option @ B))) @ N) @ N2) => (((((inf_inf @ (set @ A)) @ (((dom @ A) @ B) @ M7)) @ (((dom @ A) @ B) @ N2)) = (bot_bot @ (set @ A))) => (((ord_less_eq @ (A > (option @ B))) @ ((((map_add @ A) @ B) @ M2) @ N)) @ ((((map_add @ A) @ B) @ M7) @ N2))))))))))).
% 9.43/3.45 thf(fact_3421_map__mmupd__update__less, axiom, (! [A:$tType,B:$tType,K2:(set @ A),K9:(set @ A),M2:(A > (option @ B)),V:B]: (((((ord_less_eq @ (set @ A)) @ K2) @ K9) => ((((map_le @ A) @ B) @ (((((map_mmupd @ A) @ B) @ M2) @ (((minus_minus @ (set @ A)) @ K2) @ (((dom @ A) @ B) @ M2))) @ V)) @ (((((map_mmupd @ A) @ B) @ M2) @ (((minus_minus @ (set @ A)) @ K9) @ (((dom @ A) @ B) @ M2))) @ V)))))).
% 9.43/3.45 thf(fact_3422_trancl__set__ntrancl, axiom, (! [A:$tType,Xs:(list @ ((product_prod @ A) @ A))]: ((((transitive_trancl @ A) @ ((set2 @ ((product_prod @ A) @ A)) @ Xs)) = (((transitive_ntrancl @ A) @ (((minus_minus @ nat) @ ((finite_card @ ((product_prod @ A) @ A)) @ ((set2 @ ((product_prod @ A) @ A)) @ Xs))) @ (one_one @ nat))) @ ((set2 @ ((product_prod @ A) @ A)) @ Xs)))))).
% 9.43/3.45 thf(fact_3423_pred__nat__trancl__eq__le, axiom, (! [M2:nat,N:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ M2) @ N)) @ ((transitive_rtrancl @ nat) @ pred_nat)) = (((ord_less_eq @ nat) @ M2) @ N))))).
% 9.43/3.45 thf(fact_3424_less__eq, axiom, (! [M2:nat,N:nat]: (((((member @ ((product_prod @ nat) @ nat)) @ ((((product_Pair @ nat) @ nat) @ M2) @ N)) @ ((transitive_trancl @ nat) @ pred_nat)) = (((ord_less @ nat) @ M2) @ N))))).
% 9.43/3.45 thf(fact_3425_Gcd__fin__def, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Gcd_fin @ A) = ((((bounde2362111253966948842tice_F @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ (one_one @ A))))))).
% 9.43/3.45 thf(fact_3426_relImage__def, axiom, (! [A:$tType,B:$tType]: ((((bNF_Gr4221423524335903396lImage @ B) @ A) = (^ [R6:(set @ ((product_prod @ B) @ B)),F2:(B > A)]: (((collect @ ((product_prod @ A) @ A)) @ (^ [Uu:((product_prod @ A) @ A)]: ((? [A12:B,A23:B]: (((Uu = ((((product_Pair @ A) @ A) @ (F2 @ A12)) @ (F2 @ A23))) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A12) @ A23)) @ R6))))))))))))).
% 9.43/3.45 thf(fact_3427_set__map__filter, axiom, (! [B:$tType,A:$tType,G:(B > (option @ A)),Xs:(list @ B)]: ((((set2 @ A) @ ((((map_filter @ B) @ A) @ G) @ Xs)) = ((collect @ A) @ (^ [Y3:A]: ((? [X3:B]: (((((member @ B) @ X3) @ ((set2 @ B) @ Xs)) & ((G @ X3) = ((some @ A) @ Y3)))))))))))).
% 9.43/3.45 thf(fact_3428_ran__map__of, axiom, (! [A:$tType,B:$tType,Xs:(list @ ((product_prod @ B) @ A))]: ((((ord_less_eq @ (set @ A)) @ (((ran @ B) @ A) @ (((map_of @ B) @ A) @ Xs))) @ ((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((set2 @ ((product_prod @ B) @ A)) @ Xs)))))).
% 9.43/3.45 thf(fact_3429_trancl__restrict__reachable, axiom, (! [A:$tType,U:A,V:A,E4:(set @ ((product_prod @ A) @ A)),S2:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_trancl @ A) @ E4)) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ S2)) @ S2) => ((((member @ A) @ U) @ S2) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_trancl @ A) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ E4) @ ((((product_Sigma @ A) @ A) @ S2) @ (^ [Uu:A]: (S2)))))))))))).
% 9.43/3.45 thf(fact_3430_pair__vimage__is__Image, axiom, (! [A:$tType,B:$tType,U:B,E4:(set @ ((product_prod @ B) @ A))]: ((((((vimage @ A) @ ((product_prod @ B) @ A)) @ (((product_Pair @ B) @ A) @ U)) @ E4) = ((((image @ B) @ A) @ E4) @ (((insert2 @ B) @ U) @ (bot_bot @ (set @ B)))))))).
% 9.43/3.45 thf(fact_3431_set__bind__option, axiom, (! [A:$tType,B:$tType,X:(option @ B),F:(B > (option @ A))]: ((((set_option @ A) @ ((((bind @ B) @ A) @ X) @ F)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ B) @ (set @ A)) @ (((((comp @ (option @ A)) @ (set @ A)) @ B) @ (set_option @ A)) @ F)) @ ((set_option @ B) @ X))))))).
% 9.43/3.45 thf(fact_3432_ImageI, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,R:(set @ ((product_prod @ A) @ B)),A4:(set @ A)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R) => ((((member @ A) @ A3) @ A4) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R) @ A4))))))).
% 9.43/3.45 thf(fact_3433_elem__set, axiom, (! [A:$tType,X:A,Xo:(option @ A)]: (((((member @ A) @ X) @ ((set_option @ A) @ Xo)) = (Xo = ((some @ A) @ X)))))).
% 9.43/3.45 thf(fact_3434_set__empty__eq, axiom, (! [A:$tType,Xo:(option @ A)]: (((((set_option @ A) @ Xo) = (bot_bot @ (set @ A))) = (Xo = (none @ A)))))).
% 9.43/3.45 thf(fact_3435_Image__singleton__iff, axiom, (! [A:$tType,B:$tType,B2:A,R:(set @ ((product_prod @ B) @ A)),A3:B]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R) @ (((insert2 @ B) @ A3) @ (bot_bot @ (set @ B))))) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A3) @ B2)) @ R))))).
% 9.43/3.45 thf(fact_3436_rev__ImageI, axiom, (! [B:$tType,A:$tType,A3:A,A4:(set @ A),B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ A4) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R) => (((member @ B) @ B2) @ ((((image @ A) @ B) @ R) @ A4))))))).
% 9.43/3.45 thf(fact_3437_Image__iff, axiom, (! [A:$tType,B:$tType,B2:A,R:(set @ ((product_prod @ B) @ A)),A4:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R) @ A4)) = (? [X3:B]: (((((member @ B) @ X3) @ A4) & (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ B2)) @ R)))))))).
% 9.43/3.45 thf(fact_3438_ImageE, axiom, (! [A:$tType,B:$tType,B2:A,R:(set @ ((product_prod @ B) @ A)),A4:(set @ B)]: (((((member @ A) @ B2) @ ((((image @ B) @ A) @ R) @ A4)) => (~ ((! [X2:B]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X2) @ B2)) @ R) => (~ ((((member @ B) @ X2) @ A4)))))))))))).
% 9.43/3.45 thf(fact_3439_ospec, axiom, (! [A:$tType,A4:(option @ A),P:(A > $o),X:A]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set_option @ A) @ A4)) => (P @ X2)))) => ((A4 = ((some @ A) @ X)) => (P @ X)))))).
% 9.43/3.45 thf(fact_3440_option_Oset__intros, axiom, (! [A:$tType,X22:A]: ((((member @ A) @ X22) @ ((set_option @ A) @ ((some @ A) @ X22)))))).
% 9.43/3.45 thf(fact_3441_option_Oset__cases, axiom, (! [A:$tType,E2:A,A3:(option @ A)]: (((((member @ A) @ E2) @ ((set_option @ A) @ A3)) => (A3 = ((some @ A) @ E2)))))).
% 9.43/3.45 thf(fact_3442_map__option__idI, axiom, (! [A:$tType,X:(option @ A),F:(A > A)]: (((! [Y4:A]: (((((member @ A) @ Y4) @ ((set_option @ A) @ X)) => ((F @ Y4) = Y4)))) => (((((map_option @ A) @ A) @ F) @ X) = X))))).
% 9.43/3.45 thf(fact_3443_option_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Xa:(option @ A),F:(A > B),Fa:(A > B)]: (((! [Z3:A,Za:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ X)) => ((((member @ A) @ Za) @ ((set_option @ A) @ Xa)) => (((F @ Z3) = (Fa @ Za)) => (Z3 = Za)))))) => ((((((map_option @ A) @ B) @ F) @ X) = ((((map_option @ A) @ B) @ Fa) @ Xa)) => (X = Xa)))))).
% 9.43/3.45 thf(fact_3444_option_Omap__cong0, axiom, (! [B:$tType,A:$tType,X:(option @ A),F:(A > B),G:(A > B)]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ X)) => ((F @ Z3) = (G @ Z3))))) => (((((map_option @ A) @ B) @ F) @ X) = ((((map_option @ A) @ B) @ G) @ X)))))).
% 9.43/3.45 thf(fact_3445_option_Omap__cong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Ya:(option @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ Ya)) => ((F @ Z3) = (G @ Z3))))) => (((((map_option @ A) @ B) @ F) @ X) = ((((map_option @ A) @ B) @ G) @ Ya))))))).
% 9.43/3.45 thf(fact_3446_bind__option__cong, axiom, (! [B:$tType,A:$tType,X:(option @ A),Y:(option @ A),F:(A > (option @ B)),G:(A > (option @ B))]: (((X = Y) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ Y)) => ((F @ Z3) = (G @ Z3))))) => (((((bind @ A) @ B) @ X) @ F) = ((((bind @ A) @ B) @ Y) @ G))))))).
% 9.43/3.45 thf(fact_3447_rtrancl__image__advance__rtrancl, axiom, (! [A:$tType,Q3:A,R5:(set @ ((product_prod @ A) @ A)),Q0:(set @ A),X:A]: (((((member @ A) @ Q3) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ Q0)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ X)) @ ((transitive_rtrancl @ A) @ R5)) => (((member @ A) @ X) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ Q0))))))).
% 9.43/3.45 thf(fact_3448_rtrancl__image__advance, axiom, (! [A:$tType,Q3:A,R5:(set @ ((product_prod @ A) @ A)),Q0:(set @ A),X:A]: (((((member @ A) @ Q3) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ Q0)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Q3) @ X)) @ R5) => (((member @ A) @ X) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ Q0))))))).
% 9.43/3.45 thf(fact_3449_finite__Image__subset, axiom, (! [A:$tType,B:$tType,A4:(set @ ((product_prod @ B) @ A)),B6:(set @ B),C5:(set @ ((product_prod @ B) @ A))]: ((((finite_finite2 @ A) @ ((((image @ B) @ A) @ A4) @ B6)) => ((((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ C5) @ A4) => ((finite_finite2 @ A) @ ((((image @ B) @ A) @ C5) @ B6))))))).
% 9.43/3.45 thf(fact_3450_rtrancl__image__unfold__right, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),V6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ V6))) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ V6))))).
% 9.43/3.45 thf(fact_3451_rtrancl__reachable__induct, axiom, (! [A:$tType,I4:(set @ A),INV:(set @ A),E4:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ A)) @ I4) @ INV) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ INV)) @ INV) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ I4)) @ INV)))))).
% 9.43/3.45 thf(fact_3452_trancl__Image__unfold__right, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),S2:(set @ A)]: ((((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ S2) = ((((image @ A) @ A) @ E4) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ S2)))))).
% 9.43/3.45 thf(fact_3453_trancl__Image__unfold__left, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),S2:(set @ A)]: ((((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ S2) = ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ ((((image @ A) @ A) @ E4) @ S2)))))).
% 9.43/3.45 thf(fact_3454_option_Osimps_I14_J, axiom, (! [A:$tType]: ((((set_option @ A) @ (none @ A)) = (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_3455_option_Oset__map, axiom, (! [B:$tType,A:$tType,F:(A > B),V:(option @ A)]: ((((set_option @ B) @ ((((map_option @ A) @ B) @ F) @ V)) = ((((image2 @ A) @ B) @ F) @ ((set_option @ A) @ V)))))).
% 9.43/3.45 thf(fact_3456_option_Oset__sel, axiom, (! [A:$tType,A3:(option @ A)]: (((A3 != (none @ A)) => (((member @ A) @ ((the2 @ A) @ A3)) @ ((set_option @ A) @ A3)))))).
% 9.43/3.45 thf(fact_3457_Image__empty__rtrancl__Image__id, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),V:A]: (((((((image @ A) @ A) @ R5) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A)))) = (((insert2 @ A) @ V) @ (bot_bot @ (set @ A)))))))).
% 9.43/3.45 thf(fact_3458_Image__empty__trancl__Image__empty, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),V:A]: (((((((image @ A) @ A) @ R5) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))) => (((((image @ A) @ A) @ ((transitive_trancl @ A) @ R5)) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A)))) = (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_3459_reachable__mono, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),R7:(set @ ((product_prod @ A) @ A)),X10:(set @ A),X11:(set @ A)]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ R7) => ((((ord_less_eq @ (set @ A)) @ X10) @ X11) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ X10)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R7)) @ X11))))))).
% 9.43/3.45 thf(fact_3460_Image__subset__snd__image, axiom, (! [A:$tType,B:$tType,A4:(set @ ((product_prod @ B) @ A)),B6:(set @ B)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ A4) @ B6)) @ ((((image2 @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ A4))))).
% 9.43/3.45 thf(fact_3461_trancl__image__by__rtrancl, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),Vi:(set @ A)]: (((((sup_sup @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ Vi)) @ Vi) = ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ Vi))))).
% 9.43/3.45 thf(fact_3462_weak__map__of__SomeI, axiom, (! [A:$tType,B:$tType,K:A,X:B,L:(list @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ X)) @ ((set2 @ ((product_prod @ A) @ B)) @ L)) => (? [X2:B]: ((((((map_of @ A) @ B) @ L) @ K) = ((some @ B) @ X2)))))))).
% 9.43/3.45 thf(fact_3463_map__of__SomeD, axiom, (! [A:$tType,B:$tType,Xs:(list @ ((product_prod @ B) @ A)),K:B,Y:A]: (((((((map_of @ B) @ A) @ Xs) @ K) = ((some @ A) @ Y)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ K) @ Y)) @ ((set2 @ ((product_prod @ B) @ A)) @ Xs)))))).
% 9.43/3.45 thf(fact_3464_Image__singleton, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ A)),A3:B]: ((((((image @ B) @ A) @ R) @ (((insert2 @ B) @ A3) @ (bot_bot @ (set @ B)))) = ((collect @ A) @ (^ [B5:A]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A3) @ B5)) @ R)))))))).
% 9.43/3.45 thf(fact_3465_option_Osimps_I15_J, axiom, (! [A:$tType,X22:A]: ((((set_option @ A) @ ((some @ A) @ X22)) = (((insert2 @ A) @ X22) @ (bot_bot @ (set @ A))))))).
% 9.43/3.45 thf(fact_3466_rtrancl__apply__insert, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),X:A,S2:(set @ A)]: ((((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ (((insert2 @ A) @ X) @ S2)) = (((insert2 @ A) @ X) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ (((sup_sup @ (set @ A)) @ S2) @ ((((image @ A) @ A) @ R5) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))))))))).
% 9.43/3.45 thf(fact_3467_rtrancl__Image__in__Field, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),V6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ R5)) @ V6)) @ (((sup_sup @ (set @ A)) @ ((field2 @ A) @ R5)) @ V6))))).
% 9.43/3.45 thf(fact_3468_E__closed__restr__reach__cases, axiom, (! [A:$tType,U:A,V:A,E4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ E4)) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ R5)) @ R5) => ((~ ((((member @ A) @ V) @ R5))) => (~ (((~ ((((member @ A) @ U) @ R5))) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ ((transitive_rtrancl @ A) @ (((rel_restrict @ A) @ E4) @ R5)))))))))))))).
% 9.43/3.45 thf(fact_3469_rel__restrict__tranclI, axiom, (! [A:$tType,X:A,Y:A,E4:(set @ ((product_prod @ A) @ A)),R5:(set @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ E4)) => ((~ ((((member @ A) @ X) @ R5))) => ((~ ((((member @ A) @ Y) @ R5))) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ E4) @ R5)) @ R5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ (((rel_restrict @ A) @ E4) @ R5)))))))))).
% 9.43/3.45 thf(fact_3470_refl__on__reflcl__Image, axiom, (! [A:$tType,B6:(set @ A),A4:(set @ ((product_prod @ A) @ A)),C5:(set @ A)]: (((((refl_on @ A) @ B6) @ A4) => ((((ord_less_eq @ (set @ A)) @ C5) @ B6) => (((((image @ A) @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ A4) @ (id2 @ A))) @ C5) = ((((image @ A) @ A) @ A4) @ C5))))))).
% 9.43/3.45 thf(fact_3471_map__of__eq__None__iff, axiom, (! [A:$tType,B:$tType,Xys:(list @ ((product_prod @ B) @ A)),X:B]: (((((((map_of @ B) @ A) @ Xys) @ X) = (none @ A)) = (~ ((((member @ B) @ X) @ ((((image2 @ ((product_prod @ B) @ A)) @ B) @ ((product_fst @ B) @ A)) @ ((set2 @ ((product_prod @ B) @ A)) @ Xys))))))))).
% 9.43/3.45 thf(fact_3472_finite__reachable__advance, axiom, (! [A:$tType,E4:(set @ ((product_prod @ A) @ A)),V0:A,V:A]: ((((finite_finite2 @ A) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ (((insert2 @ A) @ V0) @ (bot_bot @ (set @ A))))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ V0) @ V)) @ ((transitive_rtrancl @ A) @ E4)) => ((finite_finite2 @ A) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.45 thf(fact_3473_rtrancl__Image__advance__ss, axiom, (! [A:$tType,U:A,V:A,E4:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ E4) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ (((insert2 @ A) @ U) @ (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3474_trancl__Image__advance__ss, axiom, (! [A:$tType,U:A,V:A,E4:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ U) @ V)) @ E4) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ (((insert2 @ A) @ V) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ ((transitive_trancl @ A) @ E4)) @ (((insert2 @ A) @ U) @ (bot_bot @ (set @ A))))))))).
% 9.43/3.45 thf(fact_3475_subset__Image1__Image1__iff, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((order_preorder_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ R)))))))).
% 9.43/3.45 thf(fact_3476_map__of__distinct__upd, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ ((product_prod @ A) @ B)),Y:B]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))))) => (((((map_add @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ (^ [X3:A]: ((none @ B)))) @ X) @ ((some @ B) @ Y))) @ (((map_of @ A) @ B) @ Xs)) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ Xs)) @ X) @ ((some @ B) @ Y))))))).
% 9.43/3.45 thf(fact_3477_subset__Image__Image__iff, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B6:(set @ A)]: (((((order_preorder_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((ord_less_eq @ (set @ A)) @ A4) @ ((field2 @ A) @ R)) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((field2 @ A) @ R)) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ A4)) @ ((((image @ A) @ A) @ R) @ B6)) = (! [X3:A]: (((((member @ A) @ X3) @ A4) => (? [Y3:A]: (((((member @ A) @ Y3) @ B6) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3478_map__of__map__restrict, axiom, (! [B:$tType,A:$tType,F:(A > B),Ks:(list @ A)]: (((((map_of @ A) @ B) @ ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [K4:A]: (((((product_Pair @ A) @ B) @ K4) @ (F @ K4))))) @ Ks)) = ((((restrict_map @ A) @ B) @ (((((comp @ B) @ (option @ B)) @ A) @ (some @ B)) @ F)) @ ((set2 @ A) @ Ks)))))).
% 9.43/3.45 thf(fact_3479_map__of__mapk__SomeI, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),T2:(list @ ((product_prod @ A) @ C)),K:A,X:C]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((map_of @ A) @ C) @ T2) @ K) = ((some @ C) @ X)) => (((((map_of @ B) @ C) @ ((((map @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ C)) @ ((((product_case_prod @ A) @ C) @ ((product_prod @ B) @ C)) @ (^ [K4:A]: ((((product_Pair @ B) @ C) @ (F @ K4)))))) @ T2)) @ (F @ K)) = ((some @ C) @ X))))))).
% 9.43/3.45 thf(fact_3480_map__comp__map, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),G:(A > C)]: (((((((comp @ (list @ C)) @ (list @ B)) @ (list @ A)) @ (((map @ C) @ B) @ F)) @ (((map @ A) @ C) @ G)) = (((map @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F) @ G)))))).
% 9.43/3.45 thf(fact_3481_map__map, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),Xs:(list @ C)]: ((((((map @ B) @ A) @ F) @ ((((map @ C) @ B) @ G) @ Xs)) = ((((map @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F) @ G)) @ Xs))))).
% 9.43/3.45 thf(fact_3482_List_Omap_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > C),G:(A > B),List:(list @ A)]: ((((((map @ B) @ C) @ F) @ ((((map @ A) @ B) @ G) @ List)) = ((((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)) @ List))))).
% 9.43/3.45 thf(fact_3483_list_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(list @ A)]: ((((((map @ B) @ C) @ G) @ ((((map @ A) @ B) @ F) @ V)) = ((((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 9.43/3.45 thf(fact_3484_List_Omap_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C),G:(A > B)]: (((((((comp @ (list @ B)) @ (list @ C)) @ (list @ A)) @ (((map @ B) @ C) @ F)) @ (((map @ A) @ B) @ G)) = (((map @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)))))).
% 9.43/3.45 thf(fact_3485_map__fst__mk__snd, axiom, (! [B:$tType,A:$tType,K:B,L:(list @ A)]: ((((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ B) @ X3) @ K)))) @ L)) = L)))).
% 9.43/3.45 thf(fact_3486_map__snd__mk__fst, axiom, (! [B:$tType,A:$tType,K:B,L:(list @ A)]: ((((((map @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((map @ A) @ ((product_prod @ B) @ A)) @ (((product_Pair @ B) @ A) @ K)) @ L)) = L)))).
% 9.43/3.45 thf(fact_3487_set__oo__map__alt, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((comp @ (list @ B)) @ (set @ B)) @ (list @ A)) @ (set2 @ B)) @ (((map @ A) @ B) @ F)) = (^ [L3:(list @ A)]: (((((image2 @ A) @ B) @ F) @ ((set2 @ A) @ L3)))))))).
% 9.43/3.45 thf(fact_3488_Id__on__set, axiom, (! [A:$tType,Xs:(list @ A)]: ((((id_on @ A) @ ((set2 @ A) @ Xs)) = ((set2 @ ((product_prod @ A) @ A)) @ ((((map @ A) @ ((product_prod @ A) @ A)) @ (^ [X3:A]: (((((product_Pair @ A) @ A) @ X3) @ X3)))) @ Xs)))))).
% 9.43/3.45 thf(fact_3489_map__of__map__keys, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),M2:(A > (option @ B))]: (((((set2 @ A) @ Xs) = (((dom @ A) @ B) @ M2)) => ((((map_of @ A) @ B) @ ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [K4:A]: (((((product_Pair @ A) @ B) @ K4) @ ((the2 @ B) @ (M2 @ K4)))))) @ Xs)) = M2))))).
% 9.43/3.45 thf(fact_3490_map__of__map, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),Xs:(list @ ((product_prod @ A) @ C))]: (((((map_of @ A) @ B) @ ((((map @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ C) @ ((product_prod @ A) @ B)) @ (^ [K4:A,V2:C]: (((((product_Pair @ A) @ B) @ K4) @ (F @ V2)))))) @ Xs)) = (((((comp @ (option @ C)) @ (option @ B)) @ A) @ (((map_option @ C) @ B) @ F)) @ (((map_of @ A) @ C) @ Xs)))))).
% 9.43/3.45 thf(fact_3491_inj__on__map__inv__f, axiom, (! [B:$tType,A:$tType,L:(list @ A),A4:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ L)) @ A4) => (((((inj_on @ A) @ B) @ F) @ A4) => (((((map @ B) @ A) @ ((((inv_on @ A) @ B) @ F) @ A4)) @ ((((map @ A) @ B) @ F) @ L)) = L)))))).
% 9.43/3.45 thf(fact_3492_Misc_Oran__distinct, axiom, (! [B:$tType,A:$tType,Al:(list @ ((product_prod @ A) @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Al)) => ((((ran @ A) @ B) @ (((map_of @ A) @ B) @ Al)) = ((((image2 @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ ((set2 @ ((product_prod @ A) @ B)) @ Al))))))).
% 9.43/3.45 thf(fact_3493_map__of__eq__Some__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B)),X:A,Y:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xys)) => ((((((map_of @ A) @ B) @ Xys) @ X) = ((some @ B) @ Y)) = (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xys))))))).
% 9.43/3.45 thf(fact_3494_Some__eq__map__of__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B)),Y:B,X:A]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xys)) => ((((some @ B) @ Y) = ((((map_of @ A) @ B) @ Xys) @ X)) = (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xys))))))).
% 9.43/3.45 thf(fact_3495_map__of__is__SomeI, axiom, (! [A:$tType,B:$tType,Xys:(list @ ((product_prod @ A) @ B)),X:A,Y:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xys)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xys)) => (((((map_of @ A) @ B) @ Xys) @ X) = ((some @ B) @ Y))))))).
% 9.43/3.45 thf(fact_3496_inv__on__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),X:A]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((member @ A) @ X) @ A4) => ((((((inv_on @ A) @ B) @ F) @ A4) @ (F @ X)) = X)))))).
% 9.43/3.45 thf(fact_3497_distinct__mapI, axiom, (! [A:$tType,B:$tType,F:(B > A),L:(list @ B)]: ((((distinct @ A) @ ((((map @ B) @ A) @ F) @ L)) => ((distinct @ B) @ L))))).
% 9.43/3.45 thf(fact_3498_distinct__finite__set, axiom, (! [A:$tType,X:(set @ A)]: (((finite_finite2 @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Ys:(list @ A)]: (((((set2 @ A) @ Ys) = X) & ((distinct @ A) @ Ys))))))))).
% 9.43/3.45 thf(fact_3499_distinct__map__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),L:(list @ B),X:B,Y:B]: ((((distinct @ A) @ ((((map @ B) @ A) @ F) @ L)) => (((F @ X) = (F @ Y)) => ((((member @ B) @ X) @ ((set2 @ B) @ L)) => ((((member @ B) @ Y) @ ((set2 @ B) @ L)) => (X = Y)))))))).
% 9.43/3.45 thf(fact_3500_finite__set__image, axiom, (! [A:$tType,A4:(set @ (list @ A))]: ((((finite_finite2 @ (set @ A)) @ ((((image2 @ (list @ A)) @ (set @ A)) @ (set2 @ A)) @ A4)) => ((! [Xs2:(list @ A)]: (((((member @ (list @ A)) @ Xs2) @ A4) => ((distinct @ A) @ Xs2)))) => ((finite_finite2 @ (list @ A)) @ A4)))))).
% 9.43/3.45 thf(fact_3501_inv__on__f__range, axiom, (! [A:$tType,B:$tType,Y:A,F:(B > A),A4:(set @ B)]: (((((member @ A) @ Y) @ ((((image2 @ B) @ A) @ F) @ A4)) => (((member @ B) @ (((((inv_on @ B) @ A) @ F) @ A4) @ Y)) @ A4))))).
% 9.43/3.45 thf(fact_3502_f__inv__on__f, axiom, (! [B:$tType,A:$tType,Y:A,F:(B > A),A4:(set @ B)]: (((((member @ A) @ Y) @ ((((image2 @ B) @ A) @ F) @ A4)) => ((F @ (((((inv_on @ B) @ A) @ F) @ A4) @ Y)) = Y))))).
% 9.43/3.45 thf(fact_3503_distinct__finite__subset, axiom, (! [A:$tType,X:(set @ A)]: ((((finite_finite2 @ A) @ X) => ((finite_finite2 @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Ys:(list @ A)]: (((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Ys)) @ X) & ((distinct @ A) @ Ys)))))))))).
% 9.43/3.45 thf(fact_3504_distinct__map__fstD, axiom, (! [A:$tType,B:$tType,Xs:(list @ ((product_prod @ A) @ B)),X:A,Y:B,Z2:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xs)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Z2)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xs)) => (Y = Z2))))))).
% 9.43/3.45 thf(fact_3505_eq__key__imp__eq__value, axiom, (! [A:$tType,B:$tType,Xs:(list @ ((product_prod @ A) @ B)),K:A,V1:B,V22:B]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V1)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xs)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ K) @ V22)) @ ((set2 @ ((product_prod @ A) @ B)) @ Xs)) => (V1 = V22))))))).
% 9.43/3.45 thf(fact_3506_map__to__set__map__of, axiom, (! [B:$tType,A:$tType,L:(list @ ((product_prod @ A) @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ L)) => ((((map_to_set @ A) @ B) @ (((map_of @ A) @ B) @ L)) = ((set2 @ ((product_prod @ A) @ B)) @ L)))))).
% 9.43/3.45 thf(fact_3507_map__of__map__to__set, axiom, (! [B:$tType,A:$tType,L:(list @ ((product_prod @ A) @ B)),M2:(A > (option @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ L)) => (((((map_of @ A) @ B) @ L) = M2) = (((set2 @ ((product_prod @ A) @ B)) @ L) = (((map_to_set @ A) @ B) @ M2))))))).
% 9.43/3.45 thf(fact_3508_set__map__of__compr, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ A) @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => (((set2 @ ((product_prod @ A) @ B)) @ Xs) = ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [K4:A,V2:B]: ((((((map_of @ A) @ B) @ Xs) @ K4) = ((some @ B) @ V2))))))))))).
% 9.43/3.45 thf(fact_3509_card__lists__distinct__length__eq, axiom, (! [A:$tType,A4:(set @ A),K:nat]: ((((finite_finite2 @ A) @ A4) => ((((ord_less_eq @ nat) @ K) @ ((finite_card @ A) @ A4)) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (((distinct @ A) @ Xs3) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A4))))))) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A4)) @ K)) @ (one_one @ nat))) @ ((finite_card @ A) @ A4))))))))).
% 9.43/3.45 thf(fact_3510_card__lists__distinct__length__eq_H, axiom, (! [A:$tType,K:nat,A4:(set @ A)]: (((((ord_less @ nat) @ K) @ ((finite_card @ A) @ A4)) => (((finite_card @ (list @ A)) @ ((collect @ (list @ A)) @ (^ [Xs3:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = K) & (((distinct @ A) @ Xs3) & (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs3)) @ A4))))))) = ((((groups7121269368397514597t_prod @ nat) @ nat) @ (^ [X3:nat]: (X3))) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ (((minus_minus @ nat) @ ((finite_card @ A) @ A4)) @ K)) @ (one_one @ nat))) @ ((finite_card @ A) @ A4)))))))).
% 9.43/3.45 thf(fact_3511_distinct__foldl__invar, axiom, (! [B:$tType,A:$tType,S2:(list @ A),I4:((set @ A) > (B > $o)),Sigma_0:B,F:(B > (A > B))]: ((((distinct @ A) @ S2) => (((I4 @ ((set2 @ A) @ S2)) @ Sigma_0) => ((! [X2:A,It:(set @ A),Sigma:B]: (((((member @ A) @ X2) @ It) => ((((ord_less_eq @ (set @ A)) @ It) @ ((set2 @ A) @ S2)) => (((I4 @ It) @ Sigma) => ((I4 @ (((minus_minus @ (set @ A)) @ It) @ (((insert2 @ A) @ X2) @ (bot_bot @ (set @ A))))) @ ((F @ Sigma) @ X2))))))) => ((I4 @ (bot_bot @ (set @ A))) @ (((((foldl @ B) @ A) @ F) @ Sigma_0) @ S2)))))))).
% 9.43/3.45 thf(fact_3512_map__of__Some__filter__not__in, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ B) @ A)),K:B,V:A,P:(((product_prod @ B) @ A) > $o)]: (((((((map_of @ B) @ A) @ Xs) @ K) = ((some @ A) @ V)) => ((~ ((P @ ((((product_Pair @ B) @ A) @ K) @ V)))) => (((distinct @ B) @ ((((map @ ((product_prod @ B) @ A)) @ B) @ ((product_fst @ B) @ A)) @ Xs)) => (((((map_of @ B) @ A) @ (((filter @ ((product_prod @ B) @ A)) @ P) @ Xs)) @ K) = (none @ A)))))))).
% 9.43/3.45 thf(fact_3513_map__add__map__of__foldr, axiom, (! [B:$tType,A:$tType,M2:(A > (option @ B)),Ps:(list @ ((product_prod @ A) @ B))]: ((((((map_add @ A) @ B) @ M2) @ (((map_of @ A) @ B) @ Ps)) = (((((foldr @ ((product_prod @ A) @ B)) @ (A > (option @ B))) @ ((((product_case_prod @ A) @ B) @ ((A > (option @ B)) > (A > (option @ B)))) @ (^ [K4:A,V2:B,M:(A > (option @ B))]: ((((((fun_upd @ A) @ (option @ B)) @ M) @ K4) @ ((some @ B) @ V2)))))) @ Ps) @ M2))))).
% 9.43/3.45 thf(fact_3514_foldl__length, axiom, (! [A:$tType,L:(list @ A)]: (((((((foldl @ nat) @ A) @ (^ [I3:nat,X3:A]: ((suc @ I3)))) @ (zero_zero @ nat)) @ L) = ((size_size @ (list @ A)) @ L))))).
% 9.43/3.45 thf(fact_3515_foldl__length__aux, axiom, (! [A:$tType,A3:nat,L:(list @ A)]: (((((((foldl @ nat) @ A) @ (^ [I3:nat,X3:A]: ((suc @ I3)))) @ A3) @ L) = (((plus_plus @ nat) @ A3) @ ((size_size @ (list @ A)) @ L)))))).
% 9.43/3.45 thf(fact_3516_foldl__A1__eq, axiom, (! [A:$tType,F:(A > (A > A)),N:A,I:A,Ww:(list @ A)]: (((! [E3:A]: ((((F @ N) @ E3) = E3))) => ((! [E3:A]: ((((F @ E3) @ N) = E3))) => ((! [A5:A,B3:A,C4:A]: ((((F @ A5) @ ((F @ B3) @ C4)) = ((F @ ((F @ A5) @ B3)) @ C4)))) => ((((((foldl @ A) @ A) @ F) @ I) @ Ww) = ((F @ I) @ (((((foldl @ A) @ A) @ F) @ N) @ Ww))))))))).
% 9.43/3.45 thf(fact_3517_sum__length__filter__compl, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: (((((plus_plus @ nat) @ ((size_size @ (list @ A)) @ (((filter @ A) @ P) @ Xs))) @ ((size_size @ (list @ A)) @ (((filter @ A) @ (^ [X3:A]: ((~ ((P @ X3)))))) @ Xs))) = ((size_size @ (list @ A)) @ Xs))))).
% 9.43/3.45 thf(fact_3518_comp__fun__commute_Ofoldr__conv__foldl, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),Xs:(list @ A),A3:B]: (((((finite6289374366891150609ommute @ A) @ B) @ F) => ((((((foldr @ A) @ B) @ F) @ Xs) @ A3) = (((((foldl @ B) @ A) @ (^ [A7:B,B5:A]: (((F @ B5) @ A7)))) @ A3) @ Xs)))))).
% 9.43/3.45 thf(fact_3519_comp__fun__commute_Ofoldl__f__commute, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),A3:A,B2:B,Xs:(list @ A)]: (((((finite6289374366891150609ommute @ A) @ B) @ F) => (((F @ A3) @ (((((foldl @ B) @ A) @ (^ [A7:B,B5:A]: (((F @ B5) @ A7)))) @ B2) @ Xs)) = (((((foldl @ B) @ A) @ (^ [A7:B,B5:A]: (((F @ B5) @ A7)))) @ ((F @ A3) @ B2)) @ Xs)))))).
% 9.43/3.45 thf(fact_3520_map__of__None__filterD, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ B) @ A)),X:B,P:(((product_prod @ B) @ A) > $o)]: (((((((map_of @ B) @ A) @ Xs) @ X) = (none @ A)) => (((((map_of @ B) @ A) @ (((filter @ ((product_prod @ B) @ A)) @ P) @ Xs)) @ X) = (none @ A)))))).
% 9.43/3.45 thf(fact_3521_fst__foldl, axiom, (! [B:$tType,A:$tType,C:$tType,F:(A > (C > A)),G:(A > (B > (C > B))),A3:A,B2:B,Xs:(list @ C)]: (((((product_fst @ A) @ B) @ (((((foldl @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ (C > ((product_prod @ A) @ B))) @ (^ [A7:A,B5:B,X3:C]: (((((product_Pair @ A) @ B) @ ((F @ A7) @ X3)) @ (((G @ A7) @ B5) @ X3)))))) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ Xs)) = (((((foldl @ A) @ C) @ F) @ A3) @ Xs))))).
% 9.43/3.45 thf(fact_3522_foldl__absorb1, axiom, (! [A:$tType]: (((monoid_mult @ A) => (! [X:A,Zs:(list @ A)]: (((((times_times @ A) @ X) @ (((((foldl @ A) @ A) @ (times_times @ A)) @ (one_one @ A)) @ Zs)) = (((((foldl @ A) @ A) @ (times_times @ A)) @ X) @ Zs)))))))).
% 9.43/3.45 thf(fact_3523_foldl__un__empty__eq, axiom, (! [A:$tType,I:(set @ A),Ww:(list @ (set @ A))]: (((((((foldl @ (set @ A)) @ (set @ A)) @ (sup_sup @ (set @ A))) @ I) @ Ww) = (((sup_sup @ (set @ A)) @ I) @ (((((foldl @ (set @ A)) @ (set @ A)) @ (sup_sup @ (set @ A))) @ (bot_bot @ (set @ A))) @ Ww)))))).
% 9.43/3.45 thf(fact_3524_distinct__map__fst__filterI, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ A) @ B)),P:(((product_prod @ A) @ B) > $o)]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => ((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ (((filter @ ((product_prod @ A) @ B)) @ P) @ Xs))))))).
% 9.43/3.45 thf(fact_3525_map__of__filter__in, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ B) @ A)),K:B,Z2:A,P:(B > (A > $o))]: (((((((map_of @ B) @ A) @ Xs) @ K) = ((some @ A) @ Z2)) => (((P @ K) @ Z2) => (((((map_of @ B) @ A) @ (((filter @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ P)) @ Xs)) @ K) = ((some @ A) @ Z2))))))).
% 9.43/3.45 thf(fact_3526_distinct__length__le, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A)]: ((((distinct @ A) @ Ys2) => ((((set2 @ A) @ Ys2) = ((set2 @ A) @ Xs)) => (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ A)) @ Xs))))))).
% 9.43/3.45 thf(fact_3527_foldl__set, axiom, (! [A:$tType,L:(list @ (set @ A))]: (((((((foldl @ (set @ A)) @ (set @ A)) @ (sup_sup @ (set @ A))) @ (bot_bot @ (set @ A))) @ L) = ((complete_Sup_Sup @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [X3:(set @ A)]: ((((member @ (set @ A)) @ X3) @ ((set2 @ (set @ A)) @ L)))))))))).
% 9.43/3.45 thf(fact_3528_map__filter__map__filter, axiom, (! [A:$tType,B:$tType,F:(B > A),P:(B > $o),Xs:(list @ B)]: ((((((map @ B) @ A) @ F) @ (((filter @ B) @ P) @ Xs)) = ((((map_filter @ B) @ A) @ (^ [X3:B]: (((((if @ (option @ A)) @ (P @ X3)) @ ((some @ A) @ (F @ X3))) @ (none @ A))))) @ Xs))))).
% 9.43/3.45 thf(fact_3529_map__filter__def, axiom, (! [B:$tType,A:$tType]: ((((map_filter @ A) @ B) = (^ [F2:(A > (option @ B)),Xs3:(list @ A)]: (((((map @ A) @ B) @ (((((comp @ (option @ B)) @ B) @ A) @ (the2 @ B)) @ F2)) @ (((filter @ A) @ (^ [X3:A]: (((F2 @ X3) != (none @ B))))) @ Xs3)))))))).
% 9.43/3.45 thf(fact_3530_execute__of__list, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H3) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(array @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))) @ (((array_alloc @ A) @ Xs) @ H3)))))))))).
% 9.43/3.45 thf(fact_3531_mergesort__by__rel__split__length, axiom, (! [A:$tType,Xs1:(list @ A),Xs22:(list @ A),Xs:(list @ A)]: (((((size_size @ (list @ A)) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ Xs))) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs1)) @ (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) @ (((modulo_modulo @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) & (((size_size @ (list @ A)) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ Xs))) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs22)) @ (((divide_divide @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))).
% 9.43/3.45 thf(fact_3532_card__disjoint__shuffles, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys2)) = (bot_bot @ (set @ A))) => (((finite_card @ (list @ A)) @ (((shuffles @ A) @ Xs) @ Ys2)) = ((binomial @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys2))) @ ((size_size @ (list @ A)) @ Xs))))))).
% 9.43/3.45 thf(fact_3533_of__list__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_of_list @ A) = (^ [Xs3:(list @ A)]: (((heap_Time_heap @ (array @ A)) @ (^ [H:(heap_ext @ product_unit)]: ((((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(array @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs3))))))) @ (((array_alloc @ A) @ Xs3) @ H)))))))))))).
% 9.43/3.45 thf(fact_3534_slice__len, axiom, (! [A:$tType,From:nat,To:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ From) @ To) => ((((ord_less_eq @ nat) @ To) @ ((size_size @ (list @ A)) @ Xs)) => (((size_size @ (list @ A)) @ ((((slice @ A) @ From) @ To) @ Xs)) = (((minus_minus @ nat) @ To) @ From))))))).
% 9.43/3.45 thf(fact_3535_slice__complete, axiom, (! [A:$tType,Xs:(list @ A)]: ((((((slice @ A) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs)) @ Xs) = Xs)))).
% 9.43/3.45 thf(fact_3536_foldr__length, axiom, (! [A:$tType,L:(list @ A)]: (((((((foldr @ A) @ nat) @ (^ [X3:A]: (suc))) @ L) @ (zero_zero @ nat)) = ((size_size @ (list @ A)) @ L))))).
% 9.43/3.45 thf(fact_3537_length__shuffles, axiom, (! [A:$tType,Zs:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: (((((member @ (list @ A)) @ Zs) @ (((shuffles @ A) @ Xs) @ Ys2)) => (((size_size @ (list @ A)) @ Zs) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys2))))))).
% 9.43/3.45 thf(fact_3538_foldr__length__aux, axiom, (! [A:$tType,L:(list @ A),A3:nat]: (((((((foldr @ A) @ nat) @ (^ [X3:A]: (suc))) @ L) @ A3) = (((plus_plus @ nat) @ A3) @ ((size_size @ (list @ A)) @ L)))))).
% 9.43/3.45 thf(fact_3539_effect__of__listI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H4:(heap_ext @ product_unit),Xs:(list @ A),H3:(heap_ext @ product_unit)]: (((((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ A3) @ H4) = (((array_alloc @ A) @ Xs) @ H3)) => ((((((heap_Time_effect @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H3) @ H4) @ A3) @ (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs)))))))))).
% 9.43/3.45 thf(fact_3540_execute__make, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H3) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(array @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H3)))))))))).
% 9.43/3.45 thf(fact_3541_execute__new, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H3) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(array @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H3)))))))))).
% 9.43/3.45 thf(fact_3542_Array__Time_Omake__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_make @ A) = (^ [N5:nat,F2:(nat > A)]: (((heap_Time_heap @ (array @ A)) @ (^ [H:(heap_ext @ product_unit)]: ((((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(array @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (((plus_plus @ nat) @ N5) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F2) @ ((upt @ (zero_zero @ nat)) @ N5))) @ H)))))))))))).
% 9.43/3.45 thf(fact_3543_new__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_new @ A) = (^ [N5:nat,X3:A]: (((heap_Time_heap @ (array @ A)) @ (^ [H:(heap_ext @ product_unit)]: ((((((product_case_prod @ (array @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(array @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (((plus_plus @ nat) @ N5) @ (one_one @ nat))))))) @ (((array_alloc @ A) @ (((replicate @ A) @ N5) @ X3)) @ H)))))))))))).
% 9.43/3.45 thf(fact_3544_upt__0__eq__Nil__conv, axiom, (! [J:nat]: (((((upt @ (zero_zero @ nat)) @ J) = (nil @ nat)) = (J = (zero_zero @ nat)))))).
% 9.43/3.45 thf(fact_3545_map__of__eq__empty__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B))]: ((((((map_of @ A) @ B) @ Xys) = (^ [X3:A]: ((none @ B)))) = (Xys = (nil @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_3546_slice__Nil, axiom, (! [A:$tType,Begin:nat,End:nat]: ((((((slice @ A) @ Begin) @ End) @ (nil @ A)) = (nil @ A))))).
% 9.43/3.45 thf(fact_3547_slice__eq__bounds__empty, axiom, (! [A:$tType,I:nat,Xs:(list @ A)]: ((((((slice @ A) @ I) @ I) @ Xs) = (nil @ A))))).
% 9.43/3.45 thf(fact_3548_empty__eq__map__of__iff, axiom, (! [B:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ B))]: ((((^ [X3:A]: ((none @ B))) = (((map_of @ A) @ B) @ Xys)) = (Xys = (nil @ ((product_prod @ A) @ B))))))).
% 9.43/3.45 thf(fact_3549_length__ge__1__conv, axiom, (! [A:$tType,L:(list @ A)]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ ((size_size @ (list @ A)) @ L)) = (L != (nil @ A)))))).
% 9.43/3.45 thf(fact_3550_map__fst__mk__fst, axiom, (! [B:$tType,A:$tType,K:A,L:(list @ B)]: ((((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ ((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ K)) @ L)) = (((replicate @ A) @ ((size_size @ (list @ B)) @ L)) @ K))))).
% 9.43/3.45 thf(fact_3551_map__snd__mk__snd, axiom, (! [B:$tType,A:$tType,K:A,L:(list @ B)]: ((((((map @ ((product_prod @ B) @ A)) @ A) @ ((product_snd @ B) @ A)) @ ((((map @ B) @ ((product_prod @ B) @ A)) @ (^ [X3:B]: (((((product_Pair @ B) @ A) @ X3) @ K)))) @ L)) = (((replicate @ A) @ ((size_size @ (list @ B)) @ L)) @ K))))).
% 9.43/3.45 thf(fact_3552_map__add__upt_H, axiom, (! [Ofs:nat,A3:nat,B2:nat]: ((((((map @ nat) @ nat) @ (^ [I3:nat]: ((((plus_plus @ nat) @ I3) @ Ofs)))) @ ((upt @ A3) @ B2)) = ((upt @ (((plus_plus @ nat) @ A3) @ Ofs)) @ (((plus_plus @ nat) @ B2) @ Ofs)))))).
% 9.43/3.45 thf(fact_3553_len__greater__imp__nonempty, axiom, (! [A:$tType,X:nat,L:(list @ A)]: (((((ord_less @ nat) @ X) @ ((size_size @ (list @ A)) @ L)) => (L != (nil @ A)))))).
% 9.43/3.45 thf(fact_3554_map__add__upt, axiom, (! [N:nat,M2:nat]: ((((((map @ nat) @ nat) @ (^ [I3:nat]: ((((plus_plus @ nat) @ I3) @ N)))) @ ((upt @ (zero_zero @ nat)) @ M2)) = ((upt @ N) @ (((plus_plus @ nat) @ M2) @ N)))))).
% 9.43/3.45 thf(fact_3555_map__of__Cons__code_I1_J, axiom, (! [B:$tType,A:$tType,K:B]: ((((((map_of @ B) @ A) @ (nil @ ((product_prod @ B) @ A))) @ K) = (none @ A))))).
% 9.43/3.45 thf(fact_3556_map__of_Osimps_I1_J, axiom, (! [B:$tType,A:$tType]: (((((map_of @ A) @ B) @ (nil @ ((product_prod @ A) @ B))) = (^ [X3:A]: ((none @ B))))))).
% 9.43/3.45 thf(fact_3557_mergesort__by__rel__split_Osimps_I1_J, axiom, (! [A:$tType,Xs1:(list @ A),Xs22:(list @ A)]: (((((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ (nil @ A)) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22))))).
% 9.43/3.45 thf(fact_3558_upt__filter__extend, axiom, (! [U:nat,U5:nat,P:(nat > $o)]: (((((ord_less_eq @ nat) @ U) @ U5) => ((! [I2:nat]: ((((((ord_less_eq @ nat) @ U) @ I2) & (((ord_less @ nat) @ I2) @ U5)) => (~ ((P @ I2)))))) => ((((filter @ nat) @ P) @ ((upt @ (zero_zero @ nat)) @ U)) = (((filter @ nat) @ P) @ ((upt @ (zero_zero @ nat)) @ U5)))))))).
% 9.43/3.45 thf(fact_3559_effect__newI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H4:(heap_ext @ product_unit),N:nat,X:A,H3:(heap_ext @ product_unit)]: (((((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ A3) @ H4) = (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H3)) => ((((((heap_Time_effect @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H3) @ H4) @ A3) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))))))).
% 9.43/3.45 thf(fact_3560_effect__makeI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H4:(heap_ext @ product_unit),F:(nat > A),N:nat,H3:(heap_ext @ product_unit)]: (((((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ A3) @ H4) = (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H3)) => ((((((heap_Time_effect @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H3) @ H4) @ A3) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))))))).
% 9.43/3.45 thf(fact_3561_mergesort__by__rel_Opinduct, axiom, (! [A:$tType,A0:(A > (A > $o)),A1:(list @ A),P:((A > (A > $o)) > ((list @ A) > $o))]: (((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (mergesort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ A0) @ A1)) => ((! [R8:(A > (A > $o)),Xs2:(list @ A)]: (((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (mergesort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ R8) @ Xs2)) => (((~ ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => ((P @ R8) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs2)))) => (((~ ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs2)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => ((P @ R8) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs2)))) => ((P @ R8) @ Xs2)))))) => ((P @ A0) @ A1)))))).
% 9.43/3.45 thf(fact_3562_mergesort__by__rel_Oelims, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Y:(list @ A)]: ((((((mergesort_by_rel @ A) @ X) @ Xa) = Y) => (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xa)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) => (Y = Xa)) & ((~ ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xa)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => (Y = ((((merges9089515139780605204_merge @ A) @ X) @ (((mergesort_by_rel @ A) @ X) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xa)))) @ (((mergesort_by_rel @ A) @ X) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xa))))))))))).
% 9.43/3.45 thf(fact_3563_mergesort__by__rel_Osimps, axiom, (! [A:$tType]: (((mergesort_by_rel @ A) = (^ [R6:(A > (A > $o)),Xs3:(list @ A)]: (((((if @ (list @ A)) @ (((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ Xs3) @ ((((merges9089515139780605204_merge @ A) @ R6) @ (((mergesort_by_rel @ A) @ R6) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs3)))) @ (((mergesort_by_rel @ A) @ R6) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs3))))))))))).
% 9.43/3.45 thf(fact_3564_enumerate__replicate__eq, axiom, (! [A:$tType,N:nat,M2:nat,A3:A]: (((((enumerate @ A) @ N) @ (((replicate @ A) @ M2) @ A3)) = ((((map @ nat) @ ((product_prod @ nat) @ A)) @ (^ [Q5:nat]: (((((product_Pair @ nat) @ A) @ Q5) @ A3)))) @ ((upt @ N) @ (((plus_plus @ nat) @ N) @ M2))))))).
% 9.43/3.45 thf(fact_3565_nth__step__trancl, axiom, (! [A:$tType,Xs:(list @ A),R5:(set @ ((product_prod @ A) @ A)),N:nat,M2:nat]: (((! [N3:nat]: (((((ord_less @ nat) @ N3) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ (suc @ N3))) @ (((nth @ A) @ Xs) @ N3))) @ R5)))) => ((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ M2) @ N) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ N)) @ (((nth @ A) @ Xs) @ M2))) @ ((transitive_trancl @ A) @ R5)))))))).
% 9.43/3.45 thf(fact_3566_mergesort__by__rel__simps_I1_J, axiom, (! [A:$tType,R5:(A > (A > $o))]: (((((mergesort_by_rel @ A) @ R5) @ (nil @ A)) = (nil @ A))))).
% 9.43/3.45 thf(fact_3567_set__mergesort__by__rel, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A)]: ((((set2 @ A) @ (((mergesort_by_rel @ A) @ R5) @ Xs)) = ((set2 @ A) @ Xs))))).
% 9.43/3.45 thf(fact_3568_mergesort__by__rel__merge__simps_I3_J, axiom, (! [A:$tType,R5:(A > (A > $o)),Ys2:(list @ A)]: ((((((merges9089515139780605204_merge @ A) @ R5) @ (nil @ A)) @ Ys2) = Ys2)))).
% 9.43/3.45 thf(fact_3569_nth__upt, axiom, (! [I:nat,K:nat,J:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I) @ K)) @ J) => ((((nth @ nat) @ ((upt @ I) @ J)) @ K) = (((plus_plus @ nat) @ I) @ K)))))).
% 9.43/3.45 thf(fact_3570_set__mergesort__by__rel__merge, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A),Ys2:(list @ A)]: ((((set2 @ A) @ ((((merges9089515139780605204_merge @ A) @ R5) @ Xs) @ Ys2)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys2)))))).
% 9.43/3.45 thf(fact_3571_map__fst__enumerate, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((((map @ ((product_prod @ nat) @ A)) @ nat) @ ((product_fst @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs)) = ((upt @ N) @ (((plus_plus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))))))).
% 9.43/3.45 thf(fact_3572_nth__image__indices, axiom, (! [A:$tType,L:(list @ A)]: ((((((image2 @ nat) @ A) @ ((nth @ A) @ L)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ L))) = ((set2 @ A) @ L))))).
% 9.43/3.45 thf(fact_3573_ran__nth__set__encoding__conv, axiom, (! [A:$tType,L:(list @ A)]: (((((ran @ nat) @ A) @ (^ [I3:nat]: (((((if @ (option @ A)) @ (((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ L))) @ ((some @ A) @ (((nth @ A) @ L) @ I3))) @ (none @ A))))) = ((set2 @ A) @ L))))).
% 9.43/3.45 thf(fact_3574_mergesort__by__rel_Ocases, axiom, (! [A:$tType,X:((product_prod @ (A > (A > $o))) @ (list @ A))]: ((~ ((! [R8:(A > (A > $o)),Xs2:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ R8) @ Xs2))))))))).
% 9.43/3.45 thf(fact_3575_map__eq__nth__eq, axiom, (! [A:$tType,B:$tType,F:(B > A),L:(list @ B),L4:(list @ B),I:nat]: (((((((map @ B) @ A) @ F) @ L) = ((((map @ B) @ A) @ F) @ L4)) => ((F @ (((nth @ B) @ L) @ I)) = (F @ (((nth @ B) @ L4) @ I))))))).
% 9.43/3.45 thf(fact_3576_nth__enumerate__eq, axiom, (! [A:$tType,M2:nat,Xs:(list @ A),N:nat]: (((((ord_less @ nat) @ M2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs)) @ M2) = ((((product_Pair @ nat) @ A) @ (((plus_plus @ nat) @ N) @ M2)) @ (((nth @ A) @ Xs) @ M2))))))).
% 9.43/3.45 thf(fact_3577_mergesort__by__rel__merge__simps_I2_J, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A)]: ((((((merges9089515139780605204_merge @ A) @ R5) @ Xs) @ (nil @ A)) = Xs)))).
% 9.43/3.45 thf(fact_3578_obtain__list__from__elements, axiom, (! [A:$tType,N:nat,P:(A > (nat > $o))]: (((! [I2:nat]: (((((ord_less @ nat) @ I2) @ N) => (? [Li:A]: (((P @ Li) @ I2)))))) => (~ ((! [L5:(list @ A)]: (((((size_size @ (list @ A)) @ L5) = N) => (~ ((! [I6:nat]: (((((ord_less @ nat) @ I6) @ N) => ((P @ (((nth @ A) @ L5) @ I6)) @ I6))))))))))))))).
% 9.43/3.45 thf(fact_3579_in__set__enumerate__eq, axiom, (! [A:$tType,P5:((product_prod @ nat) @ A),N:nat,Xs:(list @ A)]: (((((member @ ((product_prod @ nat) @ A)) @ P5) @ ((set2 @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs))) = ((((ord_less_eq @ nat) @ N) @ (((product_fst @ nat) @ A) @ P5)) & ((((ord_less @ nat) @ (((product_fst @ nat) @ A) @ P5)) @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N)) & ((((nth @ A) @ Xs) @ (((minus_minus @ nat) @ (((product_fst @ nat) @ A) @ P5)) @ N)) = (((product_snd @ nat) @ A) @ P5)))))))).
% 9.43/3.45 thf(fact_3580_all__set__conv__nth, axiom, (! [A:$tType,L:(list @ A),P:(A > $o)]: (((! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ L)) => (P @ X3)))) = (! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ L)) => (P @ (((nth @ A) @ L) @ I3))))))))).
% 9.43/3.45 thf(fact_3581_set__image__eq__pointwiseI, axiom, (! [B:$tType,A:$tType,L:(list @ A),L4:(list @ A),F:(A > B)]: (((((size_size @ (list @ A)) @ L) = ((size_size @ (list @ A)) @ L4)) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ L)) => ((F @ (((nth @ A) @ L) @ I2)) = (F @ (((nth @ A) @ L4) @ I2)))))) => (((((image2 @ A) @ B) @ F) @ ((set2 @ A) @ L)) = ((((image2 @ A) @ B) @ F) @ ((set2 @ A) @ L4)))))))).
% 9.43/3.45 thf(fact_3582_in__set__image__conv__nth, axiom, (! [B:$tType,A:$tType,F:(B > A),X:B,L:(list @ B)]: (((((member @ A) @ (F @ X)) @ ((((image2 @ B) @ A) @ F) @ ((set2 @ B) @ L))) = (? [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ B)) @ L)) & ((F @ (((nth @ B) @ L) @ I3)) = (F @ X))))))))).
% 9.43/3.45 thf(fact_3583_distinct__idx, axiom, (! [B:$tType,A:$tType,F:(B > A),L:(list @ B),I:nat,J:nat]: ((((distinct @ A) @ ((((map @ B) @ A) @ F) @ L)) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ L)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ B)) @ L)) => (((F @ (((nth @ B) @ L) @ I)) = (F @ (((nth @ B) @ L) @ J))) => (I = J)))))))).
% 9.43/3.45 thf(fact_3584_array__of__list__make, axiom, (! [A:$tType]: (((heap @ A) => ((array_of_list @ A) = (^ [Xs3:(list @ A)]: ((((array_make @ A) @ ((size_size @ (list @ A)) @ Xs3)) @ ((nth @ A) @ Xs3))))))))).
% 9.43/3.45 thf(fact_3585_nth__map__upt, axiom, (! [A:$tType,I:nat,N:nat,M2:nat,F:(nat > A)]: (((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ N) @ M2)) => ((((nth @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ M2) @ N))) @ I) = (F @ (((plus_plus @ nat) @ M2) @ I))))))).
% 9.43/3.45 thf(fact_3586_mergesort__by__rel_Opsimps, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A)]: (((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (mergesort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ R5) @ Xs)) => (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) => ((((mergesort_by_rel @ A) @ R5) @ Xs) = Xs)) & ((~ ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => ((((mergesort_by_rel @ A) @ R5) @ Xs) = ((((merges9089515139780605204_merge @ A) @ R5) @ (((mergesort_by_rel @ A) @ R5) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs)))) @ (((mergesort_by_rel @ A) @ R5) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs))))))))))).
% 9.43/3.45 thf(fact_3587_mergesort__by__rel_Opelims, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Y:(list @ A)]: ((((((mergesort_by_rel @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (mergesort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (~ (((((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xa)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) => (Y = Xa)) & ((~ ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xa)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))) => (Y = ((((merges9089515139780605204_merge @ A) @ X) @ (((mergesort_by_rel @ A) @ X) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xa)))) @ (((mergesort_by_rel @ A) @ X) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xa))))))) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (mergesort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_3588_enumerate__map__upt, axiom, (! [A:$tType,N:nat,F:(nat > A),M2:nat]: (((((enumerate @ A) @ N) @ ((((map @ nat) @ A) @ F) @ ((upt @ N) @ M2))) = ((((map @ nat) @ ((product_prod @ nat) @ A)) @ (^ [K4:nat]: (((((product_Pair @ nat) @ A) @ K4) @ (F @ K4))))) @ ((upt @ N) @ M2)))))).
% 9.43/3.45 thf(fact_3589_map__upt__eqI, axiom, (! [A:$tType,Xs:(list @ A),N:nat,M2:nat,F:(nat > A)]: (((((size_size @ (list @ A)) @ Xs) = (((minus_minus @ nat) @ N) @ M2)) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I2) = (F @ (((plus_plus @ nat) @ M2) @ I2)))))) => (((((map @ nat) @ A) @ F) @ ((upt @ M2) @ N)) = Xs)))))).
% 9.43/3.45 thf(fact_3590_slice__nth, axiom, (! [A:$tType,From:nat,To:nat,Xs:(list @ A),I:nat]: (((((ord_less @ nat) @ From) @ To) => ((((ord_less_eq @ nat) @ To) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I) @ (((minus_minus @ nat) @ To) @ From)) => ((((nth @ A) @ ((((slice @ A) @ From) @ To) @ Xs)) @ I) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ From) @ I))))))))).
% 9.43/3.45 thf(fact_3591_product__nth, axiom, (! [A:$tType,B:$tType,N:nat,Xs:(list @ A),Ys2:(list @ B)]: (((((ord_less @ nat) @ N) @ (((times_times @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys2))) => ((((nth @ ((product_prod @ A) @ B)) @ ((((product @ A) @ B) @ Xs) @ Ys2)) @ N) = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ (((divide_divide @ nat) @ N) @ ((size_size @ (list @ B)) @ Ys2)))) @ (((nth @ B) @ Ys2) @ (((modulo_modulo @ nat) @ N) @ ((size_size @ (list @ B)) @ Ys2))))))))).
% 9.43/3.45 thf(fact_3592_mergesort__def, axiom, (! [A:$tType]: (((ord @ A) => ((mergesort @ A) = ((mergesort_by_rel @ A) @ (ord_less_eq @ A))))))).
% 9.43/3.45 thf(fact_3593_sorted__in__between, axiom, (! [A:$tType]: (((linorder @ A) => (! [I:nat,J:nat,L:(list @ A),X:A]: (((((ord_less_eq @ nat) @ (zero_zero @ nat)) @ I) => ((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ L)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) => ((((ord_less_eq @ A) @ (((nth @ A) @ L) @ I)) @ X) => ((((ord_less @ A) @ X) @ (((nth @ A) @ L) @ J)) => (~ ((! [K5:nat]: (((((ord_less_eq @ nat) @ I) @ K5) => ((((ord_less @ nat) @ K5) @ J) => ((((ord_less_eq @ A) @ (((nth @ A) @ L) @ K5)) @ X) => (~ ((((ord_less @ A) @ X) @ (((nth @ A) @ L) @ (((plus_plus @ nat) @ K5) @ (one_one @ nat))))))))))))))))))))))))).
% 9.43/3.45 thf(fact_3594_Un__set__drop__extend, axiom, (! [A:$tType,J:nat,L:(list @ (set @ A))]: (((((ord_less_eq @ nat) @ (suc @ (zero_zero @ nat))) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ (set @ A))) @ L)) => ((((sup_sup @ (set @ A)) @ (((nth @ (set @ A)) @ L) @ (((minus_minus @ nat) @ J) @ (suc @ (zero_zero @ nat))))) @ ((complete_Sup_Sup @ (set @ A)) @ ((set2 @ (set @ A)) @ (((drop @ (set @ A)) @ J) @ L)))) = ((complete_Sup_Sup @ (set @ A)) @ ((set2 @ (set @ A)) @ (((drop @ (set @ A)) @ (((minus_minus @ nat) @ J) @ (suc @ (zero_zero @ nat)))) @ L))))))))).
% 9.43/3.45 thf(fact_3595_mergesort__by__rel__simps_I3_J, axiom, (! [A:$tType,R5:(A > (A > $o)),X1:A,X22:A,Xs:(list @ A)]: (((((mergesort_by_rel @ A) @ R5) @ (((cons @ A) @ X1) @ (((cons @ A) @ X22) @ Xs))) = (((((product_case_prod @ (list @ A)) @ (list @ A)) @ (list @ A)) @ (^ [Xs12:(list @ A),Xs23:(list @ A)]: (((((merges9089515139780605204_merge @ A) @ R5) @ (((mergesort_by_rel @ A) @ R5) @ Xs12)) @ (((mergesort_by_rel @ A) @ R5) @ Xs23))))) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X1) @ (nil @ A))) @ (((cons @ A) @ X22) @ (nil @ A)))) @ Xs)))))).
% 9.43/3.45 thf(fact_3596_drop__drop, axiom, (! [A:$tType,N:nat,M2:nat,Xs:(list @ A)]: (((((drop @ A) @ N) @ (((drop @ A) @ M2) @ Xs)) = (((drop @ A) @ (((plus_plus @ nat) @ N) @ M2)) @ Xs))))).
% 9.43/3.45 thf(fact_3597_sorted__wrt__mergesort__by__rel__merge, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A),Ys2:(list @ A)]: (((! [X2:A,Y4:A]: ((((R5 @ X2) @ Y4) | ((R5 @ Y4) @ X2)))) => ((! [X2:A,Y4:A,Z3:A]: ((((R5 @ X2) @ Y4) => (((R5 @ Y4) @ Z3) => ((R5 @ X2) @ Z3))))) => ((((sorted_wrt @ A) @ R5) @ ((((merges9089515139780605204_merge @ A) @ R5) @ Xs) @ Ys2)) = ((((sorted_wrt @ A) @ R5) @ Xs) & (((sorted_wrt @ A) @ R5) @ Ys2)))))))).
% 9.43/3.45 thf(fact_3598_mergesort__by__rel__simps_I2_J, axiom, (! [A:$tType,R5:(A > (A > $o)),X:A]: (((((mergesort_by_rel @ A) @ R5) @ (((cons @ A) @ X) @ (nil @ A))) = (((cons @ A) @ X) @ (nil @ A)))))).
% 9.43/3.45 thf(fact_3599_enumerate__simps_I2_J, axiom, (! [B:$tType,N:nat,X:B,Xs:(list @ B)]: (((((enumerate @ B) @ N) @ (((cons @ B) @ X) @ Xs)) = (((cons @ ((product_prod @ nat) @ B)) @ ((((product_Pair @ nat) @ B) @ N) @ X)) @ (((enumerate @ B) @ (suc @ N)) @ Xs)))))).
% 9.43/3.45 thf(fact_3600_sorted__wrt__map__linord, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [L:(list @ ((product_prod @ A) @ B))]: (((((sorted_wrt @ ((product_prod @ A) @ B)) @ (^ [X3:((product_prod @ A) @ B),Y3:((product_prod @ A) @ B)]: ((((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ X3)) @ (((product_fst @ A) @ B) @ Y3))))) @ L) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ L))))))))).
% 9.43/3.45 thf(fact_3601_nth__Cons__numeral, axiom, (! [A:$tType,X:A,Xs:(list @ A),V:num]: (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ ((numeral_numeral @ nat) @ V)) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ (one_one @ nat))))))).
% 9.43/3.45 thf(fact_3602_drop__Cons__numeral, axiom, (! [A:$tType,V:num,X:A,Xs:(list @ A)]: (((((drop @ A) @ ((numeral_numeral @ nat) @ V)) @ (((cons @ A) @ X) @ Xs)) = (((drop @ A) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ (one_one @ nat))) @ Xs))))).
% 9.43/3.45 thf(fact_3603_nth__drop, axiom, (! [A:$tType,N:nat,Xs:(list @ A),I:nat]: (((((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((drop @ A) @ N) @ Xs)) @ I) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ N) @ I))))))).
% 9.43/3.45 thf(fact_3604_nth__Cons__pos, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => ((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N) @ (one_one @ nat)))))))).
% 9.43/3.45 thf(fact_3605_Misc_Ofoldr__Cons, axiom, (! [A:$tType,Xs:(list @ A)]: (((((((foldr @ A) @ (list @ A)) @ (cons @ A)) @ Xs) @ (nil @ A)) = Xs)))).
% 9.43/3.45 thf(fact_3606_map__consI_I1_J, axiom, (! [A:$tType,B:$tType,W:(list @ A),F:(B > A),Ww:(list @ B),A3:B]: (((W = ((((map @ B) @ A) @ F) @ Ww)) => ((((cons @ A) @ (F @ A3)) @ W) = ((((map @ B) @ A) @ F) @ (((cons @ B) @ A3) @ Ww))))))).
% 9.43/3.45 thf(fact_3607_map__eq__consE, axiom, (! [B:$tType,A:$tType,F:(B > A),Ls:(list @ B),Fa:A,Fl:(list @ A)]: (((((((map @ B) @ A) @ F) @ Ls) = (((cons @ A) @ Fa) @ Fl)) => (~ ((! [A5:B,L5:(list @ B)]: (((Ls = (((cons @ B) @ A5) @ L5)) => (((F @ A5) = Fa) => (((((map @ B) @ A) @ F) @ L5) != Fl))))))))))).
% 9.43/3.45 thf(fact_3608_neq__NilE, axiom, (! [A:$tType,L:(list @ A)]: (((L != (nil @ A)) => (~ ((! [X2:A,Xs2:(list @ A)]: ((L != (((cons @ A) @ X2) @ Xs2)))))))))).
% 9.43/3.45 thf(fact_3609_list__2pre__induct, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),W1:(list @ A),W22:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [E3:A,W12:(list @ A),W23:(list @ B)]: ((((P @ W12) @ W23) => ((P @ (((cons @ A) @ E3) @ W12)) @ W23)))) => ((! [E3:B,W13:(list @ A),W24:(list @ B)]: ((((P @ W13) @ W24) => ((P @ W13) @ (((cons @ B) @ E3) @ W24))))) => ((P @ W1) @ W22))))))).
% 9.43/3.45 thf(fact_3610_list__induct__first2, axiom, (! [A:$tType,P:((list @ A) > $o),Xs:(list @ A)]: (((P @ (nil @ A)) => ((! [X2:A]: ((P @ (((cons @ A) @ X2) @ (nil @ A))))) => ((! [X13:A,X23:A,Xs2:(list @ A)]: (((P @ Xs2) => (P @ (((cons @ A) @ X13) @ (((cons @ A) @ X23) @ Xs2)))))) => (P @ Xs))))))).
% 9.43/3.45 thf(fact_3611_mergesort__by__rel__merge__induct, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),R5:(A > (B > $o)),Xs:(list @ A),Ys2:(list @ B)]: (((! [Xs2:(list @ A)]: (((P @ Xs2) @ (nil @ B)))) => ((! [X_12:(list @ B)]: (((P @ (nil @ A)) @ X_12))) => ((! [X2:A,Xs2:(list @ A),Y4:B,Ys3:(list @ B)]: ((((R5 @ X2) @ Y4) => (((P @ Xs2) @ (((cons @ B) @ Y4) @ Ys3)) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ B) @ Y4) @ Ys3)))))) => ((! [X2:A,Xs2:(list @ A),Y4:B,Ys3:(list @ B)]: (((~ (((R5 @ X2) @ Y4))) => (((P @ (((cons @ A) @ X2) @ Xs2)) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ B) @ Y4) @ Ys3)))))) => ((P @ Xs) @ Ys2)))))))).
% 9.43/3.45 thf(fact_3612_sorted__wrt__mergesort__by__rel, axiom, (! [X14:$tType,R5:(X14 > (X14 > $o)),Xs:(list @ X14)]: (((! [X2:X14,Y4:X14]: ((((R5 @ X2) @ Y4) | ((R5 @ Y4) @ X2)))) => ((! [X2:X14,Y4:X14,Z3:X14]: ((((R5 @ X2) @ Y4) => (((R5 @ Y4) @ Z3) => ((R5 @ X2) @ Z3))))) => (((sorted_wrt @ X14) @ R5) @ (((mergesort_by_rel @ X14) @ R5) @ Xs))))))).
% 9.43/3.45 thf(fact_3613_list__tail__coinc, axiom, (! [A:$tType,N12:A,R1:(list @ A),N23:A,R22:(list @ A)]: ((((((cons @ A) @ N12) @ R1) = (((cons @ A) @ N23) @ R22)) => ((N12 = N23) & (R1 = R22)))))).
% 9.43/3.45 thf(fact_3614_drop__eq__ConsD, axiom, (! [A:$tType,N:nat,Xs:(list @ A),X:A,Xs4:(list @ A)]: ((((((drop @ A) @ N) @ Xs) = (((cons @ A) @ X) @ Xs4)) => ((((drop @ A) @ (suc @ N)) @ Xs) = Xs4))))).
% 9.43/3.45 thf(fact_3615_drop__Cons_H, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: ((((N = (zero_zero @ nat)) => ((((drop @ A) @ N) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ Xs))) & ((N != (zero_zero @ nat)) => ((((drop @ A) @ N) @ (((cons @ A) @ X) @ Xs)) = (((drop @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ Xs))))))).
% 9.43/3.45 thf(fact_3616_mergesort__by__rel__merge__simps_I1_J, axiom, (! [A:$tType,R5:(A > (A > $o)),X:A,Y:A,Xs:(list @ A),Ys2:(list @ A)]: (((((R5 @ X) @ Y) => (((((merges9089515139780605204_merge @ A) @ R5) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)) = (((cons @ A) @ X) @ ((((merges9089515139780605204_merge @ A) @ R5) @ Xs) @ (((cons @ A) @ Y) @ Ys2))))) & ((~ (((R5 @ X) @ Y))) => (((((merges9089515139780605204_merge @ A) @ R5) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)) = (((cons @ A) @ Y) @ ((((merges9089515139780605204_merge @ A) @ R5) @ (((cons @ A) @ X) @ Xs)) @ Ys2)))))))).
% 9.43/3.45 thf(fact_3617_sorted__filter_H, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((filter @ A) @ P) @ L))))))))).
% 9.43/3.45 thf(fact_3618_sorted__mergesort__by__rel, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((mergesort_by_rel @ A) @ (ord_less_eq @ A)) @ Xs)))))))).
% 9.43/3.45 thf(fact_3619_map__filter__simps_I1_J, axiom, (! [A:$tType,B:$tType,F:(B > (option @ A)),X:B,Xs:(list @ B)]: ((((((map_filter @ B) @ A) @ F) @ (((cons @ B) @ X) @ Xs)) = (((((case_option @ (list @ A)) @ A) @ ((((map_filter @ B) @ A) @ F) @ Xs)) @ (^ [Y3:A]: ((((cons @ A) @ Y3) @ ((((map_filter @ B) @ A) @ F) @ Xs))))) @ (F @ X)))))).
% 9.43/3.45 thf(fact_3620_zipf_Ocases, axiom, (! [C:$tType,A:$tType,B:$tType,X:((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B)))]: (((! [F4:(A > (B > C))]: ((X != ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ F4) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B)))))) => ((! [F4:(A > (B > C)),A5:A,As:(list @ A),B3:B,Bs:(list @ B)]: ((X != ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ F4) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B3) @ Bs)))))) => ((! [A5:(A > (B > C)),V4:A,Va:(list @ A)]: ((X != ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ A5) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ B)))))) => (~ ((! [A5:(A > (B > C)),V4:B,Va:(list @ B)]: ((X != ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ A5) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (((cons @ B) @ V4) @ Va)))))))))))))).
% 9.43/3.45 thf(fact_3621_merge_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:((product_prod @ (list @ A)) @ (list @ A))]: (((! [L22:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ L22)))) => ((! [V4:A,Va:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ A))))) => (~ ((! [X13:A,L12:(list @ A),X23:A,L22:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X13) @ L12)) @ (((cons @ A) @ X23) @ L22))))))))))))))).
% 9.43/3.45 thf(fact_3622_list__all__zip_Ocases, axiom, (! [A:$tType,B:$tType,X:((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))]: (((! [P7:(A > (B > $o))]: ((X != ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ P7) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B)))))) => ((! [P7:(A > (B > $o)),A5:A,As:(list @ A),B3:B,Bs:(list @ B)]: ((X != ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ P7) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B3) @ Bs)))))) => ((! [P7:(A > (B > $o)),V4:A,Va:(list @ A)]: ((X != ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ P7) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ B)))))) => (~ ((! [P7:(A > (B > $o)),V4:B,Va:(list @ B)]: ((X != ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ P7) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (((cons @ B) @ V4) @ Va)))))))))))))).
% 9.43/3.45 thf(fact_3623_partition__rev_Ocases, axiom, (! [A:$tType,X:((product_prod @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)))]: (((! [P7:(A > $o),Yes:(list @ A),No:(list @ A)]: ((X != ((((product_Pair @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ P7) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) @ (nil @ A)))))) => (~ ((! [P7:(A > $o),Yes:(list @ A),No:(list @ A),X2:A,Xs2:(list @ A)]: ((X != ((((product_Pair @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ P7) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) @ (((cons @ A) @ X2) @ Xs2)))))))))))).
% 9.43/3.45 thf(fact_3624_quicksort__by__rel_Ocases, axiom, (! [A:$tType,X:((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))]: (((! [R8:(A > (A > $o)),Sl:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Sl) @ (nil @ A)))))) => (~ ((! [R8:(A > (A > $o)),Sl:(list @ A),X2:A,Xs2:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Sl) @ (((cons @ A) @ X2) @ Xs2)))))))))))).
% 9.43/3.45 thf(fact_3625_mergesort__by__rel__merge_Ocases, axiom, (! [A:$tType,X:((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))]: (((! [R8:(A > (A > $o)),X2:A,Xs2:(list @ A),Y4:A,Ys3:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y4) @ Ys3)))))) => ((! [R8:(A > (A > $o)),Xs2:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs2) @ (nil @ A)))))) => (~ ((! [R8:(A > (A > $o)),V4:A,Va:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (((cons @ A) @ V4) @ Va))))))))))))).
% 9.43/3.45 thf(fact_3626_mergesort__by__rel__split_Ocases, axiom, (! [A:$tType,X:((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))]: (((! [Xs13:(list @ A),Xs24:(list @ A)]: ((X != ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) @ (nil @ A))))) => ((! [Xs13:(list @ A),Xs24:(list @ A),X2:A]: ((X != ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) @ (((cons @ A) @ X2) @ (nil @ A)))))) => (~ ((! [Xs13:(list @ A),Xs24:(list @ A),X13:A,X23:A,Xs2:(list @ A)]: ((X != ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) @ (((cons @ A) @ X13) @ (((cons @ A) @ X23) @ Xs2))))))))))))).
% 9.43/3.45 thf(fact_3627_shuffles_Ocases, axiom, (! [A:$tType,X:((product_prod @ (list @ A)) @ (list @ A))]: (((! [Ys3:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys3)))) => ((! [Xs2:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs2) @ (nil @ A))))) => (~ ((! [X2:A,Xs2:(list @ A),Y4:A,Ys3:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y4) @ Ys3)))))))))))).
% 9.43/3.45 thf(fact_3628_map__tailrec__rev_Ocases, axiom, (! [A:$tType,B:$tType,X:((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B)))]: (((! [F4:(A > B),Bs:(list @ B)]: ((X != ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ F4) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ Bs))))) => (~ ((! [F4:(A > B),A5:A,As:(list @ A),Bs:(list @ B)]: ((X != ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ F4) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ Bs))))))))))).
% 9.43/3.45 thf(fact_3629_foldr_Ofoldr__Cons, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A,Xs:(list @ A)]: ((((((foldr @ A) @ B) @ F) @ (((cons @ A) @ X) @ Xs)) = (((((comp @ B) @ B) @ B) @ (F @ X)) @ ((((foldr @ A) @ B) @ F) @ Xs)))))).
% 9.43/3.45 thf(fact_3630_arg__min__list_Ocases, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [X:((product_prod @ (A > B)) @ (list @ A))]: (((! [F4:(A > B),X2:A]: ((X != ((((product_Pair @ (A > B)) @ (list @ A)) @ F4) @ (((cons @ A) @ X2) @ (nil @ A)))))) => ((! [F4:(A > B),X2:A,Y4:A,Zs2:(list @ A)]: ((X != ((((product_Pair @ (A > B)) @ (list @ A)) @ F4) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Zs2)))))) => (~ ((! [A5:(A > B)]: ((X != ((((product_Pair @ (A > B)) @ (list @ A)) @ A5) @ (nil @ A))))))))))))))).
% 9.43/3.45 thf(fact_3631_successively_Ocases, axiom, (! [A:$tType,X:((product_prod @ (A > (A > $o))) @ (list @ A))]: (((! [P7:(A > (A > $o))]: ((X != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P7) @ (nil @ A))))) => ((! [P7:(A > (A > $o)),X2:A]: ((X != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P7) @ (((cons @ A) @ X2) @ (nil @ A)))))) => (~ ((! [P7:(A > (A > $o)),X2:A,Y4:A,Xs2:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P7) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2))))))))))))).
% 9.43/3.45 thf(fact_3632_sorted__wrt_Ocases, axiom, (! [A:$tType,X:((product_prod @ (A > (A > $o))) @ (list @ A))]: (((! [P7:(A > (A > $o))]: ((X != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P7) @ (nil @ A))))) => (~ ((! [P7:(A > (A > $o)),X2:A,Ys3:(list @ A)]: ((X != ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ P7) @ (((cons @ A) @ X2) @ Ys3))))))))))).
% 9.43/3.45 thf(fact_3633_mergesort__by__rel__merge_Oelims, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Xb:(list @ A),Y:(list @ A)]: (((((((merges9089515139780605204_merge @ A) @ X) @ Xa) @ Xb) = Y) => ((! [X2:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ Xs2)) => (! [Y4:A,Ys3:(list @ A)]: (((Xb = (((cons @ A) @ Y4) @ Ys3)) => (~ (((((X @ X2) @ Y4) => (Y = (((cons @ A) @ X2) @ ((((merges9089515139780605204_merge @ A) @ X) @ Xs2) @ (((cons @ A) @ Y4) @ Ys3))))) & ((~ (((X @ X2) @ Y4))) => (Y = (((cons @ A) @ Y4) @ ((((merges9089515139780605204_merge @ A) @ X) @ (((cons @ A) @ X2) @ Xs2)) @ Ys3))))))))))))) => (((Xb = (nil @ A)) => (Y != Xa)) => (~ (((Xa = (nil @ A)) => (! [V4:A,Va:(list @ A)]: (((Xb = (((cons @ A) @ V4) @ Va)) => (Y != (((cons @ A) @ V4) @ Va)))))))))))))).
% 9.43/3.45 thf(fact_3634_mergesort__by__rel__merge_Osimps_I3_J, axiom, (! [A:$tType,R5:(A > (A > $o)),V:A,Va2:(list @ A)]: ((((((merges9089515139780605204_merge @ A) @ R5) @ (nil @ A)) @ (((cons @ A) @ V) @ Va2)) = (((cons @ A) @ V) @ Va2))))).
% 9.43/3.45 thf(fact_3635_mergesort__by__rel__split_Osimps_I3_J, axiom, (! [A:$tType,Xs1:(list @ A),Xs22:(list @ A),X1:A,X22:A,Xs:(list @ A)]: (((((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ (((cons @ A) @ X1) @ (((cons @ A) @ X22) @ Xs))) = (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X1) @ Xs1)) @ (((cons @ A) @ X22) @ Xs22))) @ Xs))))).
% 9.43/3.45 thf(fact_3636_filter__conv__foldr, axiom, (! [A:$tType]: (((filter @ A) = (^ [P4:(A > $o),Xs3:(list @ A)]: ((((((foldr @ A) @ (list @ A)) @ (^ [X3:A,Xt:(list @ A)]: (((((if @ (list @ A)) @ (P4 @ X3)) @ (((cons @ A) @ X3) @ Xt)) @ Xt)))) @ Xs3) @ (nil @ A)))))))).
% 9.43/3.45 thf(fact_3637_sorted__wrt01, axiom, (! [A:$tType,Xs:(list @ A),P:(A > (A > $o))]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((sorted_wrt @ A) @ P) @ Xs))))).
% 9.43/3.45 thf(fact_3638_length__compl__induct, axiom, (! [A:$tType,P:((list @ A) > $o),L:(list @ A)]: (((P @ (nil @ A)) => ((! [E3:A,L5:(list @ A)]: (((! [Ll:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ll)) @ ((size_size @ (list @ A)) @ L5)) => (P @ Ll)))) => (P @ (((cons @ A) @ E3) @ L5))))) => (P @ L)))))).
% 9.43/3.45 thf(fact_3639_list__decomp__1, axiom, (! [A:$tType,L:(list @ A)]: (((((size_size @ (list @ A)) @ L) = (one_one @ nat)) => (? [A5:A]: ((L = (((cons @ A) @ A5) @ (nil @ A))))))))).
% 9.43/3.45 thf(fact_3640_map__of__Cons__code_I2_J, axiom, (! [C:$tType,B:$tType,L:B,K:B,V:C,Ps:(list @ ((product_prod @ B) @ C))]: ((((L = K) => (((((map_of @ B) @ C) @ (((cons @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ L) @ V)) @ Ps)) @ K) = ((some @ C) @ V))) & ((L != K) => (((((map_of @ B) @ C) @ (((cons @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ L) @ V)) @ Ps)) @ K) = ((((map_of @ B) @ C) @ Ps) @ K))))))).
% 9.43/3.45 thf(fact_3641_mergesort__by__rel__split_Osimps_I2_J, axiom, (! [A:$tType,Xs1:(list @ A),Xs22:(list @ A),X:A]: (((((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ (((cons @ A) @ X) @ (nil @ A))) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs1)) @ Xs22))))).
% 9.43/3.45 thf(fact_3642_mergesort__by__rel__split_Oelims, axiom, (! [A:$tType,X:((product_prod @ (list @ A)) @ (list @ A)),Xa:(list @ A),Y:((product_prod @ (list @ A)) @ (list @ A))]: ((((((merges295452479951948502_split @ A) @ X) @ Xa) = Y) => ((! [Xs13:(list @ A),Xs24:(list @ A)]: (((X = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => ((Xa = (nil @ A)) => (Y != ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)))))) => ((! [Xs13:(list @ A),Xs24:(list @ A)]: (((X = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => (! [X2:A]: (((Xa = (((cons @ A) @ X2) @ (nil @ A))) => (Y != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs13)) @ Xs24)))))))) => (~ ((! [Xs13:(list @ A),Xs24:(list @ A)]: (((X = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => (! [X13:A,X23:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X13) @ (((cons @ A) @ X23) @ Xs2))) => (Y != (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X13) @ Xs13)) @ (((cons @ A) @ X23) @ Xs24))) @ Xs2)))))))))))))))).
% 9.43/3.45 thf(fact_3643_sorted01, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs)))))))).
% 9.43/3.45 thf(fact_3644_list_Osize_I4_J, axiom, (! [A:$tType,X21:A,X222:(list @ A)]: ((((size_size @ (list @ A)) @ (((cons @ A) @ X21) @ X222)) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ X222)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.45 thf(fact_3645_nth__Cons_H, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: ((((N = (zero_zero @ nat)) => ((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = X)) & ((N != (zero_zero @ nat)) => ((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))))))))).
% 9.43/3.45 thf(fact_3646_upt__eq__Cons__conv, axiom, (! [I:nat,J:nat,X:nat,Xs:(list @ nat)]: (((((upt @ I) @ J) = (((cons @ nat) @ X) @ Xs)) = ((((ord_less @ nat) @ I) @ J) & ((I = X) & (((upt @ (((plus_plus @ nat) @ I) @ (one_one @ nat))) @ J) = Xs))))))).
% 9.43/3.45 thf(fact_3647_in__set__drop__conv__nth, axiom, (! [A:$tType,X:A,N:nat,L:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ (((drop @ A) @ N) @ L))) = (? [I3:nat]: (((((ord_less_eq @ nat) @ N) @ I3) & ((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ L)) & (X = (((nth @ A) @ L) @ I3)))))))))).
% 9.43/3.45 thf(fact_3648_set__drop__conv, axiom, (! [A:$tType,N:nat,L:(list @ A)]: ((((set2 @ A) @ (((drop @ A) @ N) @ L)) = ((collect @ A) @ (^ [Uu:A]: ((? [I3:nat]: (((Uu = (((nth @ A) @ L) @ I3)) & ((((ord_less_eq @ nat) @ N) @ I3) & (((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ L))))))))))))).
% 9.43/3.45 thf(fact_3649_list__decomp__2, axiom, (! [A:$tType,L:(list @ A)]: (((((size_size @ (list @ A)) @ L) = ((numeral_numeral @ nat) @ (bit0 @ one2))) => (? [A5:A,B3:A]: ((L = (((cons @ A) @ A5) @ (((cons @ A) @ B3) @ (nil @ A)))))))))).
% 9.43/3.45 thf(fact_3650_nth__non__equal__first__eq, axiom, (! [A:$tType,X:A,Y:A,Xs:(list @ A),N:nat]: (((X != Y) => (((((nth @ A) @ (((cons @ A) @ X) @ Xs)) @ N) = Y) = (((((nth @ A) @ Xs) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) = Y) & (((ord_less @ nat) @ (zero_zero @ nat)) @ N))))))).
% 9.43/3.45 thf(fact_3651_Cons__replicate__eq, axiom, (! [A:$tType,X:A,Xs:(list @ A),N:nat,Y:A]: ((((((cons @ A) @ X) @ Xs) = (((replicate @ A) @ N) @ Y)) = ((X = Y) & ((((ord_less @ nat) @ (zero_zero @ nat)) @ N) & (Xs = (((replicate @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ X)))))))).
% 9.43/3.45 thf(fact_3652_distinct__sorted__mono, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A),I:nat,J:nat]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) => (((distinct @ A) @ L) => ((((ord_less @ nat) @ I) @ J) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ L)) => (((ord_less @ A) @ (((nth @ A) @ L) @ I)) @ (((nth @ A) @ L) @ J)))))))))))).
% 9.43/3.45 thf(fact_3653_distinct__sorted__strict__mono__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A),I:nat,J:nat]: ((((distinct @ A) @ L) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ L)) => ((((ord_less @ A) @ (((nth @ A) @ L) @ I)) @ (((nth @ A) @ L) @ J)) = (((ord_less @ nat) @ I) @ J)))))))))))).
% 9.43/3.45 thf(fact_3654_distinct__sorted__mono__iff, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A),I:nat,J:nat]: ((((distinct @ A) @ L) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ L)) => ((((ord_less_eq @ A) @ (((nth @ A) @ L) @ I)) @ (((nth @ A) @ L) @ J)) = (((ord_less_eq @ nat) @ I) @ J)))))))))))).
% 9.43/3.45 thf(fact_3655_map__of_Osimps_I2_J, axiom, (! [B:$tType,A:$tType,P5:((product_prod @ A) @ B),Ps:(list @ ((product_prod @ A) @ B))]: (((((map_of @ A) @ B) @ (((cons @ ((product_prod @ A) @ B)) @ P5) @ Ps)) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ Ps)) @ (((product_fst @ A) @ B) @ P5)) @ ((some @ B) @ (((product_snd @ A) @ B) @ P5))))))).
% 9.43/3.45 thf(fact_3656_slice__Cons, axiom, (! [A:$tType,Begin:nat,End:nat,X:A,Xs:(list @ A)]: (((((Begin = (zero_zero @ nat)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ End)) => (((((slice @ A) @ Begin) @ End) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ ((((slice @ A) @ Begin) @ (((minus_minus @ nat) @ End) @ (one_one @ nat))) @ Xs)))) & ((~ (((Begin = (zero_zero @ nat)) & (((ord_less @ nat) @ (zero_zero @ nat)) @ End)))) => (((((slice @ A) @ Begin) @ End) @ (((cons @ A) @ X) @ Xs)) = ((((slice @ A) @ (((minus_minus @ nat) @ Begin) @ (one_one @ nat))) @ (((minus_minus @ nat) @ End) @ (one_one @ nat))) @ Xs))))))).
% 9.43/3.45 thf(fact_3657_mergesort__by__rel__split_Opelims, axiom, (! [A:$tType,X:((product_prod @ (list @ A)) @ (list @ A)),Xa:(list @ A),Y:((product_prod @ (list @ A)) @ (list @ A))]: ((((((merges295452479951948502_split @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ (merges7066485432131860899it_rel @ A)) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ X) @ Xa)) => ((! [Xs13:(list @ A),Xs24:(list @ A)]: (((X = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => ((Xa = (nil @ A)) => ((Y = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => (~ ((((accp @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ (merges7066485432131860899it_rel @ A)) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) @ (nil @ A)))))))))) => ((! [Xs13:(list @ A),Xs24:(list @ A)]: (((X = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => (! [X2:A]: (((Xa = (((cons @ A) @ X2) @ (nil @ A))) => ((Y = ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs13)) @ Xs24)) => (~ ((((accp @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ (merges7066485432131860899it_rel @ A)) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) @ (((cons @ A) @ X2) @ (nil @ A))))))))))))) => (~ ((! [Xs13:(list @ A),Xs24:(list @ A)]: (((X = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) => (! [X13:A,X23:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X13) @ (((cons @ A) @ X23) @ Xs2))) => ((Y = (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X13) @ Xs13)) @ (((cons @ A) @ X23) @ Xs24))) @ Xs2)) => (~ ((((accp @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ (merges7066485432131860899it_rel @ A)) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs13) @ Xs24)) @ (((cons @ A) @ X13) @ (((cons @ A) @ X23) @ Xs2)))))))))))))))))))))).
% 9.43/3.45 thf(fact_3658_list__decode_Opelims, axiom, (! [X:nat,Y:(list @ nat)]: ((((nat_list_decode @ X) = Y) => ((((accp @ nat) @ nat_list_decode_rel) @ X) => (((X = (zero_zero @ nat)) => ((Y = (nil @ nat)) => (~ ((((accp @ nat) @ nat_list_decode_rel) @ (zero_zero @ nat)))))) => (~ ((! [N3:nat]: (((X = (suc @ N3)) => ((Y = (((((product_case_prod @ nat) @ nat) @ (list @ nat)) @ (^ [X3:nat,Y3:nat]: ((((cons @ nat) @ X3) @ (nat_list_decode @ Y3))))) @ (nat_prod_decode @ N3))) => (~ ((((accp @ nat) @ nat_list_decode_rel) @ (suc @ N3)))))))))))))))).
% 9.43/3.45 thf(fact_3659_mergesort__by__rel__merge_Opelims, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Xb:(list @ A),Y:(list @ A)]: (((((((merges9089515139780605204_merge @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (merges2244889521215249637ge_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xa) @ Xb))) => ((! [X2:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ Xs2)) => (! [Y4:A,Ys3:(list @ A)]: (((Xb = (((cons @ A) @ Y4) @ Ys3)) => (((((X @ X2) @ Y4) => (Y = (((cons @ A) @ X2) @ ((((merges9089515139780605204_merge @ A) @ X) @ Xs2) @ (((cons @ A) @ Y4) @ Ys3))))) & ((~ (((X @ X2) @ Y4))) => (Y = (((cons @ A) @ Y4) @ ((((merges9089515139780605204_merge @ A) @ X) @ (((cons @ A) @ X2) @ Xs2)) @ Ys3))))) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (merges2244889521215249637ge_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y4) @ Ys3))))))))))))) => (((Xb = (nil @ A)) => ((Y = Xa) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (merges2244889521215249637ge_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xa) @ (nil @ A)))))))) => (~ (((Xa = (nil @ A)) => (! [V4:A,Va:(list @ A)]: (((Xb = (((cons @ A) @ V4) @ Va)) => ((Y = (((cons @ A) @ V4) @ Va)) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (merges2244889521215249637ge_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (((cons @ A) @ V4) @ Va)))))))))))))))))))).
% 9.43/3.45 thf(fact_3660_subset__eq__mset__impl_Ocases, axiom, (! [A:$tType,X:((product_prod @ (list @ A)) @ (list @ A))]: (((! [Ys3:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys3)))) => (~ ((! [X2:A,Xs2:(list @ A),Ys3:(list @ A)]: ((X != ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Ys3)))))))))).
% 9.43/3.45 thf(fact_3661_drop__upt, axiom, (! [M2:nat,I:nat,J:nat]: (((((drop @ nat) @ M2) @ ((upt @ I) @ J)) = ((upt @ (((plus_plus @ nat) @ I) @ M2)) @ J))))).
% 9.43/3.45 thf(fact_3662_natural__zero__minus__one, axiom, ((((minus_minus @ code_natural) @ (zero_zero @ code_natural)) @ (one_one @ code_natural)) = (zero_zero @ code_natural))).
% 9.43/3.45 thf(fact_3663_list__decode_Opsimps_I1_J, axiom, ((((accp @ nat) @ nat_list_decode_rel) @ (zero_zero @ nat)) => ((nat_list_decode @ (zero_zero @ nat)) = (nil @ nat)))).
% 9.43/3.45 thf(fact_3664_list__decode_Opsimps_I2_J, axiom, (! [N:nat]: (((((accp @ nat) @ nat_list_decode_rel) @ (suc @ N)) => ((nat_list_decode @ (suc @ N)) = (((((product_case_prod @ nat) @ nat) @ (list @ nat)) @ (^ [X3:nat,Y3:nat]: ((((cons @ nat) @ X3) @ (nat_list_decode @ Y3))))) @ (nat_prod_decode @ N))))))).
% 9.43/3.45 thf(fact_3665_list__encode_Oelims, axiom, (! [X:(list @ nat),Y:nat]: ((((nat_list_encode @ X) = Y) => (((X = (nil @ nat)) => (Y != (zero_zero @ nat))) => (~ ((! [X2:nat,Xs2:(list @ nat)]: (((X = (((cons @ nat) @ X2) @ Xs2)) => (Y != (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X2) @ (nat_list_encode @ Xs2))))))))))))))).
% 9.43/3.45 thf(fact_3666_mergesort__remdups__correct, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: ((((distinct @ A) @ ((mergesort_remdups @ A) @ L)) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((mergesort_remdups @ A) @ L)) & (((set2 @ A) @ ((mergesort_remdups @ A) @ L)) = ((set2 @ A) @ L)))))))))).
% 9.43/3.45 thf(fact_3667_filter__upt__last, axiom, (! [A:$tType,P:(A > $o),L:(list @ A),Js:(list @ nat),J:nat,I:nat]: ((((((filter @ nat) @ (^ [K4:nat]: ((P @ (((nth @ A) @ L) @ K4))))) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ L))) = (((append @ nat) @ Js) @ (((cons @ nat) @ J) @ (nil @ nat)))) => ((((ord_less @ nat) @ J) @ I) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => (~ ((P @ (((nth @ A) @ L) @ I)))))))))).
% 9.43/3.45 thf(fact_3668_empty__append__eq__id, axiom, (! [A:$tType]: ((((append @ A) @ (nil @ A)) = (^ [X3:(list @ A)]: (X3)))))).
% 9.43/3.45 thf(fact_3669_op__conc__empty__img__id, axiom, (! [A:$tType,L6:(set @ (list @ A))]: ((((((image2 @ (list @ A)) @ (list @ A)) @ ((append @ A) @ (nil @ A))) @ L6) = L6)))).
% 9.43/3.45 thf(fact_3670_list__e__eq__lel_I2_J, axiom, (! [A:$tType,L1:(list @ A),E6:A,L2:(list @ A),E2:A]: ((((((append @ A) @ L1) @ (((cons @ A) @ E6) @ L2)) = (((cons @ A) @ E2) @ (nil @ A))) = ((L1 = (nil @ A)) & ((E6 = E2) & (L2 = (nil @ A)))))))).
% 9.43/3.45 thf(fact_3671_list__e__eq__lel_I1_J, axiom, (! [A:$tType,E2:A,L1:(list @ A),E6:A,L2:(list @ A)]: ((((((cons @ A) @ E2) @ (nil @ A)) = (((append @ A) @ L1) @ (((cons @ A) @ E6) @ L2))) = ((L1 = (nil @ A)) & ((E6 = E2) & (L2 = (nil @ A)))))))).
% 9.43/3.45 thf(fact_3672_list__se__match_I4_J, axiom, (! [A:$tType,L2:(list @ A),A3:A,L1:(list @ A)]: (((L2 != (nil @ A)) => (((((cons @ A) @ A3) @ (nil @ A)) = (((append @ A) @ L1) @ L2)) = ((L1 = (nil @ A)) & (L2 = (((cons @ A) @ A3) @ (nil @ A))))))))).
% 9.43/3.45 thf(fact_3673_list__se__match_I3_J, axiom, (! [A:$tType,L1:(list @ A),A3:A,L2:(list @ A)]: (((L1 != (nil @ A)) => (((((cons @ A) @ A3) @ (nil @ A)) = (((append @ A) @ L1) @ L2)) = ((L1 = (((cons @ A) @ A3) @ (nil @ A))) & (L2 = (nil @ A)))))))).
% 9.43/3.45 thf(fact_3674_list__se__match_I2_J, axiom, (! [A:$tType,L2:(list @ A),L1:(list @ A),A3:A]: (((L2 != (nil @ A)) => (((((append @ A) @ L1) @ L2) = (((cons @ A) @ A3) @ (nil @ A))) = ((L1 = (nil @ A)) & (L2 = (((cons @ A) @ A3) @ (nil @ A))))))))).
% 9.43/3.45 thf(fact_3675_list__se__match_I1_J, axiom, (! [A:$tType,L1:(list @ A),L2:(list @ A),A3:A]: (((L1 != (nil @ A)) => (((((append @ A) @ L1) @ L2) = (((cons @ A) @ A3) @ (nil @ A))) = ((L1 = (((cons @ A) @ A3) @ (nil @ A))) & (L2 = (nil @ A)))))))).
% 9.43/3.45 thf(fact_3676_list__ee__eq__leel_I2_J, axiom, (! [A:$tType,L1:(list @ A),E12:A,E23:A,L2:(list @ A),E1:A,E22:A]: ((((((append @ A) @ L1) @ (((cons @ A) @ E12) @ (((cons @ A) @ E23) @ L2))) = (((cons @ A) @ E1) @ (((cons @ A) @ E22) @ (nil @ A)))) = ((L1 = (nil @ A)) & ((E1 = E12) & ((E22 = E23) & (L2 = (nil @ A))))))))).
% 9.43/3.45 thf(fact_3677_list__ee__eq__leel_I1_J, axiom, (! [A:$tType,E1:A,E22:A,L1:(list @ A),E12:A,E23:A,L2:(list @ A)]: ((((((cons @ A) @ E1) @ (((cons @ A) @ E22) @ (nil @ A))) = (((append @ A) @ L1) @ (((cons @ A) @ E12) @ (((cons @ A) @ E23) @ L2)))) = ((L1 = (nil @ A)) & ((E1 = E12) & ((E22 = E23) & (L2 = (nil @ A))))))))).
% 9.43/3.45 thf(fact_3678_length__append, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((size_size @ (list @ A)) @ (((append @ A) @ Xs) @ Ys2)) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys2)))))).
% 9.43/3.45 thf(fact_3679_upt__merge, axiom, (! [I:nat,J:nat,K:nat]: ((((((ord_less_eq @ nat) @ I) @ J) & (((ord_less_eq @ nat) @ J) @ K)) => ((((append @ nat) @ ((upt @ I) @ J)) @ ((upt @ J) @ K)) = ((upt @ I) @ K)))))).
% 9.43/3.45 thf(fact_3680_nth__append__first, axiom, (! [A:$tType,I:nat,L:(list @ A),L4:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((((nth @ A) @ (((append @ A) @ L) @ L4)) @ I) = (((nth @ A) @ L) @ I)))))).
% 9.43/3.45 thf(fact_3681_nth__append__length__plus, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),N:nat]: (((((nth @ A) @ (((append @ A) @ Xs) @ Ys2)) @ (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ N)) = (((nth @ A) @ Ys2) @ N))))).
% 9.43/3.45 thf(fact_3682_list__match__lel__lel, axiom, (! [A:$tType,C1:(list @ A),Qs:A,C22:(list @ A),C12:(list @ A),Qs2:A,C23:(list @ A)]: ((((((append @ A) @ C1) @ (((cons @ A) @ Qs) @ C22)) = (((append @ A) @ C12) @ (((cons @ A) @ Qs2) @ C23))) => ((! [C21:(list @ A)]: (((C1 = (((append @ A) @ C12) @ (((cons @ A) @ Qs2) @ C21))) => (C23 != (((append @ A) @ C21) @ (((cons @ A) @ Qs) @ C22)))))) => (((C12 = C1) => ((Qs2 = Qs) => (C23 != C22))) => (~ ((! [C212:(list @ A)]: (((C12 = (((append @ A) @ C1) @ (((cons @ A) @ Qs) @ C212))) => (C22 != (((append @ A) @ C212) @ (((cons @ A) @ Qs2) @ C23)))))))))))))).
% 9.43/3.45 thf(fact_3683_foldl__conc__empty__eq, axiom, (! [A:$tType,I:(list @ A),Ww:(list @ (list @ A))]: (((((((foldl @ (list @ A)) @ (list @ A)) @ (append @ A)) @ I) @ Ww) = (((append @ A) @ I) @ (((((foldl @ (list @ A)) @ (list @ A)) @ (append @ A)) @ (nil @ A)) @ Ww)))))).
% 9.43/3.45 thf(fact_3684_Misc_Omap__eq__append__conv, axiom, (! [A:$tType,B:$tType,F:(B > A),Ls:(list @ B),Fl:(list @ A),Fl2:(list @ A)]: (((((((map @ B) @ A) @ F) @ Ls) = (((append @ A) @ Fl) @ Fl2)) = (? [L3:(list @ B),L7:(list @ B)]: (((Ls = (((append @ B) @ L3) @ L7)) & ((((((map @ B) @ A) @ F) @ L3) = Fl) & (((((map @ B) @ A) @ F) @ L7) = Fl2))))))))).
% 9.43/3.45 thf(fact_3685_Misc_Oappend__eq__map__conv, axiom, (! [A:$tType,B:$tType,Fl:(list @ A),Fl2:(list @ A),F:(B > A),Ls:(list @ B)]: ((((((append @ A) @ Fl) @ Fl2) = ((((map @ B) @ A) @ F) @ Ls)) = (? [L3:(list @ B),L7:(list @ B)]: (((Ls = (((append @ B) @ L3) @ L7)) & ((((((map @ B) @ A) @ F) @ L3) = Fl) & (((((map @ B) @ A) @ F) @ L7) = Fl2))))))))).
% 9.43/3.45 thf(fact_3686_map__eq__appendE, axiom, (! [B:$tType,A:$tType,F:(B > A),Ls:(list @ B),Fl:(list @ A),Fl2:(list @ A)]: (((((((map @ B) @ A) @ F) @ Ls) = (((append @ A) @ Fl) @ Fl2)) => (~ ((! [L5:(list @ B),L8:(list @ B)]: (((Ls = (((append @ B) @ L5) @ L8)) => ((((((map @ B) @ A) @ F) @ L5) = Fl) => (((((map @ B) @ A) @ F) @ L8) != Fl2))))))))))).
% 9.43/3.45 thf(fact_3687_append__eq__mapE, axiom, (! [B:$tType,A:$tType,Fl:(list @ A),Fl2:(list @ A),F:(B > A),Ls:(list @ B)]: ((((((append @ A) @ Fl) @ Fl2) = ((((map @ B) @ A) @ F) @ Ls)) => (~ ((! [L5:(list @ B),L8:(list @ B)]: (((Ls = (((append @ B) @ L5) @ L8)) => ((((((map @ B) @ A) @ F) @ L5) = Fl) => (((((map @ B) @ A) @ F) @ L8) != Fl2))))))))))).
% 9.43/3.45 thf(fact_3688_enumerate__append__eq, axiom, (! [A:$tType,N:nat,Xs:(list @ A),Ys2:(list @ A)]: (((((enumerate @ A) @ N) @ (((append @ A) @ Xs) @ Ys2)) = (((append @ ((product_prod @ nat) @ A)) @ (((enumerate @ A) @ N) @ Xs)) @ (((enumerate @ A) @ (((plus_plus @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs))) @ Ys2)))))).
% 9.43/3.45 thf(fact_3689_neq__Nil__revE, axiom, (! [A:$tType,L:(list @ A)]: (((L != (nil @ A)) => (~ ((! [Ll2:(list @ A),E3:A]: ((L != (((append @ A) @ Ll2) @ (((cons @ A) @ E3) @ (nil @ A)))))))))))).
% 9.43/3.45 thf(fact_3690_rev__induct2_H, axiom, (! [A:$tType,B:$tType,P:((list @ A) > ((list @ B) > $o)),Xs:(list @ A),Ys2:(list @ B)]: ((((P @ (nil @ A)) @ (nil @ B)) => ((! [X2:A,Xs2:(list @ A)]: (((P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X2) @ (nil @ A)))) @ (nil @ B)))) => ((! [Y4:B,Ys3:(list @ B)]: (((P @ (nil @ A)) @ (((append @ B) @ Ys3) @ (((cons @ B) @ Y4) @ (nil @ B)))))) => ((! [X2:A,Xs2:(list @ A),Y4:B,Ys3:(list @ B)]: ((((P @ Xs2) @ Ys3) => ((P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X2) @ (nil @ A)))) @ (((append @ B) @ Ys3) @ (((cons @ B) @ Y4) @ (nil @ B))))))) => ((P @ Xs) @ Ys2)))))))).
% 9.43/3.45 thf(fact_3691_neq__Nil__rev__conv, axiom, (! [A:$tType,L:(list @ A)]: (((L != (nil @ A)) = (? [Xs3:(list @ A),X3:A]: ((L = (((append @ A) @ Xs3) @ (((cons @ A) @ X3) @ (nil @ A)))))))))).
% 9.43/3.45 thf(fact_3692_rev__nonempty__induct2_H, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),P:((list @ A) > ((list @ B) > $o))]: (((Xs != (nil @ A)) => ((Ys2 != (nil @ B)) => ((! [X2:A,Y4:B]: (((P @ (((cons @ A) @ X2) @ (nil @ A))) @ (((cons @ B) @ Y4) @ (nil @ B))))) => ((! [X2:A,Xs2:(list @ A),Y4:B]: (((Xs2 != (nil @ A)) => ((P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X2) @ (nil @ A)))) @ (((cons @ B) @ Y4) @ (nil @ B)))))) => ((! [X2:A,Y4:B,Ys3:(list @ B)]: (((Ys3 != (nil @ B)) => ((P @ (((cons @ A) @ X2) @ (nil @ A))) @ (((append @ B) @ Ys3) @ (((cons @ B) @ Y4) @ (nil @ B))))))) => ((! [X2:A,Xs2:(list @ A),Y4:B,Ys3:(list @ B)]: ((((P @ Xs2) @ Ys3) => ((Xs2 != (nil @ A)) => ((Ys3 != (nil @ B)) => ((P @ (((append @ A) @ Xs2) @ (((cons @ A) @ X2) @ (nil @ A)))) @ (((append @ B) @ Ys3) @ (((cons @ B) @ Y4) @ (nil @ B))))))))) => ((P @ Xs) @ Ys2)))))))))).
% 9.43/3.45 thf(fact_3693_list__Cons__eq__append__cases, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: ((((((cons @ A) @ X) @ Xs) = (((append @ A) @ Ys2) @ Zs)) => (((Ys2 = (nil @ A)) => (Zs != (((cons @ A) @ X) @ Xs))) => (~ ((! [Ys4:(list @ A)]: (((Ys2 = (((cons @ A) @ X) @ Ys4)) => ((((append @ A) @ Ys4) @ Zs) != Xs))))))))))).
% 9.43/3.45 thf(fact_3694_list__append__eq__Cons__cases, axiom, (! [A:$tType,Ys2:(list @ A),Zs:(list @ A),X:A,Xs:(list @ A)]: ((((((append @ A) @ Ys2) @ Zs) = (((cons @ A) @ X) @ Xs)) => (((Ys2 = (nil @ A)) => (Zs != (((cons @ A) @ X) @ Xs))) => (~ ((! [Ys4:(list @ A)]: (((Ys2 = (((cons @ A) @ X) @ Ys4)) => ((((append @ A) @ Ys4) @ Zs) != Xs))))))))))).
% 9.43/3.45 thf(fact_3695_in__set__list__format, axiom, (! [A:$tType,E2:A,L:(list @ A)]: (((((member @ A) @ E2) @ ((set2 @ A) @ L)) => (~ ((! [L12:(list @ A),L22:(list @ A)]: ((L != (((append @ A) @ L12) @ (((cons @ A) @ E2) @ L22))))))))))).
% 9.43/3.45 thf(fact_3696_xy__in__set__cases, axiom, (! [A:$tType,X:A,L:(list @ A),Y:A]: (((((member @ A) @ X) @ ((set2 @ A) @ L)) => ((((member @ A) @ Y) @ ((set2 @ A) @ L)) => (((X = Y) => (! [L12:(list @ A),L22:(list @ A)]: ((L != (((append @ A) @ L12) @ (((cons @ A) @ Y) @ L22)))))) => (((X != Y) => (! [L12:(list @ A),L22:(list @ A),L32:(list @ A)]: ((L != (((append @ A) @ L12) @ (((cons @ A) @ X) @ (((append @ A) @ L22) @ (((cons @ A) @ Y) @ L32)))))))) => (~ (((X != Y) => (! [L12:(list @ A),L22:(list @ A),L32:(list @ A)]: ((L != (((append @ A) @ L12) @ (((cons @ A) @ Y) @ (((append @ A) @ L22) @ (((cons @ A) @ X) @ L32))))))))))))))))).
% 9.43/3.45 thf(fact_3697_map__consI_I2_J, axiom, (! [B:$tType,A:$tType,W:(list @ A),L:(list @ A),F:(B > A),Ww:(list @ B),A3:B]: ((((((append @ A) @ W) @ L) = (((append @ A) @ ((((map @ B) @ A) @ F) @ Ww)) @ L)) => ((((cons @ A) @ (F @ A3)) @ (((append @ A) @ W) @ L)) = (((append @ A) @ ((((map @ B) @ A) @ F) @ (((cons @ B) @ A3) @ Ww))) @ L)))))).
% 9.43/3.45 thf(fact_3698_list__rest__coinc, axiom, (! [A:$tType,S22:(list @ A),S1:(list @ A),R1:(list @ A),R22:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ S22)) @ ((size_size @ (list @ A)) @ S1)) => (((((append @ A) @ S1) @ R1) = (((append @ A) @ S22) @ R22)) => (? [R1p:(list @ A)]: ((R22 = (((append @ A) @ R1p) @ R1))))))))).
% 9.43/3.45 thf(fact_3699_set__union__code, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((sup_sup @ (set @ A)) @ ((set2 @ A) @ Xs)) @ ((set2 @ A) @ Ys2)) = ((set2 @ A) @ (((append @ A) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3700_distinct__match, axiom, (! [A:$tType,Al:(list @ A),E2:A,Bl:(list @ A),Al2:(list @ A),Bl2:(list @ A)]: ((((distinct @ A) @ (((append @ A) @ Al) @ (((cons @ A) @ E2) @ Bl))) => (((((append @ A) @ Al) @ (((cons @ A) @ E2) @ Bl)) = (((append @ A) @ Al2) @ (((cons @ A) @ E2) @ Bl2))) = ((Al = Al2) & (Bl = Bl2))))))).
% 9.43/3.45 thf(fact_3701_upt__eq__append__conv, axiom, (! [I:nat,J:nat,Xs:(list @ nat),Ys2:(list @ nat)]: (((((ord_less_eq @ nat) @ I) @ J) => ((((upt @ I) @ J) = (((append @ nat) @ Xs) @ Ys2)) = (? [K4:nat]: (((((ord_less_eq @ nat) @ I) @ K4) & ((((ord_less_eq @ nat) @ K4) @ J) & ((((upt @ I) @ K4) = Xs) & (((upt @ K4) @ J) = Ys2))))))))))).
% 9.43/3.45 thf(fact_3702_replicate__add, axiom, (! [A:$tType,N:nat,M2:nat,X:A]: (((((replicate @ A) @ (((plus_plus @ nat) @ N) @ M2)) @ X) = (((append @ A) @ (((replicate @ A) @ N) @ X)) @ (((replicate @ A) @ M2) @ X)))))).
% 9.43/3.45 thf(fact_3703_upt__append, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => ((((append @ nat) @ ((upt @ (zero_zero @ nat)) @ I)) @ ((upt @ I) @ J)) = ((upt @ (zero_zero @ nat)) @ J)))))).
% 9.43/3.45 thf(fact_3704_upt__add__eq__append, axiom, (! [I:nat,J:nat,K:nat]: (((((ord_less_eq @ nat) @ I) @ J) => (((upt @ I) @ (((plus_plus @ nat) @ J) @ K)) = (((append @ nat) @ ((upt @ I) @ J)) @ ((upt @ J) @ (((plus_plus @ nat) @ J) @ K)))))))).
% 9.43/3.45 thf(fact_3705_foldl__rule, axiom, (! [Sigma2:$tType,A:$tType,I4:(Sigma2 > ((list @ A) > ((list @ A) > $o))),Sigma_0:Sigma2,L0:(list @ A),F:(Sigma2 > (A > Sigma2))]: (((((I4 @ Sigma_0) @ (nil @ A)) @ L0) => ((! [L12:(list @ A),L22:(list @ A),X2:A,Sigma:Sigma2]: (((L0 = (((append @ A) @ L12) @ (((cons @ A) @ X2) @ L22))) => ((((I4 @ Sigma) @ L12) @ (((cons @ A) @ X2) @ L22)) => (((I4 @ ((F @ Sigma) @ X2)) @ (((append @ A) @ L12) @ (((cons @ A) @ X2) @ (nil @ A)))) @ L22))))) => (((I4 @ (((((foldl @ Sigma2) @ A) @ F) @ Sigma_0) @ L0)) @ L0) @ (nil @ A))))))).
% 9.43/3.45 thf(fact_3706_foldl__rule__P, axiom, (! [Sigma2:$tType,A:$tType,I4:(Sigma2 > ((list @ A) > ((list @ A) > $o))),Sigma_0:Sigma2,L0:(list @ A),F:(Sigma2 > (A > Sigma2)),P:(Sigma2 > $o)]: (((((I4 @ Sigma_0) @ (nil @ A)) @ L0) => ((! [L12:(list @ A),L22:(list @ A),X2:A,Sigma:Sigma2]: (((L0 = (((append @ A) @ L12) @ (((cons @ A) @ X2) @ L22))) => ((((I4 @ Sigma) @ L12) @ (((cons @ A) @ X2) @ L22)) => (((I4 @ ((F @ Sigma) @ X2)) @ (((append @ A) @ L12) @ (((cons @ A) @ X2) @ (nil @ A)))) @ L22))))) => ((! [Sigma:Sigma2]: (((((I4 @ Sigma) @ L0) @ (nil @ A)) => (P @ Sigma)))) => (P @ (((((foldl @ Sigma2) @ A) @ F) @ Sigma_0) @ L0)))))))).
% 9.43/3.45 thf(fact_3707_foldl__rule__aux, axiom, (! [Sigma2:$tType,A:$tType,I4:(Sigma2 > ((list @ A) > $o)),Sigma_0:Sigma2,L0:(list @ A),F:(Sigma2 > (A > Sigma2))]: ((((I4 @ Sigma_0) @ L0) => ((! [L12:(list @ A),L22:(list @ A),X2:A,Sigma:Sigma2]: (((L0 = (((append @ A) @ L12) @ (((cons @ A) @ X2) @ L22))) => (((I4 @ Sigma) @ (((cons @ A) @ X2) @ L22)) => ((I4 @ ((F @ Sigma) @ X2)) @ L22))))) => ((I4 @ (((((foldl @ Sigma2) @ A) @ F) @ Sigma_0) @ L0)) @ (nil @ A))))))).
% 9.43/3.45 thf(fact_3708_foldl__rule__aux__P, axiom, (! [Sigma2:$tType,A:$tType,I4:(Sigma2 > ((list @ A) > $o)),Sigma_0:Sigma2,L0:(list @ A),F:(Sigma2 > (A > Sigma2)),P:(Sigma2 > $o)]: ((((I4 @ Sigma_0) @ L0) => ((! [L12:(list @ A),L22:(list @ A),X2:A,Sigma:Sigma2]: (((L0 = (((append @ A) @ L12) @ (((cons @ A) @ X2) @ L22))) => (((I4 @ Sigma) @ (((cons @ A) @ X2) @ L22)) => ((I4 @ ((F @ Sigma) @ X2)) @ L22))))) => ((! [Sigma:Sigma2]: ((((I4 @ Sigma) @ (nil @ A)) => (P @ Sigma)))) => (P @ (((((foldl @ Sigma2) @ A) @ F) @ Sigma_0) @ L0)))))))).
% 9.43/3.45 thf(fact_3709_map__by__foldl, axiom, (! [B:$tType,A:$tType,F:(A > B),L:(list @ A)]: (((((((foldl @ (list @ B)) @ A) @ (^ [L3:(list @ B),X3:A]: ((((append @ B) @ L3) @ (((cons @ B) @ (F @ X3)) @ (nil @ B)))))) @ (nil @ B)) @ L) = ((((map @ A) @ B) @ F) @ L))))).
% 9.43/3.45 thf(fact_3710_length__Suc__rev__conv, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((size_size @ (list @ A)) @ Xs) = (suc @ N)) = (? [Ys:(list @ A),Y3:A]: (((Xs = (((append @ A) @ Ys) @ (((cons @ A) @ Y3) @ (nil @ A)))) & (((size_size @ (list @ A)) @ Ys) = N)))))))).
% 9.43/3.45 thf(fact_3711_length__compl__rev__induct, axiom, (! [A:$tType,P:((list @ A) > $o),L:(list @ A)]: (((P @ (nil @ A)) => ((! [L5:(list @ A),E3:A]: (((! [Ll:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ll)) @ ((size_size @ (list @ A)) @ L5)) => (P @ Ll)))) => (P @ (((append @ A) @ L5) @ (((cons @ A) @ E3) @ (nil @ A))))))) => (P @ L)))))).
% 9.43/3.45 thf(fact_3712_not__distinct__split__distinct, axiom, (! [A:$tType,Xs:(list @ A)]: (((~ (((distinct @ A) @ Xs))) => (~ ((! [Y4:A,Ys3:(list @ A)]: ((((distinct @ A) @ Ys3) => ((((member @ A) @ Y4) @ ((set2 @ A) @ Ys3)) => (! [Zs2:(list @ A)]: ((Xs != (((append @ A) @ Ys3) @ (((append @ A) @ (((cons @ A) @ Y4) @ (nil @ A))) @ Zs2))))))))))))))).
% 9.43/3.45 thf(fact_3713_filter__eq__snocD, axiom, (! [A:$tType,P:(A > $o),L:(list @ A),L4:(list @ A),X:A]: ((((((filter @ A) @ P) @ L) = (((append @ A) @ L4) @ (((cons @ A) @ X) @ (nil @ A)))) => ((((member @ A) @ X) @ ((set2 @ A) @ L)) & (P @ X)))))).
% 9.43/3.45 thf(fact_3714_replicate__Suc__conv__snoc, axiom, (! [A:$tType,N:nat,X:A]: (((((replicate @ A) @ (suc @ N)) @ X) = (((append @ A) @ (((replicate @ A) @ N) @ X)) @ (((cons @ A) @ X) @ (nil @ A))))))).
% 9.43/3.45 thf(fact_3715_slice__prepend, axiom, (! [A:$tType,I:nat,K:nat,Xs:(list @ A),Ys2:(list @ A)]: (((((ord_less_eq @ nat) @ I) @ K) => ((((ord_less_eq @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((((slice @ A) @ I) @ K) @ Xs) = ((((slice @ A) @ (((plus_plus @ nat) @ I) @ ((size_size @ (list @ A)) @ Ys2))) @ (((plus_plus @ nat) @ K) @ ((size_size @ (list @ A)) @ Ys2))) @ (((append @ A) @ Ys2) @ Xs)))))))).
% 9.43/3.45 thf(fact_3716_sorted__append__bigger, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),Y:A]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ X2) @ Y)))) => (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((append @ A) @ Xs) @ (((cons @ A) @ Y) @ (nil @ A)))))))))))).
% 9.43/3.45 thf(fact_3717_upt__eq__lel__conv, axiom, (! [L:nat,H3:nat,Is1:(list @ nat),I:nat,Is2:(list @ nat)]: (((((upt @ L) @ H3) = (((append @ nat) @ Is1) @ (((cons @ nat) @ I) @ Is2))) = ((Is1 = ((upt @ L) @ I)) & ((Is2 = ((upt @ (suc @ I)) @ H3)) & ((((ord_less_eq @ nat) @ L) @ I) & (((ord_less @ nat) @ I) @ H3)))))))).
% 9.43/3.45 thf(fact_3718_list__encode_Osimps_I2_J, axiom, (! [X:nat,Xs:(list @ nat)]: (((nat_list_encode @ (((cons @ nat) @ X) @ Xs)) = (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X) @ (nat_list_encode @ Xs)))))))).
% 9.43/3.45 thf(fact_3719_list__encode_Opelims, axiom, (! [X:(list @ nat),Y:nat]: ((((nat_list_encode @ X) = Y) => ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ X) => (((X = (nil @ nat)) => ((Y = (zero_zero @ nat)) => (~ ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ (nil @ nat)))))) => (~ ((! [X2:nat,Xs2:(list @ nat)]: (((X = (((cons @ nat) @ X2) @ Xs2)) => ((Y = (suc @ (nat_prod_encode @ ((((product_Pair @ nat) @ nat) @ X2) @ (nat_list_encode @ Xs2))))) => (~ ((((accp @ (list @ nat)) @ nat_list_encode_rel) @ (((cons @ nat) @ X2) @ Xs2)))))))))))))))).
% 9.43/3.45 thf(fact_3720_map__upds__append1, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B),M2:(A > (option @ B)),X:A]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys2)) => ((((((map_upds @ A) @ B) @ M2) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) @ Ys2) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ M2) @ Xs) @ Ys2)) @ X) @ ((some @ B) @ (((nth @ B) @ Ys2) @ ((size_size @ (list @ A)) @ Xs))))))))).
% 9.43/3.45 thf(fact_3721_lexn__conv, axiom, (! [A:$tType]: (((lexn @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),N5:nat]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs3:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = N5) & ((((size_size @ (list @ A)) @ Ys) = N5) & (? [Xys2:(list @ A),X3:A,Y3:A,Xs5:(list @ A),Ys5:(list @ A)]: (((Xs3 = (((append @ A) @ Xys2) @ (((cons @ A) @ X3) @ Xs5))) & ((Ys = (((append @ A) @ Xys2) @ (((cons @ A) @ Y3) @ Ys5))) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3))))))))))))))))).
% 9.43/3.45 thf(fact_3722_drop__last__conv, axiom, (! [A:$tType,L:(list @ A)]: (((L != (nil @ A)) => ((((drop @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ L)) @ (suc @ (zero_zero @ nat)))) @ L) = (((cons @ A) @ ((last @ A) @ L)) @ (nil @ A))))))).
% 9.43/3.45 thf(fact_3723_Misc_Olast__in__set, axiom, (! [A:$tType,L:(list @ A)]: (((L != (nil @ A)) => (((member @ A) @ ((last @ A) @ L)) @ ((set2 @ A) @ L)))))).
% 9.43/3.45 thf(fact_3724_last__upt, axiom, (! [I:nat,J:nat]: (((((ord_less @ nat) @ I) @ J) => (((last @ nat) @ ((upt @ I) @ J)) = (((minus_minus @ nat) @ J) @ (one_one @ nat))))))).
% 9.43/3.45 thf(fact_3725_map__upds__Cons, axiom, (! [A:$tType,B:$tType,M2:(A > (option @ B)),A3:A,As2:(list @ A),B2:B,Bs2:(list @ B)]: (((((((map_upds @ A) @ B) @ M2) @ (((cons @ A) @ A3) @ As2)) @ (((cons @ B) @ B2) @ Bs2)) = (((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ A3) @ ((some @ B) @ B2))) @ As2) @ Bs2))))).
% 9.43/3.45 thf(fact_3726_map__upds__twist, axiom, (! [A:$tType,B:$tType,A3:A,As2:(list @ A),M2:(A > (option @ B)),B2:B,Bs2:(list @ B)]: (((~ ((((member @ A) @ A3) @ ((set2 @ A) @ As2)))) => ((((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ M2) @ A3) @ ((some @ B) @ B2))) @ As2) @ Bs2) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ M2) @ As2) @ Bs2)) @ A3) @ ((some @ B) @ B2))))))).
% 9.43/3.45 thf(fact_3727_last__filter, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o)]: (((Xs != (nil @ A)) => ((P @ ((last @ A) @ Xs)) => (((last @ A) @ (((filter @ A) @ P) @ Xs)) = ((last @ A) @ Xs))))))).
% 9.43/3.45 thf(fact_3728_lexn__length, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A)),N:nat]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ (((lexn @ A) @ R) @ N)) => ((((size_size @ (list @ A)) @ Xs) = N) & (((size_size @ (list @ A)) @ Ys2) = N)))))).
% 9.43/3.45 thf(fact_3729_product_Osimps_I2_J, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ A),Ys2:(list @ B)]: ((((((product @ A) @ B) @ (((cons @ A) @ X) @ Xs)) @ Ys2) = (((append @ ((product_prod @ A) @ B)) @ ((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X)) @ Ys2)) @ ((((product @ A) @ B) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3730_map__of__Some__split, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ B) @ A)),K:B,V:A]: (((((((map_of @ B) @ A) @ Xs) @ K) = ((some @ A) @ V)) => (? [Ys3:(list @ ((product_prod @ B) @ A)),Zs2:(list @ ((product_prod @ B) @ A))]: (((Xs = (((append @ ((product_prod @ B) @ A)) @ Ys3) @ (((cons @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ K) @ V)) @ Zs2))) & (((((map_of @ B) @ A) @ Ys3) @ K) = (none @ A))))))))).
% 9.43/3.45 thf(fact_3731_last__conv__nth, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((last @ A) @ Xs) = (((nth @ A) @ Xs) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))))).
% 9.43/3.45 thf(fact_3732_map__of__distinct__lookup, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ ((product_prod @ A) @ B)),Ys2:(list @ ((product_prod @ A) @ B)),Y:B]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))))) => ((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Ys2))))) => (((((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ Ys2))) @ X) = ((some @ B) @ Y))))))).
% 9.43/3.45 thf(fact_3733_map__of__distinct__upd3, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ ((product_prod @ A) @ B)),Ys2:(list @ ((product_prod @ A) @ B)),Y:B,Y7:B]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))))) => ((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Ys2))))) => ((((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ Ys2))) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y7)) @ Ys2)))) @ X) @ ((some @ B) @ Y)))))))).
% 9.43/3.45 thf(fact_3734_map__of__distinct__upd2, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ ((product_prod @ A) @ B)),Ys2:(list @ ((product_prod @ A) @ B)),Y:B]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))))) => ((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Ys2))))) => ((((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ Ys2))) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ Ys2))) @ X) @ ((some @ B) @ Y)))))))).
% 9.43/3.45 thf(fact_3735_map__of__distinct__upd4, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ ((product_prod @ A) @ B)),Ys2:(list @ ((product_prod @ A) @ B)),Y:B]: (((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))))) => ((~ ((((member @ A) @ X) @ ((set2 @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Ys2))))) => ((((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ Ys2)) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ (((append @ ((product_prod @ A) @ B)) @ Xs) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ Ys2)))) @ X) @ (none @ B)))))))).
% 9.43/3.45 thf(fact_3736_lex__conv, axiom, (! [A:$tType]: (((lex @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs3:(list @ A),Ys:(list @ A)]: (((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys)) & (? [Xys2:(list @ A),X3:A,Y3:A,Xs5:(list @ A),Ys5:(list @ A)]: (((Xs3 = (((append @ A) @ Xys2) @ (((cons @ A) @ X3) @ Xs5))) & ((Ys = (((append @ A) @ Xys2) @ (((cons @ A) @ Y3) @ Ys5))) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3)))))))))))))))).
% 9.43/3.45 thf(fact_3737_shuffles_Opelims, axiom, (! [A:$tType,X:(list @ A),Xa:(list @ A),Y:(set @ (list @ A))]: ((((((shuffles @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Xa)) => (((X = (nil @ A)) => ((Y = (((insert2 @ (list @ A)) @ Xa) @ (bot_bot @ (set @ (list @ A))))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xa)))))) => (((Xa = (nil @ A)) => ((Y = (((insert2 @ (list @ A)) @ X) @ (bot_bot @ (set @ (list @ A))))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ (nil @ A))))))) => (~ ((! [X2:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ Xs2)) => (! [Y4:A,Ys3:(list @ A)]: (((Xa = (((cons @ A) @ Y4) @ Ys3)) => ((Y = (((sup_sup @ (set @ (list @ A))) @ ((((image2 @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X2)) @ (((shuffles @ A) @ Xs2) @ (((cons @ A) @ Y4) @ Ys3)))) @ ((((image2 @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ Y4)) @ (((shuffles @ A) @ (((cons @ A) @ X2) @ Xs2)) @ Ys3)))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y4) @ Ys3))))))))))))))))))))).
% 9.43/3.45 thf(fact_3738_set__relcomp, axiom, (! [B:$tType,C:$tType,A:$tType,Xys:(list @ ((product_prod @ A) @ C)),Yzs:(list @ ((product_prod @ C) @ B))]: (((((((relcomp @ A) @ C) @ B) @ ((set2 @ ((product_prod @ A) @ C)) @ Xys)) @ ((set2 @ ((product_prod @ C) @ B)) @ Yzs)) = ((set2 @ ((product_prod @ A) @ B)) @ ((concat @ ((product_prod @ A) @ B)) @ ((((map @ ((product_prod @ A) @ C)) @ (list @ ((product_prod @ A) @ B))) @ (^ [Xy:((product_prod @ A) @ C)]: (((concat @ ((product_prod @ A) @ B)) @ ((((map @ ((product_prod @ C) @ B)) @ (list @ ((product_prod @ A) @ B))) @ (^ [Yz:((product_prod @ C) @ B)]: (((((if @ (list @ ((product_prod @ A) @ B))) @ ((((product_snd @ A) @ C) @ Xy) = (((product_fst @ C) @ B) @ Yz))) @ (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Xy)) @ (((product_snd @ C) @ B) @ Yz))) @ (nil @ ((product_prod @ A) @ B)))) @ (nil @ ((product_prod @ A) @ B)))))) @ Yzs))))) @ Xys))))))).
% 9.43/3.45 thf(fact_3739_natural_Osimps_I4_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T)]: ((((((code_case_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 9.43/3.45 thf(fact_3740_Cons__in__lex, axiom, (! [A:$tType,X:A,Xs:(list @ A),Y:A,Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))) @ ((lex @ A) @ R)) = (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) & (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys2))) | ((X = Y) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lex @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3741_concat__filter__neq__Nil, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((concat @ A) @ (((filter @ (list @ A)) @ (^ [Ys:(list @ A)]: ((Ys != (nil @ A))))) @ Xs)) = ((concat @ A) @ Xs))))).
% 9.43/3.45 thf(fact_3742_Nil__notin__lex, axiom, (! [A:$tType,Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys2)) @ ((lex @ A) @ R))))))).
% 9.43/3.45 thf(fact_3743_Nil2__notin__lex, axiom, (! [A:$tType,Xs:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (nil @ A))) @ ((lex @ A) @ R))))))).
% 9.43/3.45 thf(fact_3744_lex__append__leftI, axiom, (! [A:$tType,Ys2:(list @ A),Zs:(list @ A),R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ ((lex @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys2)) @ (((append @ A) @ Xs) @ Zs))) @ ((lex @ A) @ R)))))).
% 9.43/3.45 thf(fact_3745_foldl__foldl__conv__concat, axiom, (! [A:$tType,B:$tType,F:(A > (B > A)),A3:A,Xs:(list @ (list @ B))]: (((((((foldl @ A) @ (list @ B)) @ (((foldl @ A) @ B) @ F)) @ A3) @ Xs) = (((((foldl @ A) @ B) @ F) @ A3) @ ((concat @ B) @ Xs)))))).
% 9.43/3.45 thf(fact_3746_product__concat__map, axiom, (! [B:$tType,A:$tType]: ((((product @ A) @ B) = (^ [Xs3:(list @ A),Ys:(list @ B)]: (((concat @ ((product_prod @ A) @ B)) @ ((((map @ A) @ (list @ ((product_prod @ A) @ B))) @ (^ [X3:A]: (((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X3)) @ Ys)))) @ Xs3)))))))).
% 9.43/3.45 thf(fact_3747_lex__append__left__iff, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: (((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ R))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys2)) @ (((append @ A) @ Xs) @ Zs))) @ ((lex @ A) @ R)) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ ((lex @ A) @ R))))))).
% 9.43/3.45 thf(fact_3748_lex__append__leftD, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: (((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ R))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys2)) @ (((append @ A) @ Xs) @ Zs))) @ ((lex @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ ((lex @ A) @ R))))))).
% 9.43/3.45 thf(fact_3749_lex__append__rightI, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A)),Vs:(list @ A),Us:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lex @ A) @ R)) => ((((size_size @ (list @ A)) @ Vs) = ((size_size @ (list @ A)) @ Us)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Ys2) @ Vs))) @ ((lex @ A) @ R))))))).
% 9.43/3.45 thf(fact_3750_map__of__concat, axiom, (! [B:$tType,A:$tType,Xss:(list @ (list @ ((product_prod @ A) @ B)))]: (((((map_of @ A) @ B) @ ((concat @ ((product_prod @ A) @ B)) @ Xss)) = (((((foldr @ (list @ ((product_prod @ A) @ B))) @ (A > (option @ B))) @ (^ [Xs3:(list @ ((product_prod @ A) @ B)),F2:(A > (option @ B))]: (((((map_add @ A) @ B) @ F2) @ (((map_of @ A) @ B) @ Xs3))))) @ Xss) @ (^ [X3:A]: ((none @ B)))))))).
% 9.43/3.45 thf(fact_3751_comm__append__is__replicate, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((Xs != (nil @ A)) => ((Ys2 != (nil @ A)) => (((((append @ A) @ Xs) @ Ys2) = (((append @ A) @ Ys2) @ Xs)) => (? [N3:nat,Zs2:(list @ A)]: (((((ord_less @ nat) @ (one_one @ nat)) @ N3) & (((concat @ A) @ (((replicate @ (list @ A)) @ N3) @ Zs2)) = (((append @ A) @ Xs) @ Ys2))))))))))).
% 9.43/3.45 thf(fact_3752_shuffles_Opinduct, axiom, (! [A:$tType,A0:(list @ A),A1:(list @ A),P:((list @ A) > ((list @ A) > $o))]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ A0) @ A1)) => ((! [Ys3:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys3)) => ((P @ (nil @ A)) @ Ys3)))) => ((! [Xs2:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs2) @ (nil @ A))) => ((P @ Xs2) @ (nil @ A))))) => ((! [X2:A,Xs2:(list @ A),Y4:A,Ys3:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y4) @ Ys3))) => (((P @ Xs2) @ (((cons @ A) @ Y4) @ Ys3)) => (((P @ (((cons @ A) @ X2) @ Xs2)) @ Ys3) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ (((cons @ A) @ Y4) @ Ys3))))))) => ((P @ A0) @ A1)))))))).
% 9.43/3.45 thf(fact_3753_shuffles_Opsimps_I1_J, axiom, (! [A:$tType,Ys2:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys2)) => ((((shuffles @ A) @ (nil @ A)) @ Ys2) = (((insert2 @ (list @ A)) @ Ys2) @ (bot_bot @ (set @ (list @ A))))))))).
% 9.43/3.45 thf(fact_3754_shuffles_Opsimps_I2_J, axiom, (! [A:$tType,Xs:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (nil @ A))) => ((((shuffles @ A) @ Xs) @ (nil @ A)) = (((insert2 @ (list @ A)) @ Xs) @ (bot_bot @ (set @ (list @ A))))))))).
% 9.43/3.45 thf(fact_3755_distinct__concat_H, axiom, (! [A:$tType,Xs:(list @ (list @ A))]: ((((distinct @ (list @ A)) @ (((filter @ (list @ A)) @ (^ [Ys:(list @ A)]: ((Ys != (nil @ A))))) @ Xs)) => ((! [Ys3:(list @ A)]: (((((member @ (list @ A)) @ Ys3) @ ((set2 @ (list @ A)) @ Xs)) => ((distinct @ A) @ Ys3)))) => ((! [Ys3:(list @ A),Zs2:(list @ A)]: (((((member @ (list @ A)) @ Ys3) @ ((set2 @ (list @ A)) @ Xs)) => ((((member @ (list @ A)) @ Zs2) @ ((set2 @ (list @ A)) @ Xs)) => ((Ys3 != Zs2) => ((((inf_inf @ (set @ A)) @ ((set2 @ A) @ Ys3)) @ ((set2 @ A) @ Zs2)) = (bot_bot @ (set @ A)))))))) => ((distinct @ A) @ ((concat @ A) @ Xs)))))))).
% 9.43/3.45 thf(fact_3756_shuffles_Opsimps_I3_J, axiom, (! [A:$tType,X:A,Xs:(list @ A),Y:A,Ys2:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (shuffles_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))) => ((((shuffles @ A) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2)) = (((sup_sup @ (set @ (list @ A))) @ ((((image2 @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ X)) @ (((shuffles @ A) @ Xs) @ (((cons @ A) @ Y) @ Ys2)))) @ ((((image2 @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ Y)) @ (((shuffles @ A) @ (((cons @ A) @ X) @ Xs)) @ Ys2)))))))).
% 9.43/3.45 thf(fact_3757_product__code, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B)]: ((((((product_product @ A) @ B) @ ((set2 @ A) @ Xs)) @ ((set2 @ B) @ Ys2)) = ((set2 @ ((product_prod @ A) @ B)) @ ((concat @ ((product_prod @ A) @ B)) @ ((((map @ A) @ (list @ ((product_prod @ A) @ B))) @ (^ [X3:A]: (((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X3)) @ Ys2)))) @ Xs))))))).
% 9.43/3.45 thf(fact_3758_lenlex__conv, axiom, (! [A:$tType]: (((lenlex @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs3:(list @ A),Ys:(list @ A)]: (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ ((size_size @ (list @ A)) @ Ys)) | ((((size_size @ (list @ A)) @ Xs3) = ((size_size @ (list @ A)) @ Ys)) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ Ys)) @ ((lex @ A) @ R3)))))))))))))).
% 9.43/3.45 thf(fact_3759_nth__sorted__list__of__set__greaterThanLessThan, axiom, (! [N:nat,J:nat,I:nat]: (((((ord_less @ nat) @ N) @ (((minus_minus @ nat) @ J) @ (suc @ I))) => ((((nth @ nat) @ ((linord4507533701916653071of_set @ nat) @ (((set_or5935395276787703475ssThan @ nat) @ I) @ J))) @ N) = (suc @ (((plus_plus @ nat) @ I) @ N))))))).
% 9.43/3.45 thf(fact_3760_splice_Opinduct, axiom, (! [A:$tType,A0:(list @ A),A1:(list @ A),P:((list @ A) > ((list @ A) > $o))]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ A0) @ A1)) => ((! [Ys3:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys3)) => ((P @ (nil @ A)) @ Ys3)))) => ((! [X2:A,Xs2:(list @ A),Ys3:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Ys3)) => (((P @ Ys3) @ Xs2) => ((P @ (((cons @ A) @ X2) @ Xs2)) @ Ys3))))) => ((P @ A0) @ A1))))))).
% 9.43/3.45 thf(fact_3761_Nil__lenlex__iff1, axiom, (! [A:$tType,Ns:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ns)) @ ((lenlex @ A) @ R)) = (Ns != (nil @ A)))))).
% 9.43/3.45 thf(fact_3762_lenlex__irreflexive, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ R))))) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Xs)) @ ((lenlex @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3763_Nil__lenlex__iff2, axiom, (! [A:$tType,Ns:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ns) @ (nil @ A))) @ ((lenlex @ A) @ R))))))).
% 9.43/3.45 thf(fact_3764_lenlex__length, axiom, (! [A:$tType,Ms:(list @ A),Ns:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ms) @ Ns)) @ ((lenlex @ A) @ R)) => (((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ms)) @ ((size_size @ (list @ A)) @ Ns)))))).
% 9.43/3.45 thf(fact_3765_lenlex__append1, axiom, (! [A:$tType,Us:(list @ A),Xs:(list @ A),R5:(set @ ((product_prod @ A) @ A)),Vs:(list @ A),Ys2:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Us) @ Xs)) @ ((lenlex @ A) @ R5)) => ((((size_size @ (list @ A)) @ Vs) = ((size_size @ (list @ A)) @ Ys2)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Us) @ Vs)) @ (((append @ A) @ Xs) @ Ys2))) @ ((lenlex @ A) @ R5))))))).
% 9.43/3.45 thf(fact_3766_member__product, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),A4:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_product @ A) @ B) @ A4) @ B6)) = (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))))))).
% 9.43/3.45 thf(fact_3767_Product__Type_Oproduct__def, axiom, (! [B:$tType,A:$tType]: ((((product_product @ A) @ B) = (^ [A9:(set @ A),B9:(set @ B)]: (((((product_Sigma @ A) @ B) @ A9) @ (^ [Uu:A]: (B9))))))))).
% 9.43/3.45 thf(fact_3768_lenlex__def, axiom, (! [A:$tType]: (((lenlex @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((((inv_image @ ((product_prod @ nat) @ (list @ A))) @ (list @ A)) @ ((((lex_prod @ nat) @ (list @ A)) @ less_than) @ ((lex @ A) @ R3))) @ (^ [Xs3:(list @ A)]: (((((product_Pair @ nat) @ (list @ A)) @ ((size_size @ (list @ A)) @ Xs3)) @ Xs3)))))))))).
% 9.43/3.45 thf(fact_3769_Cons__lenlex__iff, axiom, (! [A:$tType,M2:A,Ms:(list @ A),N:A,Ns:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ M2) @ Ms)) @ (((cons @ A) @ N) @ Ns))) @ ((lenlex @ A) @ R)) = ((((ord_less @ nat) @ ((size_size @ (list @ A)) @ Ms)) @ ((size_size @ (list @ A)) @ Ns)) | (((((size_size @ (list @ A)) @ Ms) = ((size_size @ (list @ A)) @ Ns)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ M2) @ N)) @ R)) | ((M2 = N) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ms) @ Ns)) @ ((lenlex @ A) @ R))))))))).
% 9.43/3.45 thf(fact_3770_nth__sorted__list__of__set__greaterThanAtMost, axiom, (! [N:nat,J:nat,I:nat]: (((((ord_less @ nat) @ N) @ (((minus_minus @ nat) @ J) @ I)) => ((((nth @ nat) @ ((linord4507533701916653071of_set @ nat) @ (((set_or3652927894154168847AtMost @ nat) @ I) @ J))) @ N) = (suc @ (((plus_plus @ nat) @ I) @ N))))))).
% 9.43/3.45 thf(fact_3771_splice_Opelims, axiom, (! [A:$tType,X:(list @ A),Xa:(list @ A),Y:(list @ A)]: ((((((splice @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Xa)) => (((X = (nil @ A)) => ((Y = Xa) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xa)))))) => (~ ((! [X2:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ Xs2)) => ((Y = (((cons @ A) @ X2) @ (((splice @ A) @ Xa) @ Xs2))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xa)))))))))))))))).
% 9.43/3.45 thf(fact_3772_sorted__find__Min, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A),P:(A > $o)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => ((? [X7:A]: (((((member @ A) @ X7) @ ((set2 @ A) @ Xs)) & (P @ X7)))) => ((((find @ A) @ P) @ Xs) = ((some @ A) @ ((lattic643756798350308766er_Min @ A) @ ((collect @ A) @ (^ [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (P @ X3)))))))))))))))).
% 9.43/3.45 thf(fact_3773_part__def, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (((linorder_part @ B) @ A) = (^ [F2:(B > A),Pivot:A,Xs3:(list @ B)]: (((((product_Pair @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))) @ (((filter @ B) @ (^ [X3:B]: ((((ord_less @ A) @ (F2 @ X3)) @ Pivot)))) @ Xs3)) @ ((((product_Pair @ (list @ B)) @ (list @ B)) @ (((filter @ B) @ (^ [X3:B]: (((F2 @ X3) = Pivot)))) @ Xs3)) @ (((filter @ B) @ (^ [X3:B]: ((((ord_less @ A) @ Pivot) @ (F2 @ X3))))) @ Xs3)))))))))).
% 9.43/3.45 thf(fact_3774_merge__list__correct, axiom, (! [A:$tType]: (((linorder @ A) => (! [Ls:(list @ (list @ A)),As2:(list @ (list @ A))]: (((! [L5:(list @ A)]: (((((member @ (list @ A)) @ L5) @ ((set2 @ (list @ A)) @ Ls)) => (((distinct @ A) @ L5) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L5))))) => ((! [L5:(list @ A)]: (((((member @ (list @ A)) @ L5) @ ((set2 @ (list @ A)) @ As2)) => (((distinct @ A) @ L5) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L5))))) => (((distinct @ A) @ (((merge_list @ A) @ As2) @ Ls)) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((merge_list @ A) @ As2) @ Ls)) & (((set2 @ A) @ (((merge_list @ A) @ As2) @ Ls)) = ((set2 @ A) @ ((concat @ A) @ (((append @ (list @ A)) @ As2) @ Ls)))))))))))))).
% 9.43/3.45 thf(fact_3775_length__splice, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((size_size @ (list @ A)) @ (((splice @ A) @ Xs) @ Ys2)) = (((plus_plus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ A)) @ Ys2)))))).
% 9.43/3.45 thf(fact_3776_splice__replicate, axiom, (! [A:$tType,M2:nat,X:A,N:nat]: (((((splice @ A) @ (((replicate @ A) @ M2) @ X)) @ (((replicate @ A) @ N) @ X)) = (((replicate @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ X))))).
% 9.43/3.45 thf(fact_3777_merge__list_Osimps_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [La:(list @ A),Acc22:(list @ (list @ A)),L:(list @ A)]: (((((merge_list @ A) @ (((cons @ (list @ A)) @ La) @ Acc22)) @ (((cons @ (list @ A)) @ L) @ (nil @ (list @ A)))) = (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L) @ (((cons @ (list @ A)) @ La) @ Acc22)))))))))).
% 9.43/3.45 thf(fact_3778_merge__list_Osimps_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [La:(list @ A),Acc22:(list @ (list @ A))]: (((((merge_list @ A) @ (((cons @ (list @ A)) @ La) @ Acc22)) @ (nil @ (list @ A))) = (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ La) @ Acc22))))))))).
% 9.43/3.45 thf(fact_3779_merge__list_Osimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: (((((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L) @ (nil @ (list @ A)))) = L))))))).
% 9.43/3.45 thf(fact_3780_merge__list_Osimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => ((((merge_list @ A) @ (nil @ (list @ A))) @ (nil @ (list @ A))) = (nil @ A)))))).
% 9.43/3.45 thf(fact_3781_find__SomeD_I1_J, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),X:A]: ((((((find @ A) @ P) @ Xs) = ((some @ A) @ X)) => (P @ X))))).
% 9.43/3.45 thf(fact_3782_find_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(list @ A)]: ((((P @ X) => ((((find @ A) @ P) @ (((cons @ A) @ X) @ Xs)) = ((some @ A) @ X))) & ((~ ((P @ X))) => ((((find @ A) @ P) @ (((cons @ A) @ X) @ Xs)) = (((find @ A) @ P) @ Xs))))))).
% 9.43/3.45 thf(fact_3783_find_Osimps_I1_J, axiom, (! [A:$tType,Uu2:(A > $o)]: (((((find @ A) @ Uu2) @ (nil @ A)) = (none @ A))))).
% 9.43/3.45 thf(fact_3784_find__SomeD_I2_J, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),X:A]: ((((((find @ A) @ P) @ Xs) = ((some @ A) @ X)) => (((member @ A) @ X) @ ((set2 @ A) @ Xs)))))).
% 9.43/3.45 thf(fact_3785_find__None__iff2, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((none @ A) = (((find @ A) @ P) @ Xs)) = (~ ((? [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (P @ X3)))))))))).
% 9.43/3.45 thf(fact_3786_find__None__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A)]: ((((((find @ A) @ P) @ Xs) = (none @ A)) = (~ ((? [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (P @ X3)))))))))).
% 9.43/3.45 thf(fact_3787_part__code_I1_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Pivot2:A]: (((((((linorder_part @ B) @ A) @ F) @ Pivot2) @ (nil @ B)) = ((((product_Pair @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))) @ (nil @ B)) @ ((((product_Pair @ (list @ B)) @ (list @ B)) @ (nil @ B)) @ (nil @ B)))))))))).
% 9.43/3.45 thf(fact_3788_find__Some__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),X:A]: ((((((find @ A) @ P) @ Xs) = ((some @ A) @ X)) = (? [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & ((P @ (((nth @ A) @ Xs) @ I3)) & ((X = (((nth @ A) @ Xs) @ I3)) & (! [J2:nat]: (((((ord_less @ nat) @ J2) @ I3) => (~ ((P @ (((nth @ A) @ Xs) @ J2)))))))))))))))).
% 9.43/3.45 thf(fact_3789_find__Some__iff2, axiom, (! [A:$tType,X:A,P:(A > $o),Xs:(list @ A)]: (((((some @ A) @ X) = (((find @ A) @ P) @ Xs)) = (? [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) & ((P @ (((nth @ A) @ Xs) @ I3)) & ((X = (((nth @ A) @ Xs) @ I3)) & (! [J2:nat]: (((((ord_less @ nat) @ J2) @ I3) => (~ ((P @ (((nth @ A) @ Xs) @ J2)))))))))))))))).
% 9.43/3.45 thf(fact_3790_mergesort__remdups__def, axiom, (! [A:$tType]: (((linorder @ A) => ((mergesort_remdups @ A) = (^ [Xs3:(list @ A)]: ((((merge_list @ A) @ (nil @ (list @ A))) @ ((((map @ A) @ (list @ A)) @ (^ [X3:A]: ((((cons @ A) @ X3) @ (nil @ A))))) @ Xs3))))))))).
% 9.43/3.45 thf(fact_3791_splice_Opsimps_I2_J, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys2:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ Ys2)) => ((((splice @ A) @ (((cons @ A) @ X) @ Xs)) @ Ys2) = (((cons @ A) @ X) @ (((splice @ A) @ Ys2) @ Xs))))))).
% 9.43/3.45 thf(fact_3792_splice_Opsimps_I1_J, axiom, (! [A:$tType,Ys2:(list @ A)]: (((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (splice_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Ys2)) => ((((splice @ A) @ (nil @ A)) @ Ys2) = Ys2))))).
% 9.43/3.45 thf(fact_3793_part__code_I2_J, axiom, (! [A:$tType,B:$tType]: (((linorder @ A) => (! [F:(B > A),Pivot2:A,X:B,Xs:(list @ B)]: (((((((linorder_part @ B) @ A) @ F) @ Pivot2) @ (((cons @ B) @ X) @ Xs)) = (((((product_case_prod @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))) @ ((product_prod @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B)))) @ (^ [Lts:(list @ B)]: (((((product_case_prod @ (list @ B)) @ (list @ B)) @ ((product_prod @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B)))) @ (^ [Eqs:(list @ B),Gts:(list @ B)]: (((((if @ ((product_prod @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B)))) @ (((ord_less @ A) @ (F @ X)) @ Pivot2)) @ ((((product_Pair @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))) @ (((cons @ B) @ X) @ Lts)) @ ((((product_Pair @ (list @ B)) @ (list @ B)) @ Eqs) @ Gts))) @ ((((if @ ((product_prod @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B)))) @ (((ord_less @ A) @ Pivot2) @ (F @ X))) @ ((((product_Pair @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))) @ Lts) @ ((((product_Pair @ (list @ B)) @ (list @ B)) @ Eqs) @ (((cons @ B) @ X) @ Gts)))) @ ((((product_Pair @ (list @ B)) @ ((product_prod @ (list @ B)) @ (list @ B))) @ Lts) @ ((((product_Pair @ (list @ B)) @ (list @ B)) @ (((cons @ B) @ X) @ Eqs)) @ Gts)))))))))) @ (((((linorder_part @ B) @ A) @ F) @ Pivot2) @ Xs))))))))).
% 9.43/3.45 thf(fact_3794_extract__SomeE, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Ys2:(list @ A),Y:A,Zs:(list @ A)]: ((((((extract @ A) @ P) @ Xs) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ Ys2) @ ((((product_Pair @ A) @ (list @ A)) @ Y) @ Zs)))) => ((Xs = (((append @ A) @ Ys2) @ (((cons @ A) @ Y) @ Zs))) & ((P @ Y) & (~ ((? [X7:A]: (((((member @ A) @ X7) @ ((set2 @ A) @ Ys2)) & (P @ X7)))))))))))).
% 9.43/3.45 thf(fact_3795_extract__Some__iff, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Ys2:(list @ A),Y:A,Zs:(list @ A)]: ((((((extract @ A) @ P) @ Xs) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ Ys2) @ ((((product_Pair @ A) @ (list @ A)) @ Y) @ Zs)))) = ((Xs = (((append @ A) @ Ys2) @ (((cons @ A) @ Y) @ Zs))) & ((P @ Y) & (~ ((? [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Ys2)) & (P @ X3)))))))))))).
% 9.43/3.45 thf(fact_3796_extract__Cons__code, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(list @ A)]: ((((P @ X) => ((((extract @ A) @ P) @ (((cons @ A) @ X) @ Xs)) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (nil @ A)) @ ((((product_Pair @ A) @ (list @ A)) @ X) @ Xs))))) & ((~ ((P @ X))) => ((((extract @ A) @ P) @ (((cons @ A) @ X) @ Xs)) = (((((case_option @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ (none @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ ((((product_case_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ (^ [Ys:(list @ A)]: (((((product_case_prod @ A) @ (list @ A)) @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ (^ [Y3:A,Zs3:(list @ A)]: (((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (((cons @ A) @ X) @ Ys)) @ ((((product_Pair @ A) @ (list @ A)) @ Y3) @ Zs3)))))))))) @ (((extract @ A) @ P) @ Xs)))))))).
% 9.43/3.45 thf(fact_3797_merge__list_Ocases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))]: (((X != ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (nil @ (list @ A)))) => ((! [L5:(list @ A)]: ((X != ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A))))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A))]: ((X != ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (nil @ (list @ A)))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A)),L5:(list @ A)]: ((X != ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A))))))) => (~ ((! [Acc23:(list @ (list @ A)),L12:(list @ A),L22:(list @ A),Ls2:(list @ (list @ A))]: ((X != ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ Acc23) @ (((cons @ (list @ A)) @ L12) @ (((cons @ (list @ A)) @ L22) @ Ls2)))))))))))))))))).
% 9.43/3.45 thf(fact_3798_subset__eq__mset__impl_Oelims, axiom, (! [A:$tType,X:(list @ A),Xa:(list @ A),Y:(option @ $o)]: ((((((subset_eq_mset_impl @ A) @ X) @ Xa) = Y) => (((X = (nil @ A)) => (Y != ((some @ $o) @ (Xa != (nil @ A))))) => (~ ((! [X2:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ Xs2)) => (Y != (((((case_option @ (option @ $o)) @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ (none @ $o)) @ ((((product_case_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (option @ $o)) @ (^ [Ys1:(list @ A)]: (((((product_case_prod @ A) @ (list @ A)) @ (option @ $o)) @ (^ [Y3:A,Ys22:(list @ A)]: ((((subset_eq_mset_impl @ A) @ Xs2) @ (((append @ A) @ Ys1) @ Ys22))))))))) @ (((extract @ A) @ ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) @ X2)) @ Xa))))))))))))).
% 9.43/3.45 thf(fact_3799_lexord__def, axiom, (! [A:$tType]: (((lexord @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [X3:(list @ A),Y3:(list @ A)]: ((? [A7:A,V2:(list @ A)]: (((Y3 = (((append @ A) @ X3) @ (((cons @ A) @ A7) @ V2))) | (? [U2:(list @ A),B5:A,C3:A,W2:(list @ A),Z5:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ C3)) @ R3) & ((X3 = (((append @ A) @ U2) @ (((cons @ A) @ B5) @ W2))) & (Y3 = (((append @ A) @ U2) @ (((cons @ A) @ C3) @ Z5)))))))))))))))))))).
% 9.43/3.45 thf(fact_3800_list__collect__set__alt, axiom, (! [A:$tType,B:$tType]: ((((list_collect_set @ B) @ A) = (^ [F2:(B > (set @ A)),L3:(list @ B)]: (((complete_Sup_Sup @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [I3:nat]: (((Uu = (F2 @ (((nth @ B) @ L3) @ I3))) & (((ord_less @ nat) @ I3) @ ((size_size @ (list @ B)) @ L3))))))))))))))).
% 9.43/3.45 thf(fact_3801_subset__eq__mset__impl_Opelims, axiom, (! [A:$tType,X:(list @ A),Xa:(list @ A),Y:(option @ $o)]: ((((((subset_eq_mset_impl @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (subset751672762298770561pl_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Xa)) => (((X = (nil @ A)) => ((Y = ((some @ $o) @ (Xa != (nil @ A)))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (subset751672762298770561pl_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xa)))))) => (~ ((! [X2:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ Xs2)) => ((Y = (((((case_option @ (option @ $o)) @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ (none @ $o)) @ ((((product_case_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (option @ $o)) @ (^ [Ys1:(list @ A)]: (((((product_case_prod @ A) @ (list @ A)) @ (option @ $o)) @ (^ [Y3:A,Ys22:(list @ A)]: ((((subset_eq_mset_impl @ A) @ Xs2) @ (((append @ A) @ Ys1) @ Ys22))))))))) @ (((extract @ A) @ ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) @ X2)) @ Xa))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (subset751672762298770561pl_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xa)))))))))))))))).
% 9.43/3.45 thf(fact_3802_list__collect__set__simps_I2_J, axiom, (! [A:$tType,B:$tType,F:(B > (set @ A)),A3:B]: ((((((list_collect_set @ B) @ A) @ F) @ (((cons @ B) @ A3) @ (nil @ B))) = (F @ A3))))).
% 9.43/3.45 thf(fact_3803_list__collect__set__simps_I1_J, axiom, (! [B:$tType,A:$tType,F:(B > (set @ A))]: ((((((list_collect_set @ B) @ A) @ F) @ (nil @ B)) = (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_3804_list__collect__set__simps_I3_J, axiom, (! [A:$tType,B:$tType,F:(B > (set @ A)),A3:B,L:(list @ B)]: ((((((list_collect_set @ B) @ A) @ F) @ (((cons @ B) @ A3) @ L)) = (((sup_sup @ (set @ A)) @ (F @ A3)) @ ((((list_collect_set @ B) @ A) @ F) @ L)))))).
% 9.43/3.45 thf(fact_3805_list__collect__set__simps_I4_J, axiom, (! [A:$tType,B:$tType,F:(B > (set @ A)),L:(list @ B),L4:(list @ B)]: ((((((list_collect_set @ B) @ A) @ F) @ (((append @ B) @ L) @ L4)) = (((sup_sup @ (set @ A)) @ ((((list_collect_set @ B) @ A) @ F) @ L)) @ ((((list_collect_set @ B) @ A) @ F) @ L4)))))).
% 9.43/3.45 thf(fact_3806_lexord__cons__cons, axiom, (! [A:$tType,A3:A,X:(list @ A),B2:A,Y:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ A3) @ X)) @ (((cons @ A) @ B2) @ Y))) @ ((lexord @ A) @ R)) = ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) | ((A3 = B2) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lexord @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3807_lexord__Nil__left, axiom, (! [A:$tType,Y:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Y)) @ ((lexord @ A) @ R)) = (? [A7:A,X3:(list @ A)]: ((Y = (((cons @ A) @ A7) @ X3)))))))).
% 9.43/3.45 thf(fact_3808_list__collect__set__map__simps_I2_J, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > (set @ A)),X:(C > B),A3:C]: ((((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ (((cons @ C) @ A3) @ (nil @ C)))) = (F @ (X @ A3)))))).
% 9.43/3.45 thf(fact_3809_list__collect__set__map__simps_I1_J, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > (set @ A)),X:(C > B)]: ((((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ (nil @ C))) = (bot_bot @ (set @ A)))))).
% 9.43/3.45 thf(fact_3810_list__collect__set__map__simps_I3_J, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > (set @ A)),X:(C > B),A3:C,L:(list @ C)]: ((((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ (((cons @ C) @ A3) @ L))) = (((sup_sup @ (set @ A)) @ (F @ (X @ A3))) @ ((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ L))))))).
% 9.43/3.45 thf(fact_3811_list__collect__set__map__simps_I4_J, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > (set @ A)),X:(C > B),L:(list @ C),L4:(list @ C)]: ((((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ (((append @ C) @ L) @ L4))) = (((sup_sup @ (set @ A)) @ ((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ L))) @ ((((list_collect_set @ B) @ A) @ F) @ ((((map @ C) @ B) @ X) @ L4))))))).
% 9.43/3.45 thf(fact_3812_lexord__irreflexive, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ R))))) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Xs)) @ ((lexord @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3813_lexord__linear, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:(list @ A),Y:(list @ A)]: (((! [A5:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B3)) @ R) | ((A5 = B3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A5)) @ R))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lexord @ A) @ R)) | ((X = Y) | (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Y) @ X)) @ ((lexord @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3814_lexord__Nil__right, axiom, (! [A:$tType,X:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ (nil @ A))) @ ((lexord @ A) @ R))))))).
% 9.43/3.45 thf(fact_3815_lexord__append__leftI, axiom, (! [A:$tType,U:(list @ A),V:(list @ A),R:(set @ ((product_prod @ A) @ A)),X:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ V)) @ ((lexord @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ X) @ U)) @ (((append @ A) @ X) @ V))) @ ((lexord @ A) @ R)))))).
% 9.43/3.45 thf(fact_3816_lexord__partial__trans, axiom, (! [A:$tType,Xs:(list @ A),R:(set @ ((product_prod @ A) @ A)),Ys2:(list @ A),Zs:(list @ A)]: (((! [X2:A,Y4:A,Z3:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Z3)) @ R)))))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lexord @ A) @ R)) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ ((lexord @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Zs)) @ ((lexord @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3817_lexord__append__leftD, axiom, (! [A:$tType,X:(list @ A),U:(list @ A),V:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ X) @ U)) @ (((append @ A) @ X) @ V))) @ ((lexord @ A) @ R)) => ((! [A5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ R))))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ V)) @ ((lexord @ A) @ R))))))).
% 9.43/3.45 thf(fact_3818_lexord__append__rightI, axiom, (! [A:$tType,Y:(list @ A),X:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((? [B8:A,Z6:(list @ A)]: ((Y = (((cons @ A) @ B8) @ Z6)))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ (((append @ A) @ X) @ Y))) @ ((lexord @ A) @ R)))))).
% 9.43/3.45 thf(fact_3819_lexord__sufE, axiom, (! [A:$tType,Xs:(list @ A),Zs:(list @ A),Ys2:(list @ A),Qs:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Zs)) @ (((append @ A) @ Ys2) @ Qs))) @ ((lexord @ A) @ R)) => ((Xs != Ys2) => ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys2)) => ((((size_size @ (list @ A)) @ Zs) = ((size_size @ (list @ A)) @ Qs)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lexord @ A) @ R))))))))).
% 9.43/3.45 thf(fact_3820_list__collect__set__as__map, axiom, (! [A:$tType,B:$tType]: ((((list_collect_set @ B) @ A) = (^ [F2:(B > (set @ A)),L3:(list @ B)]: (((complete_Sup_Sup @ (set @ A)) @ ((set2 @ (set @ A)) @ ((((map @ B) @ (set @ A)) @ F2) @ L3))))))))).
% 9.43/3.45 thf(fact_3821_lexord__lex, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lex @ A) @ R)) = ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lexord @ A) @ R)) & (((size_size @ (list @ A)) @ X) = ((size_size @ (list @ A)) @ Y))))))).
% 9.43/3.45 thf(fact_3822_lexord__append__left__rightI, axiom, (! [A:$tType,A3:A,B2:A,R:(set @ ((product_prod @ A) @ A)),U:(list @ A),X:(list @ A),Y:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ U) @ (((cons @ A) @ A3) @ X))) @ (((append @ A) @ U) @ (((cons @ A) @ B2) @ Y)))) @ ((lexord @ A) @ R)))))).
% 9.43/3.45 thf(fact_3823_lexord__same__pref__iff, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys2)) @ (((append @ A) @ Xs) @ Zs))) @ ((lexord @ A) @ R)) = ((? [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ Xs)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ R)))) | (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ ((lexord @ A) @ R))))))).
% 9.43/3.45 thf(fact_3824_lexord__sufI, axiom, (! [A:$tType,U:(list @ A),W:(list @ A),R:(set @ ((product_prod @ A) @ A)),V:(list @ A),Z2:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ U) @ W)) @ ((lexord @ A) @ R)) => ((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ W)) @ ((size_size @ (list @ A)) @ U)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ U) @ V)) @ (((append @ A) @ W) @ Z2))) @ ((lexord @ A) @ R))))))).
% 9.43/3.45 thf(fact_3825_list__collect__set__def, axiom, (! [A:$tType,B:$tType]: ((((list_collect_set @ B) @ A) = (^ [F2:(B > (set @ A)),L3:(list @ B)]: (((complete_Sup_Sup @ (set @ A)) @ ((collect @ (set @ A)) @ (^ [Uu:(set @ A)]: ((? [A7:B]: (((Uu = (F2 @ A7)) & (((member @ B) @ A7) @ ((set2 @ B) @ L3))))))))))))))).
% 9.43/3.45 thf(fact_3826_subset__eq__mset__impl_Osimps_I2_J, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys2:(list @ A)]: (((((subset_eq_mset_impl @ A) @ (((cons @ A) @ X) @ Xs)) @ Ys2) = (((((case_option @ (option @ $o)) @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ (none @ $o)) @ ((((product_case_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (option @ $o)) @ (^ [Ys1:(list @ A)]: (((((product_case_prod @ A) @ (list @ A)) @ (option @ $o)) @ (^ [X3:A,Ys22:(list @ A)]: ((((subset_eq_mset_impl @ A) @ Xs) @ (((append @ A) @ Ys1) @ Ys22))))))))) @ (((extract @ A) @ ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) @ X)) @ Ys2)))))).
% 9.43/3.45 thf(fact_3827_List_Olexordp__def, axiom, (! [A:$tType]: (((lexordp @ A) = (^ [R3:(A > (A > $o)),Xs3:(list @ A),Ys:(list @ A)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ Ys)) @ ((lexord @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.45 thf(fact_3828_upto_Opelims, axiom, (! [X:int,Xa:int,Y:(list @ int)]: (((((upto @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa)) => (~ (((((((ord_less_eq @ int) @ X) @ Xa) => (Y = (((cons @ int) @ X) @ ((upto @ (((plus_plus @ int) @ X) @ (one_one @ int))) @ Xa)))) & ((~ ((((ord_less_eq @ int) @ X) @ Xa))) => (Y = (nil @ int)))) => (~ ((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_3829_upto_Opsimps, axiom, (! [I:int,J:int]: (((((accp @ ((product_prod @ int) @ int)) @ upto_rel) @ ((((product_Pair @ int) @ int) @ I) @ J)) => (((((ord_less_eq @ int) @ I) @ J) => (((upto @ I) @ J) = (((cons @ int) @ I) @ ((upto @ (((plus_plus @ int) @ I) @ (one_one @ int))) @ J)))) & ((~ ((((ord_less_eq @ int) @ I) @ J))) => (((upto @ I) @ J) = (nil @ int)))))))).
% 9.43/3.45 thf(fact_3830_listrel1__def, axiom, (! [A:$tType]: (((listrel1 @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_case_prod @ (list @ A)) @ (list @ A)) @ $o) @ (^ [Xs3:(list @ A),Ys:(list @ A)]: ((? [Us2:(list @ A),Z5:A,Z8:A,Vs2:(list @ A)]: (((Xs3 = (((append @ A) @ Us2) @ (((cons @ A) @ Z5) @ Vs2))) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z5) @ Z8)) @ R3) & (Ys = (((append @ A) @ Us2) @ (((cons @ A) @ Z8) @ Vs2))))))))))))))))).
% 9.43/3.45 thf(fact_3831_Cons__listrel1__Cons, axiom, (! [A:$tType,X:A,Xs:(list @ A),Y:A,Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))) @ ((listrel1 @ A) @ R)) = (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) & (Xs = Ys2)) | ((X = Y) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3832_nth__upto, axiom, (! [I:int,K:nat,J:int]: (((((ord_less_eq @ int) @ (((plus_plus @ int) @ I) @ ((semiring_1_of_nat @ int) @ K))) @ J) => ((((nth @ int) @ ((upto @ I) @ J)) @ K) = (((plus_plus @ int) @ I) @ ((semiring_1_of_nat @ int) @ K))))))).
% 9.43/3.45 thf(fact_3833_length__upto, axiom, (! [I:int,J:int]: ((((size_size @ (list @ int)) @ ((upto @ I) @ J)) = (nat2 @ (((plus_plus @ int) @ (((minus_minus @ int) @ J) @ I)) @ (one_one @ int))))))).
% 9.43/3.45 thf(fact_3834_upto__rec__numeral_I1_J, axiom, (! [M2:num,N:num]: ((((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N)) => (((upto @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N)) = (((cons @ int) @ ((numeral_numeral @ int) @ M2)) @ ((upto @ (((plus_plus @ int) @ ((numeral_numeral @ int) @ M2)) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N))))) & ((~ ((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N)))) => (((upto @ ((numeral_numeral @ int) @ M2)) @ ((numeral_numeral @ int) @ N)) = (nil @ int))))))).
% 9.43/3.45 thf(fact_3835_upto__rec__numeral_I2_J, axiom, (! [M2:num,N:num]: ((((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) => (((upto @ ((numeral_numeral @ int) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((cons @ int) @ ((numeral_numeral @ int) @ M2)) @ ((upto @ (((plus_plus @ int) @ ((numeral_numeral @ int) @ M2)) @ (one_one @ int))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N)))))) & ((~ ((((ord_less_eq @ int) @ ((numeral_numeral @ int) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))))) => (((upto @ ((numeral_numeral @ int) @ M2)) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (nil @ int))))))).
% 9.43/3.45 thf(fact_3836_upto__rec__numeral_I3_J, axiom, (! [M2:num,N:num]: ((((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((numeral_numeral @ int) @ N)) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((numeral_numeral @ int) @ N)) = (((cons @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((upto @ (((plus_plus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ (one_one @ int))) @ ((numeral_numeral @ int) @ N))))) & ((~ ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((numeral_numeral @ int) @ N)))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((numeral_numeral @ int) @ N)) = (nil @ int))))))).
% 9.43/3.45 thf(fact_3837_upto__rec__numeral_I4_J, axiom, (! [M2:num,N:num]: ((((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (((cons @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((upto @ (((plus_plus @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ (one_one @ int))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N)))))) & ((~ ((((ord_less_eq @ int) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))))) => (((upto @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ M2))) @ ((uminus_uminus @ int) @ ((numeral_numeral @ int) @ N))) = (nil @ int))))))).
% 9.43/3.45 thf(fact_3838_listrel1I2, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A)),X:A]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ X) @ Ys2))) @ ((listrel1 @ A) @ R)))))).
% 9.43/3.45 thf(fact_3839_rtrancl__listrel1__ConsI1, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A)),X:A]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ X) @ Ys2))) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R))))))).
% 9.43/3.45 thf(fact_3840_not__listrel1__Nil, axiom, (! [A:$tType,Xs:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (nil @ A))) @ ((listrel1 @ A) @ R))))))).
% 9.43/3.45 thf(fact_3841_not__Nil__listrel1, axiom, (! [A:$tType,Xs:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xs)) @ ((listrel1 @ A) @ R))))))).
% 9.43/3.45 thf(fact_3842_rtrancl__listrel1__eq__len, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R))) => (((size_size @ (list @ A)) @ X) = ((size_size @ (list @ A)) @ Y)))))).
% 9.43/3.45 thf(fact_3843_listrel1__eq__len, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ A)) @ Ys2)))))).
% 9.43/3.45 thf(fact_3844_append__listrel1I, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A)),Us:(list @ A),Vs:(list @ A)]: (((((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) & (Us = Vs)) | ((Xs = Ys2) & (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Us) @ Vs)) @ ((listrel1 @ A) @ R)))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Us)) @ (((append @ A) @ Ys2) @ Vs))) @ ((listrel1 @ A) @ R)))))).
% 9.43/3.45 thf(fact_3845_upto__split2, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ J)) @ ((upto @ (((plus_plus @ int) @ J) @ (one_one @ int))) @ K)))))))).
% 9.43/3.45 thf(fact_3846_upto__rec1, axiom, (! [I:int,J:int]: (((((ord_less_eq @ int) @ I) @ J) => (((upto @ I) @ J) = (((cons @ int) @ I) @ ((upto @ (((plus_plus @ int) @ I) @ (one_one @ int))) @ J))))))).
% 9.43/3.45 thf(fact_3847_upto_Osimps, axiom, (upto = (^ [I3:int,J2:int]: (((((if @ (list @ int)) @ (((ord_less_eq @ int) @ I3) @ J2)) @ (((cons @ int) @ I3) @ ((upto @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ J2))) @ (nil @ int)))))).
% 9.43/3.45 thf(fact_3848_upto_Oelims, axiom, (! [X:int,Xa:int,Y:(list @ int)]: (((((upto @ X) @ Xa) = Y) => (((((ord_less_eq @ int) @ X) @ Xa) => (Y = (((cons @ int) @ X) @ ((upto @ (((plus_plus @ int) @ X) @ (one_one @ int))) @ Xa)))) & ((~ ((((ord_less_eq @ int) @ X) @ Xa))) => (Y = (nil @ int)))))))).
% 9.43/3.45 thf(fact_3849_Cons__listrel1E2, axiom, (! [A:$tType,Xs:(list @ A),Y:A,Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ (((cons @ A) @ Y) @ Ys2))) @ ((listrel1 @ A) @ R)) => ((! [X2:A]: (((Xs = (((cons @ A) @ X2) @ Ys2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y)) @ R)))))) => (~ ((! [Zs2:(list @ A)]: (((Xs = (((cons @ A) @ Y) @ Zs2)) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Zs2) @ Ys2)) @ ((listrel1 @ A) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3850_Cons__listrel1E1, axiom, (! [A:$tType,X:A,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ Ys2)) @ ((listrel1 @ A) @ R)) => ((! [Y4:A]: (((Ys2 = (((cons @ A) @ Y4) @ Xs)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ R)))))) => (~ ((! [Zs2:(list @ A)]: (((Ys2 = (((cons @ A) @ X) @ Zs2)) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Zs2)) @ ((listrel1 @ A) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3851_listrel1I1, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Xs))) @ ((listrel1 @ A) @ R)))))).
% 9.43/3.45 thf(fact_3852_greaterThanAtMost__upto, axiom, ((set_or3652927894154168847AtMost @ int) = (^ [I3:int,J2:int]: (((set2 @ int) @ ((upto @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ J2)))))).
% 9.43/3.45 thf(fact_3853_listrel1I, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Us:(list @ A),Vs:(list @ A),Ys2:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((Xs = (((append @ A) @ Us) @ (((cons @ A) @ X) @ Vs))) => ((Ys2 = (((append @ A) @ Us) @ (((cons @ A) @ Y) @ Vs))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3854_listrel1E, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) => (~ ((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R) => (! [Us3:(list @ A),Vs3:(list @ A)]: (((Xs = (((append @ A) @ Us3) @ (((cons @ A) @ X2) @ Vs3))) => (Ys2 != (((append @ A) @ Us3) @ (((cons @ A) @ Y4) @ Vs3))))))))))))))).
% 9.43/3.45 thf(fact_3855_upto__split3, axiom, (! [I:int,J:int,K:int]: (((((ord_less_eq @ int) @ I) @ J) => ((((ord_less_eq @ int) @ J) @ K) => (((upto @ I) @ K) = (((append @ int) @ ((upto @ I) @ (((minus_minus @ int) @ J) @ (one_one @ int)))) @ (((cons @ int) @ J) @ ((upto @ (((plus_plus @ int) @ J) @ (one_one @ int))) @ K))))))))).
% 9.43/3.45 thf(fact_3856_rtrancl__listrel1__ConsI2, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys2:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ A) @ Y) @ Ys2))) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R)))))))).
% 9.43/3.45 thf(fact_3857_greaterThanLessThan__upto, axiom, ((set_or5935395276787703475ssThan @ int) = (^ [I3:int,J2:int]: (((set2 @ int) @ ((upto @ (((plus_plus @ int) @ I3) @ (one_one @ int))) @ (((minus_minus @ int) @ J2) @ (one_one @ int)))))))).
% 9.43/3.45 thf(fact_3858_snoc__listrel1__snoc__iff, axiom, (! [A:$tType,Xs:(list @ A),X:A,Ys2:(list @ A),Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) @ (((append @ A) @ Ys2) @ (((cons @ A) @ Y) @ (nil @ A))))) @ ((listrel1 @ A) @ R)) = (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) & (X = Y)) | ((Xs = Ys2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R))))))).
% 9.43/3.45 thf(fact_3859_listrel1p__def, axiom, (! [A:$tType]: (((listrel1p @ A) = (^ [R3:(A > (A > $o)),Xs3:(list @ A),Ys:(list @ A)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ Ys)) @ ((listrel1 @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.45 thf(fact_3860_sorted__wrt__map__rev__linord, axiom, (! [B:$tType,A:$tType]: (((linorder @ A) => (! [L:(list @ ((product_prod @ A) @ B))]: (((((sorted_wrt @ ((product_prod @ A) @ B)) @ (^ [X3:((product_prod @ A) @ B),Y3:((product_prod @ A) @ B)]: ((((ord_less_eq @ A) @ (((product_fst @ A) @ B) @ Y3)) @ (((product_fst @ A) @ B) @ X3))))) @ L) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ L)))))))))).
% 9.43/3.45 thf(fact_3861_set__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B)]: ((((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) = ((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [I3:nat]: (((Uu = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ I3)) @ (((nth @ B) @ Ys2) @ I3))) & (((ord_less @ nat) @ I3) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ ((size_size @ (list @ B)) @ Ys2))))))))))))).
% 9.43/3.45 thf(fact_3862_arg__min__list_Opelims, axiom, (! [B:$tType,A:$tType]: (((linorder @ B) => (! [X:(A > B),Xa:(list @ A),Y:A]: (((((((arg_min_list @ A) @ B) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ Xa)) => ((! [X2:A]: (((Xa = (((cons @ A) @ X2) @ (nil @ A))) => ((Y = X2) => (~ ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (nil @ A)))))))))) => ((! [X2:A,Y4:A,Zs2:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Zs2))) => ((Y = ((((if @ A) @ (((ord_less_eq @ B) @ (X @ X2)) @ (X @ ((((arg_min_list @ A) @ B) @ X) @ (((cons @ A) @ Y4) @ Zs2))))) @ X2) @ ((((arg_min_list @ A) @ B) @ X) @ (((cons @ A) @ Y4) @ Zs2)))) => (~ ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Zs2)))))))))) => (~ (((Xa = (nil @ A)) => ((Y = (undefined @ A)) => (~ ((((accp @ ((product_prod @ (A > B)) @ (list @ A))) @ ((arg_min_list_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ (list @ A)) @ X) @ (nil @ A))))))))))))))))))).
% 9.43/3.45 thf(fact_3863_zip__eq__zip__same__len, axiom, (! [A:$tType,B:$tType,A3:(list @ A),B2:(list @ B),A6:(list @ A),B4:(list @ B)]: (((((size_size @ (list @ A)) @ A3) = ((size_size @ (list @ B)) @ B2)) => ((((size_size @ (list @ A)) @ A6) = ((size_size @ (list @ B)) @ B4)) => ((((((zip @ A) @ B) @ A3) @ B2) = ((((zip @ A) @ B) @ A6) @ B4)) = ((A3 = A6) & (B2 = B4)))))))).
% 9.43/3.45 thf(fact_3864_sorted__wrt__rev__linord, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: (((((sorted_wrt @ A) @ (^ [X3:A,Y3:A]: ((((ord_less_eq @ A) @ Y3) @ X3)))) @ L) = (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((rev @ A) @ L))))))))).
% 9.43/3.45 thf(fact_3865_zip__Cons__Cons, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ A),Y:B,Ys2:(list @ B)]: ((((((zip @ A) @ B) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ B) @ Y) @ Ys2)) = (((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3866_map__of__rev__distinct, axiom, (! [B:$tType,A:$tType,M2:(list @ ((product_prod @ A) @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ M2)) => ((((map_of @ A) @ B) @ ((rev @ ((product_prod @ A) @ B)) @ M2)) = (((map_of @ A) @ B) @ M2)))))).
% 9.43/3.45 thf(fact_3867_zip__replicate, axiom, (! [A:$tType,B:$tType,I:nat,X:A,J:nat,Y:B]: ((((((zip @ A) @ B) @ (((replicate @ A) @ I) @ X)) @ (((replicate @ B) @ J) @ Y)) = (((replicate @ ((product_prod @ A) @ B)) @ (((ord_min @ nat) @ I) @ J)) @ ((((product_Pair @ A) @ B) @ X) @ Y)))))).
% 9.43/3.45 thf(fact_3868_map__of__zip__is__None, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B),X:A]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ X) = (none @ B)) = (~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs))))))))).
% 9.43/3.45 thf(fact_3869_nth__zip, axiom, (! [A:$tType,B:$tType,I:nat,Xs:(list @ A),Ys2:(list @ B)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ Ys2)) => ((((nth @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ I) = ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ I)) @ (((nth @ B) @ Ys2) @ I)))))))).
% 9.43/3.45 thf(fact_3870_zip__assoc, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys2:(list @ B),Zs:(list @ C)]: ((((((zip @ A) @ ((product_prod @ B) @ C)) @ Xs) @ ((((zip @ B) @ C) @ Ys2) @ Zs)) = ((((map @ ((product_prod @ ((product_prod @ A) @ B)) @ C)) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((((product_case_prod @ ((product_prod @ A) @ B)) @ C) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((((product_case_prod @ A) @ B) @ (C > ((product_prod @ A) @ ((product_prod @ B) @ C)))) @ (^ [X3:A,Y3:B,Z5:C]: (((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ X3) @ ((((product_Pair @ B) @ C) @ Y3) @ Z5))))))) @ ((((zip @ ((product_prod @ A) @ B)) @ C) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ Zs)))))).
% 9.43/3.45 thf(fact_3871_pair__list__split, axiom, (! [A:$tType,B:$tType,L:(list @ ((product_prod @ A) @ B))]: ((~ ((! [L12:(list @ A),L22:(list @ B)]: (((L = ((((zip @ A) @ B) @ L12) @ L22)) => ((((size_size @ (list @ A)) @ L12) = ((size_size @ (list @ B)) @ L22)) => (((size_size @ (list @ ((product_prod @ A) @ B))) @ L) != ((size_size @ (list @ B)) @ L22))))))))))).
% 9.43/3.45 thf(fact_3872_zip__inj, axiom, (! [A:$tType,B:$tType,A3:(list @ A),B2:(list @ B),A6:(list @ A),B4:(list @ B)]: (((((size_size @ (list @ A)) @ A3) = ((size_size @ (list @ B)) @ B2)) => ((((size_size @ (list @ A)) @ A6) = ((size_size @ (list @ B)) @ B4)) => ((((((zip @ A) @ B) @ A3) @ B2) = ((((zip @ A) @ B) @ A6) @ B4)) => ((A3 = A6) & (B2 = B4)))))))).
% 9.43/3.45 thf(fact_3873_zip__left__commute, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),Ys2:(list @ B),Zs:(list @ C)]: ((((((zip @ A) @ ((product_prod @ B) @ C)) @ Xs) @ ((((zip @ B) @ C) @ Ys2) @ Zs)) = ((((map @ ((product_prod @ B) @ ((product_prod @ A) @ C))) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ ((((product_case_prod @ B) @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ (^ [Y3:B]: (((((product_case_prod @ A) @ C) @ ((product_prod @ A) @ ((product_prod @ B) @ C))) @ (^ [X3:A,Z5:C]: (((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ X3) @ ((((product_Pair @ B) @ C) @ Y3) @ Z5))))))))) @ ((((zip @ B) @ ((product_prod @ A) @ C)) @ Ys2) @ ((((zip @ A) @ C) @ Xs) @ Zs))))))).
% 9.43/3.45 thf(fact_3874_zip__same__conv__map, axiom, (! [A:$tType,Xs:(list @ A)]: ((((((zip @ A) @ A) @ Xs) @ Xs) = ((((map @ A) @ ((product_prod @ A) @ A)) @ (^ [X3:A]: (((((product_Pair @ A) @ A) @ X3) @ X3)))) @ Xs))))).
% 9.43/3.45 thf(fact_3875_set__zip__rightD, axiom, (! [A:$tType,B:$tType,X:A,Y:B,Xs:(list @ A),Ys2:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2))) => (((member @ B) @ Y) @ ((set2 @ B) @ Ys2)))))).
% 9.43/3.45 thf(fact_3876_set__zip__leftD, axiom, (! [B:$tType,A:$tType,X:A,Y:B,Xs:(list @ A),Ys2:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2))) => (((member @ A) @ X) @ ((set2 @ A) @ Xs)))))).
% 9.43/3.45 thf(fact_3877_in__set__zipE, axiom, (! [A:$tType,B:$tType,X:A,Y:B,Xs:(list @ A),Ys2:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2))) => (~ (((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (~ ((((member @ B) @ Y) @ ((set2 @ B) @ Ys2))))))))))).
% 9.43/3.45 thf(fact_3878_zip__same, axiom, (! [A:$tType,A3:A,B2:A,Xs:(list @ A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ ((set2 @ ((product_prod @ A) @ A)) @ ((((zip @ A) @ A) @ Xs) @ Xs))) = ((((member @ A) @ A3) @ ((set2 @ A) @ Xs)) & (A3 = B2)))))).
% 9.43/3.45 thf(fact_3879_zip__eq__ConsE, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),Xy2:((product_prod @ A) @ B),Xys:(list @ ((product_prod @ A) @ B))]: (((((((zip @ A) @ B) @ Xs) @ Ys2) = (((cons @ ((product_prod @ A) @ B)) @ Xy2) @ Xys)) => (~ ((! [X2:A,Xs6:(list @ A)]: (((Xs = (((cons @ A) @ X2) @ Xs6)) => (! [Y4:B,Ys4:(list @ B)]: (((Ys2 = (((cons @ B) @ Y4) @ Ys4)) => ((Xy2 = ((((product_Pair @ A) @ B) @ X2) @ Y4)) => (Xys != ((((zip @ A) @ B) @ Xs6) @ Ys4))))))))))))))).
% 9.43/3.45 thf(fact_3880_set__zip__cart, axiom, (! [B:$tType,A:$tType,X:((product_prod @ A) @ B),L:(list @ A),L4:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ L) @ L4))) => (((member @ ((product_prod @ A) @ B)) @ X) @ ((((product_Sigma @ A) @ B) @ ((set2 @ A) @ L)) @ (^ [Uu:A]: (((set2 @ B) @ L4))))))))).
% 9.43/3.45 thf(fact_3881_zip__commute, axiom, (! [B:$tType,A:$tType]: ((((zip @ A) @ B) = (^ [Xs3:(list @ A),Ys:(list @ B)]: (((((map @ ((product_prod @ B) @ A)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ B) @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:B,Y3:A]: (((((product_Pair @ A) @ B) @ Y3) @ X3))))) @ ((((zip @ B) @ A) @ Ys) @ Xs3)))))))).
% 9.43/3.45 thf(fact_3882_last__zip, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B)]: (((Xs != (nil @ A)) => ((Ys2 != (nil @ B)) => ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => (((last @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) = ((((product_Pair @ A) @ B) @ ((last @ A) @ Xs)) @ ((last @ B) @ Ys2))))))))).
% 9.43/3.45 thf(fact_3883_in__set__impl__in__set__zip1, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),X:A]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (~ ((! [Y4:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y4)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)))))))))))))).
% 9.43/3.45 thf(fact_3884_in__set__impl__in__set__zip2, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),Y:B]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((member @ B) @ Y) @ ((set2 @ B) @ Ys2)) => (~ ((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)))))))))))))).
% 9.43/3.45 thf(fact_3885_map__of__zip__is__Some, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),X:A]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((member @ A) @ X) @ ((set2 @ A) @ Xs)) = (? [Y3:B]: ((((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ X) = ((some @ B) @ Y3))))))))).
% 9.43/3.45 thf(fact_3886_map__upds__fold__map__upd, axiom, (! [B:$tType,A:$tType]: ((((map_upds @ A) @ B) = (^ [M:(A > (option @ B)),Ks2:(list @ A),Vs2:(list @ B)]: ((((((foldl @ (A > (option @ B))) @ ((product_prod @ A) @ B)) @ (^ [N5:(A > (option @ B))]: (((((product_case_prod @ A) @ B) @ (A > (option @ B))) @ (^ [K4:A,V2:B]: ((((((fun_upd @ A) @ (option @ B)) @ N5) @ K4) @ ((some @ B) @ V2)))))))) @ M) @ ((((zip @ A) @ B) @ Ks2) @ Vs2)))))))).
% 9.43/3.45 thf(fact_3887_enumerate__eq__zip, axiom, (! [A:$tType]: (((enumerate @ A) = (^ [N5:nat,Xs3:(list @ A)]: (((((zip @ nat) @ A) @ ((upt @ N5) @ (((plus_plus @ nat) @ N5) @ ((size_size @ (list @ A)) @ Xs3)))) @ Xs3))))))).
% 9.43/3.45 thf(fact_3888_map__zip__map, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(((product_prod @ B) @ C) > A),G:(D > B),Xs:(list @ D),Ys2:(list @ C)]: ((((((map @ ((product_prod @ B) @ C)) @ A) @ F) @ ((((zip @ B) @ C) @ ((((map @ D) @ B) @ G) @ Xs)) @ Ys2)) = ((((map @ ((product_prod @ D) @ C)) @ A) @ ((((product_case_prod @ D) @ C) @ A) @ (^ [X3:D,Y3:C]: ((F @ ((((product_Pair @ B) @ C) @ (G @ X3)) @ Y3)))))) @ ((((zip @ D) @ C) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3889_map__zip__map2, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(((product_prod @ B) @ C) > A),Xs:(list @ B),G:(D > C),Ys2:(list @ D)]: ((((((map @ ((product_prod @ B) @ C)) @ A) @ F) @ ((((zip @ B) @ C) @ Xs) @ ((((map @ D) @ C) @ G) @ Ys2))) = ((((map @ ((product_prod @ B) @ D)) @ A) @ ((((product_case_prod @ B) @ D) @ A) @ (^ [X3:B,Y3:D]: ((F @ ((((product_Pair @ B) @ C) @ X3) @ (G @ Y3))))))) @ ((((zip @ B) @ D) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3890_foldl__snd__zip, axiom, (! [B:$tType,C:$tType,A:$tType,Ys2:(list @ A),Xs:(list @ B),F:(C > (A > C)),B2:C]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ B)) @ Xs)) => ((((((foldl @ C) @ ((product_prod @ B) @ A)) @ (^ [B5:C]: (((((product_case_prod @ B) @ A) @ C) @ (^ [X3:B]: ((F @ B5))))))) @ B2) @ ((((zip @ B) @ A) @ Xs) @ Ys2)) = (((((foldl @ C) @ A) @ F) @ B2) @ Ys2)))))).
% 9.43/3.45 thf(fact_3891_map__zip2, axiom, (! [A:$tType,B:$tType,K:A,L:(list @ B)]: ((((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ K)) @ L) = ((((zip @ A) @ B) @ (((replicate @ A) @ ((size_size @ (list @ B)) @ L)) @ K)) @ L))))).
% 9.43/3.45 thf(fact_3892_map__zip1, axiom, (! [A:$tType,B:$tType,K:B,L:(list @ A)]: ((((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ B) @ X3) @ K)))) @ L) = ((((zip @ A) @ B) @ L) @ (((replicate @ B) @ ((size_size @ (list @ A)) @ L)) @ K)))))).
% 9.43/3.45 thf(fact_3893_map__prod__fun__zip, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > B),Xs:(list @ C),Ys2:(list @ D)]: ((((((map @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ D) @ ((product_prod @ A) @ B)) @ (^ [X3:C,Y3:D]: (((((product_Pair @ A) @ B) @ (F @ X3)) @ (G @ Y3)))))) @ ((((zip @ C) @ D) @ Xs) @ Ys2)) = ((((zip @ A) @ B) @ ((((map @ C) @ A) @ F) @ Xs)) @ ((((map @ D) @ B) @ G) @ Ys2)))))).
% 9.43/3.45 thf(fact_3894_zip__map2, axiom, (! [B:$tType,A:$tType,C:$tType,Xs:(list @ A),F:(C > B),Ys2:(list @ C)]: ((((((zip @ A) @ B) @ Xs) @ ((((map @ C) @ B) @ F) @ Ys2)) = ((((map @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ C) @ ((product_prod @ A) @ B)) @ (^ [X3:A,Y3:C]: (((((product_Pair @ A) @ B) @ X3) @ (F @ Y3)))))) @ ((((zip @ A) @ C) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3895_zip__map1, axiom, (! [A:$tType,C:$tType,B:$tType,F:(C > A),Xs:(list @ C),Ys2:(list @ B)]: ((((((zip @ A) @ B) @ ((((map @ C) @ A) @ F) @ Xs)) @ Ys2) = ((((map @ ((product_prod @ C) @ B)) @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ C) @ B) @ ((product_prod @ A) @ B)) @ (^ [X3:C]: ((((product_Pair @ A) @ B) @ (F @ X3)))))) @ ((((zip @ C) @ B) @ Xs) @ Ys2)))))).
% 9.43/3.45 thf(fact_3896_map__of__zip__map, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F:(A > B)]: (((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ ((((map @ A) @ B) @ F) @ Xs))) = (^ [X3:A]: (((((if @ (option @ B)) @ (((member @ A) @ X3) @ ((set2 @ A) @ Xs))) @ ((some @ B) @ (F @ X3))) @ (none @ B)))))))).
% 9.43/3.45 thf(fact_3897_map__of__zip__upd, axiom, (! [A:$tType,B:$tType,Ys2:(list @ B),Xs:(list @ A),Zs:(list @ B),X:A,Y:B,Z2:B]: (((((size_size @ (list @ B)) @ Ys2) = ((size_size @ (list @ A)) @ Xs)) => ((((size_size @ (list @ B)) @ Zs) = ((size_size @ (list @ A)) @ Xs)) => ((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs)))) => (((((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys2))) @ X) @ ((some @ B) @ Y)) = (((((fun_upd @ A) @ (option @ B)) @ (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Zs))) @ X) @ ((some @ B) @ Z2))) => ((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) = (((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Zs)))))))))).
% 9.43/3.45 thf(fact_3898_map__of__zip__nth, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),I:nat]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => (((distinct @ A) @ Xs) => ((((ord_less @ nat) @ I) @ ((size_size @ (list @ B)) @ Ys2)) => (((((map_of @ A) @ B) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ (((nth @ A) @ Xs) @ I)) = ((some @ B) @ (((nth @ B) @ Ys2) @ I))))))))).
% 9.43/3.45 thf(fact_3899_foldr__snd__zip, axiom, (! [B:$tType,A:$tType,C:$tType,Ys2:(list @ A),Xs:(list @ B),F:(A > (C > C)),B2:C]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Ys2)) @ ((size_size @ (list @ B)) @ Xs)) => ((((((foldr @ ((product_prod @ B) @ A)) @ C) @ ((((product_case_prod @ B) @ A) @ (C > C)) @ (^ [X3:B]: (F)))) @ ((((zip @ B) @ A) @ Xs) @ Ys2)) @ B2) = (((((foldr @ A) @ C) @ F) @ Ys2) @ B2)))))).
% 9.43/3.45 thf(fact_3900_nths__shift__lemma, axiom, (! [A:$tType,A4:(set @ nat),Xs:(list @ A),I:nat]: ((((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((((member @ nat) @ (((product_snd @ A) @ nat) @ P6)) @ A4)))) @ ((((zip @ A) @ nat) @ Xs) @ ((upt @ I) @ (((plus_plus @ nat) @ I) @ ((size_size @ (list @ A)) @ Xs)))))) = ((((map @ ((product_prod @ A) @ nat)) @ A) @ ((product_fst @ A) @ nat)) @ (((filter @ ((product_prod @ A) @ nat)) @ (^ [P6:((product_prod @ A) @ nat)]: ((((member @ nat) @ (((plus_plus @ nat) @ (((product_snd @ A) @ nat) @ P6)) @ I)) @ A4)))) @ ((((zip @ A) @ nat) @ Xs) @ ((upt @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))))).
% 9.43/3.45 thf(fact_3901_remove__rev__alt__def, axiom, (! [A:$tType]: (((remove_rev @ A) = (^ [X3:A,Xs3:(list @ A)]: ((((filter @ A) @ (^ [Y3:A]: ((Y3 != X3)))) @ ((rev @ A) @ Xs3)))))))).
% 9.43/3.45 thf(fact_3902_zip__Cons1, axiom, (! [A:$tType,B:$tType,X:A,Xs:(list @ A),Ys2:(list @ B)]: ((((((zip @ A) @ B) @ (((cons @ A) @ X) @ Xs)) @ Ys2) = (((((case_list @ (list @ ((product_prod @ A) @ B))) @ B) @ (nil @ ((product_prod @ A) @ B))) @ (^ [Y3:B,Ys:(list @ B)]: ((((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y3)) @ ((((zip @ A) @ B) @ Xs) @ Ys))))) @ Ys2))))).
% 9.43/3.45 thf(fact_3903_zip__Cons, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Y:B,Ys2:(list @ B)]: ((((((zip @ A) @ B) @ Xs) @ (((cons @ B) @ Y) @ Ys2)) = (((((case_list @ (list @ ((product_prod @ A) @ B))) @ A) @ (nil @ ((product_prod @ A) @ B))) @ (^ [Z5:A,Zs3:(list @ A)]: ((((cons @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Z5) @ Y)) @ ((((zip @ A) @ B) @ Zs3) @ Ys2))))) @ Xs))))).
% 9.43/3.45 thf(fact_3904_filter__rev__alt, axiom, (! [A:$tType]: (((filter_rev @ A) = (^ [P4:(A > $o),L3:(list @ A)]: ((((filter @ A) @ P4) @ ((rev @ A) @ L3)))))))).
% 9.43/3.45 thf(fact_3905_remove__rev__def, axiom, (! [A:$tType]: (((remove_rev @ A) = (^ [X3:A]: (((filter_rev @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ ((^ [Y5:A,Z4:A]: ((Y5 = Z4))) @ X3))))))))).
% 9.43/3.45 thf(fact_3906_min__list_Opelims, axiom, (! [A:$tType]: (((ord @ A) => (! [X:(list @ A),Y:A]: (((((min_list @ A) @ X) = Y) => ((((accp @ (list @ A)) @ (min_list_rel @ A)) @ X) => ((! [X2:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ Xs2)) => ((Y = (((((case_list @ A) @ A) @ X2) @ (^ [A7:A,List2:(list @ A)]: ((((ord_min @ A) @ X2) @ ((min_list @ A) @ Xs2))))) @ Xs2)) => (~ ((((accp @ (list @ A)) @ (min_list_rel @ A)) @ (((cons @ A) @ X2) @ Xs2)))))))) => (~ (((X = (nil @ A)) => ((Y = (undefined @ A)) => (~ ((((accp @ (list @ A)) @ (min_list_rel @ A)) @ (nil @ A))))))))))))))))).
% 9.43/3.45 thf(fact_3907_merge__list_Opsimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L) @ (nil @ (list @ A))))) => ((((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L) @ (nil @ (list @ A)))) = L)))))))).
% 9.43/3.45 thf(fact_3908_merge__list_Opsimps_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [La:(list @ A),Acc22:(list @ (list @ A))]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La) @ Acc22)) @ (nil @ (list @ A)))) => ((((merge_list @ A) @ (((cons @ (list @ A)) @ La) @ Acc22)) @ (nil @ (list @ A))) = (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ La) @ Acc22)))))))))).
% 9.43/3.45 thf(fact_3909_merge__list_Opsimps_I4_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [La:(list @ A),Acc22:(list @ (list @ A)),L:(list @ A)]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La) @ Acc22)) @ (((cons @ (list @ A)) @ L) @ (nil @ (list @ A))))) => ((((merge_list @ A) @ (((cons @ (list @ A)) @ La) @ Acc22)) @ (((cons @ (list @ A)) @ L) @ (nil @ (list @ A)))) = (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L) @ (((cons @ (list @ A)) @ La) @ Acc22))))))))))).
% 9.43/3.45 thf(fact_3910_merge__list_Opsimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (nil @ (list @ A)))) => ((((merge_list @ A) @ (nil @ (list @ A))) @ (nil @ (list @ A))) = (nil @ A))))))).
% 9.43/3.45 thf(fact_3911_merge__list_Opelims, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ (list @ A)),Xa:(list @ (list @ A)),Y:(list @ A)]: ((((((merge_list @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ X) @ Xa)) => (((X = (nil @ (list @ A))) => ((Xa = (nil @ (list @ A))) => ((Y = (nil @ A)) => (~ ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (nil @ (list @ A))))))))) => (((X = (nil @ (list @ A))) => (! [L5:(list @ A)]: (((Xa = (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))) => ((Y = L5) => (~ ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))))))))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A))]: (((X = (((cons @ (list @ A)) @ La2) @ Acc23)) => ((Xa = (nil @ (list @ A))) => ((Y = (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23))) => (~ ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (nil @ (list @ A))))))))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A))]: (((X = (((cons @ (list @ A)) @ La2) @ Acc23)) => (! [L5:(list @ A)]: (((Xa = (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))) => ((Y = (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (((cons @ (list @ A)) @ La2) @ Acc23)))) => (~ ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))))))))))))) => (~ ((! [L12:(list @ A),L22:(list @ A),Ls2:(list @ (list @ A))]: (((Xa = (((cons @ (list @ A)) @ L12) @ (((cons @ (list @ A)) @ L22) @ Ls2))) => ((Y = (((merge_list @ A) @ (((cons @ (list @ A)) @ (((merge @ A) @ L12) @ L22)) @ X)) @ Ls2)) => (~ ((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ X) @ (((cons @ (list @ A)) @ L12) @ (((cons @ (list @ A)) @ L22) @ Ls2)))))))))))))))))))))))).
% 9.43/3.45 thf(fact_3912_merge__list_Opsimps_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Acc22:(list @ (list @ A)),L1:(list @ A),L2:(list @ A),Ls:(list @ (list @ A))]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ Acc22) @ (((cons @ (list @ A)) @ L1) @ (((cons @ (list @ A)) @ L2) @ Ls)))) => ((((merge_list @ A) @ Acc22) @ (((cons @ (list @ A)) @ L1) @ (((cons @ (list @ A)) @ L2) @ Ls))) = (((merge_list @ A) @ (((cons @ (list @ A)) @ (((merge @ A) @ L1) @ L2)) @ Acc22)) @ Ls))))))))).
% 9.43/3.45 thf(fact_3913_partition__filter__conv, axiom, (! [A:$tType]: (((partition @ A) = (^ [F2:(A > $o),Xs3:(list @ A)]: (((((product_Pair @ (list @ A)) @ (list @ A)) @ (((filter @ A) @ F2) @ Xs3)) @ (((filter @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ F2)) @ Xs3)))))))).
% 9.43/3.45 thf(fact_3914_partition__rev__filter__conv, axiom, (! [A:$tType,P:(A > $o),Yes2:(list @ A),No2:(list @ A),Xs:(list @ A)]: ((((((partition_rev @ A) @ P) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ No2)) @ Xs) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ ((rev @ A) @ (((filter @ A) @ P) @ Xs))) @ Yes2)) @ (((append @ A) @ ((rev @ A) @ (((filter @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P)) @ Xs))) @ No2)))))).
% 9.43/3.45 thf(fact_3915_merge_Osimps_I1_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [L2:(list @ A)]: (((((merge @ A) @ (nil @ A)) @ L2) = L2))))))).
% 9.43/3.45 thf(fact_3916_merge_Osimps_I3_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [X1:A,X22:A,L1:(list @ A),L2:(list @ A)]: ((((((ord_less @ A) @ X1) @ X22) => ((((merge @ A) @ (((cons @ A) @ X1) @ L1)) @ (((cons @ A) @ X22) @ L2)) = (((cons @ A) @ X1) @ (((merge @ A) @ L1) @ (((cons @ A) @ X22) @ L2))))) & ((~ ((((ord_less @ A) @ X1) @ X22))) => (((X1 = X22) => ((((merge @ A) @ (((cons @ A) @ X1) @ L1)) @ (((cons @ A) @ X22) @ L2)) = (((cons @ A) @ X1) @ (((merge @ A) @ L1) @ L2)))) & ((X1 != X22) => ((((merge @ A) @ (((cons @ A) @ X1) @ L1)) @ (((cons @ A) @ X22) @ L2)) = (((cons @ A) @ X22) @ (((merge @ A) @ (((cons @ A) @ X1) @ L1)) @ L2))))))))))))).
% 9.43/3.45 thf(fact_3917_merge_Osimps_I2_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [V:A,Va2:(list @ A)]: (((((merge @ A) @ (((cons @ A) @ V) @ Va2)) @ (nil @ A)) = (((cons @ A) @ V) @ Va2)))))))).
% 9.43/3.45 thf(fact_3918_partition__rev_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),Yes2:(list @ A),No2:(list @ A),X:A,Xs:(list @ A)]: ((((((partition_rev @ A) @ P) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ No2)) @ (((cons @ A) @ X) @ Xs)) = ((((partition_rev @ A) @ P) @ ((((if @ ((product_prod @ (list @ A)) @ (list @ A))) @ (P @ X)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Yes2)) @ No2)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ (((cons @ A) @ X) @ No2)))) @ Xs))))).
% 9.43/3.45 thf(fact_3919_partition__rev_Osimps_I1_J, axiom, (! [A:$tType,P:(A > $o),Yes2:(list @ A),No2:(list @ A)]: ((((((partition_rev @ A) @ P) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ No2)) @ (nil @ A)) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ No2))))).
% 9.43/3.45 thf(fact_3920_merge__list_Osimps_I5_J, axiom, (! [A:$tType]: (((linorder @ A) => (! [Acc22:(list @ (list @ A)),L1:(list @ A),L2:(list @ A),Ls:(list @ (list @ A))]: (((((merge_list @ A) @ Acc22) @ (((cons @ (list @ A)) @ L1) @ (((cons @ (list @ A)) @ L2) @ Ls))) = (((merge_list @ A) @ (((cons @ (list @ A)) @ (((merge @ A) @ L1) @ L2)) @ Acc22)) @ Ls)))))))).
% 9.43/3.45 thf(fact_3921_partition_Osimps_I1_J, axiom, (! [A:$tType,P:(A > $o)]: (((((partition @ A) @ P) @ (nil @ A)) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A)))))).
% 9.43/3.45 thf(fact_3922_partition__P, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Yes2:(list @ A),No2:(list @ A)]: ((((((partition @ A) @ P) @ Xs) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ No2)) => ((! [X7:A]: (((((member @ A) @ X7) @ ((set2 @ A) @ Yes2)) => (P @ X7)))) & (! [X7:A]: (((((member @ A) @ X7) @ ((set2 @ A) @ No2)) => (~ ((P @ X7))))))))))).
% 9.43/3.45 thf(fact_3923_merge_Oelims, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ A),Xa:(list @ A),Y:(list @ A)]: ((((((merge @ A) @ X) @ Xa) = Y) => (((X = (nil @ A)) => (Y != Xa)) => ((! [V4:A,Va:(list @ A)]: (((X = (((cons @ A) @ V4) @ Va)) => ((Xa = (nil @ A)) => (Y != (((cons @ A) @ V4) @ Va)))))) => (~ ((! [X13:A,L12:(list @ A)]: (((X = (((cons @ A) @ X13) @ L12)) => (! [X23:A,L22:(list @ A)]: (((Xa = (((cons @ A) @ X23) @ L22)) => (~ ((((((ord_less @ A) @ X13) @ X23) => (Y = (((cons @ A) @ X13) @ (((merge @ A) @ L12) @ (((cons @ A) @ X23) @ L22))))) & ((~ ((((ord_less @ A) @ X13) @ X23))) => (((X13 = X23) => (Y = (((cons @ A) @ X13) @ (((merge @ A) @ L12) @ L22)))) & ((X13 != X23) => (Y = (((cons @ A) @ X23) @ (((merge @ A) @ (((cons @ A) @ X13) @ L12)) @ L22)))))))))))))))))))))))))).
% 9.43/3.45 thf(fact_3924_partition_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X:A,Xs:(list @ A)]: (((((partition @ A) @ P) @ (((cons @ A) @ X) @ Xs)) = (((((product_case_prod @ (list @ A)) @ (list @ A)) @ ((product_prod @ (list @ A)) @ (list @ A))) @ (^ [Yes3:(list @ A),No3:(list @ A)]: (((((if @ ((product_prod @ (list @ A)) @ (list @ A))) @ (P @ X)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X) @ Yes3)) @ No3)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes3) @ (((cons @ A) @ X) @ No3)))))) @ (((partition @ A) @ P) @ Xs)))))).
% 9.43/3.45 thf(fact_3925_partition__rev_Oelims, axiom, (! [A:$tType,X:(A > $o),Xa:((product_prod @ (list @ A)) @ (list @ A)),Xb:(list @ A),Y:((product_prod @ (list @ A)) @ (list @ A))]: (((((((partition_rev @ A) @ X) @ Xa) @ Xb) = Y) => ((! [Yes:(list @ A),No:(list @ A)]: (((Xa = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) => ((Xb = (nil @ A)) => (Y != ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)))))) => (~ ((! [Yes:(list @ A),No:(list @ A)]: (((Xa = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) => (! [X2:A,Xs2:(list @ A)]: (((Xb = (((cons @ A) @ X2) @ Xs2)) => (Y != ((((partition_rev @ A) @ X) @ ((((if @ ((product_prod @ (list @ A)) @ (list @ A))) @ (X @ X2)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Yes)) @ No)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ (((cons @ A) @ X2) @ No)))) @ Xs2))))))))))))))).
% 9.43/3.45 thf(fact_3926_partition__set, axiom, (! [A:$tType,P:(A > $o),Xs:(list @ A),Yes2:(list @ A),No2:(list @ A)]: ((((((partition @ A) @ P) @ Xs) = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes2) @ No2)) => ((((sup_sup @ (set @ A)) @ ((set2 @ A) @ Yes2)) @ ((set2 @ A) @ No2)) = ((set2 @ A) @ Xs)))))).
% 9.43/3.45 thf(fact_3927_inv__image__partition, axiom, (! [A:$tType,Xs:(list @ A),P:(A > $o),Ys2:(list @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => (P @ X2)))) => ((! [Y4:A]: (((((member @ A) @ Y4) @ ((set2 @ A) @ Ys2)) => (~ ((P @ Y4)))))) => (((((vimage @ (list @ A)) @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((partition @ A) @ P)) @ (((insert2 @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ (bot_bot @ (set @ ((product_prod @ (list @ A)) @ (list @ A)))))) = (((shuffles @ A) @ Xs) @ Ys2))))))).
% 9.43/3.45 thf(fact_3928_merge__list_Oelims, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ (list @ A)),Xa:(list @ (list @ A)),Y:(list @ A)]: ((((((merge_list @ A) @ X) @ Xa) = Y) => (((X = (nil @ (list @ A))) => ((Xa = (nil @ (list @ A))) => (Y != (nil @ A)))) => (((X = (nil @ (list @ A))) => (! [L5:(list @ A)]: (((Xa = (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))) => (Y != L5))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A))]: (((X = (((cons @ (list @ A)) @ La2) @ Acc23)) => ((Xa = (nil @ (list @ A))) => (Y != (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23))))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A))]: (((X = (((cons @ (list @ A)) @ La2) @ Acc23)) => (! [L5:(list @ A)]: (((Xa = (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))) => (Y != (((merge_list @ A) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (((cons @ (list @ A)) @ La2) @ Acc23)))))))))) => (~ ((! [L12:(list @ A),L22:(list @ A),Ls2:(list @ (list @ A))]: (((Xa = (((cons @ (list @ A)) @ L12) @ (((cons @ (list @ A)) @ L22) @ Ls2))) => (Y != (((merge_list @ A) @ (((cons @ (list @ A)) @ (((merge @ A) @ L12) @ L22)) @ X)) @ Ls2)))))))))))))))))).
% 9.43/3.45 thf(fact_3929_merge__correct, axiom, (! [A:$tType]: (((linorder @ A) => (! [L1:(list @ A),L2:(list @ A)]: (((((distinct @ A) @ L1) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L1)) => ((((distinct @ A) @ L2) & (((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L2)) => (((distinct @ A) @ (((merge @ A) @ L1) @ L2)) & ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ (((merge @ A) @ L1) @ L2)) & (((set2 @ A) @ (((merge @ A) @ L1) @ L2)) = (((sup_sup @ (set @ A)) @ ((set2 @ A) @ L1)) @ ((set2 @ A) @ L2))))))))))))).
% 9.43/3.45 thf(fact_3930_merge__list_Opinduct, axiom, (! [A:$tType]: (((linorder @ A) => (! [A0:(list @ (list @ A)),A1:(list @ (list @ A)),P:((list @ (list @ A)) > ((list @ (list @ A)) > $o))]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ A0) @ A1)) => (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (nil @ (list @ A)))) => ((P @ (nil @ (list @ A))) @ (nil @ (list @ A)))) => ((! [L5:(list @ A)]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A))))) => ((P @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A))))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A))]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (nil @ (list @ A)))) => (((P @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) => ((P @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (nil @ (list @ A))))))) => ((! [La2:(list @ A),Acc23:(list @ (list @ A)),L5:(list @ A)]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A))))) => (((P @ (nil @ (list @ A))) @ (((cons @ (list @ A)) @ L5) @ (((cons @ (list @ A)) @ La2) @ Acc23))) => ((P @ (((cons @ (list @ A)) @ La2) @ Acc23)) @ (((cons @ (list @ A)) @ L5) @ (nil @ (list @ A)))))))) => ((! [Acc23:(list @ (list @ A)),L12:(list @ A),L22:(list @ A),Ls2:(list @ (list @ A))]: (((((accp @ ((product_prod @ (list @ (list @ A))) @ (list @ (list @ A)))) @ (merge_list_rel @ A)) @ ((((product_Pair @ (list @ (list @ A))) @ (list @ (list @ A))) @ Acc23) @ (((cons @ (list @ A)) @ L12) @ (((cons @ (list @ A)) @ L22) @ Ls2)))) => (((P @ (((cons @ (list @ A)) @ (((merge @ A) @ L12) @ L22)) @ Acc23)) @ Ls2) => ((P @ Acc23) @ (((cons @ (list @ A)) @ L12) @ (((cons @ (list @ A)) @ L22) @ Ls2))))))) => ((P @ A0) @ A1))))))))))))).
% 9.43/3.45 thf(fact_3931_partition__rev_Opelims, axiom, (! [A:$tType,X:(A > $o),Xa:((product_prod @ (list @ A)) @ (list @ A)),Xb:(list @ A),Y:((product_prod @ (list @ A)) @ (list @ A))]: (((((((partition_rev @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)))) @ (partition_rev_rel @ A)) @ ((((product_Pair @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ X) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ Xa) @ Xb))) => ((! [Yes:(list @ A),No:(list @ A)]: (((Xa = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) => ((Xb = (nil @ A)) => ((Y = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) => (~ ((((accp @ ((product_prod @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)))) @ (partition_rev_rel @ A)) @ ((((product_Pair @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ X) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) @ (nil @ A))))))))))) => (~ ((! [Yes:(list @ A),No:(list @ A)]: (((Xa = ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) => (! [X2:A,Xs2:(list @ A)]: (((Xb = (((cons @ A) @ X2) @ Xs2)) => ((Y = ((((partition_rev @ A) @ X) @ ((((if @ ((product_prod @ (list @ A)) @ (list @ A))) @ (X @ X2)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X2) @ Yes)) @ No)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ (((cons @ A) @ X2) @ No)))) @ Xs2)) => (~ ((((accp @ ((product_prod @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)))) @ (partition_rev_rel @ A)) @ ((((product_Pair @ (A > $o)) @ ((product_prod @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A))) @ X) @ ((((product_Pair @ ((product_prod @ (list @ A)) @ (list @ A))) @ (list @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Yes) @ No)) @ (((cons @ A) @ X2) @ Xs2))))))))))))))))))))).
% 9.43/3.45 thf(fact_3932_merge_Opelims, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ A),Xa:(list @ A),Y:(list @ A)]: ((((((merge @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (merge_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Xa)) => (((X = (nil @ A)) => ((Y = Xa) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (merge_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xa)))))) => ((! [V4:A,Va:(list @ A)]: (((X = (((cons @ A) @ V4) @ Va)) => ((Xa = (nil @ A)) => ((Y = (((cons @ A) @ V4) @ Va)) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (merge_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ A)))))))))) => (~ ((! [X13:A,L12:(list @ A)]: (((X = (((cons @ A) @ X13) @ L12)) => (! [X23:A,L22:(list @ A)]: (((Xa = (((cons @ A) @ X23) @ L22)) => ((((((ord_less @ A) @ X13) @ X23) => (Y = (((cons @ A) @ X13) @ (((merge @ A) @ L12) @ (((cons @ A) @ X23) @ L22))))) & ((~ ((((ord_less @ A) @ X13) @ X23))) => (((X13 = X23) => (Y = (((cons @ A) @ X13) @ (((merge @ A) @ L12) @ L22)))) & ((X13 != X23) => (Y = (((cons @ A) @ X23) @ (((merge @ A) @ (((cons @ A) @ X13) @ L12)) @ L22))))))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (merge_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ X13) @ L12)) @ (((cons @ A) @ X23) @ L22)))))))))))))))))))))))).
% 9.43/3.45 thf(fact_3933_transpose_Opinduct, axiom, (! [A:$tType,A0:(list @ (list @ A)),P:((list @ (list @ A)) > $o)]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ A0) => (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (nil @ (list @ A))) => (P @ (nil @ (list @ A)))) => ((! [Xss2:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)) => ((P @ Xss2) => (P @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)))))) => ((! [X2:A,Xs2:(list @ A),Xss2:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xss2)) => ((P @ (((cons @ (list @ A)) @ Xs2) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss2)))) => (P @ (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xss2)))))) => (P @ A0)))))))).
% 9.43/3.45 thf(fact_3934_quicksort__by__rel_Oelims, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Xb:(list @ A),Y:(list @ A)]: (((((((quicksort_by_rel @ A) @ X) @ Xa) @ Xb) = Y) => (((Xb = (nil @ A)) => (Y != Xa)) => (~ ((! [X2:A,Xs2:(list @ A)]: (((Xb = (((cons @ A) @ X2) @ Xs2)) => (Y != (((((product_case_prod @ (list @ A)) @ (list @ A)) @ (list @ A)) @ (^ [Xs_s:(list @ A),Xs_b:(list @ A)]: (((((quicksort_by_rel @ A) @ X) @ (((cons @ A) @ X2) @ ((((quicksort_by_rel @ A) @ X) @ Xa) @ Xs_b))) @ Xs_s)))) @ ((((partition_rev @ A) @ (^ [Y3:A]: (((X @ Y3) @ X2)))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs2))))))))))))).
% 9.43/3.45 thf(fact_3935_set__quicksort__by__rel, axiom, (! [A:$tType,R5:(A > (A > $o)),Sl2:(list @ A),Xs:(list @ A)]: ((((set2 @ A) @ ((((quicksort_by_rel @ A) @ R5) @ Sl2) @ Xs)) = ((set2 @ A) @ (((append @ A) @ Xs) @ Sl2)))))).
% 9.43/3.45 thf(fact_3936_quicksort__by__rel_Osimps_I1_J, axiom, (! [A:$tType,R5:(A > (A > $o)),Sl2:(list @ A)]: ((((((quicksort_by_rel @ A) @ R5) @ Sl2) @ (nil @ A)) = Sl2)))).
% 9.43/3.45 thf(fact_3937_quicksort__by__rel__remove__acc, axiom, (! [A:$tType]: (((quicksort_by_rel @ A) = (^ [R6:(A > (A > $o)),Sl3:(list @ A),Xs3:(list @ A)]: ((((append @ A) @ ((((quicksort_by_rel @ A) @ R6) @ (nil @ A)) @ Xs3)) @ Sl3))))))).
% 9.43/3.45 thf(fact_3938_quicksort__by__rel__remove__acc__guared, axiom, (! [A:$tType,Sl2:(list @ A),R5:(A > (A > $o)),Xs:(list @ A)]: (((Sl2 != (nil @ A)) => (((((quicksort_by_rel @ A) @ R5) @ Sl2) @ Xs) = (((append @ A) @ ((((quicksort_by_rel @ A) @ R5) @ (nil @ A)) @ Xs)) @ Sl2)))))).
% 9.43/3.45 thf(fact_3939_sorted__wrt__quicksort__by__rel, axiom, (! [X14:$tType,R5:(X14 > (X14 > $o)),Xs:(list @ X14)]: (((! [X2:X14,Y4:X14]: ((((R5 @ X2) @ Y4) | ((R5 @ Y4) @ X2)))) => ((! [X2:X14,Y4:X14,Z3:X14]: ((((R5 @ X2) @ Y4) => (((R5 @ Y4) @ Z3) => ((R5 @ X2) @ Z3))))) => (((sorted_wrt @ X14) @ R5) @ ((((quicksort_by_rel @ X14) @ R5) @ (nil @ X14)) @ Xs))))))).
% 9.43/3.45 thf(fact_3940_sorted__quicksort__by__rel, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ ((((quicksort_by_rel @ A) @ (ord_less_eq @ A)) @ (nil @ A)) @ Xs)))))))).
% 9.43/3.45 thf(fact_3941_quicksort__by__rel_Osimps_I2_J, axiom, (! [A:$tType,R5:(A > (A > $o)),Sl2:(list @ A),X:A,Xs:(list @ A)]: ((((((quicksort_by_rel @ A) @ R5) @ Sl2) @ (((cons @ A) @ X) @ Xs)) = (((((product_case_prod @ (list @ A)) @ (list @ A)) @ (list @ A)) @ (^ [Xs_s:(list @ A),Xs_b:(list @ A)]: (((((quicksort_by_rel @ A) @ R5) @ (((cons @ A) @ X) @ ((((quicksort_by_rel @ A) @ R5) @ Sl2) @ Xs_b))) @ Xs_s)))) @ ((((partition_rev @ A) @ (^ [Y3:A]: (((R5 @ Y3) @ X)))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs)))))).
% 9.43/3.45 thf(fact_3942_transpose_Opsimps_I3_J, axiom, (! [A:$tType,X:A,Xs:(list @ A),Xss:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ Xss)) => (((transpose @ A) @ (((cons @ (list @ A)) @ (((cons @ A) @ X) @ Xs)) @ Xss)) = (((cons @ (list @ A)) @ (((cons @ A) @ X) @ ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss)))) @ ((transpose @ A) @ (((cons @ (list @ A)) @ Xs) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss)))))))))).
% 9.43/3.45 thf(fact_3943_transpose_Opelims, axiom, (! [A:$tType,X:(list @ (list @ A)),Y:(list @ (list @ A))]: (((((transpose @ A) @ X) = Y) => ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ X) => (((X = (nil @ (list @ A))) => ((Y = (nil @ (list @ A))) => (~ ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (nil @ (list @ A))))))) => ((! [Xss2:(list @ (list @ A))]: (((X = (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)) => ((Y = ((transpose @ A) @ Xss2)) => (~ ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss2)))))))) => (~ ((! [X2:A,Xs2:(list @ A),Xss2:(list @ (list @ A))]: (((X = (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xss2)) => ((Y = (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ ((concat @ A) @ ((((map @ (list @ A)) @ (list @ A)) @ ((((case_list @ (list @ A)) @ A) @ (nil @ A)) @ (^ [H:A,T3:(list @ A)]: ((((cons @ A) @ H) @ (nil @ A)))))) @ Xss2)))) @ ((transpose @ A) @ (((cons @ (list @ A)) @ Xs2) @ ((concat @ (list @ A)) @ ((((map @ (list @ A)) @ (list @ (list @ A))) @ ((((case_list @ (list @ (list @ A))) @ A) @ (nil @ (list @ A))) @ (^ [H:A,T3:(list @ A)]: ((((cons @ (list @ A)) @ T3) @ (nil @ (list @ A))))))) @ Xss2)))))) => (~ ((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (((cons @ A) @ X2) @ Xs2)) @ Xss2))))))))))))))))).
% 9.43/3.45 thf(fact_3944_quicksort__by__rel_Opsimps_I2_J, axiom, (! [A:$tType,R5:(A > (A > $o)),Sl2:(list @ A),X:A,Xs:(list @ A)]: (((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R5) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Sl2) @ (((cons @ A) @ X) @ Xs)))) => (((((quicksort_by_rel @ A) @ R5) @ Sl2) @ (((cons @ A) @ X) @ Xs)) = (((((product_case_prod @ (list @ A)) @ (list @ A)) @ (list @ A)) @ (^ [Xs_s:(list @ A),Xs_b:(list @ A)]: (((((quicksort_by_rel @ A) @ R5) @ (((cons @ A) @ X) @ ((((quicksort_by_rel @ A) @ R5) @ Sl2) @ Xs_b))) @ Xs_s)))) @ ((((partition_rev @ A) @ (^ [Y3:A]: (((R5 @ Y3) @ X)))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs))))))).
% 9.43/3.45 thf(fact_3945_quicksort__by__rel_Opelims, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Xb:(list @ A),Y:(list @ A)]: (((((((quicksort_by_rel @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xa) @ Xb))) => (((Xb = (nil @ A)) => ((Y = Xa) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xa) @ (nil @ A)))))))) => (~ ((! [X2:A,Xs2:(list @ A)]: (((Xb = (((cons @ A) @ X2) @ Xs2)) => ((Y = (((((product_case_prod @ (list @ A)) @ (list @ A)) @ (list @ A)) @ (^ [Xs_s:(list @ A),Xs_b:(list @ A)]: (((((quicksort_by_rel @ A) @ X) @ (((cons @ A) @ X2) @ ((((quicksort_by_rel @ A) @ X) @ Xa) @ Xs_b))) @ Xs_s)))) @ ((((partition_rev @ A) @ (^ [Y3:A]: (((X @ Y3) @ X2)))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs2))) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xa) @ (((cons @ A) @ X2) @ Xs2)))))))))))))))))).
% 9.43/3.45 thf(fact_3946_transpose_Opsimps_I1_J, axiom, (! [A:$tType]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (nil @ (list @ A))) => (((transpose @ A) @ (nil @ (list @ A))) = (nil @ (list @ A))))))).
% 9.43/3.45 thf(fact_3947_transpose_Opsimps_I2_J, axiom, (! [A:$tType,Xss:(list @ (list @ A))]: (((((accp @ (list @ (list @ A))) @ (transpose_rel @ A)) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss)) => (((transpose @ A) @ (((cons @ (list @ A)) @ (nil @ A)) @ Xss)) = ((transpose @ A) @ Xss)))))).
% 9.43/3.45 thf(fact_3948_quicksort__by__rel_Opsimps_I1_J, axiom, (! [A:$tType,R5:(A > (A > $o)),Sl2:(list @ A)]: (((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R5) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Sl2) @ (nil @ A)))) => (((((quicksort_by_rel @ A) @ R5) @ Sl2) @ (nil @ A)) = Sl2))))).
% 9.43/3.45 thf(fact_3949_quicksort__by__rel_Opinduct, axiom, (! [A:$tType,A0:(A > (A > $o)),A1:(list @ A),A22:(list @ A),P:((A > (A > $o)) > ((list @ A) > ((list @ A) > $o)))]: (((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ A0) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ A1) @ A22))) => ((! [R8:(A > (A > $o)),Sl:(list @ A)]: (((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Sl) @ (nil @ A)))) => (((P @ R8) @ Sl) @ (nil @ A))))) => ((! [R8:(A > (A > $o)),Sl:(list @ A),X2:A,Xs2:(list @ A)]: (((((accp @ ((product_prod @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A)))) @ (quicksort_by_rel_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ ((product_prod @ (list @ A)) @ (list @ A))) @ R8) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Sl) @ (((cons @ A) @ X2) @ Xs2)))) => ((! [Xa2:((product_prod @ (list @ A)) @ (list @ A)),Xb2:(list @ A),Y6:(list @ A)]: (((Xa2 = ((((partition_rev @ A) @ (^ [Z5:A]: (((R8 @ Z5) @ X2)))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs2)) => ((((((product_Pair @ (list @ A)) @ (list @ A)) @ Xb2) @ Y6) = Xa2) => (((P @ R8) @ Sl) @ Y6))))) => ((! [Xa2:((product_prod @ (list @ A)) @ (list @ A)),Xb2:(list @ A),Y6:(list @ A)]: (((Xa2 = ((((partition_rev @ A) @ (^ [Z5:A]: (((R8 @ Z5) @ X2)))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ (nil @ A))) @ Xs2)) => ((((((product_Pair @ (list @ A)) @ (list @ A)) @ Xb2) @ Y6) = Xa2) => (((P @ R8) @ (((cons @ A) @ X2) @ ((((quicksort_by_rel @ A) @ R8) @ Sl) @ Y6))) @ Xb2))))) => (((P @ R8) @ Sl) @ (((cons @ A) @ X2) @ Xs2))))))) => (((P @ A0) @ A1) @ A22))))))).
% 9.43/3.45 thf(fact_3950_filter__rev__aux__alt, axiom, (! [A:$tType]: (((filter_rev_aux @ A) = (^ [A7:(list @ A),P4:(A > $o),L3:(list @ A)]: ((((append @ A) @ (((filter @ A) @ P4) @ ((rev @ A) @ L3))) @ A7))))))).
% 9.43/3.45 thf(fact_3951_lexord__take__index__conv, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lexord @ A) @ R)) = (((((ord_less @ nat) @ ((size_size @ (list @ A)) @ X)) @ ((size_size @ (list @ A)) @ Y)) & ((((take @ A) @ ((size_size @ (list @ A)) @ X)) @ Y) = X)) | (? [I3:nat]: (((((ord_less @ nat) @ I3) @ (((ord_min @ nat) @ ((size_size @ (list @ A)) @ X)) @ ((size_size @ (list @ A)) @ Y))) & (((((take @ A) @ I3) @ X) = (((take @ A) @ I3) @ Y)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ X) @ I3)) @ (((nth @ A) @ Y) @ I3))) @ R)))))))))).
% 9.43/3.45 thf(fact_3952_last__take__nth__conv, axiom, (! [A:$tType,N:nat,L:(list @ A)]: (((((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ L)) => ((N != (zero_zero @ nat)) => (((last @ A) @ (((take @ A) @ N) @ L)) = (((nth @ A) @ L) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))))))))).
% 9.43/3.45 thf(fact_3953_revg__fun, axiom, (! [A:$tType]: (((revg @ A) = (^ [A7:(list @ A)]: (((append @ A) @ ((rev @ A) @ A7)))))))).
% 9.43/3.45 thf(fact_3954_take__upt, axiom, (! [I:nat,M2:nat,N:nat]: (((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I) @ M2)) @ N) => ((((take @ nat) @ M2) @ ((upt @ I) @ N)) = ((upt @ I) @ (((plus_plus @ nat) @ I) @ M2))))))).
% 9.43/3.45 thf(fact_3955_take__Cons__numeral, axiom, (! [A:$tType,V:num,X:A,Xs:(list @ A)]: (((((take @ A) @ ((numeral_numeral @ nat) @ V)) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ (((take @ A) @ (((minus_minus @ nat) @ ((numeral_numeral @ nat) @ V)) @ (one_one @ nat))) @ Xs)))))).
% 9.43/3.45 thf(fact_3956_take__drop, axiom, (! [A:$tType,N:nat,M2:nat,Xs:(list @ A)]: (((((take @ A) @ N) @ (((drop @ A) @ M2) @ Xs)) = (((drop @ A) @ M2) @ (((take @ A) @ (((plus_plus @ nat) @ N) @ M2)) @ Xs)))))).
% 9.43/3.45 thf(fact_3957_revg_Osimps_I2_J, axiom, (! [A:$tType,A3:A,As2:(list @ A),B2:(list @ A)]: (((((revg @ A) @ (((cons @ A) @ A3) @ As2)) @ B2) = (((revg @ A) @ As2) @ (((cons @ A) @ A3) @ B2)))))).
% 9.43/3.45 thf(fact_3958_revg_Osimps_I1_J, axiom, (! [A:$tType,B2:(list @ A)]: (((((revg @ A) @ (nil @ A)) @ B2) = B2)))).
% 9.43/3.45 thf(fact_3959_filter__rev__aux_Osimps_I2_J, axiom, (! [A:$tType,P:(A > $o),X:A,A3:(list @ A),Xs:(list @ A)]: ((((P @ X) => (((((filter_rev_aux @ A) @ A3) @ P) @ (((cons @ A) @ X) @ Xs)) = ((((filter_rev_aux @ A) @ (((cons @ A) @ X) @ A3)) @ P) @ Xs))) & ((~ ((P @ X))) => (((((filter_rev_aux @ A) @ A3) @ P) @ (((cons @ A) @ X) @ Xs)) = ((((filter_rev_aux @ A) @ A3) @ P) @ Xs))))))).
% 9.43/3.45 thf(fact_3960_filter__rev__aux_Osimps_I1_J, axiom, (! [A:$tType,A3:(list @ A),P:(A > $o)]: ((((((filter_rev_aux @ A) @ A3) @ P) @ (nil @ A)) = A3)))).
% 9.43/3.45 thf(fact_3961_drop__take__drop__unsplit, axiom, (! [A:$tType,I:nat,J:nat,L:(list @ A)]: (((((ord_less_eq @ nat) @ I) @ J) => ((((append @ A) @ (((drop @ A) @ I) @ (((take @ A) @ J) @ L))) @ (((drop @ A) @ J) @ L)) = (((drop @ A) @ I) @ L)))))).
% 9.43/3.45 thf(fact_3962_take__add, axiom, (! [A:$tType,I:nat,J:nat,Xs:(list @ A)]: (((((take @ A) @ (((plus_plus @ nat) @ I) @ J)) @ Xs) = (((append @ A) @ (((take @ A) @ I) @ Xs)) @ (((take @ A) @ J) @ (((drop @ A) @ I) @ Xs))))))).
% 9.43/3.45 thf(fact_3963_slice__def, axiom, (! [A:$tType]: (((slice @ A) = (^ [From2:nat,To2:nat,List2:(list @ A)]: ((((take @ A) @ (((minus_minus @ nat) @ To2) @ From2)) @ (((drop @ A) @ From2) @ List2)))))))).
% 9.43/3.45 thf(fact_3964_filter__upt__take__conv, axiom, (! [A:$tType,P:(A > $o),M2:nat,L:(list @ A),N:nat]: (((((filter @ nat) @ (^ [I3:nat]: ((P @ (((nth @ A) @ (((take @ A) @ M2) @ L)) @ I3))))) @ ((upt @ N) @ M2)) = (((filter @ nat) @ (^ [I3:nat]: ((P @ (((nth @ A) @ L) @ I3))))) @ ((upt @ N) @ M2)))))).
% 9.43/3.45 thf(fact_3965_zip__replicate1, axiom, (! [A:$tType,B:$tType,N:nat,X:A,Ys2:(list @ B)]: ((((((zip @ A) @ B) @ (((replicate @ A) @ N) @ X)) @ Ys2) = ((((map @ B) @ ((product_prod @ A) @ B)) @ (((product_Pair @ A) @ B) @ X)) @ (((take @ B) @ N) @ Ys2)))))).
% 9.43/3.45 thf(fact_3966_revg_Oelims, axiom, (! [A:$tType,X:(list @ A),Xa:(list @ A),Y:(list @ A)]: ((((((revg @ A) @ X) @ Xa) = Y) => (((X = (nil @ A)) => (Y != Xa)) => (~ ((! [A5:A,As:(list @ A)]: (((X = (((cons @ A) @ A5) @ As)) => (Y != (((revg @ A) @ As) @ (((cons @ A) @ A5) @ Xa))))))))))))).
% 9.43/3.45 thf(fact_3967_zip__replicate2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),N:nat,Y:B]: ((((((zip @ A) @ B) @ Xs) @ (((replicate @ B) @ N) @ Y)) = ((((map @ A) @ ((product_prod @ A) @ B)) @ (^ [X3:A]: (((((product_Pair @ A) @ B) @ X3) @ Y)))) @ (((take @ A) @ N) @ Xs)))))).
% 9.43/3.45 thf(fact_3968_take__Cons_H, axiom, (! [A:$tType,N:nat,X:A,Xs:(list @ A)]: ((((N = (zero_zero @ nat)) => ((((take @ A) @ N) @ (((cons @ A) @ X) @ Xs)) = (nil @ A))) & ((N != (zero_zero @ nat)) => ((((take @ A) @ N) @ (((cons @ A) @ X) @ Xs)) = (((cons @ A) @ X) @ (((take @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ Xs)))))))).
% 9.43/3.45 thf(fact_3969_filter__nth__ex__nth, axiom, (! [A:$tType,N:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ (((filter @ A) @ P) @ Xs))) => (? [M5:nat]: (((((ord_less_eq @ nat) @ N) @ M5) & ((((ord_less @ nat) @ M5) @ ((size_size @ (list @ A)) @ Xs)) & (((((nth @ A) @ (((filter @ A) @ P) @ Xs)) @ N) = (((nth @ A) @ Xs) @ M5)) & ((((filter @ A) @ P) @ (((take @ A) @ M5) @ Xs)) = (((take @ A) @ N) @ (((filter @ A) @ P) @ Xs)))))))))))).
% 9.43/3.45 thf(fact_3970_map__upd__upds__conv__if, axiom, (! [A:$tType,B:$tType,X:A,Ys2:(list @ B),Xs:(list @ A),F:(A > (option @ B)),Y:B]: ((((((member @ A) @ X) @ ((set2 @ A) @ (((take @ A) @ ((size_size @ (list @ B)) @ Ys2)) @ Xs))) => ((((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ ((some @ B) @ Y))) @ Xs) @ Ys2) = (((((map_upds @ A) @ B) @ F) @ Xs) @ Ys2))) & ((~ ((((member @ A) @ X) @ ((set2 @ A) @ (((take @ A) @ ((size_size @ (list @ B)) @ Ys2)) @ Xs))))) => ((((((map_upds @ A) @ B) @ (((((fun_upd @ A) @ (option @ B)) @ F) @ X) @ ((some @ B) @ Y))) @ Xs) @ Ys2) = (((((fun_upd @ A) @ (option @ B)) @ (((((map_upds @ A) @ B) @ F) @ Xs) @ Ys2)) @ X) @ ((some @ B) @ Y)))))))).
% 9.43/3.45 thf(fact_3971_Union__take__drop__id, axiom, (! [A:$tType,N:nat,L:(list @ (set @ A))]: (((((sup_sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((set2 @ (set @ A)) @ (((drop @ (set @ A)) @ N) @ L)))) @ ((complete_Sup_Sup @ (set @ A)) @ ((set2 @ (set @ A)) @ (((take @ (set @ A)) @ N) @ L)))) = ((complete_Sup_Sup @ (set @ A)) @ ((set2 @ (set @ A)) @ L)))))).
% 9.43/3.45 thf(fact_3972_lex__take__index, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lex @ A) @ R)) => (~ ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Xs)) => ((((ord_less @ nat) @ I2) @ ((size_size @ (list @ A)) @ Ys2)) => (((((take @ A) @ I2) @ Xs) = (((take @ A) @ I2) @ Ys2)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ I2)) @ (((nth @ A) @ Ys2) @ I2))) @ R)))))))))))))).
% 9.43/3.45 thf(fact_3973_map__nth__upt__drop__take__conv, axiom, (! [A:$tType,N6:nat,L:(list @ A),M3:nat]: (((((ord_less_eq @ nat) @ N6) @ ((size_size @ (list @ A)) @ L)) => (((((map @ nat) @ A) @ ((nth @ A) @ L)) @ ((upt @ M3) @ N6)) = (((drop @ A) @ M3) @ (((take @ A) @ N6) @ L))))))).
% 9.43/3.45 thf(fact_3974_filter__rev__def, axiom, (! [A:$tType]: (((filter_rev @ A) = ((filter_rev_aux @ A) @ (nil @ A)))))).
% 9.43/3.45 thf(fact_3975_revg_Opelims, axiom, (! [A:$tType,X:(list @ A),Xa:(list @ A),Y:(list @ A)]: ((((((revg @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (revg_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Xa)) => (((X = (nil @ A)) => ((Y = Xa) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (revg_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (nil @ A)) @ Xa)))))) => (~ ((! [A5:A,As:(list @ A)]: (((X = (((cons @ A) @ A5) @ As)) => ((Y = (((revg @ A) @ As) @ (((cons @ A) @ A5) @ Xa))) => (~ ((((accp @ ((product_prod @ (list @ A)) @ (list @ A))) @ (revg_rel @ A)) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((cons @ A) @ A5) @ As)) @ Xa)))))))))))))))).
% 9.43/3.45 thf(fact_3976_take__butlast__conv, axiom, (! [A:$tType,L:(list @ A)]: (((((take @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ L)) @ (suc @ (zero_zero @ nat)))) @ L) = ((butlast @ A) @ L))))).
% 9.43/3.45 thf(fact_3977_take__minus__one__conv__butlast, axiom, (! [A:$tType,N:nat,L:(list @ A)]: (((((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ L)) => ((((take @ A) @ (((minus_minus @ nat) @ N) @ (suc @ (zero_zero @ nat)))) @ L) = ((butlast @ A) @ (((take @ A) @ N) @ L))))))).
% 9.43/3.45 thf(fact_3978_take__update__last, axiom, (! [A:$tType,N:nat,List:(list @ A),X:A]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ List)) => (((((list_update @ A) @ (((take @ A) @ (suc @ N)) @ List)) @ N) @ X) = (((append @ A) @ (((take @ A) @ N) @ List)) @ (((cons @ A) @ X) @ (nil @ A)))))))).
% 9.43/3.45 thf(fact_3979_take__update, axiom, (! [A:$tType,N:nat,L:(list @ A),I:nat,X:A]: (((((take @ A) @ N) @ ((((list_update @ A) @ L) @ I) @ X)) = ((((list_update @ A) @ (((take @ A) @ N) @ L)) @ I) @ X))))).
% 9.43/3.45 thf(fact_3980_drop__upd__irrelevant, axiom, (! [A:$tType,M2:nat,N:nat,L:(list @ A),X:A]: (((((ord_less @ nat) @ M2) @ N) => ((((drop @ A) @ N) @ ((((list_update @ A) @ L) @ M2) @ X)) = (((drop @ A) @ N) @ L)))))).
% 9.43/3.45 thf(fact_3981_nth__update__invalid, axiom, (! [A:$tType,I:nat,L:(list @ A),J:nat,X:A]: (((~ ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)))) => ((((nth @ A) @ ((((list_update @ A) @ L) @ J) @ X)) @ I) = (((nth @ A) @ L) @ I)))))).
% 9.43/3.45 thf(fact_3982_length__butlast, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((butlast @ A) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))).
% 9.43/3.45 thf(fact_3983_butlast__upd__last__eq, axiom, (! [A:$tType,L:(list @ A),X:A]: (((((ord_less_eq @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ ((size_size @ (list @ A)) @ L)) => (((((list_update @ A) @ ((butlast @ A) @ L)) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ L)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ X) = (((append @ A) @ (((take @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ L)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ L)) @ (((cons @ A) @ X) @ (nil @ A)))))))).
% 9.43/3.45 thf(fact_3984_butlast__update_H, axiom, (! [A:$tType,L:(list @ A),I:nat,X:A]: ((((((list_update @ A) @ ((butlast @ A) @ L)) @ I) @ X) = ((butlast @ A) @ ((((list_update @ A) @ L) @ I) @ X)))))).
% 9.43/3.45 thf(fact_3985_distinct__butlast__swap, axiom, (! [A:$tType,Pq:(list @ A),I:nat]: ((((distinct @ A) @ Pq) => ((distinct @ A) @ ((butlast @ A) @ ((((list_update @ A) @ Pq) @ I) @ ((last @ A) @ Pq)))))))).
% 9.43/3.45 thf(fact_3986_zip__update, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),I:nat,X:A,Ys2:(list @ B),Y:B]: ((((((zip @ A) @ B) @ ((((list_update @ A) @ Xs) @ I) @ X)) @ ((((list_update @ B) @ Ys2) @ I) @ Y)) = ((((list_update @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) @ I) @ ((((product_Pair @ A) @ B) @ X) @ Y)))))).
% 9.43/3.45 thf(fact_3987_butlast__list__update, axiom, (! [A:$tType,K:nat,Xs:(list @ A),X:A]: ((((K = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((butlast @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = ((butlast @ A) @ Xs))) & ((K != (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((butlast @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = ((((list_update @ A) @ ((butlast @ A) @ Xs)) @ K) @ X))))))).
% 9.43/3.45 thf(fact_3988_in__set__upd__eq__aux, axiom, (! [A:$tType,I:nat,L:(list @ A),X:A,Y:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((((member @ A) @ X) @ ((set2 @ A) @ ((((list_update @ A) @ L) @ I) @ Y))) = ((X = Y) | (! [Y3:A]: ((((member @ A) @ X) @ ((set2 @ A) @ ((((list_update @ A) @ L) @ I) @ Y3))))))))))).
% 9.43/3.45 thf(fact_3989_in__set__upd__cases, axiom, (! [A:$tType,X:A,L:(list @ A),I:nat,Y:A]: (((((member @ A) @ X) @ ((set2 @ A) @ ((((list_update @ A) @ L) @ I) @ Y))) => (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => (X != Y)) => (((member @ A) @ X) @ ((set2 @ A) @ L))))))).
% 9.43/3.45 thf(fact_3990_in__set__upd__eq, axiom, (! [A:$tType,I:nat,L:(list @ A),X:A,Y:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((((member @ A) @ X) @ ((set2 @ A) @ ((((list_update @ A) @ L) @ I) @ Y))) = ((X = Y) | ((((member @ A) @ X) @ ((set2 @ A) @ L)) & (! [Y3:A]: ((((member @ A) @ X) @ ((set2 @ A) @ ((((list_update @ A) @ L) @ I) @ Y3)))))))))))).
% 9.43/3.45 thf(fact_3991_nth__list__update_H, axiom, (! [A:$tType,I:nat,J:nat,L:(list @ A),X:A]: (((((I = J) & (((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L))) => ((((nth @ A) @ ((((list_update @ A) @ L) @ I) @ X)) @ J) = X)) & ((~ (((I = J) & (((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L))))) => ((((nth @ A) @ ((((list_update @ A) @ L) @ I) @ X)) @ J) = (((nth @ A) @ L) @ J))))))).
% 9.43/3.45 thf(fact_3992_butlast__subset, axiom, (! [A:$tType,Xs:(list @ A),A4:(set @ A)]: (((Xs != (nil @ A)) => ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ A4) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ ((butlast @ A) @ Xs))) @ A4)))))).
% 9.43/3.45 thf(fact_3993_butlast__eq__cons__conv, axiom, (! [A:$tType,L:(list @ A),X:A,Xs:(list @ A)]: (((((butlast @ A) @ L) = (((cons @ A) @ X) @ Xs)) = (? [Xl:A]: ((L = (((cons @ A) @ X) @ (((append @ A) @ Xs) @ (((cons @ A) @ Xl) @ (nil @ A))))))))))).
% 9.43/3.45 thf(fact_3994_butlast__eq__consE, axiom, (! [A:$tType,L:(list @ A),X:A,Xs:(list @ A)]: (((((butlast @ A) @ L) = (((cons @ A) @ X) @ Xs)) => (~ ((! [Xl2:A]: ((L != (((cons @ A) @ X) @ (((append @ A) @ Xs) @ (((cons @ A) @ Xl2) @ (nil @ A))))))))))))).
% 9.43/3.45 thf(fact_3995_butlast__upt, axiom, (! [M2:nat,N:nat]: ((((butlast @ nat) @ ((upt @ M2) @ N)) = ((upt @ M2) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))))))).
% 9.43/3.45 thf(fact_3996_map__upd__eq, axiom, (! [B:$tType,A:$tType,I:nat,L:(list @ A),F:(A > B),X:A]: ((((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((F @ (((nth @ A) @ L) @ I)) = (F @ X))) => (((((map @ A) @ B) @ F) @ ((((list_update @ A) @ L) @ I) @ X)) = ((((map @ A) @ B) @ F) @ L)))))).
% 9.43/3.45 thf(fact_3997_snoc__eq__iff__butlast_H, axiom, (! [A:$tType,Ys2:(list @ A),Xs:(list @ A),X:A]: (((Ys2 = (((append @ A) @ Xs) @ (((cons @ A) @ X) @ (nil @ A)))) = ((Ys2 != (nil @ A)) & ((((butlast @ A) @ Ys2) = Xs) & (((last @ A) @ Ys2) = X))))))).
% 9.43/3.45 thf(fact_3998_butlast__conv__take, axiom, (! [A:$tType]: (((butlast @ A) = (^ [Xs3:(list @ A)]: ((((take @ A) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs3)) @ (one_one @ nat))) @ Xs3))))))).
% 9.43/3.45 thf(fact_3999_insert__swap__set__eq, axiom, (! [A:$tType,I:nat,L:(list @ A),X:A]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => ((((insert2 @ A) @ (((nth @ A) @ L) @ I)) @ ((set2 @ A) @ ((((list_update @ A) @ L) @ I) @ X))) = (((insert2 @ A) @ X) @ ((set2 @ A) @ L))))))).
% 9.43/3.45 thf(fact_4000_rev__update, axiom, (! [A:$tType,K:nat,Xs:(list @ A),Y:A]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((rev @ A) @ ((((list_update @ A) @ Xs) @ K) @ Y)) = ((((list_update @ A) @ ((rev @ A) @ Xs)) @ (((minus_minus @ nat) @ (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ K)) @ (one_one @ nat))) @ Y)))))).
% 9.43/3.45 thf(fact_4001_last__list__update, axiom, (! [A:$tType,Xs:(list @ A),K:nat,X:A]: (((Xs != (nil @ A)) => (((K = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((last @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = X)) & ((K != (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat))) => (((last @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = ((last @ A) @ Xs)))))))).
% 9.43/3.45 thf(fact_4002_butlast__take, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => (((butlast @ A) @ (((take @ A) @ N) @ Xs)) = (((take @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ Xs)))))).
% 9.43/3.45 thf(fact_4003_listrel1__iff__update, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) = (? [Y3:A,N5:nat]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((nth @ A) @ Xs) @ N5)) @ Y3)) @ R) & ((((ord_less @ nat) @ N5) @ ((size_size @ (list @ A)) @ Xs)) & (Ys2 = ((((list_update @ A) @ Xs) @ N5) @ Y3)))))))))).
% 9.43/3.45 thf(fact_4004_map__distinct__upd__conv, axiom, (! [B:$tType,A:$tType,I:nat,L:(list @ A),F:(A > B),X:B]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => (((distinct @ A) @ L) => (((((list_update @ B) @ ((((map @ A) @ B) @ F) @ L)) @ I) @ X) = ((((map @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ (((nth @ A) @ L) @ I)) @ X)) @ L))))))).
% 9.43/3.45 thf(fact_4005_foldl__list__update, axiom, (! [B:$tType,A:$tType,N:nat,Xs:(list @ A),F:(B > (A > B)),A3:B,X:A]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((((foldl @ B) @ A) @ F) @ A3) @ ((((list_update @ A) @ Xs) @ N) @ X)) = (((((foldl @ B) @ A) @ F) @ ((F @ (((((foldl @ B) @ A) @ F) @ A3) @ (((take @ A) @ N) @ Xs))) @ X)) @ (((drop @ A) @ (suc @ N)) @ Xs))))))).
% 9.43/3.45 thf(fact_4006_case__natural__def, axiom, (! [T:$tType]: (((code_case_natural @ T) = (^ [F12:T,F23:(code_natural > T)]: ((((code_rec_natural @ T) @ F12) @ (^ [X12:code_natural,X24:T]: ((F23 @ X12)))))))))).
% 9.43/3.45 thf(fact_4007_nth__rotate, axiom, (! [A:$tType,N:nat,Xs:(list @ A),M2:nat]: (((((ord_less @ nat) @ N) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ (((rotate @ A) @ M2) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (((modulo_modulo @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ ((size_size @ (list @ A)) @ Xs)))))))).
% 9.43/3.45 thf(fact_4008_list_Osize__gen_I2_J, axiom, (! [A:$tType,X:(A > nat),X21:A,X222:(list @ A)]: (((((size_list @ A) @ X) @ (((cons @ A) @ X21) @ X222)) = (((plus_plus @ nat) @ (((plus_plus @ nat) @ (X @ X21)) @ (((size_list @ A) @ X) @ X222))) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.45 thf(fact_4009_rotate1__length01, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => (((rotate1 @ A) @ Xs) = Xs))))).
% 9.43/3.45 thf(fact_4010_natural_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T))]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (zero_zero @ code_natural)) = F1)))).
% 9.43/3.45 thf(fact_4011_size__list__append, axiom, (! [A:$tType,F:(A > nat),Xs:(list @ A),Ys2:(list @ A)]: (((((size_list @ A) @ F) @ (((append @ A) @ Xs) @ Ys2)) = (((plus_plus @ nat) @ (((size_list @ A) @ F) @ Xs)) @ (((size_list @ A) @ F) @ Ys2)))))).
% 9.43/3.45 thf(fact_4012_rotate__length01, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((((ord_less_eq @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)) => ((((rotate @ A) @ N) @ Xs) = Xs))))).
% 9.43/3.45 thf(fact_4013_rotate__rotate, axiom, (! [A:$tType,M2:nat,N:nat,Xs:(list @ A)]: (((((rotate @ A) @ M2) @ (((rotate @ A) @ N) @ Xs)) = (((rotate @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ Xs))))).
% 9.43/3.45 thf(fact_4014_rotate__add, axiom, (! [A:$tType,M2:nat,N:nat]: ((((rotate @ A) @ (((plus_plus @ nat) @ M2) @ N)) = (((((comp @ (list @ A)) @ (list @ A)) @ (list @ A)) @ ((rotate @ A) @ M2)) @ ((rotate @ A) @ N)))))).
% 9.43/3.45 thf(fact_4015_rec__natural__def, axiom, (! [T:$tType]: (((code_rec_natural @ T) = (^ [F12:T,F23:(code_natural > (T > T)),X3:code_natural]: (((the @ T) @ ((((code_rec_set_natural @ T) @ F12) @ F23) @ X3)))))))).
% 9.43/3.45 thf(fact_4016_lenlex__append2, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),Us:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: ((((irrefl @ A) @ R5) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Us) @ Xs)) @ (((append @ A) @ Us) @ Ys2))) @ ((lenlex @ A) @ R5)) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((lenlex @ A) @ R5))))))).
% 9.43/3.45 thf(fact_4017_zipf__zip, axiom, (! [A:$tType,B:$tType,L1:(list @ A),L2:(list @ B)]: (((((size_size @ (list @ A)) @ L1) = ((size_size @ (list @ B)) @ L2)) => (((((((zipf @ A) @ B) @ ((product_prod @ A) @ B)) @ ((product_Pair @ A) @ B)) @ L1) @ L2) = ((((zip @ A) @ B) @ L1) @ L2)))))).
% 9.43/3.45 thf(fact_4018_map__tailrec__rev_Opelims, axiom, (! [A:$tType,B:$tType,X:(A > B),Xa:(list @ A),Xb:(list @ B),Y:(list @ B)]: ((((((((map_tailrec_rev @ A) @ B) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((map_tailrec_rev_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xa) @ Xb))) => (((Xa = (nil @ A)) => ((Y = Xb) => (~ ((((accp @ ((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((map_tailrec_rev_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ Xb))))))) => (~ ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => ((Y = (((((map_tailrec_rev @ A) @ B) @ X) @ As) @ (((cons @ B) @ (X @ A5)) @ Xb))) => (~ ((((accp @ ((product_prod @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((map_tailrec_rev_rel @ A) @ B)) @ ((((product_Pair @ (A > B)) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ Xb))))))))))))))))).
% 9.43/3.45 thf(fact_4019_lexord__same__pref__if__irrefl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),Xs:(list @ A),Ys2:(list @ A),Zs:(list @ A)]: ((((irrefl @ A) @ R) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ (((append @ A) @ Xs) @ Ys2)) @ (((append @ A) @ Xs) @ Zs))) @ ((lexord @ A) @ R)) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ ((lexord @ A) @ R))))))).
% 9.43/3.45 thf(fact_4020_irreflI, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((! [A5:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ A5)) @ R5))))) => ((irrefl @ A) @ R5))))).
% 9.43/3.45 thf(fact_4021_irrefl__def, axiom, (! [A:$tType]: (((irrefl @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [A7:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ A7)) @ R3))))))))))).
% 9.43/3.45 thf(fact_4022_zipf_Osimps_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > (B > C)),A3:A,As2:(list @ A),B2:B,Bs2:(list @ B)]: ((((((((zipf @ A) @ B) @ C) @ F) @ (((cons @ A) @ A3) @ As2)) @ (((cons @ B) @ B2) @ Bs2)) = (((cons @ C) @ ((F @ A3) @ B2)) @ ((((((zipf @ A) @ B) @ C) @ F) @ As2) @ Bs2)))))).
% 9.43/3.45 thf(fact_4023_zipf_Osimps_I1_J, axiom, (! [B:$tType,A:$tType,C:$tType,F:(A > (B > C))]: ((((((((zipf @ A) @ B) @ C) @ F) @ (nil @ A)) @ (nil @ B)) = (nil @ C))))).
% 9.43/3.45 thf(fact_4024_lexl__not__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:(list @ A)]: ((((irrefl @ A) @ R) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ X)) @ ((lex @ A) @ R)))))))).
% 9.43/3.45 thf(fact_4025_zipf_Oelims, axiom, (! [B:$tType,A:$tType,C:$tType,X:(A > (B > C)),Xa:(list @ A),Xb:(list @ B),Y:(list @ C)]: (((((((((zipf @ A) @ B) @ C) @ X) @ Xa) @ Xb) = Y) => (((Xa = (nil @ A)) => ((Xb = (nil @ B)) => (Y != (nil @ C)))) => ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => (Y != (((cons @ C) @ ((X @ A5) @ B3)) @ ((((((zipf @ A) @ B) @ C) @ X) @ As) @ Bs))))))))) => (((? [V4:A,Va:(list @ A)]: ((Xa = (((cons @ A) @ V4) @ Va)))) => ((Xb = (nil @ B)) => (Y != (undefined @ (list @ C))))) => (~ (((Xa = (nil @ A)) => ((? [V4:B,Va:(list @ B)]: ((Xb = (((cons @ B) @ V4) @ Va)))) => (Y != (undefined @ (list @ C)))))))))))))).
% 9.43/3.45 thf(fact_4026_zipf_Opelims, axiom, (! [C:$tType,A:$tType,B:$tType,X:(A > (B > C)),Xa:(list @ A),Xb:(list @ B),Y:(list @ C)]: (((((((((zipf @ A) @ B) @ C) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ (((zipf_rel @ A) @ B) @ C)) @ ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xa) @ Xb))) => (((Xa = (nil @ A)) => ((Xb = (nil @ B)) => ((Y = (nil @ C)) => (~ ((((accp @ ((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ (((zipf_rel @ A) @ B) @ C)) @ ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B))))))))) => ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => ((Y = (((cons @ C) @ ((X @ A5) @ B3)) @ ((((((zipf @ A) @ B) @ C) @ X) @ As) @ Bs))) => (~ ((((accp @ ((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ (((zipf_rel @ A) @ B) @ C)) @ ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B3) @ Bs))))))))))))) => ((! [V4:A,Va:(list @ A)]: (((Xa = (((cons @ A) @ V4) @ Va)) => ((Xb = (nil @ B)) => ((Y = (undefined @ (list @ C))) => (~ ((((accp @ ((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ (((zipf_rel @ A) @ B) @ C)) @ ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ B))))))))))) => (~ (((Xa = (nil @ A)) => (! [V4:B,Va:(list @ B)]: (((Xb = (((cons @ B) @ V4) @ Va)) => ((Y = (undefined @ (list @ C))) => (~ ((((accp @ ((product_prod @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ (((zipf_rel @ A) @ B) @ C)) @ ((((product_Pair @ (A > (B > C))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (((cons @ B) @ V4) @ Va))))))))))))))))))))).
% 9.43/3.45 thf(fact_4027_count__list_Osimps_I2_J, axiom, (! [A:$tType,X:A,Y:A,Xs:(list @ A)]: ((((X = Y) => ((((count_list @ A) @ (((cons @ A) @ X) @ Xs)) @ Y) = (((plus_plus @ nat) @ (((count_list @ A) @ Xs) @ Y)) @ (one_one @ nat)))) & ((X != Y) => ((((count_list @ A) @ (((cons @ A) @ X) @ Xs)) @ Y) = (((count_list @ A) @ Xs) @ Y))))))).
% 9.43/3.45 thf(fact_4028_effect__makeE, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(array @ A),N2:nat]: ((((((((heap_Time_effect @ (array @ A)) @ (((array_make @ A) @ N) @ F)) @ H3) @ H4) @ R) @ N2) => (~ (((R = (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H3))) => ((H4 = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) @ H3))) => (((((array_get @ A) @ H4) @ R) = ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N))) => ((((array_present @ A) @ H4) @ R) => ((~ ((((array_present @ A) @ H3) @ R))) => ((((plus_plus @ nat) @ N) @ (one_one @ nat)) != N2))))))))))))))).
% 9.43/3.45 thf(fact_4029_listrel__iff__nth, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys2)) @ (((listrel @ A) @ B) @ R)) = ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) & (! [N5:nat]: (((((ord_less @ nat) @ N5) @ ((size_size @ (list @ A)) @ Xs)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ (((nth @ A) @ Xs) @ N5)) @ (((nth @ B) @ Ys2) @ N5))) @ R))))))))).
% 9.43/3.45 thf(fact_4030_listrel__rtrancl__refl, axiom, (! [A:$tType,Xs:(list @ A),R:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Xs)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R)))))).
% 9.43/3.45 thf(fact_4031_Array__Time_Oget__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H3:(heap_ext @ product_unit),Ys2:(list @ A)]: (((((array_get @ A) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ys2) @ H3))) = Xs))))))).
% 9.43/3.45 thf(fact_4032_Array__Time_Opresent__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H3:(heap_ext @ product_unit)]: ((((array_present @ A) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))))))))).
% 9.43/3.45 thf(fact_4033_Array__Time_Onot__present__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [H3:(heap_ext @ product_unit),Xs:(list @ A)]: ((~ ((((array_present @ A) @ H3) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))))))))))).
% 9.43/3.45 thf(fact_4034_listrel_ONil, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B))) @ (((listrel @ A) @ B) @ R))))).
% 9.43/3.45 thf(fact_4035_listrel__Nil1, axiom, (! [A:$tType,B:$tType,Xs:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ Xs)) @ (((listrel @ A) @ B) @ R)) => (Xs = (nil @ B)))))).
% 9.43/3.45 thf(fact_4036_listrel__Nil2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ (nil @ B))) @ (((listrel @ A) @ B) @ R)) => (Xs = (nil @ A)))))).
% 9.43/3.45 thf(fact_4037_listrel__eq__len, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys2)) @ (((listrel @ A) @ B) @ R)) => (((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)))))).
% 9.43/3.45 thf(fact_4038_listrel__rtrancl__trans, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A)),Zs:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R))) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Ys2) @ Zs)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R))) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Zs)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R)))))))).
% 9.43/3.45 thf(fact_4039_listrel_OCons, axiom, (! [B:$tType,A:$tType,X:A,Y:B,R:(set @ ((product_prod @ A) @ B)),Xs:(list @ A),Ys2:(list @ B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ R) => ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys2)) @ (((listrel @ A) @ B) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ X) @ Xs)) @ (((cons @ B) @ Y) @ Ys2))) @ (((listrel @ A) @ B) @ R))))))).
% 9.43/3.45 thf(fact_4040_listrel__Cons1, axiom, (! [B:$tType,A:$tType,Y:A,Ys2:(list @ A),Xs:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ Y) @ Ys2)) @ Xs)) @ (((listrel @ A) @ B) @ R)) => (~ ((! [Y4:B,Ys3:(list @ B)]: (((Xs = (((cons @ B) @ Y4) @ Ys3)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Ys2) @ Ys3)) @ (((listrel @ A) @ B) @ R)))))))))))))).
% 9.43/3.45 thf(fact_4041_listrel__Cons2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Y:B,Ys2:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ (((cons @ B) @ Y) @ Ys2))) @ (((listrel @ A) @ B) @ R)) => (~ ((! [X2:A,Xs2:(list @ A)]: (((Xs = (((cons @ A) @ X2) @ Xs2)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y)) @ R) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs2) @ Ys2)) @ (((listrel @ A) @ B) @ R)))))))))))))).
% 9.43/3.45 thf(fact_4042_listrel__reflcl__if__listrel1, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((listrel1 @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ (((listrel @ A) @ A) @ ((transitive_rtrancl @ A) @ R))))))).
% 9.43/3.45 thf(fact_4043_rtrancl__listrel1__if__listrel, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ (((listrel @ A) @ A) @ R)) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs) @ Ys2)) @ ((transitive_rtrancl @ (list @ A)) @ ((listrel1 @ A) @ R))))))).
% 9.43/3.45 thf(fact_4044_effect__of__listE, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(array @ A),N2:nat]: ((((((((heap_Time_effect @ (array @ A)) @ ((array_of_list @ A) @ Xs)) @ H3) @ H4) @ R) @ N2) => (~ (((R = (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))) => ((H4 = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))) => (((((array_get @ A) @ H4) @ R) = Xs) => ((((array_present @ A) @ H4) @ R) => ((~ ((((array_present @ A) @ H3) @ R))) => (N2 != (((plus_plus @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs))))))))))))))))).
% 9.43/3.45 thf(fact_4045_effect__newE, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(array @ A),N2:nat]: ((((((((heap_Time_effect @ (array @ A)) @ (((array_new @ A) @ N) @ X)) @ H3) @ H4) @ R) @ N2) => (~ (((R = (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H3))) => ((H4 = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ (((replicate @ A) @ N) @ X)) @ H3))) => (((((array_get @ A) @ H4) @ R) = (((replicate @ A) @ N) @ X)) => ((((array_present @ A) @ H4) @ R) => ((~ ((((array_present @ A) @ H3) @ R))) => ((((plus_plus @ nat) @ N) @ (one_one @ nat)) != N2))))))))))))))).
% 9.43/3.45 thf(fact_4046_listrel_Ocases, axiom, (! [B:$tType,A:$tType,A1:(list @ A),A22:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ A1) @ A22)) @ (((listrel @ A) @ B) @ R)) => (((A1 = (nil @ A)) => (A22 != (nil @ B))) => (~ ((! [X2:A,Y4:B,Xs2:(list @ A)]: (((A1 = (((cons @ A) @ X2) @ Xs2)) => (! [Ys3:(list @ B)]: (((A22 = (((cons @ B) @ Y4) @ Ys3)) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ R) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs2) @ Ys3)) @ (((listrel @ A) @ B) @ R)))))))))))))))))).
% 9.43/3.45 thf(fact_4047_listrel_Osimps, axiom, (! [B:$tType,A:$tType,A1:(list @ A),A22:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ A1) @ A22)) @ (((listrel @ A) @ B) @ R)) = (((A1 = (nil @ A)) & (A22 = (nil @ B))) | (? [X3:A,Y3:B,Xs3:(list @ A),Ys:(list @ B)]: (((A1 = (((cons @ A) @ X3) @ Xs3)) & ((A22 = (((cons @ B) @ Y3) @ Ys)) & ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R) & (((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs3) @ Ys)) @ (((listrel @ A) @ B) @ R)))))))))))).
% 9.43/3.45 thf(fact_4048_listrel__iff__zip, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B),R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xs) @ Ys2)) @ (((listrel @ A) @ B) @ R)) = ((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) & (! [X3:((product_prod @ A) @ B)]: (((((member @ ((product_prod @ A) @ B)) @ X3) @ ((set2 @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2))) => (((((product_case_prod @ A) @ B) @ $o) @ (^ [Y3:A,Z5:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ Y3) @ Z5)) @ R)))) @ X3))))))))).
% 9.43/3.45 thf(fact_4049_present__alloc__noteq, axiom, (! [A:$tType,B:$tType]: ((((heap @ B) & (heap @ A)) => (! [H3:(heap_ext @ product_unit),A3:(array @ A),Xs:(list @ B)]: (((((array_present @ A) @ H3) @ A3) => ((((array_noteq @ A) @ B) @ A3) @ (((product_fst @ (array @ B)) @ (heap_ext @ product_unit)) @ (((array_alloc @ B) @ Xs) @ H3)))))))))).
% 9.43/3.45 thf(fact_4050_listrel__def, axiom, (! [B:$tType,A:$tType]: ((((listrel @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_case_prod @ (list @ A)) @ (list @ B)) @ $o) @ (((listrelp @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3)))))))))))).
% 9.43/3.45 thf(fact_4051_set, axiom, (! [A:$tType]: (((heap @ A) => (! [Ls:(list @ A),H3:(heap_ext @ product_unit),New_ls:(list @ A)]: ((((((array_set @ A) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ls) @ H3))) @ New_ls) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ls) @ H3))) = (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ New_ls) @ H3))))))))).
% 9.43/3.45 thf(fact_4052_Array__Time_Oset__same, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A),X:(list @ A),Y:(list @ A),H3:(heap_ext @ product_unit)]: ((((((array_set @ A) @ R) @ X) @ ((((array_set @ A) @ R) @ Y) @ H3)) = ((((array_set @ A) @ R) @ X) @ H3)))))))).
% 9.43/3.45 thf(fact_4053_Array__Time_Ounequal, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),A6:(array @ A)]: (((A3 != A6) = ((((array_noteq @ A) @ A) @ A3) @ A6)))))))).
% 9.43/3.45 thf(fact_4054_Array__Time_Oget__set__eq, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A),X:(list @ A),H3:(heap_ext @ product_unit)]: (((((array_get @ A) @ ((((array_set @ A) @ R) @ X) @ H3)) @ R) = X))))))).
% 9.43/3.45 thf(fact_4055_Array__Time_Oget__set__neq, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [R:(array @ A),S:(array @ B),X:(list @ B),H3:(heap_ext @ product_unit)]: ((((((array_noteq @ A) @ B) @ R) @ S) => ((((array_get @ A) @ ((((array_set @ B) @ S) @ X) @ H3)) @ R) = (((array_get @ A) @ H3) @ R))))))))).
% 9.43/3.45 thf(fact_4056_Array__Time_Oset__set__swap, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [R:(array @ A),R4:(array @ B),X:(list @ A),X6:(list @ B),H3:(heap_ext @ product_unit)]: ((((((array_noteq @ A) @ B) @ R) @ R4) => (((((array_set @ A) @ R) @ X) @ ((((array_set @ B) @ R4) @ X6) @ H3)) = ((((array_set @ B) @ R4) @ X6) @ ((((array_set @ A) @ R) @ X) @ H3)))))))))).
% 9.43/3.45 thf(fact_4057_Array__Time_Onoteq__irrefl, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A)]: ((~ (((((array_noteq @ A) @ A) @ R) @ R))))))))).
% 9.43/3.45 thf(fact_4058_Array__Time_Onoteq__sym, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [A3:(array @ A),B2:(array @ B)]: ((((((array_noteq @ A) @ B) @ A3) @ B2) => ((((array_noteq @ B) @ A) @ B2) @ A3)))))))).
% 9.43/3.45 thf(fact_4059_Field__not__elem, axiom, (! [A:$tType,V:A,R5:(set @ ((product_prod @ A) @ A))]: (((~ ((((member @ A) @ V) @ ((field2 @ A) @ R5)))) => (! [X7:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X7) @ R5) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y3:A,Z5:A]: (((Y3 != V) & (Z5 != V))))) @ X7)))))))).
% 9.43/3.45 thf(fact_4060_refl__on__def_H, axiom, (! [A:$tType]: (((refl_on @ A) = (^ [A9:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: (((! [X3:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X3) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y3:A,Z5:A]: (((((member @ A) @ Y3) @ A9) & (((member @ A) @ Z5) @ A9))))) @ X3)))) & (! [X3:A]: (((((member @ A) @ X3) @ A9) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ R3))))))))))).
% 9.43/3.45 thf(fact_4061_listrelp__listrel__eq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((listrelp @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) = (^ [X3:(list @ A),Y3:(list @ B)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ B))) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ X3) @ Y3)) @ (((listrel @ A) @ B) @ R)))))))).
% 9.43/3.45 thf(fact_4062_Under__def, axiom, (! [A:$tType]: (((order_Under @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A9:(set @ A)]: (((collect @ A) @ (^ [B5:A]: (((((member @ A) @ B5) @ ((field2 @ A) @ R3)) & (! [X3:A]: (((((member @ A) @ X3) @ A9) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ X3)) @ R3)))))))))))))).
% 9.43/3.45 thf(fact_4063_UnderS__def, axiom, (! [A:$tType]: (((order_UnderS @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A9:(set @ A)]: (((collect @ A) @ (^ [B5:A]: (((((member @ A) @ B5) @ ((field2 @ A) @ R3)) & (! [X3:A]: (((((member @ A) @ X3) @ A9) => ((B5 != X3) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ X3)) @ R3))))))))))))))).
% 9.43/3.45 thf(fact_4064_sorted__wrt_Opelims_I2_J, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A)]: (((((sorted_wrt @ A) @ X) @ Xa) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (((Xa = (nil @ A)) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (nil @ A)))))) => (~ ((! [X2:A,Ys3:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ Ys3)) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ Ys3))) => (~ (((! [Xa2:A]: (((((member @ A) @ Xa2) @ ((set2 @ A) @ Ys3)) => ((X @ X2) @ Xa2)))) & (((sorted_wrt @ A) @ X) @ Ys3)))))))))))))))).
% 9.43/3.45 thf(fact_4065_sorted__wrt_Opelims_I1_J, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Y:$o]: ((((((sorted_wrt @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (((Xa = (nil @ A)) => (Y => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (nil @ A))))))) => (~ ((! [X2:A,Ys3:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ Ys3)) => ((Y = ((! [Y3:A]: (((((member @ A) @ Y3) @ ((set2 @ A) @ Ys3)) => ((X @ X2) @ Y3)))) & (((sorted_wrt @ A) @ X) @ Ys3))) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ Ys3))))))))))))))))).
% 9.43/3.45 thf(fact_4066_sorted__wrt_Opelims_I3_J, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A)]: (((~ ((((sorted_wrt @ A) @ X) @ Xa))) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (~ ((! [X2:A,Ys3:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ Ys3)) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (sorted_wrt_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ Ys3))) => ((! [Xa3:A]: (((((member @ A) @ Xa3) @ ((set2 @ A) @ Ys3)) => ((X @ X2) @ Xa3)))) & (((sorted_wrt @ A) @ X) @ Ys3))))))))))))).
% 9.43/3.45 thf(fact_4067_Above__def, axiom, (! [A:$tType]: (((order_Above @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A9:(set @ A)]: (((collect @ A) @ (^ [B5:A]: (((((member @ A) @ B5) @ ((field2 @ A) @ R3)) & (! [X3:A]: (((((member @ A) @ X3) @ A9) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ B5)) @ R3)))))))))))))).
% 9.43/3.45 thf(fact_4068_execute__nth_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A)]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((((heap_Time_execute @ A) @ (((array_nth @ A) @ A3) @ I)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (one_one @ nat)))))))))))).
% 9.43/3.45 thf(fact_4069_Array__Time_Olength__alloc, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A),H3:(heap_ext @ product_unit),Ys2:(list @ A)]: (((((array_length @ A) @ (((product_snd @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Xs) @ H3))) @ (((product_fst @ (array @ A)) @ (heap_ext @ product_unit)) @ (((array_alloc @ A) @ Ys2) @ H3))) = ((size_size @ (list @ A)) @ Xs)))))))).
% 9.43/3.45 thf(fact_4070_Array__Time_Onth__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_nth @ A) = (^ [A7:(array @ A),I3:nat]: ((((heap_Time_guard @ A) @ (^ [H:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I3) @ (((array_length @ A) @ H) @ A7))))) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((nth @ A) @ (((array_get @ A) @ H) @ A7)) @ I3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (one_one @ nat))))))))))))).
% 9.43/3.45 thf(fact_4071_success__nthI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A)]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (((heap_Time_success @ A) @ (((array_nth @ A) @ A3) @ I)) @ H3)))))))).
% 9.43/3.45 thf(fact_4072_fails__array__nth, axiom, (! [A:$tType]: (((heap @ A) => (! [P5:(array @ A),I:nat,H3:(heap_ext @ product_unit)]: (((((time_fails @ A) @ (((array_nth @ A) @ P5) @ I)) @ H3) = (~ ((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ P5))))))))))).
% 9.43/3.45 thf(fact_4073_execute__nth_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [H3:(heap_ext @ product_unit),A3:(array @ A),I:nat]: (((((ord_less_eq @ nat) @ (((array_length @ A) @ H3) @ A3)) @ I) => ((((heap_Time_execute @ A) @ (((array_nth @ A) @ A3) @ I)) @ H3) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))).
% 9.43/3.45 thf(fact_4074_length__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_length @ A) = (^ [H:(heap_ext @ product_unit),A7:(array @ A)]: (((size_size @ (list @ A)) @ (((array_get @ A) @ H) @ A7))))))))).
% 9.43/3.45 thf(fact_4075_effect__nthE, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),I:nat,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ (((array_nth @ A) @ A3) @ I)) @ H3) @ H4) @ R) @ N) => (~ (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((R = (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I)) => ((H4 = H3) => (N != (one_one @ nat)))))))))))))).
% 9.43/3.45 thf(fact_4076_effect__nthI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),H4:(heap_ext @ product_unit),R:A,N:nat]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((H4 = H3) => ((R = (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ A) @ (((array_nth @ A) @ A3) @ I)) @ H3) @ H4) @ R) @ N))))))))))).
% 9.43/3.45 thf(fact_4077_nth_H__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_nth2 @ A) = (^ [A7:(array @ A)]: ((((((comp @ nat) @ (heap_Time_Heap @ A)) @ code_integer) @ ((array_nth @ A) @ A7)) @ code_nat_of_integer)))))))).
% 9.43/3.45 thf(fact_4078_freeze__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_freeze @ A) = (^ [A7:(array @ A)]: (((heap_Time_heap @ (list @ A)) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((array_get @ A) @ H) @ A7)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H) @ (((plus_plus @ nat) @ (one_one @ nat)) @ (((array_length @ A) @ H) @ A7)))))))))))))).
% 9.43/3.45 thf(fact_4079_execute__map__entry_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [H3:(heap_ext @ product_unit),A3:(array @ A),I:nat,F:(A > A)]: (((((ord_less_eq @ nat) @ (((array_length @ A) @ H3) @ A3)) @ I) => ((((heap_Time_execute @ (array @ A)) @ ((((array_map_entry @ A) @ I) @ F) @ A3)) @ H3) = (none @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))).
% 9.43/3.45 thf(fact_4080_execute__freeze, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (list @ A)) @ ((array_freeze @ A) @ A3)) @ H3) = ((some @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((array_get @ A) @ H3) @ A3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (((plus_plus @ nat) @ (one_one @ nat)) @ (((array_length @ A) @ H3) @ A3)))))))))))).
% 9.43/3.45 thf(fact_4081_success__freezeI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ (list @ A)) @ ((array_freeze @ A) @ A3)) @ H3))))))).
% 9.43/3.45 thf(fact_4082_success__map__entryI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),F:(A > A)]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (((heap_Time_success @ (array @ A)) @ ((((array_map_entry @ A) @ I) @ F) @ A3)) @ H3)))))))).
% 9.43/3.45 thf(fact_4083_effect__freezeE, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(list @ A),N:nat]: ((((((((heap_Time_effect @ (list @ A)) @ ((array_freeze @ A) @ A3)) @ H3) @ H4) @ R) @ N) => (~ (((H4 = H3) => ((R = (((array_get @ A) @ H3) @ A3)) => (N != (((plus_plus @ nat) @ (((array_length @ A) @ H3) @ A3)) @ (one_one @ nat)))))))))))))).
% 9.43/3.45 thf(fact_4084_effect__freezeI, axiom, (! [A:$tType]: (((heap @ A) => (! [H4:(heap_ext @ product_unit),H3:(heap_ext @ product_unit),R:(list @ A),A3:(array @ A),N:nat]: (((H4 = H3) => ((R = (((array_get @ A) @ H3) @ A3)) => ((N = (((array_length @ A) @ H3) @ A3)) => ((((((heap_Time_effect @ (list @ A)) @ ((array_freeze @ A) @ A3)) @ H3) @ H4) @ R) @ (((plus_plus @ nat) @ N) @ (one_one @ nat)))))))))))).
% 9.43/3.45 thf(fact_4085_execute__map__entry_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),F:(A > A)]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((((heap_Time_execute @ (array @ A)) @ ((((array_map_entry @ A) @ I) @ F) @ A3)) @ H3) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A3) @ I) @ (F @ (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I))) @ H3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))))).
% 9.43/3.45 thf(fact_4086_map__entry__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_map_entry @ A) = (^ [I3:nat,F2:(A > A),A7:(array @ A)]: ((((heap_Time_guard @ (array @ A)) @ (^ [H:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I3) @ (((array_length @ A) @ H) @ A7))))) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A7) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A7) @ I3) @ (F2 @ (((nth @ A) @ (((array_get @ A) @ H) @ A7)) @ I3))) @ H)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))).
% 9.43/3.45 thf(fact_4087_effect__map__entryI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),H4:(heap_ext @ product_unit),F:(A > A),R:(array @ A),N:nat]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((H4 = (((((array_update @ A) @ A3) @ I) @ (F @ (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I))) @ H3)) => ((R = A3) => ((N = ((numeral_numeral @ nat) @ (bit0 @ one2))) => ((((((heap_Time_effect @ (array @ A)) @ ((((array_map_entry @ A) @ I) @ F) @ A3)) @ H3) @ H4) @ R) @ N))))))))))).
% 9.43/3.45 thf(fact_4088_effect__map__entryE, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,F:(A > A),A3:(array @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(array @ A),N:nat]: ((((((((heap_Time_effect @ (array @ A)) @ ((((array_map_entry @ A) @ I) @ F) @ A3)) @ H3) @ H4) @ R) @ N) => (~ (((R = A3) => ((H4 = (((((array_update @ A) @ A3) @ I) @ (F @ (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I))) @ H3)) => ((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (N != ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))))))).
% 9.43/3.45 thf(fact_4089_length__update, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [B2:(array @ B),I:nat,V:B,H3:(heap_ext @ product_unit)]: ((((array_length @ A) @ (((((array_update @ B) @ B2) @ I) @ V) @ H3)) = ((array_length @ A) @ H3)))))))).
% 9.43/3.45 thf(fact_4090_Array__Time_Opresent__update, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [B2:(array @ B),I:nat,V:B,H3:(heap_ext @ product_unit)]: ((((array_present @ A) @ (((((array_update @ B) @ B2) @ I) @ V) @ H3)) = ((array_present @ A) @ H3)))))))).
% 9.43/3.45 thf(fact_4091_get__update__elem__neqIndex, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,J:nat,A3:(array @ A),V:A,H3:(heap_ext @ product_unit)]: (((I != J) => ((((nth @ A) @ (((array_get @ A) @ (((((array_update @ A) @ A3) @ J) @ V) @ H3)) @ A3)) @ I) = (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I))))))))).
% 9.43/3.45 thf(fact_4092_get__update__eq, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),I:nat,V:A,H3:(heap_ext @ product_unit)]: (((((array_get @ A) @ (((((array_update @ A) @ A3) @ I) @ V) @ H3)) @ A3) = ((((list_update @ A) @ (((array_get @ A) @ H3) @ A3)) @ I) @ V)))))))).
% 9.43/3.45 thf(fact_4093_nth__update__neq, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [A3:(array @ A),B2:(array @ B),J:nat,V:B,H3:(heap_ext @ product_unit),I:nat]: ((((((array_noteq @ A) @ B) @ A3) @ B2) => ((((nth @ A) @ (((array_get @ A) @ (((((array_update @ B) @ B2) @ J) @ V) @ H3)) @ A3)) @ I) = (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I))))))))).
% 9.43/3.45 thf(fact_4094_update__swap__neqIndex, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,I7:nat,A3:(array @ A),V:A,V7:A,H3:(heap_ext @ product_unit)]: (((I != I7) => ((((((array_update @ A) @ A3) @ I) @ V) @ (((((array_update @ A) @ A3) @ I7) @ V7) @ H3)) = (((((array_update @ A) @ A3) @ I7) @ V7) @ (((((array_update @ A) @ A3) @ I) @ V) @ H3)))))))))).
% 9.43/3.45 thf(fact_4095_update__swap__neq, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (! [A3:(array @ A),A6:(array @ B),I:nat,V:A,I7:nat,V7:B,H3:(heap_ext @ product_unit)]: ((((((array_noteq @ A) @ B) @ A3) @ A6) => ((((((array_update @ A) @ A3) @ I) @ V) @ (((((array_update @ B) @ A6) @ I7) @ V7) @ H3)) = (((((array_update @ B) @ A6) @ I7) @ V7) @ (((((array_update @ A) @ A3) @ I) @ V) @ H3)))))))))).
% 9.43/3.45 thf(fact_4096_Array__Time_Oupdate__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_update @ A) = (^ [A7:(array @ A),I3:nat,X3:A,H:(heap_ext @ product_unit)]: (((((array_set @ A) @ A7) @ ((((list_update @ A) @ (((array_get @ A) @ H) @ A7)) @ I3) @ X3)) @ H)))))))).
% 9.43/3.45 thf(fact_4097_execute__swap_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),X:A]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((((heap_Time_execute @ A) @ ((((array_swap @ A) @ I) @ X) @ A3)) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A3) @ I) @ X) @ H3)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))))).
% 9.43/3.45 thf(fact_4098_swap__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_swap @ A) = (^ [I3:nat,X3:A,A7:(array @ A)]: ((((heap_Time_guard @ A) @ (^ [H:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I3) @ (((array_length @ A) @ H) @ A7))))) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((nth @ A) @ (((array_get @ A) @ H) @ A7)) @ I3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A7) @ I3) @ X3) @ H)) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))))))))))))).
% 9.43/3.45 thf(fact_4099_effect__swapI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),H4:(heap_ext @ product_unit),X:A,R:A,N:nat]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((H4 = (((((array_update @ A) @ A3) @ I) @ X) @ H3)) => ((R = (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I)) => ((N = ((numeral_numeral @ nat) @ (bit0 @ one2))) => ((((((heap_Time_effect @ A) @ ((((array_swap @ A) @ I) @ X) @ A3)) @ H3) @ H4) @ R) @ N))))))))))).
% 9.43/3.45 thf(fact_4100_effect__swapE, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,X:A,A3:(array @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: ((((((((heap_Time_effect @ A) @ ((((array_swap @ A) @ I) @ X) @ A3)) @ H3) @ H4) @ R) @ N) => (~ (((R = (((nth @ A) @ (((array_get @ A) @ H3) @ A3)) @ I)) => ((H4 = (((((array_update @ A) @ A3) @ I) @ X) @ H3)) => ((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (N != ((numeral_numeral @ nat) @ (bit0 @ one2))))))))))))))).
% 9.43/3.45 thf(fact_4101_success__swapI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),X:A]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (((heap_Time_success @ A) @ ((((array_swap @ A) @ I) @ X) @ A3)) @ H3)))))))).
% 9.43/3.45 thf(fact_4102_execute__swap_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [H3:(heap_ext @ product_unit),A3:(array @ A),I:nat,X:A]: (((((ord_less_eq @ nat) @ (((array_length @ A) @ H3) @ A3)) @ I) => ((((heap_Time_execute @ A) @ ((((array_swap @ A) @ I) @ X) @ A3)) @ H3) = (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))).
% 9.43/3.45 thf(fact_4103_execute__upd_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),X:A]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((((heap_Time_execute @ (array @ A)) @ ((((array_upd @ A) @ I) @ X) @ A3)) @ H3) = ((some @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A3) @ I) @ X) @ H3)) @ (one_one @ nat)))))))))))).
% 9.43/3.45 thf(fact_4104_upd__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_upd @ A) = (^ [I3:nat,X3:A,A7:(array @ A)]: ((((heap_Time_guard @ (array @ A)) @ (^ [H:(heap_ext @ product_unit)]: ((((ord_less @ nat) @ I3) @ (((array_length @ A) @ H) @ A7))))) @ (^ [H:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ A7) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ (((((array_update @ A) @ A7) @ I3) @ X3) @ H)) @ (one_one @ nat))))))))))))).
% 9.43/3.45 thf(fact_4105_execute__upd_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [H3:(heap_ext @ product_unit),A3:(array @ A),I:nat,X:A]: (((((ord_less_eq @ nat) @ (((array_length @ A) @ H3) @ A3)) @ I) => ((((heap_Time_execute @ (array @ A)) @ ((((array_upd @ A) @ I) @ X) @ A3)) @ H3) = (none @ ((product_prod @ (array @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))))).
% 9.43/3.45 thf(fact_4106_effect__updI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),H4:(heap_ext @ product_unit),V:A,N:nat]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => ((H4 = (((((array_update @ A) @ A3) @ I) @ V) @ H3)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ (array @ A)) @ ((((array_upd @ A) @ I) @ V) @ A3)) @ H3) @ H4) @ A3) @ N)))))))))).
% 9.43/3.45 thf(fact_4107_upd__ureturn, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,X:A,A3:(array @ A)]: ((((((heap_Time_bind @ (array @ A)) @ (array @ A)) @ ((((array_upd @ A) @ I) @ X) @ A3)) @ (^ [Uu:(array @ A)]: (((heap_Time_ureturn @ (array @ A)) @ A3)))) = ((((array_upd @ A) @ I) @ X) @ A3)))))))).
% 9.43/3.45 thf(fact_4108_success__updI, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,H3:(heap_ext @ product_unit),A3:(array @ A),X:A]: (((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (((heap_Time_success @ (array @ A)) @ ((((array_upd @ A) @ I) @ X) @ A3)) @ H3)))))))).
% 9.43/3.45 thf(fact_4109_fails__array__upd, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,X:A,P5:(array @ A),H3:(heap_ext @ product_unit)]: (((((time_fails @ (array @ A)) @ ((((array_upd @ A) @ I) @ X) @ P5)) @ H3) = (~ ((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ P5))))))))))).
% 9.43/3.45 thf(fact_4110_effect__updE, axiom, (! [A:$tType]: (((heap @ A) => (! [I:nat,V:A,A3:(array @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(array @ A),N:nat]: ((((((((heap_Time_effect @ (array @ A)) @ ((((array_upd @ A) @ I) @ V) @ A3)) @ H3) @ H4) @ R) @ N) => (~ (((R = A3) => ((H4 = (((((array_update @ A) @ A3) @ I) @ V) @ H3)) => ((((ord_less @ nat) @ I) @ (((array_length @ A) @ H3) @ A3)) => (N != (one_one @ nat)))))))))))))).
% 9.43/3.45 thf(fact_4111_upd_H__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_upd2 @ A) = (^ [A7:(array @ A),I3:code_integer,X3:A]: (((((heap_Time_bind @ (array @ A)) @ product_unit) @ ((((array_upd @ A) @ (code_nat_of_integer @ I3)) @ X3) @ A7)) @ (^ [Uu:(array @ A)]: (((heap_Time_ureturn @ product_unit) @ product_Unity))))))))))).
% 9.43/3.45 thf(fact_4112_execute__len, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ nat) @ ((array_len @ A) @ A3)) @ H3) = ((some @ ((product_prod @ nat) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ nat) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ (((array_length @ A) @ H3) @ A3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H3) @ (one_one @ nat))))))))))).
% 9.43/3.45 thf(fact_4113_in__measures_I2_J, axiom, (! [A:$tType,X:A,Y:A,F:(A > nat),Fs:(list @ (A > nat))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measures @ A) @ (((cons @ (A > nat)) @ F) @ Fs))) = ((((ord_less @ nat) @ (F @ X)) @ (F @ Y)) | (((F @ X) = (F @ Y)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measures @ A) @ Fs)))))))).
% 9.43/3.45 thf(fact_4114_mset__mergesort__by__rel__split, axiom, (! [A:$tType,Xs1:(list @ A),Xs22:(list @ A),Xs:(list @ A)]: (((((plus_plus @ (multiset @ A)) @ ((mset @ A) @ (((product_fst @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ Xs)))) @ ((mset @ A) @ (((product_snd @ (list @ A)) @ (list @ A)) @ (((merges295452479951948502_split @ A) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs1) @ Xs22)) @ Xs)))) = (((plus_plus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ ((mset @ A) @ Xs)) @ ((mset @ A) @ Xs1))) @ ((mset @ A) @ Xs22)))))).
% 9.43/3.45 thf(fact_4115_union__eq__empty, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ M3) @ N6) = (zero_zero @ (multiset @ A))) = ((M3 = (zero_zero @ (multiset @ A))) & (N6 = (zero_zero @ (multiset @ A)))))))).
% 9.43/3.45 thf(fact_4116_empty__eq__union, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A)]: ((((zero_zero @ (multiset @ A)) = (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = ((M3 = (zero_zero @ (multiset @ A))) & (N6 = (zero_zero @ (multiset @ A)))))))).
% 9.43/3.45 thf(fact_4117_subset__mset_Ozero__eq__add__iff__both__eq__0, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: ((((zero_zero @ (multiset @ A)) = (((plus_plus @ (multiset @ A)) @ X) @ Y)) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))).
% 9.43/3.45 thf(fact_4118_subset__mset_Oadd__eq__0__iff__both__eq__0, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ X) @ Y) = (zero_zero @ (multiset @ A))) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))).
% 9.43/3.45 thf(fact_4119_diff__diff__add__mset, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A),P:(multiset @ A)]: (((((minus_minus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ M3) @ N6)) @ P) = (((minus_minus @ (multiset @ A)) @ M3) @ (((plus_plus @ (multiset @ A)) @ N6) @ P)))))).
% 9.43/3.45 thf(fact_4120_mergesort__by__rel__permutes, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A)]: ((((mset @ A) @ (((mergesort_by_rel @ A) @ R5) @ Xs)) = ((mset @ A) @ Xs))))).
% 9.43/3.45 thf(fact_4121_in__measures_I1_J, axiom, (! [A:$tType,X:A,Y:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measures @ A) @ (nil @ (A > nat))))))))).
% 9.43/3.45 thf(fact_4122_mset__append, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: ((((mset @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((plus_plus @ (multiset @ A)) @ ((mset @ A) @ Xs)) @ ((mset @ A) @ Ys2)))))).
% 9.43/3.45 thf(fact_4123_quicksort__by__rel__permutes, axiom, (! [A:$tType,R5:(A > (A > $o)),Sl2:(list @ A),Xs:(list @ A)]: ((((mset @ A) @ ((((quicksort_by_rel @ A) @ R5) @ Sl2) @ Xs)) = ((mset @ A) @ (((append @ A) @ Xs) @ Sl2)))))).
% 9.43/3.45 thf(fact_4124_mset__mergesort__by__rel__merge, axiom, (! [A:$tType,R5:(A > (A > $o)),Xs:(list @ A),Ys2:(list @ A)]: ((((mset @ A) @ ((((merges9089515139780605204_merge @ A) @ R5) @ Xs) @ Ys2)) = (((plus_plus @ (multiset @ A)) @ ((mset @ A) @ Xs)) @ ((mset @ A) @ Ys2)))))).
% 9.43/3.45 thf(fact_4125_success__lenI, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H3:(heap_ext @ product_unit)]: ((((heap_Time_success @ nat) @ ((array_len @ A) @ A3)) @ H3))))))).
% 9.43/3.45 thf(fact_4126_fails__array__len, axiom, (! [A:$tType]: (((heap @ A) => (! [P5:(array @ A),H3:(heap_ext @ product_unit)]: ((~ ((((time_fails @ nat) @ ((array_len @ A) @ P5)) @ H3))))))))).
% 9.43/3.45 thf(fact_4127_mset__distrib, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),M3:(multiset @ A),N6:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ A4) @ B6) = (((plus_plus @ (multiset @ A)) @ M3) @ N6)) => (~ ((! [Am:(multiset @ A),An:(multiset @ A)]: (((A4 = (((plus_plus @ (multiset @ A)) @ Am) @ An)) => (! [Bm:(multiset @ A),Bn:(multiset @ A)]: (((B6 = (((plus_plus @ (multiset @ A)) @ Bm) @ Bn)) => ((M3 = (((plus_plus @ (multiset @ A)) @ Am) @ Bm)) => (N6 != (((plus_plus @ (multiset @ A)) @ An) @ Bn))))))))))))))).
% 9.43/3.45 thf(fact_4128_mset__shuffles, axiom, (! [A:$tType,Zs:(list @ A),Xs:(list @ A),Ys2:(list @ A)]: (((((member @ (list @ A)) @ Zs) @ (((shuffles @ A) @ Xs) @ Ys2)) => (((mset @ A) @ Zs) = (((plus_plus @ (multiset @ A)) @ ((mset @ A) @ Xs)) @ ((mset @ A) @ Ys2))))))).
% 9.43/3.45 thf(fact_4129_union__less__mono, axiom, (! [A:$tType]: (((preorder @ A) => (! [A4:(multiset @ A),C5:(multiset @ A),B6:(multiset @ A),D4:(multiset @ A)]: (((((ord_less @ (multiset @ A)) @ A4) @ C5) => ((((ord_less @ (multiset @ A)) @ B6) @ D4) => (((ord_less @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ (((plus_plus @ (multiset @ A)) @ C5) @ D4)))))))))).
% 9.43/3.45 thf(fact_4130_union__le__mono2, axiom, (! [A:$tType]: (((preorder @ A) => (! [B6:(multiset @ A),D4:(multiset @ A),C5:(multiset @ A)]: (((((ord_less @ (multiset @ A)) @ B6) @ D4) => (((ord_less @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ C5) @ B6)) @ (((plus_plus @ (multiset @ A)) @ C5) @ D4))))))))).
% 9.43/3.45 thf(fact_4131_union__le__mono1, axiom, (! [A:$tType]: (((preorder @ A) => (! [B6:(multiset @ A),D4:(multiset @ A),C5:(multiset @ A)]: (((((ord_less @ (multiset @ A)) @ B6) @ D4) => (((ord_less @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ B6) @ C5)) @ (((plus_plus @ (multiset @ A)) @ D4) @ C5))))))))).
% 9.43/3.45 thf(fact_4132_multi__union__self__other__eq, axiom, (! [A:$tType,A4:(multiset @ A),X10:(multiset @ A),Y9:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ A4) @ X10) = (((plus_plus @ (multiset @ A)) @ A4) @ Y9)) => (X10 = Y9))))).
% 9.43/3.45 thf(fact_4133_union__right__cancel, axiom, (! [A:$tType,M3:(multiset @ A),K2:(multiset @ A),N6:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ M3) @ K2) = (((plus_plus @ (multiset @ A)) @ N6) @ K2)) = (M3 = N6))))).
% 9.43/3.45 thf(fact_4134_union__left__cancel, axiom, (! [A:$tType,K2:(multiset @ A),M3:(multiset @ A),N6:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ K2) @ M3) = (((plus_plus @ (multiset @ A)) @ K2) @ N6)) = (M3 = N6))))).
% 9.43/3.45 thf(fact_4135_union__commute, axiom, (! [A:$tType]: (((plus_plus @ (multiset @ A)) = (^ [M8:(multiset @ A),N9:(multiset @ A)]: ((((plus_plus @ (multiset @ A)) @ N9) @ M8))))))).
% 9.43/3.45 thf(fact_4136_union__lcomm, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A),K2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ M3) @ (((plus_plus @ (multiset @ A)) @ N6) @ K2)) = (((plus_plus @ (multiset @ A)) @ N6) @ (((plus_plus @ (multiset @ A)) @ M3) @ K2)))))).
% 9.43/3.45 thf(fact_4137_union__assoc, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A),K2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) @ K2) = (((plus_plus @ (multiset @ A)) @ M3) @ (((plus_plus @ (multiset @ A)) @ N6) @ K2)))))).
% 9.43/3.45 thf(fact_4138_empty__neutral_I1_J, axiom, (! [A:$tType,X:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (zero_zero @ (multiset @ A))) @ X) = X)))).
% 9.43/3.45 thf(fact_4139_empty__neutral_I2_J, axiom, (! [A:$tType,X:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ X) @ (zero_zero @ (multiset @ A))) = X)))).
% 9.43/3.45 thf(fact_4140_diff__union__cancelR, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A)]: (((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) @ N6) = M3)))).
% 9.43/3.45 thf(fact_4141_diff__union__cancelL, axiom, (! [A:$tType,N6:(multiset @ A),M3:(multiset @ A)]: (((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ N6) @ M3)) @ N6) = M3)))).
% 9.43/3.45 thf(fact_4142_Multiset_Odiff__add, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A),Q:(multiset @ A)]: (((((minus_minus @ (multiset @ A)) @ M3) @ (((plus_plus @ (multiset @ A)) @ N6) @ Q)) = (((minus_minus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ M3) @ N6)) @ Q))))).
% 9.43/3.45 thf(fact_4143_union__diff__assoc, axiom, (! [A:$tType,C5:(multiset @ A),B6:(multiset @ A),A4:(multiset @ A)]: ((((((minus_minus @ (multiset @ A)) @ C5) @ B6) = (zero_zero @ (multiset @ A))) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ C5) = (((plus_plus @ (multiset @ A)) @ A4) @ (((minus_minus @ (multiset @ A)) @ B6) @ C5))))))).
% 9.43/3.45 thf(fact_4144_comp__fun__commute__plus__mset, axiom, (! [A:$tType]: ((((finite6289374366891150609ommute @ (multiset @ A)) @ (multiset @ A)) @ (plus_plus @ (multiset @ A)))))).
% 9.43/3.45 thf(fact_4145_len__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_len @ A) = (^ [A7:(array @ A)]: (((heap_Time_tap @ nat) @ (^ [H:(heap_ext @ product_unit)]: ((((array_length @ A) @ H) @ A7))))))))))).
% 9.43/3.45 thf(fact_4146_measures__less, axiom, (! [A:$tType,F:(A > nat),X:A,Y:A,Fs:(list @ (A > nat))]: (((((ord_less @ nat) @ (F @ X)) @ (F @ Y)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measures @ A) @ (((cons @ (A > nat)) @ F) @ Fs))))))).
% 9.43/3.45 thf(fact_4147_measures__lesseq, axiom, (! [A:$tType,F:(A > nat),X:A,Y:A,Fs:(list @ (A > nat))]: (((((ord_less_eq @ nat) @ (F @ X)) @ (F @ Y)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measures @ A) @ Fs)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((measures @ A) @ (((cons @ (A > nat)) @ F) @ Fs)))))))).
% 9.43/3.45 thf(fact_4148_effect__lengthE, axiom, (! [A:$tType]: (((heap @ A) => (! [A3:(array @ A),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:nat,N:nat]: ((((((((heap_Time_effect @ nat) @ ((array_len @ A) @ A3)) @ H3) @ H4) @ R) @ N) => (~ (((R = (((array_length @ A) @ H4) @ A3)) => ((H4 = H3) => (N != (one_one @ nat))))))))))))).
% 9.43/3.45 thf(fact_4149_effect__lengthI, axiom, (! [A:$tType]: (((heap @ A) => (! [H4:(heap_ext @ product_unit),H3:(heap_ext @ product_unit),R:nat,A3:(array @ A),N:nat]: (((H4 = H3) => ((R = (((array_length @ A) @ H3) @ A3)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ nat) @ ((array_len @ A) @ A3)) @ H3) @ H4) @ R) @ N)))))))))).
% 9.43/3.45 thf(fact_4150_len_H__def, axiom, (! [A:$tType,B:$tType]: ((((heap @ B) & (semiring_1 @ A)) => (((array_len2 @ B) @ A) = (^ [A7:(array @ B)]: (((((heap_Time_bind @ nat) @ A) @ ((array_len @ B) @ A7)) @ (^ [N5:nat]: (((heap_Time_ureturn @ A) @ ((semiring_1_of_nat @ A) @ N5)))))))))))).
% 9.43/3.45 thf(fact_4151_mset__zip__take__Cons__drop__twice, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B),J:nat,X:A,Y:B]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((((ord_less_eq @ nat) @ J) @ ((size_size @ (list @ A)) @ Xs)) => (((mset @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ (((append @ A) @ (((take @ A) @ J) @ Xs)) @ (((cons @ A) @ X) @ (((drop @ A) @ J) @ Xs)))) @ (((append @ B) @ (((take @ B) @ J) @ Ys2)) @ (((cons @ B) @ Y) @ (((drop @ B) @ J) @ Ys2))))) = (((add_mset @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ ((mset @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2))))))))).
% 9.43/3.45 thf(fact_4152_image__mset__map__of, axiom, (! [B:$tType,A:$tType,Xs:(list @ ((product_prod @ A) @ B))]: ((((distinct @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs)) => (((((image_mset @ A) @ B) @ (^ [I3:A]: (((the2 @ B) @ ((((map_of @ A) @ B) @ Xs) @ I3))))) @ ((mset @ A) @ ((((map @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Xs))) = ((mset @ B) @ ((((map @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Xs))))))).
% 9.43/3.45 thf(fact_4153_mset__set__Union, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ A) @ B6) => (((((inf_inf @ (set @ A)) @ A4) @ B6) = (bot_bot @ (set @ A))) => (((mset_set @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((plus_plus @ (multiset @ A)) @ ((mset_set @ A) @ A4)) @ ((mset_set @ A) @ B6))))))))).
% 9.43/3.45 thf(fact_4154_union__mset__add__mset__right, axiom, (! [A:$tType,A4:(multiset @ A),A3:A,B6:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ A4) @ (((add_mset @ A) @ A3) @ B6)) = (((add_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_4155_union__mset__add__mset__left, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B6:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ A4)) @ B6) = (((add_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)))))).
% 9.43/3.45 thf(fact_4156_image__mset__union, axiom, (! [A:$tType,B:$tType,F:(B > A),M3:(multiset @ B),N6:(multiset @ B)]: ((((((image_mset @ B) @ A) @ F) @ (((plus_plus @ (multiset @ B)) @ M3) @ N6)) = (((plus_plus @ (multiset @ A)) @ ((((image_mset @ B) @ A) @ F) @ M3)) @ ((((image_mset @ B) @ A) @ F) @ N6)))))).
% 9.43/3.45 thf(fact_4157_mset__map__split__orig, axiom, (! [B:$tType,A:$tType,F:(B > A),P:(multiset @ B),M13:(multiset @ A),M24:(multiset @ A)]: (((((((image_mset @ B) @ A) @ F) @ P) = (((plus_plus @ (multiset @ A)) @ M13) @ M24)) => (~ ((! [P1:(multiset @ B),P22:(multiset @ B)]: (((P = (((plus_plus @ (multiset @ B)) @ P1) @ P22)) => ((((((image_mset @ B) @ A) @ F) @ P1) = M13) => (((((image_mset @ B) @ A) @ F) @ P22) != M24))))))))))).
% 9.43/3.45 thf(fact_4158_add__mset__add__single, axiom, (! [A:$tType]: (((add_mset @ A) = (^ [A7:A,A9:(multiset @ A)]: ((((plus_plus @ (multiset @ A)) @ A9) @ (((add_mset @ A) @ A7) @ (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4159_union__is__single, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A),A3:A]: ((((((plus_plus @ (multiset @ A)) @ M3) @ N6) = (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) = (((M3 = (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) & (N6 = (zero_zero @ (multiset @ A)))) | ((M3 = (zero_zero @ (multiset @ A))) & (N6 = (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4160_single__is__union, axiom, (! [A:$tType,A3:A,M3:(multiset @ A),N6:(multiset @ A)]: ((((((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) = (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = ((((((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) = M3) & (N6 = (zero_zero @ (multiset @ A)))) | ((M3 = (zero_zero @ (multiset @ A))) & ((((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) = N6))))))).
% 9.43/3.45 thf(fact_4161_add__mset__replicate__mset__safe, axiom, (! [A:$tType,B:$tType,M3:(multiset @ B),A3:B]: ((((((nO_MATCH @ (multiset @ A)) @ (multiset @ B)) @ (zero_zero @ (multiset @ A))) @ M3) => ((((add_mset @ B) @ A3) @ M3) = (((plus_plus @ (multiset @ B)) @ (((add_mset @ B) @ A3) @ (zero_zero @ (multiset @ B)))) @ M3)))))).
% 9.43/3.45 thf(fact_4162_mset__single__cases, axiom, (! [A:$tType,S:A,C2:(multiset @ A),R4:A,C6:(multiset @ A)]: ((((((add_mset @ A) @ S) @ C2) = (((add_mset @ A) @ R4) @ C6)) => (((S = R4) => (C2 != C6)) => (~ (((C6 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ C6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))) => ((C2 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ C2) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))))) => ((((minus_minus @ (multiset @ A)) @ C2) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))) != (((minus_minus @ (multiset @ A)) @ C6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))))))))))).
% 9.43/3.45 thf(fact_4163_mset__single__cases_H, axiom, (! [A:$tType,S:A,C2:(multiset @ A),R4:A,C6:(multiset @ A)]: ((((((add_mset @ A) @ S) @ C2) = (((add_mset @ A) @ R4) @ C6)) => (((S = R4) => (C2 != C6)) => (~ ((! [Cc:(multiset @ A)]: (((C6 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ Cc)) => ((C2 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))) @ Cc)) => (((((minus_minus @ (multiset @ A)) @ C6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) = Cc) => ((((minus_minus @ (multiset @ A)) @ C2) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))) != Cc))))))))))))).
% 9.43/3.45 thf(fact_4164_mset__single__cases2, axiom, (! [A:$tType,S:A,C2:(multiset @ A),R4:A,C6:(multiset @ A)]: ((((((add_mset @ A) @ S) @ C2) = (((add_mset @ A) @ R4) @ C6)) => (((S = R4) => (C2 != C6)) => (~ (((C6 = (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ C6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A))))) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A))))) => ((C2 = (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ C2) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A))))) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A))))) => ((((minus_minus @ (multiset @ A)) @ C2) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))) != (((minus_minus @ (multiset @ A)) @ C6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))))))))))).
% 9.43/3.45 thf(fact_4165_mset__single__cases2_H, axiom, (! [A:$tType,S:A,C2:(multiset @ A),R4:A,C6:(multiset @ A)]: ((((((add_mset @ A) @ S) @ C2) = (((add_mset @ A) @ R4) @ C6)) => (((S = R4) => (C2 != C6)) => (~ ((! [Cc:(multiset @ A)]: (((C6 = (((plus_plus @ (multiset @ A)) @ Cc) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A))))) => ((C2 = (((plus_plus @ (multiset @ A)) @ Cc) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A))))) => (((((minus_minus @ (multiset @ A)) @ C6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) = Cc) => ((((minus_minus @ (multiset @ A)) @ C2) @ (((add_mset @ A) @ R4) @ (zero_zero @ (multiset @ A)))) != Cc))))))))))))).
% 9.43/3.45 thf(fact_4166_mset__unplusm__dist__cases, axiom, (! [A:$tType,S:A,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ A4) = (((plus_plus @ (multiset @ A)) @ B6) @ C5)) => (((B6 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ B6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))) => (A4 != (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ B6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A))))) @ C5))) => (~ (((C5 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ C5) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))) => (A4 != (((plus_plus @ (multiset @ A)) @ B6) @ (((minus_minus @ (multiset @ A)) @ C5) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))))))))))).
% 9.43/3.45 thf(fact_4167_mset__unplusm__dist__cases2, axiom, (! [A:$tType,B6:(multiset @ A),C5:(multiset @ A),S:A,A4:(multiset @ A)]: ((((((plus_plus @ (multiset @ A)) @ B6) @ C5) = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ A4)) => (((B6 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ B6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))) => (A4 != (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ B6) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A))))) @ C5))) => (~ (((C5 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ C5) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))) => (A4 != (((plus_plus @ (multiset @ A)) @ B6) @ (((minus_minus @ (multiset @ A)) @ C5) @ (((add_mset @ A) @ S) @ (zero_zero @ (multiset @ A)))))))))))))).
% 9.43/3.45 thf(fact_4168_image__mset_Ocompositionality, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > C),G:(A > B),Multiset:(multiset @ A)]: ((((((image_mset @ B) @ C) @ F) @ ((((image_mset @ A) @ B) @ G) @ Multiset)) = ((((image_mset @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)) @ Multiset))))).
% 9.43/3.45 thf(fact_4169_multiset_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(multiset @ A)]: ((((((image_mset @ B) @ C) @ G) @ ((((image_mset @ A) @ B) @ F) @ V)) = ((((image_mset @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 9.43/3.45 thf(fact_4170_image__mset_Ocomp, axiom, (! [C:$tType,B:$tType,A:$tType,F:(B > C),G:(A > B)]: (((((((comp @ (multiset @ B)) @ (multiset @ C)) @ (multiset @ A)) @ (((image_mset @ B) @ C) @ F)) @ (((image_mset @ A) @ B) @ G)) = (((image_mset @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F) @ G)))))).
% 9.43/3.45 thf(fact_4171_mset__map__id, axiom, (! [B:$tType,A:$tType,F:(B > A),G:(A > B),X10:(multiset @ A)]: (((! [X2:A]: (((F @ (G @ X2)) = X2))) => (((((image_mset @ B) @ A) @ F) @ ((((image_mset @ A) @ B) @ G) @ X10)) = X10))))).
% 9.43/3.45 thf(fact_4172_mset__union__2__elem, axiom, (! [A:$tType,A3:A,B2:A,C2:A,M3:(multiset @ A)]: ((((((add_mset @ A) @ A3) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) = (((add_mset @ A) @ C2) @ M3)) => ((((((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) = M3) & (B2 = C2)) | ((A3 = C2) & ((((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))) = M3))))))).
% 9.43/3.45 thf(fact_4173_wcount__add__mset, axiom, (! [A:$tType,F:(A > nat),X:A,M3:(multiset @ A),A3:A]: ((((((wcount @ A) @ F) @ (((add_mset @ A) @ X) @ M3)) @ A3) = (((plus_plus @ nat) @ ((((if @ nat) @ (X = A3)) @ (suc @ (F @ A3))) @ (zero_zero @ nat))) @ ((((wcount @ A) @ F) @ M3) @ A3)))))).
% 9.43/3.45 thf(fact_4174_subset__mset_Osum__list__update, axiom, (! [A:$tType,K:nat,Xs:(list @ (multiset @ A)),X:(multiset @ A)]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ (multiset @ A))) @ Xs)) => (((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ ((((list_update @ (multiset @ A)) @ Xs) @ K) @ X)) = (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ ((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Xs)) @ X)) @ (((nth @ (multiset @ A)) @ Xs) @ K))))))).
% 9.43/3.45 thf(fact_4175_size__multiset__add__mset, axiom, (! [A:$tType,F:(A > nat),A3:A,M3:(multiset @ A)]: (((((size_multiset @ A) @ F) @ (((add_mset @ A) @ A3) @ M3)) = (((plus_plus @ nat) @ (suc @ (F @ A3))) @ (((size_multiset @ A) @ F) @ M3)))))).
% 9.43/3.45 thf(fact_4176_subset__mset_Osum__eq__0__iff, axiom, (! [A:$tType,B:$tType,F6:(set @ B),F:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ F6) => ((((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ F6) = (zero_zero @ (multiset @ A))) = (! [X3:B]: (((((member @ B) @ X3) @ F6) => ((F @ X3) = (zero_zero @ (multiset @ A))))))))))).
% 9.43/3.45 thf(fact_4177_size__multiset__union, axiom, (! [A:$tType,F:(A > nat),M3:(multiset @ A),N6:(multiset @ A)]: (((((size_multiset @ A) @ F) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((plus_plus @ nat) @ (((size_multiset @ A) @ F) @ M3)) @ (((size_multiset @ A) @ F) @ N6)))))).
% 9.43/3.45 thf(fact_4178_subset__mset_Osum__list__eq__0__iff, axiom, (! [A:$tType,Ns:(list @ (multiset @ A))]: (((((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Ns) = (zero_zero @ (multiset @ A))) = (! [X3:(multiset @ A)]: (((((member @ (multiset @ A)) @ X3) @ ((set2 @ (multiset @ A)) @ Ns)) => (X3 = (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4179_wcount__union, axiom, (! [A:$tType,F:(A > nat),M3:(multiset @ A),N6:(multiset @ A),A3:A]: ((((((wcount @ A) @ F) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) @ A3) = (((plus_plus @ nat) @ ((((wcount @ A) @ F) @ M3) @ A3)) @ ((((wcount @ A) @ F) @ N6) @ A3)))))).
% 9.43/3.45 thf(fact_4180_subset__mset_Oelem__le__sum__list, axiom, (! [A:$tType,K:nat,Ns:(list @ (multiset @ A))]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ (multiset @ A))) @ Ns)) => (((subseteq_mset @ A) @ (((nth @ (multiset @ A)) @ Ns) @ K)) @ ((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Ns)))))).
% 9.43/3.45 thf(fact_4181_subset__mset_Osum__mono2, axiom, (! [A:$tType,B:$tType,B6:(set @ B),A4:(set @ B),F:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ B6) => ((((ord_less_eq @ (set @ B)) @ A4) @ B6) => ((! [B3:B]: (((((member @ B) @ B3) @ (((minus_minus @ (set @ B)) @ B6) @ A4)) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ B3))))) => (((subseteq_mset @ A) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ A4)) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ B6)))))))).
% 9.43/3.45 thf(fact_4182_next_Osimps, axiom, (! [V:code_natural,W:code_natural]: (((next @ ((((product_Pair @ code_natural) @ code_natural) @ V) @ W)) = ((((product_Pair @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ (((plus_plus @ code_natural) @ (((minus_shift @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ (((minus_shift @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ (((times_times @ code_natural) @ (((modulo_modulo @ code_natural) @ V) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ one2)))))))))))))))))) @ (((times_times @ code_natural) @ (((divide_divide @ code_natural) @ V) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ one2))))))))))))))))) @ (((plus_plus @ code_natural) @ (((minus_shift @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ (((times_times @ code_natural) @ (((modulo_modulo @ code_natural) @ W) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ one2)))))))))))))))))) @ (((times_times @ code_natural) @ (((divide_divide @ code_natural) @ W) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ one2))))))))))))))) @ (one_one @ code_natural)))) @ (one_one @ code_natural))) @ ((((product_Pair @ code_natural) @ code_natural) @ (((minus_shift @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ (((times_times @ code_natural) @ (((modulo_modulo @ code_natural) @ V) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ one2)))))))))))))))))) @ (((times_times @ code_natural) @ (((divide_divide @ code_natural) @ V) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ one2))))))))))))))))) @ (((minus_shift @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ (((times_times @ code_natural) @ (((modulo_modulo @ code_natural) @ W) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ one2)))))))))))))))))) @ (((times_times @ code_natural) @ (((divide_divide @ code_natural) @ W) @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ one2)))))))))))))))))) @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ one2)))))))))))))))))))).
% 9.43/3.45 thf(fact_4183_find__dropWhile, axiom, (! [A:$tType]: (((find @ A) = (^ [P4:(A > $o),Xs3:(list @ A)]: ((((((case_list @ (option @ A)) @ A) @ (none @ A)) @ (^ [X3:A,Xa4:(list @ A)]: (((some @ A) @ X3)))) @ (((dropWhile @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P4)) @ Xs3)))))))).
% 9.43/3.45 thf(fact_4184_mset__subset__eq__mono__add__right__cancel, axiom, (! [A:$tType,A4:(multiset @ A),C5:(multiset @ A),B6:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ C5)) @ (((plus_plus @ (multiset @ A)) @ B6) @ C5)) = (((subseteq_mset @ A) @ A4) @ B6))))).
% 9.43/3.45 thf(fact_4185_mset__subset__eq__mono__add__left__cancel, axiom, (! [A:$tType,C5:(multiset @ A),A4:(multiset @ A),B6:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C5) @ A4)) @ (((plus_plus @ (multiset @ A)) @ C5) @ B6)) = (((subseteq_mset @ A) @ A4) @ B6))))).
% 9.43/3.45 thf(fact_4186_subset__mset_Oadd__le__cancel__right, axiom, (! [A:$tType,A3:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) = (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4187_subset__mset_Oadd__le__cancel__left, axiom, (! [A:$tType,C2:(multiset @ A),A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) = (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4188_subset__mset_Ole__add__same__cancel2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ B2) @ A3)) = (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2))))).
% 9.43/3.45 thf(fact_4189_subset__mset_Ole__add__same__cancel1, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) = (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2))))).
% 9.43/3.45 thf(fact_4190_subset__mset_Oadd__le__same__cancel2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) @ B2) = (((subseteq_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))))).
% 9.43/3.45 thf(fact_4191_subset__mset_Oadd__le__same__cancel1, axiom, (! [A:$tType,B2:(multiset @ A),A3:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ B2) @ A3)) @ B2) = (((subseteq_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))))).
% 9.43/3.45 thf(fact_4192_mset__subset__eq__multiset__union__diff__commute, axiom, (! [A:$tType,B6:(multiset @ A),A4:(multiset @ A),C5:(multiset @ A)]: (((((subseteq_mset @ A) @ B6) @ A4) => ((((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ A4) @ B6)) @ C5) = (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ C5)) @ B6)))))).
% 9.43/3.45 thf(fact_4193_subset__mset_Oadd__diff__assoc2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) @ C2) = (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) @ A3)))))).
% 9.43/3.45 thf(fact_4194_subset__mset_Oadd__diff__assoc, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((plus_plus @ (multiset @ A)) @ C2) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) = (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) @ A3)))))).
% 9.43/3.45 thf(fact_4195_subset__mset_Odiff__add, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) @ A3) = B2))))).
% 9.43/3.45 thf(fact_4196_mset__le__add__mset__decr__left1, axiom, (! [A:$tType,C2:A,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ C2) @ A3)) @ B2) => (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4197_mset__le__incr__right2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => (((subseteq_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)))))).
% 9.43/3.45 thf(fact_4198_mset__le__incr__right1, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => (((subseteq_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)))))).
% 9.43/3.45 thf(fact_4199_mset__le__decr__left2, axiom, (! [A:$tType,C2:(multiset @ A),A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ B2) => (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4200_mset__le__decr__left1, axiom, (! [A:$tType,A3:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ B2) => (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4201_mset__union__subset, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ C5) => ((((subseteq_mset @ A) @ A4) @ C5) & (((subseteq_mset @ A) @ B6) @ C5)))))).
% 9.43/3.45 thf(fact_4202_mset__le__distrib, axiom, (! [A:$tType,X10:(multiset @ A),A4:(multiset @ A),B6:(multiset @ A)]: (((((subseteq_mset @ A) @ X10) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) => (~ ((! [Xa5:(multiset @ A),Xb4:(multiset @ A)]: (((X10 = (((plus_plus @ (multiset @ A)) @ Xa5) @ Xb4)) => ((((subseteq_mset @ A) @ Xa5) @ A4) => (~ ((((subseteq_mset @ A) @ Xb4) @ B6))))))))))))).
% 9.43/3.45 thf(fact_4203_mset__le__addE, axiom, (! [A:$tType,Xs:(multiset @ A),Ys2:(multiset @ A)]: (((((subseteq_mset @ A) @ Xs) @ Ys2) => (~ ((! [Zs2:(multiset @ A)]: ((Ys2 != (((plus_plus @ (multiset @ A)) @ Xs) @ Zs2)))))))))).
% 9.43/3.45 thf(fact_4204_mset__subset__eq__exists__conv, axiom, (! [A:$tType]: (((subseteq_mset @ A) = (^ [A9:(multiset @ A),B9:(multiset @ A)]: ((? [C7:(multiset @ A)]: ((B9 = (((plus_plus @ (multiset @ A)) @ A9) @ C7)))))))))).
% 9.43/3.45 thf(fact_4205_mset__subset__eq__add__right, axiom, (! [A:$tType,B6:(multiset @ A),A4:(multiset @ A)]: ((((subseteq_mset @ A) @ B6) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))))).
% 9.43/3.45 thf(fact_4206_mset__subset__eq__mono__add, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A),D4:(multiset @ A)]: (((((subseteq_mset @ A) @ A4) @ B6) => ((((subseteq_mset @ A) @ C5) @ D4) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ C5)) @ (((plus_plus @ (multiset @ A)) @ B6) @ D4))))))).
% 9.43/3.45 thf(fact_4207_mset__subset__eq__add__left, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A)]: ((((subseteq_mset @ A) @ A4) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))))).
% 9.43/3.45 thf(fact_4208_subset__mset_Oadd__le__imp__le__right, axiom, (! [A:$tType,A3:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) => (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4209_subset__mset_Oadd__le__imp__le__left, axiom, (! [A:$tType,C2:(multiset @ A),A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) => (((subseteq_mset @ A) @ A3) @ B2))))).
% 9.43/3.45 thf(fact_4210_subset__mset_Oadd__right__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)))))).
% 9.43/3.45 thf(fact_4211_subset__mset_Oadd__left__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)))))).
% 9.43/3.45 thf(fact_4212_subset__mset_Ole__iff__add, axiom, (! [A:$tType]: (((subseteq_mset @ A) = (^ [A7:(multiset @ A),B5:(multiset @ A)]: ((? [C3:(multiset @ A)]: ((B5 = (((plus_plus @ (multiset @ A)) @ A7) @ C3)))))))))).
% 9.43/3.45 thf(fact_4213_subset__mset_Oless__eqE, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => (~ ((! [C4:(multiset @ A)]: ((B2 != (((plus_plus @ (multiset @ A)) @ A3) @ C4)))))))))).
% 9.43/3.45 thf(fact_4214_subset__mset_Oadd__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A),D3:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((subseteq_mset @ A) @ C2) @ D3) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ D3))))))).
% 9.43/3.45 thf(fact_4215_full__exhaustive__natural_H_Ocases, axiom, (! [X:((product_prod @ (((product_prod @ code_natural) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural))]: ((~ ((! [F4:(((product_prod @ code_natural) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term)))),D2:code_natural,I2:code_natural]: ((X != ((((product_Pair @ (((product_prod @ code_natural) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural)) @ F4) @ ((((product_Pair @ code_natural) @ code_natural) @ D2) @ I2)))))))))).
% 9.43/3.45 thf(fact_4216_full__exhaustive__fun_H_Ocases, axiom, (! [A:$tType,B:$tType]: ((((quickc3360725361186068524ustive @ B) & ((cl_HOL_Oequal @ A) & (quickc3360725361186068524ustive @ A))) => (! [X:((product_prod @ (((product_prod @ (A > B)) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural))]: ((~ ((! [F4:(((product_prod @ (A > B)) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term)))),I2:code_natural,D2:code_natural]: ((X != ((((product_Pair @ (((product_prod @ (A > B)) @ (product_unit > code_term)) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural)) @ F4) @ ((((product_Pair @ code_natural) @ code_natural) @ I2) @ D2))))))))))))).
% 9.43/3.45 thf(fact_4217_exhaustive__natural_H_Ocases, axiom, (! [X:((product_prod @ (code_natural > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural))]: ((~ ((! [F4:(code_natural > (option @ ((product_prod @ $o) @ (list @ code_term)))),D2:code_natural,I2:code_natural]: ((X != ((((product_Pair @ (code_natural > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural)) @ F4) @ ((((product_Pair @ code_natural) @ code_natural) @ D2) @ I2)))))))))).
% 9.43/3.45 thf(fact_4218_exhaustive__fun_H_Ocases, axiom, (! [A:$tType,B:$tType]: ((((quickc658316121487927005ustive @ B) & ((cl_HOL_Oequal @ A) & (quickc658316121487927005ustive @ A))) => (! [X:((product_prod @ ((A > B) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural))]: ((~ ((! [F4:((A > B) > (option @ ((product_prod @ $o) @ (list @ code_term)))),I2:code_natural,D2:code_natural]: ((X != ((((product_Pair @ ((A > B) > (option @ ((product_prod @ $o) @ (list @ code_term))))) @ ((product_prod @ code_natural) @ code_natural)) @ F4) @ ((((product_Pair @ code_natural) @ code_natural) @ I2) @ D2))))))))))))).
% 9.43/3.45 thf(fact_4219_Lazy__Sequence_Oiterate__upto_Ocases, axiom, (! [A:$tType,X:((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural))]: ((~ ((! [F4:(code_natural > A),N3:code_natural,M5:code_natural]: ((X != ((((product_Pair @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) @ F4) @ ((((product_Pair @ code_natural) @ code_natural) @ N3) @ M5)))))))))).
% 9.43/3.45 thf(fact_4220_iter_Ocases, axiom, (! [A:$tType,X:((product_prod @ (((product_prod @ code_natural) @ code_natural) > ((product_prod @ ((product_prod @ A) @ (product_unit > code_term))) @ ((product_prod @ code_natural) @ code_natural)))) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural)))]: ((~ ((! [Random:(((product_prod @ code_natural) @ code_natural) > ((product_prod @ ((product_prod @ A) @ (product_unit > code_term))) @ ((product_prod @ code_natural) @ code_natural))),Nrandom:code_natural,Seed:((product_prod @ code_natural) @ code_natural)]: ((X != ((((product_Pair @ (((product_prod @ code_natural) @ code_natural) > ((product_prod @ ((product_prod @ A) @ (product_unit > code_term))) @ ((product_prod @ code_natural) @ code_natural)))) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural))) @ Random) @ ((((product_Pair @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ Nrandom) @ Seed)))))))))).
% 9.43/3.45 thf(fact_4221_log_Ocases, axiom, (! [X:((product_prod @ code_natural) @ code_natural)]: ((~ ((! [B3:code_natural,I2:code_natural]: ((X != ((((product_Pair @ code_natural) @ code_natural) @ B3) @ I2))))))))).
% 9.43/3.45 thf(fact_4222_mset__le__subtract, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: (((((subseteq_mset @ A) @ A4) @ B6) => (((subseteq_mset @ A) @ (((minus_minus @ (multiset @ A)) @ A4) @ C5)) @ (((minus_minus @ (multiset @ A)) @ B6) @ C5)))))).
% 9.43/3.45 thf(fact_4223_mset__le__add__mset__decr__left2, axiom, (! [A:$tType,C2:A,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ C2) @ A3)) @ B2) => (((subseteq_mset @ A) @ (((add_mset @ A) @ C2) @ (zero_zero @ (multiset @ A)))) @ B2))))).
% 9.43/3.45 thf(fact_4224_mset__le__single__cases, axiom, (! [A:$tType,M3:(multiset @ A),A3:A]: (((((subseteq_mset @ A) @ M3) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) => ((M3 != (zero_zero @ (multiset @ A))) => (M3 = (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))))))).
% 9.43/3.45 thf(fact_4225_mset__le__add__mset, axiom, (! [A:$tType,X:A,B6:(multiset @ A),C5:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ X) @ B6)) @ C5) => ((((subseteq_mset @ A) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))) @ C5) & (((subseteq_mset @ A) @ B6) @ C5)))))).
% 9.43/3.45 thf(fact_4226_subset__mset_Oadd__nonpos__eq__0__iff, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: (((((subseteq_mset @ A) @ X) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ Y) @ (zero_zero @ (multiset @ A))) => (((((plus_plus @ (multiset @ A)) @ X) @ Y) = (zero_zero @ (multiset @ A))) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4227_subset__mset_Oadd__nonneg__eq__0__iff, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ X) => ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ Y) => (((((plus_plus @ (multiset @ A)) @ X) @ Y) = (zero_zero @ (multiset @ A))) = ((X = (zero_zero @ (multiset @ A))) & (Y = (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4228_subset__mset_Oadd__nonpos__nonpos, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) @ (zero_zero @ (multiset @ A)))))))).
% 9.43/3.45 thf(fact_4229_subset__mset_Oadd__nonneg__nonneg, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2))))))).
% 9.43/3.45 thf(fact_4230_subset__mset_Oadd__increasing2, axiom, (! [A:$tType,C2:(multiset @ A),B2:(multiset @ A),A3:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ C2) => ((((subseteq_mset @ A) @ B2) @ A3) => (((subseteq_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2))))))).
% 9.43/3.45 thf(fact_4231_subset__mset_Oadd__decreasing2, axiom, (! [A:$tType,C2:(multiset @ A),A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ C2) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ A3) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ B2)))))).
% 9.43/3.45 thf(fact_4232_subset__mset_Oadd__increasing, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subseteq_mset @ A) @ B2) @ C2) => (((subseteq_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2))))))).
% 9.43/3.45 thf(fact_4233_subset__mset_Oadd__decreasing, axiom, (! [A:$tType,A3:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ C2) @ B2) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ B2)))))).
% 9.43/3.45 thf(fact_4234_mset__le__subtract__left, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),X10:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ X10) => ((((subseteq_mset @ A) @ B6) @ (((minus_minus @ (multiset @ A)) @ X10) @ A4)) & (((subseteq_mset @ A) @ A4) @ X10)))))).
% 9.43/3.45 thf(fact_4235_mset__le__subtract__right, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),X10:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ X10) => ((((subseteq_mset @ A) @ A4) @ (((minus_minus @ (multiset @ A)) @ X10) @ B6)) & (((subseteq_mset @ A) @ B6) @ X10)))))).
% 9.43/3.45 thf(fact_4236_multiset__diff__union__assoc, axiom, (! [A:$tType,C5:(multiset @ A),B6:(multiset @ A),A4:(multiset @ A)]: (((((subseteq_mset @ A) @ C5) @ B6) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ C5) = (((plus_plus @ (multiset @ A)) @ A4) @ (((minus_minus @ (multiset @ A)) @ B6) @ C5))))))).
% 9.43/3.45 thf(fact_4237_subset__eq__diff__conv, axiom, (! [A:$tType,A4:(multiset @ A),C5:(multiset @ A),B6:(multiset @ A)]: (((((subseteq_mset @ A) @ (((minus_minus @ (multiset @ A)) @ A4) @ C5)) @ B6) = (((subseteq_mset @ A) @ A4) @ (((plus_plus @ (multiset @ A)) @ B6) @ C5)))))).
% 9.43/3.45 thf(fact_4238_subset__mset_Ole__imp__diff__is__add, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((subseteq_mset @ A) @ A3) @ B2) => (((((minus_minus @ (multiset @ A)) @ B2) @ A3) = C2) = (B2 = (((plus_plus @ (multiset @ A)) @ C2) @ A3)))))))).
% 9.43/3.45 thf(fact_4239_subset__mset_Oadd__diff__inverse, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((plus_plus @ (multiset @ A)) @ A3) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) = B2))))).
% 9.43/3.45 thf(fact_4240_subset__mset_Odiff__diff__right, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((minus_minus @ (multiset @ A)) @ C2) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) = (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ B2)))))).
% 9.43/3.45 thf(fact_4241_subset__mset_Odiff__add__assoc2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) @ A3) = (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) @ C2)))))).
% 9.43/3.45 thf(fact_4242_subset__mset_Odiff__add__assoc, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) @ A3) = (((plus_plus @ (multiset @ A)) @ C2) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3))))))).
% 9.43/3.45 thf(fact_4243_subset__mset_Ole__diff__conv2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((subseteq_mset @ A) @ C2) @ (((minus_minus @ (multiset @ A)) @ B2) @ A3)) = (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ B2)))))).
% 9.43/3.45 thf(fact_4244_subset__mset_Ole__add__diff, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => (((subseteq_mset @ A) @ C2) @ (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) @ A3)))))).
% 9.43/3.45 thf(fact_4245_mset__map__split__orig__le, axiom, (! [B:$tType,A:$tType,F:(B > A),P:(multiset @ B),M13:(multiset @ A),M24:(multiset @ A)]: (((((subseteq_mset @ A) @ ((((image_mset @ B) @ A) @ F) @ P)) @ (((plus_plus @ (multiset @ A)) @ M13) @ M24)) => (~ ((! [P1:(multiset @ B),P22:(multiset @ B)]: (((P = (((plus_plus @ (multiset @ B)) @ P1) @ P22)) => ((((subseteq_mset @ A) @ ((((image_mset @ B) @ A) @ F) @ P1)) @ M13) => (~ ((((subseteq_mset @ A) @ ((((image_mset @ B) @ A) @ F) @ P22)) @ M24))))))))))))).
% 9.43/3.45 thf(fact_4246_mset__le__subtract__add__mset__right, axiom, (! [A:$tType,X:A,B6:(multiset @ A),X10:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ X) @ B6)) @ X10) => ((((subseteq_mset @ A) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ X10) @ B6)) & (((subseteq_mset @ A) @ B6) @ X10)))))).
% 9.43/3.45 thf(fact_4247_mset__le__subtract__add__mset__left, axiom, (! [A:$tType,X:A,B6:(multiset @ A),X10:(multiset @ A)]: (((((subseteq_mset @ A) @ (((add_mset @ A) @ X) @ B6)) @ X10) => ((((subseteq_mset @ A) @ B6) @ (((minus_minus @ (multiset @ A)) @ X10) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A))))) & (((subseteq_mset @ A) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))) @ X10)))))).
% 9.43/3.45 thf(fact_4248_subset__mset_Osum__nonneg, axiom, (! [A:$tType,B:$tType,A4:(set @ B),F:(B > (multiset @ A))]: (((! [X2:B]: (((((member @ B) @ X2) @ A4) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ X2))))) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ A4)))))).
% 9.43/3.45 thf(fact_4249_subset__mset_Osum__nonpos, axiom, (! [B:$tType,A:$tType,A4:(set @ B),F:(B > (multiset @ A))]: (((! [X2:B]: (((((member @ B) @ X2) @ A4) => (((subseteq_mset @ A) @ (F @ X2)) @ (zero_zero @ (multiset @ A)))))) => (((subseteq_mset @ A) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ A4)) @ (zero_zero @ (multiset @ A))))))).
% 9.43/3.45 thf(fact_4250_subset__mset_Osum__list__nonneg, axiom, (! [A:$tType,Xs:(list @ (multiset @ A))]: (((! [X2:(multiset @ A)]: (((((member @ (multiset @ A)) @ X2) @ ((set2 @ (multiset @ A)) @ Xs)) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ X2)))) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ ((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Xs)))))).
% 9.43/3.45 thf(fact_4251_subset__mset_Osum__list__nonpos, axiom, (! [A:$tType,Xs:(list @ (multiset @ A))]: (((! [X2:(multiset @ A)]: (((((member @ (multiset @ A)) @ X2) @ ((set2 @ (multiset @ A)) @ Xs)) => (((subseteq_mset @ A) @ X2) @ (zero_zero @ (multiset @ A)))))) => (((subseteq_mset @ A) @ ((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Xs)) @ (zero_zero @ (multiset @ A))))))).
% 9.43/3.45 thf(fact_4252_subset__mset_Omember__le__sum__list, axiom, (! [A:$tType,X:(multiset @ A),Xs:(list @ (multiset @ A))]: (((((member @ (multiset @ A)) @ X) @ ((set2 @ (multiset @ A)) @ Xs)) => (((subseteq_mset @ A) @ X) @ ((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Xs)))))).
% 9.43/3.45 thf(fact_4253_subset__mset_Osum__list__nonneg__eq__0__iff, axiom, (! [A:$tType,Xs:(list @ (multiset @ A))]: (((! [X2:(multiset @ A)]: (((((member @ (multiset @ A)) @ X2) @ ((set2 @ (multiset @ A)) @ Xs)) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ X2)))) => ((((((groups4543113879258116180m_list @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ Xs) = (zero_zero @ (multiset @ A))) = (! [X3:(multiset @ A)]: (((((member @ (multiset @ A)) @ X3) @ ((set2 @ (multiset @ A)) @ Xs)) => (X3 = (zero_zero @ (multiset @ A))))))))))).
% 9.43/3.45 thf(fact_4254_subset__mset_Osum__mono, axiom, (! [A:$tType,B:$tType,K2:(set @ B),F:(B > (multiset @ A)),G:(B > (multiset @ A))]: (((! [I2:B]: (((((member @ B) @ I2) @ K2) => (((subseteq_mset @ A) @ (F @ I2)) @ (G @ I2))))) => (((subseteq_mset @ A) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ K2)) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ G) @ K2)))))).
% 9.43/3.45 thf(fact_4255_subset__mset_Osum__le__included, axiom, (! [B:$tType,A:$tType,C:$tType,S:(set @ B),T2:(set @ C),G:(C > (multiset @ A)),I:(C > B),F:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ S) => (((finite_finite2 @ C) @ T2) => ((! [X2:C]: (((((member @ C) @ X2) @ T2) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (G @ X2))))) => ((! [X2:B]: (((((member @ B) @ X2) @ S) => (? [Xa2:C]: (((((member @ C) @ Xa2) @ T2) & (((I @ Xa2) = X2) & (((subseteq_mset @ A) @ (F @ X2)) @ (G @ Xa2))))))))) => (((subseteq_mset @ A) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ S)) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ C) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ G) @ T2))))))))).
% 9.43/3.45 thf(fact_4256_subset__mset_Osum__nonneg__eq__0__iff, axiom, (! [A:$tType,B:$tType,A4:(set @ B),F:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ A4) => ((! [X2:B]: (((((member @ B) @ X2) @ A4) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ X2))))) => ((((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ A4) = (zero_zero @ (multiset @ A))) = (! [X3:B]: (((((member @ B) @ X3) @ A4) => ((F @ X3) = (zero_zero @ (multiset @ A)))))))))))).
% 9.43/3.45 thf(fact_4257_subset__mset_Osum__nonneg__0, axiom, (! [B:$tType,A:$tType,S:(set @ B),F:(B > (multiset @ A)),I:B]: ((((finite_finite2 @ B) @ S) => ((! [I2:B]: (((((member @ B) @ I2) @ S) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ I2))))) => ((((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ S) = (zero_zero @ (multiset @ A))) => ((((member @ B) @ I) @ S) => ((F @ I) = (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4258_subset__mset_Osum__nonneg__leq__bound, axiom, (! [B:$tType,A:$tType,S:(set @ B),F:(B > (multiset @ A)),B6:(multiset @ A),I:B]: ((((finite_finite2 @ B) @ S) => ((! [I2:B]: (((((member @ B) @ I2) @ S) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ I2))))) => ((((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ S) = B6) => ((((member @ B) @ I) @ S) => (((subseteq_mset @ A) @ (F @ I)) @ B6)))))))).
% 9.43/3.45 thf(fact_4259_minus__shift__def, axiom, (minus_shift = (^ [R3:code_natural,K4:code_natural,L3:code_natural]: (((((if @ code_natural) @ (((ord_less @ code_natural) @ K4) @ L3)) @ (((minus_minus @ code_natural) @ (((plus_plus @ code_natural) @ R3) @ K4)) @ L3)) @ (((minus_minus @ code_natural) @ K4) @ L3)))))).
% 9.43/3.45 thf(fact_4260_split__seed__def, axiom, (split_seed = (^ [S4:((product_prod @ code_natural) @ code_natural)]: ((((((product_case_prod @ code_natural) @ code_natural) @ ((product_prod @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ code_natural))) @ (^ [V2:code_natural,W2:code_natural]: ((((((product_case_prod @ code_natural) @ code_natural) @ ((product_prod @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ code_natural))) @ (^ [V5:code_natural,W3:code_natural]: (((((product_Pair @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ code_natural)) @ ((((product_Pair @ code_natural) @ code_natural) @ ((inc_shift @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ V2)) @ W3)) @ ((((product_Pair @ code_natural) @ code_natural) @ V5) @ ((inc_shift @ ((numeral_numeral @ code_natural) @ (bit0 @ (bit1 @ (bit1 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ W2)))))) @ (((product_snd @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ (next @ S4)))))) @ S4))))).
% 9.43/3.45 thf(fact_4261_log_Osimps, axiom, (log = (^ [B5:code_natural,I3:code_natural]: (((((if @ code_natural) @ ((((ord_less_eq @ code_natural) @ B5) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ I3) @ B5))) @ (one_one @ code_natural)) @ (((plus_plus @ code_natural) @ (one_one @ code_natural)) @ ((log @ B5) @ (((divide_divide @ code_natural) @ I3) @ B5)))))))).
% 9.43/3.45 thf(fact_4262_log_Oelims, axiom, (! [X:code_natural,Xa:code_natural,Y:code_natural]: (((((log @ X) @ Xa) = Y) => ((((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa) @ X)) => (Y = (one_one @ code_natural))) & ((~ (((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa) @ X)))) => (Y = (((plus_plus @ code_natural) @ (one_one @ code_natural)) @ ((log @ X) @ (((divide_divide @ code_natural) @ Xa) @ X)))))))))).
% 9.43/3.45 thf(fact_4263_extract__def, axiom, (! [A:$tType]: (((extract @ A) = (^ [P4:(A > $o),Xs3:(list @ A)]: ((((((case_list @ (option @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ A) @ (none @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A))))) @ (^ [Y3:A,Ys:(list @ A)]: (((some @ ((product_prod @ (list @ A)) @ ((product_prod @ A) @ (list @ A)))) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ A) @ (list @ A))) @ (((takeWhile @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P4)) @ Xs3)) @ ((((product_Pair @ A) @ (list @ A)) @ Y3) @ Ys)))))) @ (((dropWhile @ A) @ (((((comp @ $o) @ $o) @ A) @ (~)) @ P4)) @ Xs3)))))))).
% 9.43/3.45 thf(fact_4264_drop__takeWhile, axiom, (! [A:$tType,I:nat,P:(A > $o),L:(list @ A)]: (((((ord_less_eq @ nat) @ I) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ L))) => ((((drop @ A) @ I) @ (((takeWhile @ A) @ P) @ L)) = (((takeWhile @ A) @ P) @ (((drop @ A) @ I) @ L))))))).
% 9.43/3.45 thf(fact_4265_eq__len__takeWhile__conv, axiom, (! [A:$tType,I:nat,P:(A > $o),L:(list @ A)]: (((I = ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ L))) = ((((ord_less_eq @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) & ((! [J2:nat]: (((((ord_less @ nat) @ J2) @ I) => (P @ (((nth @ A) @ L) @ J2))))) & ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) => (~ ((P @ (((nth @ A) @ L) @ I))))))))))).
% 9.43/3.45 thf(fact_4266_less__length__takeWhile__conv, axiom, (! [A:$tType,I:nat,P:(A > $o),L:(list @ A)]: (((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ L))) = ((((ord_less @ nat) @ I) @ ((size_size @ (list @ A)) @ L)) & (! [J2:nat]: (((((ord_less_eq @ nat) @ J2) @ I) => (P @ (((nth @ A) @ L) @ J2)))))))))).
% 9.43/3.45 thf(fact_4267_length__dropWhile__takeWhile, axiom, (! [A:$tType,X:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ X) @ ((size_size @ (list @ A)) @ (((dropWhile @ A) @ P) @ Xs))) => (((ord_less @ nat) @ (((plus_plus @ nat) @ X) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs)))) @ ((size_size @ (list @ A)) @ Xs)))))).
% 9.43/3.45 thf(fact_4268_dropWhile__nth, axiom, (! [A:$tType,J:nat,P:(A > $o),Xs:(list @ A)]: (((((ord_less @ nat) @ J) @ ((size_size @ (list @ A)) @ (((dropWhile @ A) @ P) @ Xs))) => ((((nth @ A) @ (((dropWhile @ A) @ P) @ Xs)) @ J) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ J) @ ((size_size @ (list @ A)) @ (((takeWhile @ A) @ P) @ Xs))))))))).
% 9.43/3.45 thf(fact_4269_inc__shift__def, axiom, (inc_shift = (^ [V2:code_natural,K4:code_natural]: (((((if @ code_natural) @ (V2 = K4)) @ (one_one @ code_natural)) @ (((plus_plus @ code_natural) @ K4) @ (one_one @ code_natural))))))).
% 9.43/3.45 thf(fact_4270_log_Opelims, axiom, (! [X:code_natural,Xa:code_natural,Y:code_natural]: (((((log @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ code_natural) @ code_natural)) @ log_rel) @ ((((product_Pair @ code_natural) @ code_natural) @ X) @ Xa)) => (~ ((((((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa) @ X)) => (Y = (one_one @ code_natural))) & ((~ (((((ord_less_eq @ code_natural) @ X) @ (one_one @ code_natural)) | (((ord_less @ code_natural) @ Xa) @ X)))) => (Y = (((plus_plus @ code_natural) @ (one_one @ code_natural)) @ ((log @ X) @ (((divide_divide @ code_natural) @ Xa) @ X)))))) => (~ ((((accp @ ((product_prod @ code_natural) @ code_natural)) @ log_rel) @ ((((product_Pair @ code_natural) @ code_natural) @ X) @ Xa)))))))))))).
% 9.43/3.45 thf(fact_4271_Random_Orange__def, axiom, (range = (^ [K4:code_natural]: (((((((product_scomp @ ((product_prod @ code_natural) @ code_natural)) @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural))) @ (((((iterate @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ ((log @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))) @ K4)) @ (^ [L3:code_natural]: (((((((product_scomp @ ((product_prod @ code_natural) @ code_natural)) @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural))) @ next) @ (^ [V2:code_natural]: ((((product_Pair @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ (((plus_plus @ code_natural) @ V2) @ (((times_times @ code_natural) @ L3) @ ((numeral_numeral @ code_natural) @ (bit1 @ (bit0 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit0 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ (bit1 @ one2)))))))))))))))))))))))))))))))))))))))) @ (one_one @ code_natural))) @ (^ [V2:code_natural]: ((((product_Pair @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ (((modulo_modulo @ code_natural) @ V2) @ K4))))))))).
% 9.43/3.45 thf(fact_4272_Predicate_Oiterate__upto_Opinduct, axiom, (! [A:$tType,A0:(code_natural > A),A1:code_natural,A22:code_natural,P:((code_natural > A) > (code_natural > (code_natural > $o)))]: (((((accp @ ((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural))) @ (iterate_upto_rel @ A)) @ ((((product_Pair @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) @ A0) @ ((((product_Pair @ code_natural) @ code_natural) @ A1) @ A22))) => ((! [F4:(code_natural > A),N3:code_natural,M5:code_natural]: (((((accp @ ((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural))) @ (iterate_upto_rel @ A)) @ ((((product_Pair @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) @ F4) @ ((((product_Pair @ code_natural) @ code_natural) @ N3) @ M5))) => ((! [X7:product_unit]: (((~ ((((ord_less @ code_natural) @ M5) @ N3))) => (((P @ F4) @ (((plus_plus @ code_natural) @ N3) @ (one_one @ code_natural))) @ M5)))) => (((P @ F4) @ N3) @ M5))))) => (((P @ A0) @ A1) @ A22)))))).
% 9.43/3.45 thf(fact_4273_mset__size2elem, axiom, (! [A:$tType,P:(multiset @ A),Q3:A,Q4:A]: (((((ord_less_eq @ nat) @ ((size_size @ (multiset @ A)) @ P)) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) => ((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ Q3) @ (zero_zero @ (multiset @ A)))) @ (((add_mset @ A) @ Q4) @ (zero_zero @ (multiset @ A))))) @ P) => (P = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ Q3) @ (zero_zero @ (multiset @ A)))) @ (((add_mset @ A) @ Q4) @ (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4274_scomp__apply, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType]: ((((((product_scomp @ B) @ C) @ D) @ A) = (^ [F2:(B > ((product_prod @ C) @ D)),G2:(C > (D > A)),X3:B]: ((((((product_case_prod @ C) @ D) @ A) @ G2) @ (F2 @ X3)))))))).
% 9.43/3.45 thf(fact_4275_size__union, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A)]: ((((size_size @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((plus_plus @ nat) @ ((size_size @ (multiset @ A)) @ M3)) @ ((size_size @ (multiset @ A)) @ N6)))))).
% 9.43/3.45 thf(fact_4276_scomp__scomp, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F5:$tType,E:$tType,F:(A > ((product_prod @ E) @ F5)),G:(E > (F5 > ((product_prod @ C) @ D))),H3:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ ((((((product_scomp @ A) @ E) @ F5) @ ((product_prod @ C) @ D)) @ F) @ G)) @ H3) = ((((((product_scomp @ A) @ E) @ F5) @ B) @ F) @ (^ [X3:E]: (((((((product_scomp @ F5) @ C) @ D) @ B) @ (G @ X3)) @ H3)))))))).
% 9.43/3.45 thf(fact_4277_scomp__Pair, axiom, (! [C:$tType,B:$tType,A:$tType,X:(A > ((product_prod @ B) @ C))]: ((((((((product_scomp @ A) @ B) @ C) @ ((product_prod @ B) @ C)) @ X) @ ((product_Pair @ B) @ C)) = X)))).
% 9.43/3.45 thf(fact_4278_Pair__scomp, axiom, (! [A:$tType,B:$tType,C:$tType,X:C,F:(C > (A > B))]: ((((((((product_scomp @ A) @ C) @ A) @ B) @ (((product_Pair @ C) @ A) @ X)) @ F) = (F @ X))))).
% 9.43/3.45 thf(fact_4279_scomp__def, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F2:(A > ((product_prod @ B) @ C)),G2:(B > (C > D)),X3:A]: ((((((product_case_prod @ B) @ C) @ D) @ G2) @ (F2 @ X3)))))))).
% 9.43/3.45 thf(fact_4280_size__1__singleton__mset, axiom, (! [A:$tType,M3:(multiset @ A)]: (((((size_size @ (multiset @ A)) @ M3) = (one_one @ nat)) => (? [A5:A]: ((M3 = (((add_mset @ A) @ A5) @ (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4281_size__single, axiom, (! [A:$tType,B2:A]: ((((size_size @ (multiset @ A)) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) = (one_one @ nat))))).
% 9.43/3.45 thf(fact_4282_iterate_Oelims, axiom, (! [A:$tType,B:$tType,X:code_natural,Xa:(B > (A > ((product_prod @ B) @ A))),Xb:B,Y:(A > ((product_prod @ B) @ A))]: ((((((((iterate @ B) @ A) @ X) @ Xa) @ Xb) = Y) => (((X = (zero_zero @ code_natural)) => (Y = (((product_Pair @ B) @ A) @ Xb))) & ((X != (zero_zero @ code_natural)) => (Y = ((((((product_scomp @ A) @ B) @ A) @ ((product_prod @ B) @ A)) @ (Xa @ Xb)) @ ((((iterate @ B) @ A) @ (((minus_minus @ code_natural) @ X) @ (one_one @ code_natural))) @ Xa))))))))).
% 9.43/3.45 thf(fact_4283_iterate_Osimps, axiom, (! [B:$tType,A:$tType]: ((((iterate @ B) @ A) = (^ [K4:code_natural,F2:(B > (A > ((product_prod @ B) @ A))),X3:B]: (((((if @ (A > ((product_prod @ B) @ A))) @ (K4 = (zero_zero @ code_natural))) @ (((product_Pair @ B) @ A) @ X3)) @ ((((((product_scomp @ A) @ B) @ A) @ ((product_prod @ B) @ A)) @ (F2 @ X3)) @ ((((iterate @ B) @ A) @ (((minus_minus @ code_natural) @ K4) @ (one_one @ code_natural))) @ F2))))))))).
% 9.43/3.45 thf(fact_4284_size__mset__SucE, axiom, (! [A:$tType,A4:(multiset @ A),N:nat]: (((((size_size @ (multiset @ A)) @ A4) = (suc @ N)) => (~ ((! [A5:A,B10:(multiset @ A)]: (((A4 = (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A5) @ (zero_zero @ (multiset @ A)))) @ B10)) => (((size_size @ (multiset @ A)) @ B10) != N)))))))))).
% 9.43/3.45 thf(fact_4285_scomp__unfold, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType]: ((((((product_scomp @ A) @ B) @ C) @ D) = (^ [F2:(A > ((product_prod @ B) @ C)),G2:(B > (C > D)),X3:A]: (((G2 @ (((product_fst @ B) @ C) @ (F2 @ X3))) @ (((product_snd @ B) @ C) @ (F2 @ X3))))))))).
% 9.43/3.45 thf(fact_4286_mset__size__le1__cases, axiom, (! [A:$tType,M3:(multiset @ A)]: (((((ord_less_eq @ nat) @ ((size_size @ (multiset @ A)) @ M3)) @ (suc @ (zero_zero @ nat))) => ((M3 != (zero_zero @ (multiset @ A))) => (~ ((! [M5:A]: ((M3 != (((add_mset @ A) @ M5) @ (zero_zero @ (multiset @ A))))))))))))).
% 9.43/3.45 thf(fact_4287_iterate_Opelims, axiom, (! [A:$tType,B:$tType,X:code_natural,Xa:(B > (A > ((product_prod @ B) @ A))),Xb:B,Y:(A > ((product_prod @ B) @ A))]: ((((((((iterate @ B) @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B))) @ ((iterate_rel @ B) @ A)) @ ((((product_Pair @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B)) @ X) @ ((((product_Pair @ (B > (A > ((product_prod @ B) @ A)))) @ B) @ Xa) @ Xb))) => (~ (((((X = (zero_zero @ code_natural)) => (Y = (((product_Pair @ B) @ A) @ Xb))) & ((X != (zero_zero @ code_natural)) => (Y = ((((((product_scomp @ A) @ B) @ A) @ ((product_prod @ B) @ A)) @ (Xa @ Xb)) @ ((((iterate @ B) @ A) @ (((minus_minus @ code_natural) @ X) @ (one_one @ code_natural))) @ Xa))))) => (~ ((((accp @ ((product_prod @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B))) @ ((iterate_rel @ B) @ A)) @ ((((product_Pair @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B)) @ X) @ ((((product_Pair @ (B > (A > ((product_prod @ B) @ A)))) @ B) @ Xa) @ Xb))))))))))))).
% 9.43/3.45 thf(fact_4288_size__Diff__singleton__if, axiom, (! [A:$tType,X:A,A4:(multiset @ A)]: ((((((member @ A) @ X) @ ((set_mset @ A) @ A4)) => (((size_size @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ A4) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A))))) = (((minus_minus @ nat) @ ((size_size @ (multiset @ A)) @ A4)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X) @ ((set_mset @ A) @ A4)))) => (((size_size @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ A4) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A))))) = ((size_size @ (multiset @ A)) @ A4))))))).
% 9.43/3.45 thf(fact_4289_size__Diff__singleton, axiom, (! [A:$tType,X:A,M3:(multiset @ A)]: (((((member @ A) @ X) @ ((set_mset @ A) @ M3)) => (((size_size @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ M3) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A))))) = (((minus_minus @ nat) @ ((size_size @ (multiset @ A)) @ M3)) @ (one_one @ nat))))))).
% 9.43/3.45 thf(fact_4290_size__diff__se, axiom, (! [A:$tType,T2:A,S2:(multiset @ A)]: (((((member @ A) @ T2) @ ((set_mset @ A) @ S2)) => (((size_size @ (multiset @ A)) @ S2) = (((plus_plus @ nat) @ ((size_size @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ S2) @ (((add_mset @ A) @ T2) @ (zero_zero @ (multiset @ A)))))) @ (one_one @ nat))))))).
% 9.43/3.45 thf(fact_4291_set__mset__union, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A)]: ((((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((sup_sup @ (set @ A)) @ ((set_mset @ A) @ M3)) @ ((set_mset @ A) @ N6)))))).
% 9.43/3.45 thf(fact_4292_mset__diff__cancel1elem, axiom, (! [A:$tType,A3:A,B6:(multiset @ A)]: (((~ ((((member @ A) @ A3) @ ((set_mset @ A) @ B6)))) => ((((minus_minus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) @ B6) = (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))))))).
% 9.43/3.45 thf(fact_4293_ex__Melem__conv, axiom, (! [A:$tType,A4:(multiset @ A)]: (((? [X3:A]: ((((member @ A) @ X3) @ ((set_mset @ A) @ A4)))) = (A4 != (zero_zero @ (multiset @ A))))))).
% 9.43/3.45 thf(fact_4294_union__iff, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B6:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))) = ((((member @ A) @ A3) @ ((set_mset @ A) @ A4)) | (((member @ A) @ A3) @ ((set_mset @ A) @ B6))))))).
% 9.43/3.45 thf(fact_4295_mset__un__cases, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B6:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))) => ((~ ((((member @ A) @ A3) @ ((set_mset @ A) @ A4)))) => (((member @ A) @ A3) @ ((set_mset @ A) @ B6))))))).
% 9.43/3.45 thf(fact_4296_mset__left__cancel__union, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B6:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))) => ((~ ((((member @ A) @ A3) @ ((set_mset @ A) @ A4)))) => (((member @ A) @ A3) @ ((set_mset @ A) @ B6))))))).
% 9.43/3.45 thf(fact_4297_mset__right__cancel__union, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B6:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))) => ((~ ((((member @ A) @ A3) @ ((set_mset @ A) @ B6)))) => (((member @ A) @ A3) @ ((set_mset @ A) @ A4))))))).
% 9.43/3.45 thf(fact_4298_iterate_Ocases, axiom, (! [A:$tType,B:$tType,X:((product_prod @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B))]: ((~ ((! [K5:code_natural,F4:(B > (A > ((product_prod @ B) @ A))),X2:B]: ((X != ((((product_Pair @ code_natural) @ ((product_prod @ (B > (A > ((product_prod @ B) @ A)))) @ B)) @ K5) @ ((((product_Pair @ (B > (A > ((product_prod @ B) @ A)))) @ B) @ F4) @ X2)))))))))).
% 9.43/3.45 thf(fact_4299_image__mset__cong__pair, axiom, (! [C:$tType,B:$tType,A:$tType,M3:(multiset @ ((product_prod @ A) @ B)),F:(A > (B > C)),G:(A > (B > C))]: (((! [X2:A,Y4:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ ((set_mset @ ((product_prod @ A) @ B)) @ M3)) => (((F @ X2) @ Y4) = ((G @ X2) @ Y4))))) => (((((image_mset @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ F)) @ M3) = ((((image_mset @ ((product_prod @ A) @ B)) @ C) @ ((((product_case_prod @ A) @ B) @ C) @ G)) @ M3)))))).
% 9.43/3.45 thf(fact_4300_multi__member__skip, axiom, (! [A:$tType,X:A,XS:(multiset @ A),Y:A]: (((((member @ A) @ X) @ ((set_mset @ A) @ XS)) => (((member @ A) @ X) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ Y) @ (zero_zero @ (multiset @ A)))) @ XS))))))).
% 9.43/3.45 thf(fact_4301_multi__member__this, axiom, (! [A:$tType,X:A,XS:(multiset @ A)]: ((((member @ A) @ X) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))) @ XS)))))).
% 9.43/3.45 thf(fact_4302_mset__right__cancel__elem, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B2:A]: (((((member @ A) @ A3) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))))) => ((A3 != B2) => (((member @ A) @ A3) @ ((set_mset @ A) @ A4))))))).
% 9.43/3.45 thf(fact_4303_mset__left__cancel__elem, axiom, (! [A:$tType,A3:A,B2:A,A4:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) @ A4))) => ((A3 != B2) => (((member @ A) @ A3) @ ((set_mset @ A) @ A4))))))).
% 9.43/3.45 thf(fact_4304_mset__le__mono__add__single, axiom, (! [A:$tType,A3:A,Ys2:(multiset @ A),B2:A,Ws:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ Ys2)) => ((((member @ A) @ B2) @ ((set_mset @ A) @ Ws)) => (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))))) @ (((plus_plus @ (multiset @ A)) @ Ys2) @ Ws))))))).
% 9.43/3.45 thf(fact_4305_mset__union__subset__s, axiom, (! [A:$tType,A3:A,B6:(multiset @ A),C5:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) @ B6)) @ C5) => ((((member @ A) @ A3) @ ((set_mset @ A) @ C5)) & (((subseteq_mset @ A) @ B6) @ C5)))))).
% 9.43/3.45 thf(fact_4306_mset__2dist2__cases, axiom, (! [A:$tType,A3:A,B2:A,A4:(multiset @ A),B6:(multiset @ A)]: (((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))))) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) => ((~ ((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))))) @ A4))) => ((~ ((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))))) @ B6))) => (((((member @ A) @ A3) @ ((set_mset @ A) @ A4)) => (~ ((((member @ A) @ B2) @ ((set_mset @ A) @ B6))))) => (~ (((((member @ A) @ A3) @ ((set_mset @ A) @ B6)) => (~ ((((member @ A) @ B2) @ ((set_mset @ A) @ A4)))))))))))))).
% 9.43/3.45 thf(fact_4307_insert__DiffM2, axiom, (! [A:$tType,X:A,M3:(multiset @ A)]: (((((member @ A) @ X) @ ((set_mset @ A) @ M3)) => ((((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ M3) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A))))) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))) = M3))))).
% 9.43/3.45 thf(fact_4308_diff__union__single__conv, axiom, (! [A:$tType,A3:A,J4:(multiset @ A),I4:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ J4)) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ I4) @ J4)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) = (((plus_plus @ (multiset @ A)) @ I4) @ (((minus_minus @ (multiset @ A)) @ J4) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4309_mset__un__single__un__cases, axiom, (! [A:$tType,A3:A,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: ((((((add_mset @ A) @ A3) @ A4) = (((plus_plus @ (multiset @ A)) @ B6) @ C5)) => (((((member @ A) @ A3) @ ((set_mset @ A) @ B6)) => (A4 != (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ B6) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))) @ C5))) => (~ (((((member @ A) @ A3) @ ((set_mset @ A) @ C5)) => (A4 != (((plus_plus @ (multiset @ A)) @ B6) @ (((minus_minus @ (multiset @ A)) @ C5) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))))))))))))).
% 9.43/3.45 thf(fact_4310_diff__union__single__conv2, axiom, (! [A:$tType,A3:A,J4:(multiset @ A),I4:(multiset @ A)]: (((((member @ A) @ A3) @ ((set_mset @ A) @ J4)) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ J4) @ I4)) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A)))) = (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ J4) @ (((add_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))) @ I4)))))).
% 9.43/3.45 thf(fact_4311_mset__union__diff__comm, axiom, (! [A:$tType,T2:A,S2:(multiset @ A),T6:(multiset @ A)]: (((((member @ A) @ T2) @ ((set_mset @ A) @ S2)) => ((((plus_plus @ (multiset @ A)) @ T6) @ (((minus_minus @ (multiset @ A)) @ S2) @ (((add_mset @ A) @ T2) @ (zero_zero @ (multiset @ A))))) = (((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ T6) @ S2)) @ (((add_mset @ A) @ T2) @ (zero_zero @ (multiset @ A))))))))).
% 9.43/3.45 thf(fact_4312_mset__contains__eq, axiom, (! [A:$tType,M2:A,M3:(multiset @ A)]: (((((member @ A) @ M2) @ ((set_mset @ A) @ M3)) = ((((plus_plus @ (multiset @ A)) @ (((add_mset @ A) @ M2) @ (zero_zero @ (multiset @ A)))) @ (((minus_minus @ (multiset @ A)) @ M3) @ (((add_mset @ A) @ M2) @ (zero_zero @ (multiset @ A))))) = M3))))).
% 9.43/3.45 thf(fact_4313_mset__size1elem, axiom, (! [A:$tType,P:(multiset @ A),Q3:A]: (((((ord_less_eq @ nat) @ ((size_size @ (multiset @ A)) @ P)) @ (one_one @ nat)) => ((((member @ A) @ Q3) @ ((set_mset @ A) @ P)) => (P = (((add_mset @ A) @ Q3) @ (zero_zero @ (multiset @ A))))))))).
% 9.43/3.45 thf(fact_4314_subset__mset_Osum__mset__mono, axiom, (! [A:$tType,B:$tType,K2:(multiset @ B),F:(B > (multiset @ A)),G:(B > (multiset @ A))]: (((! [I2:B]: (((((member @ B) @ I2) @ ((set_mset @ B) @ K2)) => (((subseteq_mset @ A) @ (F @ I2)) @ (G @ I2))))) => (((subseteq_mset @ A) @ ((((comm_monoid_sum_mset @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ ((((image_mset @ B) @ (multiset @ A)) @ F) @ K2))) @ ((((comm_monoid_sum_mset @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ ((((image_mset @ B) @ (multiset @ A)) @ G) @ K2))))))).
% 9.43/3.45 thf(fact_4315_subset__mset_Osum__mset__0__iff, axiom, (! [A:$tType,M3:(multiset @ (multiset @ A))]: (((((((comm_monoid_sum_mset @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ M3) = (zero_zero @ (multiset @ A))) = (! [X3:(multiset @ A)]: (((((member @ (multiset @ A)) @ X3) @ ((set_mset @ (multiset @ A)) @ M3)) => (X3 = (zero_zero @ (multiset @ A)))))))))).
% 9.43/3.45 thf(fact_4316_wmsI, axiom, (! [A4:(multiset @ ((product_prod @ nat) @ nat)),B6:(multiset @ ((product_prod @ nat) @ nat)),Z9:(multiset @ ((product_prod @ nat) @ nat))]: ((((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ A4)) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ B6))) @ fun_max_strict) | ((A4 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))) & (B6 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))))) => (((member @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ A4)) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ B6))) @ ms_weak))))).
% 9.43/3.45 thf(fact_4317_smsI, axiom, (! [A4:(multiset @ ((product_prod @ nat) @ nat)),B6:(multiset @ ((product_prod @ nat) @ nat)),Z9:(multiset @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ A4)) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ B6))) @ fun_max_strict) => (((member @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ A4)) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ B6))) @ ms_strict))))).
% 9.43/3.45 thf(fact_4318_ms__reduction__pair, axiom, ((fun_reduction_pair @ (multiset @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ (set @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))))) @ (set @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))))) @ ms_strict) @ ms_weak))).
% 9.43/3.45 thf(fact_4319_ms__strictI, axiom, (! [Z9:(multiset @ ((product_prod @ nat) @ nat)),Z10:(multiset @ ((product_prod @ nat) @ nat)),A4:(multiset @ ((product_prod @ nat) @ nat)),B6:(multiset @ ((product_prod @ nat) @ nat))]: ((((pw_leq @ Z9) @ Z10) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ A4)) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ B6))) @ fun_max_strict) => (((member @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ A4)) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z10) @ B6))) @ ms_strict)))))).
% 9.43/3.45 thf(fact_4320_ms__weakI1, axiom, (! [Z9:(multiset @ ((product_prod @ nat) @ nat)),Z10:(multiset @ ((product_prod @ nat) @ nat)),A4:(multiset @ ((product_prod @ nat) @ nat)),B6:(multiset @ ((product_prod @ nat) @ nat))]: ((((pw_leq @ Z9) @ Z10) => ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ A4)) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ B6))) @ fun_max_strict) => (((member @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ A4)) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z10) @ B6))) @ ms_weak)))))).
% 9.43/3.45 thf(fact_4321_pw__leq__split, axiom, (! [X10:(multiset @ ((product_prod @ nat) @ nat)),Y9:(multiset @ ((product_prod @ nat) @ nat))]: ((((pw_leq @ X10) @ Y9) => (? [A15:(multiset @ ((product_prod @ nat) @ nat)),B10:(multiset @ ((product_prod @ nat) @ nat)),Z11:(multiset @ ((product_prod @ nat) @ nat))]: (((X10 = (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ A15) @ Z11)) & ((Y9 = (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ B10) @ Z11)) & ((((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ A15)) @ ((set_mset @ ((product_prod @ nat) @ nat)) @ B10))) @ fun_max_strict) | ((B10 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))) & (A15 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))))))))))))).
% 9.43/3.45 thf(fact_4322_ms__weakI2, axiom, (! [Z9:(multiset @ ((product_prod @ nat) @ nat)),Z10:(multiset @ ((product_prod @ nat) @ nat))]: ((((pw_leq @ Z9) @ Z10) => (((member @ ((product_prod @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (multiset @ ((product_prod @ nat) @ nat))) @ (multiset @ ((product_prod @ nat) @ nat))) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z9) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ Z10) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))))) @ ms_weak))))).
% 9.43/3.45 thf(fact_4323_pw__leq__lstep, axiom, (! [X:((product_prod @ nat) @ nat),Y:((product_prod @ nat) @ nat)]: (((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y)) @ fun_pair_leq) => ((pw_leq @ (((add_mset @ ((product_prod @ nat) @ nat)) @ X) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ Y) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))))))).
% 9.43/3.45 thf(fact_4324_pw__leq__step, axiom, (! [X:((product_prod @ nat) @ nat),Y:((product_prod @ nat) @ nat),X10:(multiset @ ((product_prod @ nat) @ nat)),Y9:(multiset @ ((product_prod @ nat) @ nat))]: (((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X) @ Y)) @ fun_pair_leq) => (((pw_leq @ X10) @ Y9) => ((pw_leq @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ X) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ X10)) @ (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ Y) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ Y9))))))).
% 9.43/3.45 thf(fact_4325_pw__leq_Osimps, axiom, (pw_leq = (^ [A12:(multiset @ ((product_prod @ nat) @ nat)),A23:(multiset @ ((product_prod @ nat) @ nat))]: ((((A12 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))) & (A23 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) | (? [X3:((product_prod @ nat) @ nat),Y3:((product_prod @ nat) @ nat),X8:(multiset @ ((product_prod @ nat) @ nat)),Y10:(multiset @ ((product_prod @ nat) @ nat))]: (((A12 = (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ X3) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ X8)) & ((A23 = (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ Y3) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ Y10)) & ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X3) @ Y3)) @ fun_pair_leq) & ((pw_leq @ X8) @ Y10))))))))))).
% 9.43/3.45 thf(fact_4326_pw__leq_Ocases, axiom, (! [A1:(multiset @ ((product_prod @ nat) @ nat)),A22:(multiset @ ((product_prod @ nat) @ nat))]: ((((pw_leq @ A1) @ A22) => (((A1 = (zero_zero @ (multiset @ ((product_prod @ nat) @ nat)))) => (A22 != (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) => (~ ((! [X2:((product_prod @ nat) @ nat),Y4:((product_prod @ nat) @ nat),X15:(multiset @ ((product_prod @ nat) @ nat))]: (((A1 = (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ X2) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ X15)) => (! [Y11:(multiset @ ((product_prod @ nat) @ nat))]: (((A22 = (((plus_plus @ (multiset @ ((product_prod @ nat) @ nat))) @ (((add_mset @ ((product_prod @ nat) @ nat)) @ Y4) @ (zero_zero @ (multiset @ ((product_prod @ nat) @ nat))))) @ Y11)) => ((((member @ ((product_prod @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat))) @ ((((product_Pair @ ((product_prod @ nat) @ nat)) @ ((product_prod @ nat) @ nat)) @ X2) @ Y4)) @ fun_pair_leq) => (~ (((pw_leq @ X15) @ Y11))))))))))))))))).
% 9.43/3.45 thf(fact_4327_mult1E, axiom, (! [A:$tType,N6:(multiset @ A),M3:(multiset @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ N6) @ M3)) @ ((mult1 @ A) @ R)) => (~ ((! [A5:A,M0:(multiset @ A)]: (((M3 = (((add_mset @ A) @ A5) @ M0)) => (! [K10:(multiset @ A)]: (((N6 = (((plus_plus @ (multiset @ A)) @ M0) @ K10)) => (~ ((! [B8:A]: (((((member @ A) @ B8) @ ((set_mset @ A) @ K10)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B8) @ A5)) @ R)))))))))))))))))).
% 9.43/3.45 thf(fact_4328_mult1I, axiom, (! [A:$tType,M3:(multiset @ A),A3:A,M02:(multiset @ A),N6:(multiset @ A),K2:(multiset @ A),R:(set @ ((product_prod @ A) @ A))]: (((M3 = (((add_mset @ A) @ A3) @ M02)) => ((N6 = (((plus_plus @ (multiset @ A)) @ M02) @ K2)) => ((! [B3:A]: (((((member @ A) @ B3) @ ((set_mset @ A) @ K2)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A3)) @ R)))) => (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ N6) @ M3)) @ ((mult1 @ A) @ R)))))))).
% 9.43/3.45 thf(fact_4329_less__add, axiom, (! [A:$tType,N6:(multiset @ A),A3:A,M02:(multiset @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ N6) @ (((add_mset @ A) @ A3) @ M02))) @ ((mult1 @ A) @ R)) => ((? [M9:(multiset @ A)]: (((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ M9) @ M02)) @ ((mult1 @ A) @ R)) & (N6 = (((add_mset @ A) @ A3) @ M9))))) | (? [K10:(multiset @ A)]: (((! [B8:A]: (((((member @ A) @ B8) @ ((set_mset @ A) @ K10)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B8) @ A3)) @ R)))) & (N6 = (((plus_plus @ (multiset @ A)) @ M02) @ K10)))))))))).
% 9.43/3.45 thf(fact_4330_one__step__implies__mult, axiom, (! [A:$tType,J4:(multiset @ A),K2:(multiset @ A),R:(set @ ((product_prod @ A) @ A)),I4:(multiset @ A)]: (((J4 != (zero_zero @ (multiset @ A))) => ((! [X2:A]: (((((member @ A) @ X2) @ ((set_mset @ A) @ K2)) => (? [Xa2:A]: (((((member @ A) @ Xa2) @ ((set_mset @ A) @ J4)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Xa2)) @ R))))))) => (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ I4) @ K2)) @ (((plus_plus @ (multiset @ A)) @ I4) @ J4))) @ ((mult @ A) @ R))))))).
% 9.43/3.45 thf(fact_4331_not__less__empty, axiom, (! [A:$tType,M3:(multiset @ A),R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ M3) @ (zero_zero @ (multiset @ A)))) @ ((mult1 @ A) @ R))))))).
% 9.43/3.45 thf(fact_4332_mult1__union, axiom, (! [A:$tType,B6:(multiset @ A),D4:(multiset @ A),R:(set @ ((product_prod @ A) @ A)),C5:(multiset @ A)]: (((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ B6) @ D4)) @ ((mult1 @ A) @ R)) => (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ C5) @ B6)) @ (((plus_plus @ (multiset @ A)) @ C5) @ D4))) @ ((mult1 @ A) @ R)))))).
% 9.43/3.45 thf(fact_4333_mult1__lessE, axiom, (! [A:$tType,N6:(multiset @ A),M3:(multiset @ A),R:(A > (A > $o))]: (((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ N6) @ M3)) @ ((mult1 @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R)))) => (((asymp @ A) @ R) => (~ ((! [A5:A,M0:(multiset @ A)]: (((M3 = (((add_mset @ A) @ A5) @ M0)) => (! [K10:(multiset @ A)]: (((N6 = (((plus_plus @ (multiset @ A)) @ M0) @ K10)) => ((~ ((((member @ A) @ A5) @ ((set_mset @ A) @ K10)))) => (~ ((! [B8:A]: (((((member @ A) @ B8) @ ((set_mset @ A) @ K10)) => ((R @ B8) @ A5)))))))))))))))))))).
% 9.43/3.45 thf(fact_4334_in__mset__fold__plus__iff, axiom, (! [A:$tType,X:A,M3:(multiset @ A),NN:(multiset @ (multiset @ A))]: (((((member @ A) @ X) @ ((set_mset @ A) @ (((((fold_mset @ (multiset @ A)) @ (multiset @ A)) @ (plus_plus @ (multiset @ A))) @ M3) @ NN))) = ((((member @ A) @ X) @ ((set_mset @ A) @ M3)) | (? [N9:(multiset @ A)]: (((((member @ (multiset @ A)) @ N9) @ ((set_mset @ (multiset @ A)) @ NN)) & (((member @ A) @ X) @ ((set_mset @ A) @ N9)))))))))).
% 9.43/3.45 thf(fact_4335_mult__implies__one__step, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),M3:(multiset @ A),N6:(multiset @ A)]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ M3) @ N6)) @ ((mult @ A) @ R)) => (? [I8:(multiset @ A),J5:(multiset @ A)]: (((N6 = (((plus_plus @ (multiset @ A)) @ I8) @ J5)) & (? [K10:(multiset @ A)]: (((M3 = (((plus_plus @ (multiset @ A)) @ I8) @ K10)) & ((J5 != (zero_zero @ (multiset @ A))) & (! [X7:A]: (((((member @ A) @ X7) @ ((set_mset @ A) @ K10)) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ ((set_mset @ A) @ J5)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X7) @ Xa3)) @ R))))))))))))))))))).
% 9.43/3.45 thf(fact_4336_mult1__def, axiom, (! [A:$tType]: (((mult1 @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_case_prod @ (multiset @ A)) @ (multiset @ A)) @ $o) @ (^ [N9:(multiset @ A),M8:(multiset @ A)]: ((? [A7:A,M03:(multiset @ A),K3:(multiset @ A)]: (((M8 = (((add_mset @ A) @ A7) @ M03)) & ((N9 = (((plus_plus @ (multiset @ A)) @ M03) @ K3)) & (! [B5:A]: (((((member @ A) @ B5) @ ((set_mset @ A) @ K3)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A7)) @ R3)))))))))))))))))).
% 9.43/3.45 thf(fact_4337_trans__def, axiom, (! [A:$tType]: (((trans @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X3:A,Y3:A,Z5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Z5)) @ R3) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Z5)) @ R3))))))))))).
% 9.43/3.45 thf(fact_4338_transI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [X2:A,Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Z3)) @ R))))) => ((trans @ A) @ R))))).
% 9.43/3.45 thf(fact_4339_transE, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z2:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ R))))))).
% 9.43/3.45 thf(fact_4340_transD, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z2:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ R) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ R))))))).
% 9.43/3.45 thf(fact_4341_lexord__trans, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A),R:(set @ ((product_prod @ A) @ A)),Z2:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lexord @ A) @ R)) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Y) @ Z2)) @ ((lexord @ A) @ R)) => (((trans @ A) @ R) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Z2)) @ ((lexord @ A) @ R)))))))).
% 9.43/3.45 thf(fact_4342_lenlex__trans, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A),R:(set @ ((product_prod @ A) @ A)),Z2:(list @ A)]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((lenlex @ A) @ R)) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Y) @ Z2)) @ ((lenlex @ A) @ R)) => (((trans @ A) @ R) => (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Z2)) @ ((lenlex @ A) @ R)))))))).
% 9.43/3.45 thf(fact_4343_union__fold__mset__add__mset, axiom, (! [A:$tType]: (((plus_plus @ (multiset @ A)) = (((fold_mset @ A) @ (multiset @ A)) @ (add_mset @ A)))))).
% 9.43/3.45 thf(fact_4344_trans__singleton, axiom, (! [A:$tType,A3:A]: (((trans @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A3)) @ (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.45 thf(fact_4345_trans__rtrancl__eq__reflcl, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A))]: ((((trans @ A) @ A4) => (((transitive_rtrancl @ A) @ A4) = (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ A4) @ (id2 @ A))))))).
% 9.43/3.46 thf(fact_4346_comp__fun__commute_Ofold__mset__union, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),S:B,M3:(multiset @ A),N6:(multiset @ A)]: (((((finite6289374366891150609ommute @ A) @ B) @ F) => ((((((fold_mset @ A) @ B) @ F) @ S) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((((fold_mset @ A) @ B) @ F) @ (((((fold_mset @ A) @ B) @ F) @ S) @ M3)) @ N6)))))).
% 9.43/3.46 thf(fact_4347_trans__join, axiom, (! [A:$tType]: (((trans @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X3:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ X3) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y3:A,Y13:A]: ((! [Z5:((product_prod @ A) @ A)]: (((((member @ ((product_prod @ A) @ A)) @ Z5) @ R3) => (((((product_case_prod @ A) @ A) @ $o) @ (^ [Y23:A,Aa2:A]: (((Y13 = Y23) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ Aa2)) @ R3))))) @ Z5))))))) @ X3)))))))))).
% 9.43/3.46 thf(fact_4348_Image__absorb__rtrancl, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),B6:(set @ A),C5:(set @ A)]: ((((trans @ A) @ A4) => ((((refl_on @ A) @ B6) @ A4) => ((((ord_less_eq @ (set @ A)) @ C5) @ B6) => (((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ A4)) @ C5) = ((((image @ A) @ A) @ A4) @ C5)))))))).
% 9.43/3.46 thf(fact_4349_wf__finite__segments, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((irrefl @ A) @ R) => (((trans @ A) @ R) => ((! [X2:A]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X2)) @ R))))))) => ((wf @ A) @ R))))))).
% 9.43/3.46 thf(fact_4350_mult__cancel__add__mset, axiom, (! [A:$tType,S:(set @ ((product_prod @ A) @ A)),Uu2:A,X10:(multiset @ A),Y9:(multiset @ A)]: ((((trans @ A) @ S) => (((irrefl @ A) @ S) => ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ (((add_mset @ A) @ Uu2) @ X10)) @ (((add_mset @ A) @ Uu2) @ Y9))) @ ((mult @ A) @ S)) = (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ X10) @ Y9)) @ ((mult @ A) @ S)))))))).
% 9.43/3.46 thf(fact_4351_mult__cancel, axiom, (! [A:$tType,S:(set @ ((product_prod @ A) @ A)),X10:(multiset @ A),Z9:(multiset @ A),Y9:(multiset @ A)]: ((((trans @ A) @ S) => (((irrefl @ A) @ S) => ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ X10) @ Z9)) @ (((plus_plus @ (multiset @ A)) @ Y9) @ Z9))) @ ((mult @ A) @ S)) = (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ X10) @ Y9)) @ ((mult @ A) @ S)))))))).
% 9.43/3.46 thf(fact_4352_mult__cancel__max, axiom, (! [A:$tType,S:(set @ ((product_prod @ A) @ A)),X10:(multiset @ A),Y9:(multiset @ A)]: ((((trans @ A) @ S) => (((irrefl @ A) @ S) => ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ X10) @ Y9)) @ ((mult @ A) @ S)) = (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ X10) @ Y9)) @ (((minus_minus @ (multiset @ A)) @ Y9) @ X10))) @ ((mult @ A) @ S)))))))).
% 9.43/3.46 thf(fact_4353_multeqp__code__iff__reflcl__mult, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),P:(A > (A > $o)),N6:(multiset @ A),M3:(multiset @ A)]: ((((irrefl @ A) @ R5) => (((trans @ A) @ R5) => ((! [X2:A,Y4:A]: ((((P @ X2) @ Y4) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R5)))) => (((((multeqp_code @ A) @ P) @ N6) @ M3) = (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ N6) @ M3)) @ (((sup_sup @ (set @ ((product_prod @ (multiset @ A)) @ (multiset @ A)))) @ ((mult @ A) @ R5)) @ (id2 @ (multiset @ A))))))))))).
% 9.43/3.46 thf(fact_4354_multp__code__iff__mult, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),P:(A > (A > $o)),N6:(multiset @ A),M3:(multiset @ A)]: ((((irrefl @ A) @ R5) => (((trans @ A) @ R5) => ((! [X2:A,Y4:A]: ((((P @ X2) @ Y4) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R5)))) => (((((multp_code @ A) @ P) @ N6) @ M3) = (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ N6) @ M3)) @ ((mult @ A) @ R5))))))))).
% 9.43/3.46 thf(fact_4355_max__ext__eq, axiom, (! [A:$tType]: (((max_ext @ A) = (^ [R6:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ ((((product_case_prod @ (set @ A)) @ (set @ A)) @ $o) @ (^ [X8:(set @ A),Y10:(set @ A)]: ((((finite_finite2 @ A) @ X8) & (((finite_finite2 @ A) @ Y10) & ((Y10 != (bot_bot @ (set @ A))) & (! [X3:A]: (((((member @ A) @ X3) @ X8) => (? [Y3:A]: (((((member @ A) @ Y3) @ Y10) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R6)))))))))))))))))))).
% 9.43/3.46 thf(fact_4356_init__seg__of__def, axiom, (! [A:$tType]: (((init_seg_of @ A) = ((collect @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_case_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ $o) @ (^ [R3:(set @ ((product_prod @ A) @ A)),S4:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R3) @ S4) & (! [A7:A,B5:A,C3:A]: ((((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ S4) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ C3)) @ R3)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R3))))))))))))).
% 9.43/3.46 thf(fact_4357_refl__on__init__seg__of, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R)) @ (init_seg_of @ A))))).
% 9.43/3.46 thf(fact_4358_antisym__init__seg__of, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ S)) @ (init_seg_of @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ S) @ R)) @ (init_seg_of @ A)) => (R = S)))))).
% 9.43/3.46 thf(fact_4359_trans__init__seg__of, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A)),T2:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ S)) @ (init_seg_of @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ S) @ T2)) @ (init_seg_of @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ T2)) @ (init_seg_of @ A))))))).
% 9.43/3.46 thf(fact_4360_Image__def, axiom, (! [B:$tType,A:$tType]: ((((image @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B)),S4:(set @ A)]: (((collect @ B) @ (^ [Y3:B]: ((? [X3:A]: (((((member @ A) @ X3) @ S4) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3))))))))))))).
% 9.43/3.46 thf(fact_4361_initial__segment__of__Diff, axiom, (! [A:$tType,P5:(set @ ((product_prod @ A) @ A)),Q3:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P5) @ Q3)) @ (init_seg_of @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ P5) @ S)) @ (((minus_minus @ (set @ ((product_prod @ A) @ A))) @ Q3) @ S))) @ (init_seg_of @ A)))))).
% 9.43/3.46 thf(fact_4362_min__ext__def, axiom, (! [A:$tType]: (((min_ext @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ (set @ A)) @ (set @ A))) @ (^ [Uu:((product_prod @ (set @ A)) @ (set @ A))]: ((? [X8:(set @ A),Y10:(set @ A)]: (((Uu = ((((product_Pair @ (set @ A)) @ (set @ A)) @ X8) @ Y10)) & ((X8 != (bot_bot @ (set @ A))) & (! [X3:A]: (((((member @ A) @ X3) @ Y10) => (? [Y3:A]: (((((member @ A) @ Y3) @ X8) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3)))))))))))))))))))).
% 9.43/3.46 thf(fact_4363_map__project__def, axiom, (! [B:$tType,A:$tType]: ((((map_project @ A) @ B) = (^ [F2:(A > (option @ B)),A9:(set @ A)]: (((collect @ B) @ (^ [B5:B]: ((? [X3:A]: (((((member @ A) @ X3) @ A9) & ((F2 @ X3) = ((some @ B) @ B5)))))))))))))).
% 9.43/3.46 thf(fact_4364_Chains__init__seg__of__Union, axiom, (! [A:$tType,R5:(set @ (set @ ((product_prod @ A) @ A))),R:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ (set @ ((product_prod @ A) @ A)))) @ R5) @ ((chains @ (set @ ((product_prod @ A) @ A))) @ (init_seg_of @ A))) => ((((member @ (set @ ((product_prod @ A) @ A))) @ R) @ R5) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ ((complete_Sup_Sup @ (set @ ((product_prod @ A) @ A))) @ R5))) @ (init_seg_of @ A))))))).
% 9.43/3.46 thf(fact_4365_mult__cancel__max0, axiom, (! [A:$tType,S:(set @ ((product_prod @ A) @ A)),X10:(multiset @ A),Y9:(multiset @ A)]: ((((trans @ A) @ S) => (((irrefl @ A) @ S) => ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ X10) @ Y9)) @ ((mult @ A) @ S)) = (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ X10) @ (((inter_mset @ A) @ X10) @ Y9))) @ (((minus_minus @ (multiset @ A)) @ Y9) @ (((inter_mset @ A) @ X10) @ Y9)))) @ ((mult @ A) @ S)))))))).
% 9.43/3.46 thf(fact_4366_sum__mset_Oremove, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [X:A,A4:(multiset @ A)]: (((((member @ A) @ X) @ ((set_mset @ A) @ A4)) => (((comm_m7189776963980413722m_mset @ A) @ A4) = (((plus_plus @ A) @ X) @ ((comm_m7189776963980413722m_mset @ A) @ (((minus_minus @ (multiset @ A)) @ A4) @ (((add_mset @ A) @ X) @ (zero_zero @ (multiset @ A)))))))))))))).
% 9.43/3.46 thf(fact_4367_sum__mset_Oadd__mset, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [X:A,N6:(multiset @ A)]: ((((comm_m7189776963980413722m_mset @ A) @ (((add_mset @ A) @ X) @ N6)) = (((plus_plus @ A) @ X) @ ((comm_m7189776963980413722m_mset @ A) @ N6))))))))).
% 9.43/3.46 thf(fact_4368_sum__mset_Ounion, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (! [M3:(multiset @ A),N6:(multiset @ A)]: ((((comm_m7189776963980413722m_mset @ A) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((plus_plus @ A) @ ((comm_m7189776963980413722m_mset @ A) @ M3)) @ ((comm_m7189776963980413722m_mset @ A) @ N6))))))))).
% 9.43/3.46 thf(fact_4369_sum__mset_Oinsert, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),X:B,A4:(multiset @ B)]: ((((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ G) @ (((add_mset @ B) @ X) @ A4))) = (((plus_plus @ A) @ (G @ X)) @ ((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ G) @ A4)))))))))).
% 9.43/3.46 thf(fact_4370_inter__union__distrib__right, axiom, (! [A:$tType,C5:(multiset @ A),A4:(multiset @ A),B6:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ C5) @ (((inter_mset @ A) @ A4) @ B6)) = (((inter_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C5) @ A4)) @ (((plus_plus @ (multiset @ A)) @ C5) @ B6)))))).
% 9.43/3.46 thf(fact_4371_inter__union__distrib__left, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((inter_mset @ A) @ A4) @ B6)) @ C5) = (((inter_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ C5)) @ (((plus_plus @ (multiset @ A)) @ B6) @ C5)))))).
% 9.43/3.46 thf(fact_4372_inter__subset__eq__union, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A)]: ((((subseteq_mset @ A) @ (((inter_mset @ A) @ A4) @ B6)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6))))).
% 9.43/3.46 thf(fact_4373_inter__mset__empty__distrib__left, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: ((((((inter_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ C5) = (zero_zero @ (multiset @ A))) = (((((inter_mset @ A) @ A4) @ C5) = (zero_zero @ (multiset @ A))) & ((((inter_mset @ A) @ B6) @ C5) = (zero_zero @ (multiset @ A)))))))).
% 9.43/3.46 thf(fact_4374_inter__mset__empty__distrib__right, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: ((((((inter_mset @ A) @ A4) @ (((plus_plus @ (multiset @ A)) @ B6) @ C5)) = (zero_zero @ (multiset @ A))) = (((((inter_mset @ A) @ A4) @ B6) = (zero_zero @ (multiset @ A))) & ((((inter_mset @ A) @ A4) @ C5) = (zero_zero @ (multiset @ A)))))))).
% 9.43/3.46 thf(fact_4375_sum__mset_Odistrib, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [G:(B > A),H3:(B > A),A4:(multiset @ B)]: ((((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ (^ [X3:B]: ((((plus_plus @ A) @ (G @ X3)) @ (H3 @ X3))))) @ A4)) = (((plus_plus @ A) @ ((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ G) @ A4))) @ ((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ H3) @ A4)))))))))).
% 9.43/3.46 thf(fact_4376_size__eq__sum__mset, axiom, (! [A:$tType]: (((size_size @ (multiset @ A)) = (^ [M8:(multiset @ A)]: (((comm_m7189776963980413722m_mset @ nat) @ ((((image_mset @ A) @ nat) @ (^ [A7:A]: ((one_one @ nat)))) @ M8)))))))).
% 9.43/3.46 thf(fact_4377_multiset__union__diff__commute, axiom, (! [A:$tType,B6:(multiset @ A),C5:(multiset @ A),A4:(multiset @ A)]: ((((((inter_mset @ A) @ B6) @ C5) = (zero_zero @ (multiset @ A))) => ((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ C5) = (((plus_plus @ (multiset @ A)) @ (((minus_minus @ (multiset @ A)) @ A4) @ C5)) @ B6)))))).
% 9.43/3.46 thf(fact_4378_sum__mset_Oeq__fold, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => ((comm_m7189776963980413722m_mset @ A) = ((((fold_mset @ A) @ A) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 9.43/3.46 thf(fact_4379_Chains__def, axiom, (! [A:$tType]: (((chains @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ (set @ A)) @ (^ [C7:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ C7) => (! [Y3:A]: (((((member @ A) @ Y3) @ C7) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3))))))))))))))))).
% 9.43/3.46 thf(fact_4380_Chains__subset_H, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ (top_top @ (set @ A))) @ R) => (((ord_less_eq @ (set @ (set @ A))) @ ((collect @ (set @ A)) @ (((pred_chain @ A) @ (top_top @ (set @ A))) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))))) @ ((chains @ A) @ R)))))).
% 9.43/3.46 thf(fact_4381_Chains__subset, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((ord_less_eq @ (set @ (set @ A))) @ ((chains @ A) @ R)) @ ((collect @ (set @ A)) @ (((pred_chain @ A) @ (top_top @ (set @ A))) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R))))))))).
% 9.43/3.46 thf(fact_4382_Chains__alt__def, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((refl_on @ A) @ (top_top @ (set @ A))) @ R) => (((chains @ A) @ R) = ((collect @ (set @ A)) @ (((pred_chain @ A) @ (top_top @ (set @ A))) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))))))))).
% 9.43/3.46 thf(fact_4383_iter_H_Ocases, axiom, (! [A:$tType]: (((quickcheck_random @ A) => (! [X:((product_prod @ (itself @ A)) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural))))]: ((~ ((! [T4:(itself @ A),Nrandom:code_natural,Sz:code_natural,Seed:((product_prod @ code_natural) @ code_natural)]: ((X != ((((product_Pair @ (itself @ A)) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural)))) @ T4) @ ((((product_Pair @ code_natural) @ ((product_prod @ code_natural) @ ((product_prod @ code_natural) @ code_natural))) @ Nrandom) @ ((((product_Pair @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ Sz) @ Seed)))))))))))))).
% 9.43/3.46 thf(fact_4384_multp__def, axiom, (! [A:$tType]: (((multp @ A) = (^ [R3:(A > (A > $o)),M8:(multiset @ A),N9:(multiset @ A)]: ((((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ M8) @ N9)) @ ((mult @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.46 thf(fact_4385_AboveS__def, axiom, (! [A:$tType]: (((order_AboveS @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A9:(set @ A)]: (((collect @ A) @ (^ [B5:A]: (((((member @ A) @ B5) @ ((field2 @ A) @ R3)) & (! [X3:A]: (((((member @ A) @ X3) @ A9) => ((B5 != X3) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ B5)) @ R3))))))))))))))).
% 9.43/3.46 thf(fact_4386_lexordp__conv__lexord, axiom, (! [A:$tType]: (((linorder @ A) => ((ord_lexordp @ A) = (^ [Xs3:(list @ A),Ys:(list @ A)]: ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ Xs3) @ Ys)) @ ((lexord @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ (ord_less @ A)))))))))))).
% 9.43/3.46 thf(fact_4387_select__weight__cons__zero, axiom, (! [A:$tType,X:A,Xs:(list @ ((product_prod @ code_natural) @ A))]: ((((select_weight @ A) @ (((cons @ ((product_prod @ code_natural) @ A)) @ ((((product_Pair @ code_natural) @ A) @ (zero_zero @ code_natural)) @ X)) @ Xs)) = ((select_weight @ A) @ Xs))))).
% 9.43/3.46 thf(fact_4388_one__step__implies__multp, axiom, (! [A:$tType,J4:(multiset @ A),K2:(multiset @ A),R:(A > (A > $o)),I4:(multiset @ A)]: (((J4 != (zero_zero @ (multiset @ A))) => ((! [X2:A]: (((((member @ A) @ X2) @ ((set_mset @ A) @ K2)) => (? [Xa2:A]: (((((member @ A) @ Xa2) @ ((set_mset @ A) @ J4)) & ((R @ X2) @ Xa2))))))) => ((((multp @ A) @ R) @ (((plus_plus @ (multiset @ A)) @ I4) @ K2)) @ (((plus_plus @ (multiset @ A)) @ I4) @ J4))))))).
% 9.43/3.46 thf(fact_4389_select__weight__select, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((select_weight @ A) @ ((((map @ A) @ ((product_prod @ code_natural) @ A)) @ (((product_Pair @ code_natural) @ A) @ (one_one @ code_natural))) @ Xs)) = ((select @ A) @ Xs)))))).
% 9.43/3.46 thf(fact_4390_wo__rel_Osuc__greater, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),B6:(set @ A),B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((field2 @ A) @ R)) => (((((order_AboveS @ A) @ R) @ B6) != (bot_bot @ (set @ A))) => ((((member @ A) @ B2) @ B6) => (((((bNF_Wellorder_wo_suc @ A) @ R) @ B6) != B2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ (((bNF_Wellorder_wo_suc @ A) @ R) @ B6))) @ R))))))))).
% 9.43/3.46 thf(fact_4391_wo__rel_Oequals__suc__AboveS, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),B6:(set @ A),A3:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((ord_less_eq @ (set @ A)) @ B6) @ ((field2 @ A) @ R)) => ((((member @ A) @ A3) @ (((order_AboveS @ A) @ R) @ B6)) => ((! [A16:A]: (((((member @ A) @ A16) @ (((order_AboveS @ A) @ R) @ B6)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ A16)) @ R)))) => (A3 = (((bNF_Wellorder_wo_suc @ A) @ R) @ B6))))))))).
% 9.43/3.46 thf(fact_4392_wo__rel_Osuc__least__AboveS, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B6:(set @ A)]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((member @ A) @ A3) @ (((order_AboveS @ A) @ R) @ B6)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (((bNF_Wellorder_wo_suc @ A) @ R) @ B6)) @ A3)) @ R)))))).
% 9.43/3.46 thf(fact_4393_wo__rel_Osuc__ofilter__in, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => ((((order_ofilter @ A) @ R) @ A4) => (((((order_AboveS @ A) @ R) @ A4) != (bot_bot @ (set @ A))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ (((bNF_Wellorder_wo_suc @ A) @ R) @ A4))) @ R) => ((B2 != (((bNF_Wellorder_wo_suc @ A) @ R) @ A4)) => (((member @ A) @ B2) @ A4))))))))).
% 9.43/3.46 thf(fact_4394_slice__head, axiom, (! [A:$tType,From:nat,To:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ From) @ To) => ((((ord_less_eq @ nat) @ To) @ ((size_size @ (list @ A)) @ Xs)) => (((hd @ A) @ ((((slice @ A) @ From) @ To) @ Xs)) = (((nth @ A) @ Xs) @ From))))))).
% 9.43/3.46 thf(fact_4395_gen__length__def, axiom, (! [A:$tType]: (((gen_length @ A) = (^ [N5:nat,Xs3:(list @ A)]: ((((plus_plus @ nat) @ N5) @ ((size_size @ (list @ A)) @ Xs3)))))))).
% 9.43/3.46 thf(fact_4396_subset__mset_Osum__pos2, axiom, (! [A:$tType,B:$tType,I4:(set @ B),I:B,F:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ I4) => ((((member @ B) @ I) @ I4) => ((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ I)) => ((! [I2:B]: (((((member @ B) @ I2) @ I4) => (((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ I2))))) => (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ I4))))))))).
% 9.43/3.46 thf(fact_4397_subset__mset_Oadd__less__cancel__right, axiom, (! [A:$tType,A3:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) = (((subset_mset @ A) @ A3) @ B2))))).
% 9.43/3.46 thf(fact_4398_subset__mset_Oadd__less__cancel__left, axiom, (! [A:$tType,C2:(multiset @ A),A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) = (((subset_mset @ A) @ A3) @ B2))))).
% 9.43/3.46 thf(fact_4399_subset__mset_Oadd__less__same__cancel1, axiom, (! [A:$tType,B2:(multiset @ A),A3:(multiset @ A)]: (((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ B2) @ A3)) @ B2) = (((subset_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))))).
% 9.43/3.46 thf(fact_4400_subset__mset_Oadd__less__same__cancel2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) @ B2) = (((subset_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))))))).
% 9.43/3.46 thf(fact_4401_subset__mset_Oless__add__same__cancel1, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) = (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2))))).
% 9.43/3.46 thf(fact_4402_subset__mset_Oless__add__same__cancel2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ (((plus_plus @ (multiset @ A)) @ B2) @ A3)) = (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2))))).
% 9.43/3.46 thf(fact_4403_subset__mset_Oadd__less__imp__less__right, axiom, (! [A:$tType,A3:(multiset @ A),C2:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)) => (((subset_mset @ A) @ A3) @ B2))))).
% 9.43/3.46 thf(fact_4404_subset__mset_Oadd__less__imp__less__left, axiom, (! [A:$tType,C2:(multiset @ A),A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)) => (((subset_mset @ A) @ A3) @ B2))))).
% 9.43/3.46 thf(fact_4405_subset__mset_Oadd__strict__right__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ B2) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ C2)))))).
% 9.43/3.46 thf(fact_4406_subset__mset_Oadd__strict__left__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ B2) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C2) @ A3)) @ (((plus_plus @ (multiset @ A)) @ C2) @ B2)))))).
% 9.43/3.46 thf(fact_4407_subset__mset_Oadd__strict__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A),D3:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ B2) => ((((subset_mset @ A) @ C2) @ D3) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ D3))))))).
% 9.43/3.46 thf(fact_4408_hd__zip, axiom, (! [A:$tType,B:$tType,Xs:(list @ A),Ys2:(list @ B)]: (((Xs != (nil @ A)) => ((Ys2 != (nil @ B)) => (((hd @ ((product_prod @ A) @ B)) @ ((((zip @ A) @ B) @ Xs) @ Ys2)) = ((((product_Pair @ A) @ B) @ ((hd @ A) @ Xs)) @ ((hd @ B) @ Ys2)))))))).
% 9.43/3.46 thf(fact_4409_subset__mset_Oadd__le__less__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A),D3:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ B2) => ((((subset_mset @ A) @ C2) @ D3) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ D3))))))).
% 9.43/3.46 thf(fact_4410_subset__mset_Oadd__less__le__mono, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A),D3:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ B2) => ((((subseteq_mset @ A) @ C2) @ D3) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2)) @ (((plus_plus @ (multiset @ A)) @ B2) @ D3))))))).
% 9.43/3.46 thf(fact_4411_subset__mset_OlessE, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ B2) => (~ ((! [C4:(multiset @ A)]: (((B2 = (((plus_plus @ (multiset @ A)) @ A3) @ C4)) => (C4 = (zero_zero @ (multiset @ A)))))))))))).
% 9.43/3.46 thf(fact_4412_subset__mset_Oadd__neg__neg, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) => ((((subset_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) @ (zero_zero @ (multiset @ A)))))))).
% 9.43/3.46 thf(fact_4413_subset__mset_Oadd__pos__pos, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2) => (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2))))))).
% 9.43/3.46 thf(fact_4414_subset__mset_Opos__add__strict, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subset_mset @ A) @ B2) @ C2) => (((subset_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2))))))).
% 9.43/3.46 thf(fact_4415_subset__implies__mult, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),R:(set @ ((product_prod @ A) @ A))]: (((((subset_mset @ A) @ A4) @ B6) => (((member @ ((product_prod @ (multiset @ A)) @ (multiset @ A))) @ ((((product_Pair @ (multiset @ A)) @ (multiset @ A)) @ A4) @ B6)) @ ((mult @ A) @ R)))))).
% 9.43/3.46 thf(fact_4416_subset__mset_Oadd__strict__increasing2, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subset_mset @ A) @ B2) @ C2) => (((subset_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2))))))).
% 9.43/3.46 thf(fact_4417_subset__mset_Oadd__strict__increasing, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A),C2:(multiset @ A)]: (((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subseteq_mset @ A) @ B2) @ C2) => (((subset_mset @ A) @ B2) @ (((plus_plus @ (multiset @ A)) @ A3) @ C2))))))).
% 9.43/3.46 thf(fact_4418_subset__mset_Oadd__pos__nonneg, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2) => (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2))))))).
% 9.43/3.46 thf(fact_4419_subset__mset_Oadd__nonpos__neg, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) => ((((subset_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) @ (zero_zero @ (multiset @ A)))))))).
% 9.43/3.46 thf(fact_4420_subset__mset_Oadd__nonneg__pos, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subseteq_mset @ A) @ (zero_zero @ (multiset @ A))) @ A3) => ((((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ B2) => (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2))))))).
% 9.43/3.46 thf(fact_4421_subset__mset_Oadd__neg__nonpos, axiom, (! [A:$tType,A3:(multiset @ A),B2:(multiset @ A)]: (((((subset_mset @ A) @ A3) @ (zero_zero @ (multiset @ A))) => ((((subseteq_mset @ A) @ B2) @ (zero_zero @ (multiset @ A))) => (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A3) @ B2)) @ (zero_zero @ (multiset @ A)))))))).
% 9.43/3.46 thf(fact_4422_sorted__hd__min, axiom, (! [A:$tType]: (((linorder @ A) => (! [Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ Xs) => (! [X7:A]: (((((member @ A) @ X7) @ ((set2 @ A) @ Xs)) => (((ord_less_eq @ A) @ ((hd @ A) @ Xs)) @ X7)))))))))))).
% 9.43/3.46 thf(fact_4423_sorted__hd__last, axiom, (! [A:$tType]: (((linorder @ A) => (! [L:(list @ A)]: (((((sorted_wrt @ A) @ (ord_less_eq @ A)) @ L) => ((L != (nil @ A)) => (((ord_less_eq @ A) @ ((hd @ A) @ L)) @ ((last @ A) @ L)))))))))).
% 9.43/3.46 thf(fact_4424_hd__last__singletonI, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((hd @ A) @ Xs) = ((last @ A) @ Xs)) => (((distinct @ A) @ Xs) => (Xs = (((cons @ A) @ ((hd @ A) @ Xs)) @ (nil @ A))))))))).
% 9.43/3.46 thf(fact_4425_hd__butlast, axiom, (! [A:$tType,Xs:(list @ A)]: (((((ord_less @ nat) @ (one_one @ nat)) @ ((size_size @ (list @ A)) @ Xs)) => (((hd @ A) @ ((butlast @ A) @ Xs)) = ((hd @ A) @ Xs)))))).
% 9.43/3.46 thf(fact_4426_subset__mset_Osum__pos, axiom, (! [A:$tType,B:$tType,I4:(set @ B),F:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ I4) => ((I4 != (bot_bot @ (set @ B))) => ((! [I2:B]: (((((member @ B) @ I2) @ I4) => (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ (F @ I2))))) => (((subset_mset @ A) @ (zero_zero @ (multiset @ A))) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ I4)))))))).
% 9.43/3.46 thf(fact_4427_subset__mset_Osum__strict__mono, axiom, (! [A:$tType,B:$tType,A4:(set @ B),F:(B > (multiset @ A)),G:(B > (multiset @ A))]: ((((finite_finite2 @ B) @ A4) => ((A4 != (bot_bot @ (set @ B))) => ((! [X2:B]: (((((member @ B) @ X2) @ A4) => (((subset_mset @ A) @ (F @ X2)) @ (G @ X2))))) => (((subset_mset @ A) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ F) @ A4)) @ ((((((groups3894954378712506084id_sum @ (multiset @ A)) @ B) @ (plus_plus @ (multiset @ A))) @ (zero_zero @ (multiset @ A))) @ G) @ A4)))))))).
% 9.43/3.46 thf(fact_4428_ofilter__ordLess, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A4) => ((((ord_less @ (set @ A)) @ A4) @ ((field2 @ A) @ R)) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4))))) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ A)))))))).
% 9.43/3.46 thf(fact_4429_ofilter__subset__ordLess, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B6:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A4) => ((((order_ofilter @ A) @ R) @ B6) => ((((ord_less @ (set @ A)) @ A4) @ B6) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4))))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ B6) @ (^ [Uu:A]: (B6)))))) @ ((bNF_We4044943003108391690rdLess @ A) @ A))))))))).
% 9.43/3.46 thf(fact_4430_ofilter__subset__ordLeq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B6:(set @ A)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_ofilter @ A) @ R) @ A4) => ((((order_ofilter @ A) @ R) @ B6) => ((((ord_less_eq @ (set @ A)) @ A4) @ B6) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4))))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ B6) @ (^ [Uu:A]: (B6)))))) @ ((bNF_Wellorder_ordLeq @ A) @ A))))))))).
% 9.43/3.46 thf(fact_4431_Nitpick_Osize__list__simp_I1_J, axiom, (! [A:$tType]: (((size_list @ A) = (^ [F2:(A > nat),Xs3:(list @ A)]: (((((if @ nat) @ (Xs3 = (nil @ A))) @ (zero_zero @ nat)) @ (suc @ (((plus_plus @ nat) @ (F2 @ ((hd @ A) @ Xs3))) @ (((size_list @ A) @ F2) @ ((tl @ A) @ Xs3))))))))))).
% 9.43/3.46 thf(fact_4432_length__tl, axiom, (! [A:$tType,Xs:(list @ A)]: ((((size_size @ (list @ A)) @ ((tl @ A) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))))).
% 9.43/3.46 thf(fact_4433_in__hd__or__tl__conv, axiom, (! [A:$tType,L:(list @ A),X:A]: (((L != (nil @ A)) => (((X = ((hd @ A) @ L)) | (((member @ A) @ X) @ ((set2 @ A) @ ((tl @ A) @ L)))) = (((member @ A) @ X) @ ((set2 @ A) @ L))))))).
% 9.43/3.46 thf(fact_4434_tl__replicate, axiom, (! [A:$tType,N:nat,X:A]: ((((tl @ A) @ (((replicate @ A) @ N) @ X)) = (((replicate @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ X))))).
% 9.43/3.46 thf(fact_4435_rev__split__conv, axiom, (! [A:$tType,L:(list @ A)]: (((L != (nil @ A)) => ((((append @ A) @ ((rev @ A) @ ((tl @ A) @ L))) @ (((cons @ A) @ ((hd @ A) @ L)) @ (nil @ A))) = ((rev @ A) @ L)))))).
% 9.43/3.46 thf(fact_4436_ordLess__irreflexive, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ A))))))).
% 9.43/3.46 thf(fact_4437_ordLess__transitive, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_We4044943003108391690rdLess @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_We4044943003108391690rdLess @ A) @ C))))))).
% 9.43/3.46 thf(fact_4438_ordLeq__transitive, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_Wellorder_ordLeq @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_Wellorder_ordLeq @ A) @ C))))))).
% 9.43/3.46 thf(fact_4439_not__ordLess__ordLeq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => (~ ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))))).
% 9.43/3.46 thf(fact_4440_ordLess__imp__ordLeq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))).
% 9.43/3.46 thf(fact_4441_ordLeq__ordLess__trans, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_We4044943003108391690rdLess @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_We4044943003108391690rdLess @ A) @ C))))))).
% 9.43/3.46 thf(fact_4442_ordLess__ordLeq__trans, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_Wellorder_ordLeq @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_We4044943003108391690rdLess @ A) @ C))))))).
% 9.43/3.46 thf(fact_4443_not__ordLess__iff__ordLeq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_We4044943003108391690rdLess @ B) @ A)))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))))).
% 9.43/3.46 thf(fact_4444_not__ordLeq__iff__ordLess, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)))))))).
% 9.43/3.46 thf(fact_4445_ordLess__or__ordLeq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) | (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))))).
% 9.43/3.46 thf(fact_4446_in__set__tlD, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((((member @ A) @ X) @ ((set2 @ A) @ ((tl @ A) @ Xs))) => (((member @ A) @ X) @ ((set2 @ A) @ Xs)))))).
% 9.43/3.46 thf(fact_4447_tl__obtain__elem, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((tl @ A) @ Xs) = (nil @ A)) => (~ ((! [E3:A]: ((Xs != (((cons @ A) @ E3) @ (nil @ A)))))))))))).
% 9.43/3.46 thf(fact_4448_not__hd__in__tl, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: (((X != ((hd @ A) @ Xs)) => ((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((member @ A) @ X) @ ((set2 @ A) @ ((tl @ A) @ Xs)))))))).
% 9.43/3.46 thf(fact_4449_tl__last, axiom, (! [A:$tType,Xs:(list @ A)]: (((((tl @ A) @ Xs) != (nil @ A)) => (((last @ A) @ Xs) = ((last @ A) @ ((tl @ A) @ Xs))))))).
% 9.43/3.46 thf(fact_4450_rev__butlast__is__tl__rev, axiom, (! [A:$tType,L:(list @ A)]: ((((rev @ A) @ ((butlast @ A) @ L)) = ((tl @ A) @ ((rev @ A) @ L)))))).
% 9.43/3.46 thf(fact_4451_tl__subset, axiom, (! [A:$tType,Xs:(list @ A),A4:(set @ A)]: (((Xs != (nil @ A)) => ((((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ Xs)) @ A4) => (((ord_less_eq @ (set @ A)) @ ((set2 @ A) @ ((tl @ A) @ Xs))) @ A4)))))).
% 9.43/3.46 thf(fact_4452_Misc_Onth__tl, axiom, (! [A:$tType,Xs:(list @ A),N:nat]: (((Xs != (nil @ A)) => ((((nth @ A) @ ((tl @ A) @ Xs)) @ N) = (((nth @ A) @ Xs) @ (suc @ N))))))).
% 9.43/3.46 thf(fact_4453_list__take__induct__tl2, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),Ys2:(list @ B),P:(B > (A > $o))]: (((((size_size @ (list @ A)) @ Xs) = ((size_size @ (list @ B)) @ Ys2)) => ((! [N3:nat]: (((((ord_less @ nat) @ N3) @ ((size_size @ (list @ A)) @ Xs)) => ((P @ (((nth @ B) @ Ys2) @ N3)) @ (((nth @ A) @ Xs) @ N3))))) => (! [N10:nat]: (((((ord_less @ nat) @ N10) @ ((size_size @ (list @ A)) @ ((tl @ A) @ Xs))) => ((P @ (((nth @ B) @ ((tl @ B) @ Ys2)) @ N10)) @ (((nth @ A) @ ((tl @ A) @ Xs)) @ N10)))))))))).
% 9.43/3.46 thf(fact_4454_exists__minim__Well__order, axiom, (! [A:$tType,R5:(set @ (set @ ((product_prod @ A) @ A)))]: (((R5 != (bot_bot @ (set @ (set @ ((product_prod @ A) @ A))))) => ((! [X2:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ ((product_prod @ A) @ A))) @ X2) @ R5) => (((order_well_order_on @ A) @ ((field2 @ A) @ X2)) @ X2)))) => (? [X2:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ ((product_prod @ A) @ A))) @ X2) @ R5) & (! [Xa2:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ ((product_prod @ A) @ A))) @ Xa2) @ R5) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ X2) @ Xa2)) @ ((bNF_Wellorder_ordLeq @ A) @ A))))))))))))).
% 9.43/3.46 thf(fact_4455_ordLeq__total, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) | (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))))).
% 9.43/3.46 thf(fact_4456_ordLeq__reflexive, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))).
% 9.43/3.46 thf(fact_4457_ordLeq__Well__order__simp, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) & (((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4)))))).
% 9.43/3.46 thf(fact_4458_tl__take, axiom, (! [A:$tType,N:nat,Xs:(list @ A)]: ((((tl @ A) @ (((take @ A) @ N) @ Xs)) = (((take @ A) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ ((tl @ A) @ Xs)))))).
% 9.43/3.46 thf(fact_4459_distinct__hd__tl, axiom, (! [A:$tType,Xs:(list @ A),X:A]: ((((distinct @ A) @ Xs) => ((X = ((hd @ A) @ Xs)) => (~ ((((member @ A) @ X) @ ((set2 @ A) @ ((tl @ A) @ Xs)))))))))).
% 9.43/3.46 thf(fact_4460_butlast__rev__tl, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => (((butlast @ A) @ ((rev @ A) @ Xs)) = ((rev @ A) @ ((tl @ A) @ Xs))))))).
% 9.43/3.46 thf(fact_4461_finite__ordLess__infinite, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => (((finite_finite2 @ A) @ ((field2 @ A) @ R)) => ((~ (((finite_finite2 @ B) @ ((field2 @ B) @ R4)))) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B))))))))).
% 9.43/3.46 thf(fact_4462_ordLeq__iff__ordLess__Restr, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) = (! [X3:A]: (((((member @ A) @ X3) @ ((field2 @ A) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ (((order_underS @ A) @ R) @ X3)) @ (^ [Uu:A]: ((((order_underS @ A) @ R) @ X3)))))) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B))))))))))).
% 9.43/3.46 thf(fact_4463_underS__Restr__ordLess, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((field2 @ A) @ R) != (bot_bot @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ (((order_underS @ A) @ R) @ A3)) @ (^ [Uu:A]: ((((order_underS @ A) @ R) @ A3)))))) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ A))))))).
% 9.43/3.46 thf(fact_4464_horner__sum__append, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_1 @ A) => (! [F:(B > A),A3:A,Xs:(list @ B),Ys2:(list @ B)]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ (((append @ B) @ Xs) @ Ys2)) = (((plus_plus @ A) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ Xs)) @ (((times_times @ A) @ (((power_power @ A) @ A3) @ ((size_size @ (list @ B)) @ Xs))) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ Ys2)))))))))).
% 9.43/3.46 thf(fact_4465_Code__Numeral_Osub__code_I4_J, axiom, (! [N:num]: ((((code_sub @ one2) @ (bit0 @ N)) = (code_Neg @ (bitM @ N)))))).
% 9.43/3.46 thf(fact_4466_horner__sum__simps_I2_J, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (! [F:(B > A),A3:A,X:B,Xs:(list @ B)]: (((((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ (((cons @ B) @ X) @ Xs)) = (((plus_plus @ A) @ (F @ X)) @ (((times_times @ A) @ A3) @ (((((groups4207007520872428315er_sum @ B) @ A) @ F) @ A3) @ Xs)))))))))).
% 9.43/3.46 thf(fact_4467_underS__E, axiom, (! [A:$tType,I:A,R5:(set @ ((product_prod @ A) @ A)),J:A]: (((((member @ A) @ I) @ (((order_underS @ A) @ R5) @ J)) => ((I != J) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R5)))))).
% 9.43/3.46 thf(fact_4468_underS__I, axiom, (! [A:$tType,I:A,J:A,R5:(set @ ((product_prod @ A) @ A))]: (((I != J) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I) @ J)) @ R5) => (((member @ A) @ I) @ (((order_underS @ A) @ R5) @ J))))))).
% 9.43/3.46 thf(fact_4469_underS__def, axiom, (! [A:$tType]: (((order_underS @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A7:A]: (((collect @ A) @ (^ [B5:A]: (((B5 != A7) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A7)) @ R3))))))))))).
% 9.43/3.46 thf(fact_4470_Code__Numeral_Odup__code_I3_J, axiom, (! [N:num]: (((code_dup @ (code_Neg @ N)) = (code_Neg @ (bit0 @ N)))))).
% 9.43/3.46 thf(fact_4471_less__integer__code_I3_J, axiom, (! [L:num]: ((~ ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (code_Neg @ L))))))).
% 9.43/3.46 thf(fact_4472_less__integer__code_I7_J, axiom, (! [K:num]: ((((ord_less @ code_integer) @ (code_Neg @ K)) @ (zero_zero @ code_integer))))).
% 9.43/3.46 thf(fact_4473_plus__integer__code_I6_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ code_integer) @ (code_Neg @ M2)) @ (code_Neg @ N)) = (code_Neg @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4474_less__eq__integer__code_I7_J, axiom, (! [K:num]: ((((ord_less_eq @ code_integer) @ (code_Neg @ K)) @ (zero_zero @ code_integer))))).
% 9.43/3.46 thf(fact_4475_less__eq__integer__code_I3_J, axiom, (! [L:num]: ((~ ((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ (code_Neg @ L))))))).
% 9.43/3.46 thf(fact_4476_less__integer__code_I9_J, axiom, (! [K:num,L:num]: (((((ord_less @ code_integer) @ (code_Neg @ K)) @ (code_Neg @ L)) = (((ord_less @ num) @ L) @ K))))).
% 9.43/3.46 thf(fact_4477_less__eq__integer__code_I9_J, axiom, (! [K:num,L:num]: (((((ord_less_eq @ code_integer) @ (code_Neg @ K)) @ (code_Neg @ L)) = (((ord_less_eq @ num) @ L) @ K))))).
% 9.43/3.46 thf(fact_4478_minus__integer__code_I6_J, axiom, (! [M2:num,N:num]: (((((minus_minus @ code_integer) @ (code_Neg @ M2)) @ (code_Neg @ N)) = ((code_sub @ N) @ M2))))).
% 9.43/3.46 thf(fact_4479_divmod__abs__code_I2_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Neg @ K)) @ (code_Neg @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 9.43/3.46 thf(fact_4480_horner__sum__foldr, axiom, (! [A:$tType,B:$tType]: (((comm_semiring_0 @ A) => (((groups4207007520872428315er_sum @ B) @ A) = (^ [F2:(B > A),A7:A,Xs3:(list @ B)]: ((((((foldr @ B) @ A) @ (^ [X3:B,B5:A]: ((((plus_plus @ A) @ (F2 @ X3)) @ (((times_times @ A) @ A7) @ B5))))) @ Xs3) @ (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_4481_underS__incl__iff, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((order_679001287576687338der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => ((((member @ A) @ B2) @ ((field2 @ A) @ R)) => ((((ord_less_eq @ (set @ A)) @ (((order_underS @ A) @ R) @ A3)) @ (((order_underS @ A) @ R) @ B2)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)))))))).
% 9.43/3.46 thf(fact_4482_Code__Numeral_Osub__code_I5_J, axiom, (! [N:num]: ((((code_sub @ one2) @ (bit1 @ N)) = (code_Neg @ (bit0 @ N)))))).
% 9.43/3.46 thf(fact_4483_ordLess__iff__ordIso__Restr, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_We4044943003108391690rdLess @ B) @ A)) = (? [X3:A]: (((((member @ A) @ X3) @ ((field2 @ A) @ R)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ (((order_underS @ A) @ R) @ X3)) @ (^ [Uu:A]: ((((order_underS @ A) @ R) @ X3))))))) @ ((bNF_Wellorder_ordIso @ B) @ A))))))))))).
% 9.43/3.46 thf(fact_4484_sum__list__update, axiom, (! [A:$tType]: (((ordere1170586879665033532d_diff @ A) => (! [K:nat,Xs:(list @ A),X:A]: (((((ord_less @ nat) @ K) @ ((size_size @ (list @ A)) @ Xs)) => (((groups8242544230860333062m_list @ A) @ ((((list_update @ A) @ Xs) @ K) @ X)) = (((minus_minus @ A) @ (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ X)) @ (((nth @ A) @ Xs) @ K)))))))))).
% 9.43/3.46 thf(fact_4485_length__product__lists, axiom, (! [B:$tType,Xss:(list @ (list @ B))]: ((((size_size @ (list @ (list @ B))) @ ((product_lists @ B) @ Xss)) = (((((foldr @ nat) @ nat) @ (times_times @ nat)) @ ((((map @ (list @ B)) @ nat) @ (size_size @ (list @ B))) @ Xss)) @ (one_one @ nat)))))).
% 9.43/3.46 thf(fact_4486_horner__sum__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType]: ((((comm_semiring_0 @ B) & (comm_semiring_0 @ A)) => (! [A4:(A > (B > $o)),B6:(C > (D > $o))]: ((((A4 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A4) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A4) @ A4)) @ (plus_plus @ A)) @ (plus_plus @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A4) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A4) @ A4)) @ (times_times @ A)) @ (times_times @ B)) => ((((((((bNF_rel_fun @ (C > A)) @ (D > B)) @ (A > ((list @ C) > A))) @ (B > ((list @ D) > B))) @ ((((((bNF_rel_fun @ C) @ D) @ A) @ B) @ B6) @ A4)) @ ((((((bNF_rel_fun @ A) @ B) @ ((list @ C) > A)) @ ((list @ D) > B)) @ A4) @ ((((((bNF_rel_fun @ (list @ C)) @ (list @ D)) @ A) @ B) @ (((list_all2 @ C) @ D) @ B6)) @ A4))) @ ((groups4207007520872428315er_sum @ C) @ A)) @ ((groups4207007520872428315er_sum @ D) @ B))))))))))).
% 9.43/3.46 thf(fact_4487_sum__list_OCons, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [X:A,Xs:(list @ A)]: ((((groups8242544230860333062m_list @ A) @ (((cons @ A) @ X) @ Xs)) = (((plus_plus @ A) @ X) @ ((groups8242544230860333062m_list @ A) @ Xs))))))))).
% 9.43/3.46 thf(fact_4488_sum__list__append, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [Xs:(list @ A),Ys2:(list @ A)]: ((((groups8242544230860333062m_list @ A) @ (((append @ A) @ Xs) @ Ys2)) = (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ Xs)) @ ((groups8242544230860333062m_list @ A) @ Ys2))))))))).
% 9.43/3.46 thf(fact_4489_ordIso__transitive, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_Wellorder_ordIso @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_Wellorder_ordIso @ A) @ C))))))).
% 9.43/3.46 thf(fact_4490_ordIso__symmetric, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordIso @ B) @ A)))))).
% 9.43/3.46 thf(fact_4491_sum__list__transfer, axiom, (! [A:$tType,B:$tType]: ((((monoid_add @ B) & (monoid_add @ A)) => (! [A4:(A > (B > $o))]: ((((A4 @ (zero_zero @ A)) @ (zero_zero @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A4) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A4) @ A4)) @ (plus_plus @ A)) @ (plus_plus @ B)) => ((((((((bNF_rel_fun @ (list @ A)) @ (list @ B)) @ A) @ B) @ (((list_all2 @ A) @ B) @ A4)) @ A4) @ (groups8242544230860333062m_list @ A)) @ (groups8242544230860333062m_list @ B)))))))))).
% 9.43/3.46 thf(fact_4492_Misc_Olist__all2__induct, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),L:(list @ A),L4:(list @ B),Q:((list @ A) > ((list @ B) > $o))]: (((((((list_all2 @ A) @ B) @ P) @ L) @ L4) => (((Q @ (nil @ A)) @ (nil @ B)) => ((! [X2:A,X5:B,Ls2:(list @ A),Ls3:(list @ B)]: ((((P @ X2) @ X5) => ((((((list_all2 @ A) @ B) @ P) @ Ls2) @ Ls3) => (((Q @ Ls2) @ Ls3) => ((Q @ (((cons @ A) @ X2) @ Ls2)) @ (((cons @ B) @ X5) @ Ls3))))))) => ((Q @ L) @ L4))))))).
% 9.43/3.46 thf(fact_4493_ordIso__iff__ordLeq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) = ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A))))))).
% 9.43/3.46 thf(fact_4494_ordIso__imp__ordLeq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))).
% 9.43/3.46 thf(fact_4495_ordIso__ordLeq__trans, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_Wellorder_ordLeq @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_Wellorder_ordLeq @ A) @ C))))))).
% 9.43/3.46 thf(fact_4496_ordLeq__ordIso__trans, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_Wellorder_ordIso @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_Wellorder_ordLeq @ A) @ C))))))).
% 9.43/3.46 thf(fact_4497_not__ordLess__ordIso, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => (~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)))))))).
% 9.43/3.46 thf(fact_4498_ordIso__ordLess__trans, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_We4044943003108391690rdLess @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_We4044943003108391690rdLess @ A) @ C))))))).
% 9.43/3.46 thf(fact_4499_ordLess__ordIso__trans, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),R9:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R4) @ R9)) @ ((bNF_Wellorder_ordIso @ B) @ C)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ C) @ C))) @ R) @ R9)) @ ((bNF_We4044943003108391690rdLess @ A) @ C))))))).
% 9.43/3.46 thf(fact_4500_sum__list__addf, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [F:(B > A),G:(B > A),Xs:(list @ B)]: ((((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (^ [X3:B]: ((((plus_plus @ A) @ (F @ X3)) @ (G @ X3))))) @ Xs)) = (((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ Xs))) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ G) @ Xs)))))))))).
% 9.43/3.46 thf(fact_4501_finite__well__order__on__ordIso, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ A) @ A4) => ((((order_well_order_on @ A) @ A4) @ R) => ((((order_well_order_on @ A) @ A4) @ R4) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ A)))))))).
% 9.43/3.46 thf(fact_4502_ordIso__reflexive, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R)) @ ((bNF_Wellorder_ordIso @ A) @ A)))))).
% 9.43/3.46 thf(fact_4503_uminus__sum__list__map, axiom, (! [A:$tType,B:$tType]: (((ab_group_add @ A) => (! [F:(B > A),Xs:(list @ B)]: ((((uminus_uminus @ A) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ Xs))) = ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ (((((comp @ A) @ A) @ B) @ (uminus_uminus @ A)) @ F)) @ Xs))))))))).
% 9.43/3.46 thf(fact_4504_ordLeq__iff__ordLess__or__ordIso, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) = ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) | (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B))))))).
% 9.43/3.46 thf(fact_4505_sum__list_Oeq__foldr, axiom, (! [A:$tType]: (((monoid_add @ A) => ((groups8242544230860333062m_list @ A) = (^ [Xs3:(list @ A)]: ((((((foldr @ A) @ A) @ (plus_plus @ A)) @ Xs3) @ (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_4506_internalize__ordLeq, axiom, (! [A:$tType,B:$tType,R4:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) = (? [P6:(set @ ((product_prod @ B) @ B))]: (((((ord_less_eq @ (set @ B)) @ ((field2 @ B) @ P6)) @ ((field2 @ B) @ R)) & ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R4) @ P6)) @ ((bNF_Wellorder_ordIso @ A) @ B)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ P6) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ B)))))))))).
% 9.43/3.46 thf(fact_4507_internalize__ordLess, axiom, (! [A:$tType,B:$tType,R4:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R4) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) = (? [P6:(set @ ((product_prod @ B) @ B))]: (((((ord_less @ (set @ B)) @ ((field2 @ B) @ P6)) @ ((field2 @ B) @ R)) & ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R4) @ P6)) @ ((bNF_Wellorder_ordIso @ A) @ B)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ P6) @ R)) @ ((bNF_We4044943003108391690rdLess @ B) @ B)))))))))).
% 9.43/3.46 thf(fact_4508_card__length__sum__list__rec, axiom, (! [M2:nat,N6:nat]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ M2) => (((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = M2) & (((groups8242544230860333062m_list @ nat) @ L3) = N6)))))) = (((plus_plus @ nat) @ ((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = (((minus_minus @ nat) @ M2) @ (one_one @ nat))) & (((groups8242544230860333062m_list @ nat) @ L3) = N6))))))) @ ((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = M2) & ((((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ L3)) @ (one_one @ nat)) = N6)))))))))))).
% 9.43/3.46 thf(fact_4509_size__list__conv__sum__list, axiom, (! [B:$tType]: (((size_list @ B) = (^ [F2:(B > nat),Xs3:(list @ B)]: ((((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ B) @ nat) @ F2) @ Xs3))) @ ((size_size @ (list @ B)) @ Xs3)))))))).
% 9.43/3.46 thf(fact_4510_card__length__sum__list, axiom, (! [M2:nat,N6:nat]: ((((finite_card @ (list @ nat)) @ ((collect @ (list @ nat)) @ (^ [L3:(list @ nat)]: (((((size_size @ (list @ nat)) @ L3) = M2) & (((groups8242544230860333062m_list @ nat) @ L3) = N6)))))) = ((binomial @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ N6) @ M2)) @ (one_one @ nat))) @ N6))))).
% 9.43/3.46 thf(fact_4511_sum__list__Suc, axiom, (! [A:$tType,F:(A > nat),Xs:(list @ A)]: ((((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ (^ [X3:A]: ((suc @ (F @ X3))))) @ Xs)) = (((plus_plus @ nat) @ ((groups8242544230860333062m_list @ nat) @ ((((map @ A) @ nat) @ F) @ Xs))) @ ((size_size @ (list @ A)) @ Xs)))))).
% 9.43/3.46 thf(fact_4512_select__weight__def, axiom, (! [A:$tType]: (((select_weight @ A) = (^ [Xs3:(list @ ((product_prod @ code_natural) @ A))]: (((((((product_scomp @ ((product_prod @ code_natural) @ code_natural)) @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ A) @ ((product_prod @ code_natural) @ code_natural))) @ (range @ ((groups8242544230860333062m_list @ code_natural) @ ((((map @ ((product_prod @ code_natural) @ A)) @ code_natural) @ ((product_fst @ code_natural) @ A)) @ Xs3)))) @ (^ [K4:code_natural]: ((((product_Pair @ A) @ ((product_prod @ code_natural) @ code_natural)) @ (((pick @ A) @ Xs3) @ K4))))))))))).
% 9.43/3.46 thf(fact_4513_prod__list__transfer, axiom, (! [A:$tType,B:$tType]: ((((monoid_mult @ B) & (monoid_mult @ A)) => (! [A4:(A > (B > $o))]: ((((A4 @ (one_one @ A)) @ (one_one @ B)) => (((((((((bNF_rel_fun @ A) @ B) @ (A > A)) @ (B > B)) @ A4) @ ((((((bNF_rel_fun @ A) @ B) @ A) @ B) @ A4) @ A4)) @ (times_times @ A)) @ (times_times @ B)) => ((((((((bNF_rel_fun @ (list @ A)) @ (list @ B)) @ A) @ B) @ (((list_all2 @ A) @ B) @ A4)) @ A4) @ (groups5270119922927024881d_list @ A)) @ (groups5270119922927024881d_list @ B)))))))))).
% 9.43/3.46 thf(fact_4514_dir__image__ordIso, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(A > B)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => (((((inj_on @ A) @ B) @ F) @ ((field2 @ A) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ ((((bNF_We2720479622203943262_image @ A) @ B) @ R) @ F))) @ ((bNF_Wellorder_ordIso @ A) @ B))))))).
% 9.43/3.46 thf(fact_4515_underS__incr, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((trans @ A) @ R) => (((antisym @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((ord_less_eq @ (set @ A)) @ (((order_underS @ A) @ R) @ A3)) @ (((order_underS @ A) @ R) @ B2)))))))).
% 9.43/3.46 thf(fact_4516_prod__list_ONil, axiom, (! [A:$tType]: (((monoid_mult @ A) => (((groups5270119922927024881d_list @ A) @ (nil @ A)) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_4517_antisymD, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((antisym @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ R) => (A3 = B2))))))).
% 9.43/3.46 thf(fact_4518_antisymI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [X2:A,Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X2)) @ R) => (X2 = Y4))))) => ((antisym @ A) @ R))))).
% 9.43/3.46 thf(fact_4519_antisym__def, axiom, (! [A:$tType]: (((antisym @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R3) => (X3 = Y3))))))))))).
% 9.43/3.46 thf(fact_4520_dir__image__def, axiom, (! [A2:$tType,A:$tType]: ((((bNF_We2720479622203943262_image @ A) @ A2) = (^ [R3:(set @ ((product_prod @ A) @ A)),F2:(A > A2)]: (((collect @ ((product_prod @ A2) @ A2)) @ (^ [Uu:((product_prod @ A2) @ A2)]: ((? [A7:A,B5:A]: (((Uu = ((((product_Pair @ A2) @ A2) @ (F2 @ A7)) @ (F2 @ B5))) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R3))))))))))))).
% 9.43/3.46 thf(fact_4521_prod__list_Oeq__foldr, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((groups5270119922927024881d_list @ A) = (^ [Xs3:(list @ A)]: ((((((foldr @ A) @ A) @ (times_times @ A)) @ Xs3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4522_pick__same, axiom, (! [A:$tType,L:nat,Xs:(list @ A)]: (((((ord_less @ nat) @ L) @ ((size_size @ (list @ A)) @ Xs)) => ((((pick @ A) @ ((((map @ A) @ ((product_prod @ code_natural) @ A)) @ (((product_Pair @ code_natural) @ A) @ (one_one @ code_natural))) @ Xs)) @ (code_natural_of_nat @ L)) = (((nth @ A) @ Xs) @ L)))))).
% 9.43/3.46 thf(fact_4523_flip__pred, axiom, (! [A:$tType,B:$tType,A4:(set @ ((product_prod @ A) @ B)),R5:(B > (A > $o))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ A4) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (((conversep @ B) @ A) @ R5)))) => (((ord_less_eq @ (set @ ((product_prod @ B) @ A))) @ ((((image2 @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ A) @ B) @ ((product_prod @ B) @ A)) @ (^ [X3:A,Y3:B]: (((((product_Pair @ B) @ A) @ Y3) @ X3))))) @ A4)) @ ((collect @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ R5))))))).
% 9.43/3.46 thf(fact_4524_prod__set__simps_I1_J, axiom, (! [B:$tType,A:$tType,X:A,Y:B]: (((((basic_fsts @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = (((insert2 @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_4525_prod__set__simps_I2_J, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: (((((basic_snds @ A) @ B) @ ((((product_Pair @ A) @ B) @ X) @ Y)) = (((insert2 @ B) @ Y) @ (bot_bot @ (set @ B))))))).
% 9.43/3.46 thf(fact_4526_natural__of__nat__of__nat, axiom, (code_natural_of_nat = (semiring_1_of_nat @ code_natural))).
% 9.43/3.46 thf(fact_4527_division__segment__natural_Oabs__eq, axiom, (! [X:nat]: ((((euclid7384307370059645450egment @ code_natural) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ ((euclid7384307370059645450egment @ nat) @ X)))))).
% 9.43/3.46 thf(fact_4528_set__bit__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se5668285175392031749et_bit @ code_natural) @ Xa) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se5668285175392031749et_bit @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4529_flip__bit__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se8732182000553998342ip_bit @ code_natural) @ Xa) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se8732182000553998342ip_bit @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4530_unset__bit__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se2638667681897837118et_bit @ code_natural) @ Xa) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se2638667681897837118et_bit @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4531_euclidean__size__natural_Oabs__eq, axiom, (! [X:nat]: ((((euclid6346220572633701492n_size @ code_natural) @ (code_natural_of_nat @ X)) = ((euclid6346220572633701492n_size @ nat) @ X))))).
% 9.43/3.46 thf(fact_4532_bit__natural_Oabs__eq, axiom, (! [X:nat]: ((((bit_se5641148757651400278ts_bit @ code_natural) @ (code_natural_of_nat @ X)) = ((bit_se5641148757651400278ts_bit @ nat) @ X))))).
% 9.43/3.46 thf(fact_4533_natural__of__nat__cases, axiom, (! [X:code_natural]: ((~ ((! [Y4:nat]: (((X = (code_natural_of_nat @ Y4)) => (~ ((((member @ nat) @ Y4) @ (top_top @ (set @ nat))))))))))))).
% 9.43/3.46 thf(fact_4534_natural__of__nat__induct, axiom, (! [P:(code_natural > $o),X:code_natural]: (((! [Y4:nat]: (((((member @ nat) @ Y4) @ (top_top @ (set @ nat))) => (P @ (code_natural_of_nat @ Y4))))) => (P @ X))))).
% 9.43/3.46 thf(fact_4535_natural__of__nat__inject, axiom, (! [X:nat,Y:nat]: (((((member @ nat) @ X) @ (top_top @ (set @ nat))) => ((((member @ nat) @ Y) @ (top_top @ (set @ nat))) => (((code_natural_of_nat @ X) = (code_natural_of_nat @ Y)) = (X = Y))))))).
% 9.43/3.46 thf(fact_4536_take__bit__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se2584673776208193580ke_bit @ code_natural) @ Xa) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se2584673776208193580ke_bit @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4537_and__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se5824344872417868541ns_and @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se5824344872417868541ns_and @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4538_xor__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se5824344971392196577ns_xor @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se5824344971392196577ns_xor @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4539_or__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se1065995026697491101ons_or @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se1065995026697491101ons_or @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4540_mask__natural_Oabs__eq, axiom, ((bit_se2239418461657761734s_mask @ code_natural) = (^ [X3:nat]: ((code_natural_of_nat @ ((bit_se2239418461657761734s_mask @ nat) @ X3)))))).
% 9.43/3.46 thf(fact_4541_push__bit__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se4730199178511100633sh_bit @ code_natural) @ Xa) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se4730199178511100633sh_bit @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4542_drop__bit__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((bit_se4197421643247451524op_bit @ code_natural) @ Xa) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((bit_se4197421643247451524op_bit @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4543_zero__natural__def, axiom, ((zero_zero @ code_natural) = (code_natural_of_nat @ (zero_zero @ nat)))).
% 9.43/3.46 thf(fact_4544_one__natural__def, axiom, ((one_one @ code_natural) = (code_natural_of_nat @ (one_one @ nat)))).
% 9.43/3.46 thf(fact_4545_less__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((ord_less @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (((ord_less @ nat) @ Xa) @ X))))).
% 9.43/3.46 thf(fact_4546_plus__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((plus_plus @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((plus_plus @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4547_minus__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((minus_minus @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((minus_minus @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4548_less__eq__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((ord_less_eq @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (((ord_less_eq @ nat) @ Xa) @ X))))).
% 9.43/3.46 thf(fact_4549_times__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((times_times @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((times_times @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4550_divide__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((divide_divide @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((divide_divide @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4551_mask__natural__def, axiom, ((bit_se2239418461657761734s_mask @ code_natural) = (((((((map_fun @ nat) @ nat) @ nat) @ code_natural) @ (id @ nat)) @ code_natural_of_nat) @ (bit_se2239418461657761734s_mask @ nat)))).
% 9.43/3.46 thf(fact_4552_modulo__natural_Oabs__eq, axiom, (! [Xa:nat,X:nat]: (((((modulo_modulo @ code_natural) @ (code_natural_of_nat @ Xa)) @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (((modulo_modulo @ nat) @ Xa) @ X)))))).
% 9.43/3.46 thf(fact_4553_select__def, axiom, (! [A:$tType]: (((select @ A) = (^ [Xs3:(list @ A)]: (((((((product_scomp @ ((product_prod @ code_natural) @ code_natural)) @ code_natural) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ A) @ ((product_prod @ code_natural) @ code_natural))) @ (range @ (code_natural_of_nat @ ((size_size @ (list @ A)) @ Xs3)))) @ (^ [K4:code_natural]: ((((product_Pair @ A) @ ((product_prod @ code_natural) @ code_natural)) @ (((nth @ A) @ Xs3) @ (code_nat_of_natural @ K4)))))))))))).
% 9.43/3.46 thf(fact_4554_Nat_Oabs__eq, axiom, (! [X:int]: (((code_Nat @ (code_integer_of_int @ X)) = (code_natural_of_nat @ (nat2 @ X)))))).
% 9.43/3.46 thf(fact_4555_Nat__def, axiom, (code_Nat = (((((((map_fun @ code_integer) @ int) @ nat) @ code_natural) @ code_int_of_integer) @ code_natural_of_nat) @ nat2))).
% 9.43/3.46 thf(fact_4556_natural__of__integer_Oabs__eq, axiom, (! [X:int]: (((code_n4118661773612635043nteger @ (code_integer_of_int @ X)) = (code_natural_of_nat @ (nat2 @ X)))))).
% 9.43/3.46 thf(fact_4557_natural__of__nat__of__natural__inverse, axiom, (! [N:code_natural]: (((code_natural_of_nat @ (code_nat_of_natural @ N)) = N)))).
% 9.43/3.46 thf(fact_4558_nat__of__natural__of__nat__inverse, axiom, (! [N:nat]: (((code_nat_of_natural @ (code_natural_of_nat @ N)) = N)))).
% 9.43/3.46 thf(fact_4559_nat__of__natural__inverse, axiom, (! [X:code_natural]: (((code_natural_of_nat @ (code_nat_of_natural @ X)) = X)))).
% 9.43/3.46 thf(fact_4560_nat__of__natural__min, axiom, (! [K:code_natural,L:code_natural]: (((code_nat_of_natural @ (((ord_min @ code_natural) @ K) @ L)) = (((ord_min @ nat) @ (code_nat_of_natural @ K)) @ (code_nat_of_natural @ L)))))).
% 9.43/3.46 thf(fact_4561_nat__of__natural__max, axiom, (! [K:code_natural,L:code_natural]: (((code_nat_of_natural @ (((ord_max @ code_natural) @ K) @ L)) = (((ord_max @ nat) @ (code_nat_of_natural @ K)) @ (code_nat_of_natural @ L)))))).
% 9.43/3.46 thf(fact_4562_of__nat__of__natural, axiom, (! [N:code_natural]: ((((semiring_1_of_nat @ code_natural) @ (code_nat_of_natural @ N)) = N)))).
% 9.43/3.46 thf(fact_4563_nat__of__natural__of__nat, axiom, (! [N:nat]: (((code_nat_of_natural @ ((semiring_1_of_nat @ code_natural) @ N)) = N)))).
% 9.43/3.46 thf(fact_4564_zero__natural_Orep__eq, axiom, ((code_nat_of_natural @ (zero_zero @ code_natural)) = (zero_zero @ nat))).
% 9.43/3.46 thf(fact_4565_one__natural_Orep__eq, axiom, ((code_nat_of_natural @ (one_one @ code_natural)) = (one_one @ nat))).
% 9.43/3.46 thf(fact_4566_plus__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((plus_plus @ code_natural) @ X) @ Xa)) = (((plus_plus @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4567_minus__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((minus_minus @ code_natural) @ X) @ Xa)) = (((minus_minus @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4568_nat__of__natural__numeral, axiom, (! [K:num]: (((code_nat_of_natural @ ((numeral_numeral @ code_natural) @ K)) = ((numeral_numeral @ nat) @ K))))).
% 9.43/3.46 thf(fact_4569_times__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((times_times @ code_natural) @ X) @ Xa)) = (((times_times @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4570_divide__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((divide_divide @ code_natural) @ X) @ Xa)) = (((divide_divide @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4571_modulo__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((modulo_modulo @ code_natural) @ X) @ Xa)) = (((modulo_modulo @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4572_division__segment__natural_Orep__eq, axiom, (! [X:code_natural]: (((code_nat_of_natural @ ((euclid7384307370059645450egment @ code_natural) @ X)) = ((euclid7384307370059645450egment @ nat) @ (code_nat_of_natural @ X)))))).
% 9.43/3.46 thf(fact_4573_euclidean__size__natural_Orep__eq, axiom, ((euclid6346220572633701492n_size @ code_natural) = (^ [X3:code_natural]: (((euclid6346220572633701492n_size @ nat) @ (code_nat_of_natural @ X3)))))).
% 9.43/3.46 thf(fact_4574_unset__bit__natural_Orep__eq, axiom, (! [X:nat,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se2638667681897837118et_bit @ code_natural) @ X) @ Xa)) = (((bit_se2638667681897837118et_bit @ nat) @ X) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4575_flip__bit__natural_Orep__eq, axiom, (! [X:nat,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se8732182000553998342ip_bit @ code_natural) @ X) @ Xa)) = (((bit_se8732182000553998342ip_bit @ nat) @ X) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4576_set__bit__natural_Orep__eq, axiom, (! [X:nat,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se5668285175392031749et_bit @ code_natural) @ X) @ Xa)) = (((bit_se5668285175392031749et_bit @ nat) @ X) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4577_Nat_Orep__eq, axiom, (! [X:code_integer]: (((code_nat_of_natural @ (code_Nat @ X)) = (nat2 @ (code_int_of_integer @ X)))))).
% 9.43/3.46 thf(fact_4578_push__bit__natural_Orep__eq, axiom, (! [X:nat,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se4730199178511100633sh_bit @ code_natural) @ X) @ Xa)) = (((bit_se4730199178511100633sh_bit @ nat) @ X) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4579_mask__natural_Orep__eq, axiom, (! [X:nat]: (((code_nat_of_natural @ ((bit_se2239418461657761734s_mask @ code_natural) @ X)) = ((bit_se2239418461657761734s_mask @ nat) @ X))))).
% 9.43/3.46 thf(fact_4580_and__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se5824344872417868541ns_and @ code_natural) @ X) @ Xa)) = (((bit_se5824344872417868541ns_and @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4581_bit__natural_Orep__eq, axiom, ((bit_se5641148757651400278ts_bit @ code_natural) = (^ [X3:code_natural]: (((bit_se5641148757651400278ts_bit @ nat) @ (code_nat_of_natural @ X3)))))).
% 9.43/3.46 thf(fact_4582_nat__of__natural__inject, axiom, (! [X:code_natural,Y:code_natural]: ((((code_nat_of_natural @ X) = (code_nat_of_natural @ Y)) = (X = Y))))).
% 9.43/3.46 thf(fact_4583_natural__eqI, axiom, (! [M2:code_natural,N:code_natural]: ((((code_nat_of_natural @ M2) = (code_nat_of_natural @ N)) => (M2 = N))))).
% 9.43/3.46 thf(fact_4584_natural__eq__iff, axiom, ((^ [Y5:code_natural,Z4:code_natural]: ((Y5 = Z4))) = (^ [M:code_natural,N5:code_natural]: (((code_nat_of_natural @ M) = (code_nat_of_natural @ N5)))))).
% 9.43/3.46 thf(fact_4585_natural__cases, axiom, (! [M2:code_natural]: ((~ ((! [N3:nat]: ((M2 != ((semiring_1_of_nat @ code_natural) @ N3))))))))).
% 9.43/3.46 thf(fact_4586_nat__of__natural, axiom, (! [X:code_natural]: ((((member @ nat) @ (code_nat_of_natural @ X)) @ (top_top @ (set @ nat)))))).
% 9.43/3.46 thf(fact_4587_nat__of__natural__cases, axiom, (! [Y:nat]: (((((member @ nat) @ Y) @ (top_top @ (set @ nat))) => (~ ((! [X2:code_natural]: ((Y != (code_nat_of_natural @ X2)))))))))).
% 9.43/3.46 thf(fact_4588_nat__of__natural__induct, axiom, (! [Y:nat,P:(nat > $o)]: (((((member @ nat) @ Y) @ (top_top @ (set @ nat))) => ((! [X2:code_natural]: ((P @ (code_nat_of_natural @ X2)))) => (P @ Y)))))).
% 9.43/3.46 thf(fact_4589_take__bit__natural_Orep__eq, axiom, (! [X:nat,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se2584673776208193580ke_bit @ code_natural) @ X) @ Xa)) = (((bit_se2584673776208193580ke_bit @ nat) @ X) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4590_or__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se1065995026697491101ons_or @ code_natural) @ X) @ Xa)) = (((bit_se1065995026697491101ons_or @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4591_xor__natural_Orep__eq, axiom, (! [X:code_natural,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se5824344971392196577ns_xor @ code_natural) @ X) @ Xa)) = (((bit_se5824344971392196577ns_xor @ nat) @ (code_nat_of_natural @ X)) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4592_drop__bit__natural_Orep__eq, axiom, (! [X:nat,Xa:code_natural]: (((code_nat_of_natural @ (((bit_se4197421643247451524op_bit @ code_natural) @ X) @ Xa)) = (((bit_se4197421643247451524op_bit @ nat) @ X) @ (code_nat_of_natural @ Xa)))))).
% 9.43/3.46 thf(fact_4593_natural__of__integer_Orep__eq, axiom, (! [X:code_integer]: (((code_nat_of_natural @ (code_n4118661773612635043nteger @ X)) = (nat2 @ (code_int_of_integer @ X)))))).
% 9.43/3.46 thf(fact_4594_less__natural_Orep__eq, axiom, ((ord_less @ code_natural) = (^ [X3:code_natural,Xa4:code_natural]: ((((ord_less @ nat) @ (code_nat_of_natural @ X3)) @ (code_nat_of_natural @ Xa4)))))).
% 9.43/3.46 thf(fact_4595_natural__of__nat__inverse, axiom, (! [Y:nat]: (((((member @ nat) @ Y) @ (top_top @ (set @ nat))) => ((code_nat_of_natural @ (code_natural_of_nat @ Y)) = Y))))).
% 9.43/3.46 thf(fact_4596_type__definition__natural, axiom, (((((type_definition @ code_natural) @ nat) @ code_nat_of_natural) @ code_natural_of_nat) @ (top_top @ (set @ nat)))).
% 9.43/3.46 thf(fact_4597_or__natural__def, axiom, ((bit_se1065995026697491101ons_or @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se1065995026697491101ons_or @ nat)))).
% 9.43/3.46 thf(fact_4598_xor__natural__def, axiom, ((bit_se5824344971392196577ns_xor @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se5824344971392196577ns_xor @ nat)))).
% 9.43/3.46 thf(fact_4599_and__natural__def, axiom, ((bit_se5824344872417868541ns_and @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se5824344872417868541ns_and @ nat)))).
% 9.43/3.46 thf(fact_4600_less__eq__natural_Orep__eq, axiom, ((ord_less_eq @ code_natural) = (^ [X3:code_natural,Xa4:code_natural]: ((((ord_less_eq @ nat) @ (code_nat_of_natural @ X3)) @ (code_nat_of_natural @ Xa4)))))).
% 9.43/3.46 thf(fact_4601_division__segment__natural__def, axiom, ((euclid7384307370059645450egment @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat) @ (euclid7384307370059645450egment @ nat)))).
% 9.43/3.46 thf(fact_4602_euclidean__size__natural__def, axiom, ((euclid6346220572633701492n_size @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ nat) @ nat) @ code_nat_of_natural) @ (id @ nat)) @ (euclid6346220572633701492n_size @ nat)))).
% 9.43/3.46 thf(fact_4603_plus__natural__def, axiom, ((plus_plus @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (plus_plus @ nat)))).
% 9.43/3.46 thf(fact_4604_minus__natural__def, axiom, ((minus_minus @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (minus_minus @ nat)))).
% 9.43/3.46 thf(fact_4605_bit__natural__def, axiom, ((bit_se5641148757651400278ts_bit @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > $o)) @ (nat > $o)) @ code_nat_of_natural) @ (id @ (nat > $o))) @ (bit_se5641148757651400278ts_bit @ nat)))).
% 9.43/3.46 thf(fact_4606_less__natural__def, axiom, ((ord_less @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > $o)) @ (code_natural > $o)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ $o) @ $o) @ code_nat_of_natural) @ (id @ $o))) @ (ord_less @ nat)))).
% 9.43/3.46 thf(fact_4607_times__natural__def, axiom, ((times_times @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (times_times @ nat)))).
% 9.43/3.46 thf(fact_4608_divide__natural__def, axiom, ((divide_divide @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (divide_divide @ nat)))).
% 9.43/3.46 thf(fact_4609_take__bit__natural__def, axiom, ((bit_se2584673776208193580ke_bit @ code_natural) = (((((((map_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (id @ nat)) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se2584673776208193580ke_bit @ nat)))).
% 9.43/3.46 thf(fact_4610_push__bit__natural__def, axiom, ((bit_se4730199178511100633sh_bit @ code_natural) = (((((((map_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (id @ nat)) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se4730199178511100633sh_bit @ nat)))).
% 9.43/3.46 thf(fact_4611_less__eq__natural__def, axiom, ((ord_less_eq @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > $o)) @ (code_natural > $o)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ $o) @ $o) @ code_nat_of_natural) @ (id @ $o))) @ (ord_less_eq @ nat)))).
% 9.43/3.46 thf(fact_4612_modulo__natural__def, axiom, ((modulo_modulo @ code_natural) = (((((((map_fun @ code_natural) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ code_nat_of_natural) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (modulo_modulo @ nat)))).
% 9.43/3.46 thf(fact_4613_drop__bit__natural__def, axiom, ((bit_se4197421643247451524op_bit @ code_natural) = (((((((map_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (id @ nat)) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se4197421643247451524op_bit @ nat)))).
% 9.43/3.46 thf(fact_4614_unset__bit__natural__def, axiom, ((bit_se2638667681897837118et_bit @ code_natural) = (((((((map_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (id @ nat)) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se2638667681897837118et_bit @ nat)))).
% 9.43/3.46 thf(fact_4615_flip__bit__natural__def, axiom, ((bit_se8732182000553998342ip_bit @ code_natural) = (((((((map_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (id @ nat)) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se8732182000553998342ip_bit @ nat)))).
% 9.43/3.46 thf(fact_4616_set__bit__natural__def, axiom, ((bit_se5668285175392031749et_bit @ code_natural) = (((((((map_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (id @ nat)) @ ((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat)) @ (bit_se5668285175392031749et_bit @ nat)))).
% 9.43/3.46 thf(fact_4617_natural__decr, axiom, (! [N:code_natural]: (((N != (zero_zero @ code_natural)) => (((ord_less @ nat) @ (((minus_minus @ nat) @ (code_nat_of_natural @ N)) @ (suc @ (zero_zero @ nat)))) @ (code_nat_of_natural @ N)))))).
% 9.43/3.46 thf(fact_4618_natural__of__integer__def, axiom, (code_n4118661773612635043nteger = (((((((map_fun @ code_integer) @ int) @ nat) @ code_natural) @ code_int_of_integer) @ code_natural_of_nat) @ nat2))).
% 9.43/3.46 thf(fact_4619_integer__of__natural__of__integer, axiom, (! [K:code_integer]: (((code_i5400310926305786745atural @ (code_n4118661773612635043nteger @ K)) = (((ord_max @ code_integer) @ (zero_zero @ code_integer)) @ K))))).
% 9.43/3.46 thf(fact_4620_integer__of__natural_Orep__eq, axiom, (! [X:code_natural]: (((code_int_of_integer @ (code_i5400310926305786745atural @ X)) = ((semiring_1_of_nat @ int) @ (code_nat_of_natural @ X)))))).
% 9.43/3.46 thf(fact_4621_int__of__integer__of__natural, axiom, (! [N:code_natural]: (((code_int_of_integer @ (code_i5400310926305786745atural @ N)) = ((semiring_1_of_nat @ int) @ (code_nat_of_natural @ N)))))).
% 9.43/3.46 thf(fact_4622_integer__of__natural__of__nat, axiom, (! [N:nat]: (((code_i5400310926305786745atural @ ((semiring_1_of_nat @ code_natural) @ N)) = ((semiring_1_of_nat @ code_integer) @ N))))).
% 9.43/3.46 thf(fact_4623_natural__of__integer__of__natural, axiom, (! [N:code_natural]: (((code_n4118661773612635043nteger @ (code_i5400310926305786745atural @ N)) = N)))).
% 9.43/3.46 thf(fact_4624_integer__of__natural__def, axiom, (code_i5400310926305786745atural = (((((((map_fun @ code_natural) @ nat) @ int) @ code_integer) @ code_nat_of_natural) @ code_integer_of_int) @ (semiring_1_of_nat @ int)))).
% 9.43/3.46 thf(fact_4625_integer__of__natural_Oabs__eq, axiom, (! [X:nat]: (((code_i5400310926305786745atural @ (code_natural_of_nat @ X)) = (code_integer_of_int @ ((semiring_1_of_nat @ int) @ X)))))).
% 9.43/3.46 thf(fact_4626_Code__Numeral_Osize__nat__def, axiom, (code_size_nat = code_nat_of_natural)).
% 9.43/3.46 thf(fact_4627_Code__Numeral_OSuc__def, axiom, (code_Suc = (((((((map_fun @ code_natural) @ nat) @ nat) @ code_natural) @ code_nat_of_natural) @ code_natural_of_nat) @ suc))).
% 9.43/3.46 thf(fact_4628_sum__mset_Ounion__disjoint, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [A4:(multiset @ B),B6:(multiset @ B),G:(B > A)]: ((((((inter_mset @ B) @ A4) @ B6) = (zero_zero @ (multiset @ B))) => (((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ G) @ (((union_mset @ B) @ A4) @ B6))) = (((plus_plus @ A) @ ((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ G) @ A4))) @ ((comm_m7189776963980413722m_mset @ A) @ ((((image_mset @ B) @ A) @ G) @ B6))))))))))).
% 9.43/3.46 thf(fact_4629_natural_Oinject, axiom, (! [Natural:code_natural,Natural2:code_natural]: ((((code_Suc @ Natural) = (code_Suc @ Natural2)) = (Natural = Natural2))))).
% 9.43/3.46 thf(fact_4630_natural_Osimps_I7_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > (T > T)),Natural:code_natural]: ((((((code_rec_natural @ T) @ F1) @ F22) @ (code_Suc @ Natural)) = ((F22 @ Natural) @ ((((code_rec_natural @ T) @ F1) @ F22) @ Natural)))))).
% 9.43/3.46 thf(fact_4631_natural_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:(code_natural > T),Natural:code_natural]: ((((((code_case_natural @ T) @ F1) @ F22) @ (code_Suc @ Natural)) = (F22 @ Natural))))).
% 9.43/3.46 thf(fact_4632_Suc_Orep__eq, axiom, (! [X:code_natural]: (((code_nat_of_natural @ (code_Suc @ X)) = (suc @ (code_nat_of_natural @ X)))))).
% 9.43/3.46 thf(fact_4633_union__sup__distrib__right, axiom, (! [A:$tType,C5:(multiset @ A),A4:(multiset @ A),B6:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ C5) @ (((union_mset @ A) @ A4) @ B6)) = (((union_mset @ A) @ (((plus_plus @ (multiset @ A)) @ C5) @ A4)) @ (((plus_plus @ (multiset @ A)) @ C5) @ B6)))))).
% 9.43/3.46 thf(fact_4634_sup__union__distrib__left, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A),C5:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((union_mset @ A) @ A4) @ B6)) @ C5) = (((union_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ C5)) @ (((plus_plus @ (multiset @ A)) @ B6) @ C5)))))).
% 9.43/3.46 thf(fact_4635_natural_Oexhaust, axiom, (! [Y:code_natural]: (((Y != (zero_zero @ code_natural)) => (~ ((! [Natural3:code_natural]: ((Y != (code_Suc @ Natural3)))))))))).
% 9.43/3.46 thf(fact_4636_natural_Odistinct_I1_J, axiom, (! [Natural2:code_natural]: (((zero_zero @ code_natural) != (code_Suc @ Natural2))))).
% 9.43/3.46 thf(fact_4637_natural_Odistinct_I2_J, axiom, (! [Natural4:code_natural]: (((code_Suc @ Natural4) != (zero_zero @ code_natural))))).
% 9.43/3.46 thf(fact_4638_union__mset__def, axiom, (! [A:$tType]: (((union_mset @ A) = (^ [A9:(multiset @ A),B9:(multiset @ A)]: ((((plus_plus @ (multiset @ A)) @ A9) @ (((minus_minus @ (multiset @ A)) @ B9) @ A9)))))))).
% 9.43/3.46 thf(fact_4639_union__diff__inter__eq__sup, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A)]: (((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ (((inter_mset @ A) @ A4) @ B6)) = (((union_mset @ A) @ A4) @ B6))))).
% 9.43/3.46 thf(fact_4640_union__diff__sup__eq__inter, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A)]: (((((minus_minus @ (multiset @ A)) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) @ (((union_mset @ A) @ A4) @ B6)) = (((inter_mset @ A) @ A4) @ B6))))).
% 9.43/3.46 thf(fact_4641_size__Un__Int, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A)]: (((((plus_plus @ nat) @ ((size_size @ (multiset @ A)) @ A4)) @ ((size_size @ (multiset @ A)) @ B6)) = (((plus_plus @ nat) @ ((size_size @ (multiset @ A)) @ (((union_mset @ A) @ A4) @ B6))) @ ((size_size @ (multiset @ A)) @ (((inter_mset @ A) @ A4) @ B6))))))).
% 9.43/3.46 thf(fact_4642_Suc_Oabs__eq, axiom, (! [X:nat]: (((code_Suc @ (code_natural_of_nat @ X)) = (code_natural_of_nat @ (suc @ X)))))).
% 9.43/3.46 thf(fact_4643_Suc__natural__minus__one, axiom, (! [N:code_natural]: (((((minus_minus @ code_natural) @ (code_Suc @ N)) @ (one_one @ code_natural)) = N)))).
% 9.43/3.46 thf(fact_4644_size__Un__disjoint, axiom, (! [A:$tType,A4:(multiset @ A),B6:(multiset @ A)]: ((((((inter_mset @ A) @ A4) @ B6) = (zero_zero @ (multiset @ A))) => (((size_size @ (multiset @ A)) @ (((union_mset @ A) @ A4) @ B6)) = (((plus_plus @ nat) @ ((size_size @ (multiset @ A)) @ A4)) @ ((size_size @ (multiset @ A)) @ B6))))))).
% 9.43/3.46 thf(fact_4645_random__aux__set_Ocases, axiom, (! [X:((product_prod @ code_natural) @ code_natural)]: (((! [J3:code_natural]: ((X != ((((product_Pair @ code_natural) @ code_natural) @ (zero_zero @ code_natural)) @ J3)))) => (~ ((! [I2:code_natural,J3:code_natural]: ((X != ((((product_Pair @ code_natural) @ code_natural) @ (code_Suc @ I2)) @ J3)))))))))).
% 9.43/3.46 thf(fact_4646_Eps__Opt__def, axiom, (! [A:$tType]: (((eps_Opt @ A) = (^ [P4:(A > $o)]: (((((if @ (option @ A)) @ (? [X8:A]: ((P4 @ X8)))) @ ((some @ A) @ ((fChoice @ A) @ P4))) @ (none @ A)))))))).
% 9.43/3.46 thf(fact_4647_acyclic__insert, axiom, (! [A:$tType,Y:A,X:A,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_acyclic @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ R)) = (((transitive_acyclic @ A) @ R) & (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R))))))))).
% 9.43/3.46 thf(fact_4648_acyclic__empty, axiom, (! [A:$tType]: (((transitive_acyclic @ A) @ (bot_bot @ (set @ ((product_prod @ A) @ A))))))).
% 9.43/3.46 thf(fact_4649_Eps__case__prod__eq, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: ((((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [X9:A,Y8:B]: (((X = X9) & (Y = Y8)))))) = ((((product_Pair @ A) @ B) @ X) @ Y))))).
% 9.43/3.46 thf(fact_4650_some__insert__self, axiom, (! [A:$tType,S2:(set @ A)]: (((S2 != (bot_bot @ (set @ A))) => ((((insert2 @ A) @ ((fChoice @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ S2))))) @ S2) = S2))))).
% 9.43/3.46 thf(fact_4651_some__elem, axiom, (! [A:$tType,S2:(set @ A)]: (((S2 != (bot_bot @ (set @ A))) => (((member @ A) @ ((fChoice @ A) @ (^ [X3:A]: ((((member @ A) @ X3) @ S2))))) @ S2))))).
% 9.43/3.46 thf(fact_4652_cyclic__subset, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),S2:(set @ ((product_prod @ A) @ A))]: (((~ (((transitive_acyclic @ A) @ R5))) => ((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ S2) => (~ (((transitive_acyclic @ A) @ S2)))))))).
% 9.43/3.46 thf(fact_4653_acyclic__union_I1_J, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),B6:(set @ ((product_prod @ A) @ A))]: ((((transitive_acyclic @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ A4) @ B6)) => ((transitive_acyclic @ A) @ A4))))).
% 9.43/3.46 thf(fact_4654_acyclic__union_I2_J, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A)),B6:(set @ ((product_prod @ A) @ A))]: ((((transitive_acyclic @ A) @ (((sup_sup @ (set @ ((product_prod @ A) @ A))) @ A4) @ B6)) => ((transitive_acyclic @ A) @ B6))))).
% 9.43/3.46 thf(fact_4655_inv__on__def, axiom, (! [B:$tType,A:$tType]: ((((inv_on @ A) @ B) = (^ [F2:(A > B),A9:(set @ A),X3:B]: (((fChoice @ A) @ (^ [Y3:A]: (((((member @ A) @ Y3) @ A9) & ((F2 @ Y3) = X3))))))))))).
% 9.43/3.46 thf(fact_4656_split__paired__Eps, axiom, (! [B:$tType,A:$tType]: (((fChoice @ ((product_prod @ A) @ B)) = (^ [P4:(((product_prod @ A) @ B) > $o)]: (((fChoice @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [A7:A,B5:B]: ((P4 @ ((((product_Pair @ A) @ B) @ A7) @ B5)))))))))))).
% 9.43/3.46 thf(fact_4657_acyclicI__order, axiom, (! [A:$tType,B:$tType]: (((preorder @ A) => (! [R:(set @ ((product_prod @ B) @ B)),F:(B > A)]: (((! [A5:B,B3:B]: (((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A5) @ B3)) @ R) => (((ord_less @ A) @ (F @ B3)) @ (F @ A5))))) => ((transitive_acyclic @ B) @ R)))))))).
% 9.43/3.46 thf(fact_4658_subset__mset_OacyclicI__order, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ B)),F:(B > (multiset @ A))]: (((! [A5:B,B3:B]: (((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ A5) @ B3)) @ R) => (((subset_mset @ A) @ (F @ B3)) @ (F @ A5))))) => ((transitive_acyclic @ B) @ R))))).
% 9.43/3.46 thf(fact_4659_acyclicI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ ((transitive_trancl @ A) @ R)))))) => ((transitive_acyclic @ A) @ R))))).
% 9.43/3.46 thf(fact_4660_acyclic__def, axiom, (! [A:$tType]: (((transitive_acyclic @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [X3:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ X3)) @ ((transitive_trancl @ A) @ R3)))))))))))).
% 9.43/3.46 thf(fact_4661_cyclicE, axiom, (! [A:$tType,G:(set @ ((product_prod @ A) @ A))]: (((~ (((transitive_acyclic @ A) @ G))) => (~ ((! [X2:A]: ((~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ X2)) @ ((transitive_trancl @ A) @ G)))))))))))).
% 9.43/3.46 thf(fact_4662_some__theI, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o))]: (((? [A17:A,X_1:B]: (((P @ A17) @ X_1))) => ((! [B13:B,B24:B]: (((? [A5:A]: (((P @ A5) @ B13))) => ((? [A5:A]: (((P @ A5) @ B24))) => (B13 = B24))))) => ((P @ ((fChoice @ A) @ (^ [A7:A]: ((? [X8:B]: (((P @ A7) @ X8))))))) @ ((the @ B) @ (^ [B5:B]: ((? [A7:A]: (((P @ A7) @ B5)))))))))))).
% 9.43/3.46 thf(fact_4663_acyclic__insert__cyclic, axiom, (! [A:$tType,G:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: ((((transitive_acyclic @ A) @ G) => ((~ (((transitive_acyclic @ A) @ (((insert2 @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ G)))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ ((transitive_rtrancl @ A) @ G))))))).
% 9.43/3.46 thf(fact_4664_fun__of__rel__def, axiom, (! [A:$tType,B:$tType]: ((((fun_of_rel @ B) @ A) = (^ [R6:(set @ ((product_prod @ B) @ A)),X3:B]: (((fChoice @ A) @ (^ [Y3:A]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y3)) @ R6)))))))))).
% 9.43/3.46 thf(fact_4665_irrefl__tranclI, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A]: ((((((inf_inf @ (set @ ((product_prod @ A) @ A))) @ (((converse @ A) @ A) @ R)) @ ((transitive_rtrancl @ A) @ R)) = (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ ((transitive_trancl @ A) @ R)))))))).
% 9.43/3.46 thf(fact_4666_quicksort_Opelims, axiom, (! [A:$tType]: (((linorder @ A) => (! [X:(list @ A),Y:(list @ A)]: (((((linorder_quicksort @ A) @ X) = Y) => ((((accp @ (list @ A)) @ (linord6200660962353139674rt_rel @ A)) @ X) => (((X = (nil @ A)) => ((Y = (nil @ A)) => (~ ((((accp @ (list @ A)) @ (linord6200660962353139674rt_rel @ A)) @ (nil @ A)))))) => (~ ((! [X2:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ Xs2)) => ((Y = (((append @ A) @ ((linorder_quicksort @ A) @ (((filter @ A) @ (^ [Y3:A]: ((~ ((((ord_less_eq @ A) @ X2) @ Y3)))))) @ Xs2))) @ (((append @ A) @ (((cons @ A) @ X2) @ (nil @ A))) @ ((linorder_quicksort @ A) @ (((filter @ A) @ ((ord_less_eq @ A) @ X2)) @ Xs2))))) => (~ ((((accp @ (list @ A)) @ (linord6200660962353139674rt_rel @ A)) @ (((cons @ A) @ X2) @ Xs2))))))))))))))))))).
% 9.43/3.46 thf(fact_4667_converse__iff, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,R:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ (((converse @ B) @ A) @ R)) = (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A3)) @ R))))).
% 9.43/3.46 thf(fact_4668_pair__set__inverse, axiom, (! [B:$tType,A:$tType,P:(B > (A > $o))]: (((((converse @ B) @ A) @ ((collect @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ P))) = ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [B5:A,A7:B]: (((P @ A7) @ B5))))))))).
% 9.43/3.46 thf(fact_4669_below__Id__inv, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ (((converse @ A) @ A) @ R5)) @ (id2 @ A)) = (((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ (id2 @ A)))))).
% 9.43/3.46 thf(fact_4670_converseI, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A3)) @ (((converse @ A) @ B) @ R)))))).
% 9.43/3.46 thf(fact_4671_converseE, axiom, (! [A:$tType,B:$tType,Yx:((product_prod @ A) @ B),R:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ Yx) @ (((converse @ B) @ A) @ R)) => (~ ((! [X2:B,Y4:A]: (((Yx = ((((product_Pair @ A) @ B) @ Y4) @ X2)) => (~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X2) @ Y4)) @ R)))))))))))).
% 9.43/3.46 thf(fact_4672_converseD, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,R:(set @ ((product_prod @ B) @ A))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ (((converse @ B) @ A) @ R)) => (((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ B2) @ A3)) @ R))))).
% 9.43/3.46 thf(fact_4673_converse_Osimps, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R)) = (? [A7:A,B5:B]: (((A1 = B5) & ((A22 = A7) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B5)) @ R))))))))).
% 9.43/3.46 thf(fact_4674_converse_Ocases, axiom, (! [B:$tType,A:$tType,A1:B,A22:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A1) @ A22)) @ (((converse @ A) @ B) @ R)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A22) @ A1)) @ R))))).
% 9.43/3.46 thf(fact_4675_in__listrel1__converse, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A),R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ ((listrel1 @ A) @ (((converse @ A) @ A) @ R))) = (((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ X) @ Y)) @ (((converse @ (list @ A)) @ (list @ A)) @ ((listrel1 @ A) @ R))))))).
% 9.43/3.46 thf(fact_4676_rtrancl__converseI, axiom, (! [A:$tType,Y:A,X:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ (((converse @ A) @ A) @ R))))))).
% 9.43/3.46 thf(fact_4677_rtrancl__converseD, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ (((converse @ A) @ A) @ R))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ ((transitive_rtrancl @ A) @ R)))))).
% 9.43/3.46 thf(fact_4678_trancl__converseI, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((converse @ A) @ A) @ ((transitive_trancl @ A) @ R))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ (((converse @ A) @ A) @ R))))))).
% 9.43/3.46 thf(fact_4679_trancl__converseD, axiom, (! [A:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_trancl @ A) @ (((converse @ A) @ A) @ R))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((converse @ A) @ A) @ ((transitive_trancl @ A) @ R))))))).
% 9.43/3.46 thf(fact_4680_finite__wf__eq__wf__converse, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ ((product_prod @ A) @ A)) @ R5) => (((wf @ A) @ (((converse @ A) @ A) @ R5)) = ((wf @ A) @ R5)))))).
% 9.43/3.46 thf(fact_4681_converse__unfold, axiom, (! [A:$tType,B:$tType]: ((((converse @ B) @ A) = (^ [R3:(set @ ((product_prod @ B) @ A))]: (((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ (^ [Y3:A,X3:B]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y3)) @ R3))))))))))).
% 9.43/3.46 thf(fact_4682_conversep__converse__eq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((conversep @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) = (^ [X3:B,Y3:A]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ X3) @ Y3)) @ (((converse @ A) @ B) @ R)))))))).
% 9.43/3.46 thf(fact_4683_converse__def, axiom, (! [B:$tType,A:$tType]: ((((converse @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ ((product_prod @ B) @ A)) @ ((((product_case_prod @ B) @ A) @ $o) @ (((conversep @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3)))))))))))).
% 9.43/3.46 thf(fact_4684_is__singleton__altdef, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A9:(set @ A)]: ((((finite_card @ A) @ A9) = (one_one @ nat)))))))).
% 9.43/3.46 thf(fact_4685_Code__Numeral_Osub__code_I2_J, axiom, (! [M2:num]: ((((code_sub @ (bit0 @ M2)) @ one2) = (code_Pos @ (bitM @ M2)))))).
% 9.43/3.46 thf(fact_4686_su__rel__fun__def, axiom, (! [B:$tType,A:$tType]: ((((su_rel_fun @ A) @ B) = (^ [F7:(set @ ((product_prod @ A) @ B)),F2:(A > B)]: (((! [A9:A,B9:B,B11:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A9) @ B9)) @ F7) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A9) @ B11)) @ F7) => (B9 = B11))))) & ((! [A9:A,P4:$o]: (((! [B9:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A9) @ B9)) @ F7) => P4))) => P4))) & (! [A9:A]: (((F2 @ A9) = ((the @ B) @ (^ [B9:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A9) @ B9)) @ F7))))))))))))))).
% 9.43/3.46 thf(fact_4687_Code__Numeral_OPos__def, axiom, (code_Pos = (numeral_numeral @ code_integer))).
% 9.43/3.46 thf(fact_4688_su__rel__fun_Osurjective, axiom, (! [B:$tType,A:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B),A4:A]: ((((((su_rel_fun @ A) @ B) @ F6) @ F) => (~ ((! [B10:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B10)) @ F6))))))))))).
% 9.43/3.46 thf(fact_4689_su__rel__fun_Ounique, axiom, (! [A:$tType,B:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B),A4:A,B6:B,B14:B]: ((((((su_rel_fun @ A) @ B) @ F6) @ F) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B6)) @ F6) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B14)) @ F6) => (B6 = B14))))))).
% 9.43/3.46 thf(fact_4690_su__rel__fun_Orepr2, axiom, (! [B:$tType,A:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B),A4:A,B6:B]: ((((((su_rel_fun @ A) @ B) @ F6) @ F) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B6)) @ F6) => (B6 = (F @ A4))))))).
% 9.43/3.46 thf(fact_4691_su__rel__fun_Orepr1, axiom, (! [B:$tType,A:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B),A4:A]: ((((((su_rel_fun @ A) @ B) @ F6) @ F) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ (F @ A4))) @ F6))))).
% 9.43/3.46 thf(fact_4692_su__rel__fun_Orepr, axiom, (! [B:$tType,A:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B),A4:A,B6:B]: ((((((su_rel_fun @ A) @ B) @ F6) @ F) => (((F @ A4) = B6) = (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B6)) @ F6)))))).
% 9.43/3.46 thf(fact_4693_Pos__fold_I2_J, axiom, (! [K:num]: ((((numeral_numeral @ code_integer) @ (bit0 @ K)) = (code_Pos @ (bit0 @ K)))))).
% 9.43/3.46 thf(fact_4694_Pos__fold_I1_J, axiom, (((numeral_numeral @ code_integer) @ one2) = (code_Pos @ one2))).
% 9.43/3.46 thf(fact_4695_Pos__fold_I3_J, axiom, (! [K:num]: ((((numeral_numeral @ code_integer) @ (bit1 @ K)) = (code_Pos @ (bit1 @ K)))))).
% 9.43/3.46 thf(fact_4696_one__integer__code, axiom, ((one_one @ code_integer) = (code_Pos @ one2))).
% 9.43/3.46 thf(fact_4697_Code__Numeral_Odup__code_I2_J, axiom, (! [N:num]: (((code_dup @ (code_Pos @ N)) = (code_Pos @ (bit0 @ N)))))).
% 9.43/3.46 thf(fact_4698_less__integer__code_I4_J, axiom, (! [K:num]: ((~ ((((ord_less @ code_integer) @ (code_Pos @ K)) @ (zero_zero @ code_integer))))))).
% 9.43/3.46 thf(fact_4699_less__integer__code_I2_J, axiom, (! [L:num]: ((((ord_less @ code_integer) @ (zero_zero @ code_integer)) @ (code_Pos @ L))))).
% 9.43/3.46 thf(fact_4700_plus__integer__code_I3_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ code_integer) @ (code_Pos @ M2)) @ (code_Pos @ N)) = (code_Pos @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4701_less__eq__integer__code_I2_J, axiom, (! [L:num]: ((((ord_less_eq @ code_integer) @ (zero_zero @ code_integer)) @ (code_Pos @ L))))).
% 9.43/3.46 thf(fact_4702_less__eq__integer__code_I4_J, axiom, (! [K:num]: ((~ ((((ord_less_eq @ code_integer) @ (code_Pos @ K)) @ (zero_zero @ code_integer))))))).
% 9.43/3.46 thf(fact_4703_less__integer__code_I8_J, axiom, (! [K:num,L:num]: ((((ord_less @ code_integer) @ (code_Neg @ K)) @ (code_Pos @ L))))).
% 9.43/3.46 thf(fact_4704_less__integer__code_I6_J, axiom, (! [K:num,L:num]: ((~ ((((ord_less @ code_integer) @ (code_Pos @ K)) @ (code_Neg @ L))))))).
% 9.43/3.46 thf(fact_4705_uminus__integer__code_I3_J, axiom, (! [M2:num]: ((((uminus_uminus @ code_integer) @ (code_Neg @ M2)) = (code_Pos @ M2))))).
% 9.43/3.46 thf(fact_4706_uminus__integer__code_I2_J, axiom, (! [M2:num]: ((((uminus_uminus @ code_integer) @ (code_Pos @ M2)) = (code_Neg @ M2))))).
% 9.43/3.46 thf(fact_4707_Code__Numeral_ONeg__def, axiom, (code_Neg = (^ [N5:num]: (((uminus_uminus @ code_integer) @ (code_Pos @ N5)))))).
% 9.43/3.46 thf(fact_4708_less__eq__integer__code_I8_J, axiom, (! [K:num,L:num]: ((((ord_less_eq @ code_integer) @ (code_Neg @ K)) @ (code_Pos @ L))))).
% 9.43/3.46 thf(fact_4709_less__eq__integer__code_I6_J, axiom, (! [K:num,L:num]: ((~ ((((ord_less_eq @ code_integer) @ (code_Pos @ K)) @ (code_Neg @ L))))))).
% 9.43/3.46 thf(fact_4710_times__integer__code_I3_J, axiom, (! [M2:num,N:num]: (((((times_times @ code_integer) @ (code_Pos @ M2)) @ (code_Pos @ N)) = (code_Pos @ (((times_times @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4711_less__integer__code_I5_J, axiom, (! [K:num,L:num]: (((((ord_less @ code_integer) @ (code_Pos @ K)) @ (code_Pos @ L)) = (((ord_less @ num) @ K) @ L))))).
% 9.43/3.46 thf(fact_4712_less__eq__integer__code_I5_J, axiom, (! [K:num,L:num]: (((((ord_less_eq @ code_integer) @ (code_Pos @ K)) @ (code_Pos @ L)) = (((ord_less_eq @ num) @ K) @ L))))).
% 9.43/3.46 thf(fact_4713_minus__integer__code_I3_J, axiom, (! [M2:num,N:num]: (((((minus_minus @ code_integer) @ (code_Pos @ M2)) @ (code_Pos @ N)) = ((code_sub @ M2) @ N))))).
% 9.43/3.46 thf(fact_4714_divmod__abs__code_I1_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Pos @ K)) @ (code_Pos @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 9.43/3.46 thf(fact_4715_minus__integer__code_I4_J, axiom, (! [M2:num,N:num]: (((((minus_minus @ code_integer) @ (code_Pos @ M2)) @ (code_Neg @ N)) = (code_Pos @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4716_minus__integer__code_I5_J, axiom, (! [M2:num,N:num]: (((((minus_minus @ code_integer) @ (code_Neg @ M2)) @ (code_Pos @ N)) = (code_Neg @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4717_times__integer__code_I4_J, axiom, (! [M2:num,N:num]: (((((times_times @ code_integer) @ (code_Pos @ M2)) @ (code_Neg @ N)) = (code_Neg @ (((times_times @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4718_times__integer__code_I5_J, axiom, (! [M2:num,N:num]: (((((times_times @ code_integer) @ (code_Neg @ M2)) @ (code_Pos @ N)) = (code_Neg @ (((times_times @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4719_times__integer__code_I6_J, axiom, (! [M2:num,N:num]: (((((times_times @ code_integer) @ (code_Neg @ M2)) @ (code_Neg @ N)) = (code_Pos @ (((times_times @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_4720_plus__integer__code_I4_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ code_integer) @ (code_Pos @ M2)) @ (code_Neg @ N)) = ((code_sub @ M2) @ N))))).
% 9.43/3.46 thf(fact_4721_plus__integer__code_I5_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ code_integer) @ (code_Neg @ M2)) @ (code_Pos @ N)) = ((code_sub @ N) @ M2))))).
% 9.43/3.46 thf(fact_4722_divmod__abs__code_I3_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Neg @ K)) @ (code_Pos @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 9.43/3.46 thf(fact_4723_divmod__abs__code_I4_J, axiom, (! [K:num,L:num]: ((((code_divmod_abs @ (code_Pos @ K)) @ (code_Neg @ L)) = (((unique8689654367752047608divmod @ code_integer) @ K) @ L))))).
% 9.43/3.46 thf(fact_4724_Code__Numeral_Osub__code_I3_J, axiom, (! [M2:num]: ((((code_sub @ (bit1 @ M2)) @ one2) = (code_Pos @ (bit0 @ M2)))))).
% 9.43/3.46 thf(fact_4725_su__rel__fun_Of__def, axiom, (! [A:$tType,B:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B),A4:A]: ((((((su_rel_fun @ A) @ B) @ F6) @ F) => ((F @ A4) = ((the @ B) @ (^ [B9:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A4) @ B9)) @ F6))))))))).
% 9.43/3.46 thf(fact_4726_su__rel__fun_Ointro, axiom, (! [B:$tType,A:$tType,F6:(set @ ((product_prod @ A) @ B)),F:(A > B)]: (((! [A15:A,B10:B,B15:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A15) @ B10)) @ F6) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A15) @ B15)) @ F6) => (B10 = B15))))) => ((! [A15:A,P7:$o]: (((! [B16:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A15) @ B16)) @ F6) => P7))) => P7))) => ((! [A15:A]: (((F @ A15) = ((the @ B) @ (^ [B9:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A15) @ B9)) @ F6))))))) => ((((su_rel_fun @ A) @ B) @ F6) @ F))))))).
% 9.43/3.46 thf(fact_4727_aboveS__def, axiom, (! [A:$tType]: (((order_aboveS @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A7:A]: (((collect @ A) @ (^ [B5:A]: (((B5 != A7) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R3))))))))))).
% 9.43/3.46 thf(fact_4728_relInvImage__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Gr7122648621184425601vImage @ A) @ B) = (^ [A9:(set @ A),R6:(set @ ((product_prod @ B) @ B)),F2:(A > B)]: (((collect @ ((product_prod @ A) @ A)) @ (^ [Uu:((product_prod @ A) @ A)]: ((? [A12:A,A23:A]: (((Uu = ((((product_Pair @ A) @ A) @ A12) @ A23)) & ((((member @ A) @ A12) @ A9) & ((((member @ A) @ A23) @ A9) & (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F2 @ A12)) @ (F2 @ A23))) @ R6))))))))))))))).
% 9.43/3.46 thf(fact_4729_prod__list__def, axiom, (! [A:$tType]: (((monoid_mult @ A) => ((groups5270119922927024881d_list @ A) = (((groups_monoid_F @ A) @ (times_times @ A)) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_4730_sum__list__def, axiom, (! [A:$tType]: (((monoid_add @ A) => ((groups8242544230860333062m_list @ A) = (((groups_monoid_F @ A) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 9.43/3.46 thf(fact_4731_sum__list__map__remove1, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_add @ A) => (! [X:B,Xs:(list @ B),F:(B > A)]: (((((member @ B) @ X) @ ((set2 @ B) @ Xs)) => (((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ Xs)) = (((plus_plus @ A) @ (F @ X)) @ ((groups8242544230860333062m_list @ A) @ ((((map @ B) @ A) @ F) @ (((remove1 @ B) @ X) @ Xs)))))))))))).
% 9.43/3.46 thf(fact_4732_normalize__stable, axiom, (! [Q3:int,P5:int]: (((((ord_less @ int) @ (zero_zero @ int)) @ Q3) => ((((algebr8660921524188924756oprime @ int) @ P5) @ Q3) => ((normalize @ ((((product_Pair @ int) @ int) @ P5) @ Q3)) = ((((product_Pair @ int) @ int) @ P5) @ Q3))))))).
% 9.43/3.46 thf(fact_4733_distinct__count__atmost__1, axiom, (! [A:$tType]: (((distinct @ A) = (^ [X3:(list @ A)]: ((! [A7:A]: ((((((member @ A) @ A7) @ ((set2 @ A) @ X3)) => ((((count @ A) @ ((mset @ A) @ X3)) @ A7) = (one_one @ nat))) & ((~ ((((member @ A) @ A7) @ ((set2 @ A) @ X3)))) => ((((count @ A) @ ((mset @ A) @ X3)) @ A7) = (zero_zero @ nat)))))))))))).
% 9.43/3.46 thf(fact_4734_coprime__self, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ A3) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4735_mset__empty__count, axiom, (! [A:$tType,M3:(multiset @ A)]: (((! [P6:A]: (((((count @ A) @ M3) @ P6) = (zero_zero @ nat)))) = (M3 = (zero_zero @ (multiset @ A))))))).
% 9.43/3.46 thf(fact_4736_coprime__imp__gcd__eq__1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => ((((gcd_gcd @ A) @ A3) @ B2) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4737_count__union, axiom, (! [A:$tType,M3:(multiset @ A),N6:(multiset @ A),A3:A]: (((((count @ A) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) @ A3) = (((plus_plus @ nat) @ (((count @ A) @ M3) @ A3)) @ (((count @ A) @ N6) @ A3)))))).
% 9.43/3.46 thf(fact_4738_coprime__0__right__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ (zero_zero @ A)) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4739_coprime__0__left__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: (((((algebr8660921524188924756oprime @ A) @ (zero_zero @ A)) @ A3) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4740_coprime__mult__self__right__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,C2:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ (((times_times @ A) @ A3) @ C2)) @ (((times_times @ A) @ B2) @ C2)) = ((((dvd_dvd @ A) @ C2) @ (one_one @ A)) & (((algebr8660921524188924756oprime @ A) @ A3) @ B2))))))))).
% 9.43/3.46 thf(fact_4741_coprime__mult__self__left__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ (((times_times @ A) @ C2) @ A3)) @ (((times_times @ A) @ C2) @ B2)) = ((((dvd_dvd @ A) @ C2) @ (one_one @ A)) & (((algebr8660921524188924756oprime @ A) @ A3) @ B2))))))))).
% 9.43/3.46 thf(fact_4742_is__unit__gcd, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) @ (one_one @ A)) = (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 9.43/3.46 thf(fact_4743_count__greater__eq__one__iff, axiom, (! [A:$tType,M3:(multiset @ A),X:A]: (((((ord_less_eq @ nat) @ (one_one @ nat)) @ (((count @ A) @ M3) @ X)) = (((member @ A) @ X) @ ((set_mset @ A) @ M3)))))).
% 9.43/3.46 thf(fact_4744_count__mset__set_I1_J, axiom, (! [A:$tType,A4:(set @ A),X:A]: ((((finite_finite2 @ A) @ A4) => ((((member @ A) @ X) @ A4) => ((((count @ A) @ ((mset_set @ A) @ A4)) @ X) = (one_one @ nat))))))).
% 9.43/3.46 thf(fact_4745_coprime__1__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ (one_one @ A)) @ A3))))))).
% 9.43/3.46 thf(fact_4746_coprime__1__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ A3) @ (one_one @ A)))))))).
% 9.43/3.46 thf(fact_4747_coprime__add__one__left, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ (((plus_plus @ A) @ A3) @ (one_one @ A))) @ A3))))))).
% 9.43/3.46 thf(fact_4748_coprime__add__one__right, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ A3) @ (((plus_plus @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4749_coprime__doff__one__right, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ A3) @ (((minus_minus @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4750_coprime__diff__one__left, axiom, (! [A:$tType]: (((ring_gcd @ A) => (! [A3:A]: ((((algebr8660921524188924756oprime @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ A3))))))).
% 9.43/3.46 thf(fact_4751_is__unit__right__imp__coprime, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 9.43/3.46 thf(fact_4752_is__unit__left__imp__coprime, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 9.43/3.46 thf(fact_4753_coprime__common__divisor, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A,C2:A]: (((((algebr8660921524188924756oprime @ A) @ A3) @ B2) => ((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => (((dvd_dvd @ A) @ C2) @ (one_one @ A))))))))))).
% 9.43/3.46 thf(fact_4754_coprime__absorb__right, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [Y:A,X:A]: (((((dvd_dvd @ A) @ Y) @ X) => ((((algebr8660921524188924756oprime @ A) @ X) @ Y) = (((dvd_dvd @ A) @ Y) @ (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_4755_coprime__imp__coprime, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,D3:A,A3:A,B2:A]: (((((algebr8660921524188924756oprime @ A) @ C2) @ D3) => ((! [E3:A]: (((~ ((((dvd_dvd @ A) @ E3) @ (one_one @ A)))) => ((((dvd_dvd @ A) @ E3) @ A3) => ((((dvd_dvd @ A) @ E3) @ B2) => (((dvd_dvd @ A) @ E3) @ C2)))))) => ((! [E3:A]: (((~ ((((dvd_dvd @ A) @ E3) @ (one_one @ A)))) => ((((dvd_dvd @ A) @ E3) @ A3) => ((((dvd_dvd @ A) @ E3) @ B2) => (((dvd_dvd @ A) @ E3) @ D3)))))) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))))).
% 9.43/3.46 thf(fact_4756_coprime__absorb__left, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [X:A,Y:A]: (((((dvd_dvd @ A) @ X) @ Y) => ((((algebr8660921524188924756oprime @ A) @ X) @ Y) = (((dvd_dvd @ A) @ X) @ (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_4757_not__coprimeI, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [C2:A,A3:A,B2:A]: (((((dvd_dvd @ A) @ C2) @ A3) => ((((dvd_dvd @ A) @ C2) @ B2) => ((~ ((((dvd_dvd @ A) @ C2) @ (one_one @ A)))) => (~ ((((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))))))).
% 9.43/3.46 thf(fact_4758_not__coprimeE, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((~ ((((algebr8660921524188924756oprime @ A) @ A3) @ B2))) => (~ ((! [C4:A]: (((((dvd_dvd @ A) @ C4) @ A3) => ((((dvd_dvd @ A) @ C4) @ B2) => (((dvd_dvd @ A) @ C4) @ (one_one @ A))))))))))))))).
% 9.43/3.46 thf(fact_4759_coprime__def, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => ((algebr8660921524188924756oprime @ A) = (^ [A7:A,B5:A]: ((! [C3:A]: (((((dvd_dvd @ A) @ C3) @ A7) => ((((dvd_dvd @ A) @ C3) @ B5) => (((dvd_dvd @ A) @ C3) @ (one_one @ A))))))))))))).
% 9.43/3.46 thf(fact_4760_coprimeI, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A3:A,B2:A]: (((! [C4:A]: (((((dvd_dvd @ A) @ C4) @ A3) => ((((dvd_dvd @ A) @ C4) @ B2) => (((dvd_dvd @ A) @ C4) @ (one_one @ A)))))) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 9.43/3.46 thf(fact_4761_plus__multiset_Orep__eq, axiom, (! [A:$tType,X:(multiset @ A),Xa:(multiset @ A)]: ((((count @ A) @ (((plus_plus @ (multiset @ A)) @ X) @ Xa)) = (^ [A7:A]: ((((plus_plus @ nat) @ (((count @ A) @ X) @ A7)) @ (((count @ A) @ Xa) @ A7)))))))).
% 9.43/3.46 thf(fact_4762_coprime__iff__gcd__eq__1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((algebr8660921524188924756oprime @ A) = (^ [A7:A,B5:A]: (((((gcd_gcd @ A) @ A7) @ B5) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4763_gcd__eq__1__imp__coprime, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((((gcd_gcd @ A) @ A3) @ B2) = (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ B2)))))))).
% 9.43/3.46 thf(fact_4764_quotient__of__coprime, axiom, (! [R:rat,P5:int,Q3:int]: ((((quotient_of @ R) = ((((product_Pair @ int) @ int) @ P5) @ Q3)) => (((algebr8660921524188924756oprime @ int) @ P5) @ Q3))))).
% 9.43/3.46 thf(fact_4765_normalize__coprime, axiom, (! [R:((product_prod @ int) @ int),P5:int,Q3:int]: ((((normalize @ R) = ((((product_Pair @ int) @ int) @ P5) @ Q3)) => (((algebr8660921524188924756oprime @ int) @ P5) @ Q3))))).
% 9.43/3.46 thf(fact_4766_invertible__coprime, axiom, (! [A:$tType]: (((euclid8851590272496341667cancel @ A) => (! [A3:A,B2:A,C2:A]: ((((((modulo_modulo @ A) @ (((times_times @ A) @ A3) @ B2)) @ C2) = (one_one @ A)) => (((algebr8660921524188924756oprime @ A) @ A3) @ C2)))))))).
% 9.43/3.46 thf(fact_4767_count__ne__remove, axiom, (! [A:$tType,X:A,T2:A,S2:(multiset @ A)]: (((X != T2) => ((((count @ A) @ S2) @ X) = (((count @ A) @ (((minus_minus @ (multiset @ A)) @ S2) @ (((add_mset @ A) @ T2) @ (zero_zero @ (multiset @ A))))) @ X)))))).
% 9.43/3.46 thf(fact_4768_count__in__diffI, axiom, (! [A:$tType,N6:(multiset @ A),X:A,M3:(multiset @ A)]: (((! [N3:nat]: (((((count @ A) @ N6) @ X) != (((plus_plus @ nat) @ N3) @ (((count @ A) @ M3) @ X))))) => (((member @ A) @ X) @ ((set_mset @ A) @ (((minus_minus @ (multiset @ A)) @ M3) @ N6))))))).
% 9.43/3.46 thf(fact_4769_count__single, axiom, (! [A:$tType,B2:A,A3:A]: ((((B2 = A3) => ((((count @ A) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) @ A3) = (one_one @ nat))) & ((B2 != A3) => ((((count @ A) @ (((add_mset @ A) @ B2) @ (zero_zero @ (multiset @ A)))) @ A3) = (zero_zero @ nat))))))).
% 9.43/3.46 thf(fact_4770_in__diff__countE, axiom, (! [A:$tType,X:A,M3:(multiset @ A),N6:(multiset @ A)]: (((((member @ A) @ X) @ ((set_mset @ A) @ (((minus_minus @ (multiset @ A)) @ M3) @ N6))) => (~ ((! [N3:nat]: (((((count @ A) @ M3) @ X) != (((plus_plus @ nat) @ (suc @ N3)) @ (((count @ A) @ N6) @ X))))))))))).
% 9.43/3.46 thf(fact_4771_remove1__tl, axiom, (! [A:$tType,Xs:(list @ A)]: (((Xs != (nil @ A)) => ((((remove1 @ A) @ ((hd @ A) @ Xs)) @ Xs) = ((tl @ A) @ Xs)))))).
% 9.43/3.46 thf(fact_4772_count__mset__set__finite__iff, axiom, (! [A:$tType,S2:(set @ A),A3:A]: ((((finite_finite2 @ A) @ S2) => (((((member @ A) @ A3) @ S2) => ((((count @ A) @ ((mset_set @ A) @ S2)) @ A3) = (one_one @ nat))) & ((~ ((((member @ A) @ A3) @ S2))) => ((((count @ A) @ ((mset_set @ A) @ S2)) @ A3) = (zero_zero @ nat)))))))).
% 9.43/3.46 thf(fact_4773_count__mset__set_H, axiom, (! [A:$tType,A4:(set @ A),X:A]: ((((((finite_finite2 @ A) @ A4) & (((member @ A) @ X) @ A4)) => ((((count @ A) @ ((mset_set @ A) @ A4)) @ X) = (one_one @ nat))) & ((~ ((((finite_finite2 @ A) @ A4) & (((member @ A) @ X) @ A4)))) => ((((count @ A) @ ((mset_set @ A) @ A4)) @ X) = (zero_zero @ nat))))))).
% 9.43/3.46 thf(fact_4774_length__remove1, axiom, (! [A:$tType,X:A,Xs:(list @ A)]: ((((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((size_size @ (list @ A)) @ (((remove1 @ A) @ X) @ Xs)) = (((minus_minus @ nat) @ ((size_size @ (list @ A)) @ Xs)) @ (one_one @ nat)))) & ((~ ((((member @ A) @ X) @ ((set2 @ A) @ Xs)))) => (((size_size @ (list @ A)) @ (((remove1 @ A) @ X) @ Xs)) = ((size_size @ (list @ A)) @ Xs))))))).
% 9.43/3.46 thf(fact_4775_prod__mset__delta_H, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [Y:B,C2:A,A4:(multiset @ B)]: ((((comm_m9189036328036947845d_mset @ A) @ ((((image_mset @ B) @ A) @ (^ [X3:B]: (((((if @ A) @ (Y = X3)) @ C2) @ (one_one @ A))))) @ A4)) = (((power_power @ A) @ C2) @ (((count @ B) @ A4) @ Y))))))))).
% 9.43/3.46 thf(fact_4776_prod__mset__delta, axiom, (! [A:$tType,B:$tType]: (((comm_monoid_mult @ A) => (! [Y:B,C2:A,A4:(multiset @ B)]: ((((comm_m9189036328036947845d_mset @ A) @ ((((image_mset @ B) @ A) @ (^ [X3:B]: (((((if @ A) @ (X3 = Y)) @ C2) @ (one_one @ A))))) @ A4)) = (((power_power @ A) @ C2) @ (((count @ B) @ A4) @ Y))))))))).
% 9.43/3.46 thf(fact_4777_prod_OatLeastLessThan__reindex, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_mult @ A) & (ord @ B)) => (! [H3:(nat > B),M2:nat,N:nat,G:(B > A)]: (((((((bij_betw @ nat) @ B) @ H3) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) @ (((set_or7035219750837199246ssThan @ B) @ (H3 @ M2)) @ (H3 @ N))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ (H3 @ M2)) @ (H3 @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ B) @ A) @ nat) @ G) @ H3)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4778_bij__betw__id, axiom, (! [A:$tType,A4:(set @ A)]: ((((((bij_betw @ A) @ A) @ (id @ A)) @ A4) @ A4)))).
% 9.43/3.46 thf(fact_4779_bij__betw__add, axiom, (! [A:$tType]: (((cancel_semigroup_add @ A) => (! [A3:A,A4:(set @ A),B6:(set @ A)]: (((((((bij_betw @ A) @ A) @ ((plus_plus @ A) @ A3)) @ A4) @ B6) = (((((image2 @ A) @ A) @ ((plus_plus @ A) @ A3)) @ A4) = B6)))))))).
% 9.43/3.46 thf(fact_4780_prod__mset__empty, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((comm_m9189036328036947845d_mset @ A) @ (zero_zero @ (multiset @ A))) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_4781_prod__mset_Ounion, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [M3:(multiset @ A),N6:(multiset @ A)]: ((((comm_m9189036328036947845d_mset @ A) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((times_times @ A) @ ((comm_m9189036328036947845d_mset @ A) @ M3)) @ ((comm_m9189036328036947845d_mset @ A) @ N6))))))))).
% 9.43/3.46 thf(fact_4782_prod__mset__Un, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A4:(multiset @ A),B6:(multiset @ A)]: ((((comm_m9189036328036947845d_mset @ A) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) = (((times_times @ A) @ ((comm_m9189036328036947845d_mset @ A) @ A4)) @ ((comm_m9189036328036947845d_mset @ A) @ B6))))))))).
% 9.43/3.46 thf(fact_4783_bij__betw__of__nat, axiom, (! [A:$tType]: (((semiring_char_0 @ A) => (! [N6:(set @ nat),A4:(set @ A)]: (((((((bij_betw @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N6) @ A4) = (((((image2 @ nat) @ A) @ (semiring_1_of_nat @ A)) @ N6) = A4)))))))).
% 9.43/3.46 thf(fact_4784_prod__mset_Oneutral__const, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (! [A4:(multiset @ B)]: ((((comm_m9189036328036947845d_mset @ A) @ ((((image_mset @ B) @ A) @ (^ [Uu:B]: ((one_one @ A)))) @ A4)) = (one_one @ A)))))))).
% 9.43/3.46 thf(fact_4785_bij__betw__imp__surj__on, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (((((image2 @ A) @ B) @ F) @ A4) = B6))))).
% 9.43/3.46 thf(fact_4786_f__the__inv__into__f__bij__betw, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ B),X:B]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (((member @ B) @ X) @ B6)) => ((F @ (((((the_inv_into @ A) @ B) @ A4) @ F) @ X)) = X)))))).
% 9.43/3.46 thf(fact_4787_bij__betw__the__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (((((bij_betw @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A4) @ F)) @ B6) @ A4))))).
% 9.43/3.46 thf(fact_4788_bij__betw__id__iff, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((((bij_betw @ A) @ A) @ (id @ A)) @ A4) @ B6) = (A4 = B6))))).
% 9.43/3.46 thf(fact_4789_bij__betwE, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (! [X7:A]: (((((member @ A) @ X7) @ A4) => (((member @ B) @ (F @ X7)) @ B6)))))))).
% 9.43/3.46 thf(fact_4790_bij__betw__inv, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (? [G4:(B > A)]: ((((((bij_betw @ B) @ A) @ G4) @ B6) @ A4))))))).
% 9.43/3.46 thf(fact_4791_bij__betw__cong, axiom, (! [A:$tType,B:$tType,A4:(set @ A),F:(A > B),G:(A > B),A18:(set @ B)]: (((! [A5:A]: (((((member @ A) @ A5) @ A4) => ((F @ A5) = (G @ A5))))) => ((((((bij_betw @ A) @ B) @ F) @ A4) @ A18) = (((((bij_betw @ A) @ B) @ G) @ A4) @ A18)))))).
% 9.43/3.46 thf(fact_4792_bij__betw__apply, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ B),A3:A]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => ((((member @ A) @ A3) @ A4) => (((member @ B) @ (F @ A3)) @ B6)))))).
% 9.43/3.46 thf(fact_4793_bij__betw__iff__bijections, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F2:(A > B),A9:(set @ A),B9:(set @ B)]: ((? [G2:(B > A)]: (((! [X3:A]: (((((member @ A) @ X3) @ A9) => ((((member @ B) @ (F2 @ X3)) @ B9) & ((G2 @ (F2 @ X3)) = X3))))) & (! [X3:B]: (((((member @ B) @ X3) @ B9) => ((((member @ A) @ (G2 @ X3)) @ A9) & ((F2 @ (G2 @ X3)) = X3)))))))))))))).
% 9.43/3.46 thf(fact_4794_involuntory__imp__bij, axiom, (! [A:$tType,F:(A > A)]: (((! [X2:A]: (((F @ (F @ X2)) = X2))) => (((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 9.43/3.46 thf(fact_4795_bij__pointE, axiom, (! [B:$tType,A:$tType,F:(A > B),Y:B]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (~ ((! [X2:A]: (((Y = (F @ X2)) => (~ ((! [X16:A]: (((Y = (F @ X16)) => (X16 = X2))))))))))))))).
% 9.43/3.46 thf(fact_4796_bij__iff, axiom, (! [A:$tType,B:$tType,F:(A > B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (! [X3:B]: ((? [Y3:A]: ((((F @ Y3) = X3) & (! [Z5:A]: ((((F @ Z5) = X3) => (Z5 = Y3))))))))))))).
% 9.43/3.46 thf(fact_4797_bij__betw__imp__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => ((((inj_on @ A) @ B) @ F) @ A4))))).
% 9.43/3.46 thf(fact_4798_bij__betw__trans, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),A4:(set @ A),B6:(set @ B),G:(B > C),C5:(set @ C)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => ((((((bij_betw @ B) @ C) @ G) @ B6) @ C5) => (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ A4) @ C5)))))).
% 9.43/3.46 thf(fact_4799_bij__betw__comp__iff, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),A4:(set @ A),A18:(set @ B),F3:(B > C),A19:(set @ C)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ A18) => ((((((bij_betw @ B) @ C) @ F3) @ A18) @ A19) = (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F3) @ F)) @ A4) @ A19)))))).
% 9.43/3.46 thf(fact_4800_bij__betw__empty2, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ (bot_bot @ (set @ B))) => (A4 = (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_4801_bij__betw__empty1, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ (bot_bot @ (set @ A))) @ A4) => (A4 = (bot_bot @ (set @ B))))))).
% 9.43/3.46 thf(fact_4802_bij__betw__byWitness, axiom, (! [A:$tType,B:$tType,A4:(set @ A),F3:(B > A),F:(A > B),A18:(set @ B)]: (((! [X2:A]: (((((member @ A) @ X2) @ A4) => ((F3 @ (F @ X2)) = X2)))) => ((! [X2:B]: (((((member @ B) @ X2) @ A18) => ((F @ (F3 @ X2)) = X2)))) => ((((ord_less_eq @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4)) @ A18) => ((((ord_less_eq @ (set @ A)) @ ((((image2 @ B) @ A) @ F3) @ A18)) @ A4) => (((((bij_betw @ A) @ B) @ F) @ A4) @ A18)))))))).
% 9.43/3.46 thf(fact_4803_bij__betw__subset, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),A18:(set @ B),B6:(set @ A),B14:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ A18) => ((((ord_less_eq @ (set @ A)) @ B6) @ A4) => ((((((image2 @ A) @ B) @ F) @ B6) = B14) => (((((bij_betw @ A) @ B) @ F) @ B6) @ B14))))))).
% 9.43/3.46 thf(fact_4804_bij__betw__imp__surj, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ (top_top @ (set @ B))) => (((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))))))).
% 9.43/3.46 thf(fact_4805_bij__is__surj, axiom, (! [A:$tType,B:$tType,F:(A > B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))))))).
% 9.43/3.46 thf(fact_4806_bij__uminus, axiom, (! [A:$tType]: (((ab_group_add @ A) => (((((bij_betw @ A) @ A) @ (uminus_uminus @ A)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))))))).
% 9.43/3.46 thf(fact_4807_bij__comp, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),G:(B > C)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((((bij_betw @ B) @ C) @ G) @ (top_top @ (set @ B))) @ (top_top @ (set @ C))) => (((((bij_betw @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ (top_top @ (set @ A))) @ (top_top @ (set @ C)))))))).
% 9.43/3.46 thf(fact_4808_inj__on__imp__bij__betw, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((((bij_betw @ A) @ B) @ F) @ A4) @ ((((image2 @ A) @ B) @ F) @ A4)))))).
% 9.43/3.46 thf(fact_4809_bij__betw__imageI, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((((image2 @ A) @ B) @ F) @ A4) = B6) => (((((bij_betw @ A) @ B) @ F) @ A4) @ B6)))))).
% 9.43/3.46 thf(fact_4810_bij__betw__def, axiom, (! [B:$tType,A:$tType]: ((((bij_betw @ A) @ B) = (^ [F2:(A > B),A9:(set @ A),B9:(set @ B)]: ((((((inj_on @ A) @ B) @ F2) @ A9) & (((((image2 @ A) @ B) @ F2) @ A9) = B9)))))))).
% 9.43/3.46 thf(fact_4811_bij__betw__iff__card, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ B) @ B6) => ((? [F2:(A > B)]: ((((((bij_betw @ A) @ B) @ F2) @ A4) @ B6))) = (((finite_card @ A) @ A4) = ((finite_card @ B) @ B6)))))))).
% 9.43/3.46 thf(fact_4812_finite__same__card__bij, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ B) @ B6) => ((((finite_card @ A) @ A4) = ((finite_card @ B) @ B6)) => (? [H2:(A > B)]: ((((((bij_betw @ A) @ B) @ H2) @ A4) @ B6))))))))).
% 9.43/3.46 thf(fact_4813_bij__is__inj, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))))))).
% 9.43/3.46 thf(fact_4814_bij__id, axiom, (! [A:$tType]: ((((((bij_betw @ A) @ A) @ (id @ A)) @ (top_top @ (set @ A))) @ (top_top @ (set @ A)))))).
% 9.43/3.46 thf(fact_4815_coprime__common__divisor__nat, axiom, (! [A3:nat,B2:nat,X:nat]: (((((algebr8660921524188924756oprime @ nat) @ A3) @ B2) => ((((dvd_dvd @ nat) @ X) @ A3) => ((((dvd_dvd @ nat) @ X) @ B2) => (X = (one_one @ nat)))))))).
% 9.43/3.46 thf(fact_4816_prod__mset_Oneutral, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (! [A4:(multiset @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set_mset @ A) @ A4)) => (X2 = (one_one @ A))))) => (((comm_m9189036328036947845d_mset @ A) @ A4) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_4817_bij__betw__comp__iff2, axiom, (! [C:$tType,A:$tType,B:$tType,F3:(A > B),A18:(set @ A),A19:(set @ B),F:(C > A),A4:(set @ C)]: (((((((bij_betw @ A) @ B) @ F3) @ A18) @ A19) => ((((ord_less_eq @ (set @ A)) @ ((((image2 @ C) @ A) @ F) @ A4)) @ A18) => ((((((bij_betw @ C) @ A) @ F) @ A4) @ A18) = (((((bij_betw @ C) @ B) @ (((((comp @ A) @ B) @ C) @ F3) @ F)) @ A4) @ A19))))))).
% 9.43/3.46 thf(fact_4818_notIn__Un__bij__betw, axiom, (! [A:$tType,B:$tType,B2:A,A4:(set @ A),F:(A > B),A18:(set @ B)]: (((~ ((((member @ A) @ B2) @ A4))) => ((~ ((((member @ B) @ (F @ B2)) @ A18))) => ((((((bij_betw @ A) @ B) @ F) @ A4) @ A18) => (((((bij_betw @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) @ (((sup_sup @ (set @ B)) @ A18) @ (((insert2 @ B) @ (F @ B2)) @ (bot_bot @ (set @ B))))))))))).
% 9.43/3.46 thf(fact_4819_notIn__Un__bij__betw3, axiom, (! [A:$tType,B:$tType,B2:A,A4:(set @ A),F:(A > B),A18:(set @ B)]: (((~ ((((member @ A) @ B2) @ A4))) => ((~ ((((member @ B) @ (F @ B2)) @ A18))) => ((((((bij_betw @ A) @ B) @ F) @ A4) @ A18) = (((((bij_betw @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) @ (((sup_sup @ (set @ B)) @ A18) @ (((insert2 @ B) @ (F @ B2)) @ (bot_bot @ (set @ B))))))))))).
% 9.43/3.46 thf(fact_4820_bijI, axiom, (! [A:$tType,B:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => (((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 9.43/3.46 thf(fact_4821_bij__def, axiom, (! [A:$tType,B:$tType,F:(A > B)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) = (((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) & (((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B)))))))).
% 9.43/3.46 thf(fact_4822_bij__betw__partition, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),C5:(set @ A),B6:(set @ B),D4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ C5)) @ (((sup_sup @ (set @ B)) @ B6) @ D4)) => ((((((bij_betw @ A) @ B) @ F) @ C5) @ D4) => (((((inf_inf @ (set @ A)) @ A4) @ C5) = (bot_bot @ (set @ A))) => (((((inf_inf @ (set @ B)) @ B6) @ D4) = (bot_bot @ (set @ B))) => (((((bij_betw @ A) @ B) @ F) @ A4) @ B6)))))))).
% 9.43/3.46 thf(fact_4823_bij__betw__combine, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),B6:(set @ B),C5:(set @ A),D4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => ((((((bij_betw @ A) @ B) @ F) @ C5) @ D4) => (((((inf_inf @ (set @ B)) @ B6) @ D4) = (bot_bot @ (set @ B))) => (((((bij_betw @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A4) @ C5)) @ (((sup_sup @ (set @ B)) @ B6) @ D4)))))))).
% 9.43/3.46 thf(fact_4824_o__bij, axiom, (! [A:$tType,B:$tType,G:(B > A),F:(A > B)]: ((((((((comp @ B) @ A) @ A) @ G) @ F) = (id @ A)) => (((((((comp @ A) @ B) @ B) @ F) @ G) = (id @ B)) => (((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B)))))))).
% 9.43/3.46 thf(fact_4825_bij__image__Compl__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => (((((image2 @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ A4)) = ((uminus_uminus @ (set @ B)) @ ((((image2 @ A) @ B) @ F) @ A4))))))).
% 9.43/3.46 thf(fact_4826_Well__order__iso__copy, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),F:(A > B),A18:(set @ B)]: (((((order_well_order_on @ A) @ A4) @ R) => ((((((bij_betw @ A) @ B) @ F) @ A4) @ A18) => (? [R10:(set @ ((product_prod @ B) @ B))]: (((((order_well_order_on @ B) @ A18) @ R10) & (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R10)) @ ((bNF_Wellorder_ordIso @ A) @ B)))))))))).
% 9.43/3.46 thf(fact_4827_bij__betw__disjoint__Un, axiom, (! [A:$tType,B:$tType,F:(A > B),A4:(set @ A),C5:(set @ B),G:(A > B),B6:(set @ A),D4:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ C5) => ((((((bij_betw @ A) @ B) @ G) @ B6) @ D4) => (((((inf_inf @ (set @ A)) @ A4) @ B6) = (bot_bot @ (set @ A))) => (((((inf_inf @ (set @ B)) @ C5) @ D4) = (bot_bot @ (set @ B))) => (((((bij_betw @ A) @ B) @ (^ [X3:A]: (((((if @ B) @ (((member @ A) @ X3) @ A4)) @ (F @ X3)) @ (G @ X3))))) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) @ (((sup_sup @ (set @ B)) @ C5) @ D4))))))))).
% 9.43/3.46 thf(fact_4828_is__unit__prod__mset__iff, axiom, (! [A:$tType]: (((algebraic_semidom @ A) => (! [A4:(multiset @ A)]: (((((dvd_dvd @ A) @ ((comm_m9189036328036947845d_mset @ A) @ A4)) @ (one_one @ A)) = (! [X3:A]: (((((member @ A) @ X3) @ ((set_mset @ A) @ A4)) => (((dvd_dvd @ A) @ X3) @ (one_one @ A)))))))))))).
% 9.43/3.46 thf(fact_4829_prod__mset_Oeq__fold, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => ((comm_m9189036328036947845d_mset @ A) = ((((fold_mset @ A) @ A) @ (times_times @ A)) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_4830_vimage__subset__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),B6:(set @ B),A4:(set @ A)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((ord_less_eq @ (set @ A)) @ ((((vimage @ A) @ B) @ F) @ B6)) @ A4) = (((ord_less_eq @ (set @ B)) @ B6) @ ((((image2 @ A) @ B) @ F) @ A4))))))).
% 9.43/3.46 thf(fact_4831_ex__bij__betw__nat__finite, axiom, (! [A:$tType,M3:(set @ A)]: ((((finite_finite2 @ A) @ M3) => (? [H2:(nat > A)]: ((((((bij_betw @ nat) @ A) @ H2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ M3))) @ M3))))))).
% 9.43/3.46 thf(fact_4832_ex__bij__betw__nat__finite__1, axiom, (! [A:$tType,M3:(set @ A)]: ((((finite_finite2 @ A) @ M3) => (? [H2:(nat > A)]: ((((((bij_betw @ nat) @ A) @ H2) @ (((set_or1337092689740270186AtMost @ nat) @ (one_one @ nat)) @ ((finite_card @ A) @ M3))) @ M3))))))).
% 9.43/3.46 thf(fact_4833_coprime__diff__one__right__nat, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((algebr8660921524188924756oprime @ nat) @ N) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))))))).
% 9.43/3.46 thf(fact_4834_coprime__diff__one__left__nat, axiom, (! [N:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ N) => (((algebr8660921524188924756oprime @ nat) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) @ N))))).
% 9.43/3.46 thf(fact_4835_prod_Oreindex__bij__betw__not__neutral, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_mult @ A) => (! [S7:(set @ B),T5:(set @ C),H3:(B > C),S2:(set @ B),T6:(set @ C),G:(C > A)]: ((((finite_finite2 @ B) @ S7) => (((finite_finite2 @ C) @ T5) => ((((((bij_betw @ B) @ C) @ H3) @ (((minus_minus @ (set @ B)) @ S2) @ S7)) @ (((minus_minus @ (set @ C)) @ T6) @ T5)) => ((! [A5:B]: (((((member @ B) @ A5) @ S7) => ((G @ (H3 @ A5)) = (one_one @ A))))) => ((! [B3:C]: (((((member @ C) @ B3) @ T5) => ((G @ B3) = (one_one @ A))))) => (((((groups7121269368397514597t_prod @ B) @ A) @ (^ [X3:B]: ((G @ (H3 @ X3))))) @ S2) = ((((groups7121269368397514597t_prod @ C) @ A) @ G) @ T6))))))))))))).
% 9.43/3.46 thf(fact_4836_sum_OatLeastAtMost__reindex, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_add @ A) & (ord @ B)) => (! [H3:(nat > B),M2:nat,N:nat,G:(B > A)]: (((((((bij_betw @ nat) @ B) @ H3) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) @ (((set_or1337092689740270186AtMost @ B) @ (H3 @ M2)) @ (H3 @ N))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((set_or1337092689740270186AtMost @ B) @ (H3 @ M2)) @ (H3 @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ B) @ A) @ nat) @ G) @ H3)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4837_sum_OatLeastLessThan__reindex, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_add @ A) & (ord @ B)) => (! [H3:(nat > B),M2:nat,N:nat,G:(B > A)]: (((((((bij_betw @ nat) @ B) @ H3) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) @ (((set_or7035219750837199246ssThan @ B) @ (H3 @ M2)) @ (H3 @ N))) => (((((groups7311177749621191930dd_sum @ B) @ A) @ G) @ (((set_or7035219750837199246ssThan @ B) @ (H3 @ M2)) @ (H3 @ N))) = ((((groups7311177749621191930dd_sum @ nat) @ A) @ (((((comp @ B) @ A) @ nat) @ G) @ H3)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4838_prod_OatLeastAtMost__reindex, axiom, (! [B:$tType,A:$tType]: ((((comm_monoid_mult @ A) & (ord @ B)) => (! [H3:(nat > B),M2:nat,N:nat,G:(B > A)]: (((((((bij_betw @ nat) @ B) @ H3) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) @ (((set_or1337092689740270186AtMost @ B) @ (H3 @ M2)) @ (H3 @ N))) => (((((groups7121269368397514597t_prod @ B) @ A) @ G) @ (((set_or1337092689740270186AtMost @ B) @ (H3 @ M2)) @ (H3 @ N))) = ((((groups7121269368397514597t_prod @ nat) @ A) @ (((((comp @ B) @ A) @ nat) @ G) @ H3)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4839_antisymp__antisym__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((antisymp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = ((antisym @ A) @ R))))).
% 9.43/3.46 thf(fact_4840_convol__expand__snd_H, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > ((product_prod @ B) @ C)),G:(A > B),H3:(A > C)]: ((((((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ F) = G) => ((H3 = (((((comp @ ((product_prod @ B) @ C)) @ C) @ A) @ ((product_snd @ B) @ C)) @ F)) = ((((((bNF_convol @ A) @ B) @ C) @ G) @ H3) = F)))))).
% 9.43/3.46 thf(fact_4841_convol__expand__snd, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > ((product_prod @ B) @ C)),G:(A > B)]: ((((((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ F) = G) => ((((((bNF_convol @ A) @ B) @ C) @ G) @ (((((comp @ ((product_prod @ B) @ C)) @ C) @ A) @ ((product_snd @ B) @ C)) @ F)) = F))))).
% 9.43/3.46 thf(fact_4842_convol__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_convol @ A) @ B) @ C) = (^ [F2:(A > B),G2:(A > C),A7:A]: (((((product_Pair @ B) @ C) @ (F2 @ A7)) @ (G2 @ A7)))))))).
% 9.43/3.46 thf(fact_4843_bij__swap, axiom, (! [A:$tType,B:$tType]: ((((((bij_betw @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)) @ ((product_swap @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) @ (top_top @ (set @ ((product_prod @ B) @ A))))))).
% 9.43/3.46 thf(fact_4844_fst__convol, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > B),G:(A > C)]: (((((((comp @ ((product_prod @ B) @ C)) @ B) @ A) @ ((product_fst @ B) @ C)) @ (((((bNF_convol @ A) @ B) @ C) @ F) @ G)) = F)))).
% 9.43/3.46 thf(fact_4845_snd__convol, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > C),G:(A > B)]: (((((((comp @ ((product_prod @ C) @ B)) @ B) @ A) @ ((product_snd @ C) @ B)) @ (((((bNF_convol @ A) @ C) @ B) @ F) @ G)) = G)))).
% 9.43/3.46 thf(fact_4846_convol__o, axiom, (! [B:$tType,C:$tType,D:$tType,A:$tType,F:(D > B),G:(D > C),H3:(A > D)]: (((((((comp @ D) @ ((product_prod @ B) @ C)) @ A) @ (((((bNF_convol @ D) @ B) @ C) @ F) @ G)) @ H3) = (((((bNF_convol @ A) @ B) @ C) @ (((((comp @ D) @ B) @ A) @ F) @ H3)) @ (((((comp @ D) @ C) @ A) @ G) @ H3)))))).
% 9.43/3.46 thf(fact_4847_ex__bij__betw__finite__nat, axiom, (! [A:$tType,M3:(set @ A)]: ((((finite_finite2 @ A) @ M3) => (? [H2:(A > nat)]: ((((((bij_betw @ A) @ nat) @ H2) @ M3) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((finite_card @ A) @ M3))))))))).
% 9.43/3.46 thf(fact_4848_map__prod__o__convol__id, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > A),G:(C > B),X:C]: ((((((((comp @ ((product_prod @ C) @ B)) @ ((product_prod @ A) @ B)) @ C) @ ((((((product_map_prod @ C) @ A) @ B) @ B) @ F) @ (id @ B))) @ (((((bNF_convol @ C) @ C) @ B) @ (id @ C)) @ G)) @ X) = ((((((bNF_convol @ C) @ A) @ B) @ (((((comp @ A) @ A) @ C) @ (id @ A)) @ F)) @ G) @ X))))).
% 9.43/3.46 thf(fact_4849_Heap__lub__empty, axiom, (! [A:$tType]: ((((heap_Time_Heap_lub @ A) @ (bot_bot @ (set @ (heap_Time_Heap @ A)))) = ((heap_Time_Heap2 @ A) @ (^ [X3:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 9.43/3.46 thf(fact_4850_and_Osemilattice__neutr__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((semilattice_neutr @ A) @ (bit_se5824344872417868541ns_and @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_4851_map__prod__ident, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X3:A]: (X3))) @ (^ [Y3:B]: (Y3))) = (^ [Z5:((product_prod @ A) @ B)]: (Z5)))))).
% 9.43/3.46 thf(fact_4852_map__prod__simp, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,F:(C > A),G:(D > B),A3:C,B2:D]: (((((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ ((((product_Pair @ C) @ D) @ A3) @ B2)) = ((((product_Pair @ A) @ B) @ (F @ A3)) @ (G @ B2)))))).
% 9.43/3.46 thf(fact_4853_fst__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > A),G:(D > B),X:((product_prod @ C) @ D)]: (((((product_fst @ A) @ B) @ (((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G) @ X)) = (F @ (((product_fst @ C) @ D) @ X)))))).
% 9.43/3.46 thf(fact_4854_snd__map__prod, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(C > B),G:(D > A),X:((product_prod @ C) @ D)]: (((((product_snd @ B) @ A) @ (((((((product_map_prod @ C) @ B) @ D) @ A) @ F) @ G) @ X)) = (G @ (((product_snd @ C) @ D) @ X)))))).
% 9.43/3.46 thf(fact_4855_map__prod__imageI, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,A3:A,B2:B,R5:(set @ ((product_prod @ A) @ B)),F:(A > C),G:(B > D)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R5) => (((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ (F @ A3)) @ (G @ B2))) @ ((((image2 @ ((product_prod @ A) @ B)) @ ((product_prod @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G)) @ R5)))))).
% 9.43/3.46 thf(fact_4856_fst__comp__map__prod, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F:(A > C),G:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ C) @ ((product_prod @ A) @ B)) @ ((product_fst @ C) @ D)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ F) @ G)) = (((((comp @ A) @ C) @ ((product_prod @ A) @ B)) @ F) @ ((product_fst @ A) @ B)))))).
% 9.43/3.46 thf(fact_4857_snd__comp__map__prod, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,F:(A > D),G:(B > C)]: (((((((comp @ ((product_prod @ D) @ C)) @ C) @ ((product_prod @ A) @ B)) @ ((product_snd @ D) @ C)) @ ((((((product_map_prod @ A) @ D) @ B) @ C) @ F) @ G)) = (((((comp @ B) @ C) @ ((product_prod @ A) @ B)) @ G) @ ((product_snd @ A) @ B)))))).
% 9.43/3.46 thf(fact_4858_prod_Omap__comp, axiom, (! [D:$tType,F5:$tType,E:$tType,C:$tType,B:$tType,A:$tType,G1:(C > E),G22:(D > F5),F1:(A > C),F22:(B > D),V:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F5) @ G1) @ G22) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ F1) @ F22) @ V)) = (((((((product_map_prod @ A) @ E) @ B) @ F5) @ (((((comp @ C) @ E) @ A) @ G1) @ F1)) @ (((((comp @ D) @ F5) @ B) @ G22) @ F22)) @ V))))).
% 9.43/3.46 thf(fact_4859_map__prod_Ocompositionality, axiom, (! [D:$tType,F5:$tType,E:$tType,C:$tType,B:$tType,A:$tType,F:(C > E),G:(D > F5),H3:(A > C),I:(B > D),Prod:((product_prod @ A) @ B)]: (((((((((product_map_prod @ C) @ E) @ D) @ F5) @ F) @ G) @ (((((((product_map_prod @ A) @ C) @ B) @ D) @ H3) @ I) @ Prod)) = (((((((product_map_prod @ A) @ E) @ B) @ F5) @ (((((comp @ C) @ E) @ A) @ F) @ H3)) @ (((((comp @ D) @ F5) @ B) @ G) @ I)) @ Prod))))).
% 9.43/3.46 thf(fact_4860_map__prod_Oidentity, axiom, (! [B:$tType,A:$tType]: ((((((((product_map_prod @ A) @ A) @ B) @ B) @ (^ [X3:A]: (X3))) @ (^ [X3:B]: (X3))) = (id @ ((product_prod @ A) @ B)))))).
% 9.43/3.46 thf(fact_4861_prod__fun__imageE, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,C2:((product_prod @ A) @ B),F:(C > A),G:(D > B),R5:(set @ ((product_prod @ C) @ D))]: (((((member @ ((product_prod @ A) @ B)) @ C2) @ ((((image2 @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ A) @ D) @ B) @ F) @ G)) @ R5)) => (~ ((! [X2:C,Y4:D]: (((C2 = ((((product_Pair @ A) @ B) @ (F @ X2)) @ (G @ Y4))) => (~ ((((member @ ((product_prod @ C) @ D)) @ ((((product_Pair @ C) @ D) @ X2) @ Y4)) @ R5)))))))))))).
% 9.43/3.46 thf(fact_4862_apfst__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((product_apfst @ A) @ C) @ B) = (^ [F2:(A > C)]: (((((((product_map_prod @ A) @ C) @ B) @ B) @ F2) @ (id @ B)))))))).
% 9.43/3.46 thf(fact_4863_apsnd__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((product_apsnd @ B) @ C) @ A) = (((((product_map_prod @ A) @ A) @ B) @ C) @ (id @ A)))))).
% 9.43/3.46 thf(fact_4864_map__prod__def, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType]: ((((((product_map_prod @ A) @ C) @ B) @ D) = (^ [F2:(A > C),G2:(B > D)]: (((((product_case_prod @ A) @ B) @ ((product_prod @ C) @ D)) @ (^ [X3:A,Y3:B]: (((((product_Pair @ C) @ D) @ (F2 @ X3)) @ (G2 @ Y3))))))))))).
% 9.43/3.46 thf(fact_4865_map__prod__compose, axiom, (! [D:$tType,C:$tType,A:$tType,E:$tType,F5:$tType,B:$tType,F1:(E > C),F22:(A > E),G1:(F5 > D),G22:(B > F5)]: ((((((((product_map_prod @ A) @ C) @ B) @ D) @ (((((comp @ E) @ C) @ A) @ F1) @ F22)) @ (((((comp @ F5) @ D) @ B) @ G1) @ G22)) = (((((comp @ ((product_prod @ E) @ F5)) @ ((product_prod @ C) @ D)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ E) @ C) @ F5) @ D) @ F1) @ G1)) @ ((((((product_map_prod @ A) @ E) @ B) @ F5) @ F22) @ G22)))))).
% 9.43/3.46 thf(fact_4866_map__prod_Ocomp, axiom, (! [A:$tType,C:$tType,E:$tType,F5:$tType,D:$tType,B:$tType,F:(C > E),G:(D > F5),H3:(A > C),I:(B > D)]: (((((((comp @ ((product_prod @ C) @ D)) @ ((product_prod @ E) @ F5)) @ ((product_prod @ A) @ B)) @ ((((((product_map_prod @ C) @ E) @ D) @ F5) @ F) @ G)) @ ((((((product_map_prod @ A) @ C) @ B) @ D) @ H3) @ I)) = ((((((product_map_prod @ A) @ E) @ B) @ F5) @ (((((comp @ C) @ E) @ A) @ F) @ H3)) @ (((((comp @ D) @ F5) @ B) @ G) @ I)))))).
% 9.43/3.46 thf(fact_4867_map__prod__surj__on, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(B > A),A4:(set @ B),A18:(set @ A),G:(D > C),B6:(set @ D),B14:(set @ C)]: (((((((image2 @ B) @ A) @ F) @ A4) = A18) => ((((((image2 @ D) @ C) @ G) @ B6) = B14) => (((((image2 @ ((product_prod @ B) @ D)) @ ((product_prod @ A) @ C)) @ ((((((product_map_prod @ B) @ A) @ D) @ C) @ F) @ G)) @ ((((product_Sigma @ B) @ D) @ A4) @ (^ [Uu:B]: (B6)))) = ((((product_Sigma @ A) @ C) @ A18) @ (^ [Uu:A]: (B14))))))))).
% 9.43/3.46 thf(fact_4868_map__prod__o__convol, axiom, (! [D:$tType,B:$tType,C:$tType,E:$tType,A:$tType,H1:(D > B),H22:(E > C),F:(A > D),G:(A > E)]: (((((((comp @ ((product_prod @ D) @ E)) @ ((product_prod @ B) @ C)) @ A) @ ((((((product_map_prod @ D) @ B) @ E) @ C) @ H1) @ H22)) @ (((((bNF_convol @ A) @ D) @ E) @ F) @ G)) = (((((bNF_convol @ A) @ B) @ C) @ (((((comp @ D) @ B) @ A) @ H1) @ F)) @ (((((comp @ E) @ C) @ A) @ H22) @ G)))))).
% 9.43/3.46 thf(fact_4869_map__prod__inj__on, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,F:(A > B),A4:(set @ A),G:(C > D),B6:(set @ C)]: ((((((inj_on @ A) @ B) @ F) @ A4) => (((((inj_on @ C) @ D) @ G) @ B6) => ((((inj_on @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G)) @ ((((product_Sigma @ A) @ C) @ A4) @ (^ [Uu:A]: (B6))))))))).
% 9.43/3.46 thf(fact_4870_map__prod__surj, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(A > B),G:(C > D)]: (((((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((((((image2 @ C) @ D) @ G) @ (top_top @ (set @ C))) = (top_top @ (set @ D))) => (((((image2 @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ ((((((product_map_prod @ A) @ B) @ C) @ D) @ F) @ G)) @ (top_top @ (set @ ((product_prod @ A) @ C)))) = (top_top @ (set @ ((product_prod @ B) @ D))))))))).
% 9.43/3.46 thf(fact_4871_Zorns__po__lemma, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_7125193373082350890der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((! [C8:(set @ A)]: (((((member @ (set @ A)) @ C8) @ ((chains @ A) @ R)) => (? [X7:A]: (((((member @ A) @ X7) @ ((field2 @ A) @ R)) & (! [Xa3:A]: (((((member @ A) @ Xa3) @ C8) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Xa3) @ X7)) @ R)))))))))) => (? [X2:A]: (((((member @ A) @ X2) @ ((field2 @ A) @ R)) & (! [Xa2:A]: (((((member @ A) @ Xa2) @ ((field2 @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Xa2)) @ R) => (Xa2 = X2))))))))))))).
% 9.43/3.46 thf(fact_4872_assert__def, axiom, (! [A:$tType]: (((heap_Time_assert @ A) = (^ [P4:(A > $o),X3:A]: (((((if @ (heap_Time_Heap @ A)) @ (P4 @ X3)) @ ((heap_Time_return @ A) @ X3)) @ ((heap_Time_raise @ A) @ (((cons @ char) @ ((((((((char2 @ $true) @ $false) @ $false) @ $false) @ $false) @ $true) @ $true) @ $false)) @ (((cons @ char) @ ((((((((char2 @ $true) @ $true) @ $false) @ $false) @ $true) @ $true) @ $true) @ $false)) @ (((cons @ char) @ ((((((((char2 @ $true) @ $true) @ $false) @ $false) @ $true) @ $true) @ $true) @ $false)) @ (((cons @ char) @ ((((((((char2 @ $true) @ $false) @ $true) @ $false) @ $false) @ $true) @ $true) @ $false)) @ (((cons @ char) @ ((((((((char2 @ $false) @ $true) @ $false) @ $false) @ $true) @ $true) @ $true) @ $false)) @ (((cons @ char) @ ((((((((char2 @ $false) @ $false) @ $true) @ $false) @ $true) @ $true) @ $true) @ $false)) @ (nil @ char))))))))))))))).
% 9.43/3.46 thf(fact_4873_power__int__add__1, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M2:int]: ((((X != (zero_zero @ A)) | (M2 != ((uminus_uminus @ int) @ (one_one @ int)))) => ((((power_int @ A) @ X) @ (((plus_plus @ int) @ M2) @ (one_one @ int))) = (((times_times @ A) @ (((power_int @ A) @ X) @ M2)) @ X))))))))).
% 9.43/3.46 thf(fact_4874_power__int__1__left, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int]: (((((power_int @ A) @ (one_one @ A)) @ N) = (one_one @ A)))))))).
% 9.43/3.46 thf(fact_4875_power__int__0__right, axiom, (! [B:$tType]: ((((inverse @ B) & (power @ B)) => (! [X:B]: (((((power_int @ B) @ X) @ (zero_zero @ int)) = (one_one @ B)))))))).
% 9.43/3.46 thf(fact_4876_power__int__minus__one__mult__self, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M2:int]: (((((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M2)) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M2)) = (one_one @ A)))))))).
% 9.43/3.46 thf(fact_4877_power__int__minus__one__mult__self_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M2:int,B2:A]: (((((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M2)) @ (((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ M2)) @ B2)) = B2))))))).
% 9.43/3.46 thf(fact_4878_power__int__add__numeral, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M2:num,N:num]: (((((times_times @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ M2))) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ N))) = (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ (((plus_plus @ num) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4879_power__int__add__numeral2, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M2:num,N:num,B2:A]: (((((times_times @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ M2))) @ (((times_times @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ N))) @ B2)) = (((times_times @ A) @ (((power_int @ A) @ X) @ ((numeral_numeral @ int) @ (((plus_plus @ num) @ M2) @ N)))) @ B2)))))))).
% 9.43/3.46 thf(fact_4880_power__int__one__over, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,N:int]: (((((power_int @ A) @ (((divide_divide @ A) @ (one_one @ A)) @ X)) @ N) = (((divide_divide @ A) @ (one_one @ A)) @ (((power_int @ A) @ X) @ N))))))))).
% 9.43/3.46 thf(fact_4881_power__int__0__left__If, axiom, (! [A:$tType]: (((division_ring @ A) => (! [M2:int]: ((((M2 = (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M2) = (one_one @ A))) & ((M2 != (zero_zero @ int)) => ((((power_int @ A) @ (zero_zero @ A)) @ M2) = (zero_zero @ A)))))))))).
% 9.43/3.46 thf(fact_4882_power__int__increasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N:int,N6:int,A3:A]: (((((ord_less_eq @ int) @ N) @ N6) => ((((ord_less_eq @ A) @ (one_one @ A)) @ A3) => (((ord_less_eq @ A) @ (((power_int @ A) @ A3) @ N)) @ (((power_int @ A) @ A3) @ N6)))))))))).
% 9.43/3.46 thf(fact_4883_power__int__strict__increasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N:int,N6:int,A3:A]: (((((ord_less @ int) @ N) @ N6) => ((((ord_less @ A) @ (one_one @ A)) @ A3) => (((ord_less @ A) @ (((power_int @ A) @ A3) @ N)) @ (((power_int @ A) @ A3) @ N6)))))))))).
% 9.43/3.46 thf(fact_4884_power__int__minus__one__minus, axiom, (! [A:$tType]: (((division_ring @ A) => (! [N:int]: (((((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ ((uminus_uminus @ int) @ N)) = (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)))))))).
% 9.43/3.46 thf(fact_4885_power__int__minus__one__diff__commute, axiom, (! [A:$tType]: (((division_ring @ A) => (! [A3:int,B2:int]: (((((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((minus_minus @ int) @ A3) @ B2)) = (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ (((minus_minus @ int) @ B2) @ A3))))))))).
% 9.43/3.46 thf(fact_4886_power__int__strict__decreasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N:int,N6:int,A3:A]: (((((ord_less @ int) @ N) @ N6) => ((((ord_less @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less @ A) @ A3) @ (one_one @ A)) => (((ord_less @ A) @ (((power_int @ A) @ A3) @ N6)) @ (((power_int @ A) @ A3) @ N))))))))))).
% 9.43/3.46 thf(fact_4887_one__le__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,N:int]: (((((ord_less_eq @ A) @ (one_one @ A)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ A) @ (one_one @ A)) @ (((power_int @ A) @ X) @ N)))))))))).
% 9.43/3.46 thf(fact_4888_one__less__power__int, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [A3:A,N:int]: (((((ord_less @ A) @ (one_one @ A)) @ A3) => ((((ord_less @ int) @ (zero_zero @ int)) @ N) => (((ord_less @ A) @ (one_one @ A)) @ (((power_int @ A) @ A3) @ N)))))))))).
% 9.43/3.46 thf(fact_4889_power__int__add, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M2:int,N:int]: ((((X != (zero_zero @ A)) | ((((plus_plus @ int) @ M2) @ N) != (zero_zero @ int))) => ((((power_int @ A) @ X) @ (((plus_plus @ int) @ M2) @ N)) = (((times_times @ A) @ (((power_int @ A) @ X) @ M2)) @ (((power_int @ A) @ X) @ N)))))))))).
% 9.43/3.46 thf(fact_4890_power__int__minus__left__distrib, axiom, (! [C:$tType,B:$tType,A:$tType]: ((((division_ring @ A) & ((one @ B) & (uminus @ B))) => (! [X:C,A3:A,N:int]: ((((((nO_MATCH @ B) @ C) @ ((uminus_uminus @ B) @ (one_one @ B))) @ X) => ((((power_int @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = (((times_times @ A) @ (((power_int @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N)) @ (((power_int @ A) @ A3) @ N)))))))))).
% 9.43/3.46 thf(fact_4891_power__int__decreasing, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [N:int,N6:int,A3:A]: (((((ord_less_eq @ int) @ N) @ N6) => ((((ord_less_eq @ A) @ (zero_zero @ A)) @ A3) => ((((ord_less_eq @ A) @ A3) @ (one_one @ A)) => (((A3 != (zero_zero @ A)) | ((N6 != (zero_zero @ int)) | (N = (zero_zero @ int)))) => (((ord_less_eq @ A) @ (((power_int @ A) @ A3) @ N6)) @ (((power_int @ A) @ A3) @ N)))))))))))).
% 9.43/3.46 thf(fact_4892_power__int__le__one, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,N:int]: (((((ord_less_eq @ A) @ (zero_zero @ A)) @ X) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => ((((ord_less_eq @ A) @ X) @ (one_one @ A)) => (((ord_less_eq @ A) @ (((power_int @ A) @ X) @ N)) @ (one_one @ A))))))))))).
% 9.43/3.46 thf(fact_4893_power__int__le__imp__le__exp, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,M2:int,N:int]: (((((ord_less @ A) @ (one_one @ A)) @ X) => ((((ord_less_eq @ A) @ (((power_int @ A) @ X) @ M2)) @ (((power_int @ A) @ X) @ N)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => (((ord_less_eq @ int) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4894_power__int__le__imp__less__exp, axiom, (! [A:$tType]: (((linordered_field @ A) => (! [X:A,M2:int,N:int]: (((((ord_less @ A) @ (one_one @ A)) @ X) => ((((ord_less @ A) @ (((power_int @ A) @ X) @ M2)) @ (((power_int @ A) @ X) @ N)) => ((((ord_less_eq @ int) @ (zero_zero @ int)) @ N) => (((ord_less @ int) @ M2) @ N)))))))))).
% 9.43/3.46 thf(fact_4895_power__int__add__1_H, axiom, (! [A:$tType]: (((division_ring @ A) => (! [X:A,M2:int]: ((((X != (zero_zero @ A)) | (M2 != ((uminus_uminus @ int) @ (one_one @ int)))) => ((((power_int @ A) @ X) @ (((plus_plus @ int) @ M2) @ (one_one @ int))) = (((times_times @ A) @ X) @ (((power_int @ A) @ X) @ M2)))))))))).
% 9.43/3.46 thf(fact_4896_integer__of__char__code, axiom, (! [B0:$o,B1:$o,B22:$o,B32:$o,B42:$o,B52:$o,B62:$o,B72:$o]: (((integer_of_char @ ((((((((char2 @ B0) @ B1) @ B22) @ B32) @ B42) @ B52) @ B62) @ B72)) = (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ (((plus_plus @ code_integer) @ (((times_times @ code_integer) @ ((zero_neq_one_of_bool @ code_integer) @ B72)) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B62))) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B52))) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B42))) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B32))) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B22))) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B1))) @ ((numeral_numeral @ code_integer) @ (bit0 @ one2)))) @ ((zero_neq_one_of_bool @ code_integer) @ B0)))))).
% 9.43/3.46 thf(fact_4897_char__of__def, axiom, (! [A:$tType]: (((bit_un5681908812861735899ations @ A) => ((unique5772411509450598832har_of @ A) = (^ [N5:A]: (((((((((char2 @ (~ ((((dvd_dvd @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N5)))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ (one_one @ nat))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ one2)))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ ((numeral_numeral @ nat) @ (bit1 @ one2)))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ (bit0 @ one2))))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ ((numeral_numeral @ nat) @ (bit1 @ (bit0 @ one2))))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ ((numeral_numeral @ nat) @ (bit0 @ (bit1 @ one2))))) @ (((bit_se5641148757651400278ts_bit @ A) @ N5) @ ((numeral_numeral @ nat) @ (bit1 @ (bit1 @ one2)))))))))))).
% 9.43/3.46 thf(fact_4898_numeral__unfold__funpow, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((numeral_numeral @ A) = (^ [K4:num]: (((((compow @ (A > A)) @ ((numeral_numeral @ nat) @ K4)) @ ((plus_plus @ A) @ (one_one @ A))) @ (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_4899_relpowp__1, axiom, (! [A:$tType,P:(A > (A > $o))]: (((((compow @ (A > (A > $o))) @ (one_one @ nat)) @ P) = P)))).
% 9.43/3.46 thf(fact_4900_Suc__funpow, axiom, (! [N:nat]: (((((compow @ (nat > nat)) @ N) @ suc) = ((plus_plus @ nat) @ N))))).
% 9.43/3.46 thf(fact_4901_comp__funpow, axiom, (! [B:$tType,A:$tType,N:nat,F:(A > A)]: (((((compow @ ((B > A) > (B > A))) @ N) @ ((((comp @ A) @ A) @ B) @ F)) = ((((comp @ A) @ A) @ B) @ (((compow @ (A > A)) @ N) @ F)))))).
% 9.43/3.46 thf(fact_4902_funpow_Osimps_I2_J, axiom, (! [A:$tType,N:nat,F:(A > A)]: (((((compow @ (A > A)) @ (suc @ N)) @ F) = (((((comp @ A) @ A) @ A) @ F) @ (((compow @ (A > A)) @ N) @ F)))))).
% 9.43/3.46 thf(fact_4903_funpow__Suc__right, axiom, (! [A:$tType,N:nat,F:(A > A)]: (((((compow @ (A > A)) @ (suc @ N)) @ F) = (((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ F))))).
% 9.43/3.46 thf(fact_4904_funpow__add, axiom, (! [A:$tType,M2:nat,N:nat,F:(A > A)]: (((((compow @ (A > A)) @ (((plus_plus @ nat) @ M2) @ N)) @ F) = (((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ M2) @ F)) @ (((compow @ (A > A)) @ N) @ F)))))).
% 9.43/3.46 thf(fact_4905_of__nat__def, axiom, (! [A:$tType]: (((semiring_1 @ A) => ((semiring_1_of_nat @ A) = (^ [N5:nat]: (((((compow @ (A > A)) @ N5) @ ((plus_plus @ A) @ (one_one @ A))) @ (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_4906_numeral__add__unfold__funpow, axiom, (! [A:$tType]: (((semiring_numeral @ A) => (! [K:num,A3:A]: (((((plus_plus @ A) @ ((numeral_numeral @ A) @ K)) @ A3) = ((((compow @ (A > A)) @ ((numeral_numeral @ nat) @ K)) @ ((plus_plus @ A) @ (one_one @ A))) @ A3)))))))).
% 9.43/3.46 thf(fact_4907_inj__on__funpow__least, axiom, (! [A:$tType,N:nat,F:(A > A),S:A]: (((((((compow @ (A > A)) @ N) @ F) @ S) = S) => ((! [M5:nat]: (((((ord_less @ nat) @ (zero_zero @ nat)) @ M5) => ((((ord_less @ nat) @ M5) @ N) => (((((compow @ (A > A)) @ M5) @ F) @ S) != S))))) => ((((inj_on @ nat) @ A) @ (^ [K4:nat]: (((((compow @ (A > A)) @ K4) @ F) @ S)))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))).
% 9.43/3.46 thf(fact_4908_card__Un__disjnt, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ A) @ B6) => ((((disjnt @ A) @ A4) @ B6) => (((finite_card @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6)) = (((plus_plus @ nat) @ ((finite_card @ A) @ A4)) @ ((finite_card @ A) @ B6))))))))).
% 9.43/3.46 thf(fact_4909_plus__multiset__def, axiom, (! [A:$tType]: (((plus_plus @ (multiset @ A)) = (((((((map_fun @ (multiset @ A)) @ (A > nat)) @ ((A > nat) > (A > nat))) @ ((multiset @ A) > (multiset @ A))) @ (count @ A)) @ ((((((map_fun @ (multiset @ A)) @ (A > nat)) @ (A > nat)) @ (multiset @ A)) @ (count @ A)) @ (abs_multiset @ A))) @ (^ [M8:(A > nat),N9:(A > nat),A7:A]: ((((plus_plus @ nat) @ (M8 @ A7)) @ (N9 @ A7))))))))).
% 9.43/3.46 thf(fact_4910_mset__subset__add__iff2, axiom, (! [A:$tType,I:nat,J:nat,U:(multiset @ A),M2:(multiset @ A),N:(multiset @ A)]: (((((ord_less_eq @ nat) @ I) @ J) => ((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ J) @ U)) @ N)) = (((subset_mset @ A) @ M2) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 9.43/3.46 thf(fact_4911_repeat__mset__distrib2, axiom, (! [A:$tType,N:nat,A4:(multiset @ A),B6:(multiset @ A)]: (((((repeat_mset @ A) @ N) @ (((plus_plus @ (multiset @ A)) @ A4) @ B6)) = (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ N) @ A4)) @ (((repeat_mset @ A) @ N) @ B6)))))).
% 9.43/3.46 thf(fact_4912_repeat__mset__Suc, axiom, (! [A:$tType,N:nat,M3:(multiset @ A)]: (((((repeat_mset @ A) @ (suc @ N)) @ M3) = (((plus_plus @ (multiset @ A)) @ M3) @ (((repeat_mset @ A) @ N) @ M3)))))).
% 9.43/3.46 thf(fact_4913_disjnt__Times1__iff, axiom, (! [A:$tType,B:$tType,C5:(set @ A),A4:(set @ B),B6:(set @ B)]: (((((disjnt @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu:A]: (A4)))) @ ((((product_Sigma @ A) @ B) @ C5) @ (^ [Uu:A]: (B6)))) = ((C5 = (bot_bot @ (set @ A))) | (((disjnt @ B) @ A4) @ B6)))))).
% 9.43/3.46 thf(fact_4914_disjnt__Times2__iff, axiom, (! [B:$tType,A:$tType,A4:(set @ A),C5:(set @ B),B6:(set @ A)]: (((((disjnt @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (C5)))) @ ((((product_Sigma @ A) @ B) @ B6) @ (^ [Uu:A]: (C5)))) = ((C5 = (bot_bot @ (set @ B))) | (((disjnt @ A) @ A4) @ B6)))))).
% 9.43/3.46 thf(fact_4915_disjnt__Sigma__iff, axiom, (! [B:$tType,A:$tType,A4:(set @ A),C5:(A > (set @ B)),B6:(set @ A)]: (((((disjnt @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ C5)) @ ((((product_Sigma @ A) @ B) @ B6) @ C5)) = ((! [X3:A]: (((((member @ A) @ X3) @ (((inf_inf @ (set @ A)) @ A4) @ B6)) => ((C5 @ X3) = (bot_bot @ (set @ B)))))) | (((disjnt @ A) @ A4) @ B6)))))).
% 9.43/3.46 thf(fact_4916_left__add__mult__distrib__mset, axiom, (! [A:$tType,I:nat,U:(multiset @ A),J:nat,K:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ I) @ U)) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ J) @ U)) @ K)) = (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ (((plus_plus @ nat) @ I) @ J)) @ U)) @ K))))).
% 9.43/3.46 thf(fact_4917_repeat__mset__distrib, axiom, (! [A:$tType,M2:nat,N:nat,A4:(multiset @ A)]: (((((repeat_mset @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ A4) = (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ M2) @ A4)) @ (((repeat_mset @ A) @ N) @ A4)))))).
% 9.43/3.46 thf(fact_4918_mset__subseteq__add__iff1, axiom, (! [A:$tType,J:nat,I:nat,U:(multiset @ A),M2:(multiset @ A),N:(multiset @ A)]: (((((ord_less_eq @ nat) @ J) @ I) => ((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ J) @ U)) @ N)) = (((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N)))))).
% 9.43/3.46 thf(fact_4919_mset__subseteq__add__iff2, axiom, (! [A:$tType,I:nat,J:nat,U:(multiset @ A),M2:(multiset @ A),N:(multiset @ A)]: (((((ord_less_eq @ nat) @ I) @ J) => ((((subseteq_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ J) @ U)) @ N)) = (((subseteq_mset @ A) @ M2) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N))))))).
% 9.43/3.46 thf(fact_4920_mset__subset__add__iff1, axiom, (! [A:$tType,J:nat,I:nat,U:(multiset @ A),M2:(multiset @ A),N:(multiset @ A)]: (((((ord_less_eq @ nat) @ J) @ I) => ((((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ J) @ U)) @ N)) = (((subset_mset @ A) @ (((plus_plus @ (multiset @ A)) @ (((repeat_mset @ A) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N)))))).
% 9.43/3.46 thf(fact_4921_plus__multiset_Oabs__eq, axiom, (! [A:$tType,Xa:(A > nat),X:(A > nat)]: ((((((bNF_eq_onp @ (A > nat)) @ (^ [F2:(A > nat)]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X3))))))))) @ Xa) @ Xa) => (((((bNF_eq_onp @ (A > nat)) @ (^ [F2:(A > nat)]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X3))))))))) @ X) @ X) => ((((plus_plus @ (multiset @ A)) @ ((abs_multiset @ A) @ Xa)) @ ((abs_multiset @ A) @ X)) = ((abs_multiset @ A) @ (^ [A7:A]: ((((plus_plus @ nat) @ (Xa @ A7)) @ (X @ A7))))))))))).
% 9.43/3.46 thf(fact_4922_repeat__mset__distrib__add__mset, axiom, (! [A:$tType,N:nat,A3:A,A4:(multiset @ A)]: (((((repeat_mset @ A) @ N) @ (((add_mset @ A) @ A3) @ A4)) = (((plus_plus @ (multiset @ A)) @ (((replicate_mset @ A) @ N) @ A3)) @ (((repeat_mset @ A) @ N) @ A4)))))).
% 9.43/3.46 thf(fact_4923_o__prs_I2_J, axiom, (! [F5:$tType,E:$tType,C:$tType,D:$tType,A:$tType,B:$tType,H8:$tType,G3:$tType,R12:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R23:(C > (C > $o)),Abs22:(C > D),Rep22:(D > C),R32:(E > (E > $o)),Abs32:(E > F5),Rep32:(F5 > E)]: (((((((quotient3 @ A) @ B) @ R12) @ Abs1) @ Rep1) => ((((((quotient3 @ C) @ D) @ R23) @ Abs22) @ Rep22) => ((((((quotient3 @ E) @ F5) @ R32) @ Abs32) @ Rep32) => ((((((((map_fun @ (G3 > H8)) @ (G3 > H8)) @ ((A > G3) > (A > H8))) @ ((B > G3) > (B > H8))) @ (id @ (G3 > H8))) @ ((((((map_fun @ (B > G3)) @ (A > G3)) @ (A > H8)) @ (B > H8)) @ ((((((map_fun @ A) @ B) @ G3) @ G3) @ Abs1) @ (id @ G3))) @ ((((((map_fun @ B) @ A) @ H8) @ H8) @ Rep1) @ (id @ H8)))) @ (((comp @ G3) @ H8) @ A)) = (((comp @ G3) @ H8) @ B)))))))).
% 9.43/3.46 thf(fact_4924_abs__o__rep, axiom, (! [A:$tType,B:$tType,R5:(A > (A > $o)),Abs:(A > B),Rep:(B > A)]: (((((((quotient3 @ A) @ B) @ R5) @ Abs) @ Rep) => ((((((comp @ A) @ B) @ B) @ Abs) @ Rep) = (id @ B)))))).
% 9.43/3.46 thf(fact_4925_o__prs_I1_J, axiom, (! [C:$tType,E:$tType,A:$tType,B:$tType,F5:$tType,D:$tType,R12:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R23:(C > (C > $o)),Abs22:(C > D),Rep22:(D > C),R32:(E > (E > $o)),Abs32:(E > F5),Rep32:(F5 > E)]: (((((((quotient3 @ A) @ B) @ R12) @ Abs1) @ Rep1) => ((((((quotient3 @ C) @ D) @ R23) @ Abs22) @ Rep22) => ((((((quotient3 @ E) @ F5) @ R32) @ Abs32) @ Rep32) => ((((((((map_fun @ (D > F5)) @ (C > E)) @ ((A > C) > (A > E))) @ ((B > D) > (B > F5))) @ ((((((map_fun @ C) @ D) @ F5) @ E) @ Abs22) @ Rep32)) @ ((((((map_fun @ (B > D)) @ (A > C)) @ (A > E)) @ (B > F5)) @ ((((((map_fun @ A) @ B) @ D) @ C) @ Abs1) @ Rep22)) @ ((((((map_fun @ B) @ A) @ E) @ F5) @ Rep1) @ Abs32))) @ (((comp @ C) @ E) @ A)) = (((comp @ D) @ F5) @ B)))))))).
% 9.43/3.46 thf(fact_4926_plus__multiset_Orsp, axiom, (! [A:$tType]: (((((((((bNF_rel_fun @ (A > nat)) @ (A > nat)) @ ((A > nat) > (A > nat))) @ ((A > nat) > (A > nat))) @ ((bNF_eq_onp @ (A > nat)) @ (^ [F2:(A > nat)]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X3)))))))))) @ ((((((bNF_rel_fun @ (A > nat)) @ (A > nat)) @ (A > nat)) @ (A > nat)) @ ((bNF_eq_onp @ (A > nat)) @ (^ [F2:(A > nat)]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X3)))))))))) @ ((bNF_eq_onp @ (A > nat)) @ (^ [F2:(A > nat)]: (((finite_finite2 @ A) @ ((collect @ A) @ (^ [X3:A]: ((((ord_less @ nat) @ (zero_zero @ nat)) @ (F2 @ X3))))))))))) @ (^ [M8:(A > nat),N9:(A > nat),A7:A]: ((((plus_plus @ nat) @ (M8 @ A7)) @ (N9 @ A7))))) @ (^ [M8:(A > nat),N9:(A > nat),A7:A]: ((((plus_plus @ nat) @ (M8 @ A7)) @ (N9 @ A7)))))))).
% 9.43/3.46 thf(fact_4927_or__not__num__neg_Opelims, axiom, (! [X:num,Xa:num,Y:num]: (((((bit_or_not_num_neg @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ X) @ Xa)) => (((X = one2) => ((Xa = one2) => ((Y = one2) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ one2) @ one2))))))) => (((X = one2) => (! [M5:num]: (((Xa = (bit0 @ M5)) => ((Y = (bit1 @ M5)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ M5)))))))))) => (((X = one2) => (! [M5:num]: (((Xa = (bit1 @ M5)) => ((Y = (bit1 @ M5)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ M5)))))))))) => ((! [N3:num]: (((X = (bit0 @ N3)) => ((Xa = one2) => ((Y = (bit0 @ one2)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ N3)) @ one2))))))))) => ((! [N3:num]: (((X = (bit0 @ N3)) => (! [M5:num]: (((Xa = (bit0 @ M5)) => ((Y = (bitM @ ((bit_or_not_num_neg @ N3) @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ N3)) @ (bit0 @ M5)))))))))))) => ((! [N3:num]: (((X = (bit0 @ N3)) => (! [M5:num]: (((Xa = (bit1 @ M5)) => ((Y = (bit0 @ ((bit_or_not_num_neg @ N3) @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ N3)) @ (bit1 @ M5)))))))))))) => ((! [N3:num]: (((X = (bit1 @ N3)) => ((Xa = one2) => ((Y = one2) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ N3)) @ one2))))))))) => ((! [N3:num]: (((X = (bit1 @ N3)) => (! [M5:num]: (((Xa = (bit0 @ M5)) => ((Y = (bitM @ ((bit_or_not_num_neg @ N3) @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ N3)) @ (bit0 @ M5)))))))))))) => (~ ((! [N3:num]: (((X = (bit1 @ N3)) => (! [M5:num]: (((Xa = (bit1 @ M5)) => ((Y = (bitM @ ((bit_or_not_num_neg @ N3) @ M5))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_or3848514188828904588eg_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ N3)) @ (bit1 @ M5))))))))))))))))))))))))))).
% 9.43/3.46 thf(fact_4928_cr__natural__def, axiom, (code_cr_natural = (^ [X3:nat,Y3:code_natural]: ((X3 = (code_nat_of_natural @ Y3)))))).
% 9.43/3.46 thf(fact_4929_inv__o__cancel, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((comp @ B) @ A) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ F) = (id @ A)))))).
% 9.43/3.46 thf(fact_4930_o__inv__o__cancel, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),G:(A > C)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((comp @ B) @ C) @ A) @ (((((comp @ A) @ C) @ B) @ G) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F))) @ F) = G))))).
% 9.43/3.46 thf(fact_4931_inj__map__inv__f, axiom, (! [B:$tType,A:$tType,F:(A > B),L:(list @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((map @ B) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ ((((map @ A) @ B) @ F) @ L)) = L))))).
% 9.43/3.46 thf(fact_4932_inv__into__comp, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > B),G:(C > A),A4:(set @ C),X:B]: ((((((inj_on @ A) @ B) @ F) @ ((((image2 @ C) @ A) @ G) @ A4)) => (((((inj_on @ C) @ A) @ G) @ A4) => ((((member @ B) @ X) @ ((((image2 @ A) @ B) @ F) @ ((((image2 @ C) @ A) @ G) @ A4))) => ((((((hilbert_inv_into @ C) @ B) @ A4) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ X) = ((((((comp @ A) @ C) @ B) @ ((((hilbert_inv_into @ C) @ A) @ A4) @ G)) @ ((((hilbert_inv_into @ A) @ B) @ ((((image2 @ C) @ A) @ G) @ A4)) @ F)) @ X)))))))).
% 9.43/3.46 thf(fact_4933_inv__unique__comp, axiom, (! [B:$tType,A:$tType,F:(B > A),G:(A > B)]: ((((((((comp @ B) @ A) @ A) @ F) @ G) = (id @ A)) => (((((((comp @ A) @ B) @ B) @ G) @ F) = (id @ B)) => (((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F) = G)))))).
% 9.43/3.46 thf(fact_4934_o__inv__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > B),G:(C > A)]: (((((((bij_betw @ A) @ B) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ B))) => ((((((bij_betw @ C) @ A) @ G) @ (top_top @ (set @ C))) @ (top_top @ (set @ A))) => (((((hilbert_inv_into @ C) @ B) @ (top_top @ (set @ C))) @ (((((comp @ A) @ B) @ C) @ F) @ G)) = (((((comp @ A) @ C) @ B) @ ((((hilbert_inv_into @ C) @ A) @ (top_top @ (set @ C))) @ G)) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)))))))).
% 9.43/3.46 thf(fact_4935_surj__iff, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image2 @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = ((((((comp @ B) @ A) @ A) @ F) @ ((((hilbert_inv_into @ B) @ A) @ (top_top @ (set @ B))) @ F)) = (id @ A)))))).
% 9.43/3.46 thf(fact_4936_inj__iff, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) = ((((((comp @ B) @ A) @ A) @ ((((hilbert_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F)) @ F) = (id @ A)))))).
% 9.43/3.46 thf(fact_4937_fn__o__inv__fn__is__id, axiom, (! [A:$tType,F:(A > A),N:nat]: (((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => ((((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ N) @ F)) @ (((compow @ (A > A)) @ N) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F))) = (^ [X3:A]: (X3))))))).
% 9.43/3.46 thf(fact_4938_inv__fn__o__fn__is__id, axiom, (! [A:$tType,F:(A > A),N:nat]: (((((((bij_betw @ A) @ A) @ F) @ (top_top @ (set @ A))) @ (top_top @ (set @ A))) => ((((((comp @ A) @ A) @ A) @ (((compow @ (A > A)) @ N) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F))) @ (((compow @ (A > A)) @ N) @ F)) = (^ [X3:A]: (X3))))))).
% 9.43/3.46 thf(fact_4939_bijection_Oinv__comp__right, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => ((((((comp @ A) @ A) @ A) @ F) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F)) = (id @ A)))))).
% 9.43/3.46 thf(fact_4940_bijection_Oinv__comp__left, axiom, (! [A:$tType,F:(A > A)]: ((((hilbert_bijection @ A) @ F) => ((((((comp @ A) @ A) @ A) @ ((((hilbert_inv_into @ A) @ A) @ (top_top @ (set @ A))) @ F)) @ F) = (id @ A)))))).
% 9.43/3.46 thf(fact_4941_finite__reachable__restrictedI, axiom, (! [A:$tType,Q:(set @ A),I4:(set @ A),E4:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ A) @ Q) => ((((ord_less_eq @ (set @ A)) @ I4) @ Q) => ((((ord_less_eq @ (set @ A)) @ (((range2 @ A) @ A) @ E4)) @ Q) => ((finite_finite2 @ A) @ ((((image @ A) @ A) @ ((transitive_rtrancl @ A) @ E4)) @ I4)))))))).
% 9.43/3.46 thf(fact_4942_Range__insert, axiom, (! [A:$tType,B:$tType,A3:B,B2:A,R:(set @ ((product_prod @ B) @ A))]: (((((range2 @ B) @ A) @ (((insert2 @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A3) @ B2)) @ R)) = (((insert2 @ A) @ B2) @ (((range2 @ B) @ A) @ R)))))).
% 9.43/3.46 thf(fact_4943_Range__iff, axiom, (! [A:$tType,B:$tType,A3:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ A3) @ (((range2 @ B) @ A) @ R)) = (? [Y3:B]: ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ Y3) @ A3)) @ R))))))).
% 9.43/3.46 thf(fact_4944_RangeE, axiom, (! [A:$tType,B:$tType,B2:A,R:(set @ ((product_prod @ B) @ A))]: (((((member @ A) @ B2) @ (((range2 @ B) @ A) @ R)) => (~ ((! [A5:B]: ((~ ((((member @ ((product_prod @ B) @ A)) @ ((((product_Pair @ B) @ A) @ A5) @ B2)) @ R))))))))))).
% 9.43/3.46 thf(fact_4945_Range_Ointros, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R) => (((member @ B) @ B2) @ (((range2 @ A) @ B) @ R)))))).
% 9.43/3.46 thf(fact_4946_Range_Osimps, axiom, (! [B:$tType,A:$tType,A3:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A3) @ (((range2 @ A) @ B) @ R)) = (? [A7:A,B5:B]: (((A3 = B5) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B5)) @ R)))))))).
% 9.43/3.46 thf(fact_4947_Range_Ocases, axiom, (! [B:$tType,A:$tType,A3:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ B) @ A3) @ (((range2 @ A) @ B) @ R)) => (~ ((! [A5:A]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ A3)) @ R))))))))))).
% 9.43/3.46 thf(fact_4948_trancl__Image__in__Range, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),V6:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ ((transitive_trancl @ A) @ R5)) @ V6)) @ (((range2 @ A) @ A) @ R5))))).
% 9.43/3.46 thf(fact_4949_Range__rel__restrict, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((range2 @ A) @ A) @ (((rel_restrict @ A) @ R5) @ A4))) @ (((minus_minus @ (set @ A)) @ (((range2 @ A) @ A) @ R5)) @ A4))))).
% 9.43/3.46 thf(fact_4950_dom__ran__disj__comp, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((((inf_inf @ (set @ A)) @ (((domain @ A) @ A) @ R5)) @ (((range2 @ A) @ A) @ R5)) = (bot_bot @ (set @ A))) => ((((((relcomp @ A) @ A) @ A) @ R5) @ R5) = (bot_bot @ (set @ ((product_prod @ A) @ A)))))))).
% 9.43/3.46 thf(fact_4951_wf__max, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ (((converse @ A) @ A) @ R5)) => ((R5 != (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (~ ((! [M5:A]: ((~ ((((member @ A) @ M5) @ (((minus_minus @ (set @ A)) @ (((range2 @ A) @ A) @ R5)) @ (((domain @ A) @ A) @ R5)))))))))))))).
% 9.43/3.46 thf(fact_4952_Rangep__Range__eq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((rangep @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) = (^ [X3:B]: ((((member @ B) @ X3) @ (((range2 @ A) @ B) @ R)))))))).
% 9.43/3.46 thf(fact_4953_Domain__insert, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((domain @ A) @ B) @ (((insert2 @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R)) = (((insert2 @ A) @ A3) @ (((domain @ A) @ B) @ R)))))).
% 9.43/3.46 thf(fact_4954_Domain_Ocases, axiom, (! [B:$tType,A:$tType,A3:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R)) => (~ ((! [B3:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R))))))))))).
% 9.43/3.46 thf(fact_4955_Domain_Osimps, axiom, (! [B:$tType,A:$tType,A3:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R)) = (? [A7:A,B5:B]: (((A3 = A7) & (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A7) @ B5)) @ R)))))))).
% 9.43/3.46 thf(fact_4956_Domain_ODomainI, axiom, (! [B:$tType,A:$tType,A3:A,B2:B,R:(set @ ((product_prod @ A) @ B))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ R) => (((member @ A) @ A3) @ (((domain @ A) @ B) @ R)))))).
% 9.43/3.46 thf(fact_4957_DomainE, axiom, (! [B:$tType,A:$tType,A3:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R)) => (~ ((! [B3:B]: ((~ ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B3)) @ R))))))))))).
% 9.43/3.46 thf(fact_4958_Domain__iff, axiom, (! [A:$tType,B:$tType,A3:A,R:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ A3) @ (((domain @ A) @ B) @ R)) = (? [Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ Y3)) @ R))))))).
% 9.43/3.46 thf(fact_4959_Not__Domain__rtrancl, axiom, (! [A:$tType,X:A,R5:(set @ ((product_prod @ A) @ A)),Y:A]: (((~ ((((member @ A) @ X) @ (((domain @ A) @ A) @ R5)))) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R5)) = (X = Y)))))).
% 9.43/3.46 thf(fact_4960_Domain__unfold, axiom, (! [B:$tType,A:$tType]: ((((domain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ A) @ (^ [X3:A]: ((? [Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3)))))))))))).
% 9.43/3.46 thf(fact_4961_for__in__RI, axiom, (! [B:$tType,A:$tType,X:A,R5:(set @ ((product_prod @ A) @ B))]: (((((member @ A) @ X) @ (((domain @ A) @ B) @ R5)) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ ((((fun_of_rel @ A) @ B) @ R5) @ X))) @ R5))))).
% 9.43/3.46 thf(fact_4962_Domain__rel__restrict, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A4:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((domain @ A) @ A) @ (((rel_restrict @ A) @ R5) @ A4))) @ (((minus_minus @ (set @ A)) @ (((domain @ A) @ A) @ R5)) @ A4))))).
% 9.43/3.46 thf(fact_4963_wf__no__path, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((((inf_inf @ (set @ A)) @ (((domain @ A) @ A) @ R5)) @ (((range2 @ A) @ A) @ R5)) = (bot_bot @ (set @ A))) => ((wf @ A) @ R5))))).
% 9.43/3.46 thf(fact_4964_wf__min, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((wf @ A) @ R5) => ((R5 != (bot_bot @ (set @ ((product_prod @ A) @ A)))) => (~ ((! [M5:A]: ((~ ((((member @ A) @ M5) @ (((minus_minus @ (set @ A)) @ (((domain @ A) @ A) @ R5)) @ (((range2 @ A) @ A) @ R5)))))))))))))).
% 9.43/3.46 thf(fact_4965_Range__def, axiom, (! [B:$tType,A:$tType]: ((((range2 @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ B) @ (((rangep @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3))))))))))).
% 9.43/3.46 thf(fact_4966_and_Omonoid__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((monoid @ A) @ (bit_se5824344872417868541ns_and @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_4967_cr__integer__def, axiom, (code_cr_integer = (^ [X3:int,Y3:code_integer]: ((X3 = (code_int_of_integer @ Y3)))))).
% 9.43/3.46 thf(fact_4968_wo__rel_OisMinim__def, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ A),B2:A]: ((((bNF_Wellorder_wo_rel @ A) @ R) => (((((bNF_We4791949203932849705sMinim @ A) @ R) @ A4) @ B2) = ((((member @ A) @ B2) @ A4) & (! [X3:A]: (((((member @ A) @ X3) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ X3)) @ R)))))))))).
% 9.43/3.46 thf(fact_4969_monoid_Oleft__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,A3:A]: (((((monoid @ A) @ F) @ Z2) => (((F @ Z2) @ A3) = A3))))).
% 9.43/3.46 thf(fact_4970_monoid_Oright__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,A3:A]: (((((monoid @ A) @ F) @ Z2) => (((F @ A3) @ Z2) = A3))))).
% 9.43/3.46 thf(fact_4971_add_Omonoid__axioms, axiom, (! [A:$tType]: (((monoid_add @ A) => (((monoid @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_4972_mult_Omonoid__axioms, axiom, (! [A:$tType]: (((monoid_mult @ A) => (((monoid @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_4973_integer_Opcr__cr__eq, axiom, (code_pcr_integer = code_cr_integer)).
% 9.43/3.46 thf(fact_4974_multp__implies__one__step, axiom, (! [A:$tType,R:(A > (A > $o)),M3:(multiset @ A),N6:(multiset @ A)]: ((((transp @ A) @ R) => (((((multp @ A) @ R) @ M3) @ N6) => (? [I8:(multiset @ A),J5:(multiset @ A)]: (((N6 = (((plus_plus @ (multiset @ A)) @ I8) @ J5)) & (? [K10:(multiset @ A)]: (((M3 = (((plus_plus @ (multiset @ A)) @ I8) @ K10)) & ((J5 != (zero_zero @ (multiset @ A))) & (! [X7:A]: (((((member @ A) @ X7) @ ((set_mset @ A) @ K10)) => (? [Xa3:A]: (((((member @ A) @ Xa3) @ ((set_mset @ A) @ J5)) & ((R @ X7) @ Xa3))))))))))))))))))).
% 9.43/3.46 thf(fact_4975_relpow__fun__conv, axiom, (! [A:$tType,A3:A,B2:A,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) = (? [F2:(nat > A)]: ((((F2 @ (zero_zero @ nat)) = A3) & (((F2 @ N) = B2) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ N) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ (F2 @ I3)) @ (F2 @ (suc @ I3)))) @ R5)))))))))))).
% 9.43/3.46 thf(fact_4976_irreflp__irrefl__eq, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((irreflp @ A) @ (^ [A7:A,B5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R5)))) = ((irrefl @ A) @ R5))))).
% 9.43/3.46 thf(fact_4977_relpow__1, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (one_one @ nat)) @ R5) = R5)))).
% 9.43/3.46 thf(fact_4978_multp__cancel, axiom, (! [A:$tType,R:(A > (A > $o)),X10:(multiset @ A),Z9:(multiset @ A),Y9:(multiset @ A)]: ((((transp @ A) @ R) => (((irreflp @ A) @ R) => (((((multp @ A) @ R) @ (((plus_plus @ (multiset @ A)) @ X10) @ Z9)) @ (((plus_plus @ (multiset @ A)) @ Y9) @ Z9)) = ((((multp @ A) @ R) @ X10) @ Y9))))))).
% 9.43/3.46 thf(fact_4979_relpow__Suc__D2_H, axiom, (! [A:$tType,N:nat,R5:(set @ ((product_prod @ A) @ A)),X7:A,Y6:A,Z6:A]: ((((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X7) @ Y6)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ Z6)) @ R5)) => (? [W4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X7) @ W4)) @ R5) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ W4) @ Z6)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5))))))))).
% 9.43/3.46 thf(fact_4980_relpow__0__E, axiom, (! [A:$tType,X:A,Y:A,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R5)) => (X = Y))))).
% 9.43/3.46 thf(fact_4981_relpow__0__I, axiom, (! [A:$tType,X:A,R5:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (zero_zero @ nat)) @ R5))))).
% 9.43/3.46 thf(fact_4982_relpow__Suc__E, axiom, (! [A:$tType,X:A,Z2:A,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R5)) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ R5)))))))))))).
% 9.43/3.46 thf(fact_4983_relpow__Suc__I, axiom, (! [A:$tType,X:A,Y:A,N:nat,R5:(set @ ((product_prod @ A) @ A)),Z2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ R5) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R5))))))).
% 9.43/3.46 thf(fact_4984_relpow__Suc__D2, axiom, (! [A:$tType,X:A,Z2:A,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R5)) => (? [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ R5) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5))))))))).
% 9.43/3.46 thf(fact_4985_relpow__Suc__E2, axiom, (! [A:$tType,X:A,Z2:A,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R5)) => (~ ((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ R5) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5))))))))))))).
% 9.43/3.46 thf(fact_4986_relpow__Suc__I2, axiom, (! [A:$tType,X:A,Y:A,R5:(set @ ((product_prod @ A) @ A)),Z2:A,N:nat]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ (suc @ N)) @ R5))))))).
% 9.43/3.46 thf(fact_4987_relpow__add, axiom, (! [A:$tType,M2:nat,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((compow @ (set @ ((product_prod @ A) @ A))) @ (((plus_plus @ nat) @ M2) @ N)) @ R5) = (((((relcomp @ A) @ A) @ A) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M2) @ R5)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)))))).
% 9.43/3.46 thf(fact_4988_transp__trans__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((transp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = ((trans @ A) @ R))))).
% 9.43/3.46 thf(fact_4989_relpowp__relpow__eq, axiom, (! [A:$tType,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((compow @ (A > (A > $o))) @ N) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R5)))) = (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)))))))).
% 9.43/3.46 thf(fact_4990_relpow__E2, axiom, (! [A:$tType,X:A,Z2:A,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) => (((N = (zero_zero @ nat)) => (X != Z2)) => (~ ((! [Y4:A,M5:nat]: (((N = (suc @ M5)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ R5) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M5) @ R5))))))))))))))).
% 9.43/3.46 thf(fact_4991_relpow__E, axiom, (! [A:$tType,X:A,Z2:A,N:nat,R5:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ N) @ R5)) => (((N = (zero_zero @ nat)) => (X != Z2)) => (~ ((! [Y4:A,M5:nat]: (((N = (suc @ M5)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y4)) @ (((compow @ (set @ ((product_prod @ A) @ A))) @ M5) @ R5)) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z2)) @ R5)))))))))))))).
% 9.43/3.46 thf(fact_4992_nths__drop, axiom, (! [A:$tType,N:nat,Xs:(list @ A),I4:(set @ nat)]: (((((nths @ A) @ (((drop @ A) @ N) @ Xs)) @ I4) = (((nths @ A) @ Xs) @ ((((image2 @ nat) @ nat) @ ((plus_plus @ nat) @ N)) @ I4)))))).
% 9.43/3.46 thf(fact_4993_nths__append, axiom, (! [A:$tType,L:(list @ A),L4:(list @ A),A4:(set @ nat)]: (((((nths @ A) @ (((append @ A) @ L) @ L4)) @ A4) = (((append @ A) @ (((nths @ A) @ L) @ A4)) @ (((nths @ A) @ L4) @ ((collect @ nat) @ (^ [J2:nat]: ((((member @ nat) @ (((plus_plus @ nat) @ J2) @ ((size_size @ (list @ A)) @ L))) @ A4)))))))))).
% 9.43/3.46 thf(fact_4994_asymp__asym__eq, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: ((((asymp @ A) @ (^ [A7:A,B5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R5)))) = ((asym @ A) @ R5))))).
% 9.43/3.46 thf(fact_4995_asym_Ocases, axiom, (! [A:$tType,A3:(set @ ((product_prod @ A) @ A))]: ((((asym @ A) @ A3) => (! [A17:A,B8:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A17) @ B8)) @ A3) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B8) @ A17)) @ A3)))))))))).
% 9.43/3.46 thf(fact_4996_asym_Osimps, axiom, (! [A:$tType]: (((asym @ A) = (^ [A7:(set @ ((product_prod @ A) @ A))]: ((? [R6:(set @ ((product_prod @ A) @ A))]: (((A7 = R6) & (! [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R6) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R6))))))))))))))).
% 9.43/3.46 thf(fact_4997_asym_Ointros, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((! [A5:A,B3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A5) @ B3)) @ R5) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B3) @ A5)) @ R5)))))) => ((asym @ A) @ R5))))).
% 9.43/3.46 thf(fact_4998_asymD, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: ((((asym @ A) @ R5) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R5) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ X)) @ R5)))))))).
% 9.43/3.46 thf(fact_4999_asym__iff, axiom, (! [A:$tType]: (((asym @ A) = (^ [R6:(set @ ((product_prod @ A) @ A))]: ((! [X3:A,Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R6) => (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R6)))))))))))).
% 9.43/3.46 thf(fact_5000_lexord__asymmetric, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A)),A3:(list @ A),B2:(list @ A)]: ((((asym @ A) @ R5) => ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ A3) @ B2)) @ ((lexord @ A) @ R5)) => (~ ((((member @ ((product_prod @ (list @ A)) @ (list @ A))) @ ((((product_Pair @ (list @ A)) @ (list @ A)) @ B2) @ A3)) @ ((lexord @ A) @ R5))))))))).
% 9.43/3.46 thf(fact_5001_ordLess__iff, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) = ((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) & ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) & (~ ((? [X8:(B > A)]: ((((((bNF_Wellorder_embed @ B) @ A) @ R4) @ R) @ X8))))))))))).
% 9.43/3.46 thf(fact_5002_option_Oin__rel, axiom, (! [B:$tType,A:$tType]: ((((rel_option @ A) @ B) = (^ [R6:(A > (B > $o)),A7:(option @ A),B5:(option @ B)]: ((? [Z5:(option @ ((product_prod @ A) @ B))]: (((((member @ (option @ ((product_prod @ A) @ B))) @ Z5) @ ((collect @ (option @ ((product_prod @ A) @ B))) @ (^ [X3:(option @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((set_option @ ((product_prod @ A) @ B)) @ X3)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) & ((((((map_option @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Z5) = A7) & (((((map_option @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Z5) = B5))))))))))).
% 9.43/3.46 thf(fact_5003_bit__double__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((times_times @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ A3)) @ N) = ((((bit_se5641148757651400278ts_bit @ A) @ A3) @ (((minus_minus @ nat) @ N) @ (one_one @ nat))) & ((N != (zero_zero @ nat)) & (((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N)))))))))).
% 9.43/3.46 thf(fact_5004_rel__option__None2, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o)),X:(option @ A)]: (((((((rel_option @ A) @ B) @ P) @ X) @ (none @ B)) = (X = (none @ A)))))).
% 9.43/3.46 thf(fact_5005_rel__option__None1, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),X:(option @ B)]: (((((((rel_option @ A) @ B) @ P) @ (none @ A)) @ X) = (X = (none @ B)))))).
% 9.43/3.46 thf(fact_5006_bit__minus__1__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((uminus_uminus @ A) @ (one_one @ A))) @ N) = (((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N)))))))).
% 9.43/3.46 thf(fact_5007_rel__option__iff, axiom, (! [B:$tType,A:$tType]: ((((rel_option @ A) @ B) = (^ [R6:(A > (B > $o)),X3:(option @ A),Y3:(option @ B)]: ((((((product_case_prod @ (option @ A)) @ (option @ B)) @ $o) @ (^ [A7:(option @ A),B5:(option @ B)]: ((((((case_option @ $o) @ A) @ (((((case_option @ $o) @ B) @ $true) @ (^ [C3:B]: ($false))) @ B5)) @ (^ [Z5:A]: ((((((case_option @ $o) @ B) @ $false) @ (R6 @ Z5)) @ B5)))) @ A7)))) @ ((((product_Pair @ (option @ A)) @ (option @ B)) @ X3) @ Y3)))))))).
% 9.43/3.46 thf(fact_5008_option_Orel__mono, axiom, (! [B:$tType,A:$tType,R5:(A > (B > $o)),Ra2:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R5) @ Ra2) => (((ord_less_eq @ ((option @ A) > ((option @ B) > $o))) @ (((rel_option @ A) @ B) @ R5)) @ (((rel_option @ A) @ B) @ Ra2)))))).
% 9.43/3.46 thf(fact_5009_option_Orel__inject_I2_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X22:A,Y2:B]: (((((((rel_option @ A) @ B) @ R5) @ ((some @ A) @ X22)) @ ((some @ B) @ Y2)) = ((R5 @ X22) @ Y2))))).
% 9.43/3.46 thf(fact_5010_option_Orel__intros_I2_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X22:A,Y2:B]: ((((R5 @ X22) @ Y2) => (((((rel_option @ A) @ B) @ R5) @ ((some @ A) @ X22)) @ ((some @ B) @ Y2)))))).
% 9.43/3.46 thf(fact_5011_option__rel__Some1, axiom, (! [A:$tType,B:$tType,A4:(A > (B > $o)),X:A,Y:(option @ B)]: (((((((rel_option @ A) @ B) @ A4) @ ((some @ A) @ X)) @ Y) = (? [Y8:B]: (((Y = ((some @ B) @ Y8)) & ((A4 @ X) @ Y8)))))))).
% 9.43/3.46 thf(fact_5012_option__rel__Some2, axiom, (! [B:$tType,A:$tType,A4:(A > (B > $o)),X:(option @ A),Y:B]: (((((((rel_option @ A) @ B) @ A4) @ X) @ ((some @ B) @ Y)) = (? [X9:A]: (((X = ((some @ A) @ X9)) & ((A4 @ X9) @ Y)))))))).
% 9.43/3.46 thf(fact_5013_option_Octr__transfer_I1_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: ((((((rel_option @ A) @ B) @ R5) @ (none @ A)) @ (none @ B))))).
% 9.43/3.46 thf(fact_5014_rel__option__inf, axiom, (! [B:$tType,A:$tType,A4:(A > (B > $o)),B6:(A > (B > $o))]: (((((inf_inf @ ((option @ A) > ((option @ B) > $o))) @ (((rel_option @ A) @ B) @ A4)) @ (((rel_option @ A) @ B) @ B6)) = (((rel_option @ A) @ B) @ (((inf_inf @ (A > (B > $o))) @ A4) @ B6)))))).
% 9.43/3.46 thf(fact_5015_option_Orel__refl, axiom, (! [B:$tType,Ra2:(B > (B > $o)),X:(option @ B)]: (((! [X2:B]: (((Ra2 @ X2) @ X2))) => (((((rel_option @ B) @ B) @ Ra2) @ X) @ X))))).
% 9.43/3.46 thf(fact_5016_option_Orel__eq, axiom, (! [A:$tType]: (((((rel_option @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) = (^ [Y5:(option @ A),Z4:(option @ A)]: ((Y5 = Z4))))))).
% 9.43/3.46 thf(fact_5017_option_Orel__cong, axiom, (! [A:$tType,B:$tType,X:(option @ A),Ya:(option @ A),Y:(option @ B),Xa:(option @ B),R5:(A > (B > $o)),Ra2:(A > (B > $o))]: (((X = Ya) => ((Y = Xa) => ((! [Z3:A,Yb:B]: (((((member @ A) @ Z3) @ ((set_option @ A) @ Ya)) => ((((member @ B) @ Yb) @ ((set_option @ B) @ Xa)) => (((R5 @ Z3) @ Yb) = ((Ra2 @ Z3) @ Yb)))))) => ((((((rel_option @ A) @ B) @ R5) @ X) @ Y) = (((((rel_option @ A) @ B) @ Ra2) @ Ya) @ Xa)))))))).
% 9.43/3.46 thf(fact_5018_option_Orel__mono__strong, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X:(option @ A),Y:(option @ B),Ra2:(A > (B > $o))]: (((((((rel_option @ A) @ B) @ R5) @ X) @ Y) => ((! [Z3:A,Yb:B]: (((((member @ A) @ Z3) @ ((set_option @ A) @ X)) => ((((member @ B) @ Yb) @ ((set_option @ B) @ Y)) => (((R5 @ Z3) @ Yb) => ((Ra2 @ Z3) @ Yb)))))) => (((((rel_option @ A) @ B) @ Ra2) @ X) @ Y)))))).
% 9.43/3.46 thf(fact_5019_option_Orel__refl__strong, axiom, (! [A:$tType,X:(option @ A),Ra2:(A > (A > $o))]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ X)) => ((Ra2 @ Z3) @ Z3)))) => (((((rel_option @ A) @ A) @ Ra2) @ X) @ X))))).
% 9.43/3.46 thf(fact_5020_rel__option__reflI, axiom, (! [A:$tType,Y:(option @ A),P:(A > (A > $o))]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set_option @ A) @ Y)) => ((P @ X2) @ X2)))) => (((((rel_option @ A) @ A) @ P) @ Y) @ Y))))).
% 9.43/3.46 thf(fact_5021_option_Orel__map_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,Sb:(C > (B > $o)),I:(A > C),X:(option @ A),Y:(option @ B)]: (((((((rel_option @ C) @ B) @ Sb) @ ((((map_option @ A) @ C) @ I) @ X)) @ Y) = (((((rel_option @ A) @ B) @ (^ [X3:A]: ((Sb @ (I @ X3))))) @ X) @ Y))))).
% 9.43/3.46 thf(fact_5022_option_Orel__map_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,Sa:(A > (C > $o)),X:(option @ A),G:(B > C),Y:(option @ B)]: (((((((rel_option @ A) @ C) @ Sa) @ X) @ ((((map_option @ B) @ C) @ G) @ Y)) = (((((rel_option @ A) @ B) @ (^ [X3:A,Y3:B]: (((Sa @ X3) @ (G @ Y3))))) @ X) @ Y))))).
% 9.43/3.46 thf(fact_5023_option_Orel__flip, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),A3:(option @ B),B2:(option @ A)]: (((((((rel_option @ B) @ A) @ (((conversep @ A) @ B) @ R5)) @ A3) @ B2) = (((((rel_option @ A) @ B) @ R5) @ B2) @ A3))))).
% 9.43/3.46 thf(fact_5024_option_Orel__conversep, axiom, (! [B:$tType,A:$tType,R5:(A > (B > $o))]: (((((rel_option @ B) @ A) @ (((conversep @ A) @ B) @ R5)) = (((conversep @ (option @ A)) @ (option @ B)) @ (((rel_option @ A) @ B) @ R5)))))).
% 9.43/3.46 thf(fact_5025_option_Octr__transfer_I2_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ (option @ A)) @ (option @ B)) @ R5) @ (((rel_option @ A) @ B) @ R5)) @ (some @ A)) @ (some @ B))))).
% 9.43/3.46 thf(fact_5026_option_Odisc__transfer_I2_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ $o) @ $o) @ (((rel_option @ A) @ B) @ R5)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))) @ (^ [Option3:(option @ A)]: ((Option3 != (none @ A))))) @ (^ [Option3:(option @ B)]: ((Option3 != (none @ B)))))))).
% 9.43/3.46 thf(fact_5027_option_Odisc__transfer_I1_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ $o) @ $o) @ (((rel_option @ A) @ B) @ R5)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))) @ (^ [Option3:(option @ A)]: ((Option3 = (none @ A))))) @ (^ [Option3:(option @ B)]: ((Option3 = (none @ B)))))))).
% 9.43/3.46 thf(fact_5028_option__bind__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A4:(A > (B > $o)),B6:(C > (D > $o))]: (((((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ ((A > (option @ C)) > (option @ C))) @ ((B > (option @ D)) > (option @ D))) @ (((rel_option @ A) @ B) @ A4)) @ ((((((bNF_rel_fun @ (A > (option @ C))) @ (B > (option @ D))) @ (option @ C)) @ (option @ D)) @ ((((((bNF_rel_fun @ A) @ B) @ (option @ C)) @ (option @ D)) @ A4) @ (((rel_option @ C) @ D) @ B6))) @ (((rel_option @ C) @ D) @ B6))) @ ((bind @ A) @ C)) @ ((bind @ B) @ D))))).
% 9.43/3.46 thf(fact_5029_option_Orel__transp, axiom, (! [A:$tType,R5:(A > (A > $o))]: ((((transp @ A) @ R5) => ((transp @ (option @ A)) @ (((rel_option @ A) @ A) @ R5)))))).
% 9.43/3.46 thf(fact_5030_option_Orel__distinct_I2_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),Y2:A]: ((~ ((((((rel_option @ A) @ B) @ R5) @ ((some @ A) @ Y2)) @ (none @ B))))))).
% 9.43/3.46 thf(fact_5031_option_Orel__distinct_I1_J, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),Y2:B]: ((~ ((((((rel_option @ A) @ B) @ R5) @ (none @ A)) @ ((some @ B) @ Y2))))))).
% 9.43/3.46 thf(fact_5032_option_Orel__cases, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),A3:(option @ A),B2:(option @ B)]: (((((((rel_option @ A) @ B) @ R5) @ A3) @ B2) => (((A3 = (none @ A)) => (B2 != (none @ B))) => (~ ((! [X2:A]: (((A3 = ((some @ A) @ X2)) => (! [Y4:B]: (((B2 = ((some @ B) @ Y4)) => (~ (((R5 @ X2) @ Y4)))))))))))))))).
% 9.43/3.46 thf(fact_5033_option_Orel__induct, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X:(option @ A),Y:(option @ B),Q:((option @ A) > ((option @ B) > $o))]: (((((((rel_option @ A) @ B) @ R5) @ X) @ Y) => (((Q @ (none @ A)) @ (none @ B)) => ((! [A24:A,B24:B]: ((((R5 @ A24) @ B24) => ((Q @ ((some @ A) @ A24)) @ ((some @ B) @ B24))))) => ((Q @ X) @ Y))))))).
% 9.43/3.46 thf(fact_5034_option_Omap__transfer, axiom, (! [A:$tType,B:$tType,F5:$tType,E:$tType,Rb2:(A > (E > $o)),Sd:(B > (F5 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (E > F5)) @ ((option @ A) > (option @ B))) @ ((option @ E) > (option @ F5))) @ ((((((bNF_rel_fun @ A) @ E) @ B) @ F5) @ Rb2) @ Sd)) @ ((((((bNF_rel_fun @ (option @ A)) @ (option @ E)) @ (option @ B)) @ (option @ F5)) @ (((rel_option @ A) @ E) @ Rb2)) @ (((rel_option @ B) @ F5) @ Sd))) @ ((map_option @ A) @ B)) @ ((map_option @ E) @ F5))))).
% 9.43/3.46 thf(fact_5035_option_Ocase__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,S2:(C > (D > $o)),R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ C) @ D) @ ((A > C) > ((option @ A) > C))) @ ((B > D) > ((option @ B) > D))) @ S2) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ ((option @ A) > C)) @ ((option @ B) > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R5) @ S2)) @ ((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ C) @ D) @ (((rel_option @ A) @ B) @ R5)) @ S2))) @ ((case_option @ C) @ A)) @ ((case_option @ D) @ B))))).
% 9.43/3.46 thf(fact_5036_option_Orel__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,Sa:(A > (C > $o)),Sc:(B > (D > $o))]: (((((((((bNF_rel_fun @ (A > (B > $o))) @ (C > (D > $o))) @ ((option @ A) > ((option @ B) > $o))) @ ((option @ C) > ((option @ D) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ (B > $o)) @ (D > $o)) @ Sa) @ ((((((bNF_rel_fun @ B) @ D) @ $o) @ $o) @ Sc) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))))) @ ((((((bNF_rel_fun @ (option @ A)) @ (option @ C)) @ ((option @ B) > $o)) @ ((option @ D) > $o)) @ (((rel_option @ A) @ C) @ Sa)) @ ((((((bNF_rel_fun @ (option @ B)) @ (option @ D)) @ $o) @ $o) @ (((rel_option @ B) @ D) @ Sc)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))))) @ ((rel_option @ A) @ B)) @ ((rel_option @ C) @ D))))).
% 9.43/3.46 thf(fact_5037_bit__minus__iff, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (! [A3:A,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((uminus_uminus @ A) @ A3)) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (~ ((((bit_se5641148757651400278ts_bit @ A) @ (((minus_minus @ A) @ A3) @ (one_one @ A))) @ N))))))))))).
% 9.43/3.46 thf(fact_5038_option_Orel__sel, axiom, (! [B:$tType,A:$tType]: ((((rel_option @ A) @ B) = (^ [R6:(A > (B > $o)),A7:(option @ A),B5:(option @ B)]: ((((A7 = (none @ A)) = (B5 = (none @ B))) & ((A7 != (none @ A)) => ((B5 != (none @ B)) => ((R6 @ ((the2 @ A) @ A7)) @ ((the2 @ B) @ B5))))))))))).
% 9.43/3.46 thf(fact_5039_ordLess__not__embed, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => (~ ((? [X_1:(B > A)]: ((((((bNF_Wellorder_embed @ B) @ A) @ R4) @ R) @ X_1))))))))).
% 9.43/3.46 thf(fact_5040_option_Orec__transfer, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,S2:(C > (D > $o)),R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ C) @ D) @ ((A > C) > ((option @ A) > C))) @ ((B > D) > ((option @ B) > D))) @ S2) @ ((((((bNF_rel_fun @ (A > C)) @ (B > D)) @ ((option @ A) > C)) @ ((option @ B) > D)) @ ((((((bNF_rel_fun @ A) @ B) @ C) @ D) @ R5) @ S2)) @ ((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ C) @ D) @ (((rel_option @ A) @ B) @ R5)) @ S2))) @ ((rec_option @ C) @ A)) @ ((rec_option @ D) @ B))))).
% 9.43/3.46 thf(fact_5041_bit__2__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ (one_one @ nat)) & (N = (one_one @ nat)))))))))).
% 9.43/3.46 thf(fact_5042_BNF__Wellorder__Constructions_OordLess__Field, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R1) @ R22)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => ((((((bNF_Wellorder_embed @ A) @ B) @ R1) @ R22) @ F) => (((((image2 @ A) @ B) @ F) @ ((field2 @ A) @ R1)) != ((field2 @ B) @ R22))))))).
% 9.43/3.46 thf(fact_5043_bit__mask__sub__iff, axiom, (! [A:$tType]: (((bit_semiring_bits @ A) => (! [M2:nat,N:nat]: (((((bit_se5641148757651400278ts_bit @ A) @ (((minus_minus @ A) @ (((power_power @ A) @ ((numeral_numeral @ A) @ (bit0 @ one2))) @ M2)) @ (one_one @ A))) @ N) = ((((bit_se6407376104438227557le_bit @ A) @ (type2 @ A)) @ N) & (((ord_less @ nat) @ N) @ M2))))))))).
% 9.43/3.46 thf(fact_5044_embed__ordLess__ofilterIncl, axiom, (! [B:$tType,A:$tType,C:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),R33:(set @ ((product_prod @ C) @ C)),F13:(A > C),F232:(B > C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R1) @ R22)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ R22) @ R33)) @ ((bNF_We4044943003108391690rdLess @ B) @ C)) => ((((((bNF_Wellorder_embed @ A) @ C) @ R1) @ R33) @ F13) => ((((((bNF_Wellorder_embed @ B) @ C) @ R22) @ R33) @ F232) => (((member @ ((product_prod @ (set @ C)) @ (set @ C))) @ ((((product_Pair @ (set @ C)) @ (set @ C)) @ ((((image2 @ A) @ C) @ F13) @ ((field2 @ A) @ R1))) @ ((((image2 @ B) @ C) @ F232) @ ((field2 @ B) @ R22)))) @ ((bNF_We413866401316099525erIncl @ C) @ R33))))))))).
% 9.43/3.46 thf(fact_5045_comp__embed, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((((bNF_Wellorder_embed @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_embed @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embed @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))).
% 9.43/3.46 thf(fact_5046_override__on__insert, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B),X:A,X10:(set @ A)]: (((((((override_on @ A) @ B) @ F) @ G) @ (((insert2 @ A) @ X) @ X10)) = (((((fun_upd @ A) @ B) @ (((((override_on @ A) @ B) @ F) @ G) @ X10)) @ X) @ (G @ X)))))).
% 9.43/3.46 thf(fact_5047_override__on__apply__in, axiom, (! [B:$tType,A:$tType,A3:A,A4:(set @ A),F:(A > B),G:(A > B)]: (((((member @ A) @ A3) @ A4) => (((((((override_on @ A) @ B) @ F) @ G) @ A4) @ A3) = (G @ A3)))))).
% 9.43/3.46 thf(fact_5048_override__on__apply__notin, axiom, (! [B:$tType,A:$tType,A3:A,A4:(set @ A),F:(A > B),G:(A > B)]: (((~ ((((member @ A) @ A3) @ A4))) => (((((((override_on @ A) @ B) @ F) @ G) @ A4) @ A3) = (F @ A3)))))).
% 9.43/3.46 thf(fact_5049_override__on__emptyset, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((((((override_on @ A) @ B) @ F) @ G) @ (bot_bot @ (set @ A))) = F)))).
% 9.43/3.46 thf(fact_5050_override__on__def, axiom, (! [B:$tType,A:$tType]: ((((override_on @ A) @ B) = (^ [F2:(A > B),G2:(A > B),A9:(set @ A),A7:A]: (((((if @ B) @ (((member @ A) @ A7) @ A9)) @ (G2 @ A7)) @ (F2 @ A7)))))))).
% 9.43/3.46 thf(fact_5051_override__on__insert_H, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B),X:A,X10:(set @ A)]: (((((((override_on @ A) @ B) @ F) @ G) @ (((insert2 @ A) @ X) @ X10)) = (((((override_on @ A) @ B) @ (((((fun_upd @ A) @ B) @ F) @ X) @ (G @ X))) @ G) @ X10))))).
% 9.43/3.46 thf(fact_5052_embedS__comp__embed, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((((bNF_Wellorder_embedS @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_embed @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embedS @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))))).
% 9.43/3.46 thf(fact_5053_embed__comp__embedS, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((((bNF_Wellorder_embed @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_embedS @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embedS @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))))).
% 9.43/3.46 thf(fact_5054_pred__option__parametric, axiom, (! [A:$tType,B:$tType,A4:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((option @ A) > $o)) @ ((option @ B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ A4) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ $o) @ $o) @ (((rel_option @ A) @ B) @ A4)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (pred_option @ A)) @ (pred_option @ B))))).
% 9.43/3.46 thf(fact_5055_option_Opred__inject_I2_J, axiom, (! [A:$tType,P:(A > $o),A3:A]: (((((pred_option @ A) @ P) @ ((some @ A) @ A3)) = (P @ A3))))).
% 9.43/3.46 thf(fact_5056_option_Opred__mono, axiom, (! [A:$tType,P:(A > $o),Pa:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ P) @ Pa) => (((ord_less_eq @ ((option @ A) > $o)) @ ((pred_option @ A) @ P)) @ ((pred_option @ A) @ Pa)))))).
% 9.43/3.46 thf(fact_5057_option_Opred__inject_I1_J, axiom, (! [A:$tType,P:(A > $o)]: ((((pred_option @ A) @ P) @ (none @ A))))).
% 9.43/3.46 thf(fact_5058_option_Opred__True, axiom, (! [A:$tType]: ((((pred_option @ A) @ (^ [Uu:A]: ($true))) = (^ [Uu:(option @ A)]: ($true)))))).
% 9.43/3.46 thf(fact_5059_option_Opred__mono__strong, axiom, (! [A:$tType,P:(A > $o),X:(option @ A),Pa:(A > $o)]: (((((pred_option @ A) @ P) @ X) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ X)) => ((P @ Z3) => (Pa @ Z3))))) => (((pred_option @ A) @ Pa) @ X)))))).
% 9.43/3.46 thf(fact_5060_option_Opred__cong, axiom, (! [A:$tType,X:(option @ A),Ya:(option @ A),P:(A > $o),Pa:(A > $o)]: (((X = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((set_option @ A) @ Ya)) => ((P @ Z3) = (Pa @ Z3))))) => ((((pred_option @ A) @ P) @ X) = (((pred_option @ A) @ Pa) @ Ya))))))).
% 9.43/3.46 thf(fact_5061_option_Omap__cong__pred, axiom, (! [B:$tType,A:$tType,X:(option @ A),Ya:(option @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((((pred_option @ A) @ (^ [Z5:A]: (((F @ Z5) = (G @ Z5))))) @ Ya) => (((((map_option @ A) @ B) @ F) @ X) = ((((map_option @ A) @ B) @ G) @ Ya))))))).
% 9.43/3.46 thf(fact_5062_option_Opred__set, axiom, (! [A:$tType]: (((pred_option @ A) = (^ [P4:(A > $o),X3:(option @ A)]: ((! [Y3:A]: (((((member @ A) @ Y3) @ ((set_option @ A) @ X3)) => (P4 @ Y3)))))))))).
% 9.43/3.46 thf(fact_5063_option_Opred__rel, axiom, (! [A:$tType]: (((pred_option @ A) = (^ [P4:(A > $o),X3:(option @ A)]: ((((((rel_option @ A) @ A) @ ((bNF_eq_onp @ A) @ P4)) @ X3) @ X3))))))).
% 9.43/3.46 thf(fact_5064_option_Orel__eq__onp, axiom, (! [A:$tType,P:(A > $o)]: (((((rel_option @ A) @ A) @ ((bNF_eq_onp @ A) @ P)) = ((bNF_eq_onp @ (option @ A)) @ ((pred_option @ A) @ P)))))).
% 9.43/3.46 thf(fact_5065_option_Opred__map, axiom, (! [B:$tType,A:$tType,Q:(B > $o),F:(A > B),X:(option @ A)]: (((((pred_option @ B) @ Q) @ ((((map_option @ A) @ B) @ F) @ X)) = (((pred_option @ A) @ (((((comp @ B) @ $o) @ A) @ Q) @ F)) @ X))))).
% 9.43/3.46 thf(fact_5066_option_Opred__transfer, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((option @ A) > $o)) @ ((option @ B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R5) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((((bNF_rel_fun @ (option @ A)) @ (option @ B)) @ $o) @ $o) @ (((rel_option @ A) @ B) @ R5)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (pred_option @ A)) @ (pred_option @ B))))).
% 9.43/3.46 thf(fact_5067_iso__comp__embedS, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((order_well_order_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((((bNF_Wellorder_iso @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_embedS @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embedS @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))))).
% 9.43/3.46 thf(fact_5068_cut__def, axiom, (! [B:$tType,A:$tType]: ((((cut @ A) @ B) = (^ [F2:(A > B),R6:(set @ ((product_prod @ A) @ A)),X3:A,Y3:A]: (((((if @ B) @ (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R6)) @ (F2 @ Y3)) @ (undefined @ B)))))))).
% 9.43/3.46 thf(fact_5069_above__def, axiom, (! [A:$tType]: (((order_above @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A7:A]: (((collect @ A) @ (^ [B5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R3)))))))))).
% 9.43/3.46 thf(fact_5070_comp__iso, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((((bNF_Wellorder_iso @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_iso @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_iso @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))).
% 9.43/3.46 thf(fact_5071_iso__forward, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((((((bNF_Wellorder_iso @ A) @ B) @ R) @ R4) @ F) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F @ X)) @ (F @ Y))) @ R4)))))).
% 9.43/3.46 thf(fact_5072_cut__apply, axiom, (! [B:$tType,A:$tType,X:A,A3:A,R5:(set @ ((product_prod @ A) @ A)),F:(A > B)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ A3)) @ R5) => (((((((cut @ A) @ B) @ F) @ R5) @ A3) @ X) = (F @ X)))))).
% 9.43/3.46 thf(fact_5073_cuts__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),R5:(set @ ((product_prod @ A) @ A)),X:A,G:(A > B)]: ((((((((cut @ A) @ B) @ F) @ R5) @ X) = (((((cut @ A) @ B) @ G) @ R5) @ X)) = (! [Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X)) @ R5) => ((F @ Y3) = (G @ Y3))))))))).
% 9.43/3.46 thf(fact_5074_iso__comp__embed, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((((bNF_Wellorder_iso @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_embed @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embed @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))).
% 9.43/3.46 thf(fact_5075_embed__comp__iso, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((((bNF_Wellorder_embed @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_iso @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embed @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))).
% 9.43/3.46 thf(fact_5076_embedS__comp__iso, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),F:(A > B),R9:(set @ ((product_prod @ C) @ C)),F3:(B > C)]: (((((((bNF_Wellorder_embedS @ A) @ B) @ R) @ R4) @ F) => ((((((bNF_Wellorder_iso @ B) @ C) @ R4) @ R9) @ F3) => (((((bNF_Wellorder_embedS @ A) @ C) @ R) @ R9) @ (((((comp @ B) @ C) @ A) @ F3) @ F))))))).
% 9.43/3.46 thf(fact_5077_iso__iff2, axiom, (! [B:$tType,A:$tType]: ((((bNF_Wellorder_iso @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),R11:(set @ ((product_prod @ B) @ B)),F2:(A > B)]: (((((((bij_betw @ A) @ B) @ F2) @ ((field2 @ A) @ R3)) @ ((field2 @ B) @ R11)) & (! [X3:A]: (((((member @ A) @ X3) @ ((field2 @ A) @ R3)) => (! [Y3:A]: (((((member @ A) @ Y3) @ ((field2 @ A) @ R3)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3) = (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (F2 @ X3)) @ (F2 @ Y3))) @ R11))))))))))))))).
% 9.43/3.46 thf(fact_5078_iso__backward, axiom, (! [A:$tType,B:$tType,X:A,Y:A,R4:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B)),F:(B > A)]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R4) => ((((((bNF_Wellorder_iso @ B) @ A) @ R) @ R4) @ F) => (((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ (((((hilbert_inv_into @ B) @ A) @ ((field2 @ B) @ R)) @ F) @ X)) @ (((((hilbert_inv_into @ B) @ A) @ ((field2 @ B) @ R)) @ F) @ Y))) @ R)))))).
% 9.43/3.46 thf(fact_5079_lists__of__len__fin1, axiom, (! [A:$tType,P:(set @ A),N:nat]: ((((finite_finite2 @ A) @ P) => ((finite_finite2 @ (list @ A)) @ (((inf_inf @ (set @ (list @ A))) @ ((lists @ A) @ P)) @ ((collect @ (list @ A)) @ (^ [L3:(list @ A)]: ((((size_size @ (list @ A)) @ L3) = N)))))))))).
% 9.43/3.46 thf(fact_5080_lists__of__len__fin2, axiom, (! [A:$tType,P:(set @ A),N:nat]: ((((finite_finite2 @ A) @ P) => ((finite_finite2 @ (list @ A)) @ (((inf_inf @ (set @ (list @ A))) @ ((lists @ A) @ P)) @ ((collect @ (list @ A)) @ (^ [L3:(list @ A)]: ((N = ((size_size @ (list @ A)) @ L3))))))))))).
% 9.43/3.46 thf(fact_5081_Random__Pred_Ounion__def, axiom, (! [A:$tType]: (((random_union @ A) = (^ [R13:(((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))),R24:(((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))),S4:((product_prod @ code_natural) @ code_natural)]: ((((((product_case_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))) @ (^ [P12:(pred @ A),S8:((product_prod @ code_natural) @ code_natural)]: ((((((product_case_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))) @ (^ [P23:(pred @ A)]: ((((product_Pair @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)) @ (((sup_sup @ (pred @ A)) @ P12) @ P23))))) @ (R24 @ S8))))) @ (R13 @ S4)))))))).
% 9.43/3.46 thf(fact_5082_lists__image__witness, axiom, (! [A:$tType,B:$tType,X:(list @ A),F:(B > A),Q:(set @ B)]: (((((member @ (list @ A)) @ X) @ ((lists @ A) @ ((((image2 @ B) @ A) @ F) @ Q))) => (~ ((! [Xo2:(list @ B)]: (((((member @ (list @ B)) @ Xo2) @ ((lists @ B) @ Q)) => (X != ((((map @ B) @ A) @ F) @ Xo2))))))))))).
% 9.43/3.46 thf(fact_5083_Random__Pred_Oempty__def, axiom, (! [A:$tType]: (((random_empty @ A) = (((product_Pair @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)) @ (bot_bot @ (pred @ A))))))).
% 9.43/3.46 thf(fact_5084_Random__Pred_Obind__def, axiom, (! [B:$tType,A:$tType]: ((((random_bind @ A) @ B) = (^ [R6:(((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural))),F2:(A > (((product_prod @ code_natural) @ code_natural) > ((product_prod @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural)))),S4:((product_prod @ code_natural) @ code_natural)]: ((((((product_case_prod @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural))) @ (^ [P4:(pred @ A),S8:((product_prod @ code_natural) @ code_natural)]: ((((((product_case_prod @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ code_natural) @ code_natural)) @ ((product_prod @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural))) @ (^ [S12:((product_prod @ code_natural) @ code_natural)]: ((((product_Pair @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural)) @ ((((bind2 @ A) @ B) @ P4) @ (^ [A7:A]: ((((product_fst @ (pred @ B)) @ ((product_prod @ code_natural) @ code_natural)) @ ((F2 @ A7) @ S12))))))))) @ (split_seed @ S8))))) @ (R6 @ S4)))))))).
% 9.43/3.46 thf(fact_5085_card__Plus__conv__if, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: ((((((finite_finite2 @ A) @ A4) & ((finite_finite2 @ B) @ B6)) => (((finite_card @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6)) = (((plus_plus @ nat) @ ((finite_card @ A) @ A4)) @ ((finite_card @ B) @ B6)))) & ((~ ((((finite_finite2 @ A) @ A4) & ((finite_finite2 @ B) @ B6)))) => (((finite_card @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6)) = (zero_zero @ nat))))))).
% 9.43/3.46 thf(fact_5086_card__Plus, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: ((((finite_finite2 @ A) @ A4) => (((finite_finite2 @ B) @ B6) => (((finite_card @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6)) = (((plus_plus @ nat) @ ((finite_card @ A) @ A4)) @ ((finite_card @ B) @ B6)))))))).
% 9.43/3.46 thf(fact_5087_option_Orel__compp__Grp, axiom, (! [B:$tType,A:$tType]: ((((rel_option @ A) @ B) = (^ [R6:(A > (B > $o))]: ((((((relcompp @ (option @ A)) @ (option @ ((product_prod @ A) @ B))) @ (option @ B)) @ (((conversep @ (option @ ((product_prod @ A) @ B))) @ (option @ A)) @ ((((bNF_Grp @ (option @ ((product_prod @ A) @ B))) @ (option @ A)) @ ((collect @ (option @ ((product_prod @ A) @ B))) @ (^ [X3:(option @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((set_option @ ((product_prod @ A) @ B)) @ X3)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) @ (((map_option @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B))))) @ ((((bNF_Grp @ (option @ ((product_prod @ A) @ B))) @ (option @ B)) @ ((collect @ (option @ ((product_prod @ A) @ B))) @ (^ [X3:(option @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((set_option @ ((product_prod @ A) @ B)) @ X3)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) @ (((map_option @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)))))))))).
% 9.43/3.46 thf(fact_5088_mult__one__div__unit__factor, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((times_times @ A) @ A3) @ (((divide_divide @ A) @ (one_one @ A)) @ ((unit_f5069060285200089521factor @ A) @ B2))) = (((divide_divide @ A) @ A3) @ ((unit_f5069060285200089521factor @ A) @ B2))))))))).
% 9.43/3.46 thf(fact_5089_unit__factor__mult__unit__left, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((unit_f5069060285200089521factor @ A) @ (((times_times @ A) @ A3) @ B2)) = (((times_times @ A) @ A3) @ ((unit_f5069060285200089521factor @ A) @ B2)))))))))).
% 9.43/3.46 thf(fact_5090_unit__factor__simps_I2_J, axiom, (! [N:nat]: ((((unit_f5069060285200089521factor @ nat) @ (suc @ N)) = (one_one @ nat))))).
% 9.43/3.46 thf(fact_5091_unit__factor__1, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (((unit_f5069060285200089521factor @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_5092_inv__unit__factor__eq__0__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: ((((((divide_divide @ A) @ (one_one @ A)) @ ((unit_f5069060285200089521factor @ A) @ A3)) = (zero_zero @ A)) = (A3 = (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_5093_unit__factor__mult__unit__right, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((unit_f5069060285200089521factor @ A) @ (((times_times @ A) @ B2) @ A3)) = (((times_times @ A) @ ((unit_f5069060285200089521factor @ A) @ B2)) @ A3))))))))).
% 9.43/3.46 thf(fact_5094_option_Orel__compp, axiom, (! [A:$tType,C:$tType,B:$tType,R5:(A > (B > $o)),S2:(B > (C > $o))]: (((((rel_option @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R5) @ S2)) = (((((relcompp @ (option @ A)) @ (option @ B)) @ (option @ C)) @ (((rel_option @ A) @ B) @ R5)) @ (((rel_option @ B) @ C) @ S2)))))).
% 9.43/3.46 thf(fact_5095_OOO__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,R12:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),R23:(B > (B > $o)),Abs22:(B > C),Rep22:(C > B),R25:(A > (A > $o))]: (((((((quotient3 @ A) @ B) @ R12) @ Abs1) @ Rep1) => ((((((quotient3 @ B) @ C) @ R23) @ Abs22) @ Rep22) => ((! [X2:A,Y4:A]: ((((R25 @ X2) @ Y4) => (((R12 @ X2) @ X2) => (((R12 @ Y4) @ Y4) => ((R23 @ (Abs1 @ X2)) @ (Abs1 @ Y4))))))) => ((! [X2:B,Y4:B]: ((((R23 @ X2) @ Y4) => ((R25 @ (Rep1 @ X2)) @ (Rep1 @ Y4))))) => (((((quotient3 @ A) @ C) @ (((((relcompp @ A) @ A) @ A) @ R12) @ (((((relcompp @ A) @ A) @ A) @ R25) @ R12))) @ (((((comp @ B) @ C) @ A) @ Abs22) @ Abs1)) @ (((((comp @ B) @ A) @ C) @ Rep1) @ Rep22))))))))).
% 9.43/3.46 thf(fact_5096_OOO__eq__quotient3, axiom, (! [A:$tType,B:$tType,C:$tType,R12:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),Abs22:(B > C),Rep22:(C > B)]: (((((((quotient3 @ A) @ B) @ R12) @ Abs1) @ Rep1) => ((((((quotient3 @ B) @ C) @ (^ [Y5:B,Z4:B]: ((Y5 = Z4)))) @ Abs22) @ Rep22) => (((((quotient3 @ A) @ C) @ (((((relcompp @ A) @ A) @ A) @ R12) @ (((((relcompp @ A) @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) @ R12))) @ (((((comp @ B) @ C) @ A) @ Abs22) @ Abs1)) @ (((((comp @ B) @ A) @ C) @ Rep1) @ Rep22))))))).
% 9.43/3.46 thf(fact_5097_relpowp__add, axiom, (! [A:$tType,M2:nat,N:nat,P:(A > (A > $o))]: (((((compow @ (A > (A > $o))) @ (((plus_plus @ nat) @ M2) @ N)) @ P) = (((((relcompp @ A) @ A) @ A) @ (((compow @ (A > (A > $o))) @ M2) @ P)) @ (((compow @ (A > (A > $o))) @ N) @ P)))))).
% 9.43/3.46 thf(fact_5098_relcompp__relcomp__eq, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S:(set @ ((product_prod @ B) @ C))]: (((((((relcompp @ A) @ B) @ C) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) @ (^ [X3:B,Y3:C]: ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ X3) @ Y3)) @ S)))) = (^ [X3:A,Y3:C]: ((((member @ ((product_prod @ A) @ C)) @ ((((product_Pair @ A) @ C) @ X3) @ Y3)) @ (((((relcomp @ A) @ B) @ C) @ R) @ S)))))))).
% 9.43/3.46 thf(fact_5099_is__unit__unit__factor, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((unit_f5069060285200089521factor @ A) @ A3) = A3)))))))).
% 9.43/3.46 thf(fact_5100_unit__factor__nat__def, axiom, ((unit_f5069060285200089521factor @ nat) = (^ [N5:nat]: (((((if @ nat) @ (N5 = (zero_zero @ nat))) @ (zero_zero @ nat)) @ (one_one @ nat)))))).
% 9.43/3.46 thf(fact_5101_option_Orel__Grp, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F:(A > B)]: (((((rel_option @ A) @ B) @ ((((bNF_Grp @ A) @ B) @ A4) @ F)) = ((((bNF_Grp @ (option @ A)) @ (option @ B)) @ ((collect @ (option @ A)) @ (^ [X3:(option @ A)]: ((((ord_less_eq @ (set @ A)) @ ((set_option @ A) @ X3)) @ A4))))) @ (((map_option @ A) @ B) @ F)))))).
% 9.43/3.46 thf(fact_5102_fun_Orel__Grp, axiom, (! [D:$tType,B:$tType,A:$tType,A4:(set @ A),F:(A > B)]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ ((((bNF_Grp @ A) @ B) @ A4) @ F)) = ((((bNF_Grp @ (D > A)) @ (D > B)) @ ((collect @ (D > A)) @ (^ [X3:(D > A)]: ((((ord_less_eq @ (set @ A)) @ ((((image2 @ D) @ A) @ X3) @ (top_top @ (set @ D)))) @ A4))))) @ ((((comp @ A) @ B) @ D) @ F)))))).
% 9.43/3.46 thf(fact_5103_relcomp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((relcomp @ A) @ B) @ C) = (^ [R3:(set @ ((product_prod @ A) @ B)),S4:(set @ ((product_prod @ B) @ C))]: (((collect @ ((product_prod @ A) @ C)) @ ((((product_case_prod @ A) @ C) @ $o) @ (((((relcompp @ A) @ B) @ C) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3)))) @ (^ [X3:B,Y3:C]: ((((member @ ((product_prod @ B) @ C)) @ ((((product_Pair @ B) @ C) @ X3) @ Y3)) @ S4)))))))))))).
% 9.43/3.46 thf(fact_5104_unit__factor__is__unit, axiom, (! [A:$tType]: (((semido2269285787275462019factor @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((dvd_dvd @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5105_unit__factor__gcd, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((A3 = (zero_zero @ A)) & (B2 = (zero_zero @ A))) => (((unit_f5069060285200089521factor @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) = (zero_zero @ A))) & ((~ (((A3 = (zero_zero @ A)) & (B2 = (zero_zero @ A))))) => (((unit_f5069060285200089521factor @ A) @ (((gcd_gcd @ A) @ A3) @ B2)) = (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5106_type__copy__vimage2p__Grp__Abs, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Rep:(A > B),Abs:(B > A),G:(D > C),P:(C > $o),H3:(C > A)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => (((((((((bNF_vimage2p @ D) @ C) @ B) @ A) @ $o) @ G) @ Abs) @ ((((bNF_Grp @ C) @ A) @ ((collect @ C) @ P)) @ H3)) = ((((bNF_Grp @ D) @ B) @ ((collect @ D) @ (^ [X3:D]: ((P @ (G @ X3)))))) @ (((((comp @ C) @ B) @ D) @ (((((comp @ A) @ B) @ C) @ Rep) @ H3)) @ G))))))).
% 9.43/3.46 thf(fact_5107_type__copy__vimage2p__Grp__Rep, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,Rep:(A > B),Abs:(B > A),F:(C > D),P:(D > $o),H3:(D > B)]: (((((((type_definition @ A) @ B) @ Rep) @ Abs) @ (top_top @ (set @ B))) => (((((((((bNF_vimage2p @ C) @ D) @ A) @ B) @ $o) @ F) @ Rep) @ ((((bNF_Grp @ D) @ B) @ ((collect @ D) @ P)) @ H3)) = ((((bNF_Grp @ C) @ A) @ ((collect @ C) @ (^ [X3:C]: ((P @ (F @ X3)))))) @ (((((comp @ D) @ A) @ C) @ (((((comp @ B) @ A) @ D) @ Abs) @ H3)) @ F))))))).
% 9.43/3.46 thf(fact_5108_Nat_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ code_natural) @ code_pcr_integer) @ code_pcr_natural) @ nat2) @ code_Nat)).
% 9.43/3.46 thf(fact_5109_pcr__natural__def, axiom, (code_pcr_natural = (((((relcompp @ nat) @ nat) @ code_natural) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ code_cr_natural))).
% 9.43/3.46 thf(fact_5110_natural_Orep__transfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ nat) @ code_pcr_natural) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (^ [X3:nat]: (X3))) @ code_nat_of_natural)).
% 9.43/3.46 thf(fact_5111_natural_Oid__abs__transfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ code_natural) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ code_pcr_natural) @ (^ [X3:nat]: (X3))) @ code_natural_of_nat)).
% 9.43/3.46 thf(fact_5112_natural_Opcr__cr__eq, axiom, (code_pcr_natural = code_cr_natural)).
% 9.43/3.46 thf(fact_5113_zero__natural_Otransfer, axiom, ((code_pcr_natural @ (zero_zero @ nat)) @ (zero_zero @ code_natural))).
% 9.43/3.46 thf(fact_5114_one__natural_Otransfer, axiom, ((code_pcr_natural @ (one_one @ nat)) @ (one_one @ code_natural))).
% 9.43/3.46 thf(fact_5115_less__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > $o)) @ (code_natural > $o)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ $o) @ $o) @ code_pcr_natural) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less @ nat)) @ (ord_less @ code_natural))).
% 9.43/3.46 thf(fact_5116_vimage2p__comp, axiom, (! [E:$tType,D:$tType,F5:$tType,A:$tType,C:$tType,B:$tType,G3:$tType,F1:(F5 > A),F22:(D > F5),G1:(G3 > B),G22:(E > G3)]: (((((((((bNF_vimage2p @ D) @ A) @ E) @ B) @ C) @ (((((comp @ F5) @ A) @ D) @ F1) @ F22)) @ (((((comp @ G3) @ B) @ E) @ G1) @ G22)) = (((((comp @ (F5 > (G3 > C))) @ (D > (E > C))) @ (A > (B > C))) @ (((((((bNF_vimage2p @ D) @ F5) @ E) @ G3) @ C) @ F22) @ G22)) @ (((((((bNF_vimage2p @ F5) @ A) @ G3) @ B) @ C) @ F1) @ G1)))))).
% 9.43/3.46 thf(fact_5117_plus__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (plus_plus @ nat)) @ (plus_plus @ code_natural))).
% 9.43/3.46 thf(fact_5118_Suc_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural) @ suc) @ code_Suc)).
% 9.43/3.46 thf(fact_5119_minus__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (minus_minus @ nat)) @ (minus_minus @ code_natural))).
% 9.43/3.46 thf(fact_5120_pcr__integer__def, axiom, (code_pcr_integer = (((((relcompp @ int) @ int) @ code_integer) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ code_cr_integer))).
% 9.43/3.46 thf(fact_5121_less__eq__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > $o)) @ (code_natural > $o)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ $o) @ $o) @ code_pcr_natural) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (ord_less_eq @ nat)) @ (ord_less_eq @ code_natural))).
% 9.43/3.46 thf(fact_5122_times__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (times_times @ nat)) @ (times_times @ code_natural))).
% 9.43/3.46 thf(fact_5123_divide__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (divide_divide @ nat)) @ (divide_divide @ code_natural))).
% 9.43/3.46 thf(fact_5124_modulo__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (modulo_modulo @ nat)) @ (modulo_modulo @ code_natural))).
% 9.43/3.46 thf(fact_5125_take__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se2584673776208193580ke_bit @ nat)) @ (bit_se2584673776208193580ke_bit @ code_natural))).
% 9.43/3.46 thf(fact_5126_xor__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se5824344971392196577ns_xor @ nat)) @ (bit_se5824344971392196577ns_xor @ code_natural))).
% 9.43/3.46 thf(fact_5127_or__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se1065995026697491101ons_or @ nat)) @ (bit_se1065995026697491101ons_or @ code_natural))).
% 9.43/3.46 thf(fact_5128_and__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > nat)) @ (code_natural > code_natural)) @ code_pcr_natural) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se5824344872417868541ns_and @ nat)) @ (bit_se5824344872417868541ns_and @ code_natural))).
% 9.43/3.46 thf(fact_5129_bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ (nat > $o)) @ (nat > $o)) @ code_pcr_natural) @ (^ [Y5:(nat > $o),Z4:(nat > $o)]: ((Y5 = Z4)))) @ (bit_se5641148757651400278ts_bit @ nat)) @ (bit_se5641148757651400278ts_bit @ code_natural))).
% 9.43/3.46 thf(fact_5130_push__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se4730199178511100633sh_bit @ nat)) @ (bit_se4730199178511100633sh_bit @ code_natural))).
% 9.43/3.46 thf(fact_5131_mask__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ nat) @ code_natural) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ code_pcr_natural) @ (bit_se2239418461657761734s_mask @ nat)) @ (bit_se2239418461657761734s_mask @ code_natural))).
% 9.43/3.46 thf(fact_5132_drop__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se4197421643247451524op_bit @ nat)) @ (bit_se4197421643247451524op_bit @ code_natural))).
% 9.43/3.46 thf(fact_5133_division__segment__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural) @ (euclid7384307370059645450egment @ nat)) @ (euclid7384307370059645450egment @ code_natural))).
% 9.43/3.46 thf(fact_5134_flip__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se8732182000553998342ip_bit @ nat)) @ (bit_se8732182000553998342ip_bit @ code_natural))).
% 9.43/3.46 thf(fact_5135_set__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se5668285175392031749et_bit @ nat)) @ (bit_se5668285175392031749et_bit @ code_natural))).
% 9.43/3.46 thf(fact_5136_unset__bit__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ nat) @ (nat > nat)) @ (code_natural > code_natural)) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ ((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ code_natural) @ code_pcr_natural) @ code_pcr_natural)) @ (bit_se2638667681897837118et_bit @ nat)) @ (bit_se2638667681897837118et_bit @ code_natural))).
% 9.43/3.46 thf(fact_5137_euclidean__size__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ nat) @ nat) @ code_pcr_natural) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ (euclid6346220572633701492n_size @ nat)) @ (euclid6346220572633701492n_size @ code_natural))).
% 9.43/3.46 thf(fact_5138_integer__of__natural_Otransfer, axiom, ((((((((bNF_rel_fun @ nat) @ code_natural) @ int) @ code_integer) @ code_pcr_natural) @ code_pcr_integer) @ (semiring_1_of_nat @ int)) @ code_i5400310926305786745atural)).
% 9.43/3.46 thf(fact_5139_natural__of__integer_Otransfer, axiom, ((((((((bNF_rel_fun @ int) @ code_integer) @ nat) @ code_natural) @ code_pcr_integer) @ code_pcr_natural) @ nat2) @ code_n4118661773612635043nteger)).
% 9.43/3.46 thf(fact_5140_fun_Orel__compp__Grp, axiom, (! [D:$tType,B:$tType,A:$tType,R5:(A > (B > $o))]: ((((((((bNF_rel_fun @ D) @ D) @ A) @ B) @ (^ [Y5:D,Z4:D]: ((Y5 = Z4)))) @ R5) = (((((relcompp @ (D > A)) @ (D > ((product_prod @ A) @ B))) @ (D > B)) @ (((conversep @ (D > ((product_prod @ A) @ B))) @ (D > A)) @ ((((bNF_Grp @ (D > ((product_prod @ A) @ B))) @ (D > A)) @ ((collect @ (D > ((product_prod @ A) @ B))) @ (^ [X3:(D > ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ D) @ ((product_prod @ A) @ B)) @ X3) @ (top_top @ (set @ D)))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R5))))))) @ ((((comp @ ((product_prod @ A) @ B)) @ A) @ D) @ ((product_fst @ A) @ B))))) @ ((((bNF_Grp @ (D > ((product_prod @ A) @ B))) @ (D > B)) @ ((collect @ (D > ((product_prod @ A) @ B))) @ (^ [X3:(D > ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((((image2 @ D) @ ((product_prod @ A) @ B)) @ X3) @ (top_top @ (set @ D)))) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R5))))))) @ ((((comp @ ((product_prod @ A) @ B)) @ B) @ D) @ ((product_snd @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5141_plus__num__def, axiom, ((plus_plus @ num) = (^ [M:num,N5:num]: ((num_of_nat @ (((plus_plus @ nat) @ (nat_of_num @ M)) @ (nat_of_num @ N5))))))).
% 9.43/3.46 thf(fact_5142_Lcm__no__units, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => ((gcd_Lcm @ A) = (^ [A9:(set @ A)]: (((gcd_Lcm @ A) @ (((minus_minus @ (set @ A)) @ A9) @ ((collect @ A) @ (^ [A7:A]: ((((dvd_dvd @ A) @ A7) @ (one_one @ A)))))))))))))).
% 9.43/3.46 thf(fact_5143_remdups__adj__altdef, axiom, (! [A:$tType,Xs:(list @ A),Ys2:(list @ A)]: (((((remdups_adj @ A) @ Xs) = Ys2) = (? [F2:(nat > nat)]: (((((order_mono @ nat) @ nat) @ F2) & ((((((image2 @ nat) @ nat) @ F2) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Xs))) = (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ ((size_size @ (list @ A)) @ Ys2))) & ((! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ Xs)) => ((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Ys2) @ (F2 @ I3)))))) & (! [I3:nat]: (((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ (one_one @ nat))) @ ((size_size @ (list @ A)) @ Xs)) => (((((nth @ A) @ Xs) @ I3) = (((nth @ A) @ Xs) @ (((plus_plus @ nat) @ I3) @ (one_one @ nat)))) = ((F2 @ I3) = (F2 @ (((plus_plus @ nat) @ I3) @ (one_one @ nat))))))))))))))))).
% 9.43/3.46 thf(fact_5144_Lcm__empty, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Lcm @ A) @ (bot_bot @ (set @ A))) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_5145_Lcm__1__iff, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A4:(set @ A)]: (((((gcd_Lcm @ A) @ A4) = (one_one @ A)) = (! [X3:A]: (((((member @ A) @ X3) @ A4) => (((dvd_dvd @ A) @ X3) @ (one_one @ A)))))))))))).
% 9.43/3.46 thf(fact_5146_mono__add, axiom, (! [A:$tType]: (((ordere6658533253407199908up_add @ A) => (! [A3:A]: ((((order_mono @ A) @ A) @ ((plus_plus @ A) @ A3)))))))).
% 9.43/3.46 thf(fact_5147_Lcm__nat__empty, axiom, (((gcd_Lcm @ nat) @ (bot_bot @ (set @ nat))) = (one_one @ nat))).
% 9.43/3.46 thf(fact_5148_nat__of__num_Osimps_I2_J, axiom, (! [X:num]: (((nat_of_num @ (bit0 @ X)) = (((plus_plus @ nat) @ (nat_of_num @ X)) @ (nat_of_num @ X)))))).
% 9.43/3.46 thf(fact_5149_nat__of__num__code_I1_J, axiom, ((nat_of_num @ one2) = (one_one @ nat))).
% 9.43/3.46 thf(fact_5150_nat__of__num__add, axiom, (! [X:num,Y:num]: (((nat_of_num @ (((plus_plus @ num) @ X) @ Y)) = (((plus_plus @ nat) @ (nat_of_num @ X)) @ (nat_of_num @ Y)))))).
% 9.43/3.46 thf(fact_5151_nat__of__num__code_I2_J, axiom, (! [N:num]: (((nat_of_num @ (bit0 @ N)) = (((plus_plus @ nat) @ (nat_of_num @ N)) @ (nat_of_num @ N)))))).
% 9.43/3.46 thf(fact_5152_mono__image__least, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [F:(A > B),M2:A,N:A,M7:B,N2:B]: (((((order_mono @ A) @ B) @ F) => ((((((image2 @ A) @ B) @ F) @ (((set_or7035219750837199246ssThan @ A) @ M2) @ N)) = (((set_or7035219750837199246ssThan @ B) @ M7) @ N2)) => ((((ord_less @ A) @ M2) @ N) => ((F @ M2) = M7)))))))))).
% 9.43/3.46 thf(fact_5153_nat__of__num_Osimps_I3_J, axiom, (! [X:num]: (((nat_of_num @ (bit1 @ X)) = (suc @ (((plus_plus @ nat) @ (nat_of_num @ X)) @ (nat_of_num @ X))))))).
% 9.43/3.46 thf(fact_5154_unit__factor__Lcm, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A4:(set @ A)]: ((((((gcd_Lcm @ A) @ A4) = (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Lcm @ A) @ A4)) = (zero_zero @ A))) & ((((gcd_Lcm @ A) @ A4) != (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Lcm @ A) @ A4)) = (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5155_nat__of__num__code_I3_J, axiom, (! [N:num]: (((nat_of_num @ (bit1 @ N)) = (suc @ (((plus_plus @ nat) @ (nat_of_num @ N)) @ (nat_of_num @ N))))))).
% 9.43/3.46 thf(fact_5156_remdups__adj_Opelims, axiom, (! [A:$tType,X:(list @ A),Y:(list @ A)]: (((((remdups_adj @ A) @ X) = Y) => ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ X) => (((X = (nil @ A)) => ((Y = (nil @ A)) => (~ ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ (nil @ A)))))) => ((! [X2:A]: (((X = (((cons @ A) @ X2) @ (nil @ A))) => ((Y = (((cons @ A) @ X2) @ (nil @ A))) => (~ ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ (((cons @ A) @ X2) @ (nil @ A))))))))) => (~ ((! [X2:A,Y4:A,Xs2:(list @ A)]: (((X = (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2))) => ((((X2 = Y4) => (Y = ((remdups_adj @ A) @ (((cons @ A) @ X2) @ Xs2)))) & ((X2 != Y4) => (Y = (((cons @ A) @ X2) @ ((remdups_adj @ A) @ (((cons @ A) @ Y4) @ Xs2)))))) => (~ ((((accp @ (list @ A)) @ (remdups_adj_rel @ A)) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2)))))))))))))))))).
% 9.43/3.46 thf(fact_5157_relChain__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (((bNF_Ca3754400796208372196lChain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ A)),As3:(A > B)]: ((! [I3:A,J2:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ I3) @ J2)) @ R3) => (((ord_less_eq @ B) @ (As3 @ I3)) @ (As3 @ J2)))))))))))).
% 9.43/3.46 thf(fact_5158_unit__factor__Gcd, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A4:(set @ A)]: ((((((gcd_Gcd @ A) @ A4) = (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Gcd @ A) @ A4)) = (zero_zero @ A))) & ((((gcd_Gcd @ A) @ A4) != (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((gcd_Gcd @ A) @ A4)) = (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5159_Gcd__UNIV, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (((gcd_Gcd @ A) @ (top_top @ (set @ A))) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_5160_Gcd__1, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A4:(set @ A)]: (((((member @ A) @ (one_one @ A)) @ A4) => (((gcd_Gcd @ A) @ A4) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5161_Gcd__nat__eq__one, axiom, (! [N6:(set @ nat)]: (((((member @ nat) @ (one_one @ nat)) @ N6) => (((gcd_Gcd @ nat) @ N6) = (one_one @ nat)))))).
% 9.43/3.46 thf(fact_5162_Gcd__eq__1__I, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [A3:A,A4:(set @ A)]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((member @ A) @ A3) @ A4) => (((gcd_Gcd @ A) @ A4) = (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5163_cofinal__def, axiom, (! [A:$tType]: (((bNF_Ca7293521722713021262ofinal @ A) = (^ [A9:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: ((! [X3:A]: (((((member @ A) @ X3) @ ((field2 @ A) @ R3)) => (? [Y3:A]: (((((member @ A) @ Y3) @ A9) & ((X3 != Y3) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3)))))))))))))).
% 9.43/3.46 thf(fact_5164_card__of__UNION__ordLeq__infinite, axiom, (! [B:$tType,A:$tType,C:$tType,B6:(set @ A),I4:(set @ B),A4:(B > (set @ C))]: (((~ (((finite_finite2 @ A) @ B6))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ I4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((! [X2:B]: (((((member @ B) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ X2))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ C) @ A))))) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ B) @ (set @ C)) @ A4) @ I4)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ C) @ A)))))))).
% 9.43/3.46 thf(fact_5165_card__of__ordLess, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((~ ((? [F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ A4) & (((ord_less_eq @ (set @ B)) @ ((((image2 @ A) @ B) @ F2) @ A4)) @ B6)))))) = (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_We4044943003108391690rdLess @ B) @ A)))))).
% 9.43/3.46 thf(fact_5166_card__of__Times3, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ A)) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4)))))) @ ((bNF_Wellorder_ordLeq @ A) @ ((product_prod @ A) @ A)))))).
% 9.43/3.46 thf(fact_5167_card__of__Sigma__mono1, axiom, (! [C:$tType,B:$tType,A:$tType,I4:(set @ A),A4:(A > (set @ B)),B6:(A > (set @ C))]: (((! [X2:A]: (((((member @ A) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (A4 @ X2))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (B6 @ X2)))) @ ((bNF_Wellorder_ordLeq @ B) @ C))))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ I4) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ C)) @ ((((product_Sigma @ A) @ C) @ I4) @ B6)))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))))))).
% 9.43/3.46 thf(fact_5168_card__of__Times__mono1, axiom, (! [B:$tType,C:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ C)) @ ((((product_Sigma @ A) @ C) @ A4) @ (^ [Uu:A]: (C5))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ C)) @ ((((product_Sigma @ B) @ C) @ B6) @ (^ [Uu:B]: (C5)))))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ C))))))).
% 9.43/3.46 thf(fact_5169_card__of__Times__mono2, axiom, (! [B:$tType,A:$tType,C:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ C) @ A)) @ ((((product_Sigma @ C) @ A) @ C5) @ (^ [Uu:C]: (A4))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ C) @ B)) @ ((((product_Sigma @ C) @ B) @ C5) @ (^ [Uu:C]: (B6)))))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ B))))))).
% 9.43/3.46 thf(fact_5170_infinite__iff__card__of__nat, axiom, (! [A:$tType,A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) = (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ nat) @ (top_top @ (set @ nat)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ nat) @ A)))))).
% 9.43/3.46 thf(fact_5171_card__of__UNION__Sigma, axiom, (! [B:$tType,A:$tType,A4:(B > (set @ A)),I4:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ B) @ (set @ A)) @ A4) @ I4)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ I4) @ A4)))) @ ((bNF_Wellorder_ordLeq @ A) @ ((product_prod @ B) @ A)))))).
% 9.43/3.46 thf(fact_5172_card__of__Times__commute, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (A4)))))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)))))).
% 9.43/3.46 thf(fact_5173_card__of__Times__same__infinite, axiom, (! [A:$tType,A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ A)) @ ((((product_Sigma @ A) @ A) @ A4) @ (^ [Uu:A]: (A4))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ A)) @ A)))))).
% 9.43/3.46 thf(fact_5174_card__of__Times1, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((A4 != (bot_bot @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (A4)))))) @ ((bNF_Wellorder_ordLeq @ B) @ ((product_prod @ B) @ A))))))).
% 9.43/3.46 thf(fact_5175_card__of__Times2, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((A4 != (bot_bot @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))))) @ ((bNF_Wellorder_ordLeq @ B) @ ((product_prod @ A) @ B))))))).
% 9.43/3.46 thf(fact_5176_card__of__bool, axiom, (! [A:$tType,A1:A,A22:A]: (((A1 != A22) => (((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ $o) @ (top_top @ (set @ $o)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (((insert2 @ A) @ A1) @ (((insert2 @ A) @ A22) @ (bot_bot @ (set @ A))))))) @ ((bNF_Wellorder_ordIso @ $o) @ A)))))).
% 9.43/3.46 thf(fact_5177_card__of__Plus__empty1, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ (bot_bot @ (set @ B)))))) @ ((bNF_Wellorder_ordIso @ A) @ ((sum_sum @ A) @ B)))))).
% 9.43/3.46 thf(fact_5178_card__of__Plus__empty2, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ (bot_bot @ (set @ B))) @ A4)))) @ ((bNF_Wellorder_ordIso @ A) @ ((sum_sum @ B) @ A)))))).
% 9.43/3.46 thf(fact_5179_card__of__Plus__Times__bool, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A)))) @ (set @ ((product_prod @ ((product_prod @ A) @ $o)) @ ((product_prod @ A) @ $o))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A)))) @ (set @ ((product_prod @ ((product_prod @ A) @ $o)) @ ((product_prod @ A) @ $o)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ A)) @ ((((sum_Plus @ A) @ A) @ A4) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ $o)) @ ((((product_Sigma @ A) @ $o) @ A4) @ (^ [Uu:A]: ((top_top @ (set @ $o)))))))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ A)) @ ((product_prod @ A) @ $o)))))).
% 9.43/3.46 thf(fact_5180_card__of__Plus__commute, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ B6) @ A4)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ ((sum_sum @ B) @ A)))))).
% 9.43/3.46 thf(fact_5181_card__of__Plus__cong2, axiom, (! [B:$tType,A:$tType,C:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ A)) @ ((((sum_Plus @ C) @ A) @ C5) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ B)) @ ((((sum_Plus @ C) @ B) @ C5) @ B6)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ B))))))).
% 9.43/3.46 thf(fact_5182_card__of__Plus__cong1, axiom, (! [B:$tType,C:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ A4) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ B6) @ C5)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5183_card__of__Plus__assoc, axiom, (! [C:$tType,B:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((((sum_Plus @ ((sum_sum @ A) @ B)) @ C) @ ((((sum_Plus @ A) @ B) @ A4) @ B6)) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) @ ((((sum_Plus @ A) @ ((sum_sum @ B) @ C)) @ A4) @ ((((sum_Plus @ B) @ C) @ B6) @ C5))))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5184_card__of__Plus__cong, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C),D4:(set @ D)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ C5)) @ ((bNF_Ca6860139660246222851ard_of @ D) @ D4))) @ ((bNF_Wellorder_ordIso @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ A4) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ D)) @ ((((sum_Plus @ B) @ D) @ B6) @ D4)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5185_card__of__Times__Plus__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ ((sum_sum @ B) @ C))) @ ((product_prod @ A) @ ((sum_sum @ B) @ C))))) @ (set @ ((product_prod @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))) @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ ((sum_sum @ B) @ C))) @ ((product_prod @ A) @ ((sum_sum @ B) @ C))))) @ (set @ ((product_prod @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))) @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ ((sum_sum @ B) @ C))) @ ((((product_Sigma @ A) @ ((sum_sum @ B) @ C)) @ A4) @ (^ [Uu:A]: (((((sum_Plus @ B) @ C) @ B6) @ C5)))))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))) @ ((((sum_Plus @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))) @ ((((product_Sigma @ A) @ C) @ A4) @ (^ [Uu:A]: (C5))))))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ ((sum_sum @ B) @ C))) @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))))))).
% 9.43/3.46 thf(fact_5186_card__of__Plus__mono2, axiom, (! [B:$tType,A:$tType,C:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ A)) @ ((((sum_Plus @ C) @ A) @ C5) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ B)) @ ((((sum_Plus @ C) @ B) @ C5) @ B6)))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ B))))))).
% 9.43/3.46 thf(fact_5187_card__of__Plus__mono1, axiom, (! [B:$tType,C:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ A4) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ B6) @ C5)))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5188_card__of__Plus__mono, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C),D4:(set @ D)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ C5)) @ ((bNF_Ca6860139660246222851ard_of @ D) @ D4))) @ ((bNF_Wellorder_ordLeq @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ A4) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ D)) @ ((((sum_Plus @ B) @ D) @ B6) @ D4)))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5189_card__of__Plus2, axiom, (! [B:$tType,A:$tType,B6:(set @ A),A4:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ A4) @ B6)))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ B) @ A)))))).
% 9.43/3.46 thf(fact_5190_card__of__Plus1, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6)))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ A) @ B)))))).
% 9.43/3.46 thf(fact_5191_card__of__Un__Plus__ordLeq, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ A)) @ ((((sum_Plus @ A) @ A) @ A4) @ B6)))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ A) @ A)))))).
% 9.43/3.46 thf(fact_5192_card__of__Plus__infinite, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ A)) & (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ B6) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ B) @ A)) @ A)))))))).
% 9.43/3.46 thf(fact_5193_card__of__Plus__infinite1, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ A))))))).
% 9.43/3.46 thf(fact_5194_card__of__Plus__infinite2, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ B6) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ B) @ A)) @ A))))))).
% 9.43/3.46 thf(fact_5195_card__of__Plus__ordLess__infinite, axiom, (! [A:$tType,C:$tType,B:$tType,C5:(set @ A),A4:(set @ B),B6:(set @ C)]: (((~ (((finite_finite2 @ A) @ C5))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ C5))) @ ((bNF_We4044943003108391690rdLess @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ C5))) @ ((bNF_We4044943003108391690rdLess @ C) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ C5))) @ ((bNF_We4044943003108391690rdLess @ ((sum_sum @ B) @ C)) @ A)))))))).
% 9.43/3.46 thf(fact_5196_card__of__Sigma__ordLeq__infinite, axiom, (! [A:$tType,C:$tType,B:$tType,B6:(set @ A),I4:(set @ B),A4:(B > (set @ C))]: (((~ (((finite_finite2 @ A) @ B6))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ I4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((! [X2:B]: (((((member @ B) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ X2))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ C) @ A))))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ C)) @ ((((product_Sigma @ B) @ C) @ I4) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ B) @ C)) @ A)))))))).
% 9.43/3.46 thf(fact_5197_card__of__refl, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ A) @ A))))).
% 9.43/3.46 thf(fact_5198_card__of__Plus__Times__aux, axiom, (! [B:$tType,A:$tType,A1:A,A22:A,A4:(set @ A),B6:(set @ B)]: ((((A1 != A22) & (((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ A1) @ (((insert2 @ A) @ A22) @ (bot_bot @ (set @ A))))) @ A4)) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5199_card__of__Times__infinite__simps_I4_J, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((B6 != (bot_bot @ (set @ B))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (A4)))))) @ ((bNF_Wellorder_ordIso @ A) @ ((product_prod @ B) @ A))))))))).
% 9.43/3.46 thf(fact_5200_card__of__Times__infinite__simps_I3_J, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((B6 != (bot_bot @ (set @ B))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (A4))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ B) @ A)) @ A)))))))).
% 9.43/3.46 thf(fact_5201_card__of__Times__infinite__simps_I2_J, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((B6 != (bot_bot @ (set @ B))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))))) @ ((bNF_Wellorder_ordIso @ A) @ ((product_prod @ A) @ B))))))))).
% 9.43/3.46 thf(fact_5202_card__of__Times__infinite__simps_I1_J, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((B6 != (bot_bot @ (set @ B))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ B)) @ A)))))))).
% 9.43/3.46 thf(fact_5203_card__of__Times__infinite, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ A4))) => ((B6 != (bot_bot @ (set @ B))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ B)) @ A)) & (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ B6) @ (^ [Uu:B]: (A4))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ B) @ A)) @ A))))))))).
% 9.43/3.46 thf(fact_5204_card__of__Plus__Times, axiom, (! [B:$tType,A:$tType,A1:A,A22:A,A4:(set @ A),B1:B,B22:B,B6:(set @ B)]: ((((A1 != A22) & (((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ A1) @ (((insert2 @ A) @ A22) @ (bot_bot @ (set @ A))))) @ A4)) => (((B1 != B22) & (((ord_less_eq @ (set @ B)) @ (((insert2 @ B) @ B1) @ (((insert2 @ B) @ B22) @ (bot_bot @ (set @ B))))) @ B6)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ B)) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5205_card__of__image, axiom, (! [B:$tType,A:$tType,F:(B > A),A4:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((((image2 @ B) @ A) @ F) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4))) @ ((bNF_Wellorder_ordLeq @ A) @ B))))).
% 9.43/3.46 thf(fact_5206_card__of__empty, axiom, (! [B:$tType,A:$tType,A4:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (bot_bot @ (set @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4))) @ ((bNF_Wellorder_ordLeq @ A) @ B))))).
% 9.43/3.46 thf(fact_5207_card__of__empty3, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (bot_bot @ (set @ B))))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (A4 = (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_5208_card__of__ordLeq__finite, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((finite_finite2 @ B) @ B6) => ((finite_finite2 @ A) @ A4)))))).
% 9.43/3.46 thf(fact_5209_card__of__ordLeq__infinite, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((~ (((finite_finite2 @ A) @ A4))) => (~ (((finite_finite2 @ B) @ B6)))))))).
% 9.43/3.46 thf(fact_5210_card__of__mono1, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A4) @ B6) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))).
% 9.43/3.46 thf(fact_5211_card__of__empty__ordIso, axiom, (! [B:$tType,A:$tType]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (bot_bot @ (set @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (bot_bot @ (set @ B))))) @ ((bNF_Wellorder_ordIso @ A) @ B))))).
% 9.43/3.46 thf(fact_5212_card__of__empty2, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (bot_bot @ (set @ B))))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (A4 = (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_5213_card__of__ordIso__finite, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B6:(set @ B)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((finite_finite2 @ A) @ A4) = ((finite_finite2 @ B) @ B6)))))).
% 9.43/3.46 thf(fact_5214_card__of__mono2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ ((field2 @ B) @ R4)))) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))).
% 9.43/3.46 thf(fact_5215_card__of__cong, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ ((field2 @ B) @ R4)))) @ ((bNF_Wellorder_ordIso @ A) @ B)))))).
% 9.43/3.46 thf(fact_5216_card__of__ordLeqI, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((! [A5:A]: (((((member @ A) @ A5) @ A4) => (((member @ B) @ (F @ A5)) @ B6)))) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B))))))).
% 9.43/3.46 thf(fact_5217_ex__bij__betw, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (? [F4:(B > A),B10:(set @ B)]: ((((((bij_betw @ B) @ A) @ F4) @ B10) @ A4))))))).
% 9.43/3.46 thf(fact_5218_card__of__least, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ A4) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))).
% 9.43/3.46 thf(fact_5219_BNF__Cardinal__Order__Relation_OordLess__Field, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R))) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)))))).
% 9.43/3.46 thf(fact_5220_card__of__ordIsoI, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A),B6:(set @ B)]: (((((((bij_betw @ A) @ B) @ F) @ A4) @ B6) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ B)))))).
% 9.43/3.46 thf(fact_5221_card__of__ordIso, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((? [F2:(A > B)]: ((((((bij_betw @ A) @ B) @ F2) @ A4) @ B6))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ B)))))).
% 9.43/3.46 thf(fact_5222_type__copy__set__bd, axiom, (! [A:$tType,D:$tType,C:$tType,B:$tType,S2:(A > (set @ B)),Bd:(set @ ((product_prod @ C) @ C)),Rep:(D > A),X:D]: (((! [Y4:A]: ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (S2 @ Y4))) @ Bd)) @ ((bNF_Wellorder_ordLeq @ B) @ C)))) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ ((((((comp @ A) @ (set @ B)) @ D) @ S2) @ Rep) @ X))) @ Bd)) @ ((bNF_Wellorder_ordLeq @ B) @ C)))))).
% 9.43/3.46 thf(fact_5223_card__of__ordLeq2, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((A4 != (bot_bot @ (set @ A))) => ((? [G2:(B > A)]: ((((((image2 @ B) @ A) @ G2) @ B6) = A4))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B))))))).
% 9.43/3.46 thf(fact_5224_surj__imp__ordLeq, axiom, (! [B:$tType,A:$tType,B6:(set @ A),F:(B > A),A4:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B6) @ ((((image2 @ B) @ A) @ F) @ A4)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4))) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))).
% 9.43/3.46 thf(fact_5225_card__of__singl__ordLeq, axiom, (! [A:$tType,B:$tType,A4:(set @ A),B2:B]: (((A4 != (bot_bot @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (((insert2 @ B) @ B2) @ (bot_bot @ (set @ B))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))).
% 9.43/3.46 thf(fact_5226_card__of__ordLess2, axiom, (! [A:$tType,B:$tType,B6:(set @ A),A4:(set @ B)]: (((B6 != (bot_bot @ (set @ A))) => ((~ ((? [F2:(B > A)]: ((((((image2 @ B) @ A) @ F2) @ A4) = B6))))) = (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_We4044943003108391690rdLess @ B) @ A))))))).
% 9.43/3.46 thf(fact_5227_internalize__card__of__ordLeq2, axiom, (! [A:$tType,B:$tType,A4:(set @ A),C5:(set @ B)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ C5))) @ ((bNF_Wellorder_ordLeq @ A) @ B)) = (? [B9:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B9) @ C5) & ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B9))) @ ((bNF_Wellorder_ordIso @ A) @ B)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B9)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ C5))) @ ((bNF_Wellorder_ordLeq @ B) @ B)))))))))).
% 9.43/3.46 thf(fact_5228_card__of__Field__ordLess, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R))) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))).
% 9.43/3.46 thf(fact_5229_ordLeq__Plus__mono, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),P5:(set @ ((product_prod @ C) @ C)),P8:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P5) @ P8)) @ ((bNF_Wellorder_ordLeq @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ ((field2 @ A) @ R)) @ ((field2 @ C) @ P5)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ D)) @ ((((sum_Plus @ B) @ D) @ ((field2 @ B) @ R4)) @ ((field2 @ D) @ P8))))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5230_ordLeq__Plus__mono1, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ ((field2 @ A) @ R)) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ ((field2 @ B) @ R4)) @ C5)))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5231_ordLeq__Plus__mono2, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),A4:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ A)) @ ((((sum_Plus @ C) @ A) @ A4) @ ((field2 @ A) @ R)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ B)) @ ((((sum_Plus @ C) @ B) @ A4) @ ((field2 @ B) @ R4))))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ B))))))).
% 9.43/3.46 thf(fact_5232_ordIso__Plus__cong, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),P5:(set @ ((product_prod @ C) @ C)),P8:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P5) @ P8)) @ ((bNF_Wellorder_ordIso @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ ((field2 @ A) @ R)) @ ((field2 @ C) @ P5)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ D)) @ ((((sum_Plus @ B) @ D) @ ((field2 @ B) @ R4)) @ ((field2 @ D) @ P8))))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5233_ordIso__Plus__cong1, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ C)) @ ((((sum_Plus @ A) @ C) @ ((field2 @ A) @ R)) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ ((field2 @ B) @ R4)) @ C5)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5234_ordIso__Plus__cong2, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),A4:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ A)) @ ((((sum_Plus @ C) @ A) @ A4) @ ((field2 @ A) @ R)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ C) @ B)) @ ((((sum_Plus @ C) @ B) @ A4) @ ((field2 @ B) @ R4))))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ B))))))).
% 9.43/3.46 thf(fact_5235_ordLeq__Times__mono1, axiom, (! [B:$tType,C:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),C5:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ C)) @ ((((product_Sigma @ A) @ C) @ ((field2 @ A) @ R)) @ (^ [Uu:A]: (C5))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ C)) @ ((((product_Sigma @ B) @ C) @ ((field2 @ B) @ R4)) @ (^ [Uu:B]: (C5)))))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ C))))))).
% 9.43/3.46 thf(fact_5236_ordLeq__Times__mono2, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B)),A4:(set @ C)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ C) @ A)) @ ((((product_Sigma @ C) @ A) @ A4) @ (^ [Uu:C]: (((field2 @ A) @ R)))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ C) @ B)) @ ((((product_Sigma @ C) @ B) @ A4) @ (^ [Uu:C]: (((field2 @ B) @ R4))))))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ B))))))).
% 9.43/3.46 thf(fact_5237_card__of__ordLeq, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: (((? [F2:(A > B)]: ((((((inj_on @ A) @ B) @ F2) @ A4) & (((ord_less_eq @ (set @ B)) @ ((((image2 @ A) @ B) @ F2) @ A4)) @ B6)))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6))) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))).
% 9.43/3.46 thf(fact_5238_internalize__card__of__ordLeq, axiom, (! [A:$tType,B:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) = (? [B9:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B9) @ ((field2 @ B) @ R)) & ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B9))) @ ((bNF_Wellorder_ordIso @ A) @ B)) & (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B9)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ B)))))))))).
% 9.43/3.46 thf(fact_5239_regularCard__def, axiom, (! [A:$tType]: (((bNF_Ca7133664381575040944arCard @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: ((! [K3:(set @ A)]: ((((((ord_less_eq @ (set @ A)) @ K3) @ ((field2 @ A) @ R3)) & (((bNF_Ca7293521722713021262ofinal @ A) @ K3) @ R3)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ K3)) @ R3)) @ ((bNF_Wellorder_ordIso @ A) @ A))))))))))).
% 9.43/3.46 thf(fact_5240_card__of__ordIso__subst, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: (((A4 = B6) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6))) @ ((bNF_Wellorder_ordIso @ A) @ A)))))).
% 9.43/3.46 thf(fact_5241_comp__set__bd__Union__o__collect, axiom, (! [C:$tType,B:$tType,A:$tType,X:C,X10:(set @ (C > (set @ (set @ A)))),Hbd:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((complete_Sup_Sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ (set @ A))) @ ((((image2 @ (C > (set @ (set @ A)))) @ (set @ (set @ A))) @ (^ [F2:(C > (set @ (set @ A)))]: ((F2 @ X)))) @ X10))))) @ Hbd)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((((((comp @ (set @ (set @ A))) @ (set @ A)) @ C) @ (complete_Sup_Sup @ (set @ A))) @ (((bNF_collect @ C) @ (set @ A)) @ X10)) @ X))) @ Hbd)) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))).
% 9.43/3.46 thf(fact_5242_card__of__Plus__ordLess__infinite__Field, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B),B6:(set @ C)]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4)) @ R)) @ ((bNF_We4044943003108391690rdLess @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ B6)) @ R)) @ ((bNF_We4044943003108391690rdLess @ C) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ A4) @ B6))) @ R)) @ ((bNF_We4044943003108391690rdLess @ ((sum_sum @ B) @ C)) @ A))))))))).
% 9.43/3.46 thf(fact_5243_card__of__Times__ordLeq__infinite__Field, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B),B6:(set @ C)]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ B6)) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ A)) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ C)) @ ((((product_Sigma @ B) @ C) @ A4) @ (^ [Uu:B]: (B6))))) @ R)) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ B) @ C)) @ A))))))))).
% 9.43/3.46 thf(fact_5244_card__of__Sigma__ordLeq__infinite__Field, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),I4:(set @ B),A4:(B > (set @ C))]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ I4)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((! [X2:B]: (((((member @ B) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ X2))) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ A))))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ C)) @ ((((product_Sigma @ B) @ C) @ I4) @ A4))) @ R)) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ B) @ C)) @ A))))))))).
% 9.43/3.46 thf(fact_5245_Card__order__trans, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z2:A]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((X != Y) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => ((Y != Z2) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ R) => ((X != Z2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ R)))))))))).
% 9.43/3.46 thf(fact_5246_card__order__on__ordIso, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ A4) @ R) => ((((bNF_Ca8970107618336181345der_on @ A) @ A4) @ R4) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ A))))))).
% 9.43/3.46 thf(fact_5247_infinite__Card__order__limit, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => (? [X2:A]: (((((member @ A) @ X2) @ ((field2 @ A) @ R)) & ((A3 != X2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ X2)) @ R))))))))))).
% 9.43/3.46 thf(fact_5248_ordLeq__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))).
% 9.43/3.46 thf(fact_5249_Card__order__ordIso, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordIso @ B) @ A)) => (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4)))))).
% 9.43/3.46 thf(fact_5250_Card__order__ordIso2, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4)))))).
% 9.43/3.46 thf(fact_5251_ordIso__refl, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ R)) @ ((bNF_Wellorder_ordIso @ A) @ A)))))).
% 9.43/3.46 thf(fact_5252_card__of__unique, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ A4) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordIso @ A) @ A)))))).
% 9.43/3.46 thf(fact_5253_card__order__on__def, axiom, (! [A:$tType]: (((bNF_Ca8970107618336181345der_on @ A) = (^ [A9:(set @ A),R3:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ A9) @ R3) & (! [R11:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ A9) @ R11) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R3) @ R11)) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))))))))).
% 9.43/3.46 thf(fact_5254_Card__order__iff__ordLeq__card__of, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R)))) @ ((bNF_Wellorder_ordLeq @ A) @ A)))))).
% 9.43/3.46 thf(fact_5255_ordIso__card__of__imp__Card__order, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R))))).
% 9.43/3.46 thf(fact_5256_Card__order__iff__ordIso__card__of, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R)))) @ ((bNF_Wellorder_ordIso @ A) @ A)))))).
% 9.43/3.46 thf(fact_5257_card__of__Field__ordIso, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R))) @ R)) @ ((bNF_Wellorder_ordIso @ A) @ A)))))).
% 9.43/3.46 thf(fact_5258_dir__image, axiom, (! [B:$tType,A:$tType,F:(A > B),R:(set @ ((product_prod @ A) @ A))]: (((! [X2:A,Y4:A]: ((((F @ X2) = (F @ Y4)) = (X2 = Y4)))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ ((((bNF_We2720479622203943262_image @ A) @ B) @ R) @ F))) @ ((bNF_Wellorder_ordIso @ A) @ B))))))).
% 9.43/3.46 thf(fact_5259_exists__minim__Card__order, axiom, (! [A:$tType,R5:(set @ (set @ ((product_prod @ A) @ A)))]: (((R5 != (bot_bot @ (set @ (set @ ((product_prod @ A) @ A))))) => ((! [X2:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ ((product_prod @ A) @ A))) @ X2) @ R5) => (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ X2)) @ X2)))) => (? [X2:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ ((product_prod @ A) @ A))) @ X2) @ R5) & (! [Xa2:(set @ ((product_prod @ A) @ A))]: (((((member @ (set @ ((product_prod @ A) @ A))) @ Xa2) @ R5) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ X2) @ Xa2)) @ ((bNF_Wellorder_ordLeq @ A) @ A))))))))))))).
% 9.43/3.46 thf(fact_5260_Card__order__empty, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (bot_bot @ (set @ B)))) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))).
% 9.43/3.46 thf(fact_5261_card__of__ordIso__finite__Field, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((finite_finite2 @ A) @ ((field2 @ A) @ R)) = ((finite_finite2 @ B) @ A4))))))).
% 9.43/3.46 thf(fact_5262_card__of__underS, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ A) @ A3) @ ((field2 @ A) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (((order_underS @ A) @ R) @ A3))) @ R)) @ ((bNF_We4044943003108391690rdLess @ A) @ A))))))).
% 9.43/3.46 thf(fact_5263_Card__order__Plus1, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),B6:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ ((field2 @ A) @ R)) @ B6)))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ A) @ B))))))).
% 9.43/3.46 thf(fact_5264_Card__order__Plus2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ A4) @ ((field2 @ A) @ R))))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ B) @ A))))))).
% 9.43/3.46 thf(fact_5265_Card__order__singl__ordLeq, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),B2:B]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((field2 @ A) @ R) != (bot_bot @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (((insert2 @ B) @ B2) @ (bot_bot @ (set @ B))))) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A))))))).
% 9.43/3.46 thf(fact_5266_card__of__Un__ordLeq__infinite__Field, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B),B6:(set @ B)]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (((sup_sup @ (set @ B)) @ A4) @ B6))) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A))))))))).
% 9.43/3.46 thf(fact_5267_card__of__empty1, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) | (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (bot_bot @ (set @ B)))) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))).
% 9.43/3.46 thf(fact_5268_Card__order__Times1, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),B6:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((B6 != (bot_bot @ (set @ B))) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ ((field2 @ A) @ R)) @ (^ [Uu:A]: (B6)))))) @ ((bNF_Wellorder_ordLeq @ A) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5269_Card__order__Times2, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((A4 != (bot_bot @ (set @ B))) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ A4) @ (^ [Uu:B]: (((field2 @ A) @ R))))))) @ ((bNF_Wellorder_ordLeq @ A) @ ((product_prod @ B) @ A)))))))).
% 9.43/3.46 thf(fact_5270_Card__order__Times__same__infinite, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ A)) @ ((((product_Sigma @ A) @ A) @ ((field2 @ A) @ R)) @ (^ [Uu:A]: (((field2 @ A) @ R)))))) @ R)) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ A)) @ A))))))).
% 9.43/3.46 thf(fact_5271_card__of__UNION__ordLeq__infinite__Field, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ A) @ A)),I4:(set @ B),A4:(B > (set @ C))]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ I4)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((! [X2:B]: (((((member @ B) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ X2))) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ A))))) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ B) @ (set @ C)) @ A4) @ I4)))) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ A))))))))).
% 9.43/3.46 thf(fact_5272_Card__order__Plus__infinite, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),P5:(set @ ((product_prod @ B) @ B))]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ P5) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ ((field2 @ A) @ R)) @ ((field2 @ B) @ P5)))) @ R)) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ A)) & (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ A)) @ ((((sum_Plus @ B) @ A) @ ((field2 @ B) @ P5)) @ ((field2 @ A) @ R)))) @ R)) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ B) @ A)) @ A))))))))).
% 9.43/3.46 thf(fact_5273_Card__order__iff__Restr__underS, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((order_well_order_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) = (! [X3:A]: (((((member @ A) @ X3) @ ((field2 @ A) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ (((inf_inf @ (set @ ((product_prod @ A) @ A))) @ R) @ ((((product_Sigma @ A) @ A) @ (((order_underS @ A) @ R) @ X3)) @ (^ [Uu:A]: ((((order_underS @ A) @ R) @ X3)))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ ((field2 @ A) @ R)))) @ ((bNF_We4044943003108391690rdLess @ A) @ A)))))))))).
% 9.43/3.46 thf(fact_5274_regularCard__UNION, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),As4:(A > (set @ B)),B6:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((bNF_Ca7133664381575040944arCard @ A) @ R) => (((((bNF_Ca3754400796208372196lChain @ A) @ (set @ B)) @ R) @ As4) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image2 @ A) @ (set @ B)) @ As4) @ ((field2 @ A) @ R)))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ R)) @ ((bNF_We4044943003108391690rdLess @ B) @ A)) => (? [X2:A]: (((((member @ A) @ X2) @ ((field2 @ A) @ R)) & (((ord_less_eq @ (set @ B)) @ B6) @ (As4 @ X2))))))))))))).
% 9.43/3.46 thf(fact_5275_card__of__Plus__ordLeq__infinite__Field, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),A4:(set @ B),B6:(set @ C)]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ B6)) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ A)) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ B) @ C)) @ ((((sum_Plus @ B) @ C) @ A4) @ B6))) @ R)) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ B) @ C)) @ A))))))))).
% 9.43/3.46 thf(fact_5276_Card__order__Times__infinite, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),P5:(set @ ((product_prod @ B) @ B))]: (((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((field2 @ B) @ P5) != (bot_bot @ (set @ B))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ P5) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ A) @ B)) @ ((((product_Sigma @ A) @ B) @ ((field2 @ A) @ R)) @ (^ [Uu:A]: (((field2 @ B) @ P5)))))) @ R)) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ B)) @ A)) & (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ B) @ A)) @ ((((product_Sigma @ B) @ A) @ ((field2 @ B) @ P5)) @ (^ [Uu:B]: (((field2 @ A) @ R)))))) @ R)) @ ((bNF_Wellorder_ordIso @ ((product_prod @ B) @ A)) @ A)))))))))).
% 9.43/3.46 thf(fact_5277_ex__toCard__pred, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R) => (? [X_12:(A > B)]: ((((((bNF_Gr1419584066657907630d_pred @ A) @ B) @ A4) @ R) @ X_12)))))))).
% 9.43/3.46 thf(fact_5278_cardSuc__UNION, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),As4:((set @ A) > (set @ B)),B6:(set @ B)]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((~ (((finite_finite2 @ A) @ ((field2 @ A) @ R)))) => (((((bNF_Ca3754400796208372196lChain @ (set @ A)) @ (set @ B)) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ As4) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image2 @ (set @ A)) @ (set @ B)) @ As4) @ ((field2 @ (set @ A)) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (? [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ ((field2 @ (set @ A)) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))) & (((ord_less_eq @ (set @ B)) @ B6) @ (As4 @ X2))))))))))))).
% 9.43/3.46 thf(fact_5279_toCard__inj, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B)),X:A,Y:A]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => (((((((bNF_Greatest_toCard @ A) @ B) @ A4) @ R) @ X) = (((((bNF_Greatest_toCard @ A) @ B) @ A4) @ R) @ Y)) = (X = Y))))))))).
% 9.43/3.46 thf(fact_5280_cardSuc__greater, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))) @ ((bNF_We4044943003108391690rdLess @ A) @ (set @ A))))))).
% 9.43/3.46 thf(fact_5281_cardSuc__ordLeq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))) @ ((bNF_Wellorder_ordLeq @ A) @ (set @ A))))))).
% 9.43/3.46 thf(fact_5282_toCard__pred__toCard, axiom, (! [A:$tType,B:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R) => (((((bNF_Gr1419584066657907630d_pred @ A) @ B) @ A4) @ R) @ ((((bNF_Greatest_toCard @ A) @ B) @ A4) @ R))))))).
% 9.43/3.46 thf(fact_5283_cardSuc__least__aux, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ (set @ A)) @ (set @ A)))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ (set @ A)) @ ((field2 @ (set @ A)) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ R4)) @ ((bNF_Wellorder_ordLeq @ (set @ A)) @ (set @ A))))))))).
% 9.43/3.46 thf(fact_5284_cardSuc__ordLeq__ordLess, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R4) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))) @ ((bNF_We4044943003108391690rdLess @ B) @ (set @ A))) = (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R4) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)))))))).
% 9.43/3.46 thf(fact_5285_cardSuc__ordLess__ordLeq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) = (((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ R4)) @ ((bNF_Wellorder_ordLeq @ (set @ A)) @ B)))))))).
% 9.43/3.46 thf(fact_5286_cardSuc__least, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_We4044943003108391690rdLess @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ R4)) @ ((bNF_Wellorder_ordLeq @ (set @ A)) @ B)))))))).
% 9.43/3.46 thf(fact_5287_cardSuc__mono__ordLeq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ B)) @ (set @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ B)) @ (set @ B)))) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ ((bNF_Ca8387033319878233205ardSuc @ B) @ R4))) @ ((bNF_Wellorder_ordLeq @ (set @ A)) @ (set @ B))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5288_cardSuc__invar__ordIso, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),R4:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R4)) @ R4) => ((((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ B)) @ (set @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ B)) @ (set @ B)))) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ ((bNF_Ca8387033319878233205ardSuc @ B) @ R4))) @ ((bNF_Wellorder_ordIso @ (set @ A)) @ (set @ B))) = (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ R4)) @ ((bNF_Wellorder_ordIso @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5289_fromCard__toCard, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B)),B2:A]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R) => ((((member @ A) @ B2) @ A4) => ((((((bNF_Gr5436034075474128252omCard @ A) @ B) @ A4) @ R) @ (((((bNF_Greatest_toCard @ A) @ B) @ A4) @ R) @ B2)) = B2))))))).
% 9.43/3.46 thf(fact_5290_cardSuc__UNION__Cinfinite, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),As4:((set @ A) > (set @ B)),B6:(set @ B)]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((((bNF_Ca3754400796208372196lChain @ (set @ A)) @ (set @ B)) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R)) @ As4) => ((((ord_less_eq @ (set @ B)) @ B6) @ ((complete_Sup_Sup @ (set @ B)) @ ((((image2 @ (set @ A)) @ (set @ B)) @ As4) @ ((field2 @ (set @ A)) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))))) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (? [X2:(set @ A)]: (((((member @ (set @ A)) @ X2) @ ((field2 @ (set @ A)) @ ((bNF_Ca8387033319878233205ardSuc @ A) @ R))) & (((ord_less_eq @ (set @ B)) @ B6) @ (As4 @ X2)))))))))))).
% 9.43/3.46 thf(fact_5291_isCardSuc__def, axiom, (! [A:$tType]: (((bNF_Ca6246979054910435723ardSuc @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),R11:(set @ ((product_prod @ (set @ A)) @ (set @ A)))]: (((((bNF_Ca8970107618336181345der_on @ (set @ A)) @ ((field2 @ (set @ A)) @ R11)) @ R11) & ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R3) @ R11)) @ ((bNF_We4044943003108391690rdLess @ A) @ (set @ A))) & (! [R14:(set @ ((product_prod @ (set @ A)) @ (set @ A)))]: ((((((bNF_Ca8970107618336181345der_on @ (set @ A)) @ ((field2 @ (set @ A)) @ R14)) @ R14) & (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R3) @ R14)) @ ((bNF_We4044943003108391690rdLess @ A) @ (set @ A)))) => (((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R11) @ R14)) @ ((bNF_Wellorder_ordLeq @ (set @ A)) @ (set @ A)))))))))))))).
% 9.43/3.46 thf(fact_5292_cinfinite__mono, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R1) @ R22)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((bNF_Ca4139267488887388095finite @ A) @ R1) => ((bNF_Ca4139267488887388095finite @ B) @ R22)))))).
% 9.43/3.46 thf(fact_5293_Cinfinite__limit2, axiom, (! [A:$tType,X1:A,R:(set @ ((product_prod @ A) @ A)),X22:A]: (((((member @ A) @ X1) @ ((field2 @ A) @ R)) => ((((member @ A) @ X22) @ ((field2 @ A) @ R)) => ((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (? [X2:A]: (((((member @ A) @ X2) @ ((field2 @ A) @ R)) & ((X1 != X2) & ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X1) @ X2)) @ R) & ((X22 != X2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X22) @ X2)) @ R))))))))))))).
% 9.43/3.46 thf(fact_5294_Cinfinite__limit, axiom, (! [A:$tType,X:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ X) @ ((field2 @ A) @ R)) => ((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (? [X2:A]: (((((member @ A) @ X2) @ ((field2 @ A) @ R)) & ((X != X2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ X2)) @ R)))))))))).
% 9.43/3.46 thf(fact_5295_Cinfinite__cong, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R1) @ R22)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((bNF_Ca4139267488887388095finite @ A) @ R1) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R1)) @ R1)) => (((bNF_Ca4139267488887388095finite @ B) @ R22) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R22)) @ R22))))))).
% 9.43/3.46 thf(fact_5296_Cinfinite__limit__finite, axiom, (! [A:$tType,X10:(set @ A),R:(set @ ((product_prod @ A) @ A))]: ((((finite_finite2 @ A) @ X10) => ((((ord_less_eq @ (set @ A)) @ X10) @ ((field2 @ A) @ R)) => ((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (? [X2:A]: (((((member @ A) @ X2) @ ((field2 @ A) @ R)) & (! [Xa2:A]: (((((member @ A) @ Xa2) @ X10) => ((Xa2 != X2) & (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Xa2) @ X2)) @ R)))))))))))))).
% 9.43/3.46 thf(fact_5297_Un__Cinfinite__bound, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ B) @ B)),B6:(set @ A)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca4139267488887388095finite @ B) @ R) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6))) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5298_UNION__Cinfinite__bound, axiom, (! [A:$tType,B:$tType,C:$tType,I4:(set @ A),R:(set @ ((product_prod @ B) @ B)),A4:(A > (set @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ I4)) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((! [X2:A]: (((((member @ A) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ X2))) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ B))))) => ((((bNF_Ca4139267488887388095finite @ B) @ R) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ A) @ (set @ C)) @ A4) @ I4)))) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ B)))))))).
% 9.43/3.46 thf(fact_5299_Cfinite__ordLess__Cinfinite, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((bNF_Cardinal_cfinite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => ((((bNF_Ca4139267488887388095finite @ B) @ S) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ S)) @ S)) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ S)) @ ((bNF_We4044943003108391690rdLess @ A) @ B))))))).
% 9.43/3.46 thf(fact_5300_comp__single__set__bd, axiom, (! [B:$tType,D:$tType,A:$tType,E:$tType,C:$tType,Fbd:(set @ ((product_prod @ A) @ A)),Fset:(B > (set @ C)),Gset:(D > (set @ B)),Gbd:(set @ ((product_prod @ E) @ E)),X:D]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ Fbd)) @ Fbd) => ((! [X2:B]: ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (Fset @ X2))) @ Fbd)) @ ((bNF_Wellorder_ordLeq @ C) @ A)))) => ((! [X2:D]: ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ E) @ E)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ E) @ E))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (Gset @ X2))) @ Gbd)) @ ((bNF_Wellorder_ordLeq @ B) @ E)))) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ ((product_prod @ E) @ A)) @ ((product_prod @ E) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ ((product_prod @ E) @ A)) @ ((product_prod @ E) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ B) @ (set @ C)) @ Fset) @ (Gset @ X))))) @ ((((bNF_Cardinal_cprod @ E) @ A) @ Gbd) @ Fbd))) @ ((bNF_Wellorder_ordLeq @ C) @ ((product_prod @ E) @ A))))))))).
% 9.43/3.46 thf(fact_5301_cprod__dup, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),P5:(set @ ((product_prod @ B) @ B)),P8:(set @ ((product_prod @ C) @ C))]: ((((bNF_Ca4139267488887388095finite @ A) @ R) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ ((((bNF_Cardinal_cprod @ B) @ C) @ P5) @ P8)) @ ((((bNF_Cardinal_cprod @ A) @ A) @ R) @ R))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ B) @ C)) @ ((product_prod @ A) @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_cprod @ B) @ C) @ P5) @ P8)) @ R)) @ ((bNF_Wellorder_ordIso @ ((product_prod @ B) @ C)) @ A)))))))).
% 9.43/3.46 thf(fact_5302_cprod__com, axiom, (! [B:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),P24:(set @ ((product_prod @ B) @ B))]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ ((((bNF_Cardinal_cprod @ A) @ B) @ P13) @ P24)) @ ((((bNF_Cardinal_cprod @ B) @ A) @ P24) @ P13))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ B)) @ ((product_prod @ B) @ A)))))).
% 9.43/3.46 thf(fact_5303_Cfinite__cprod__Cinfinite, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B))]: (((((bNF_Cardinal_cfinite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => ((((bNF_Ca4139267488887388095finite @ B) @ S) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ S)) @ S)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ B) @ B))) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R) @ S)) @ S)) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ B)) @ B))))))).
% 9.43/3.46 thf(fact_5304_cprod__mono, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ B) @ D)))) @ ((((bNF_Cardinal_cprod @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_cprod @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5305_cprod__mono1, axiom, (! [B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ ((((bNF_Cardinal_cprod @ A) @ C) @ P13) @ Q3)) @ ((((bNF_Cardinal_cprod @ B) @ C) @ R1) @ Q3))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ C))))))).
% 9.43/3.46 thf(fact_5306_cprod__mono2, axiom, (! [B:$tType,A:$tType,C:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B)))) @ ((((bNF_Cardinal_cprod @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_cprod @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ B))))))).
% 9.43/3.46 thf(fact_5307_cprod__cong, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordIso @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ D)) @ ((product_prod @ B) @ D)))) @ ((((bNF_Cardinal_cprod @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_cprod @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5308_cprod__cong1, axiom, (! [B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ ((((bNF_Cardinal_cprod @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_cprod @ B) @ C) @ R1) @ P24))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ C))))))).
% 9.43/3.46 thf(fact_5309_cprod__cong2, axiom, (! [B:$tType,A:$tType,C:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ A)))) @ (set @ ((product_prod @ ((product_prod @ C) @ B)) @ ((product_prod @ C) @ B)))) @ ((((bNF_Cardinal_cprod @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_cprod @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordIso @ ((product_prod @ C) @ A)) @ ((product_prod @ C) @ B))))))).
% 9.43/3.46 thf(fact_5310_cprod__infinite, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ A)) @ ((product_prod @ A) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_cprod @ A) @ A) @ R) @ R)) @ R)) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ A)) @ A)))))).
% 9.43/3.46 thf(fact_5311_cprod__cinfinite__bound, axiom, (! [B:$tType,C:$tType,A:$tType,P5:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P5) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B))) @ Q3) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ B)) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P5)) @ P5) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => ((((bNF_Ca4139267488887388095finite @ B) @ R) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ (set @ ((product_prod @ B) @ B))) @ ((((bNF_Cardinal_cprod @ A) @ C) @ P5) @ Q3)) @ R)) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ C)) @ B)))))))))).
% 9.43/3.46 thf(fact_5312_card__of__Csum__Times_H, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),I4:(set @ B),A4:(B > (set @ C))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((! [X2:B]: (((((member @ B) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ X2))) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ A))))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ C)))) @ (set @ ((product_prod @ ((product_prod @ B) @ A)) @ ((product_prod @ B) @ A)))) @ ((((bNF_Cardinal_Csum @ B) @ C) @ ((bNF_Ca6860139660246222851ard_of @ B) @ I4)) @ (^ [I3:B]: (((bNF_Ca6860139660246222851ard_of @ C) @ (A4 @ I3)))))) @ ((((bNF_Cardinal_cprod @ B) @ A) @ ((bNF_Ca6860139660246222851ard_of @ B) @ I4)) @ R))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ B) @ C)) @ ((product_prod @ B) @ A)))))))).
% 9.43/3.46 thf(fact_5313_card__of__Csum__Times, axiom, (! [C:$tType,B:$tType,A:$tType,I4:(set @ A),A4:(A > (set @ B)),B6:(set @ C)]: (((! [X2:A]: (((((member @ A) @ X2) @ I4) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ C) @ C))) @ ((bNF_Ca6860139660246222851ard_of @ B) @ (A4 @ X2))) @ ((bNF_Ca6860139660246222851ard_of @ C) @ B6))) @ ((bNF_Wellorder_ordLeq @ B) @ C))))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ((product_prod @ A) @ C)) @ ((product_prod @ A) @ C)))) @ ((((bNF_Cardinal_Csum @ A) @ B) @ ((bNF_Ca6860139660246222851ard_of @ A) @ I4)) @ (^ [I3:A]: (((bNF_Ca6860139660246222851ard_of @ B) @ (A4 @ I3)))))) @ ((((bNF_Cardinal_cprod @ A) @ C) @ ((bNF_Ca6860139660246222851ard_of @ A) @ I4)) @ ((bNF_Ca6860139660246222851ard_of @ C) @ B6)))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))))))).
% 9.43/3.46 thf(fact_5314_cprod__infinite1_H, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),P5:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((~ ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ P5) @ (bNF_Cardinal_czero @ B))) @ ((bNF_Wellorder_ordIso @ B) @ B)))) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ P5)) @ P5)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ P5) @ R)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R) @ P5)) @ R)) @ ((bNF_Wellorder_ordIso @ ((product_prod @ A) @ B)) @ A)))))))).
% 9.43/3.46 thf(fact_5315_czero__ordIso, axiom, (! [B:$tType,A:$tType]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ (bNF_Cardinal_czero @ A)) @ (bNF_Cardinal_czero @ B))) @ ((bNF_Wellorder_ordIso @ A) @ B))))).
% 9.43/3.46 thf(fact_5316_cinfinite__not__czero, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ B) @ B))]: ((((bNF_Ca4139267488887388095finite @ B) @ R) => (~ ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ B) @ A)))))))).
% 9.43/3.46 thf(fact_5317_czeroE, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ (bNF_Cardinal_czero @ B))) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((field2 @ A) @ R) = (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_5318_card__of__ordIso__czero__iff__empty, axiom, (! [B:$tType,A:$tType,A4:(set @ A)]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ (bNF_Cardinal_czero @ B))) @ ((bNF_Wellorder_ordIso @ A) @ B)) = (A4 = (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_5319_czeroI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((field2 @ A) @ R) = (bot_bot @ (set @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ (bNF_Cardinal_czero @ B))) @ ((bNF_Wellorder_ordIso @ A) @ B))))))).
% 9.43/3.46 thf(fact_5320_Cnotzero__imp__not__empty, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((field2 @ A) @ R) != (bot_bot @ (set @ A))))))).
% 9.43/3.46 thf(fact_5321_Cnotzero__mono, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),Q3:(set @ ((product_prod @ B) @ B))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ Q3)) @ Q3) => ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ R) @ Q3)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ B) @ B))) @ Q3) @ (bNF_Cardinal_czero @ B))) @ ((bNF_Wellorder_ordIso @ B) @ B)))) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ Q3)) @ Q3)))))))).
% 9.43/3.46 thf(fact_5322_Cinfinite__Cnotzero, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)))))).
% 9.43/3.46 thf(fact_5323_Cnotzero__UNIV, axiom, (! [A:$tType]: (((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (top_top @ (set @ A)))) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (top_top @ (set @ A))))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (top_top @ (set @ A)))))))).
% 9.43/3.46 thf(fact_5324_cinfinite__cprod2, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R1) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R1)) @ R1)) => ((((bNF_Ca4139267488887388095finite @ B) @ R22) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R22)) @ R22)) => ((bNF_Ca4139267488887388095finite @ ((product_prod @ A) @ B)) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R1) @ R22))))))).
% 9.43/3.46 thf(fact_5325_Cinfinite__cprod2, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R1) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R1)) @ R1)) => ((((bNF_Ca4139267488887388095finite @ B) @ R22) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R22)) @ R22)) => (((bNF_Ca4139267488887388095finite @ ((product_prod @ A) @ B)) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R1) @ R22)) & (((bNF_Ca8970107618336181345der_on @ ((product_prod @ A) @ B)) @ ((field2 @ ((product_prod @ A) @ B)) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R1) @ R22))) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R1) @ R22)))))))).
% 9.43/3.46 thf(fact_5326_ordLeq__cprod2, axiom, (! [A:$tType,B:$tType,P13:(set @ ((product_prod @ A) @ A)),P24:(set @ ((product_prod @ B) @ B))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P13) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P13)) @ P13)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ P24)) @ P24) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ P24) @ ((((bNF_Cardinal_cprod @ A) @ B) @ P13) @ P24))) @ ((bNF_Wellorder_ordLeq @ B) @ ((product_prod @ A) @ B)))))))).
% 9.43/3.46 thf(fact_5327_cone__ordLeq__Cnotzero, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ product_unit) @ product_unit))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ product_unit) @ product_unit))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_cone) @ R)) @ ((bNF_Wellorder_ordLeq @ product_unit) @ A)))))).
% 9.43/3.46 thf(fact_5328_csum__dup, axiom, (! [A:$tType,C:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),P5:(set @ ((product_prod @ B) @ B)),P8:(set @ ((product_prod @ C) @ C))]: ((((bNF_Ca4139267488887388095finite @ A) @ R) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A)))) @ ((((bNF_Cardinal_csum @ B) @ C) @ P5) @ P8)) @ ((((bNF_Cardinal_csum @ A) @ A) @ R) @ R))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ B) @ C)) @ ((sum_sum @ A) @ A))) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_csum @ B) @ C) @ P5) @ P8)) @ R)) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ B) @ C)) @ A)))))))).
% 9.43/3.46 thf(fact_5329_csum__Cnotzero1, axiom, (! [A:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R1) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R1)) @ R1)) => ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ ((((bNF_Cardinal_csum @ A) @ B) @ R1) @ R22)) @ (bNF_Cardinal_czero @ ((sum_sum @ A) @ B)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) & (((bNF_Ca8970107618336181345der_on @ ((sum_sum @ A) @ B)) @ ((field2 @ ((sum_sum @ A) @ B)) @ ((((bNF_Cardinal_csum @ A) @ B) @ R1) @ R22))) @ ((((bNF_Cardinal_csum @ A) @ B) @ R1) @ R22))))))).
% 9.43/3.46 thf(fact_5330_csum__csum, axiom, (! [D:$tType,C:$tType,B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),R33:(set @ ((product_prod @ C) @ C)),R42:(set @ ((product_prod @ D) @ D))]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ B)) @ ((sum_sum @ C) @ D))) @ ((sum_sum @ ((sum_sum @ A) @ B)) @ ((sum_sum @ C) @ D))))) @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D))) @ ((sum_sum @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ B)) @ ((sum_sum @ C) @ D))) @ ((sum_sum @ ((sum_sum @ A) @ B)) @ ((sum_sum @ C) @ D))))) @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D))) @ ((sum_sum @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D))))) @ ((((bNF_Cardinal_csum @ ((sum_sum @ A) @ B)) @ ((sum_sum @ C) @ D)) @ ((((bNF_Cardinal_csum @ A) @ B) @ R1) @ R22)) @ ((((bNF_Cardinal_csum @ C) @ D) @ R33) @ R42))) @ ((((bNF_Cardinal_csum @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)) @ ((((bNF_Cardinal_csum @ A) @ C) @ R1) @ R33)) @ ((((bNF_Cardinal_csum @ B) @ D) @ R22) @ R42)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ ((sum_sum @ A) @ B)) @ ((sum_sum @ C) @ D))) @ ((sum_sum @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D))))))).
% 9.43/3.46 thf(fact_5331_csum__assoc, axiom, (! [C:$tType,B:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),P24:(set @ ((product_prod @ B) @ B)),P32:(set @ ((product_prod @ C) @ C))]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) @ ((sum_sum @ A) @ ((sum_sum @ B) @ C)))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))) @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))) @ ((((bNF_Cardinal_csum @ ((sum_sum @ A) @ B)) @ C) @ ((((bNF_Cardinal_csum @ A) @ B) @ P13) @ P24)) @ P32)) @ ((((bNF_Cardinal_csum @ A) @ ((sum_sum @ B) @ C)) @ P13) @ ((((bNF_Cardinal_csum @ B) @ C) @ P24) @ P32)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ ((sum_sum @ A) @ B)) @ C)) @ ((sum_sum @ A) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5332_cprod__csum__distrib1, axiom, (! [C:$tType,B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),R33:(set @ ((product_prod @ C) @ C))]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))) @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))))) @ (set @ ((product_prod @ ((product_prod @ A) @ ((sum_sum @ B) @ C))) @ ((product_prod @ A) @ ((sum_sum @ B) @ C)))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))) @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))))) @ (set @ ((product_prod @ ((product_prod @ A) @ ((sum_sum @ B) @ C))) @ ((product_prod @ A) @ ((sum_sum @ B) @ C))))) @ ((((bNF_Cardinal_csum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C)) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R1) @ R22)) @ ((((bNF_Cardinal_cprod @ A) @ C) @ R1) @ R33))) @ ((((bNF_Cardinal_cprod @ A) @ ((sum_sum @ B) @ C)) @ R1) @ ((((bNF_Cardinal_csum @ B) @ C) @ R22) @ R33)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ C))) @ ((product_prod @ A) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5333_cone__not__czero, axiom, (! [A:$tType]: ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ product_unit) @ product_unit))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ product_unit) @ product_unit))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_cone) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ product_unit) @ A))))))).
% 9.43/3.46 thf(fact_5334_csum__com, axiom, (! [B:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),P24:(set @ ((product_prod @ B) @ B))]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ ((((bNF_Cardinal_csum @ A) @ B) @ P13) @ P24)) @ ((((bNF_Cardinal_csum @ B) @ A) @ P24) @ P13))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ ((sum_sum @ B) @ A)))))).
% 9.43/3.46 thf(fact_5335_Un__csum, axiom, (! [A:$tType,A4:(set @ A),B6:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ A)) @ ((sum_sum @ A) @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ (((sup_sup @ (set @ A)) @ A4) @ B6))) @ ((((bNF_Cardinal_csum @ A) @ A) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ A) @ B6)))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ A) @ A)))))).
% 9.43/3.46 thf(fact_5336_ordLeq__csum1, axiom, (! [B:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),P24:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P13)) @ P13) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ P13) @ ((((bNF_Cardinal_csum @ A) @ B) @ P13) @ P24))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ A) @ B))))))).
% 9.43/3.46 thf(fact_5337_ordLeq__csum2, axiom, (! [B:$tType,A:$tType,P24:(set @ ((product_prod @ A) @ A)),P13:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P24)) @ P24) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ P24) @ ((((bNF_Cardinal_csum @ B) @ A) @ P13) @ P24))) @ ((bNF_Wellorder_ordLeq @ A) @ ((sum_sum @ B) @ A))))))).
% 9.43/3.46 thf(fact_5338_csum__cong, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordIso @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D)))) @ ((((bNF_Cardinal_csum @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_csum @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5339_csum__cong1, axiom, (! [B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ ((((bNF_Cardinal_csum @ A) @ C) @ P13) @ Q3)) @ ((((bNF_Cardinal_csum @ B) @ C) @ R1) @ Q3))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5340_csum__cong2, axiom, (! [B:$tType,A:$tType,C:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B)))) @ ((((bNF_Cardinal_csum @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_csum @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ B))))))).
% 9.43/3.46 thf(fact_5341_csum__mono, axiom, (! [D:$tType,B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ C) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ D)) @ ((sum_sum @ B) @ D)))) @ ((((bNF_Cardinal_csum @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_csum @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ D)))))))).
% 9.43/3.46 thf(fact_5342_csum__mono1, axiom, (! [B:$tType,C:$tType,A:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ ((sum_sum @ B) @ C)) @ ((sum_sum @ B) @ C)))) @ ((((bNF_Cardinal_csum @ A) @ C) @ P13) @ Q3)) @ ((((bNF_Cardinal_csum @ B) @ C) @ R1) @ Q3))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ ((sum_sum @ B) @ C))))))).
% 9.43/3.46 thf(fact_5343_csum__mono2, axiom, (! [B:$tType,A:$tType,C:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ A)))) @ (set @ ((product_prod @ ((sum_sum @ C) @ B)) @ ((sum_sum @ C) @ B)))) @ ((((bNF_Cardinal_csum @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_csum @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ C) @ A)) @ ((sum_sum @ C) @ B))))))).
% 9.43/3.46 thf(fact_5344_Plus__csum, axiom, (! [B:$tType,A:$tType,A4:(set @ A),B6:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B))))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ ((bNF_Ca6860139660246222851ard_of @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A4) @ B6))) @ ((((bNF_Cardinal_csum @ A) @ B) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ B) @ B6)))) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))))).
% 9.43/3.46 thf(fact_5345_csum__absorb2_H, axiom, (! [A:$tType,B:$tType,R22:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R22)) @ R22) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R1) @ R22)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((bNF_Ca4139267488887388095finite @ B) @ R1) | ((bNF_Ca4139267488887388095finite @ A) @ R22)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ B) @ A)) @ ((sum_sum @ B) @ A)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_csum @ B) @ A) @ R1) @ R22)) @ R22)) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ B) @ A)) @ A)))))))).
% 9.43/3.46 thf(fact_5346_csum__absorb1_H, axiom, (! [B:$tType,A:$tType,R22:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R22)) @ R22) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R1) @ R22)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => ((((bNF_Ca4139267488887388095finite @ B) @ R1) | ((bNF_Ca4139267488887388095finite @ A) @ R22)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_csum @ A) @ B) @ R22) @ R1)) @ R22)) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ A)))))))).
% 9.43/3.46 thf(fact_5347_csum__absorb1, axiom, (! [B:$tType,A:$tType,R22:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca4139267488887388095finite @ A) @ R22) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R22)) @ R22)) => ((((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ A) @ A))) @ R1) @ R22)) @ ((bNF_Wellorder_ordLeq @ B) @ A)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ B)) @ ((sum_sum @ A) @ B)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_csum @ A) @ B) @ R22) @ R1)) @ R22)) @ ((bNF_Wellorder_ordIso @ ((sum_sum @ A) @ B)) @ A))))))).
% 9.43/3.46 thf(fact_5348_csum__cinfinite__bound, axiom, (! [B:$tType,C:$tType,A:$tType,P5:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P5) @ R)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B))) @ Q3) @ R)) @ ((bNF_Wellorder_ordLeq @ C) @ B)) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P5)) @ P5) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => ((((bNF_Ca4139267488887388095finite @ B) @ R) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ ((sum_sum @ A) @ C)) @ ((sum_sum @ A) @ C)))) @ (set @ ((product_prod @ B) @ B))) @ ((((bNF_Cardinal_csum @ A) @ C) @ P5) @ Q3)) @ R)) @ ((bNF_Wellorder_ordLeq @ ((sum_sum @ A) @ C)) @ B)))))))))).
% 9.43/3.46 thf(fact_5349_natLeq__ordLeq__cinfinite, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Ca8665028551170535155natLeq) @ R)) @ ((bNF_Wellorder_ordLeq @ nat) @ A)))))).
% 9.43/3.46 thf(fact_5350_cexp__mono2__Cnotzero, axiom, (! [B:$tType,C:$tType,A:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => (((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ P24) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P24)) @ P24)) => (((member @ ((product_prod @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C)))) @ ((((bNF_Cardinal_cexp @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_cexp @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordLeq @ (A > C)) @ (B > C))))))))).
% 9.43/3.46 thf(fact_5351_cexp__mono2, axiom, (! [D:$tType,E:$tType,B:$tType,C:$tType,A:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ (bNF_Cardinal_czero @ D))) @ ((bNF_Wellorder_ordIso @ A) @ D)) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ E) @ E)))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ E) @ E))) @ R22) @ (bNF_Cardinal_czero @ E))) @ ((bNF_Wellorder_ordIso @ B) @ E))) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P24)) @ P24) => (((member @ ((product_prod @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C)))) @ ((((bNF_Cardinal_cexp @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_cexp @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordLeq @ (A > C)) @ (B > C)))))))))).
% 9.43/3.46 thf(fact_5352_cprod__cexp, axiom, (! [C:$tType,B:$tType,A:$tType,R:(set @ ((product_prod @ B) @ B)),S:(set @ ((product_prod @ C) @ C)),T2:(set @ ((product_prod @ A) @ A))]: ((((member @ ((product_prod @ (set @ ((product_prod @ (A > ((product_prod @ B) @ C))) @ (A > ((product_prod @ B) @ C))))) @ (set @ ((product_prod @ ((product_prod @ (A > B)) @ (A > C))) @ ((product_prod @ (A > B)) @ (A > C)))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > ((product_prod @ B) @ C))) @ (A > ((product_prod @ B) @ C))))) @ (set @ ((product_prod @ ((product_prod @ (A > B)) @ (A > C))) @ ((product_prod @ (A > B)) @ (A > C))))) @ ((((bNF_Cardinal_cexp @ ((product_prod @ B) @ C)) @ A) @ ((((bNF_Cardinal_cprod @ B) @ C) @ R) @ S)) @ T2)) @ ((((bNF_Cardinal_cprod @ (A > B)) @ (A > C)) @ ((((bNF_Cardinal_cexp @ B) @ A) @ R) @ T2)) @ ((((bNF_Cardinal_cexp @ C) @ A) @ S) @ T2)))) @ ((bNF_Wellorder_ordIso @ (A > ((product_prod @ B) @ C))) @ ((product_prod @ (A > B)) @ (A > C))))))).
% 9.43/3.46 thf(fact_5353_cexp__cprod, axiom, (! [A:$tType,C:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ C) @ C)),R33:(set @ ((product_prod @ B) @ B))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R1)) @ R1) => (((member @ ((product_prod @ (set @ ((product_prod @ (B > (C > A))) @ (B > (C > A))))) @ (set @ ((product_prod @ (((product_prod @ C) @ B) > A)) @ (((product_prod @ C) @ B) > A))))) @ ((((product_Pair @ (set @ ((product_prod @ (B > (C > A))) @ (B > (C > A))))) @ (set @ ((product_prod @ (((product_prod @ C) @ B) > A)) @ (((product_prod @ C) @ B) > A)))) @ ((((bNF_Cardinal_cexp @ (C > A)) @ B) @ ((((bNF_Cardinal_cexp @ A) @ C) @ R1) @ R22)) @ R33)) @ ((((bNF_Cardinal_cexp @ A) @ ((product_prod @ C) @ B)) @ R1) @ ((((bNF_Cardinal_cprod @ C) @ B) @ R22) @ R33)))) @ ((bNF_Wellorder_ordIso @ (B > (C > A))) @ (((product_prod @ C) @ B) > A))))))).
% 9.43/3.46 thf(fact_5354_cprod__cexp__csum__cexp__Cinfinite, axiom, (! [C:$tType,B:$tType,A:$tType,T2:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B)),S:(set @ ((product_prod @ C) @ C))]: (((((bNF_Ca4139267488887388095finite @ A) @ T2) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ T2)) @ T2)) => (((member @ ((product_prod @ (set @ ((product_prod @ (A > ((product_prod @ B) @ C))) @ (A > ((product_prod @ B) @ C))))) @ (set @ ((product_prod @ (A > ((sum_sum @ B) @ C))) @ (A > ((sum_sum @ B) @ C)))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > ((product_prod @ B) @ C))) @ (A > ((product_prod @ B) @ C))))) @ (set @ ((product_prod @ (A > ((sum_sum @ B) @ C))) @ (A > ((sum_sum @ B) @ C))))) @ ((((bNF_Cardinal_cexp @ ((product_prod @ B) @ C)) @ A) @ ((((bNF_Cardinal_cprod @ B) @ C) @ R) @ S)) @ T2)) @ ((((bNF_Cardinal_cexp @ ((sum_sum @ B) @ C)) @ A) @ ((((bNF_Cardinal_csum @ B) @ C) @ R) @ S)) @ T2))) @ ((bNF_Wellorder_ordLeq @ (A > ((product_prod @ B) @ C))) @ (A > ((sum_sum @ B) @ C)))))))).
% 9.43/3.46 thf(fact_5355_card__of__Field__natLeq, axiom, (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((bNF_Ca6860139660246222851ard_of @ nat) @ ((field2 @ nat) @ bNF_Ca8665028551170535155natLeq))) @ bNF_Ca8665028551170535155natLeq)) @ ((bNF_Wellorder_ordIso @ nat) @ nat))).
% 9.43/3.46 thf(fact_5356_cexp__cone, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ (product_unit > A)) @ (product_unit > A)))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ (product_unit > A)) @ (product_unit > A)))) @ (set @ ((product_prod @ A) @ A))) @ ((((bNF_Cardinal_cexp @ A) @ product_unit) @ R) @ bNF_Cardinal_cone)) @ R)) @ ((bNF_Wellorder_ordIso @ (product_unit > A)) @ A)))))).
% 9.43/3.46 thf(fact_5357_finite__iff__ordLess__natLeq, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A9:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ nat) @ nat))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A9)) @ bNF_Ca8665028551170535155natLeq)) @ ((bNF_We4044943003108391690rdLess @ A) @ nat)))))))).
% 9.43/3.46 thf(fact_5358_cexp__cprod__ordLeq, axiom, (! [A:$tType,B:$tType,C:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),R33:(set @ ((product_prod @ C) @ C))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R1)) @ R1) => ((((bNF_Ca4139267488887388095finite @ B) @ R22) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R22)) @ R22)) => (((~ ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ C) @ C)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ C) @ C))) @ R33) @ (bNF_Cardinal_czero @ C))) @ ((bNF_Wellorder_ordIso @ C) @ C)))) & (((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ R33)) @ R33)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ B) @ B))) @ R33) @ R22)) @ ((bNF_Wellorder_ordLeq @ C) @ B)) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > (B > A))) @ (C > (B > A))))) @ (set @ ((product_prod @ (B > A)) @ (B > A))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > (B > A))) @ (C > (B > A))))) @ (set @ ((product_prod @ (B > A)) @ (B > A)))) @ ((((bNF_Cardinal_cexp @ (B > A)) @ C) @ ((((bNF_Cardinal_cexp @ A) @ B) @ R1) @ R22)) @ R33)) @ ((((bNF_Cardinal_cexp @ A) @ B) @ R1) @ R22))) @ ((bNF_Wellorder_ordIso @ (C > (B > A))) @ (B > A)))))))))).
% 9.43/3.46 thf(fact_5359_cexp__mono_H, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ C) @ D)) => (((((field2 @ C) @ P24) = (bot_bot @ (set @ C))) => (((field2 @ D) @ R22) = (bot_bot @ (set @ D)))) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (D > B)) @ (D > B))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (D > B)) @ (D > B)))) @ ((((bNF_Cardinal_cexp @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_cexp @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordLeq @ (C > A)) @ (D > B))))))))).
% 9.43/3.46 thf(fact_5360_cexp__mono1, axiom, (! [B:$tType,A:$tType,C:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (C > B)) @ (C > B))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (C > B)) @ (C > B)))) @ ((((bNF_Cardinal_cexp @ A) @ C) @ P13) @ Q3)) @ ((((bNF_Cardinal_cexp @ B) @ C) @ R1) @ Q3))) @ ((bNF_Wellorder_ordLeq @ (C > A)) @ (C > B)))))))).
% 9.43/3.46 thf(fact_5361_cexp__cong2, axiom, (! [B:$tType,C:$tType,A:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => ((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ P24)) @ P24) => (((member @ ((product_prod @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C)))) @ ((((bNF_Cardinal_cexp @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_cexp @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordIso @ (A > C)) @ (B > C))))))))).
% 9.43/3.46 thf(fact_5362_cexp__cong1, axiom, (! [B:$tType,A:$tType,C:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (C > B)) @ (C > B))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (C > B)) @ (C > B)))) @ ((((bNF_Cardinal_cexp @ A) @ C) @ P13) @ Q3)) @ ((((bNF_Cardinal_cexp @ B) @ C) @ R1) @ Q3))) @ ((bNF_Wellorder_ordIso @ (C > A)) @ (C > B)))))))).
% 9.43/3.46 thf(fact_5363_cexp__cong, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordIso @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordIso @ C) @ D)) => ((((bNF_Ca8970107618336181345der_on @ D) @ ((field2 @ D) @ R22)) @ R22) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ P24)) @ P24) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (D > B)) @ (D > B))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (D > B)) @ (D > B)))) @ ((((bNF_Cardinal_cexp @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_cexp @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordIso @ (C > A)) @ (D > B)))))))))).
% 9.43/3.46 thf(fact_5364_card__of__nat, axiom, (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ nat) @ nat))) @ ((bNF_Ca6860139660246222851ard_of @ nat) @ (top_top @ (set @ nat)))) @ bNF_Ca8665028551170535155natLeq)) @ ((bNF_Wellorder_ordIso @ nat) @ nat))).
% 9.43/3.46 thf(fact_5365_cexp__mono2_H, axiom, (! [B:$tType,C:$tType,A:$tType,P24:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),Q3:(set @ ((product_prod @ C) @ C))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ Q3)) @ Q3) => (((((field2 @ A) @ P24) = (bot_bot @ (set @ A))) => (((field2 @ B) @ R22) = (bot_bot @ (set @ B)))) => (((member @ ((product_prod @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > C)) @ (A > C)))) @ (set @ ((product_prod @ (B > C)) @ (B > C)))) @ ((((bNF_Cardinal_cexp @ C) @ A) @ Q3) @ P24)) @ ((((bNF_Cardinal_cexp @ C) @ B) @ Q3) @ R22))) @ ((bNF_Wellorder_ordLeq @ (A > C)) @ (B > C))))))))).
% 9.43/3.46 thf(fact_5366_ordLeq__cexp1, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ A)),Q3:(set @ ((product_prod @ B) @ B))]: ((((~ ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ A) @ A))) @ R) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ A) @ A)))) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ Q3)) @ Q3) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ (A > B)) @ (A > B))))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ (A > B)) @ (A > B)))) @ Q3) @ ((((bNF_Cardinal_cexp @ B) @ A) @ Q3) @ R))) @ ((bNF_Wellorder_ordLeq @ B) @ (A > B)))))))).
% 9.43/3.46 thf(fact_5367_infinite__iff__natLeq__ordLeq, axiom, (! [A:$tType,A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) != (((member @ ((product_prod @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ nat) @ nat))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Ca8665028551170535155natLeq) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4))) @ ((bNF_Wellorder_ordLeq @ nat) @ A)))))).
% 9.43/3.46 thf(fact_5368_cexp__mono, axiom, (! [E:$tType,F5:$tType,B:$tType,D:$tType,A:$tType,C:$tType,P13:(set @ ((product_prod @ A) @ A)),R1:(set @ ((product_prod @ B) @ B)),P24:(set @ ((product_prod @ C) @ C)),R22:(set @ ((product_prod @ D) @ D))]: (((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B)))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ B) @ B))) @ P13) @ R1)) @ ((bNF_Wellorder_ordLeq @ A) @ B)) => ((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ D) @ D))) @ P24) @ R22)) @ ((bNF_Wellorder_ordLeq @ C) @ D)) => (((((member @ ((product_prod @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ E) @ E)))) @ ((((product_Pair @ (set @ ((product_prod @ C) @ C))) @ (set @ ((product_prod @ E) @ E))) @ P24) @ (bNF_Cardinal_czero @ E))) @ ((bNF_Wellorder_ordIso @ C) @ E)) => (((member @ ((product_prod @ (set @ ((product_prod @ D) @ D))) @ (set @ ((product_prod @ F5) @ F5)))) @ ((((product_Pair @ (set @ ((product_prod @ D) @ D))) @ (set @ ((product_prod @ F5) @ F5))) @ R22) @ (bNF_Cardinal_czero @ F5))) @ ((bNF_Wellorder_ordIso @ D) @ F5))) => ((((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ P24)) @ P24) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (D > B)) @ (D > B))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > A)) @ (C > A)))) @ (set @ ((product_prod @ (D > B)) @ (D > B)))) @ ((((bNF_Cardinal_cexp @ A) @ C) @ P13) @ P24)) @ ((((bNF_Cardinal_cexp @ B) @ D) @ R1) @ R22))) @ ((bNF_Wellorder_ordLeq @ (C > A)) @ (D > B)))))))))).
% 9.43/3.46 thf(fact_5369_ordLeq__cexp2, axiom, (! [A:$tType,B:$tType,Q3:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_ctwo) @ Q3)) @ ((bNF_Wellorder_ordLeq @ $o) @ A)) => ((((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ (B > A)) @ (B > A))))) @ ((((product_Pair @ (set @ ((product_prod @ B) @ B))) @ (set @ ((product_prod @ (B > A)) @ (B > A)))) @ R) @ ((((bNF_Cardinal_cexp @ A) @ B) @ Q3) @ R))) @ ((bNF_Wellorder_ordLeq @ B) @ (B > A)))))))).
% 9.43/3.46 thf(fact_5370_Cfinite__cexp__Cinfinite, axiom, (! [A:$tType,B:$tType,S:(set @ ((product_prod @ A) @ A)),T2:(set @ ((product_prod @ B) @ B))]: (((((bNF_Cardinal_cfinite @ A) @ S) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ S)) @ S)) => ((((bNF_Ca4139267488887388095finite @ B) @ T2) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ T2)) @ T2)) => (((member @ ((product_prod @ (set @ ((product_prod @ (B > A)) @ (B > A)))) @ (set @ ((product_prod @ (B > $o)) @ (B > $o))))) @ ((((product_Pair @ (set @ ((product_prod @ (B > A)) @ (B > A)))) @ (set @ ((product_prod @ (B > $o)) @ (B > $o)))) @ ((((bNF_Cardinal_cexp @ A) @ B) @ S) @ T2)) @ ((((bNF_Cardinal_cexp @ $o) @ B) @ bNF_Cardinal_ctwo) @ T2))) @ ((bNF_Wellorder_ordLeq @ (B > A)) @ (B > $o)))))))).
% 9.43/3.46 thf(fact_5371_cinfinite__cexp, axiom, (! [A:$tType,B:$tType,Q3:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_ctwo) @ Q3)) @ ((bNF_Wellorder_ordLeq @ $o) @ A)) => ((((bNF_Ca4139267488887388095finite @ B) @ R) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R)) => ((bNF_Ca4139267488887388095finite @ (B > A)) @ ((((bNF_Cardinal_cexp @ A) @ B) @ Q3) @ R))))))).
% 9.43/3.46 thf(fact_5372_ctwo__Cnotzero, axiom, ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ $o) @ $o)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ $o) @ $o))) @ bNF_Cardinal_ctwo) @ (bNF_Cardinal_czero @ $o))) @ ((bNF_Wellorder_ordIso @ $o) @ $o)))) & (((bNF_Ca8970107618336181345der_on @ $o) @ ((field2 @ $o) @ bNF_Cardinal_ctwo)) @ bNF_Cardinal_ctwo))).
% 9.43/3.46 thf(fact_5373_ctwo__ordLess__natLeq, axiom, (((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ nat) @ nat)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ nat) @ nat))) @ bNF_Cardinal_ctwo) @ bNF_Ca8665028551170535155natLeq)) @ ((bNF_We4044943003108391690rdLess @ $o) @ nat))).
% 9.43/3.46 thf(fact_5374_ordLess__ctwo__cexp, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (A > $o)) @ (A > $o))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (A > $o)) @ (A > $o)))) @ R) @ ((((bNF_Cardinal_cexp @ $o) @ A) @ bNF_Cardinal_ctwo) @ R))) @ ((bNF_We4044943003108391690rdLess @ A) @ (A > $o))))))).
% 9.43/3.46 thf(fact_5375_cprod__csum__cexp, axiom, (! [B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B))]: ((((member @ ((product_prod @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ($o > ((sum_sum @ A) @ B))) @ ($o > ((sum_sum @ A) @ B)))))) @ ((((product_Pair @ (set @ ((product_prod @ ((product_prod @ A) @ B)) @ ((product_prod @ A) @ B)))) @ (set @ ((product_prod @ ($o > ((sum_sum @ A) @ B))) @ ($o > ((sum_sum @ A) @ B))))) @ ((((bNF_Cardinal_cprod @ A) @ B) @ R1) @ R22)) @ ((((bNF_Cardinal_cexp @ ((sum_sum @ A) @ B)) @ $o) @ ((((bNF_Cardinal_csum @ A) @ B) @ R1) @ R22)) @ bNF_Cardinal_ctwo))) @ ((bNF_Wellorder_ordLeq @ ((product_prod @ A) @ B)) @ ($o > ((sum_sum @ A) @ B))))))).
% 9.43/3.46 thf(fact_5376_ctwo__not__czero, axiom, (! [A:$tType]: ((~ ((((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_ctwo) @ (bNF_Cardinal_czero @ A))) @ ((bNF_Wellorder_ordIso @ $o) @ A))))))).
% 9.43/3.46 thf(fact_5377_ctwo__ordLeq__Cinfinite, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_ctwo) @ R)) @ ((bNF_Wellorder_ordLeq @ $o) @ A)))))).
% 9.43/3.46 thf(fact_5378_ctwo__ordLess__Cinfinite, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca4139267488887388095finite @ A) @ R) & (((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R)) => (((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_ctwo) @ R)) @ ((bNF_We4044943003108391690rdLess @ $o) @ A)))))).
% 9.43/3.46 thf(fact_5379_csum__Cfinite__cexp__Cinfinite, axiom, (! [B:$tType,A:$tType,C:$tType,R:(set @ ((product_prod @ A) @ A)),S:(set @ ((product_prod @ B) @ B)),T2:(set @ ((product_prod @ C) @ C))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => ((((bNF_Cardinal_cfinite @ B) @ S) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ S)) @ S)) => ((((bNF_Ca4139267488887388095finite @ C) @ T2) & (((bNF_Ca8970107618336181345der_on @ C) @ ((field2 @ C) @ T2)) @ T2)) => (((member @ ((product_prod @ (set @ ((product_prod @ (C > ((sum_sum @ A) @ B))) @ (C > ((sum_sum @ A) @ B))))) @ (set @ ((product_prod @ (C > ((sum_sum @ A) @ $o))) @ (C > ((sum_sum @ A) @ $o)))))) @ ((((product_Pair @ (set @ ((product_prod @ (C > ((sum_sum @ A) @ B))) @ (C > ((sum_sum @ A) @ B))))) @ (set @ ((product_prod @ (C > ((sum_sum @ A) @ $o))) @ (C > ((sum_sum @ A) @ $o))))) @ ((((bNF_Cardinal_cexp @ ((sum_sum @ A) @ B)) @ C) @ ((((bNF_Cardinal_csum @ A) @ B) @ R) @ S)) @ T2)) @ ((((bNF_Cardinal_cexp @ ((sum_sum @ A) @ $o)) @ C) @ ((((bNF_Cardinal_csum @ A) @ $o) @ R) @ bNF_Cardinal_ctwo)) @ T2))) @ ((bNF_Wellorder_ordLeq @ (C > ((sum_sum @ A) @ B))) @ (C > ((sum_sum @ A) @ $o)))))))))).
% 9.43/3.46 thf(fact_5380_Cinfinite__cexp, axiom, (! [A:$tType,B:$tType,Q3:(set @ ((product_prod @ A) @ A)),R:(set @ ((product_prod @ B) @ B))]: (((((member @ ((product_prod @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A)))) @ ((((product_Pair @ (set @ ((product_prod @ $o) @ $o))) @ (set @ ((product_prod @ A) @ A))) @ bNF_Cardinal_ctwo) @ Q3)) @ ((bNF_Wellorder_ordLeq @ $o) @ A)) => ((((bNF_Ca4139267488887388095finite @ B) @ R) & (((bNF_Ca8970107618336181345der_on @ B) @ ((field2 @ B) @ R)) @ R)) => (((bNF_Ca4139267488887388095finite @ (B > A)) @ ((((bNF_Cardinal_cexp @ A) @ B) @ Q3) @ R)) & (((bNF_Ca8970107618336181345der_on @ (B > A)) @ ((field2 @ (B > A)) @ ((((bNF_Cardinal_cexp @ A) @ B) @ Q3) @ R))) @ ((((bNF_Cardinal_cexp @ A) @ B) @ Q3) @ R)))))))).
% 9.43/3.46 thf(fact_5381_card__of__Func__UNIV, axiom, (! [B:$tType,A:$tType,B6:(set @ B)]: ((((member @ ((product_prod @ (set @ ((product_prod @ (A > B)) @ (A > B)))) @ (set @ ((product_prod @ (A > B)) @ (A > B))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > B)) @ (A > B)))) @ (set @ ((product_prod @ (A > B)) @ (A > B)))) @ ((bNF_Ca6860139660246222851ard_of @ (A > B)) @ ((((bNF_Wellorder_Func @ A) @ B) @ (top_top @ (set @ A))) @ B6))) @ ((bNF_Ca6860139660246222851ard_of @ (A > B)) @ ((collect @ (A > B)) @ (^ [F2:(A > B)]: ((((ord_less_eq @ (set @ B)) @ ((((image2 @ A) @ B) @ F2) @ (top_top @ (set @ A)))) @ B6))))))) @ ((bNF_Wellorder_ordIso @ (A > B)) @ (A > B)))))).
% 9.43/3.46 thf(fact_5382_Card__order__Pow, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: (((((bNF_Ca8970107618336181345der_on @ A) @ ((field2 @ A) @ R)) @ R) => (((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ R) @ ((bNF_Ca6860139660246222851ard_of @ (set @ A)) @ ((pow @ A) @ ((field2 @ A) @ R))))) @ ((bNF_We4044943003108391690rdLess @ A) @ (set @ A))))))).
% 9.43/3.46 thf(fact_5383_Func__Times__Range, axiom, (! [C:$tType,B:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: ((((member @ ((product_prod @ (set @ ((product_prod @ (A > ((product_prod @ B) @ C))) @ (A > ((product_prod @ B) @ C))))) @ (set @ ((product_prod @ ((product_prod @ (A > B)) @ (A > C))) @ ((product_prod @ (A > B)) @ (A > C)))))) @ ((((product_Pair @ (set @ ((product_prod @ (A > ((product_prod @ B) @ C))) @ (A > ((product_prod @ B) @ C))))) @ (set @ ((product_prod @ ((product_prod @ (A > B)) @ (A > C))) @ ((product_prod @ (A > B)) @ (A > C))))) @ ((bNF_Ca6860139660246222851ard_of @ (A > ((product_prod @ B) @ C))) @ ((((bNF_Wellorder_Func @ A) @ ((product_prod @ B) @ C)) @ A4) @ ((((product_Sigma @ B) @ C) @ B6) @ (^ [Uu:B]: (C5)))))) @ ((bNF_Ca6860139660246222851ard_of @ ((product_prod @ (A > B)) @ (A > C))) @ ((((product_Sigma @ (A > B)) @ (A > C)) @ ((((bNF_Wellorder_Func @ A) @ B) @ A4) @ B6)) @ (^ [Uu:(A > B)]: (((((bNF_Wellorder_Func @ A) @ C) @ A4) @ C5))))))) @ ((bNF_Wellorder_ordIso @ (A > ((product_prod @ B) @ C))) @ ((product_prod @ (A > B)) @ (A > C))))))).
% 9.43/3.46 thf(fact_5384_card__of__Func__Times, axiom, (! [C:$tType,B:$tType,A:$tType,A4:(set @ A),B6:(set @ B),C5:(set @ C)]: ((((member @ ((product_prod @ (set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ A) @ B) > C)))) @ (set @ ((product_prod @ (A > (B > C))) @ (A > (B > C)))))) @ ((((product_Pair @ (set @ ((product_prod @ (((product_prod @ A) @ B) > C)) @ (((product_prod @ A) @ B) > C)))) @ (set @ ((product_prod @ (A > (B > C))) @ (A > (B > C))))) @ ((bNF_Ca6860139660246222851ard_of @ (((product_prod @ A) @ B) > C)) @ ((((bNF_Wellorder_Func @ ((product_prod @ A) @ B)) @ C) @ ((((product_Sigma @ A) @ B) @ A4) @ (^ [Uu:A]: (B6)))) @ C5))) @ ((bNF_Ca6860139660246222851ard_of @ (A > (B > C))) @ ((((bNF_Wellorder_Func @ A) @ (B > C)) @ A4) @ ((((bNF_Wellorder_Func @ B) @ C) @ B6) @ C5))))) @ ((bNF_Wellorder_ordIso @ (((product_prod @ A) @ B) > C)) @ (A > (B > C))))))).
% 9.43/3.46 thf(fact_5385_card__of__Pow__Func, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (A > $o)) @ (A > $o))))) @ ((((product_Pair @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ (set @ ((product_prod @ (A > $o)) @ (A > $o)))) @ ((bNF_Ca6860139660246222851ard_of @ (set @ A)) @ ((pow @ A) @ A4))) @ ((bNF_Ca6860139660246222851ard_of @ (A > $o)) @ ((((bNF_Wellorder_Func @ A) @ $o) @ A4) @ (top_top @ (set @ $o)))))) @ ((bNF_Wellorder_ordIso @ (set @ A)) @ (A > $o)))))).
% 9.43/3.46 thf(fact_5386_Cantors__paradox, axiom, (! [A:$tType,A4:(set @ A)]: ((~ ((? [F8:(A > (set @ A))]: ((((((image2 @ A) @ (set @ A)) @ F8) @ A4) = ((pow @ A) @ A4))))))))).
% 9.43/3.46 thf(fact_5387_inj__on__image__Pow, axiom, (! [B:$tType,A:$tType,F:(A > B),A4:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A4) => ((((inj_on @ (set @ A)) @ (set @ B)) @ (((image2 @ A) @ B) @ F)) @ ((pow @ A) @ A4)))))).
% 9.43/3.46 thf(fact_5388_card__of__Pow, axiom, (! [A:$tType,A4:(set @ A)]: ((((member @ ((product_prod @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A))))) @ ((((product_Pair @ (set @ ((product_prod @ A) @ A))) @ (set @ ((product_prod @ (set @ A)) @ (set @ A)))) @ ((bNF_Ca6860139660246222851ard_of @ A) @ A4)) @ ((bNF_Ca6860139660246222851ard_of @ (set @ A)) @ ((pow @ A) @ A4)))) @ ((bNF_We4044943003108391690rdLess @ A) @ (set @ A)))))).
% 9.43/3.46 thf(fact_5389_lfp__induct2, axiom, (! [A:$tType,B:$tType,A3:A,B2:B,F:((set @ ((product_prod @ A) @ B)) > (set @ ((product_prod @ A) @ B))),P:(A > (B > $o))]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) @ ((complete_lattice_lfp @ (set @ ((product_prod @ A) @ B))) @ F)) => ((((order_mono @ (set @ ((product_prod @ A) @ B))) @ (set @ ((product_prod @ A) @ B))) @ F) => ((! [A5:A,B3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ (F @ (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ ((complete_lattice_lfp @ (set @ ((product_prod @ A) @ B))) @ F)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ P))))) => ((P @ A5) @ B3)))) => ((P @ A3) @ B2))))))).
% 9.43/3.46 thf(fact_5390_sort__quicksort__by__rel, axiom, (! [A:$tType]: (((linorder @ A) => ((((linorder_sort_key @ A) @ A) @ (^ [X3:A]: (X3))) = (((quicksort_by_rel @ A) @ (ord_less_eq @ A)) @ (nil @ A))))))).
% 9.43/3.46 thf(fact_5391_Gr__def, axiom, (! [B:$tType,A:$tType]: ((((bNF_Gr @ A) @ B) = (^ [A9:(set @ A),F2:(A > B)]: (((collect @ ((product_prod @ A) @ B)) @ (^ [Uu:((product_prod @ A) @ B)]: ((? [A7:A]: (((Uu = ((((product_Pair @ A) @ B) @ A7) @ (F2 @ A7))) & (((member @ A) @ A7) @ A9))))))))))))).
% 9.43/3.46 thf(fact_5392_GrD2, axiom, (! [A:$tType,B:$tType,X:A,Fx:B,A4:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A4) @ F)) => ((F @ X) = Fx))))).
% 9.43/3.46 thf(fact_5393_GrD1, axiom, (! [B:$tType,A:$tType,X:A,Fx:B,A4:(set @ A),F:(A > B)]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Fx)) @ ((((bNF_Gr @ A) @ B) @ A4) @ F)) => (((member @ A) @ X) @ A4))))).
% 9.43/3.46 thf(fact_5394_sort__mergesort, axiom, (! [A:$tType]: (((linorder @ A) => ((((linorder_sort_key @ A) @ A) @ (^ [X3:A]: (X3))) = (mergesort @ A)))))).
% 9.43/3.46 thf(fact_5395_sort__mergesort__by__rel, axiom, (! [A:$tType]: (((linorder @ A) => ((((linorder_sort_key @ A) @ A) @ (^ [X3:A]: (X3))) = ((mergesort_by_rel @ A) @ (ord_less_eq @ A))))))).
% 9.43/3.46 thf(fact_5396_remdup__sort__mergesort__remdups, axiom, (! [A:$tType]: (((linorder @ A) => ((((((comp @ (list @ A)) @ (list @ A)) @ (list @ A)) @ (remdups @ A)) @ (((linorder_sort_key @ A) @ A) @ (^ [X3:A]: (X3)))) = (mergesort_remdups @ A)))))).
% 9.43/3.46 thf(fact_5397_prod_H__def, axiom, (! [C:$tType,A:$tType]: (((comm_monoid_mult @ A) => (((groups1962203154675924110t_prod @ C) @ A) = ((((groups_comm_monoid_G @ A) @ C) @ (times_times @ A)) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_5398_prod_Ocomm__monoid__list__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((groups4802862169904069756st_set @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_5399_length__remdups__card, axiom, (! [A:$tType,L:(list @ A)]: ((((size_size @ (list @ A)) @ ((remdups @ A) @ L)) = ((finite_card @ A) @ ((set2 @ A) @ L)))))).
% 9.43/3.46 thf(fact_5400_sum_H__def, axiom, (! [C:$tType,A:$tType]: (((comm_monoid_add @ A) => (((groups1027152243600224163dd_sum @ C) @ A) = ((((groups_comm_monoid_G @ A) @ C) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 9.43/3.46 thf(fact_5401_sum_Ocomm__monoid__list__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((groups4802862169904069756st_set @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_5402_and_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((bit_ri3973907225187159222ations @ A) => (((comm_monoid @ A) @ (bit_se5824344872417868541ns_and @ A)) @ ((uminus_uminus @ A) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_5403_bijective__alt, axiom, (! [B:$tType,A:$tType]: ((((bijective @ A) @ B) = (^ [R6:(set @ ((product_prod @ A) @ B))]: (((((single_valued @ A) @ B) @ R6) & (((single_valued @ B) @ A) @ (((converse @ A) @ B) @ R6))))))))).
% 9.43/3.46 thf(fact_5404_Heap_Osize__gen__o__map, axiom, (! [B:$tType,A:$tType,F:(B > nat),G:(A > B)]: (((((((comp @ (heap_Time_Heap @ B)) @ nat) @ (heap_Time_Heap @ A)) @ ((heap_Time_size_Heap @ B) @ F)) @ (((heap_Time_map_Heap @ A) @ B) @ G)) = ((heap_Time_size_Heap @ A) @ (((((comp @ B) @ nat) @ A) @ F) @ G)))))).
% 9.43/3.46 thf(fact_5405_single__valued__inter1, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((((single_valued @ A) @ B) @ R5) => (((single_valued @ A) @ B) @ (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ R5) @ S2)))))).
% 9.43/3.46 thf(fact_5406_single__valued__inter2, axiom, (! [B:$tType,A:$tType,R5:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((((single_valued @ A) @ B) @ R5) => (((single_valued @ A) @ B) @ (((inf_inf @ (set @ ((product_prod @ A) @ B))) @ S2) @ R5)))))).
% 9.43/3.46 thf(fact_5407_comm__monoid_Ocomm__neutral, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,A3:A]: (((((comm_monoid @ A) @ F) @ Z2) => (((F @ A3) @ Z2) = A3))))).
% 9.43/3.46 thf(fact_5408_Heap_Omap__ident, axiom, (! [A:$tType,T2:(heap_Time_Heap @ A)]: ((((((heap_Time_map_Heap @ A) @ A) @ (^ [X3:A]: (X3))) @ T2) = T2)))).
% 9.43/3.46 thf(fact_5409_Heap_Omap__id0, axiom, (! [A:$tType]: (((((heap_Time_map_Heap @ A) @ A) @ (id @ A)) = (id @ (heap_Time_Heap @ A)))))).
% 9.43/3.46 thf(fact_5410_Heap_Omap__id, axiom, (! [A:$tType,T2:(heap_Time_Heap @ A)]: ((((((heap_Time_map_Heap @ A) @ A) @ (id @ A)) @ T2) = T2)))).
% 9.43/3.46 thf(fact_5411_Heap_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,G:(B > C),F:(A > B),V:(heap_Time_Heap @ A)]: ((((((heap_Time_map_Heap @ B) @ C) @ G) @ ((((heap_Time_map_Heap @ A) @ B) @ F) @ V)) = ((((heap_Time_map_Heap @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 9.43/3.46 thf(fact_5412_Heap_Omap, axiom, (! [B:$tType,A:$tType,F:(A > B),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((((heap_Time_map_Heap @ A) @ B) @ F) @ ((heap_Time_Heap2 @ A) @ X)) = ((heap_Time_Heap2 @ B) @ (((((comp @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (((map_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((product_map_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ F) @ (id @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ X)))))).
% 9.43/3.46 thf(fact_5413_single__valuedD, axiom, (! [A:$tType,B:$tType,R:(set @ ((product_prod @ A) @ B)),X:A,Y:B,Z2:B]: (((((single_valued @ A) @ B) @ R) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Y)) @ R) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X) @ Z2)) @ R) => (Y = Z2))))))).
% 9.43/3.46 thf(fact_5414_single__valuedI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((! [X2:A,Y4:B,Z3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y4)) @ R) => ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Z3)) @ R) => (Y4 = Z3))))) => (((single_valued @ A) @ B) @ R))))).
% 9.43/3.46 thf(fact_5415_single__valued__def, axiom, (! [B:$tType,A:$tType]: ((((single_valued @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: ((! [X3:A,Y3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3) => (! [Z5:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Z5)) @ R3) => (Y3 = Z5))))))))))))).
% 9.43/3.46 thf(fact_5416_add_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((comm_monoid @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_5417_mult_Ocomm__monoid__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((comm_monoid @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_5418_Heap_Oinj__map, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_map_Heap @ A) @ B) @ F)) @ (top_top @ (set @ (heap_Time_Heap @ A)))))))).
% 9.43/3.46 thf(fact_5419_single__valued__confluent, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),X:A,Y:A,Z2:A]: (((((single_valued @ A) @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Z2)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ ((transitive_rtrancl @ A) @ R)) | (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z2) @ Y)) @ ((transitive_rtrancl @ A) @ R))))))))).
% 9.43/3.46 thf(fact_5420_single__valued__below__Id, axiom, (! [A:$tType,R5:(set @ ((product_prod @ A) @ A))]: (((((ord_less_eq @ (set @ ((product_prod @ A) @ A))) @ R5) @ (id2 @ A)) => (((single_valued @ A) @ A) @ R5))))).
% 9.43/3.46 thf(fact_5421_Heap_Orec__o__map, axiom, (! [B:$tType,C:$tType,A:$tType,G:(((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > C),F:(A > B)]: (((((((comp @ (heap_Time_Heap @ B)) @ C) @ (heap_Time_Heap @ A)) @ (((heap_Time_rec_Heap @ B) @ C) @ G)) @ (((heap_Time_map_Heap @ A) @ B) @ F)) = (((heap_Time_rec_Heap @ A) @ C) @ (^ [X3:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((G @ (((((comp @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ (((map_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((product_map_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ F) @ (id @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ X3))))))))).
% 9.43/3.46 thf(fact_5422_single__valuedp__single__valued__eq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((single_valuedp @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) = (((single_valued @ A) @ B) @ R))))).
% 9.43/3.46 thf(fact_5423_disjnt__equiv__class, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((disjnt @ A) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) = (~ ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)))))))).
% 9.43/3.46 thf(fact_5424_equiv__class__eq__iff, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) = ((((((image @ A) @ A) @ R) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A))))) & ((((member @ A) @ X) @ A4) & (((member @ A) @ Y) @ A4)))))))).
% 9.43/3.46 thf(fact_5425_eq__equiv__class__iff, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => ((((((image @ A) @ A) @ R) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A))))) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R)))))))).
% 9.43/3.46 thf(fact_5426_equiv__class__eq, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.46 thf(fact_5427_eq__equiv__class, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A,A4:(set @ A)]: (((((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A)))) = ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) => ((((equiv_equiv @ A) @ A4) @ R) => ((((member @ A) @ B2) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R))))))).
% 9.43/3.46 thf(fact_5428_Heap_Orec, axiom, (! [C:$tType,A:$tType,F:(((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > C),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((((heap_Time_rec_Heap @ A) @ C) @ F) @ ((heap_Time_Heap2 @ A) @ X)) = (F @ X))))).
% 9.43/3.46 thf(fact_5429_subset__equiv__class, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),B2:A,A3:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))) => ((((member @ A) @ B2) @ A4) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R))))))).
% 9.43/3.46 thf(fact_5430_equiv__class__subset, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((ord_less_eq @ (set @ A)) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A)))))))))).
% 9.43/3.46 thf(fact_5431_equiv__class__nondisjoint, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,A3:A,B2:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ A) @ X) @ (((inf_inf @ (set @ A)) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ A3) @ (bot_bot @ (set @ A))))) @ ((((image @ A) @ A) @ R) @ (((insert2 @ A) @ B2) @ (bot_bot @ (set @ A)))))) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R)))))).
% 9.43/3.46 thf(fact_5432_proj__iff, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ X) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A))))) @ A4) => ((((((equiv_proj @ A) @ A) @ R) @ X) = ((((equiv_proj @ A) @ A) @ R) @ Y)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R))))))).
% 9.43/3.46 thf(fact_5433_in__quotient__imp__in__rel, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X10:(set @ A),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ (set @ A)) @ X10) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((ord_less_eq @ (set @ A)) @ (((insert2 @ A) @ X) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A))))) @ X10) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R))))))).
% 9.43/3.46 thf(fact_5434_congruent2__commuteI, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),F:(A > (A > B))]: (((((equiv_equiv @ A) @ A4) @ R) => ((! [Y4:A,Z3:A]: (((((member @ A) @ Y4) @ A4) => ((((member @ A) @ Z3) @ A4) => (((F @ Y4) @ Z3) = ((F @ Z3) @ Y4)))))) => ((! [Y4:A,Z3:A,W4:A]: (((((member @ A) @ W4) @ A4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => (((F @ W4) @ Y4) = ((F @ W4) @ Z3)))))) => ((((((equiv_congruent2 @ A) @ A) @ B) @ R) @ R) @ F))))))).
% 9.43/3.46 thf(fact_5435_quotient__eqI, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X10:(set @ A),Y9:(set @ A),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ (set @ A)) @ X10) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((member @ (set @ A)) @ Y9) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((member @ A) @ X) @ X10) => ((((member @ A) @ Y) @ Y9) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (X10 = Y9)))))))))).
% 9.43/3.46 thf(fact_5436_quotient__eq__iff, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X10:(set @ A),Y9:(set @ A),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ (set @ A)) @ X10) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((member @ (set @ A)) @ Y9) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((member @ A) @ X) @ X10) => ((((member @ A) @ Y) @ Y9) => ((X10 = Y9) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R)))))))))).
% 9.43/3.46 thf(fact_5437_in__quotient__imp__closed, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X10:(set @ A),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ (set @ A)) @ X10) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((member @ A) @ X) @ X10) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R) => (((member @ A) @ Y) @ X10)))))))).
% 9.43/3.46 thf(fact_5438_congruent2D, axiom, (! [A:$tType,C:$tType,B:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C)),Y1:A,Z1:A,Y2:B,Z22:B]: ((((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y1) @ Z1)) @ R1) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y2) @ Z22)) @ R22) => (((F @ Y1) @ Y2) = ((F @ Z1) @ Z22)))))))).
% 9.43/3.46 thf(fact_5439_congruent2I_H, axiom, (! [C:$tType,B:$tType,A:$tType,R1:(set @ ((product_prod @ A) @ A)),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C))]: (((! [Y12:A,Z12:A,Y22:B,Z23:B]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y12) @ Z12)) @ R1) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y22) @ Z23)) @ R22) => (((F @ Y12) @ Y22) = ((F @ Z12) @ Z23)))))) => ((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F))))).
% 9.43/3.46 thf(fact_5440_eq__equiv__class__iff2, axiom, (! [A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),X:A,Y:A]: (((((equiv_equiv @ A) @ A4) @ R) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => (((((equiv_quotient @ A) @ (((insert2 @ A) @ X) @ (bot_bot @ (set @ A)))) @ R) = (((equiv_quotient @ A) @ (((insert2 @ A) @ Y) @ (bot_bot @ (set @ A)))) @ R)) = (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X) @ Y)) @ R)))))))).
% 9.43/3.46 thf(fact_5441_congruent2I, axiom, (! [C:$tType,B:$tType,A:$tType,A110:(set @ A),R1:(set @ ((product_prod @ A) @ A)),A26:(set @ B),R22:(set @ ((product_prod @ B) @ B)),F:(A > (B > C))]: (((((equiv_equiv @ A) @ A110) @ R1) => ((((equiv_equiv @ B) @ A26) @ R22) => ((! [Y4:A,Z3:A,W4:B]: (((((member @ B) @ W4) @ A26) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R1) => (((F @ Y4) @ W4) = ((F @ Z3) @ W4)))))) => ((! [Y4:B,Z3:B,W4:A]: (((((member @ A) @ W4) @ A110) => ((((member @ ((product_prod @ B) @ B)) @ ((((product_Pair @ B) @ B) @ Y4) @ Z3)) @ R22) => (((F @ W4) @ Y4) = ((F @ W4) @ Z3)))))) => ((((((equiv_congruent2 @ A) @ B) @ C) @ R1) @ R22) @ F)))))))).
% 9.43/3.46 thf(fact_5442_UN__equiv__class__inject, axiom, (! [B:$tType,A:$tType,A4:(set @ A),R:(set @ ((product_prod @ A) @ A)),F:(A > (set @ B)),X10:(set @ A),Y9:(set @ A)]: (((((equiv_equiv @ A) @ A4) @ R) => (((((equiv_congruent @ A) @ (set @ B)) @ R) @ F) => ((((complete_Sup_Sup @ (set @ B)) @ ((((image2 @ A) @ (set @ B)) @ F) @ X10)) = ((complete_Sup_Sup @ (set @ B)) @ ((((image2 @ A) @ (set @ B)) @ F) @ Y9))) => ((((member @ (set @ A)) @ X10) @ (((equiv_quotient @ A) @ A4) @ R)) => ((((member @ (set @ A)) @ Y9) @ (((equiv_quotient @ A) @ A4) @ R)) => ((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ A4) => ((((member @ A) @ Y4) @ A4) => (((F @ X2) = (F @ Y4)) => (((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X2) @ Y4)) @ R)))))) => (X10 = Y9)))))))))).
% 9.43/3.46 thf(fact_5443_equivp__equiv, axiom, (! [A:$tType,A4:(set @ ((product_prod @ A) @ A))]: (((((equiv_equiv @ A) @ (top_top @ (set @ A))) @ A4) = ((equiv_equivp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ A4)))))))).
% 9.43/3.46 thf(fact_5444_prod__mset__def, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => ((comm_m9189036328036947845d_mset @ A) = (((comm_monoid_F @ A) @ (times_times @ A)) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_5445_congruentI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(A > B)]: (((! [Y4:A,Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ Z3)) @ R) => ((F @ Y4) = (F @ Z3))))) => ((((equiv_congruent @ A) @ B) @ R) @ F))))).
% 9.43/3.46 thf(fact_5446_congruentD, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ A)),F:(A > B),Y:A,Z2:A]: ((((((equiv_congruent @ A) @ B) @ R) @ F) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y) @ Z2)) @ R) => ((F @ Y) = (F @ Z2))))))).
% 9.43/3.46 thf(fact_5447_sum__mset__def, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => ((comm_m7189776963980413722m_mset @ A) = (((comm_monoid_F @ A) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 9.43/3.46 thf(fact_5448_or__num_Opelims, axiom, (! [X:num,Xa:num,Y:num]: (((((bit_un6697907153464112080or_num @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ X) @ Xa)) => (((X = one2) => ((Xa = one2) => ((Y = one2) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ one2))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = (bit1 @ N3)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit0 @ N3)))))))))) => (((X = one2) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = (bit1 @ N3)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ one2) @ (bit1 @ N3)))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => ((Xa = one2) => ((Y = (bit1 @ M5)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = (bit0 @ ((bit_un6697907153464112080or_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit0 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit0 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = (bit1 @ ((bit_un6697907153464112080or_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ (bit0 @ M5)) @ (bit1 @ N3)))))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => ((Xa = one2) => ((Y = (bit1 @ M5)) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ one2))))))))) => ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit0 @ N3)) => ((Y = (bit1 @ ((bit_un6697907153464112080or_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit0 @ N3)))))))))))) => (~ ((! [M5:num]: (((X = (bit1 @ M5)) => (! [N3:num]: (((Xa = (bit1 @ N3)) => ((Y = (bit1 @ ((bit_un6697907153464112080or_num @ M5) @ N3))) => (~ ((((accp @ ((product_prod @ num) @ num)) @ bit_un4773296044027857193um_rel) @ ((((product_Pair @ num) @ num) @ (bit1 @ M5)) @ (bit1 @ N3))))))))))))))))))))))))))).
% 9.43/3.46 thf(fact_5449_Heap_Oset, axiom, (! [A:$tType,X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((heap_Time_set_Heap @ A) @ ((heap_Time_Heap2 @ A) @ X)) = ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (set @ A)) @ (^ [Uu3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (set @ A)) @ ((basic_fsts @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((set_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ Uu3)))))) @ ((((image2 @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ X) @ (top_top @ (set @ (heap_ext @ product_unit)))))))))).
% 9.43/3.46 thf(fact_5450_under__incr, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A)),A3:A,B2:A]: ((((trans @ A) @ R) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((ord_less_eq @ (set @ A)) @ (((order_under @ A) @ R) @ A3)) @ (((order_under @ A) @ R) @ B2))))))).
% 9.43/3.46 thf(fact_5451_Heap_Oset__intros, axiom, (! [A:$tType,Y:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))),Ya:((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)),Yb2:A]: (((((member @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ Y) @ ((((image2 @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ X) @ (top_top @ (set @ (heap_ext @ product_unit))))) => ((((member @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ Ya) @ ((set_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ Y)) => ((((member @ A) @ Yb2) @ (((basic_fsts @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ Ya)) => (((member @ A) @ Yb2) @ ((heap_Time_set_Heap @ A) @ ((heap_Time_Heap2 @ A) @ X))))))))).
% 9.43/3.46 thf(fact_5452_Heap_Oset__cases, axiom, (! [A:$tType,E2:A,A3:(heap_Time_Heap @ A)]: (((((member @ A) @ E2) @ ((heap_Time_set_Heap @ A) @ A3)) => (~ ((! [Z3:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((A3 = ((heap_Time_Heap2 @ A) @ Z3)) => (! [Xa3:(option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))]: (((((member @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ Xa3) @ ((((image2 @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ Z3) @ (top_top @ (set @ (heap_ext @ product_unit))))) => (! [Xb3:((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))]: (((((member @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ Xb3) @ ((set_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ Xa3)) => (~ ((((member @ A) @ E2) @ (((basic_fsts @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ Xb3))))))))))))))))))).
% 9.43/3.46 thf(fact_5453_under__def, axiom, (! [A:$tType]: (((order_under @ A) = (^ [R3:(set @ ((product_prod @ A) @ A)),A7:A]: (((collect @ A) @ (^ [B5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B5) @ A7)) @ R3)))))))))).
% 9.43/3.46 thf(fact_5454_Heap_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),Xa:(heap_Time_Heap @ A),F:(A > B),Fa:(A > B)]: (((! [Z3:A,Za:A]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ X)) => ((((member @ A) @ Za) @ ((heap_Time_set_Heap @ A) @ Xa)) => (((F @ Z3) = (Fa @ Za)) => (Z3 = Za)))))) => ((((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ Fa) @ Xa)) => (X = Xa)))))).
% 9.43/3.46 thf(fact_5455_Heap_Omap__cong0, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),F:(A > B),G:(A > B)]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ X)) => ((F @ Z3) = (G @ Z3))))) => (((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ G) @ X)))))).
% 9.43/3.46 thf(fact_5456_Heap_Omap__cong, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ Ya)) => ((F @ Z3) = (G @ Z3))))) => (((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ G) @ Ya))))))).
% 9.43/3.46 thf(fact_5457_Heap_Oset__map, axiom, (! [B:$tType,A:$tType,F:(A > B),V:(heap_Time_Heap @ A)]: ((((heap_Time_set_Heap @ B) @ ((((heap_Time_map_Heap @ A) @ B) @ F) @ V)) = ((((image2 @ A) @ B) @ F) @ ((heap_Time_set_Heap @ A) @ V)))))).
% 9.43/3.46 thf(fact_5458_Heap_Orel__Grp, axiom, (! [B:$tType,A:$tType,A4:(set @ A),F:(A > B)]: (((((heap_Time_rel_Heap @ A) @ B) @ ((((bNF_Grp @ A) @ B) @ A4) @ F)) = ((((bNF_Grp @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ ((collect @ (heap_Time_Heap @ A)) @ (^ [X3:(heap_Time_Heap @ A)]: ((((ord_less_eq @ (set @ A)) @ ((heap_Time_set_Heap @ A) @ X3)) @ A4))))) @ (((heap_Time_map_Heap @ A) @ B) @ F)))))).
% 9.43/3.46 thf(fact_5459_Heap_Orel__compp__Grp, axiom, (! [B:$tType,A:$tType]: ((((heap_Time_rel_Heap @ A) @ B) = (^ [R6:(A > (B > $o))]: ((((((relcompp @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (heap_Time_Heap @ B)) @ (((conversep @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (heap_Time_Heap @ A)) @ ((((bNF_Grp @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (heap_Time_Heap @ A)) @ ((collect @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (^ [X3:(heap_Time_Heap @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((heap_Time_set_Heap @ ((product_prod @ A) @ B)) @ X3)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) @ (((heap_Time_map_Heap @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B))))) @ ((((bNF_Grp @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (heap_Time_Heap @ B)) @ ((collect @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (^ [X3:(heap_Time_Heap @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((heap_Time_set_Heap @ ((product_prod @ A) @ B)) @ X3)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) @ (((heap_Time_map_Heap @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)))))))))).
% 9.43/3.46 thf(fact_5460_Heap_Oin__rel, axiom, (! [B:$tType,A:$tType]: ((((heap_Time_rel_Heap @ A) @ B) = (^ [R6:(A > (B > $o)),A7:(heap_Time_Heap @ A),B5:(heap_Time_Heap @ B)]: ((? [Z5:(heap_Time_Heap @ ((product_prod @ A) @ B))]: (((((member @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ Z5) @ ((collect @ (heap_Time_Heap @ ((product_prod @ A) @ B))) @ (^ [X3:(heap_Time_Heap @ ((product_prod @ A) @ B))]: ((((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ ((heap_Time_set_Heap @ ((product_prod @ A) @ B)) @ X3)) @ ((collect @ ((product_prod @ A) @ B)) @ ((((product_case_prod @ A) @ B) @ $o) @ R6))))))) & ((((((heap_Time_map_Heap @ ((product_prod @ A) @ B)) @ A) @ ((product_fst @ A) @ B)) @ Z5) = A7) & (((((heap_Time_map_Heap @ ((product_prod @ A) @ B)) @ B) @ ((product_snd @ A) @ B)) @ Z5) = B5))))))))))).
% 9.43/3.46 thf(fact_5461_Heap_Orel__cong, axiom, (! [A:$tType,B:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ B),Xa:(heap_Time_Heap @ B),R5:(A > (B > $o)),Ra2:(A > (B > $o))]: (((X = Ya) => ((Y = Xa) => ((! [Z3:A,Yb:B]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ Ya)) => ((((member @ B) @ Yb) @ ((heap_Time_set_Heap @ B) @ Xa)) => (((R5 @ Z3) @ Yb) = ((Ra2 @ Z3) @ Yb)))))) => ((((((heap_Time_rel_Heap @ A) @ B) @ R5) @ X) @ Y) = (((((heap_Time_rel_Heap @ A) @ B) @ Ra2) @ Ya) @ Xa)))))))).
% 9.43/3.46 thf(fact_5462_Heap_Orel__mono__strong, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ B),Ra2:(A > (B > $o))]: (((((((heap_Time_rel_Heap @ A) @ B) @ R5) @ X) @ Y) => ((! [Z3:A,Yb:B]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ X)) => ((((member @ B) @ Yb) @ ((heap_Time_set_Heap @ B) @ Y)) => (((R5 @ Z3) @ Yb) => ((Ra2 @ Z3) @ Yb)))))) => (((((heap_Time_rel_Heap @ A) @ B) @ Ra2) @ X) @ Y)))))).
% 9.43/3.46 thf(fact_5463_Heap_Orel__refl__strong, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Ra2:(A > (A > $o))]: (((! [Z3:A]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ X)) => ((Ra2 @ Z3) @ Z3)))) => (((((heap_Time_rel_Heap @ A) @ A) @ Ra2) @ X) @ X))))).
% 9.43/3.46 thf(fact_5464_Heap_Orel__map_I2_J, axiom, (! [A:$tType,C:$tType,B:$tType,Sa:(A > (C > $o)),X:(heap_Time_Heap @ A),G:(B > C),Y:(heap_Time_Heap @ B)]: (((((((heap_Time_rel_Heap @ A) @ C) @ Sa) @ X) @ ((((heap_Time_map_Heap @ B) @ C) @ G) @ Y)) = (((((heap_Time_rel_Heap @ A) @ B) @ (^ [X3:A,Y3:B]: (((Sa @ X3) @ (G @ Y3))))) @ X) @ Y))))).
% 9.43/3.46 thf(fact_5465_Heap_Orel__map_I1_J, axiom, (! [A:$tType,C:$tType,B:$tType,Sb:(C > (B > $o)),I:(A > C),X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ B)]: (((((((heap_Time_rel_Heap @ C) @ B) @ Sb) @ ((((heap_Time_map_Heap @ A) @ C) @ I) @ X)) @ Y) = (((((heap_Time_rel_Heap @ A) @ B) @ (^ [X3:A]: ((Sb @ (I @ X3))))) @ X) @ Y))))).
% 9.43/3.46 thf(fact_5466_Heap_Orel__mono, axiom, (! [B:$tType,A:$tType,R5:(A > (B > $o)),Ra2:(A > (B > $o))]: (((((ord_less_eq @ (A > (B > $o))) @ R5) @ Ra2) => (((ord_less_eq @ ((heap_Time_Heap @ A) > ((heap_Time_Heap @ B) > $o))) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) @ (((heap_Time_rel_Heap @ A) @ B) @ Ra2)))))).
% 9.43/3.46 thf(fact_5467_Heap_Orel__transp, axiom, (! [A:$tType,R5:(A > (A > $o))]: ((((transp @ A) @ R5) => ((transp @ (heap_Time_Heap @ A)) @ (((heap_Time_rel_Heap @ A) @ A) @ R5)))))).
% 9.43/3.46 thf(fact_5468_Heap_Orel__refl, axiom, (! [B:$tType,Ra2:(B > (B > $o)),X:(heap_Time_Heap @ B)]: (((! [X2:B]: (((Ra2 @ X2) @ X2))) => (((((heap_Time_rel_Heap @ B) @ B) @ Ra2) @ X) @ X))))).
% 9.43/3.46 thf(fact_5469_Heap_Orel__eq, axiom, (! [A:$tType]: (((((heap_Time_rel_Heap @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) = (^ [Y5:(heap_Time_Heap @ A),Z4:(heap_Time_Heap @ A)]: ((Y5 = Z4))))))).
% 9.43/3.46 thf(fact_5470_Heap_Orel__conversep, axiom, (! [B:$tType,A:$tType,R5:(A > (B > $o))]: (((((heap_Time_rel_Heap @ B) @ A) @ (((conversep @ A) @ B) @ R5)) = (((conversep @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)))))).
% 9.43/3.46 thf(fact_5471_Heap_Orel__flip, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),A3:(heap_Time_Heap @ B),B2:(heap_Time_Heap @ A)]: (((((((heap_Time_rel_Heap @ B) @ A) @ (((conversep @ A) @ B) @ R5)) @ A3) @ B2) = (((((heap_Time_rel_Heap @ A) @ B) @ R5) @ B2) @ A3))))).
% 9.43/3.46 thf(fact_5472_Heap_Orel__compp, axiom, (! [A:$tType,C:$tType,B:$tType,R5:(A > (B > $o)),S2:(B > (C > $o))]: (((((heap_Time_rel_Heap @ A) @ C) @ (((((relcompp @ A) @ B) @ C) @ R5) @ S2)) = (((((relcompp @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ C)) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) @ (((heap_Time_rel_Heap @ B) @ C) @ S2)))))).
% 9.43/3.46 thf(fact_5473_Heap_Orel__transfer, axiom, (! [A:$tType,B:$tType,D:$tType,C:$tType,Sa:(A > (C > $o)),Sc:(B > (D > $o))]: (((((((((bNF_rel_fun @ (A > (B > $o))) @ (C > (D > $o))) @ ((heap_Time_Heap @ A) > ((heap_Time_Heap @ B) > $o))) @ ((heap_Time_Heap @ C) > ((heap_Time_Heap @ D) > $o))) @ ((((((bNF_rel_fun @ A) @ C) @ (B > $o)) @ (D > $o)) @ Sa) @ ((((((bNF_rel_fun @ B) @ D) @ $o) @ $o) @ Sc) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))))) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ C)) @ ((heap_Time_Heap @ B) > $o)) @ ((heap_Time_Heap @ D) > $o)) @ (((heap_Time_rel_Heap @ A) @ C) @ Sa)) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ D)) @ $o) @ $o) @ (((heap_Time_rel_Heap @ B) @ D) @ Sc)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4)))))) @ ((heap_Time_rel_Heap @ A) @ B)) @ ((heap_Time_rel_Heap @ C) @ D))))).
% 9.43/3.46 thf(fact_5474_Heap_Omap__transfer, axiom, (! [A:$tType,B:$tType,F5:$tType,E:$tType,Rb2:(A > (E > $o)),Sd:(B > (F5 > $o))]: (((((((((bNF_rel_fun @ (A > B)) @ (E > F5)) @ ((heap_Time_Heap @ A) > (heap_Time_Heap @ B))) @ ((heap_Time_Heap @ E) > (heap_Time_Heap @ F5))) @ ((((((bNF_rel_fun @ A) @ E) @ B) @ F5) @ Rb2) @ Sd)) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ E)) @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ F5)) @ (((heap_Time_rel_Heap @ A) @ E) @ Rb2)) @ (((heap_Time_rel_Heap @ B) @ F5) @ Sd))) @ ((heap_Time_map_Heap @ A) @ B)) @ ((heap_Time_map_Heap @ E) @ F5))))).
% 9.43/3.46 thf(fact_5475_those_Osimps_I2_J, axiom, (! [A:$tType,X:(option @ A),Xs:(list @ (option @ A))]: ((((those @ A) @ (((cons @ (option @ A)) @ X) @ Xs)) = (((((case_option @ (option @ (list @ A))) @ A) @ (none @ (list @ A))) @ (^ [Y3:A]: (((((map_option @ (list @ A)) @ (list @ A)) @ ((cons @ A) @ Y3)) @ ((those @ A) @ Xs))))) @ X))))).
% 9.43/3.46 thf(fact_5476_Quotient__natural, axiom, ((((((quotient @ nat) @ code_natural) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) @ code_natural_of_nat) @ code_nat_of_natural) @ code_cr_natural)).
% 9.43/3.46 thf(fact_5477_Quotient__integer, axiom, ((((((quotient @ int) @ code_integer) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) @ code_integer_of_int) @ code_int_of_integer) @ code_cr_integer)).
% 9.43/3.46 thf(fact_5478_Heap_OQuotient, axiom, (! [B:$tType,A:$tType,R5:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T6:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ R5) @ Abs) @ Rep) @ T6) => ((((((quotient @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_rel_Heap @ A) @ A) @ R5)) @ (((heap_Time_map_Heap @ A) @ B) @ Abs)) @ (((heap_Time_map_Heap @ B) @ A) @ Rep)) @ (((heap_Time_rel_Heap @ A) @ B) @ T6)))))).
% 9.43/3.46 thf(fact_5479_option_OQuotient, axiom, (! [B:$tType,A:$tType,R5:(A > (A > $o)),Abs:(A > B),Rep:(B > A),T6:(A > (B > $o))]: ((((((((quotient @ A) @ B) @ R5) @ Abs) @ Rep) @ T6) => ((((((quotient @ (option @ A)) @ (option @ B)) @ (((rel_option @ A) @ A) @ R5)) @ (((map_option @ A) @ B) @ Abs)) @ (((map_option @ B) @ A) @ Rep)) @ (((rel_option @ A) @ B) @ T6)))))).
% 9.43/3.46 thf(fact_5480_Quotient__compose, axiom, (! [A:$tType,C:$tType,B:$tType,R12:(A > (A > $o)),Abs1:(A > B),Rep1:(B > A),T1:(A > (B > $o)),R23:(B > (B > $o)),Abs22:(B > C),Rep22:(C > B),T22:(B > (C > $o))]: ((((((((quotient @ A) @ B) @ R12) @ Abs1) @ Rep1) @ T1) => (((((((quotient @ B) @ C) @ R23) @ Abs22) @ Rep22) @ T22) => ((((((quotient @ A) @ C) @ (((((relcompp @ A) @ B) @ A) @ T1) @ (((((relcompp @ B) @ B) @ A) @ R23) @ (((conversep @ A) @ B) @ T1)))) @ (((((comp @ B) @ C) @ A) @ Abs22) @ Abs1)) @ (((((comp @ B) @ A) @ C) @ Rep1) @ Rep22)) @ (((((relcompp @ A) @ B) @ C) @ T1) @ T22))))))).
% 9.43/3.46 thf(fact_5481_Heap_Opred__transfer, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ (A > $o)) @ (B > $o)) @ ((heap_Time_Heap @ A) > $o)) @ ((heap_Time_Heap @ B) > $o)) @ ((((((bNF_rel_fun @ A) @ B) @ $o) @ $o) @ R5) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ $o) @ $o) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) @ (^ [Y5:$o,Z4:$o]: ((Y5 = Z4))))) @ (heap_Time_pred_Heap @ A)) @ (heap_Time_pred_Heap @ B))))).
% 9.43/3.46 thf(fact_5482_Heap_Orec__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R5:(A > (B > $o)),S2:(C > (D > $o))]: (((((((((bNF_rel_fun @ (((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > C)) @ (((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > D)) @ ((heap_Time_Heap @ A) > C)) @ ((heap_Time_Heap @ B) > D)) @ ((((((bNF_rel_fun @ ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ ((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ C) @ D) @ ((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4))))))) @ S2)) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ C) @ D) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) @ S2)) @ ((heap_Time_rec_Heap @ A) @ C)) @ ((heap_Time_rec_Heap @ B) @ D))))).
% 9.43/3.46 thf(fact_5483_rel__prod__inject, axiom, (! [B:$tType,A:$tType,C:$tType,D:$tType,R12:(A > (B > $o)),R23:(C > (D > $o)),A3:A,B2:C,C2:B,D3:D]: ((((((((((basic_rel_prod @ A) @ B) @ C) @ D) @ R12) @ R23) @ ((((product_Pair @ A) @ C) @ A3) @ B2)) @ ((((product_Pair @ B) @ D) @ C2) @ D3)) = (((R12 @ A3) @ C2) & ((R23 @ B2) @ D3)))))).
% 9.43/3.46 thf(fact_5484_Heap_Opred__mono, axiom, (! [A:$tType,P:(A > $o),Pa:(A > $o)]: (((((ord_less_eq @ (A > $o)) @ P) @ Pa) => (((ord_less_eq @ ((heap_Time_Heap @ A) > $o)) @ ((heap_Time_pred_Heap @ A) @ P)) @ ((heap_Time_pred_Heap @ A) @ Pa)))))).
% 9.43/3.46 thf(fact_5485_rel__prod_Ointros, axiom, (! [C:$tType,A:$tType,B:$tType,D:$tType,R12:(A > (B > $o)),A3:A,B2:B,R23:(C > (D > $o)),C2:C,D3:D]: ((((R12 @ A3) @ B2) => (((R23 @ C2) @ D3) => ((((((((basic_rel_prod @ A) @ B) @ C) @ D) @ R12) @ R23) @ ((((product_Pair @ A) @ C) @ A3) @ C2)) @ ((((product_Pair @ B) @ D) @ B2) @ D3))))))).
% 9.43/3.46 thf(fact_5486_rel__prod_Osimps, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType]: ((((((basic_rel_prod @ A) @ B) @ C) @ D) = (^ [R13:(A > (B > $o)),R24:(C > (D > $o)),A12:((product_prod @ A) @ C),A23:((product_prod @ B) @ D)]: ((? [A7:A,B5:B,C3:C,D5:D]: (((A12 = ((((product_Pair @ A) @ C) @ A7) @ C3)) & ((A23 = ((((product_Pair @ B) @ D) @ B5) @ D5)) & (((R13 @ A7) @ B5) & ((R24 @ C3) @ D5)))))))))))).
% 9.43/3.46 thf(fact_5487_rel__prod_Ocases, axiom, (! [B:$tType,A:$tType,C:$tType,D:$tType,R12:(A > (B > $o)),R23:(C > (D > $o)),A1:((product_prod @ A) @ C),A22:((product_prod @ B) @ D)]: ((((((((((basic_rel_prod @ A) @ B) @ C) @ D) @ R12) @ R23) @ A1) @ A22) => (~ ((! [A5:A,B3:B,C4:C]: (((A1 = ((((product_Pair @ A) @ C) @ A5) @ C4)) => (! [D2:D]: (((A22 = ((((product_Pair @ B) @ D) @ B3) @ D2)) => (((R12 @ A5) @ B3) => (~ (((R23 @ C4) @ D2)))))))))))))))).
% 9.43/3.46 thf(fact_5488_Heap_Opred__True, axiom, (! [A:$tType]: ((((heap_Time_pred_Heap @ A) @ (^ [Uu:A]: ($true))) = (^ [Uu:(heap_Time_Heap @ A)]: ($true)))))).
% 9.43/3.46 thf(fact_5489_Heap_Opred__mono__strong, axiom, (! [A:$tType,P:(A > $o),X:(heap_Time_Heap @ A),Pa:(A > $o)]: (((((heap_Time_pred_Heap @ A) @ P) @ X) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ X)) => ((P @ Z3) => (Pa @ Z3))))) => (((heap_Time_pred_Heap @ A) @ Pa) @ X)))))).
% 9.43/3.46 thf(fact_5490_Heap_Opred__cong, axiom, (! [A:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),P:(A > $o),Pa:(A > $o)]: (((X = Ya) => ((! [Z3:A]: (((((member @ A) @ Z3) @ ((heap_Time_set_Heap @ A) @ Ya)) => ((P @ Z3) = (Pa @ Z3))))) => ((((heap_Time_pred_Heap @ A) @ P) @ X) = (((heap_Time_pred_Heap @ A) @ Pa) @ Ya))))))).
% 9.43/3.46 thf(fact_5491_Heap_Octr__transfer, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o))]: (((((((((bNF_rel_fun @ ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ ((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ ((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4))))))) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) @ (heap_Time_Heap2 @ A)) @ (heap_Time_Heap2 @ B))))).
% 9.43/3.46 thf(fact_5492_Heap_Orel__intros, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))),Y:((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4)))))) @ X) @ Y) => (((((heap_Time_rel_Heap @ A) @ B) @ R5) @ ((heap_Time_Heap2 @ A) @ X)) @ ((heap_Time_Heap2 @ B) @ Y)))))).
% 9.43/3.46 thf(fact_5493_Heap_Orel__inject, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))),Y:((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((((((heap_Time_rel_Heap @ A) @ B) @ R5) @ ((heap_Time_Heap2 @ A) @ X)) @ ((heap_Time_Heap2 @ B) @ Y)) = ((((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4)))))) @ X) @ Y))))).
% 9.43/3.46 thf(fact_5494_Heap_Orel__induct, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),X:(heap_Time_Heap @ A),Y:(heap_Time_Heap @ B),Q:((heap_Time_Heap @ A) > ((heap_Time_Heap @ B) > $o))]: (((((((heap_Time_rel_Heap @ A) @ B) @ R5) @ X) @ Y) => ((! [A5:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))),B3:((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4)))))) @ A5) @ B3) => ((Q @ ((heap_Time_Heap2 @ A) @ A5)) @ ((heap_Time_Heap2 @ B) @ B3))))) => ((Q @ X) @ Y)))))).
% 9.43/3.46 thf(fact_5495_Heap_Orel__cases, axiom, (! [A:$tType,B:$tType,R5:(A > (B > $o)),A3:(heap_Time_Heap @ A),B2:(heap_Time_Heap @ B)]: (((((((heap_Time_rel_Heap @ A) @ B) @ R5) @ A3) @ B2) => (~ ((! [Xa3:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((A3 = ((heap_Time_Heap2 @ A) @ Xa3)) => (! [Ya2:((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((B2 = ((heap_Time_Heap2 @ B) @ Ya2)) => (~ (((((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4)))))) @ Xa3) @ Ya2))))))))))))))).
% 9.43/3.46 thf(fact_5496_Heap_Omap__cong__pred, axiom, (! [B:$tType,A:$tType,X:(heap_Time_Heap @ A),Ya:(heap_Time_Heap @ A),F:(A > B),G:(A > B)]: (((X = Ya) => ((((heap_Time_pred_Heap @ A) @ (^ [Z5:A]: (((F @ Z5) = (G @ Z5))))) @ Ya) => (((((heap_Time_map_Heap @ A) @ B) @ F) @ X) = ((((heap_Time_map_Heap @ A) @ B) @ G) @ Ya))))))).
% 9.43/3.46 thf(fact_5497_Heap_Orel__eq__onp, axiom, (! [A:$tType,P:(A > $o)]: (((((heap_Time_rel_Heap @ A) @ A) @ ((bNF_eq_onp @ A) @ P)) = ((bNF_eq_onp @ (heap_Time_Heap @ A)) @ ((heap_Time_pred_Heap @ A) @ P)))))).
% 9.43/3.46 thf(fact_5498_Heap_Opred__rel, axiom, (! [A:$tType]: (((heap_Time_pred_Heap @ A) = (^ [P4:(A > $o),X3:(heap_Time_Heap @ A)]: ((((((heap_Time_rel_Heap @ A) @ A) @ ((bNF_eq_onp @ A) @ P4)) @ X3) @ X3))))))).
% 9.43/3.46 thf(fact_5499_Heap_Opred__set, axiom, (! [A:$tType]: (((heap_Time_pred_Heap @ A) = (^ [P4:(A > $o),X3:(heap_Time_Heap @ A)]: ((! [Y3:A]: (((((member @ A) @ Y3) @ ((heap_Time_set_Heap @ A) @ X3)) => (P4 @ Y3)))))))))).
% 9.43/3.46 thf(fact_5500_Pair__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,A4:(A > (B > $o)),B6:(C > (D > $o))]: (((((((((bNF_rel_fun @ A) @ B) @ (C > ((product_prod @ A) @ C))) @ (D > ((product_prod @ B) @ D))) @ A4) @ ((((((bNF_rel_fun @ C) @ D) @ ((product_prod @ A) @ C)) @ ((product_prod @ B) @ D)) @ B6) @ ((((((basic_rel_prod @ A) @ B) @ C) @ D) @ A4) @ B6))) @ ((product_Pair @ A) @ C)) @ ((product_Pair @ B) @ D))))).
% 9.43/3.46 thf(fact_5501_Heap_Opred__map, axiom, (! [B:$tType,A:$tType,Q:(B > $o),F:(A > B),X:(heap_Time_Heap @ A)]: (((((heap_Time_pred_Heap @ B) @ Q) @ ((((heap_Time_map_Heap @ A) @ B) @ F) @ X)) = (((heap_Time_pred_Heap @ A) @ (((((comp @ B) @ $o) @ A) @ Q) @ F)) @ X))))).
% 9.43/3.46 thf(fact_5502_Heap_Ocase__transfer, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,R5:(A > (B > $o)),S2:(C > (D > $o))]: (((((((((bNF_rel_fun @ (((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > C)) @ (((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > D)) @ ((heap_Time_Heap @ A) > C)) @ ((heap_Time_Heap @ B) > D)) @ ((((((bNF_rel_fun @ ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ ((heap_ext @ product_unit) > (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ C) @ D) @ ((((((bNF_rel_fun @ (heap_ext @ product_unit)) @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Y5:(heap_ext @ product_unit),Z4:(heap_ext @ product_unit)]: ((Y5 = Z4)))) @ (((rel_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((product_prod @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((((basic_rel_prod @ A) @ B) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R5) @ (^ [Y5:((product_prod @ (heap_ext @ product_unit)) @ nat),Z4:((product_prod @ (heap_ext @ product_unit)) @ nat)]: ((Y5 = Z4))))))) @ S2)) @ ((((((bNF_rel_fun @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ C) @ D) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) @ S2)) @ ((heap_Time_case_Heap @ A) @ C)) @ ((heap_Time_case_Heap @ B) @ D))))).
% 9.43/3.46 thf(fact_5503_sndOp__def, axiom, (! [B:$tType,A:$tType,C:$tType]: (((((bNF_sndOp @ C) @ A) @ B) = (^ [P4:(C > (A > $o)),Q6:(A > (B > $o)),Ac:((product_prod @ C) @ B)]: (((((product_Pair @ A) @ B) @ (((((((bNF_pick_middlep @ C) @ A) @ B) @ P4) @ Q6) @ (((product_fst @ C) @ B) @ Ac)) @ (((product_snd @ C) @ B) @ Ac))) @ (((product_snd @ C) @ B) @ Ac)))))))).
% 9.43/3.46 thf(fact_5504_fstOp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((bNF_fstOp @ A) @ B) @ C) = (^ [P4:(A > (B > $o)),Q6:(B > (C > $o)),Ac:((product_prod @ A) @ C)]: (((((product_Pair @ A) @ B) @ (((product_fst @ A) @ C) @ Ac)) @ (((((((bNF_pick_middlep @ A) @ B) @ C) @ P4) @ Q6) @ (((product_fst @ A) @ C) @ Ac)) @ (((product_snd @ A) @ C) @ Ac))))))))).
% 9.43/3.46 thf(fact_5505_Heap_Ocase__distrib, axiom, (! [C:$tType,B:$tType,A:$tType,H3:(B > C),F:(((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > B),Heap:(heap_Time_Heap @ A)]: (((H3 @ ((((heap_Time_case_Heap @ A) @ B) @ F) @ Heap)) = ((((heap_Time_case_Heap @ A) @ C) @ (^ [X3:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((H3 @ (F @ X3))))) @ Heap))))).
% 9.43/3.46 thf(fact_5506_Heap_Ocase, axiom, (! [B:$tType,A:$tType,F:(((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) > B),X:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: ((((((heap_Time_case_Heap @ A) @ B) @ F) @ ((heap_Time_Heap2 @ A) @ X)) = (F @ X))))).
% 9.43/3.46 thf(fact_5507_trancl__def, axiom, (! [A:$tType]: (((transitive_trancl @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ ((transitive_tranclp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3)))))))))))).
% 9.43/3.46 thf(fact_5508_Nitpick_Otranclp__unfold, axiom, (! [A:$tType]: (((transitive_tranclp @ A) = (^ [R3:(A > (A > $o)),A7:A,B5:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ ((transitive_trancl @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.46 thf(fact_5509_reflp__refl__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((reflp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = (((refl_on @ A) @ (top_top @ (set @ A))) @ R))))).
% 9.43/3.46 thf(fact_5510_tranclp__induct2, axiom, (! [A:$tType,B:$tType,R:(((product_prod @ A) @ B) > (((product_prod @ A) @ B) > $o)),Ax:A,Ay:B,Bx:A,By:B,P:(A > (B > $o))]: ((((((transitive_tranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By)) => ((! [A5:A,B3:B]: ((((R @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((P @ A5) @ B3)))) => ((! [A5:A,B3:B,Aa3:A,Ba:B]: ((((((transitive_tranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) => (((R @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba)) => (((P @ A5) @ B3) => ((P @ Aa3) @ Ba)))))) => ((P @ Bx) @ By))))))).
% 9.43/3.46 thf(fact_5511_Heap_Orel__reflp, axiom, (! [A:$tType,R5:(A > (A > $o))]: ((((reflp @ A) @ R5) => ((reflp @ (heap_Time_Heap @ A)) @ (((heap_Time_rel_Heap @ A) @ A) @ R5)))))).
% 9.43/3.46 thf(fact_5512_option_Orel__reflp, axiom, (! [A:$tType,R5:(A > (A > $o))]: ((((reflp @ A) @ R5) => ((reflp @ (option @ A)) @ (((rel_option @ A) @ A) @ R5)))))).
% 9.43/3.46 thf(fact_5513_tranclp__trancl__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_tranclp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ ((transitive_trancl @ A) @ R)))))))).
% 9.43/3.46 thf(fact_5514_Lcm__fin__1__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A4:(set @ A)]: (((((semiring_gcd_Lcm_fin @ A) @ A4) = (one_one @ A)) = ((! [X3:A]: (((((member @ A) @ X3) @ A4) => (((dvd_dvd @ A) @ X3) @ (one_one @ A))))) & ((finite_finite2 @ A) @ A4))))))))).
% 9.43/3.46 thf(fact_5515_heap__interpretation, axiom, (! [A:$tType]: ((((partia7178651479351089652itions @ (heap_Time_Heap @ A)) @ (heap_Time_Heap_ord @ A)) @ (heap_Time_Heap_lub @ A))))).
% 9.43/3.46 thf(fact_5516_old_Orec__bool__def, axiom, (! [T:$tType]: (((product_rec_bool @ T) = (^ [F12:T,F23:T,X3:$o]: (((the @ T) @ ((((product_rec_set_bool @ T) @ F12) @ F23) @ X3)))))))).
% 9.43/3.46 thf(fact_5517_old_Obool_Osimps_I6_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $false) = F22)))).
% 9.43/3.46 thf(fact_5518_old_Obool_Osimps_I5_J, axiom, (! [T:$tType,F1:T,F22:T]: ((((((product_rec_bool @ T) @ F1) @ F22) @ $true) = F1)))).
% 9.43/3.46 thf(fact_5519_Lcm__fin_Oempty, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((semiring_gcd_Lcm_fin @ A) @ (bot_bot @ (set @ A))) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_5520_is__unit__Lcm__fin__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A4:(set @ A)]: (((((dvd_dvd @ A) @ ((semiring_gcd_Lcm_fin @ A) @ A4)) @ (one_one @ A)) = (((semiring_gcd_Lcm_fin @ A) @ A4) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5521_Lcm__fin_Oeq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Lcm_fin @ A) = (^ [A9:(set @ A)]: (((((if @ A) @ ((finite_finite2 @ A) @ A9)) @ (((((finite_fold @ A) @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ A9)) @ (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_5522_Lcm__fin__def, axiom, (! [A:$tType]: (((semiring_gcd @ A) => ((semiring_gcd_Lcm_fin @ A) = ((((bounde2362111253966948842tice_F @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ (zero_zero @ A))))))).
% 9.43/3.46 thf(fact_5523_fun_Opred__map, axiom, (! [B:$tType,A:$tType,D:$tType,Q:(B > $o),F:(A > B),X:(D > A)]: (((((((basic_pred_fun @ D) @ B) @ (^ [Uu:D]: ($true))) @ Q) @ (((((comp @ A) @ B) @ D) @ F) @ X)) = (((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (((((comp @ B) @ $o) @ A) @ Q) @ F)) @ X))))).
% 9.43/3.46 thf(fact_5524_lcm__eq__1__iff, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: ((((((gcd_lcm @ A) @ A3) @ B2) = (one_one @ A)) = ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) & (((dvd_dvd @ A) @ B2) @ (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5525_unit__factor__lcm, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A]: (((((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A))) => (((unit_f5069060285200089521factor @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) = (zero_zero @ A))) & ((~ (((A3 = (zero_zero @ A)) | (B2 = (zero_zero @ A))))) => (((unit_f5069060285200089521factor @ A) @ (((gcd_lcm @ A) @ A3) @ B2)) = (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5526_fun_Omap__cong__pred, axiom, (! [B:$tType,A:$tType,D:$tType,X:(D > A),Ya:(D > A),F:(A > B),G:(A > B)]: (((X = Ya) => ((((((basic_pred_fun @ D) @ A) @ (^ [Uu:D]: ($true))) @ (^ [Z5:A]: (((F @ Z5) = (G @ Z5))))) @ Ya) => ((((((comp @ A) @ B) @ D) @ F) @ X) = (((((comp @ A) @ B) @ D) @ G) @ Ya))))))).
% 9.43/3.46 thf(fact_5527_lcm__mult__unit2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_lcm @ A) @ B2) @ (((times_times @ A) @ C2) @ A3)) = (((gcd_lcm @ A) @ B2) @ C2))))))))).
% 9.43/3.46 thf(fact_5528_lcm__mult__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_lcm @ A) @ (((times_times @ A) @ B2) @ A3)) @ C2) = (((gcd_lcm @ A) @ B2) @ C2))))))))).
% 9.43/3.46 thf(fact_5529_lcm__div__unit1, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_lcm @ A) @ (((divide_divide @ A) @ B2) @ A3)) @ C2) = (((gcd_lcm @ A) @ B2) @ C2))))))))).
% 9.43/3.46 thf(fact_5530_lcm__div__unit2, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A,B2:A,C2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => ((((gcd_lcm @ A) @ B2) @ (((divide_divide @ A) @ C2) @ A3)) = (((gcd_lcm @ A) @ B2) @ C2))))))))).
% 9.43/3.46 thf(fact_5531_Lcm__nat__def, axiom, ((gcd_Lcm @ nat) = (^ [M8:(set @ nat)]: (((((if @ nat) @ ((finite_finite2 @ nat) @ M8)) @ ((((lattic5214292709420241887eutr_F @ nat) @ (gcd_lcm @ nat)) @ (one_one @ nat)) @ M8)) @ (zero_zero @ nat)))))).
% 9.43/3.46 thf(fact_5532_Heap_Opred__inject, axiom, (! [A:$tType,P:(A > $o),A3:((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))]: (((((heap_Time_pred_Heap @ A) @ P) @ ((heap_Time_Heap2 @ A) @ A3)) = (((((basic_pred_fun @ (heap_ext @ product_unit)) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (^ [Uu:(heap_ext @ product_unit)]: ($true))) @ ((pred_option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((basic_pred_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ P) @ (top_top @ (((product_prod @ (heap_ext @ product_unit)) @ nat) > $o))))) @ A3))))).
% 9.43/3.46 thf(fact_5533_Lcm__fin_Oset__eq__fold, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [Xs:(list @ A)]: ((((semiring_gcd_Lcm_fin @ A) @ ((set2 @ A) @ Xs)) = (((((fold @ A) @ A) @ (gcd_lcm @ A)) @ Xs) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5534_pred__prod__inject, axiom, (! [A:$tType,B:$tType,P14:(A > $o),P25:(B > $o),A3:A,B2:B]: (((((((basic_pred_prod @ A) @ B) @ P14) @ P25) @ ((((product_Pair @ A) @ B) @ A3) @ B2)) = ((P14 @ A3) & (P25 @ B2)))))).
% 9.43/3.46 thf(fact_5535_fold__append, axiom, (! [A:$tType,B:$tType,F:(B > (A > A)),Xs:(list @ B),Ys2:(list @ B)]: ((((((fold @ B) @ A) @ F) @ (((append @ B) @ Xs) @ Ys2)) = (((((comp @ A) @ A) @ A) @ ((((fold @ B) @ A) @ F) @ Ys2)) @ ((((fold @ B) @ A) @ F) @ Xs)))))).
% 9.43/3.46 thf(fact_5536_pred__prod__split, axiom, (! [B:$tType,A:$tType,P:($o > $o),Q:(A > $o),R5:(B > $o),Xy2:((product_prod @ A) @ B)]: (((P @ (((((basic_pred_prod @ A) @ B) @ Q) @ R5) @ Xy2)) = (! [X3:A,Y3:B]: (((Xy2 = ((((product_Pair @ A) @ B) @ X3) @ Y3)) => (P @ ((Q @ X3) & (R5 @ Y3)))))))))).
% 9.43/3.46 thf(fact_5537_pred__prod_Ocases, axiom, (! [A:$tType,B:$tType,P14:(A > $o),P25:(B > $o),A3:((product_prod @ A) @ B)]: (((((((basic_pred_prod @ A) @ B) @ P14) @ P25) @ A3) => (~ ((! [A5:A,B3:B]: (((A3 = ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((P14 @ A5) => (~ ((P25 @ B3))))))))))))).
% 9.43/3.46 thf(fact_5538_pred__prod_Osimps, axiom, (! [B:$tType,A:$tType]: ((((basic_pred_prod @ A) @ B) = (^ [P12:(A > $o),P23:(B > $o),A7:((product_prod @ A) @ B)]: ((? [B5:A,C3:B]: (((A7 = ((((product_Pair @ A) @ B) @ B5) @ C3)) & ((P12 @ B5) & (P23 @ C3))))))))))).
% 9.43/3.46 thf(fact_5539_pred__prod_Ointros, axiom, (! [A:$tType,B:$tType,P14:(A > $o),A3:A,P25:(B > $o),B2:B]: (((P14 @ A3) => ((P25 @ B2) => (((((basic_pred_prod @ A) @ B) @ P14) @ P25) @ ((((product_Pair @ A) @ B) @ A3) @ B2))))))).
% 9.43/3.46 thf(fact_5540_fold__Cons, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A,Xs:(list @ A)]: ((((((fold @ A) @ B) @ F) @ (((cons @ A) @ X) @ Xs)) = (((((comp @ B) @ B) @ B) @ ((((fold @ A) @ B) @ F) @ Xs)) @ (F @ X)))))).
% 9.43/3.46 thf(fact_5541_fold__commute__apply, axiom, (! [A:$tType,C:$tType,B:$tType,Xs:(list @ A),H3:(B > C),G:(A > (B > B)),F:(A > (C > C)),S:B]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((((comp @ B) @ C) @ B) @ H3) @ (G @ X2)) = (((((comp @ C) @ C) @ B) @ (F @ X2)) @ H3))))) => ((H3 @ (((((fold @ A) @ B) @ G) @ Xs) @ S)) = (((((fold @ A) @ C) @ F) @ Xs) @ (H3 @ S))))))).
% 9.43/3.46 thf(fact_5542_fold__commute, axiom, (! [A:$tType,C:$tType,B:$tType,Xs:(list @ A),H3:(B > C),G:(A > (B > B)),F:(A > (C > C))]: (((! [X2:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((((comp @ B) @ C) @ B) @ H3) @ (G @ X2)) = (((((comp @ C) @ C) @ B) @ (F @ X2)) @ H3))))) => ((((((comp @ B) @ C) @ B) @ H3) @ ((((fold @ A) @ B) @ G) @ Xs)) = (((((comp @ C) @ C) @ B) @ ((((fold @ A) @ C) @ F) @ Xs)) @ H3)))))).
% 9.43/3.46 thf(fact_5543_Lcm__nat__set__eq__fold, axiom, (! [Xs:(list @ nat)]: ((((gcd_Lcm @ nat) @ ((set2 @ nat) @ Xs)) = (((((fold @ nat) @ nat) @ (gcd_lcm @ nat)) @ Xs) @ (one_one @ nat)))))).
% 9.43/3.46 thf(fact_5544_fold__multiset__equiv, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F:(A > (B > B)),Ys2:(list @ A)]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((member @ A) @ Y4) @ ((set2 @ A) @ Xs)) => ((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y4)) = (((((comp @ B) @ B) @ B) @ (F @ Y4)) @ (F @ X2))))))) => ((((mset @ A) @ Xs) = ((mset @ A) @ Ys2)) => (((((fold @ A) @ B) @ F) @ Xs) = ((((fold @ A) @ B) @ F) @ Ys2))))))).
% 9.43/3.46 thf(fact_5545_fold__rev, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F:(A > (B > B))]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((member @ A) @ Y4) @ ((set2 @ A) @ Xs)) => ((((((comp @ B) @ B) @ B) @ (F @ Y4)) @ (F @ X2)) = (((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y4))))))) => (((((fold @ A) @ B) @ F) @ ((rev @ A) @ Xs)) = ((((fold @ A) @ B) @ F) @ Xs)))))).
% 9.43/3.46 thf(fact_5546_foldr__fold, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F:(A > (B > B))]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((member @ A) @ Y4) @ ((set2 @ A) @ Xs)) => ((((((comp @ B) @ B) @ B) @ (F @ Y4)) @ (F @ X2)) = (((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y4))))))) => (((((foldr @ A) @ B) @ F) @ Xs) = ((((fold @ A) @ B) @ F) @ Xs)))))).
% 9.43/3.46 thf(fact_5547_fold__plus__sum__list__rev, axiom, (! [A:$tType]: (((monoid_add @ A) => (! [Xs:(list @ A)]: ((((((fold @ A) @ A) @ (plus_plus @ A)) @ Xs) = ((plus_plus @ A) @ ((groups8242544230860333062m_list @ A) @ ((rev @ A) @ Xs)))))))))).
% 9.43/3.46 thf(fact_5548_fold__remove1__split, axiom, (! [B:$tType,A:$tType,Xs:(list @ A),F:(A > (B > B)),X:A]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ ((set2 @ A) @ Xs)) => ((((member @ A) @ Y4) @ ((set2 @ A) @ Xs)) => ((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y4)) = (((((comp @ B) @ B) @ B) @ (F @ Y4)) @ (F @ X2))))))) => ((((member @ A) @ X) @ ((set2 @ A) @ Xs)) => (((((fold @ A) @ B) @ F) @ Xs) = (((((comp @ B) @ B) @ B) @ ((((fold @ A) @ B) @ F) @ (((remove1 @ A) @ X) @ Xs))) @ (F @ X)))))))).
% 9.43/3.46 thf(fact_5549_Lcm__set__eq__fold, axiom, (! [A:$tType]: (((semiring_Gcd @ A) => (! [Xs:(list @ A)]: ((((gcd_Lcm @ A) @ ((set2 @ A) @ Xs)) = (((((fold @ A) @ A) @ (gcd_lcm @ A)) @ Xs) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5550_subset__mset_Omax__add__distrib__left, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A),Z2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ ((((max @ (multiset @ A)) @ (subseteq_mset @ A)) @ X) @ Y)) @ Z2) = ((((max @ (multiset @ A)) @ (subseteq_mset @ A)) @ (((plus_plus @ (multiset @ A)) @ X) @ Z2)) @ (((plus_plus @ (multiset @ A)) @ Y) @ Z2)))))).
% 9.43/3.46 thf(fact_5551_subset__mset_Omax__add__distrib__right, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A),Z2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ X) @ ((((max @ (multiset @ A)) @ (subseteq_mset @ A)) @ Y) @ Z2)) = ((((max @ (multiset @ A)) @ (subseteq_mset @ A)) @ (((plus_plus @ (multiset @ A)) @ X) @ Y)) @ (((plus_plus @ (multiset @ A)) @ X) @ Z2)))))).
% 9.43/3.46 thf(fact_5552_prod__list_Ocomm__monoid__list__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((groups1828464146339083142d_list @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_5553_sum__list_Ocomm__monoid__list__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((groups1828464146339083142d_list @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_5554_lazI, axiom, (! [A:$tType,B:$tType,A3:(list @ A),B2:(list @ B),P:(A > (B > $o))]: (((((size_size @ (list @ A)) @ A3) = ((size_size @ (list @ B)) @ B2)) => ((! [I2:nat]: (((((ord_less @ nat) @ I2) @ ((size_size @ (list @ B)) @ B2)) => ((P @ (((nth @ A) @ A3) @ I2)) @ (((nth @ B) @ B2) @ I2))))) => (((((list_all_zip @ A) @ B) @ P) @ A3) @ B2)))))).
% 9.43/3.46 thf(fact_5555_list__all__zip__alt, axiom, (! [B:$tType,A:$tType]: ((((list_all_zip @ A) @ B) = (^ [P4:(A > (B > $o)),As5:(list @ A),Bs3:(list @ B)]: (((((size_size @ (list @ A)) @ As5) = ((size_size @ (list @ B)) @ Bs3)) & (! [I3:nat]: (((((ord_less @ nat) @ I3) @ ((size_size @ (list @ A)) @ As5)) => ((P4 @ (((nth @ A) @ As5) @ I3)) @ (((nth @ B) @ Bs3) @ I3)))))))))))).
% 9.43/3.46 thf(fact_5556_is__none__bind, axiom, (! [A:$tType,B:$tType,F:(option @ B),G:(B > (option @ A))]: ((((is_none @ A) @ ((((bind @ B) @ A) @ F) @ G)) = (((is_none @ B) @ F) | ((is_none @ A) @ (G @ ((the2 @ B) @ F)))))))).
% 9.43/3.46 thf(fact_5557_laz__conj, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),Q:(A > (B > $o)),A3:(list @ A),B2:(list @ B)]: (((((((list_all_zip @ A) @ B) @ (^ [X3:A,Y3:B]: ((((P @ X3) @ Y3) & ((Q @ X3) @ Y3))))) @ A3) @ B2) = ((((((list_all_zip @ A) @ B) @ P) @ A3) @ B2) & (((((list_all_zip @ A) @ B) @ Q) @ A3) @ B2)))))).
% 9.43/3.46 thf(fact_5558_is__none__code_I2_J, axiom, (! [B:$tType,X:B]: ((~ (((is_none @ B) @ ((some @ B) @ X))))))).
% 9.43/3.46 thf(fact_5559_is__none__code_I1_J, axiom, (! [A:$tType]: (((is_none @ A) @ (none @ A))))).
% 9.43/3.46 thf(fact_5560_is__none__map__option, axiom, (! [A:$tType,B:$tType,F:(B > A),X:(option @ B)]: ((((is_none @ A) @ ((((map_option @ B) @ A) @ F) @ X)) = ((is_none @ B) @ X))))).
% 9.43/3.46 thf(fact_5561_laz__weak__Pb, axiom, (! [A:$tType,B:$tType,P:(B > $o),A4:(list @ A),B6:(list @ B)]: (((((((list_all_zip @ A) @ B) @ (^ [A7:A]: (P))) @ A4) @ B6) = ((((size_size @ (list @ A)) @ A4) = ((size_size @ (list @ B)) @ B6)) & (! [X3:B]: (((((member @ B) @ X3) @ ((set2 @ B) @ B6)) => (P @ X3))))))))).
% 9.43/3.46 thf(fact_5562_laz__weak__Pa, axiom, (! [B:$tType,A:$tType,P:(A > $o),A4:(list @ A),B6:(list @ B)]: (((((((list_all_zip @ A) @ B) @ (^ [A7:A,B5:B]: ((P @ A7)))) @ A4) @ B6) = ((((size_size @ (list @ A)) @ A4) = ((size_size @ (list @ B)) @ B6)) & (! [X3:A]: (((((member @ A) @ X3) @ ((set2 @ A) @ A4)) => (P @ X3))))))))).
% 9.43/3.46 thf(fact_5563_list__all__zip_Osimps_I1_J, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o))]: ((((((list_all_zip @ A) @ B) @ P) @ (nil @ A)) @ (nil @ B))))).
% 9.43/3.46 thf(fact_5564_laz__eq, axiom, (! [A:$tType,A3:(list @ A),B2:(list @ A)]: (((((((list_all_zip @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))) @ A3) @ B2) = (A3 = B2))))).
% 9.43/3.46 thf(fact_5565_laz__len, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),A3:(list @ A),B2:(list @ B)]: (((((((list_all_zip @ A) @ B) @ P) @ A3) @ B2) => (((size_size @ (list @ A)) @ A3) = ((size_size @ (list @ B)) @ B2)))))).
% 9.43/3.46 thf(fact_5566_laz__swap__ex, axiom, (! [B:$tType,A:$tType,C:$tType,P:(A > (B > (C > $o))),A4:(list @ A),B6:(list @ B)]: (((((((list_all_zip @ A) @ B) @ (^ [A7:A,B5:B]: ((? [X8:C]: ((((P @ A7) @ B5) @ X8)))))) @ A4) @ B6) => (~ ((! [C8:(list @ C)]: (((((((list_all_zip @ A) @ C) @ (^ [A7:A,C3:C]: ((? [B5:B]: ((((P @ A7) @ B5) @ C3)))))) @ A4) @ C8) => (~ ((((((list_all_zip @ B) @ C) @ (^ [B5:B,C3:C]: ((? [A7:A]: ((((P @ A7) @ B5) @ C3)))))) @ B6) @ C8)))))))))))).
% 9.43/3.46 thf(fact_5567_is__none__simps_I2_J, axiom, (! [B:$tType,X:B]: ((~ (((is_none @ B) @ ((some @ B) @ X))))))).
% 9.43/3.46 thf(fact_5568_list__all__zip__map2, axiom, (! [A:$tType,B:$tType,C:$tType,P:(A > (B > $o)),As2:(list @ A),F:(C > B),Bs2:(list @ C)]: (((((((list_all_zip @ A) @ B) @ P) @ As2) @ ((((map @ C) @ B) @ F) @ Bs2)) = (((((list_all_zip @ A) @ C) @ (^ [A7:A,B5:C]: (((P @ A7) @ (F @ B5))))) @ As2) @ Bs2))))).
% 9.43/3.46 thf(fact_5569_list__all__zip__map1, axiom, (! [C:$tType,A:$tType,B:$tType,P:(A > (B > $o)),F:(C > A),As2:(list @ C),Bs2:(list @ B)]: (((((((list_all_zip @ A) @ B) @ P) @ ((((map @ C) @ A) @ F) @ As2)) @ Bs2) = (((((list_all_zip @ C) @ B) @ (^ [A7:C]: ((P @ (F @ A7))))) @ As2) @ Bs2))))).
% 9.43/3.46 thf(fact_5570_is__none__simps_I1_J, axiom, (! [A:$tType]: (((is_none @ A) @ (none @ A))))).
% 9.43/3.46 thf(fact_5571_Option_Ois__none__def, axiom, (! [A:$tType]: (((is_none @ A) = (^ [X3:(option @ A)]: ((X3 = (none @ A)))))))).
% 9.43/3.46 thf(fact_5572_list__all__zip_Osimps_I2_J, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),A3:A,As2:(list @ A),B2:B,Bs2:(list @ B)]: (((((((list_all_zip @ A) @ B) @ P) @ (((cons @ A) @ A3) @ As2)) @ (((cons @ B) @ B2) @ Bs2)) = (((P @ A3) @ B2) & (((((list_all_zip @ A) @ B) @ P) @ As2) @ Bs2)))))).
% 9.43/3.46 thf(fact_5573_list__all__zip_Oelims_I3_J, axiom, (! [A:$tType,B:$tType,X:(A > (B > $o)),Xa:(list @ A),Xb:(list @ B)]: (((~ ((((((list_all_zip @ A) @ B) @ X) @ Xa) @ Xb))) => ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => (((X @ A5) @ B3) & (((((list_all_zip @ A) @ B) @ X) @ As) @ Bs)))))))) => (((? [V4:A,Va:(list @ A)]: ((Xa = (((cons @ A) @ V4) @ Va)))) => (Xb != (nil @ B))) => (~ (((Xa = (nil @ A)) => (! [V4:B,Va:(list @ B)]: ((Xb != (((cons @ B) @ V4) @ Va))))))))))))).
% 9.43/3.46 thf(fact_5574_list__all__zip_Oelims_I2_J, axiom, (! [A:$tType,B:$tType,X:(A > (B > $o)),Xa:(list @ A),Xb:(list @ B)]: (((((((list_all_zip @ A) @ B) @ X) @ Xa) @ Xb) => (((Xa = (nil @ A)) => (Xb != (nil @ B))) => (~ ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => (~ ((((X @ A5) @ B3) & (((((list_all_zip @ A) @ B) @ X) @ As) @ Bs))))))))))))))))).
% 9.43/3.46 thf(fact_5575_list__all__zip_Oelims_I1_J, axiom, (! [B:$tType,A:$tType,X:(A > (B > $o)),Xa:(list @ A),Xb:(list @ B),Y:$o]: ((((((((list_all_zip @ A) @ B) @ X) @ Xa) @ Xb) = Y) => (((Xa = (nil @ A)) => ((Xb = (nil @ B)) => (~ (Y)))) => ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => (Y = (~ ((((X @ A5) @ B3) & (((((list_all_zip @ A) @ B) @ X) @ As) @ Bs))))))))))) => (((? [V4:A,Va:(list @ A)]: ((Xa = (((cons @ A) @ V4) @ Va)))) => ((Xb = (nil @ B)) => Y)) => (~ (((Xa = (nil @ A)) => ((? [V4:B,Va:(list @ B)]: ((Xb = (((cons @ B) @ V4) @ Va)))) => Y))))))))))).
% 9.43/3.46 thf(fact_5576_list__all__zip_Osimps_I4_J, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),V:B,Va2:(list @ B)]: ((~ ((((((list_all_zip @ A) @ B) @ P) @ (nil @ A)) @ (((cons @ B) @ V) @ Va2))))))).
% 9.43/3.46 thf(fact_5577_list__all__zip_Osimps_I3_J, axiom, (! [A:$tType,B:$tType,P:(A > (B > $o)),V:A,Va2:(list @ A)]: ((~ ((((((list_all_zip @ A) @ B) @ P) @ (((cons @ A) @ V) @ Va2)) @ (nil @ B))))))).
% 9.43/3.46 thf(fact_5578_the__map__option, axiom, (! [B:$tType,A:$tType,X:(option @ A),F:(A > B)]: (((~ (((is_none @ A) @ X))) => (((the2 @ B) @ ((((map_option @ A) @ B) @ F) @ X)) = (F @ ((the2 @ A) @ X))))))).
% 9.43/3.46 thf(fact_5579_rel__optionI, axiom, (! [A:$tType,B:$tType,X:(option @ A),Y:(option @ B),P:(A > (B > $o))]: (((((is_none @ A) @ X) = ((is_none @ B) @ Y)) => (((~ (((is_none @ A) @ X))) => ((~ (((is_none @ B) @ Y))) => ((P @ ((the2 @ A) @ X)) @ ((the2 @ B) @ Y)))) => (((((rel_option @ A) @ B) @ P) @ X) @ Y)))))).
% 9.43/3.46 thf(fact_5580_rel__option__unfold, axiom, (! [B:$tType,A:$tType]: ((((rel_option @ A) @ B) = (^ [R6:(A > (B > $o)),X3:(option @ A),Y3:(option @ B)]: (((((is_none @ A) @ X3) = ((is_none @ B) @ Y3)) & ((~ (((is_none @ A) @ X3))) => ((~ (((is_none @ B) @ Y3))) => ((R6 @ ((the2 @ A) @ X3)) @ ((the2 @ B) @ Y3))))))))))).
% 9.43/3.46 thf(fact_5581_list__all__zip_Opelims_I1_J, axiom, (! [A:$tType,B:$tType,X:(A > (B > $o)),Xa:(list @ A),Xb:(list @ B),Y:$o]: ((((((((list_all_zip @ A) @ B) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xa) @ Xb))) => (((Xa = (nil @ A)) => ((Xb = (nil @ B)) => (Y => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B))))))))) => ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => ((Y = (((X @ A5) @ B3) & (((((list_all_zip @ A) @ B) @ X) @ As) @ Bs))) => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B3) @ Bs))))))))))))) => ((! [V4:A,Va:(list @ A)]: (((Xa = (((cons @ A) @ V4) @ Va)) => ((Xb = (nil @ B)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ B))))))))))) => (~ (((Xa = (nil @ A)) => (! [V4:B,Va:(list @ B)]: (((Xb = (((cons @ B) @ V4) @ Va)) => ((~ (Y)) => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (((cons @ B) @ V4) @ Va))))))))))))))))))))).
% 9.43/3.46 thf(fact_5582_list__all__zip_Opelims_I2_J, axiom, (! [A:$tType,B:$tType,X:(A > (B > $o)),Xa:(list @ A),Xb:(list @ B)]: (((((((list_all_zip @ A) @ B) @ X) @ Xa) @ Xb) => ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xa) @ Xb))) => (((Xa = (nil @ A)) => ((Xb = (nil @ B)) => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (nil @ B)))))))) => (~ ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B3) @ Bs)))) => (~ ((((X @ A5) @ B3) & (((((list_all_zip @ A) @ B) @ X) @ As) @ Bs))))))))))))))))))).
% 9.43/3.46 thf(fact_5583_list__all__zip_Opelims_I3_J, axiom, (! [A:$tType,B:$tType,X:(A > (B > $o)),Xa:(list @ A),Xb:(list @ B)]: (((~ ((((((list_all_zip @ A) @ B) @ X) @ Xa) @ Xb))) => ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ Xa) @ Xb))) => ((! [A5:A,As:(list @ A)]: (((Xa = (((cons @ A) @ A5) @ As)) => (! [B3:B,Bs:(list @ B)]: (((Xb = (((cons @ B) @ B3) @ Bs)) => ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ A5) @ As)) @ (((cons @ B) @ B3) @ Bs)))) => (((X @ A5) @ B3) & (((((list_all_zip @ A) @ B) @ X) @ As) @ Bs))))))))) => ((! [V4:A,Va:(list @ A)]: (((Xa = (((cons @ A) @ V4) @ Va)) => ((Xb = (nil @ B)) => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (((cons @ A) @ V4) @ Va)) @ (nil @ B)))))))))) => (~ (((Xa = (nil @ A)) => (! [V4:B,Va:(list @ B)]: (((Xb = (((cons @ B) @ V4) @ Va)) => (~ ((((accp @ ((product_prod @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B)))) @ ((list_all_zip_rel @ A) @ B)) @ ((((product_Pair @ (A > (B > $o))) @ ((product_prod @ (list @ A)) @ (list @ B))) @ X) @ ((((product_Pair @ (list @ A)) @ (list @ B)) @ (nil @ A)) @ (((cons @ B) @ V4) @ Va))))))))))))))))))).
% 9.43/3.46 thf(fact_5584_plus__multiset_Otransfer, axiom, (! [A:$tType]: (((((((((bNF_rel_fun @ (A > nat)) @ (multiset @ A)) @ ((A > nat) > (A > nat))) @ ((multiset @ A) > (multiset @ A))) @ (((pcr_multiset @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4))))) @ ((((((bNF_rel_fun @ (A > nat)) @ (multiset @ A)) @ (A > nat)) @ (multiset @ A)) @ (((pcr_multiset @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4))))) @ (((pcr_multiset @ A) @ A) @ (^ [Y5:A,Z4:A]: ((Y5 = Z4)))))) @ (^ [M8:(A > nat),N9:(A > nat),A7:A]: ((((plus_plus @ nat) @ (M8 @ A7)) @ (N9 @ A7))))) @ (plus_plus @ (multiset @ A)))))).
% 9.43/3.46 thf(fact_5585_map__option__o__case__sum, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,F:(D > C),G:(A > (option @ D)),H3:(B > (option @ D))]: (((((((comp @ (option @ D)) @ (option @ C)) @ ((sum_sum @ A) @ B)) @ (((map_option @ D) @ C) @ F)) @ (((((sum_case_sum @ A) @ (option @ D)) @ B) @ G) @ H3)) = (((((sum_case_sum @ A) @ (option @ C)) @ B) @ (((((comp @ (option @ D)) @ (option @ C)) @ A) @ (((map_option @ D) @ C) @ F)) @ G)) @ (((((comp @ (option @ D)) @ (option @ C)) @ B) @ (((map_option @ D) @ C) @ F)) @ H3)))))).
% 9.43/3.46 thf(fact_5586_semilattice__set_Oeq__fold_H, axiom, (! [A:$tType,F:(A > (A > A)),A4:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => ((((lattic1715443433743089157tice_F @ A) @ F) @ A4) = ((the2 @ A) @ (((((finite_fold @ A) @ (option @ A)) @ (^ [X3:A,Y3:(option @ A)]: (((some @ A) @ (((((case_option @ A) @ A) @ X3) @ (F @ X3)) @ Y3))))) @ (none @ A)) @ A4))))))).
% 9.43/3.46 thf(fact_5587_o__case__sum, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,H3:(D > C),F:(A > D),G:(B > D)]: (((((((comp @ D) @ C) @ ((sum_sum @ A) @ B)) @ H3) @ (((((sum_case_sum @ A) @ D) @ B) @ F) @ G)) = (((((sum_case_sum @ A) @ C) @ B) @ (((((comp @ D) @ C) @ A) @ H3) @ F)) @ (((((comp @ D) @ C) @ B) @ H3) @ G)))))).
% 9.43/3.46 thf(fact_5588_semilattice__set_Oinfinite, axiom, (! [A:$tType,F:(A > (A > A)),A4:(set @ A)]: ((((lattic149705377957585745ce_set @ A) @ F) => ((~ (((finite_finite2 @ A) @ A4))) => ((((lattic1715443433743089157tice_F @ A) @ F) @ A4) = ((the2 @ A) @ (none @ A)))))))).
% 9.43/3.46 thf(fact_5589_group_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A)]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((((group_axioms @ A) @ F) @ Z2) @ Inverse))))).
% 9.43/3.46 thf(fact_5590_normalize__div, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((divide_divide @ A) @ ((normal6383669964737779283malize @ A) @ A3)) @ A3) = (((divide_divide @ A) @ (one_one @ A)) @ ((unit_f5069060285200089521factor @ A) @ A3))))))))).
% 9.43/3.46 thf(fact_5591_strict__mono__inv, axiom, (! [A:$tType,B:$tType]: ((((linorder @ B) & (linorder @ A)) => (! [F:(A > B),G:(B > A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((((image2 @ A) @ B) @ F) @ (top_top @ (set @ A))) = (top_top @ (set @ B))) => ((! [X2:A]: (((G @ (F @ X2)) = X2))) => (((order_strict_mono @ B) @ A) @ G)))))))))).
% 9.43/3.46 thf(fact_5592_gcd_Onormalize__bottom, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((normal6383669964737779283malize @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_5593_normalize__1, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (((normal6383669964737779283malize @ A) @ (one_one @ A)) = (one_one @ A)))))).
% 9.43/3.46 thf(fact_5594_lcm_Otop__left__normalize, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_lcm @ A) @ (one_one @ A)) @ A3) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 9.43/3.46 thf(fact_5595_lcm_Otop__right__normalize, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (! [A3:A]: (((((gcd_lcm @ A) @ A3) @ (one_one @ A)) = ((normal6383669964737779283malize @ A) @ A3)))))))).
% 9.43/3.46 thf(fact_5596_normalize__mult__unit__right, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [B2:A,A3:A]: (((((dvd_dvd @ A) @ B2) @ (one_one @ A)) => (((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2)) = ((normal6383669964737779283malize @ A) @ A3))))))))).
% 9.43/3.46 thf(fact_5597_normalize__mult__unit__left, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((normal6383669964737779283malize @ A) @ (((times_times @ A) @ A3) @ B2)) = ((normal6383669964737779283malize @ A) @ B2))))))))).
% 9.43/3.46 thf(fact_5598_unit__factor__normalize, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((unit_f5069060285200089521factor @ A) @ ((normal6383669964737779283malize @ A) @ A3)) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5599_normalize__unit__factor, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((A3 != (zero_zero @ A)) => (((normal6383669964737779283malize @ A) @ ((unit_f5069060285200089521factor @ A) @ A3)) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5600_associated__unit, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A,B2:A]: (((((normal6383669964737779283malize @ A) @ A3) = ((normal6383669964737779283malize @ A) @ B2)) => ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((dvd_dvd @ A) @ B2) @ (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5601_normalize__1__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((normal6383669964737779283malize @ A) @ A3) = (one_one @ A)) = (((dvd_dvd @ A) @ A3) @ (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5602_is__unit__normalize, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((dvd_dvd @ A) @ A3) @ (one_one @ A)) => (((normal6383669964737779283malize @ A) @ A3) = (one_one @ A))))))))).
% 9.43/3.46 thf(fact_5603_normalize__idem__imp__is__unit__iff, axiom, (! [A:$tType]: (((normal8620421768224518004emidom @ A) => (! [A3:A]: (((((normal6383669964737779283malize @ A) @ A3) = A3) => ((((dvd_dvd @ A) @ A3) @ (one_one @ A)) = (A3 = (one_one @ A)))))))))).
% 9.43/3.46 thf(fact_5604_group__axioms__def, axiom, (! [A:$tType]: (((group_axioms @ A) = (^ [F2:(A > (A > A)),Z5:A,Inverse2:(A > A)]: (((! [A7:A]: ((((F2 @ Z5) @ A7) = A7))) & (! [A7:A]: ((((F2 @ (Inverse2 @ A7)) @ A7) = Z5)))))))))).
% 9.43/3.46 thf(fact_5605_group__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A)]: (((! [A5:A]: ((((F @ Z2) @ A5) = A5))) => ((! [A5:A]: ((((F @ (Inverse @ A5)) @ A5) = Z2))) => ((((group_axioms @ A) @ F) @ Z2) @ Inverse)))))).
% 9.43/3.46 thf(fact_5606_strict__mono__imp__inj__on, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (order @ B)) => (! [F:(A > B),A4:(set @ A)]: (((((order_strict_mono @ A) @ B) @ F) => ((((inj_on @ A) @ B) @ F) @ A4)))))))).
% 9.43/3.46 thf(fact_5607_Lcm__fin_Obounded__quasi__semilattice__set__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde6485984586167503788ce_set @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ (zero_zero @ A)) @ (normal6383669964737779283malize @ A)))))).
% 9.43/3.46 thf(fact_5608_Gcd__fin_Obounded__quasi__semilattice__set__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde6485984586167503788ce_set @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ (one_one @ A)) @ (normal6383669964737779283malize @ A)))))).
% 9.43/3.46 thf(fact_5609_lcm_Obounded__quasi__semilattice__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde8507323023520639062attice @ A) @ (gcd_lcm @ A)) @ (one_one @ A)) @ (zero_zero @ A)) @ (normal6383669964737779283malize @ A)))))).
% 9.43/3.46 thf(fact_5610_gcd_Obounded__quasi__semilattice__axioms, axiom, (! [A:$tType]: (((semiring_gcd @ A) => (((((bounde8507323023520639062attice @ A) @ (gcd_gcd @ A)) @ (zero_zero @ A)) @ (one_one @ A)) @ (normal6383669964737779283malize @ A)))))).
% 9.43/3.46 thf(fact_5611_group__def, axiom, (! [A:$tType]: (((group @ A) = (^ [F2:(A > (A > A)),Z5:A,Inverse2:(A > A)]: ((((semigroup @ A) @ F2) & ((((group_axioms @ A) @ F2) @ Z5) @ Inverse2)))))))).
% 9.43/3.46 thf(fact_5612_group_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A)]: ((((semigroup @ A) @ F) => (((((group_axioms @ A) @ F) @ Z2) @ Inverse) => ((((group @ A) @ F) @ Z2) @ Inverse)))))).
% 9.43/3.46 thf(fact_5613_Predicate_Oiterate__upto_Opsimps, axiom, (! [A:$tType,F:(code_natural > A),N:code_natural,M2:code_natural]: (((((accp @ ((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural))) @ (iterate_upto_rel @ A)) @ ((((product_Pair @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) @ F) @ ((((product_Pair @ code_natural) @ code_natural) @ N) @ M2))) => (((((iterate_upto @ A) @ F) @ N) @ M2) = ((seq2 @ A) @ (^ [U2:product_unit]: (((((if @ (seq @ A)) @ (((ord_less @ code_natural) @ M2) @ N)) @ (empty @ A)) @ (((insert @ A) @ (F @ N)) @ ((((iterate_upto @ A) @ F) @ (((plus_plus @ code_natural) @ N) @ (one_one @ code_natural))) @ M2))))))))))).
% 9.43/3.46 thf(fact_5614_group_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,Inverse:(A > A)]: ((((((group @ A) @ F) @ Z2) @ Inverse) => ((semigroup @ A) @ F))))).
% 9.43/3.46 thf(fact_5615_monoid_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: (((((monoid @ A) @ F) @ Z2) => ((semigroup @ A) @ F))))).
% 9.43/3.46 thf(fact_5616_semigroup__def, axiom, (! [A:$tType]: (((semigroup @ A) = (^ [F2:(A > (A > A))]: ((! [A7:A,B5:A,C3:A]: ((((F2 @ ((F2 @ A7) @ B5)) @ C3) = ((F2 @ A7) @ ((F2 @ B5) @ C3))))))))))).
% 9.43/3.46 thf(fact_5617_semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A5:A,B3:A,C4:A]: ((((F @ ((F @ A5) @ B3)) @ C4) = ((F @ A5) @ ((F @ B3) @ C4))))) => ((semigroup @ A) @ F))))).
% 9.43/3.46 thf(fact_5618_semigroup_Oassoc, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A,C2:A]: ((((semigroup @ A) @ F) => (((F @ ((F @ A3) @ B2)) @ C2) = ((F @ A3) @ ((F @ B2) @ C2))))))).
% 9.43/3.46 thf(fact_5619_mult_Osemigroup__axioms, axiom, (! [A:$tType]: (((semigroup_mult @ A) => ((semigroup @ A) @ (times_times @ A)))))).
% 9.43/3.46 thf(fact_5620_Predicate_Oiterate__upto_Osimps, axiom, (! [A:$tType]: (((iterate_upto @ A) = (^ [F2:(code_natural > A),N5:code_natural,M:code_natural]: (((seq2 @ A) @ (^ [U2:product_unit]: (((((if @ (seq @ A)) @ (((ord_less @ code_natural) @ M) @ N5)) @ (empty @ A)) @ (((insert @ A) @ (F2 @ N5)) @ ((((iterate_upto @ A) @ F2) @ (((plus_plus @ code_natural) @ N5) @ (one_one @ code_natural))) @ M)))))))))))).
% 9.43/3.46 thf(fact_5621_Predicate_Oiterate__upto_Oelims, axiom, (! [A:$tType,X:(code_natural > A),Xa:code_natural,Xb:code_natural,Y:(pred @ A)]: (((((((iterate_upto @ A) @ X) @ Xa) @ Xb) = Y) => (Y = ((seq2 @ A) @ (^ [U2:product_unit]: (((((if @ (seq @ A)) @ (((ord_less @ code_natural) @ Xb) @ Xa)) @ (empty @ A)) @ (((insert @ A) @ (X @ Xa)) @ ((((iterate_upto @ A) @ X) @ (((plus_plus @ code_natural) @ Xa) @ (one_one @ code_natural))) @ Xb))))))))))).
% 9.43/3.46 thf(fact_5622_add_Osemigroup__axioms, axiom, (! [A:$tType]: (((semigroup_add @ A) => ((semigroup @ A) @ (plus_plus @ A)))))).
% 9.43/3.46 thf(fact_5623_Predicate_Oiterate__upto_Opelims, axiom, (! [A:$tType,X:(code_natural > A),Xa:code_natural,Xb:code_natural,Y:(pred @ A)]: (((((((iterate_upto @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural))) @ (iterate_upto_rel @ A)) @ ((((product_Pair @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) @ X) @ ((((product_Pair @ code_natural) @ code_natural) @ Xa) @ Xb))) => (~ (((Y = ((seq2 @ A) @ (^ [U2:product_unit]: (((((if @ (seq @ A)) @ (((ord_less @ code_natural) @ Xb) @ Xa)) @ (empty @ A)) @ (((insert @ A) @ (X @ Xa)) @ ((((iterate_upto @ A) @ X) @ (((plus_plus @ code_natural) @ Xa) @ (one_one @ code_natural))) @ Xb))))))) => (~ ((((accp @ ((product_prod @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural))) @ (iterate_upto_rel @ A)) @ ((((product_Pair @ (code_natural > A)) @ ((product_prod @ code_natural) @ code_natural)) @ X) @ ((((product_Pair @ code_natural) @ code_natural) @ Xa) @ Xb))))))))))))).
% 9.43/3.46 thf(fact_5624_Random__Pred_Oiterate__upto__def, axiom, (! [A:$tType]: (((random_iterate_upto @ A) = (^ [F2:(code_natural > A),N5:code_natural,M:code_natural]: ((((product_Pair @ (pred @ A)) @ ((product_prod @ code_natural) @ code_natural)) @ ((((iterate_upto @ A) @ F2) @ N5) @ M)))))))).
% 9.43/3.46 thf(fact_5625_monoid_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: ((((semigroup @ A) @ F) => ((((monoid_axioms @ A) @ F) @ Z2) => (((monoid @ A) @ F) @ Z2)))))).
% 9.43/3.46 thf(fact_5626_monoid__def, axiom, (! [A:$tType]: (((monoid @ A) = (^ [F2:(A > (A > A)),Z5:A]: ((((semigroup @ A) @ F2) & (((monoid_axioms @ A) @ F2) @ Z5)))))))).
% 9.43/3.46 thf(fact_5627_monoid__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: (((! [A5:A]: ((((F @ Z2) @ A5) = A5))) => ((! [A5:A]: ((((F @ A5) @ Z2) = A5))) => (((monoid_axioms @ A) @ F) @ Z2)))))).
% 9.43/3.46 thf(fact_5628_monoid__axioms__def, axiom, (! [A:$tType]: (((monoid_axioms @ A) = (^ [F2:(A > (A > A)),Z5:A]: (((! [A7:A]: ((((F2 @ Z5) @ A7) = A7))) & (! [A7:A]: ((((F2 @ A7) @ Z5) = A7)))))))))).
% 9.43/3.46 thf(fact_5629_monoid_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: (((((monoid @ A) @ F) @ Z2) => (((monoid_axioms @ A) @ F) @ Z2))))).
% 9.43/3.46 thf(fact_5630_mod__h__bot__normalize, axiom, (! [A:$tType,H3:(heap_ext @ product_unit),P:assn]: ((((((syntax7388354845996824322omatch @ A) @ (heap_ext @ product_unit)) @ (undefined @ A)) @ H3) => (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = ((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ (undefined @ (heap_ext @ product_unit))) @ (bot_bot @ (set @ nat))))))))).
% 9.43/3.46 thf(fact_5631_image__bind, axiom, (! [A:$tType,B:$tType,C:$tType,F:(B > A),A4:(set @ C),G:(C > (set @ B))]: ((((((image2 @ B) @ A) @ F) @ ((((bind3 @ C) @ B) @ A4) @ G)) = ((((bind3 @ C) @ A) @ A4) @ (((((comp @ (set @ B)) @ (set @ A)) @ C) @ (((image2 @ B) @ A) @ F)) @ G)))))).
% 9.43/3.46 thf(fact_5632_pure__assn__raw_Ocases, axiom, (! [A:$tType,B:$tType,X:((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))]: ((~ ((! [B3:$o,H2:A,As:(set @ B)]: ((X != ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ B3) @ ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)))))))))).
% 9.43/3.46 thf(fact_5633_mod__h__bot__iff_I7_J, axiom, (! [P:assn,Q:assn,H3:(heap_ext @ product_unit)]: ((((rep_assn @ (((sup_sup @ assn) @ P) @ Q)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) | ((rep_assn @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat))))))))).
% 9.43/3.46 thf(fact_5634_mod__h__bot__iff_I5_J, axiom, (! [P:assn,Q:assn,H3:(heap_ext @ product_unit)]: ((((rep_assn @ (((times_times @ assn) @ P) @ Q)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) & ((rep_assn @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat))))))))).
% 9.43/3.46 thf(fact_5635_mod__h__bot__iff_I6_J, axiom, (! [P:assn,Q:assn,H3:(heap_ext @ product_unit)]: ((((rep_assn @ (((inf_inf @ assn) @ P) @ Q)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) & ((rep_assn @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat))))))))).
% 9.43/3.46 thf(fact_5636_mod__h__bot__iff_I2_J, axiom, (! [H3:(heap_ext @ product_unit)]: (((rep_assn @ (top_top @ assn)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat))))))).
% 9.43/3.46 thf(fact_5637_times__assn__raw_Ocases, axiom, (! [X:((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))]: ((~ ((! [P7:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Q7:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),H2:(heap_ext @ product_unit),As:(set @ nat)]: ((X != ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ P7) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Q7) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))))))))).
% 9.43/3.46 thf(fact_5638_sngr__assn__raw_Ocases, axiom, (! [A:$tType]: (((heap @ A) => (! [X:((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))]: ((~ ((! [R2:(ref @ A),X2:A,H2:(heap_ext @ product_unit),As:(set @ nat)]: ((X != ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ R2) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ X2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))))))))))))).
% 9.43/3.46 thf(fact_5639_snga__assn__raw_Ocases, axiom, (! [A:$tType]: (((heap @ A) => (! [X:((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))]: ((~ ((! [R2:(array @ A),X2:(list @ A),H2:(heap_ext @ product_unit),As:(set @ nat)]: ((X != ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ R2) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ X2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))))))))))))).
% 9.43/3.46 thf(fact_5640_one__assn__raw_Ocases, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: ((~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: ((X != ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))))))).
% 9.43/3.46 thf(fact_5641_mod__star__conv, axiom, (! [A4:assn,B6:assn,H3:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: ((((rep_assn @ (((times_times @ assn) @ A4) @ B6)) @ H3) = (? [Hr:(heap_ext @ product_unit),As1:(set @ nat),As22:(set @ nat)]: (((H3 = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ Hr) @ (((sup_sup @ (set @ nat)) @ As1) @ As22))) & (((((inf_inf @ (set @ nat)) @ As1) @ As22) = (bot_bot @ (set @ nat))) & (((rep_assn @ A4) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ Hr) @ As1)) & ((rep_assn @ B6) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ Hr) @ As22))))))))))).
% 9.43/3.46 thf(fact_5642_star__assnI, axiom, (! [P:assn,H3:(heap_ext @ product_unit),As2:(set @ nat),Q:assn,As6:(set @ nat)]: ((((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (((rep_assn @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As6)) => (((((inf_inf @ (set @ nat)) @ As2) @ As6) = (bot_bot @ (set @ nat))) => ((rep_assn @ (((times_times @ assn) @ P) @ Q)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (((sup_sup @ (set @ nat)) @ As2) @ As6))))))))).
% 9.43/3.46 thf(fact_5643_mod__emp__simp, axiom, (! [H3:(heap_ext @ product_unit)]: (((rep_assn @ (one_one @ assn)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat))))))).
% 9.43/3.46 thf(fact_5644_mod__h__bot__indep, axiom, (! [P:assn,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit)]: ((((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = ((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ (bot_bot @ (set @ nat)))))))).
% 9.43/3.46 thf(fact_5645_bind__image, axiom, (! [A:$tType,B:$tType,C:$tType,F:(C > B),A4:(set @ C),G:(B > (set @ A))]: ((((((bind3 @ B) @ A) @ ((((image2 @ C) @ B) @ F) @ A4)) @ G) = ((((bind3 @ C) @ A) @ A4) @ (((((comp @ B) @ (set @ A)) @ C) @ G) @ F)))))).
% 9.43/3.46 thf(fact_5646_mod__h__bot__iff_I3_J, axiom, (! [A:$tType]: (((heap @ A) => (! [P5:(ref @ A),X:A,H3:(heap_ext @ product_unit)]: ((~ (((rep_assn @ (((sngr_assn @ A) @ P5) @ X)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))))))))))).
% 9.43/3.46 thf(fact_5647_mod__h__bot__iff_I4_J, axiom, (! [B:$tType]: (((heap @ B) => (! [Q3:(array @ B),Y:(list @ B),H3:(heap_ext @ product_unit)]: ((~ (((rep_assn @ (((snga_assn @ B) @ Q3) @ Y)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))))))))))).
% 9.43/3.46 thf(fact_5648_times__assn__raw_Osimps, axiom, (! [P:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Q:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),H3:(heap_ext @ product_unit),As2:(set @ nat)]: (((((times_assn_raw @ P) @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = (? [As1:(set @ nat),As22:(set @ nat)]: (((As2 = (((sup_sup @ (set @ nat)) @ As1) @ As22)) & (((((inf_inf @ (set @ nat)) @ As1) @ As22) = (bot_bot @ (set @ nat))) & ((P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As1)) & (Q @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As22))))))))))).
% 9.43/3.46 thf(fact_5649_times__assn__raw_Oelims_I3_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((((times_assn_raw @ X) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (? [As12:(set @ nat),As23:(set @ nat)]: (((As = (((sup_sup @ (set @ nat)) @ As12) @ As23)) & (((((inf_inf @ (set @ nat)) @ As12) @ As23) = (bot_bot @ (set @ nat))) & ((X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As12)) & (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As23)))))))))))))))).
% 9.43/3.46 thf(fact_5650_times__assn__raw_Oelims_I2_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((((times_assn_raw @ X) @ Xa) @ Xb) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (~ ((? [As13:(set @ nat),As24:(set @ nat)]: (((As = (((sup_sup @ (set @ nat)) @ As13) @ As24)) & (((((inf_inf @ (set @ nat)) @ As13) @ As24) = (bot_bot @ (set @ nat))) & ((X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As13)) & (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As24)))))))))))))))))).
% 9.43/3.46 thf(fact_5651_times__assn__raw_Oelims_I1_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((((times_assn_raw @ X) @ Xa) @ Xb) = Y) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (Y = (~ ((? [As1:(set @ nat),As22:(set @ nat)]: (((As = (((sup_sup @ (set @ nat)) @ As1) @ As22)) & (((((inf_inf @ (set @ nat)) @ As1) @ As22) = (bot_bot @ (set @ nat))) & ((X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As1)) & (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As22))))))))))))))))))).
% 9.43/3.46 thf(fact_5652_times__assn__raw_Opelims_I1_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((((times_assn_raw @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ times_assn_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((Y = (? [As1:(set @ nat),As22:(set @ nat)]: (((As = (((sup_sup @ (set @ nat)) @ As1) @ As22)) & (((((inf_inf @ (set @ nat)) @ As1) @ As22) = (bot_bot @ (set @ nat))) & ((X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As1)) & (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As22)))))))) => (~ ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ times_assn_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))))))))))))))).
% 9.43/3.46 thf(fact_5653_times__assn__raw_Opelims_I2_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((((times_assn_raw @ X) @ Xa) @ Xb) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ times_assn_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ times_assn_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (~ ((? [As13:(set @ nat),As24:(set @ nat)]: (((As = (((sup_sup @ (set @ nat)) @ As13) @ As24)) & (((((inf_inf @ (set @ nat)) @ As13) @ As24) = (bot_bot @ (set @ nat))) & ((X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As13)) & (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As24)))))))))))))))))))).
% 9.43/3.46 thf(fact_5654_times__assn__raw_Opelims_I3_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((((times_assn_raw @ X) @ Xa) @ Xb))) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ times_assn_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ times_assn_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (? [As12:(set @ nat),As23:(set @ nat)]: (((As = (((sup_sup @ (set @ nat)) @ As12) @ As23)) & (((((inf_inf @ (set @ nat)) @ As12) @ As23) = (bot_bot @ (set @ nat))) & ((X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As12)) & (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As23)))))))))))))))))).
% 9.43/3.46 thf(fact_5655_mod__h__bot__iff_I8_J, axiom, (! [C:$tType,R5:(C > assn),H3:(heap_ext @ product_unit)]: ((((rep_assn @ ((ex_assn @ C) @ R5)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = (? [X3:C]: (((rep_assn @ (R5 @ X3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))))))))).
% 9.43/3.46 thf(fact_5656_make__rule, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,F:(nat > A)]: (((((hoare_hoare_triple @ (array @ A)) @ (one_one @ assn)) @ (((array_make @ A) @ N) @ F)) @ (^ [R3:(array @ A)]: ((((snga_assn @ A) @ R3) @ ((((map @ nat) @ A) @ F) @ ((upt @ (zero_zero @ nat)) @ N)))))))))))).
% 9.43/3.46 thf(fact_5657_mod__h__bot__iff_I1_J, axiom, (! [B2:$o,H3:(heap_ext @ product_unit)]: ((((rep_assn @ (pure_assn @ B2)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat)))) = B2)))).
% 9.43/3.46 thf(fact_5658_bind__rule, axiom, (! [A:$tType,B:$tType,P:assn,F:(heap_Time_Heap @ A),R5:(A > assn),G:(A > (heap_Time_Heap @ B)),Q:(B > assn)]: ((((((hoare_hoare_triple @ A) @ P) @ F) @ R5) => ((! [X2:A]: (((((hoare_hoare_triple @ B) @ (R5 @ X2)) @ (G @ X2)) @ Q))) => ((((hoare_hoare_triple @ B) @ P) @ ((((heap_Time_bind @ A) @ B) @ F) @ G)) @ Q)))))).
% 9.43/3.46 thf(fact_5659_raise__iff, axiom, (! [A:$tType,P:assn,S:(list @ char),Q:(A > assn)]: ((((((hoare_hoare_triple @ A) @ P) @ ((heap_Time_raise @ A) @ S)) @ Q) = (P = (bot_bot @ assn)))))).
% 9.43/3.46 thf(fact_5660_raise__rule, axiom, (! [A:$tType,S:(list @ char),Q:(A > assn)]: (((((hoare_hoare_triple @ A) @ (bot_bot @ assn)) @ ((heap_Time_raise @ A) @ S)) @ Q)))).
% 9.43/3.46 thf(fact_5661_bind__rule_H, axiom, (! [A:$tType,B:$tType,C:$tType,P:assn,F:(heap_Time_Heap @ B),R5:(B > assn),G:(B > (heap_Time_Heap @ C)),Q:(C > assn)]: (((! [R2:A]: (((((hoare_hoare_triple @ B) @ P) @ F) @ (^ [S4:B]: ((((times_times @ assn) @ P) @ (R5 @ S4))))))) => ((! [X2:B]: (((((hoare_hoare_triple @ C) @ (((times_times @ assn) @ (R5 @ X2)) @ P)) @ (G @ X2)) @ Q))) => ((((hoare_hoare_triple @ C) @ P) @ ((((heap_Time_bind @ B) @ C) @ F) @ G)) @ Q)))))).
% 9.43/3.46 thf(fact_5662_case__prod__rule, axiom, (! [A:$tType,B:$tType,C:$tType,X:((product_prod @ A) @ B),P:assn,F:(A > (B > (heap_Time_Heap @ C))),Q:(C > assn)]: (((! [A5:A,B3:B]: (((X = ((((product_Pair @ A) @ B) @ A5) @ B3)) => ((((hoare_hoare_triple @ C) @ P) @ ((F @ A5) @ B3)) @ Q)))) => ((((hoare_hoare_triple @ C) @ P) @ (((((product_case_prod @ A) @ B) @ (heap_Time_Heap @ C)) @ F) @ X)) @ Q))))).
% 9.43/3.46 thf(fact_5663_hoare__triple__success, axiom, (! [A:$tType,P:assn,C2:(heap_Time_Heap @ A),Q:(A > assn),H3:(heap_ext @ product_unit),As2:(set @ nat)]: ((((((hoare_hoare_triple @ A) @ P) @ C2) @ Q) => (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (((heap_Time_success @ A) @ C2) @ H3)))))).
% 9.43/3.46 thf(fact_5664_case__option__rule, axiom, (! [A:$tType,B:$tType,V:(option @ A),P:assn,Fn:(heap_Time_Heap @ B),Q:(B > assn),Fs:(A > (heap_Time_Heap @ B))]: ((((V = (none @ A)) => ((((hoare_hoare_triple @ B) @ P) @ Fn) @ Q)) => ((! [X2:A]: (((V = ((some @ A) @ X2)) => ((((hoare_hoare_triple @ B) @ P) @ (Fs @ X2)) @ Q)))) => ((((hoare_hoare_triple @ B) @ P) @ (((((case_option @ (heap_Time_Heap @ B)) @ A) @ Fn) @ Fs) @ V)) @ Q)))))).
% 9.43/3.46 thf(fact_5665_of__list__rule, axiom, (! [A:$tType]: (((heap @ A) => (! [Xs:(list @ A)]: (((((hoare_hoare_triple @ (array @ A)) @ (one_one @ assn)) @ ((array_of_list @ A) @ Xs)) @ (^ [R3:(array @ A)]: ((((snga_assn @ A) @ R3) @ Xs)))))))))).
% 9.43/3.46 thf(fact_5666_new__rule, axiom, (! [A:$tType]: (((heap @ A) => (! [N:nat,X:A]: (((((hoare_hoare_triple @ (array @ A)) @ (one_one @ assn)) @ (((array_new @ A) @ N) @ X)) @ (^ [R3:(array @ A)]: ((((snga_assn @ A) @ R3) @ (((replicate @ A) @ N) @ X))))))))))).
% 9.43/3.46 thf(fact_5667_hoare__triple__effect, axiom, (! [A:$tType,P:assn,C2:(heap_Time_Heap @ A),Q:(A > assn),H3:(heap_ext @ product_unit),As2:(set @ nat)]: ((((((hoare_hoare_triple @ A) @ P) @ C2) @ Q) => (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (? [H6:(heap_ext @ product_unit),R2:A,T7:nat]: ((((((((heap_Time_effect @ A) @ C2) @ H3) @ H6) @ R2) @ T7) & ((rep_assn @ (Q @ R2)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ (((hoare_new_addrs @ H3) @ As2) @ H6))))))))))).
% 9.43/3.46 thf(fact_5668_hoare__triple__def, axiom, (! [A:$tType]: (((hoare_hoare_triple @ A) = (^ [P4:assn,C3:(heap_Time_Heap @ A),Q6:(A > assn)]: ((! [H:(heap_ext @ product_unit),As5:(set @ nat)]: ((((rep_assn @ P4) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H) @ As5)) => (? [R3:A,H7:(heap_ext @ product_unit)]: (((? [T3:nat]: (((((heap_Time_execute @ A) @ C3) @ H) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ T3)))))) & (((rep_assn @ (Q6 @ R3)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ (((hoare_new_addrs @ H) @ As5) @ H7))) & ((((relH @ ((collect @ nat) @ (^ [A7:nat]: (((((ord_less @ nat) @ A7) @ ((lim @ product_unit) @ H)) & (~ ((((member @ nat) @ A7) @ As5)))))))) @ H) @ H7) & (((ord_less_eq @ nat) @ ((lim @ product_unit) @ H)) @ ((lim @ product_unit) @ H7)))))))))))))))).
% 9.43/3.46 thf(fact_5669_hoare__tripleI, axiom, (! [A:$tType,P:assn,C2:(heap_Time_Heap @ A),Q:(A > assn)]: (((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: ((((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (? [R15:A,H9:(heap_ext @ product_unit),T8:nat]: ((((((heap_Time_execute @ A) @ C2) @ H2) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R15) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H9) @ T8)))) & (((rep_assn @ (Q @ R15)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ (((hoare_new_addrs @ H2) @ As) @ H9))) & ((((relH @ ((collect @ nat) @ (^ [A7:nat]: (((((ord_less @ nat) @ A7) @ ((lim @ product_unit) @ H2)) & (~ ((((member @ nat) @ A7) @ As)))))))) @ H2) @ H9) & (((ord_less_eq @ nat) @ ((lim @ product_unit) @ H2)) @ ((lim @ product_unit) @ H9)))))))))) => ((((hoare_hoare_triple @ A) @ P) @ C2) @ Q))))).
% 9.43/3.46 thf(fact_5670_mod__relH, axiom, (! [As2:(set @ nat),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),P:assn]: (((((relH @ As2) @ H3) @ H4) => (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = ((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ As2))))))).
% 9.43/3.46 thf(fact_5671_hoare__tripleE, axiom, (! [A:$tType,P:assn,C2:(heap_Time_Heap @ A),Q:(A > assn),H3:(heap_ext @ product_unit),As2:(set @ nat)]: ((((((hoare_hoare_triple @ A) @ P) @ C2) @ Q) => (((rep_assn @ P) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (~ ((! [R2:A,H6:(heap_ext @ product_unit)]: (((? [T7:nat]: (((((heap_Time_execute @ A) @ C2) @ H3) = ((some @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R2) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H6) @ T7)))))) => (((rep_assn @ (Q @ R2)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ (((hoare_new_addrs @ H3) @ As2) @ H6))) => ((((relH @ ((collect @ nat) @ (^ [A7:nat]: (((((ord_less @ nat) @ A7) @ ((lim @ product_unit) @ H3)) & (~ ((((member @ nat) @ A7) @ As2)))))))) @ H3) @ H6) => (~ ((((ord_less_eq @ nat) @ ((lim @ product_unit) @ H3)) @ ((lim @ product_unit) @ H6)))))))))))))))).
% 9.43/3.46 thf(fact_5672_Array__Time_Oalloc__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_alloc @ A) = (^ [Xs3:(list @ A),H:(heap_ext @ product_unit)]: (((((product_Pair @ (array @ A)) @ (heap_ext @ product_unit)) @ ((array2 @ A) @ ((lim @ product_unit) @ H))) @ ((((array_set @ A) @ ((array2 @ A) @ ((lim @ product_unit) @ H))) @ Xs3) @ (((lim_update @ product_unit) @ (^ [Uu:nat]: ((((plus_plus @ nat) @ ((lim @ product_unit) @ H)) @ (one_one @ nat))))) @ H)))))))))).
% 9.43/3.46 thf(fact_5673_snga__assn__raw_Opelims_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(array @ A),Xa:(list @ A),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: (((((((snga_assn_raw @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (snga_assn_raw_rel @ A)) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((Y = (((((array_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_array @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ X)) @ ((lim @ product_unit) @ H2))))) => (~ ((((accp @ ((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (snga_assn_raw_rel @ A)) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))))))))))))))))))).
% 9.43/3.46 thf(fact_5674_snga__assn__raw_Opelims_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(array @ A),Xa:(list @ A),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: ((((((snga_assn_raw @ A) @ X) @ Xa) @ Xb) => ((((accp @ ((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (snga_assn_raw_rel @ A)) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (snga_assn_raw_rel @ A)) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (~ ((((((array_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_array @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))))))).
% 9.43/3.46 thf(fact_5675_Array__Time_Opresent__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_present @ A) = (^ [H:(heap_ext @ product_unit),A7:(array @ A)]: ((((ord_less @ nat) @ ((addr_of_array @ A) @ A7)) @ ((lim @ product_unit) @ H))))))))).
% 9.43/3.46 thf(fact_5676_snga__assn__raw_Osimps, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A),X:(list @ A),H3:(heap_ext @ product_unit),As2:(set @ nat)]: ((((((snga_assn_raw @ A) @ R) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = (((((array_get @ A) @ H3) @ R) = X) & ((As2 = (((insert2 @ nat) @ ((addr_of_array @ A) @ R)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ R)) @ ((lim @ product_unit) @ H3))))))))))).
% 9.43/3.46 thf(fact_5677_snga__assn__raw_Oelims_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(array @ A),Xa:(list @ A),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: (((((((snga_assn_raw @ A) @ X) @ Xa) @ Xb) = Y) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (Y = (~ ((((((array_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_array @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ X)) @ ((lim @ product_unit) @ H2))))))))))))))))))).
% 9.43/3.46 thf(fact_5678_snga__assn__raw_Oelims_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(array @ A),Xa:(list @ A),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: ((((((snga_assn_raw @ A) @ X) @ Xa) @ Xb) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (~ ((((((array_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_array @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))))).
% 9.43/3.46 thf(fact_5679_snga__assn__raw_Oelims_I3_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(array @ A),Xa:(list @ A),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ (((((snga_assn_raw @ A) @ X) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (((((array_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_array @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))).
% 9.43/3.46 thf(fact_5680_snga__assn__raw_Opelims_I3_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(array @ A),Xa:(list @ A),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ (((((snga_assn_raw @ A) @ X) @ Xa) @ Xb))) => ((((accp @ ((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (snga_assn_raw_rel @ A)) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (snga_assn_raw_rel @ A)) @ ((((product_Pair @ (array @ A)) @ ((product_prod @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (list @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (((((array_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_array @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_array @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))))).
% 9.43/3.46 thf(fact_5681_relH__set__array, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(array @ A),As2:(set @ nat),H3:(heap_ext @ product_unit),X:(list @ A)]: (((~ ((((member @ nat) @ ((addr_of_array @ A) @ R)) @ As2))) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (((relH @ As2) @ H3) @ ((((array_set @ A) @ R) @ X) @ H3)))))))))).
% 9.43/3.46 thf(fact_5682_sngr__assn__raw_Opelims_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(ref @ A),Xa:A,Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: (((((((sngr_assn_raw @ A) @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (sngr_assn_raw_rel @ A)) @ ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((Y = (((((ref_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_ref @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ X)) @ ((lim @ product_unit) @ H2))))) => (~ ((((accp @ ((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (sngr_assn_raw_rel @ A)) @ ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))))))))))))))))))).
% 9.43/3.46 thf(fact_5683_sngr__assn__raw_Opelims_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(ref @ A),Xa:A,Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: ((((((sngr_assn_raw @ A) @ X) @ Xa) @ Xb) => ((((accp @ ((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (sngr_assn_raw_rel @ A)) @ ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (sngr_assn_raw_rel @ A)) @ ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (~ ((((((ref_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_ref @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))))))).
% 9.43/3.46 thf(fact_5684_in__range__empty, axiom, (! [H3:(heap_ext @ product_unit)]: ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (bot_bot @ (set @ nat))))))).
% 9.43/3.46 thf(fact_5685_in__range__dist__union, axiom, (! [H3:(heap_ext @ product_unit),As2:(set @ nat),As6:(set @ nat)]: (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ (((sup_sup @ (set @ nat)) @ As2) @ As6))) = ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) & (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As6))))))).
% 9.43/3.46 thf(fact_5686_relH__set__ref, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),As2:(set @ nat),H3:(heap_ext @ product_unit),X:A]: (((~ ((((member @ nat) @ ((addr_of_ref @ A) @ R)) @ As2))) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (((relH @ As2) @ H3) @ ((((ref_set @ A) @ R) @ X) @ H3)))))))))).
% 9.43/3.46 thf(fact_5687_relH__refl, axiom, (! [H3:(heap_ext @ product_unit),As2:(set @ nat)]: (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (((relH @ As2) @ H3) @ H3))))).
% 9.43/3.46 thf(fact_5688_relH__in__rangeI_I1_J, axiom, (! [As2:(set @ nat),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit)]: (((((relH @ As2) @ H3) @ H4) => (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)))))).
% 9.43/3.46 thf(fact_5689_relH__in__rangeI_I2_J, axiom, (! [As2:(set @ nat),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit)]: (((((relH @ As2) @ H3) @ H4) => (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ As2)))))).
% 9.43/3.46 thf(fact_5690_in__range__subset, axiom, (! [As2:(set @ nat),As6:(set @ nat),H3:(heap_ext @ product_unit)]: (((((ord_less_eq @ (set @ nat)) @ As2) @ As6) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As6)) => (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2))))))).
% 9.43/3.46 thf(fact_5691_in__range_Oelims_I3_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((in_range @ X))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (! [X2:nat]: (((((member @ nat) @ X2) @ As) => (((ord_less @ nat) @ X2) @ ((lim @ product_unit) @ H2)))))))))))))).
% 9.43/3.46 thf(fact_5692_in__range_Oelims_I2_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((in_range @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (~ ((! [X7:nat]: (((((member @ nat) @ X7) @ As) => (((ord_less @ nat) @ X7) @ ((lim @ product_unit) @ H2)))))))))))))))).
% 9.43/3.46 thf(fact_5693_in__range_Oelims_I1_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((in_range @ X) = Y) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (Y = (~ ((! [X3:nat]: (((((member @ nat) @ X3) @ As) => (((ord_less @ nat) @ X3) @ ((lim @ product_unit) @ H2))))))))))))))))).
% 9.43/3.46 thf(fact_5694_in__range_Osimps, axiom, (! [H3:(heap_ext @ product_unit),As2:(set @ nat)]: (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = (! [X3:nat]: (((((member @ nat) @ X3) @ As2) => (((ord_less @ nat) @ X3) @ ((lim @ product_unit) @ H3))))))))).
% 9.43/3.46 thf(fact_5695_sngr__assn__raw_Oelims_I3_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(ref @ A),Xa:A,Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ (((((sngr_assn_raw @ A) @ X) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (((((ref_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_ref @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))).
% 9.43/3.46 thf(fact_5696_sngr__assn__raw_Oelims_I2_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(ref @ A),Xa:A,Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: ((((((sngr_assn_raw @ A) @ X) @ Xa) @ Xb) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (~ ((((((ref_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_ref @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))))).
% 9.43/3.46 thf(fact_5697_sngr__assn__raw_Oelims_I1_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(ref @ A),Xa:A,Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: (((((((sngr_assn_raw @ A) @ X) @ Xa) @ Xb) = Y) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (Y = (~ ((((((ref_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_ref @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ X)) @ ((lim @ product_unit) @ H2))))))))))))))))))).
% 9.43/3.46 thf(fact_5698_sngr__assn__raw_Osimps, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),X:A,H3:(heap_ext @ product_unit),As2:(set @ nat)]: ((((((sngr_assn_raw @ A) @ R) @ X) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = (((((ref_get @ A) @ H3) @ R) = X) & ((As2 = (((insert2 @ nat) @ ((addr_of_ref @ A) @ R)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ R)) @ ((lim @ product_unit) @ H3))))))))))).
% 9.43/3.46 thf(fact_5699_sngr__assn__raw_Opelims_I3_J, axiom, (! [A:$tType]: (((heap @ A) => (! [X:(ref @ A),Xa:A,Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ (((((sngr_assn_raw @ A) @ X) @ Xa) @ Xb))) => ((((accp @ ((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (sngr_assn_raw_rel @ A)) @ ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ (sngr_assn_raw_rel @ A)) @ ((((product_Pair @ (ref @ A)) @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (((((ref_get @ A) @ H2) @ X) = Xa) & ((As = (((insert2 @ nat) @ ((addr_of_ref @ A) @ X)) @ (bot_bot @ (set @ nat)))) & (((ord_less @ nat) @ ((addr_of_ref @ A) @ X)) @ ((lim @ product_unit) @ H2)))))))))))))))))).
% 9.43/3.46 thf(fact_5700_wand__assnI, axiom, (! [H3:(heap_ext @ product_unit),As2:(set @ nat),Q:assn,R5:assn]: (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => ((! [H6:(heap_ext @ product_unit),As7:(set @ nat)]: ((((((inf_inf @ (set @ nat)) @ As2) @ As7) = (bot_bot @ (set @ nat))) => ((((relH @ As2) @ H3) @ H6) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As2)) => (((rep_assn @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As7)) => ((rep_assn @ R5) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ (((sup_sup @ (set @ nat)) @ As2) @ As7))))))))) => ((rep_assn @ ((wand_assn @ Q) @ R5)) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2))))))).
% 9.43/3.46 thf(fact_5701_wand__raw_Osimps, axiom, (! [P:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Q:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),H3:(heap_ext @ product_unit),As2:(set @ nat)]: (((((wand_raw @ P) @ Q) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) & (! [H7:(heap_ext @ product_unit),As8:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As2) @ As8) = (bot_bot @ (set @ nat))) & ((((relH @ As2) @ H3) @ H7) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As2)) & (P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As8))))) => (Q @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ (((sup_sup @ (set @ nat)) @ As2) @ As8))))))))))).
% 9.43/3.46 thf(fact_5702_wand__raw_Oelims_I1_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((((wand_raw @ X) @ Xa) @ Xb) = Y) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (Y = (~ (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) & (! [H7:(heap_ext @ product_unit),As8:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As) @ As8) = (bot_bot @ (set @ nat))) & ((((relH @ As) @ H2) @ H7) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As)) & (X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As8))))) => (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ (((sup_sup @ (set @ nat)) @ As) @ As8))))))))))))))))))).
% 9.43/3.46 thf(fact_5703_wand__raw_Oelims_I3_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((((wand_raw @ X) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) & (! [H6:(heap_ext @ product_unit),As7:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As) @ As7) = (bot_bot @ (set @ nat))) & ((((relH @ As) @ H2) @ H6) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As)) & (X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As7))))) => (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ (((sup_sup @ (set @ nat)) @ As) @ As7)))))))))))))))).
% 9.43/3.46 thf(fact_5704_wand__raw_Oelims_I2_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((((wand_raw @ X) @ Xa) @ Xb) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (~ (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) & (! [H9:(heap_ext @ product_unit),As9:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As) @ As9) = (bot_bot @ (set @ nat))) & ((((relH @ As) @ H2) @ H9) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ As)) & (X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ As9))))) => (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ (((sup_sup @ (set @ nat)) @ As) @ As9)))))))))))))))))).
% 9.43/3.46 thf(fact_5705_wand__raw_Opelims_I3_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((((wand_raw @ X) @ Xa) @ Xb))) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ wand_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ wand_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) & (! [H6:(heap_ext @ product_unit),As7:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As) @ As7) = (bot_bot @ (set @ nat))) & ((((relH @ As) @ H2) @ H6) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As)) & (X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As7))))) => (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ (((sup_sup @ (set @ nat)) @ As) @ As7)))))))))))))))))).
% 9.43/3.46 thf(fact_5706_wand__raw_Opelims_I2_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((((wand_raw @ X) @ Xa) @ Xb) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ wand_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ wand_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)))) => (~ (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) & (! [H9:(heap_ext @ product_unit),As9:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As) @ As9) = (bot_bot @ (set @ nat))) & ((((relH @ As) @ H2) @ H9) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ As)) & (X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ As9))))) => (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H9) @ (((sup_sup @ (set @ nat)) @ As) @ As9)))))))))))))))))))).
% 9.43/3.46 thf(fact_5707_wand__raw_Opelims_I1_J, axiom, (! [X:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xa:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),Xb:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((((wand_raw @ X) @ Xa) @ Xb) = Y) => ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ wand_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ Xb))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((Xb = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((Y = ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) & (! [H7:(heap_ext @ product_unit),As8:(set @ nat)]: (((((((inf_inf @ (set @ nat)) @ As) @ As8) = (bot_bot @ (set @ nat))) & ((((relH @ As) @ H2) @ H7) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As)) & (X @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As8))))) => (Xa @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ (((sup_sup @ (set @ nat)) @ As) @ As8)))))))) => (~ ((((accp @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))))) @ wand_raw_rel) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)))) @ X) @ ((((product_Pair @ (((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)) @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ Xa) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))))))))))))))).
% 9.43/3.46 thf(fact_5708_in__range_Opelims_I3_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((in_range @ X))) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ in_range_rel) @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ in_range_rel) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (! [X2:nat]: (((((member @ nat) @ X2) @ As) => (((ord_less @ nat) @ X2) @ ((lim @ product_unit) @ H2)))))))))))))))).
% 9.43/3.46 thf(fact_5709_in__range_Opelims_I2_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((in_range @ X) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ in_range_rel) @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ in_range_rel) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (~ ((! [X7:nat]: (((((member @ nat) @ X7) @ As) => (((ord_less @ nat) @ X7) @ ((lim @ product_unit) @ H2)))))))))))))))))).
% 9.43/3.46 thf(fact_5710_in__range_Opelims_I1_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((in_range @ X) = Y) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ in_range_rel) @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((Y = (! [X3:nat]: (((((member @ nat) @ X3) @ As) => (((ord_less @ nat) @ X3) @ ((lim @ product_unit) @ H2)))))) => (~ ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ in_range_rel) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))))))))))))).
% 9.43/3.46 thf(fact_5711_Ref__Time_Oalloc__def, axiom, (! [A:$tType]: (((heap @ A) => ((ref_alloc @ A) = (^ [X3:A,H:(heap_ext @ product_unit)]: (((((product_Pair @ (ref @ A)) @ (heap_ext @ product_unit)) @ ((ref2 @ A) @ ((lim @ product_unit) @ H))) @ ((((ref_set @ A) @ ((ref2 @ A) @ ((lim @ product_unit) @ H))) @ X3) @ (((lim_update @ product_unit) @ (^ [Uu:nat]: ((((plus_plus @ nat) @ ((lim @ product_unit) @ H)) @ (one_one @ nat))))) @ H)))))))))).
% 9.43/3.46 thf(fact_5712_pure__assn__raw_Osimps, axiom, (! [A:$tType,B:$tType,B2:$o,H3:A,As2:(set @ B)]: ((((((pure_assn_raw @ A) @ B) @ B2) @ ((((product_Pair @ A) @ (set @ B)) @ H3) @ As2)) = ((As2 = (bot_bot @ (set @ B))) & B2))))).
% 9.43/3.46 thf(fact_5713_pure__assn__raw_Oelims_I1_J, axiom, (! [A:$tType,B:$tType,X:$o,Xa:((product_prod @ A) @ (set @ B)),Y:$o]: (((((((pure_assn_raw @ A) @ B) @ X) @ Xa) = Y) => (~ ((! [H2:A,As:(set @ B)]: (((Xa = ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)) => (Y = (~ (((As = (bot_bot @ (set @ B))) & X))))))))))))).
% 9.43/3.46 thf(fact_5714_pure__assn__raw_Oelims_I3_J, axiom, (! [B:$tType,A:$tType,X:$o,Xa:((product_prod @ A) @ (set @ B))]: (((~ (((((pure_assn_raw @ A) @ B) @ X) @ Xa))) => (~ ((! [H2:A,As:(set @ B)]: (((Xa = ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)) => ((As = (bot_bot @ (set @ B))) & X)))))))))).
% 9.43/3.46 thf(fact_5715_pure__assn__raw_Oelims_I2_J, axiom, (! [B:$tType,A:$tType,X:$o,Xa:((product_prod @ A) @ (set @ B))]: ((((((pure_assn_raw @ A) @ B) @ X) @ Xa) => (~ ((! [H2:A,As:(set @ B)]: (((Xa = ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)) => (~ (((As = (bot_bot @ (set @ B))) & X)))))))))))).
% 9.43/3.46 thf(fact_5716_pure__assn__raw_Opelims_I3_J, axiom, (! [B:$tType,A:$tType,X:$o,Xa:((product_prod @ A) @ (set @ B))]: (((~ (((((pure_assn_raw @ A) @ B) @ X) @ Xa))) => ((((accp @ ((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))) @ ((pure_assn_raw_rel @ A) @ B)) @ ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ X) @ Xa)) => (~ ((! [H2:A,As:(set @ B)]: (((Xa = ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)) => ((((accp @ ((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))) @ ((pure_assn_raw_rel @ A) @ B)) @ ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ X) @ ((((product_Pair @ A) @ (set @ B)) @ H2) @ As))) => ((As = (bot_bot @ (set @ B))) & X)))))))))))).
% 9.43/3.46 thf(fact_5717_pure__assn__raw_Opelims_I2_J, axiom, (! [B:$tType,A:$tType,X:$o,Xa:((product_prod @ A) @ (set @ B))]: ((((((pure_assn_raw @ A) @ B) @ X) @ Xa) => ((((accp @ ((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))) @ ((pure_assn_raw_rel @ A) @ B)) @ ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ X) @ Xa)) => (~ ((! [H2:A,As:(set @ B)]: (((Xa = ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)) => ((((accp @ ((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))) @ ((pure_assn_raw_rel @ A) @ B)) @ ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ X) @ ((((product_Pair @ A) @ (set @ B)) @ H2) @ As))) => (~ (((As = (bot_bot @ (set @ B))) & X)))))))))))))).
% 9.43/3.46 thf(fact_5718_pure__assn__raw_Opelims_I1_J, axiom, (! [A:$tType,B:$tType,X:$o,Xa:((product_prod @ A) @ (set @ B)),Y:$o]: (((((((pure_assn_raw @ A) @ B) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))) @ ((pure_assn_raw_rel @ A) @ B)) @ ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ X) @ Xa)) => (~ ((! [H2:A,As:(set @ B)]: (((Xa = ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)) => ((Y = ((As = (bot_bot @ (set @ B))) & X)) => (~ ((((accp @ ((product_prod @ $o) @ ((product_prod @ A) @ (set @ B)))) @ ((pure_assn_raw_rel @ A) @ B)) @ ((((product_Pair @ $o) @ ((product_prod @ A) @ (set @ B))) @ X) @ ((((product_Pair @ A) @ (set @ B)) @ H2) @ As)))))))))))))))).
% 9.43/3.46 thf(fact_5719_effect__refI, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),H4:(heap_ext @ product_unit),V:A,H3:(heap_ext @ product_unit),N:nat]: (((((((product_Pair @ (ref @ A)) @ (heap_ext @ product_unit)) @ R) @ H4) = (((ref_alloc @ A) @ V) @ H3)) => ((N = (one_one @ nat)) => ((((((heap_Time_effect @ (ref @ A)) @ ((ref_ref @ A) @ V)) @ H3) @ H4) @ R) @ N))))))))).
% 9.43/3.46 thf(fact_5720_ref__def, axiom, (! [A:$tType]: (((heap @ A) => ((ref_ref @ A) = (^ [V2:A]: (((heap_Time_heap @ (ref @ A)) @ (^ [H:(heap_ext @ product_unit)]: ((((((product_case_prod @ (ref @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (ref @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(ref @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (ref @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (one_one @ nat)))))) @ (((ref_alloc @ A) @ V2) @ H)))))))))))).
% 9.43/3.46 thf(fact_5721_execute__ref, axiom, (! [A:$tType]: (((heap @ A) => (! [V:A,H3:(heap_ext @ product_unit)]: (((((heap_Time_execute @ (ref @ A)) @ ((ref_ref @ A) @ V)) @ H3) = ((some @ ((product_prod @ (ref @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (((((product_case_prod @ (ref @ A)) @ (heap_ext @ product_unit)) @ ((product_prod @ (ref @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))) @ (^ [R3:(ref @ A),H7:(heap_ext @ product_unit)]: (((((product_Pair @ (ref @ A)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)) @ R3) @ ((((product_Pair @ (heap_ext @ product_unit)) @ nat) @ H7) @ (one_one @ nat)))))) @ (((ref_alloc @ A) @ V) @ H3)))))))))).
% 9.43/3.46 thf(fact_5722_effect__refE, axiom, (! [A:$tType]: (((heap @ A) => (! [V:A,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:(ref @ A),N:nat]: ((((((((heap_Time_effect @ (ref @ A)) @ ((ref_ref @ A) @ V)) @ H3) @ H4) @ R) @ N) => (~ ((((((ref_get @ A) @ H4) @ R) = V) => ((((ref_present @ A) @ H4) @ R) => ((~ ((((ref_present @ A) @ H3) @ R))) => (N != (one_one @ nat)))))))))))))).
% 9.43/3.46 thf(fact_5723_next__fresh, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),H4:(heap_ext @ product_unit),X:A,H3:(heap_ext @ product_unit)]: (((((((product_Pair @ (ref @ A)) @ (heap_ext @ product_unit)) @ R) @ H4) = (((ref_alloc @ A) @ X) @ H3)) => (~ ((((ref_present @ A) @ H3) @ R)))))))))).
% 9.43/3.46 thf(fact_5724_next__present, axiom, (! [A:$tType]: (((heap @ A) => (! [R:(ref @ A),H4:(heap_ext @ product_unit),X:A,H3:(heap_ext @ product_unit)]: (((((((product_Pair @ (ref @ A)) @ (heap_ext @ product_unit)) @ R) @ H4) = (((ref_alloc @ A) @ X) @ H3)) => (((ref_present @ A) @ H4) @ R)))))))).
% 9.43/3.46 thf(fact_5725_one__assn__raw_Oelims_I3_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((one_assn_raw @ X))) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (As = (bot_bot @ (set @ nat)))))))))))).
% 9.43/3.46 thf(fact_5726_one__assn__raw_Oelims_I2_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((one_assn_raw @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (As != (bot_bot @ (set @ nat)))))))))))).
% 9.43/3.46 thf(fact_5727_one__assn__raw_Oelims_I1_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((one_assn_raw @ X) = Y) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (Y = (As != (bot_bot @ (set @ nat))))))))))))).
% 9.43/3.46 thf(fact_5728_one__assn__raw_Osimps, axiom, (! [H3:(heap_ext @ product_unit),As2:(set @ nat)]: (((one_assn_raw @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = (As2 = (bot_bot @ (set @ nat))))))).
% 9.43/3.46 thf(fact_5729_one__assn__raw_Opelims_I3_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((~ ((one_assn_raw @ X))) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ one_assn_raw_rel) @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ one_assn_raw_rel) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (As = (bot_bot @ (set @ nat)))))))))))))).
% 9.43/3.46 thf(fact_5730_one__assn__raw_Opelims_I2_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))]: (((one_assn_raw @ X) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ one_assn_raw_rel) @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ one_assn_raw_rel) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (As != (bot_bot @ (set @ nat)))))))))))))).
% 9.43/3.46 thf(fact_5731_one__assn__raw_Opelims_I1_J, axiom, (! [X:((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)),Y:$o]: ((((one_assn_raw @ X) = Y) => ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ one_assn_raw_rel) @ X) => (~ ((! [H2:(heap_ext @ product_unit),As:(set @ nat)]: (((X = ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((Y = (As = (bot_bot @ (set @ nat)))) => (~ ((((accp @ ((product_prod @ (heap_ext @ product_unit)) @ (set @ nat))) @ one_assn_raw_rel) @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))))))))))))).
% 9.43/3.46 thf(fact_5732_proper__iff, axiom, (! [P:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),As2:(set @ nat),H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit)]: (((proper @ P) => ((((relH @ As2) @ H3) @ H4) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ As2)) => ((P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) = (P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ As2))))))))).
% 9.43/3.46 thf(fact_5733_proper__def, axiom, (proper = (^ [P4:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)]: ((! [H:(heap_ext @ product_unit),H7:(heap_ext @ product_unit),As5:(set @ nat)]: ((((P4 @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H) @ As5)) => (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H) @ As5))) & (((P4 @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H) @ As5)) & ((((relH @ As5) @ H) @ H7) & (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As5)))) => (P4 @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As5)))))))))).
% 9.43/3.46 thf(fact_5734_properD2, axiom, (! [P:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),H3:(heap_ext @ product_unit),As2:(set @ nat),H4:(heap_ext @ product_unit)]: (((proper @ P) => ((P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => ((((relH @ As2) @ H3) @ H4) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ As2)) => (P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H4) @ As2))))))))).
% 9.43/3.46 thf(fact_5735_properD1, axiom, (! [P:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o),H3:(heap_ext @ product_unit),As2:(set @ nat)]: (((proper @ P) => ((P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2)) => (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H3) @ As2))))))).
% 9.43/3.46 thf(fact_5736_properI, axiom, (! [P:(((product_prod @ (heap_ext @ product_unit)) @ (set @ nat)) > $o)]: (((! [As:(set @ nat),H2:(heap_ext @ product_unit)]: (((P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => (in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As))))) => ((! [As:(set @ nat),H2:(heap_ext @ product_unit),H6:(heap_ext @ product_unit)]: (((P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H2) @ As)) => ((((relH @ As) @ H2) @ H6) => ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As)) => (P @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H6) @ As))))))) => (proper @ P)))))).
% 9.43/3.46 thf(fact_5737_Array__Time_Onoteq__def, axiom, (! [B:$tType,A:$tType]: ((((heap @ A) & (heap @ B)) => (((array_noteq @ A) @ B) = (^ [R3:(array @ A),S4:(array @ B)]: (((((typerep_typerep @ A) @ (type2 @ A)) != ((typerep_typerep @ B) @ (type2 @ B))) | (((addr_of_array @ A) @ R3) != ((addr_of_array @ B) @ S4)))))))))).
% 9.43/3.46 thf(fact_5738_comm__monoid__mset_Ounion, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M3:(multiset @ A),N6:(multiset @ A)]: (((((comm_monoid_mset @ A) @ F) @ Z2) => (((((comm_monoid_F @ A) @ F) @ Z2) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = ((F @ ((((comm_monoid_F @ A) @ F) @ Z2) @ M3)) @ ((((comm_monoid_F @ A) @ F) @ Z2) @ N6))))))).
% 9.43/3.46 thf(fact_5739_prod__list_Omonoid__list__axioms, axiom, (! [A:$tType]: (((monoid_mult @ A) => (((groups_monoid_list @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_5740_sum__mset_Ocomm__monoid__mset__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((comm_monoid_mset @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_5741_prod__mset_Ocomm__monoid__mset__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((comm_monoid_mset @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_5742_sum__list_Omonoid__list__axioms, axiom, (! [A:$tType]: (((monoid_add @ A) => (((groups_monoid_list @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_5743_Array__Time_Oset__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_set @ A) = (^ [A7:(array @ A),X3:(list @ A)]: (((arrays_update @ product_unit) @ (^ [H:(typerep > (nat > (list @ nat)))]: ((((((fun_upd @ typerep) @ (nat > (list @ nat))) @ H) @ ((typerep_typerep @ A) @ (type2 @ A))) @ (((((fun_upd @ nat) @ (list @ nat)) @ (H @ ((typerep_typerep @ A) @ (type2 @ A)))) @ ((addr_of_array @ A) @ A7)) @ ((((map @ A) @ nat) @ (to_nat @ A)) @ X3))))))))))))).
% 9.43/3.46 thf(fact_5744_Array__Time_Oget__def, axiom, (! [A:$tType]: (((heap @ A) => ((array_get @ A) = (^ [H:(heap_ext @ product_unit),A7:(array @ A)]: (((((map @ nat) @ A) @ (from_nat @ A)) @ ((((arrays @ product_unit) @ H) @ ((typerep_typerep @ A) @ (type2 @ A))) @ ((addr_of_array @ A) @ A7)))))))))).
% 9.43/3.46 thf(fact_5745_Int_Osub__code_I8_J, axiom, (! [M2:num,N:num]: ((((sub @ (bit1 @ M2)) @ (bit0 @ N)) = (((plus_plus @ int) @ (dup @ ((sub @ M2) @ N))) @ (one_one @ int)))))).
% 9.43/3.46 thf(fact_5746_Int_Odup__def, axiom, (dup = (^ [K4:int]: ((((plus_plus @ int) @ K4) @ K4))))).
% 9.43/3.46 thf(fact_5747_relH__def, axiom, (relH = (^ [As5:(set @ nat),H:(heap_ext @ product_unit),H7:(heap_ext @ product_unit)]: (((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H) @ As5)) & ((in_range @ ((((product_Pair @ (heap_ext @ product_unit)) @ (set @ nat)) @ H7) @ As5)) & (! [T3:typerep,X3:nat]: (((((member @ nat) @ X3) @ As5) => ((((((refs @ product_unit) @ H) @ T3) @ X3) = ((((refs @ product_unit) @ H7) @ T3) @ X3)) & (((((arrays @ product_unit) @ H) @ T3) @ X3) = ((((arrays @ product_unit) @ H7) @ T3) @ X3)))))))))))).
% 9.43/3.46 thf(fact_5748_case__sum__o__map__sum__id, axiom, (! [A:$tType,B:$tType,C:$tType,G:(B > A),F:(C > A),X:((sum_sum @ C) @ B)]: ((((((((comp @ ((sum_sum @ A) @ B)) @ A) @ ((sum_sum @ C) @ B)) @ (((((sum_case_sum @ A) @ A) @ B) @ (id @ A)) @ G)) @ ((((((sum_map_sum @ C) @ A) @ B) @ B) @ F) @ (id @ B))) @ X) = ((((((sum_case_sum @ C) @ A) @ B) @ (((((comp @ C) @ A) @ C) @ F) @ (id @ C))) @ G) @ X))))).
% 9.43/3.46 thf(fact_5749_comm__monoid_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: (((((comm_monoid @ A) @ F) @ Z2) => (((comm_monoid_axioms @ A) @ F) @ Z2))))).
% 9.43/3.46 thf(fact_5750_sum_Omap__comp, axiom, (! [D:$tType,F5:$tType,E:$tType,C:$tType,B:$tType,A:$tType,G1:(C > E),G22:(D > F5),F1:(A > C),F22:(B > D),V:((sum_sum @ A) @ B)]: (((((((((sum_map_sum @ C) @ E) @ D) @ F5) @ G1) @ G22) @ (((((((sum_map_sum @ A) @ C) @ B) @ D) @ F1) @ F22) @ V)) = (((((((sum_map_sum @ A) @ E) @ B) @ F5) @ (((((comp @ C) @ E) @ A) @ G1) @ F1)) @ (((((comp @ D) @ F5) @ B) @ G22) @ F22)) @ V))))).
% 9.43/3.46 thf(fact_5751_comm__monoid__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: (((! [A5:A]: ((((F @ A5) @ Z2) = A5))) => (((comm_monoid_axioms @ A) @ F) @ Z2))))).
% 9.43/3.46 thf(fact_5752_comm__monoid__axioms__def, axiom, (! [A:$tType]: (((comm_monoid_axioms @ A) = (^ [F2:(A > (A > A)),Z5:A]: ((! [A7:A]: ((((F2 @ A7) @ Z5) = A7))))))))).
% 9.43/3.46 thf(fact_5753_case__sum__map__sum, axiom, (! [C:$tType,A:$tType,B:$tType,E:$tType,D:$tType,L:(B > A),R:(C > A),F:(D > B),G:(E > C),X:((sum_sum @ D) @ E)]: ((((((((sum_case_sum @ B) @ A) @ C) @ L) @ R) @ (((((((sum_map_sum @ D) @ B) @ E) @ C) @ F) @ G) @ X)) = ((((((sum_case_sum @ D) @ A) @ E) @ (((((comp @ B) @ A) @ D) @ L) @ F)) @ (((((comp @ C) @ A) @ E) @ R) @ G)) @ X))))).
% 9.43/3.46 thf(fact_5754_case__sum__o__map__sum, axiom, (! [A:$tType,D:$tType,C:$tType,E:$tType,B:$tType,F:(D > C),G:(E > C),H1:(A > D),H22:(B > E)]: (((((((comp @ ((sum_sum @ D) @ E)) @ C) @ ((sum_sum @ A) @ B)) @ (((((sum_case_sum @ D) @ C) @ E) @ F) @ G)) @ ((((((sum_map_sum @ A) @ D) @ B) @ E) @ H1) @ H22)) = (((((sum_case_sum @ A) @ C) @ B) @ (((((comp @ D) @ C) @ A) @ F) @ H1)) @ (((((comp @ E) @ C) @ B) @ G) @ H22)))))).
% 9.43/3.46 thf(fact_5755_map__sum_Ocomp, axiom, (! [A:$tType,C:$tType,E:$tType,F5:$tType,D:$tType,B:$tType,F:(C > E),G:(D > F5),H3:(A > C),I:(B > D)]: (((((((comp @ ((sum_sum @ C) @ D)) @ ((sum_sum @ E) @ F5)) @ ((sum_sum @ A) @ B)) @ ((((((sum_map_sum @ C) @ E) @ D) @ F5) @ F) @ G)) @ ((((((sum_map_sum @ A) @ C) @ B) @ D) @ H3) @ I)) = ((((((sum_map_sum @ A) @ E) @ B) @ F5) @ (((((comp @ C) @ E) @ A) @ F) @ H3)) @ (((((comp @ D) @ F5) @ B) @ G) @ I)))))).
% 9.43/3.46 thf(fact_5756_map__sum_Ocompositionality, axiom, (! [D:$tType,F5:$tType,E:$tType,C:$tType,B:$tType,A:$tType,F:(C > E),G:(D > F5),H3:(A > C),I:(B > D),Sum:((sum_sum @ A) @ B)]: (((((((((sum_map_sum @ C) @ E) @ D) @ F5) @ F) @ G) @ (((((((sum_map_sum @ A) @ C) @ B) @ D) @ H3) @ I) @ Sum)) = (((((((sum_map_sum @ A) @ E) @ B) @ F5) @ (((((comp @ C) @ E) @ A) @ F) @ H3)) @ (((((comp @ D) @ F5) @ B) @ G) @ I)) @ Sum))))).
% 9.43/3.46 thf(fact_5757_comm__monoid__def, axiom, (! [A:$tType]: (((comm_monoid @ A) = (^ [F2:(A > (A > A)),Z5:A]: ((((abel_semigroup @ A) @ F2) & (((comm_monoid_axioms @ A) @ F2) @ Z5)))))))).
% 9.43/3.46 thf(fact_5758_mult_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((ab_semigroup_mult @ A) => ((abel_semigroup @ A) @ (times_times @ A)))))).
% 9.43/3.46 thf(fact_5759_comm__monoid_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: (((((comm_monoid @ A) @ F) @ Z2) => ((abel_semigroup @ A) @ F))))).
% 9.43/3.46 thf(fact_5760_abel__semigroup_Oleft__commute, axiom, (! [A:$tType,F:(A > (A > A)),B2:A,A3:A,C2:A]: ((((abel_semigroup @ A) @ F) => (((F @ B2) @ ((F @ A3) @ C2)) = ((F @ A3) @ ((F @ B2) @ C2))))))).
% 9.43/3.46 thf(fact_5761_abel__semigroup_Ocommute, axiom, (! [A:$tType,F:(A > (A > A)),A3:A,B2:A]: ((((abel_semigroup @ A) @ F) => (((F @ A3) @ B2) = ((F @ B2) @ A3)))))).
% 9.43/3.46 thf(fact_5762_add_Oabel__semigroup__axioms, axiom, (! [A:$tType]: (((ab_semigroup_add @ A) => ((abel_semigroup @ A) @ (plus_plus @ A)))))).
% 9.43/3.46 thf(fact_5763_abel__semigroup_Oaxioms_I1_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((semigroup @ A) @ F))))).
% 9.43/3.46 thf(fact_5764_comm__monoid_Ointro, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A]: ((((abel_semigroup @ A) @ F) => ((((comm_monoid_axioms @ A) @ F) @ Z2) => (((comm_monoid @ A) @ F) @ Z2)))))).
% 9.43/3.46 thf(fact_5765_abel__semigroup_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: ((((semigroup @ A) @ F) => (((abel_s757365448890700780axioms @ A) @ F) => ((abel_semigroup @ A) @ F)))))).
% 9.43/3.46 thf(fact_5766_abel__semigroup__def, axiom, (! [A:$tType]: (((abel_semigroup @ A) = (^ [F2:(A > (A > A))]: ((((semigroup @ A) @ F2) & ((abel_s757365448890700780axioms @ A) @ F2)))))))).
% 9.43/3.46 thf(fact_5767_curr__def, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((bNF_Wellorder_curr @ A) @ B) @ C) = (^ [A9:(set @ A),F2:(((product_prod @ A) @ B) > C),A7:A]: (((((if @ (B > C)) @ (((member @ A) @ A7) @ A9)) @ (^ [B5:B]: ((F2 @ ((((product_Pair @ A) @ B) @ A7) @ B5))))) @ (undefined @ (B > C))))))))).
% 9.43/3.46 thf(fact_5768_abel__semigroup__axioms_Ointro, axiom, (! [A:$tType,F:(A > (A > A))]: (((! [A5:A,B3:A]: ((((F @ A5) @ B3) = ((F @ B3) @ A5)))) => ((abel_s757365448890700780axioms @ A) @ F))))).
% 9.43/3.46 thf(fact_5769_abel__semigroup__axioms__def, axiom, (! [A:$tType]: (((abel_s757365448890700780axioms @ A) = (^ [F2:(A > (A > A))]: ((! [A7:A,B5:A]: ((((F2 @ A7) @ B5) = ((F2 @ B5) @ A7)))))))))).
% 9.43/3.46 thf(fact_5770_abel__semigroup_Oaxioms_I2_J, axiom, (! [A:$tType,F:(A > (A > A))]: ((((abel_semigroup @ A) @ F) => ((abel_s757365448890700780axioms @ A) @ F))))).
% 9.43/3.46 thf(fact_5771_comm__monoid__set_Ozero__middle, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,P5:nat,K:nat,G:(nat > A),H3:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ (one_one @ nat)) @ P5) => ((((ord_less_eq @ nat) @ K) @ P5) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [J2:nat]: (((((if @ A) @ (((ord_less @ nat) @ J2) @ K)) @ (G @ J2)) @ ((((if @ A) @ (J2 = K)) @ Z2) @ (H3 @ (((minus_minus @ nat) @ J2) @ (suc @ (zero_zero @ nat))))))))) @ ((set_ord_atMost @ nat) @ P5)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [J2:nat]: (((((if @ A) @ (((ord_less @ nat) @ J2) @ K)) @ (G @ J2)) @ (H3 @ J2))))) @ ((set_ord_atMost @ nat) @ (((minus_minus @ nat) @ P5) @ (suc @ (zero_zero @ nat)))))))))))).
% 9.43/3.46 thf(fact_5772_iterate__add__diff__add__eq2, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [I:nat,J:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ I) @ J) => ((((minus_minus @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (((minus_minus @ A) @ M2) @ (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N)))))))))).
% 9.43/3.46 thf(fact_5773_iterate__add__add__eq1, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [J:nat,I:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ J) @ I) => ((((minus_minus @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (((minus_minus @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N))))))))).
% 9.43/3.46 thf(fact_5774_iterate__add__simps_I2_J, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [N:nat,A3:A]: (((((iterate_add @ A) @ (suc @ N)) @ A3) = (((plus_plus @ A) @ A3) @ (((iterate_add @ A) @ N) @ A3))))))))).
% 9.43/3.46 thf(fact_5775_iterate__add__distrib, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [M2:nat,N:nat,A3:A]: (((((iterate_add @ A) @ (((plus_plus @ nat) @ M2) @ N)) @ A3) = (((plus_plus @ A) @ (((iterate_add @ A) @ M2) @ A3)) @ (((iterate_add @ A) @ N) @ A3))))))))).
% 9.43/3.46 thf(fact_5776_iterate__add__1, axiom, (! [A:$tType]: ((((cancel1802427076303600483id_add @ A) & (semiring_1 @ A)) => (! [N:nat]: (((((iterate_add @ A) @ N) @ (one_one @ A)) = ((semiring_1_of_nat @ A) @ N)))))))).
% 9.43/3.46 thf(fact_5777_prod_Ocomm__monoid__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_mult @ A) => (((groups778175481326437816id_set @ A) @ (times_times @ A)) @ (one_one @ A)))))).
% 9.43/3.46 thf(fact_5778_comm__monoid__set_Onat__group, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),K:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [M:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((times_times @ nat) @ M) @ K)) @ (((plus_plus @ nat) @ (((times_times @ nat) @ M) @ K)) @ K)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_lessThan @ nat) @ (((times_times @ nat) @ N) @ K)))))))).
% 9.43/3.46 thf(fact_5779_comm__monoid__set_Oub__add__nat, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A),P5:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (((plus_plus @ nat) @ N) @ P5))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ N) @ (one_one @ nat))) @ (((plus_plus @ nat) @ N) @ P5)))))))))).
% 9.43/3.46 thf(fact_5780_comm__monoid__set_Oivl__cong, axiom, (! [A:$tType,B:$tType]: (((ord @ B) => (! [F:(A > (A > A)),Z2:A,A3:B,C2:B,B2:B,D3:B,G:(B > A),H3:(B > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((A3 = C2) => ((B2 = D3) => ((! [X2:B]: (((((ord_less_eq @ B) @ C2) @ X2) => ((((ord_less @ B) @ X2) @ D3) => ((G @ X2) = (H3 @ X2)))))) => (((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ B) @ A3) @ B2)) = ((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z2) @ H3) @ (((set_or7035219750837199246ssThan @ B) @ C2) @ D3))))))))))))).
% 9.43/3.46 thf(fact_5781_comm__monoid__set_Ohead, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((F @ (G @ M2)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or3652927894154168847AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.46 thf(fact_5782_comm__monoid__set_Olast__plus, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((F @ (G @ N)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))).
% 9.43/3.46 thf(fact_5783_comm__monoid__set_OatLeast__Suc__atMost, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((F @ (G @ M2)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ N))))))))).
% 9.43/3.46 thf(fact_5784_comm__monoid__set_Onat__ivl__Suc_H, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ (suc @ N)) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = ((F @ (G @ (suc @ N))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.46 thf(fact_5785_comm__monoid__set_OatLeastLessThan__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,A3:nat,B2:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ A3) @ B2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ (suc @ B2))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ A3) @ B2))) @ (G @ B2)))))))).
% 9.43/3.46 thf(fact_5786_comm__monoid__set_OatLeastLessThan__concat, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,P5:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ N) => ((((ord_less_eq @ nat) @ N) @ P5) => (((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ P5))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ P5))))))))).
% 9.43/3.46 thf(fact_5787_comm__monoid__set_OSuc__reindex__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ N) => (((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ (G @ (suc @ N))) = ((F @ (G @ M2)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))).
% 9.43/3.46 thf(fact_5788_comm__monoid__set_OatLeast1__atMost__eq, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ (zero_zero @ nat))) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [K4:nat]: ((G @ (suc @ K4))))) @ ((set_ord_lessThan @ nat) @ N))))))).
% 9.43/3.46 thf(fact_5789_comm__monoid__set_OatLeastLessThan__rev, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat,M2:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ (suc @ I3)))))) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2))))))).
% 9.43/3.46 thf(fact_5790_comm__monoid__set_Onested__swap, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,A3:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (A3 @ I3)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [J2:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((A3 @ I3) @ J2)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J2)) @ N))))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))))))).
% 9.43/3.46 thf(fact_5791_comm__monoid__set_Onat__diff__reindex, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ N) @ (suc @ I3)))))) @ ((set_ord_lessThan @ nat) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_lessThan @ nat) @ N))))))).
% 9.43/3.46 thf(fact_5792_comm__monoid__set_Oshift__bounds__cl__Suc__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5793_comm__monoid__set_Oshift__bounds__Suc__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5794_comm__monoid__set_OlessThan__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_lessThan @ nat) @ N))) @ (G @ N))))))).
% 9.43/3.46 thf(fact_5795_comm__monoid__set_OatMost__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_atMost @ nat) @ N))) @ (G @ (suc @ N)))))))).
% 9.43/3.46 thf(fact_5796_comm__monoid__set_Ohead__if, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,N:nat,M2:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((ord_less @ nat) @ N) @ M2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = Z2)) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ (G @ N))))))))).
% 9.43/3.46 thf(fact_5797_comm__monoid__set_Ocl__ivl__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,N:nat,M2:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((ord_less @ nat) @ (suc @ N)) @ M2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = Z2)) & ((~ ((((ord_less @ nat) @ (suc @ N)) @ M2))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))) @ (G @ (suc @ N)))))))))).
% 9.43/3.46 thf(fact_5798_comm__monoid__set_Oop__ivl__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,N:nat,M2:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((ord_less @ nat) @ N) @ M2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N))) = Z2)) & ((~ ((((ord_less @ nat) @ N) @ M2))) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))) @ (G @ N))))))))).
% 9.43/3.46 thf(fact_5799_comm__monoid__set_OatLeast__Suc__lessThan, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less @ nat) @ M2) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = ((F @ (G @ M2)) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ N))))))))).
% 9.43/3.46 thf(fact_5800_comm__monoid__set_OatLeast0__lessThan__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ N))))))).
% 9.43/3.46 thf(fact_5801_comm__monoid__set_OatLeast0__atMost__Suc, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N))) @ (G @ (suc @ N)))))))).
% 9.43/3.46 thf(fact_5802_comm__monoid__set_OatMost__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_atMost @ nat) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_atMost @ nat) @ N)))))))).
% 9.43/3.46 thf(fact_5803_comm__monoid__set_OatLeastAtMost__rev, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat,M2:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ N) @ M2)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ N) @ M2))))))).
% 9.43/3.46 thf(fact_5804_comm__monoid__set_OlessThan__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_lessThan @ nat) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))).
% 9.43/3.46 thf(fact_5805_comm__monoid__set_Oshift__bounds__cl__nat__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,K:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5806_comm__monoid__set_Oshift__bounds__nat__ivl, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,K:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (((plus_plus @ nat) @ I3) @ K))))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5807_sum_Ocomm__monoid__set__axioms, axiom, (! [A:$tType]: (((comm_monoid_add @ A) => (((groups778175481326437816id_set @ A) @ (plus_plus @ A)) @ (zero_zero @ A)))))).
% 9.43/3.46 thf(fact_5808_comm__monoid__set_OatMost__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_atMost @ nat) @ N)) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (suc @ I3))))) @ ((set_ord_lessThan @ nat) @ N)))))))).
% 9.43/3.46 thf(fact_5809_comm__monoid__set_Onested__swap_H, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,A3:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (A3 @ I3)) @ ((set_ord_lessThan @ nat) @ I3))))) @ ((set_ord_atMost @ nat) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [J2:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((A3 @ I3) @ J2)))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ J2)) @ N))))) @ ((set_ord_lessThan @ nat) @ N))))))).
% 9.43/3.46 thf(fact_5810_comm__monoid__set_Oreindex__nontrivial, axiom, (! [C:$tType,A:$tType,B:$tType,F:(A > (A > A)),Z2:A,A4:(set @ B),H3:(B > C),G:(C > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((finite_finite2 @ B) @ A4) => ((! [X2:B,Y4:B]: (((((member @ B) @ X2) @ A4) => ((((member @ B) @ Y4) @ A4) => ((X2 != Y4) => (((H3 @ X2) = (H3 @ Y4)) => ((G @ (H3 @ X2)) = Z2))))))) => (((((((groups_comm_monoid_F @ A) @ C) @ F) @ Z2) @ G) @ ((((image2 @ B) @ C) @ H3) @ A4)) = ((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z2) @ (((((comp @ C) @ A) @ B) @ G) @ H3)) @ A4)))))))).
% 9.43/3.46 thf(fact_5811_comm__monoid__set_Oreindex, axiom, (! [C:$tType,A:$tType,B:$tType,F:(A > (A > A)),Z2:A,H3:(B > C),A4:(set @ B),G:(C > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((inj_on @ B) @ C) @ H3) @ A4) => (((((((groups_comm_monoid_F @ A) @ C) @ F) @ Z2) @ G) @ ((((image2 @ B) @ C) @ H3) @ A4)) = ((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z2) @ (((((comp @ C) @ A) @ B) @ G) @ H3)) @ A4))))))).
% 9.43/3.46 thf(fact_5812_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5813_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5814_comm__monoid__set_OatLeastAtMost__shift__bounds, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,K:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5815_comm__monoid__set_OatLeastLessThan__shift__bounds, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,K:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (((plus_plus @ nat) @ M2) @ K)) @ (((plus_plus @ nat) @ N) @ K))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ K))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5816_comm__monoid__set_OatLeastLessThan__reindex, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > (A > A)),Z2:A,H3:(nat > B),M2:nat,N:nat,G:(B > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((((bij_betw @ nat) @ B) @ H3) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) @ (((set_or7035219750837199246ssThan @ B) @ (H3 @ M2)) @ (H3 @ N))) => (((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ B) @ (H3 @ M2)) @ (H3 @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ B) @ A) @ nat) @ G) @ H3)) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))))))).
% 9.43/3.46 thf(fact_5817_comm__monoid__set_OatLeastAtMost__reindex, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > (A > A)),Z2:A,H3:(nat > B),M2:nat,N:nat,G:(B > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((((bij_betw @ nat) @ B) @ H3) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) @ (((set_or1337092689740270186AtMost @ B) @ (H3 @ M2)) @ (H3 @ N))) => (((((((groups_comm_monoid_F @ A) @ B) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ B) @ (H3 @ M2)) @ (H3 @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ B) @ A) @ nat) @ G) @ H3)) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))))))).
% 9.43/3.46 thf(fact_5818_comm__monoid__set_OatLeast0__lessThan__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 9.43/3.46 thf(fact_5819_comm__monoid__set_OatLeast0__atMost__Suc__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (suc @ N))) = ((F @ (G @ (zero_zero @ nat))) @ ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ suc)) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ N)))))))).
% 9.43/3.46 thf(fact_5820_comm__monoid__set_OatLeastLessThan__shift__0, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M2))) @ (((set_or7035219750837199246ssThan @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M2)))))))).
% 9.43/3.46 thf(fact_5821_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat,M2:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ N) @ M2)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: ((G @ (((minus_minus @ nat) @ (((plus_plus @ nat) @ M2) @ N)) @ I3))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ N)) @ M2))))))).
% 9.43/3.46 thf(fact_5822_iterate__add__eq__add__iff1, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [I:nat,J:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ I) @ J) => (((((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ M2) = (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ N)) = ((((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ M2) = N))))))))).
% 9.43/3.46 thf(fact_5823_iterate__add__eq__add__iff2, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => (! [I:nat,J:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ I) @ J) => (((((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2) = (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (M2 = (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N)))))))))).
% 9.43/3.46 thf(fact_5824_iterate__add__def, axiom, (! [A:$tType]: (((cancel1802427076303600483id_add @ A) => ((iterate_add @ A) = (^ [N5:nat,A7:A]: (((((compow @ (A > A)) @ N5) @ ((plus_plus @ A) @ A7)) @ (zero_zero @ A))))))))).
% 9.43/3.46 thf(fact_5825_comm__monoid__set_OatLeast__atMost__pred__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N5:nat]: ((((minus_minus @ nat) @ N5) @ (suc @ (zero_zero @ nat))))))) @ (((set_or1337092689740270186AtMost @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5826_comm__monoid__set_OatLeast__lessThan__pred__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ (^ [N5:nat]: ((((minus_minus @ nat) @ N5) @ (suc @ (zero_zero @ nat))))))) @ (((set_or7035219750837199246ssThan @ nat) @ (suc @ M2)) @ (suc @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5827_comm__monoid__add__class_Osum__def, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_add @ A) => (((groups7311177749621191930dd_sum @ B) @ A) = ((((groups_comm_monoid_F @ A) @ B) @ (plus_plus @ A)) @ (zero_zero @ A))))))).
% 9.43/3.46 thf(fact_5828_comm__monoid__mult__class_Oprod__def, axiom, (! [B:$tType,A:$tType]: (((comm_monoid_mult @ A) => (((groups7121269368397514597t_prod @ B) @ A) = ((((groups_comm_monoid_F @ A) @ B) @ (times_times @ A)) @ (one_one @ A))))))).
% 9.43/3.46 thf(fact_5829_comm__monoid__set_OatLeastAtMost__shift__0, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,M2:nat,N:nat,G:(nat > A)]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => ((((ord_less_eq @ nat) @ M2) @ N) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N)) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ nat) @ A) @ nat) @ G) @ ((plus_plus @ nat) @ M2))) @ (((set_or1337092689740270186AtMost @ nat) @ (zero_zero @ nat)) @ (((minus_minus @ nat) @ N) @ M2))))))))).
% 9.43/3.46 thf(fact_5830_comm__monoid__set_Oin__pairs, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ nat) @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ M2)) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((F @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5831_comm__monoid__set_Oin__pairs__0, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > A),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ G) @ ((set_ord_atMost @ nat) @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ N)))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((F @ (G @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))) @ (G @ (suc @ (((times_times @ nat) @ ((numeral_numeral @ nat) @ (bit0 @ one2))) @ I3))))))) @ ((set_ord_atMost @ nat) @ N))))))).
% 9.43/3.46 thf(fact_5832_iterate__add__less__eq__iff2, axiom, (! [A:$tType]: ((((cancel1802427076303600483id_add @ A) & (ordere2412721322843649153imp_le @ A)) => (! [I:nat,J:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (((ord_less_eq @ A) @ M2) @ (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N)))))))))).
% 9.43/3.46 thf(fact_5833_iterate__add__less__eq__iff1, axiom, (! [A:$tType]: ((((cancel1802427076303600483id_add @ A) & (ordere2412721322843649153imp_le @ A)) => (! [J:nat,I:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less_eq @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (((ord_less_eq @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N))))))))).
% 9.43/3.46 thf(fact_5834_iterate__add__less__iff1, axiom, (! [A:$tType]: ((((cancel1802427076303600483id_add @ A) & (ordere2412721322843649153imp_le @ A)) => (! [J:nat,I:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ J) @ I) => ((((ord_less @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (((ord_less @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ I) @ J)) @ U)) @ M2)) @ N))))))))).
% 9.43/3.46 thf(fact_5835_iterate__add__less__iff2, axiom, (! [A:$tType]: ((((cancel1802427076303600483id_add @ A) & (ordere2412721322843649153imp_le @ A)) => (! [I:nat,J:nat,U:A,M2:A,N:A]: (((((ord_less_eq @ nat) @ I) @ J) => ((((ord_less @ A) @ (((plus_plus @ A) @ (((iterate_add @ A) @ I) @ U)) @ M2)) @ (((plus_plus @ A) @ (((iterate_add @ A) @ J) @ U)) @ N)) = (((ord_less @ A) @ M2) @ (((plus_plus @ A) @ (((iterate_add @ A) @ (((minus_minus @ nat) @ J) @ I)) @ U)) @ N)))))))))).
% 9.43/3.46 thf(fact_5836_comm__monoid__set_Otriangle__reindex, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ ((product_prod @ nat) @ nat)) @ F) @ Z2) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J2:nat]: ((((ord_less @ nat) @ (((plus_plus @ nat) @ I3) @ J2)) @ N)))))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [K4:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K4) @ I3))))) @ ((set_ord_atMost @ nat) @ K4))))) @ ((set_ord_lessThan @ nat) @ N))))))).
% 9.43/3.46 thf(fact_5837_comm__monoid__set_Otriangle__reindex__eq, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(nat > (nat > A)),N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ ((product_prod @ nat) @ nat)) @ F) @ Z2) @ ((((product_case_prod @ nat) @ nat) @ A) @ G)) @ ((collect @ ((product_prod @ nat) @ nat)) @ ((((product_case_prod @ nat) @ nat) @ $o) @ (^ [I3:nat,J2:nat]: ((((ord_less_eq @ nat) @ (((plus_plus @ nat) @ I3) @ J2)) @ N)))))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [K4:nat]: (((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (^ [I3:nat]: (((G @ I3) @ (((minus_minus @ nat) @ K4) @ I3))))) @ ((set_ord_atMost @ nat) @ K4))))) @ ((set_ord_atMost @ nat) @ N))))))).
% 9.43/3.46 thf(fact_5838_comm__monoid__set_OatLeast__int__lessThan__int__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(int > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ int) @ F) @ Z2) @ G) @ (((set_or7035219750837199246ssThan @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or7035219750837199246ssThan @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5839_comm__monoid__set_OatLeast__int__atMost__int__shift, axiom, (! [A:$tType,F:(A > (A > A)),Z2:A,G:(int > A),M2:nat,N:nat]: (((((groups778175481326437816id_set @ A) @ F) @ Z2) => (((((((groups_comm_monoid_F @ A) @ int) @ F) @ Z2) @ G) @ (((set_or1337092689740270186AtMost @ int) @ ((semiring_1_of_nat @ int) @ M2)) @ ((semiring_1_of_nat @ int) @ N))) = ((((((groups_comm_monoid_F @ A) @ nat) @ F) @ Z2) @ (((((comp @ int) @ A) @ nat) @ G) @ (semiring_1_of_nat @ int))) @ (((set_or1337092689740270186AtMost @ nat) @ M2) @ N))))))).
% 9.43/3.46 thf(fact_5840_old_Ounit_Ocase, axiom, (! [A:$tType,F:A]: (((((product_case_unit @ A) @ F) @ product_Unity) = F)))).
% 9.43/3.46 thf(fact_5841_wfP__wf__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((wfP @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = ((wf @ A) @ R))))).
% 9.43/3.46 thf(fact_5842_subset__mset_Omin__add__distrib__left, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A),Z2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ ((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ X) @ Y)) @ Z2) = ((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ (((plus_plus @ (multiset @ A)) @ X) @ Z2)) @ (((plus_plus @ (multiset @ A)) @ Y) @ Z2)))))).
% 9.43/3.46 thf(fact_5843_subset__mset_Omin__arg__le_I1_J, axiom, (! [A:$tType,N:(multiset @ A),M2:(multiset @ A)]: (((((subseteq_mset @ A) @ N) @ ((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ M2) @ N)) = (((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ M2) @ N) = N))))).
% 9.43/3.46 thf(fact_5844_subset__mset_Omin__arg__le_I2_J, axiom, (! [A:$tType,M2:(multiset @ A),N:(multiset @ A)]: (((((subseteq_mset @ A) @ M2) @ ((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ M2) @ N)) = (((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ M2) @ N) = M2))))).
% 9.43/3.46 thf(fact_5845_wfP__accp__iff, axiom, (! [A:$tType]: (((wfP @ A) = (^ [R3:(A > (A > $o))]: ((! [X8:A]: ((((accp @ A) @ R3) @ X8))))))))).
% 9.43/3.46 thf(fact_5846_accp__wfPI, axiom, (! [A:$tType,R:(A > (A > $o))]: (((! [X_12:A]: ((((accp @ A) @ R) @ X_12))) => ((wfP @ A) @ R))))).
% 9.43/3.46 thf(fact_5847_accp__wfPD, axiom, (! [A:$tType,R:(A > (A > $o)),X:A]: ((((wfP @ A) @ R) => (((accp @ A) @ R) @ X))))).
% 9.43/3.46 thf(fact_5848_unit_Ocase__distrib, axiom, (! [A:$tType,B:$tType,H3:(A > B),F:A,Unit:product_unit]: (((H3 @ (((product_case_unit @ A) @ F) @ Unit)) = (((product_case_unit @ B) @ (H3 @ F)) @ Unit))))).
% 9.43/3.46 thf(fact_5849_subset__mset_Omin__add__distrib__right, axiom, (! [A:$tType,X:(multiset @ A),Y:(multiset @ A),Z2:(multiset @ A)]: (((((plus_plus @ (multiset @ A)) @ X) @ ((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ Y) @ Z2)) = ((((min @ (multiset @ A)) @ (subseteq_mset @ A)) @ (((plus_plus @ (multiset @ A)) @ X) @ Y)) @ (((plus_plus @ (multiset @ A)) @ X) @ Z2)))))).
% 9.43/3.46 thf(fact_5850_accp__eq__acc, axiom, (! [A:$tType]: (((accp @ A) = (^ [R3:(A > (A > $o)),X3:A]: ((((member @ A) @ X3) @ ((acc @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.46 thf(fact_5851_rtrancl__def, axiom, (! [A:$tType]: (((transitive_rtrancl @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ ((transitive_rtranclp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3)))))))))))).
% 9.43/3.46 thf(fact_5852_Enum_Ortranclp__rtrancl__eq, axiom, (! [A:$tType]: (((transitive_rtranclp @ A) = (^ [R3:(A > (A > $o)),X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ ((transitive_rtrancl @ A) @ ((collect @ ((product_prod @ A) @ A)) @ ((((product_case_prod @ A) @ A) @ $o) @ R3)))))))))).
% 9.43/3.46 thf(fact_5853_acc_Ocases, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ A3) @ ((acc @ A) @ R)) => (! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ A3)) @ R) => (((member @ A) @ Y6) @ ((acc @ A) @ R))))))))).
% 9.43/3.46 thf(fact_5854_acc_Osimps, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ A) @ A3) @ ((acc @ A) @ R)) = (? [X3:A]: (((A3 = X3) & (! [Y3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y3) @ X3)) @ R) => (((member @ A) @ Y3) @ ((acc @ A) @ R)))))))))))).
% 9.43/3.46 thf(fact_5855_acc_Ointros, axiom, (! [A:$tType,X:A,R:(set @ ((product_prod @ A) @ A))]: (((! [Y4:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y4) @ X)) @ R) => (((member @ A) @ Y4) @ ((acc @ A) @ R))))) => (((member @ A) @ X) @ ((acc @ A) @ R)))))).
% 9.43/3.46 thf(fact_5856_acc__induct, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ A) @ A3) @ ((acc @ A) @ R)) => ((! [X2:A]: (((((member @ A) @ X2) @ ((acc @ A) @ R)) => ((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R) => (P @ Y6)))) => (P @ X2))))) => (P @ A3)))))).
% 9.43/3.46 thf(fact_5857_acc__downward, axiom, (! [A:$tType,B2:A,R:(set @ ((product_prod @ A) @ A)),A3:A]: (((((member @ A) @ B2) @ ((acc @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A3) @ B2)) @ R) => (((member @ A) @ A3) @ ((acc @ A) @ R))))))).
% 9.43/3.46 thf(fact_5858_not__acc__down, axiom, (! [A:$tType,X:A,R5:(set @ ((product_prod @ A) @ A))]: (((~ ((((member @ A) @ X) @ ((acc @ A) @ R5)))) => (~ ((! [Z3:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ X)) @ R5) => (((member @ A) @ Z3) @ ((acc @ A) @ R5))))))))))).
% 9.43/3.46 thf(fact_5859_acc__induct__rule, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A)),P:(A > $o)]: (((((member @ A) @ A3) @ ((acc @ A) @ R)) => ((! [X2:A]: (((((member @ A) @ X2) @ ((acc @ A) @ R)) => ((! [Y6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Y6) @ X2)) @ R) => (P @ Y6)))) => (P @ X2))))) => (P @ A3)))))).
% 9.43/3.46 thf(fact_5860_rtranclp__induct2, axiom, (! [A:$tType,B:$tType,R:(((product_prod @ A) @ B) > (((product_prod @ A) @ B) > $o)),Ax:A,Ay:B,Bx:A,By:B,P:(A > (B > $o))]: ((((((transitive_rtranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By)) => (((P @ Ax) @ Ay) => ((! [A5:A,B3:B,Aa3:A,Ba:B]: ((((((transitive_rtranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) => (((R @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba)) => (((P @ A5) @ B3) => ((P @ Aa3) @ Ba)))))) => ((P @ Bx) @ By))))))).
% 9.43/3.46 thf(fact_5861_converse__rtranclpE2, axiom, (! [A:$tType,B:$tType,R:(((product_prod @ A) @ B) > (((product_prod @ A) @ B) > $o)),Xa:A,Xb:B,Za2:A,Zb:B]: ((((((transitive_rtranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Xa) @ Xb)) @ ((((product_Pair @ A) @ B) @ Za2) @ Zb)) => ((((((product_Pair @ A) @ B) @ Xa) @ Xb) != ((((product_Pair @ A) @ B) @ Za2) @ Zb)) => (~ ((! [A5:A,B3:B]: ((((R @ ((((product_Pair @ A) @ B) @ Xa) @ Xb)) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) => (~ (((((transitive_rtranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Za2) @ Zb)))))))))))))).
% 9.43/3.46 thf(fact_5862_converse__rtranclp__induct2, axiom, (! [A:$tType,B:$tType,R:(((product_prod @ A) @ B) > (((product_prod @ A) @ B) > $o)),Ax:A,Ay:B,Bx:A,By:B,P:(A > (B > $o))]: ((((((transitive_rtranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Ax) @ Ay)) @ ((((product_Pair @ A) @ B) @ Bx) @ By)) => (((P @ Bx) @ By) => ((! [A5:A,B3:B,Aa3:A,Ba:B]: ((((R @ ((((product_Pair @ A) @ B) @ A5) @ B3)) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba)) => (((((transitive_rtranclp @ ((product_prod @ A) @ B)) @ R) @ ((((product_Pair @ A) @ B) @ Aa3) @ Ba)) @ ((((product_Pair @ A) @ B) @ Bx) @ By)) => (((P @ Aa3) @ Ba) => ((P @ A5) @ B3)))))) => ((P @ Ax) @ Ay))))))).
% 9.43/3.46 thf(fact_5863_accp__downwards__aux, axiom, (! [A:$tType,R:(A > (A > $o)),B2:A,A3:A]: ((((((transitive_rtranclp @ A) @ R) @ B2) @ A3) => ((((accp @ A) @ R) @ A3) => (((accp @ A) @ R) @ B2)))))).
% 9.43/3.46 thf(fact_5864_accp__downwards, axiom, (! [A:$tType,R:(A > (A > $o)),A3:A,B2:A]: (((((accp @ A) @ R) @ A3) => (((((transitive_rtranclp @ A) @ R) @ B2) @ A3) => (((accp @ A) @ R) @ B2)))))).
% 9.43/3.46 thf(fact_5865_Transitive__Closure_Ortranclp__rtrancl__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((transitive_rtranclp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ ((transitive_rtrancl @ A) @ R)))))))).
% 9.43/3.46 thf(fact_5866_acc__subset__induct, axiom, (! [A:$tType,D4:(set @ A),R5:(set @ ((product_prod @ A) @ A)),X:A,P:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ D4) @ ((acc @ A) @ R5)) => ((! [X2:A,Z3:A]: (((((member @ A) @ X2) @ D4) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z3) @ X2)) @ R5) => (((member @ A) @ Z3) @ D4))))) => ((((member @ A) @ X) @ D4) => ((! [X2:A]: (((((member @ A) @ X2) @ D4) => ((! [Z6:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z6) @ X2)) @ R5) => (P @ Z6)))) => (P @ X2))))) => (P @ X)))))))).
% 9.43/3.46 thf(fact_5867_acc__downwards, axiom, (! [A:$tType,A3:A,R:(set @ ((product_prod @ A) @ A)),B2:A]: (((((member @ A) @ A3) @ ((acc @ A) @ R)) => ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ ((transitive_rtrancl @ A) @ R)) => (((member @ A) @ B2) @ ((acc @ A) @ R))))))).
% 9.43/3.46 thf(fact_5868_acc__downwards__aux, axiom, (! [A:$tType,B2:A,A3:A,R:(set @ ((product_prod @ A) @ A))]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ B2) @ A3)) @ ((transitive_rtrancl @ A) @ R)) => ((((member @ A) @ A3) @ ((acc @ A) @ R)) => (((member @ A) @ B2) @ ((acc @ A) @ R))))))).
% 9.43/3.46 thf(fact_5869_accp__acc__eq, axiom, (! [A:$tType,R:(set @ ((product_prod @ A) @ A))]: ((((accp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R)))) = (^ [X3:A]: ((((member @ A) @ X3) @ ((acc @ A) @ R)))))))).
% 9.43/3.46 thf(fact_5870_acc__def, axiom, (! [A:$tType]: (((acc @ A) = (^ [R3:(set @ ((product_prod @ A) @ A))]: (((collect @ A) @ ((accp @ A) @ (^ [X3:A,Y3:A]: ((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ X3) @ Y3)) @ R3))))))))))).
% 9.43/3.46 thf(fact_5871_scomp__fcomp, axiom, (! [A:$tType,C:$tType,B:$tType,E:$tType,D:$tType,F:(A > ((product_prod @ D) @ E)),G:(D > (E > C)),H3:(C > B)]: (((((((fcomp @ A) @ C) @ B) @ ((((((product_scomp @ A) @ D) @ E) @ C) @ F) @ G)) @ H3) = ((((((product_scomp @ A) @ D) @ E) @ B) @ F) @ (^ [X3:D]: ((((((fcomp @ E) @ C) @ B) @ (G @ X3)) @ H3)))))))).
% 9.43/3.46 thf(fact_5872_successively_Opelims_I2_J, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A)]: (((((successively @ A) @ X) @ Xa) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (((Xa = (nil @ A)) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (nil @ A)))))) => ((! [X2:A]: (((Xa = (((cons @ A) @ X2) @ (nil @ A))) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (nil @ A))))))))) => (~ ((! [X2:A,Y4:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2))) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2)))) => (~ ((((X @ X2) @ Y4) & (((successively @ A) @ X) @ (((cons @ A) @ Y4) @ Xs2)))))))))))))))))).
% 9.43/3.46 thf(fact_5873_successively_Opelims_I1_J, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A),Y:$o]: ((((((successively @ A) @ X) @ Xa) = Y) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (((Xa = (nil @ A)) => (Y => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (nil @ A))))))) => ((! [X2:A]: (((Xa = (((cons @ A) @ X2) @ (nil @ A))) => (Y => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (nil @ A)))))))))) => (~ ((! [X2:A,Y4:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2))) => ((Y = (((X @ X2) @ Y4) & (((successively @ A) @ X) @ (((cons @ A) @ Y4) @ Xs2)))) => (~ ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2))))))))))))))))))).
% 9.43/3.46 thf(fact_5874_fcomp__apply, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((fcomp @ B) @ C) @ A) = (^ [F2:(B > C),G2:(C > A),X3:B]: ((G2 @ (F2 @ X3)))))))).
% 9.43/3.46 thf(fact_5875_id__fcomp, axiom, (! [B:$tType,A:$tType,G:(A > B)]: (((((((fcomp @ A) @ A) @ B) @ (id @ A)) @ G) = G)))).
% 9.43/3.46 thf(fact_5876_fcomp__id, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((((((fcomp @ A) @ B) @ B) @ F) @ (id @ B)) = F)))).
% 9.43/3.46 thf(fact_5877_fcomp__scomp, axiom, (! [A:$tType,E:$tType,B:$tType,D:$tType,C:$tType,F:(A > E),G:(E > ((product_prod @ C) @ D)),H3:(C > (D > B))]: ((((((((product_scomp @ A) @ C) @ D) @ B) @ (((((fcomp @ A) @ E) @ ((product_prod @ C) @ D)) @ F) @ G)) @ H3) = (((((fcomp @ A) @ E) @ B) @ F) @ ((((((product_scomp @ E) @ C) @ D) @ B) @ G) @ H3)))))).
% 9.43/3.46 thf(fact_5878_fcomp__assoc, axiom, (! [A:$tType,D:$tType,B:$tType,C:$tType,F:(A > D),G:(D > C),H3:(C > B)]: (((((((fcomp @ A) @ C) @ B) @ (((((fcomp @ A) @ D) @ C) @ F) @ G)) @ H3) = (((((fcomp @ A) @ D) @ B) @ F) @ (((((fcomp @ D) @ C) @ B) @ G) @ H3)))))).
% 9.43/3.46 thf(fact_5879_fcomp__def, axiom, (! [C:$tType,B:$tType,A:$tType]: (((((fcomp @ A) @ B) @ C) = (^ [F2:(A > B),G2:(B > C),X3:A]: ((G2 @ (F2 @ X3)))))))).
% 9.43/3.46 thf(fact_5880_fcomp__comp, axiom, (! [B:$tType,C:$tType,A:$tType]: (((((fcomp @ A) @ C) @ B) = (^ [F2:(A > C),G2:(C > B)]: ((((((comp @ C) @ B) @ A) @ G2) @ F2))))))).
% 9.43/3.46 thf(fact_5881_successively_Opelims_I3_J, axiom, (! [A:$tType,X:(A > (A > $o)),Xa:(list @ A)]: (((~ ((((successively @ A) @ X) @ Xa))) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ Xa)) => (~ ((! [X2:A,Y4:A,Xs2:(list @ A)]: (((Xa = (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2))) => ((((accp @ ((product_prod @ (A > (A > $o))) @ (list @ A))) @ (successively_rel @ A)) @ ((((product_Pair @ (A > (A > $o))) @ (list @ A)) @ X) @ (((cons @ A) @ X2) @ (((cons @ A) @ Y4) @ Xs2)))) => (((X @ X2) @ Y4) & (((successively @ A) @ X) @ (((cons @ A) @ Y4) @ Xs2)))))))))))))).
% 9.43/3.46 thf(fact_5882_folding__idem__axioms_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ X2)) = (F @ X2)))) => (((finite7837460588564673216axioms @ A) @ B) @ F))))).
% 9.43/3.46 thf(fact_5883_comp__fun__idem__axioms_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ X2)) = (F @ X2)))) => (((finite1981172056906289455axioms @ A) @ B) @ F))))).
% 9.43/3.46 thf(fact_5884_folding__idem__on__axioms_Ointro, axiom, (! [B:$tType,A:$tType,S2:(set @ A),F:(A > (B > B))]: (((! [X2:A,Y4:A]: (((((member @ A) @ X2) @ S2) => ((((member @ A) @ Y4) @ S2) => ((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ X2)) = (F @ X2)))))) => ((((finite6916993218817215295axioms @ A) @ B) @ S2) @ F))))).
% 9.43/3.46 thf(fact_5885_folding__idem__on__axioms__def, axiom, (! [B:$tType,A:$tType]: ((((finite6916993218817215295axioms @ A) @ B) = (^ [S5:(set @ A),F2:(A > (B > B))]: ((! [X3:A,Y3:A]: (((((member @ A) @ X3) @ S5) => ((((member @ A) @ Y3) @ S5) => ((((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ X3)) = (F2 @ X3)))))))))))).
% 9.43/3.46 thf(fact_5886_comp__fun__idem__axioms__def, axiom, (! [B:$tType,A:$tType]: ((((finite1981172056906289455axioms @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ X3)) = (F2 @ X3)))))))))).
% 9.43/3.46 thf(fact_5887_folding__idem__axioms__def, axiom, (! [B:$tType,A:$tType]: ((((finite7837460588564673216axioms @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ X3)) = (F2 @ X3)))))))))).
% 9.43/3.46 thf(fact_5888_comp__fun__idem__on__axioms__def, axiom, (! [B:$tType,A:$tType]: ((((finite4980608107308702382axioms @ A) @ B) = (^ [S5:(set @ A),F2:(A > (B > B))]: ((! [X3:A]: (((((member @ A) @ X3) @ S5) => ((((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ X3)) = (F2 @ X3))))))))))).
% 9.43/3.46 thf(fact_5889_comp__fun__idem__on__axioms_Ointro, axiom, (! [B:$tType,A:$tType,S2:(set @ A),F:(A > (B > B))]: (((! [X2:A]: (((((member @ A) @ X2) @ S2) => ((((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ X2)) = (F @ X2))))) => ((((finite4980608107308702382axioms @ A) @ B) @ S2) @ F))))).
% 9.43/3.46 thf(fact_5890_image__mset__If, axiom, (! [A:$tType,B:$tType,P:(B > $o),F:(B > A),G:(B > A),A4:(multiset @ B)]: ((((((image_mset @ B) @ A) @ (^ [X3:B]: (((((if @ A) @ (P @ X3)) @ (F @ X3)) @ (G @ X3))))) @ A4) = (((plus_plus @ (multiset @ A)) @ ((((image_mset @ B) @ A) @ F) @ (((filter_mset @ B) @ P) @ A4))) @ ((((image_mset @ B) @ A) @ G) @ (((filter_mset @ B) @ (^ [X3:B]: ((~ ((P @ X3)))))) @ A4))))))).
% 9.43/3.46 thf(fact_5891_union__filter__mset__complement, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o),M3:(multiset @ A)]: (((! [X2:A]: (((P @ X2) = (~ ((Q @ X2)))))) => ((((plus_plus @ (multiset @ A)) @ (((filter_mset @ A) @ P) @ M3)) @ (((filter_mset @ A) @ Q) @ M3)) = M3))))).
% 9.43/3.46 thf(fact_5892_filter__union__mset, axiom, (! [A:$tType,P:(A > $o),M3:(multiset @ A),N6:(multiset @ A)]: (((((filter_mset @ A) @ P) @ (((plus_plus @ (multiset @ A)) @ M3) @ N6)) = (((plus_plus @ (multiset @ A)) @ (((filter_mset @ A) @ P) @ M3)) @ (((filter_mset @ A) @ P) @ N6)))))).
% 9.43/3.46 thf(fact_5893_multiset__partition, axiom, (! [A:$tType,M3:(multiset @ A),P:(A > $o)]: ((M3 = (((plus_plus @ (multiset @ A)) @ (((filter_mset @ A) @ P) @ M3)) @ (((filter_mset @ A) @ (^ [X3:A]: ((~ ((P @ X3)))))) @ M3)))))).
% 9.43/3.46 thf(fact_5894_heap_Olub__upper, axiom, (! [A:$tType,A4:(set @ (heap_Time_Heap @ A)),X:(heap_Time_Heap @ A)]: (((((comple1602240252501008431_chain @ (heap_Time_Heap @ A)) @ (heap_Time_Heap_ord @ A)) @ A4) => ((((member @ (heap_Time_Heap @ A)) @ X) @ A4) => (((heap_Time_Heap_ord @ A) @ X) @ ((heap_Time_Heap_lub @ A) @ A4))))))).
% 9.43/3.46 thf(fact_5895_heap_Olub__least, axiom, (! [A:$tType,A4:(set @ (heap_Time_Heap @ A)),Z2:(heap_Time_Heap @ A)]: (((((comple1602240252501008431_chain @ (heap_Time_Heap @ A)) @ (heap_Time_Heap_ord @ A)) @ A4) => ((! [X2:(heap_Time_Heap @ A)]: (((((member @ (heap_Time_Heap @ A)) @ X2) @ A4) => (((heap_Time_Heap_ord @ A) @ X2) @ Z2)))) => (((heap_Time_Heap_ord @ A) @ ((heap_Time_Heap_lub @ A) @ A4)) @ Z2)))))).
% 9.43/3.46 thf(fact_5896_plus__int__code_I6_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ int) @ (neg @ M2)) @ (neg @ N)) = (neg @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_5897_ccpo__Sup__mono, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [A4:(set @ A),B6:(set @ A)]: (((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ A4) => ((((comple1602240252501008431_chain @ A) @ (ord_less_eq @ A)) @ B6) => ((! [X2:A]: (((((member @ A) @ X2) @ A4) => (? [Xa2:A]: (((((member @ A) @ Xa2) @ B6) & (((ord_less_eq @ A) @ X2) @ Xa2))))))) => (((ord_less_eq @ A) @ ((complete_Sup_Sup @ A) @ A4)) @ ((complete_Sup_Sup @ A) @ B6))))))))))).
% 9.43/3.46 thf(fact_5898_plus__int__code_I5_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ int) @ (neg @ M2)) @ (pos @ N)) = ((sub @ N) @ M2))))).
% 9.43/3.46 thf(fact_5899_plus__int__code_I4_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ int) @ (pos @ M2)) @ (neg @ N)) = ((sub @ M2) @ N))))).
% 9.43/3.46 thf(fact_5900_minus__int__code_I4_J, axiom, (! [M2:num,N:num]: (((((minus_minus @ int) @ (pos @ M2)) @ (neg @ N)) = (pos @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_5901_plus__int__code_I3_J, axiom, (! [M2:num,N:num]: (((((plus_plus @ int) @ (pos @ M2)) @ (pos @ N)) = (pos @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_5902_minus__int__code_I5_J, axiom, (! [M2:num,N:num]: (((((minus_minus @ int) @ (neg @ M2)) @ (pos @ N)) = (neg @ (((plus_plus @ num) @ M2) @ N)))))).
% 9.43/3.46 thf(fact_5903_Unity__def, axiom, (product_Unity = (product_Abs_unit @ $true))).
% 9.43/3.46 thf(fact_5904_Domainp__Domain__eq, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B))]: (((((domainp @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R)))) = (^ [X3:A]: ((((member @ A) @ X3) @ (((domain @ A) @ B) @ R)))))))).
% 9.43/3.46 thf(fact_5905_Domain__def, axiom, (! [B:$tType,A:$tType]: ((((domain @ A) @ B) = (^ [R3:(set @ ((product_prod @ A) @ B))]: (((collect @ A) @ (((domainp @ A) @ B) @ (^ [X3:A,Y3:B]: ((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) @ R3))))))))))).
% 9.43/3.46 thf(fact_5906_Abs__unit__inject, axiom, (! [X:$o,Y:$o]: (((((member @ $o) @ X) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))) => ((((member @ $o) @ Y) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))) => (((product_Abs_unit @ X) = (product_Abs_unit @ Y)) = (X = Y))))))).
% 9.43/3.46 thf(fact_5907_Abs__unit__induct, axiom, (! [P:(product_unit > $o),X:product_unit]: (((! [Y4:$o]: (((((member @ $o) @ Y4) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))) => (P @ (product_Abs_unit @ Y4))))) => (P @ X))))).
% 9.43/3.46 thf(fact_5908_Abs__unit__cases, axiom, (! [X:product_unit]: ((~ ((! [Y4:$o]: (((X = (product_Abs_unit @ Y4)) => (~ ((((member @ $o) @ Y4) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))))))))))))).
% 9.43/3.46 thf(fact_5909_Heap_ODomainp__rel, axiom, (! [B:$tType,A:$tType,R5:(A > (B > $o))]: (((((domainp @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ B)) @ (((heap_Time_rel_Heap @ A) @ B) @ R5)) = ((heap_Time_pred_Heap @ A) @ (((domainp @ A) @ B) @ R5)))))).
% 9.43/3.46 thf(fact_5910_integer_Odomain, axiom, (! [P:(int > $o)]: ((((((domainp @ int) @ int) @ (^ [Y5:int,Z4:int]: ((Y5 = Z4)))) = P) => ((((domainp @ int) @ code_integer) @ code_pcr_integer) = P))))).
% 9.43/3.46 thf(fact_5911_natural_Odomain, axiom, (! [P:(nat > $o)]: ((((((domainp @ nat) @ nat) @ (^ [Y5:nat,Z4:nat]: ((Y5 = Z4)))) = P) => ((((domainp @ nat) @ code_natural) @ code_pcr_natural) = P))))).
% 9.43/3.46 thf(fact_5912_option_ODomainp__rel, axiom, (! [B:$tType,A:$tType,R5:(A > (B > $o))]: (((((domainp @ (option @ A)) @ (option @ B)) @ (((rel_option @ A) @ B) @ R5)) = ((pred_option @ A) @ (((domainp @ A) @ B) @ R5)))))).
% 9.43/3.46 thf(fact_5913_Abs__unit__inverse, axiom, (! [Y:$o]: (((((member @ $o) @ Y) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))) => ((product_Rep_unit @ (product_Abs_unit @ Y)) = Y))))).
% 9.43/3.46 thf(fact_5914_type__definition__unit, axiom, (((((type_definition @ product_unit) @ $o) @ product_Rep_unit) @ product_Abs_unit) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o))))).
% 9.43/3.46 thf(fact_5915_Rep__unit__inverse, axiom, (! [X:product_unit]: (((product_Abs_unit @ (product_Rep_unit @ X)) = X)))).
% 9.43/3.46 thf(fact_5916_Rep__unit__inject, axiom, (! [X:product_unit,Y:product_unit]: ((((product_Rep_unit @ X) = (product_Rep_unit @ Y)) = (X = Y))))).
% 9.43/3.46 thf(fact_5917_Rep__unit, axiom, (! [X:product_unit]: ((((member @ $o) @ (product_Rep_unit @ X)) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o))))))).
% 9.43/3.46 thf(fact_5918_Rep__unit__cases, axiom, (! [Y:$o]: (((((member @ $o) @ Y) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))) => (~ ((! [X2:product_unit]: ((Y = (~ ((product_Rep_unit @ X2)))))))))))).
% 9.43/3.46 thf(fact_5919_Rep__unit__induct, axiom, (! [Y:$o,P:($o > $o)]: (((((member @ $o) @ Y) @ (((insert2 @ $o) @ $true) @ (bot_bot @ (set @ $o)))) => ((! [X2:product_unit]: ((P @ (product_Rep_unit @ X2)))) => (P @ Y)))))).
% 9.43/3.46 thf(fact_5920_adm__wf__def, axiom, (! [B:$tType,A:$tType]: ((((adm_wf @ A) @ B) = (^ [R6:(set @ ((product_prod @ A) @ A)),F7:((A > B) > (A > B))]: ((! [F2:(A > B),G2:(A > B),X3:A]: (((! [Z5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ Z5) @ X3)) @ R6) => ((F2 @ Z5) = (G2 @ Z5))))) => (((F7 @ F2) @ X3) = ((F7 @ G2) @ X3))))))))))).
% 9.43/3.46 thf(fact_5921_folding__idem_Ocomp__fun__idem, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),X:A]: (((((finite_folding_idem @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ X)) = (F @ X)))))).
% 9.43/3.46 thf(fact_5922_compat__def, axiom, (! [A2:$tType,A:$tType]: ((((bNF_Wellorder_compat @ A) @ A2) = (^ [R3:(set @ ((product_prod @ A) @ A)),R11:(set @ ((product_prod @ A2) @ A2)),F2:(A > A2)]: ((! [A7:A,B5:A]: (((((member @ ((product_prod @ A) @ A)) @ ((((product_Pair @ A) @ A) @ A7) @ B5)) @ R3) => (((member @ ((product_prod @ A2) @ A2)) @ ((((product_Pair @ A2) @ A2) @ (F2 @ A7)) @ (F2 @ B5))) @ R11)))))))))).
% 9.43/3.46 thf(fact_5923_positive__add, axiom, (! [X:rat,Y:rat]: (((positive @ X) => ((positive @ Y) => (positive @ (((plus_plus @ rat) @ X) @ Y))))))).
% 9.43/3.46 thf(fact_5924_inj__on__iff__Uniq, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F2:(A > B),A9:(set @ A)]: ((! [X3:A]: (((((member @ A) @ X3) @ A9) => ((uniq @ A) @ (^ [Y3:A]: (((((member @ A) @ Y3) @ A9) & ((F2 @ X3) = (F2 @ Y3))))))))))))))).
% 9.43/3.46 thf(fact_5925_mono__imp__mono__on, axiom, (! [B:$tType,A:$tType]: ((((order @ A) & (order @ B)) => (! [F:(A > B),A4:(set @ A)]: (((((order_mono @ A) @ B) @ F) => ((((mono_on @ A) @ B) @ F) @ A4)))))))).
% 9.43/3.46 thf(fact_5926_mono__on__subset, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [F:(A > B),A4:(set @ A),B6:(set @ A)]: ((((((mono_on @ A) @ B) @ F) @ A4) => ((((ord_less_eq @ (set @ A)) @ B6) @ A4) => ((((mono_on @ A) @ B) @ F) @ B6))))))))).
% 9.43/3.46 thf(fact_5927_mono__on__greaterD, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (linorder @ B)) => (! [G:(A > B),A4:(set @ A),X:A,Y:A]: ((((((mono_on @ A) @ B) @ G) @ A4) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => ((((ord_less @ B) @ (G @ Y)) @ (G @ X)) => (((ord_less @ A) @ Y) @ X))))))))))).
% 9.43/3.46 thf(fact_5928_mono__onD, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [F:(A > B),A4:(set @ A),R:A,S:A]: ((((((mono_on @ A) @ B) @ F) @ A4) => ((((member @ A) @ R) @ A4) => ((((member @ A) @ S) @ A4) => ((((ord_less_eq @ A) @ R) @ S) => (((ord_less_eq @ B) @ (F @ R)) @ (F @ S)))))))))))).
% 9.43/3.46 thf(fact_5929_mono__onI, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [A4:(set @ A),F:(A > B)]: (((! [R2:A,S6:A]: (((((member @ A) @ R2) @ A4) => ((((member @ A) @ S6) @ A4) => ((((ord_less_eq @ A) @ R2) @ S6) => (((ord_less_eq @ B) @ (F @ R2)) @ (F @ S6))))))) => ((((mono_on @ A) @ B) @ F) @ A4)))))))).
% 9.43/3.46 thf(fact_5930_mono__on__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (((mono_on @ A) @ B) = (^ [F2:(A > B),A9:(set @ A)]: ((! [R3:A,S4:A]: ((((((member @ A) @ R3) @ A9) & ((((member @ A) @ S4) @ A9) & (((ord_less_eq @ A) @ R3) @ S4))) => (((ord_less_eq @ B) @ (F2 @ R3)) @ (F2 @ S4)))))))))))).
% 9.43/3.46 thf(fact_5931_strict__mono__on__imp__mono__on, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (preorder @ B)) => (! [F:(A > B),A4:(set @ A)]: ((((((strict_mono_on @ A) @ B) @ F) @ A4) => ((((mono_on @ A) @ B) @ F) @ A4)))))))).
% 9.43/3.46 thf(fact_5932_folding_Ointro, axiom, (! [B:$tType,A:$tType,F:(A > (B > B))]: (((! [Y4:A,X2:A]: (((((((comp @ B) @ B) @ B) @ (F @ Y4)) @ (F @ X2)) = (((((comp @ B) @ B) @ B) @ (F @ X2)) @ (F @ Y4))))) => (((finite_folding @ A) @ B) @ F))))).
% 9.43/3.46 thf(fact_5933_folding_Ocomp__fun__commute, axiom, (! [B:$tType,A:$tType,F:(A > (B > B)),Y:A,X:A]: (((((finite_folding @ A) @ B) @ F) => ((((((comp @ B) @ B) @ B) @ (F @ Y)) @ (F @ X)) = (((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ Y))))))).
% 9.43/3.46 thf(fact_5934_strict__mono__on__eqD, axiom, (! [D:$tType,C:$tType]: ((((linorder @ C) & (preorder @ D)) => (! [F:(C > D),A4:(set @ C),X:C,Y:C]: ((((((strict_mono_on @ C) @ D) @ F) @ A4) => (((F @ X) = (F @ Y)) => ((((member @ C) @ X) @ A4) => ((((member @ C) @ Y) @ A4) => (Y = X))))))))))).
% 9.43/3.46 thf(fact_5935_strict__mono__on__leD, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (preorder @ B)) => (! [F:(A > B),A4:(set @ A),X:A,Y:A]: ((((((strict_mono_on @ A) @ B) @ F) @ A4) => ((((member @ A) @ X) @ A4) => ((((member @ A) @ Y) @ A4) => ((((ord_less_eq @ A) @ X) @ Y) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y)))))))))))).
% 9.43/3.46 thf(fact_5936_strict__mono__on__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (((strict_mono_on @ A) @ B) = (^ [F2:(A > B),A9:(set @ A)]: ((! [R3:A,S4:A]: ((((((member @ A) @ R3) @ A9) & ((((member @ A) @ S4) @ A9) & (((ord_less @ A) @ R3) @ S4))) => (((ord_less @ B) @ (F2 @ R3)) @ (F2 @ S4)))))))))))).
% 9.43/3.46 thf(fact_5937_strict__mono__onI, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [A4:(set @ A),F:(A > B)]: (((! [R2:A,S6:A]: (((((member @ A) @ R2) @ A4) => ((((member @ A) @ S6) @ A4) => ((((ord_less @ A) @ R2) @ S6) => (((ord_less @ B) @ (F @ R2)) @ (F @ S6))))))) => ((((strict_mono_on @ A) @ B) @ F) @ A4)))))))).
% 9.43/3.46 thf(fact_5938_strict__mono__onD, axiom, (! [B:$tType,A:$tType]: ((((ord @ A) & (ord @ B)) => (! [F:(A > B),A4:(set @ A),R:A,S:A]: ((((((strict_mono_on @ A) @ B) @ F) @ A4) => ((((member @ A) @ R) @ A4) => ((((member @ A) @ S) @ A4) => ((((ord_less @ A) @ R) @ S) => (((ord_less @ B) @ (F @ R)) @ (F @ S)))))))))))).
% 9.43/3.46 thf(fact_5939_strict__mono__on__imp__inj__on, axiom, (! [B:$tType,A:$tType]: ((((linorder @ A) & (preorder @ B)) => (! [F:(A > B),A4:(set @ A)]: ((((((strict_mono_on @ A) @ B) @ F) @ A4) => ((((inj_on @ A) @ B) @ F) @ A4)))))))).
% 9.43/3.46 thf(fact_5940_folding__def, axiom, (! [B:$tType,A:$tType]: ((((finite_folding @ A) @ B) = (^ [F2:(A > (B > B))]: ((! [Y3:A,X3:A]: (((((((comp @ B) @ B) @ B) @ (F2 @ Y3)) @ (F2 @ X3)) = (((((comp @ B) @ B) @ B) @ (F2 @ X3)) @ (F2 @ Y3))))))))))).
% 9.43/3.46 thf(fact_5941_folding__idem__on_Ocomp__fun__idem__on, axiom, (! [B:$tType,A:$tType,S2:(set @ A),F:(A > (B > B)),X:A,Y:A]: ((((((finite1890593828518410140dem_on @ A) @ B) @ S2) @ F) => ((((member @ A) @ X) @ S2) => ((((member @ A) @ Y) @ S2) => ((((((comp @ B) @ B) @ B) @ (F @ X)) @ (F @ X)) = (F @ X)))))))).
% 9.43/3.46 thf(fact_5942_sum_OPlus, axiom, (! [B:$tType,A:$tType,C:$tType]: (((comm_monoid_add @ A) => (! [A4:(set @ B),B6:(set @ C),G:(((sum_sum @ B) @ C) > A)]: ((((finite_finite2 @ B) @ A4) => (((finite_finite2 @ C) @ B6) => (((((groups7311177749621191930dd_sum @ ((sum_sum @ B) @ C)) @ A) @ G) @ ((((sum_Plus @ B) @ C) @ A4) @ B6)) = (((plus_plus @ A) @ ((((groups7311177749621191930dd_sum @ B) @ A) @ (((((comp @ ((sum_sum @ B) @ C)) @ A) @ B) @ G) @ ((sum_Inl @ B) @ C))) @ A4)) @ ((((groups7311177749621191930dd_sum @ C) @ A) @ (((((comp @ ((sum_sum @ B) @ C)) @ A) @ C) @ G) @ ((sum_Inr @ C) @ B))) @ B6))))))))))).
% 9.43/3.46 thf(fact_5943_map__sum__Inl__conv, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,Fl:(C > A),Fr:(D > B),S:((sum_sum @ C) @ D),Y:A]: ((((((((((sum_map_sum @ C) @ A) @ D) @ B) @ Fl) @ Fr) @ S) = (((sum_Inl @ A) @ B) @ Y)) = (? [X3:C]: (((S = (((sum_Inl @ C) @ D) @ X3)) & (Y = (Fl @ X3))))))))).
% 9.43/3.46 thf(fact_5944_map__sum__Inr__conv, axiom, (! [A:$tType,B:$tType,C:$tType,D:$tType,Fl:(C > A),Fr:(D > B),S:((sum_sum @ C) @ D),Y:B]: ((((((((((sum_map_sum @ C) @ A) @ D) @ B) @ Fl) @ Fr) @ S) = (((sum_Inr @ B) @ A) @ Y)) = (? [X3:D]: (((S = (((sum_Inr @ D) @ C) @ X3)) & (Y = (Fr @ X3))))))))).
% 9.43/3.46 thf(fact_5945_Union__plus, axiom, (! [A:$tType,B:$tType,C:$tType,F:(((sum_sum @ B) @ C) > (set @ A)),A4:(set @ B),B6:(set @ C)]: ((((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ ((sum_sum @ B) @ C)) @ (set @ A)) @ F) @ ((((sum_Plus @ B) @ C) @ A4) @ B6))) = (((sup_sup @ (set @ A)) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ B) @ (set @ A)) @ (^ [A7:B]: ((F @ (((sum_Inl @ B) @ C) @ A7))))) @ A4))) @ ((complete_Sup_Sup @ (set @ A)) @ ((((image2 @ C) @ (set @ A)) @ (^ [B5:C]: ((F @ (((sum_Inr @ C) @ B) @ B5))))) @ B6))))))).
% 9.43/3.46 thf(fact_5946_Union__sum, axiom, (! [C:$tType,A:$tType,B:$tType,F:(((sum_sum @ A) @ B) > (set @ C))]: ((((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ ((sum_sum @ A) @ B)) @ (set @ C)) @ F) @ (top_top @ (set @ ((sum_sum @ A) @ B))))) = (((sup_sup @ (set @ C)) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ A) @ (set @ C)) @ (^ [L3:A]: ((F @ (((sum_Inl @ A) @ B) @ L3))))) @ (top_top @ (set @ A))))) @ ((complete_Sup_Sup @ (set @ C)) @ ((((image2 @ B) @ (set @ C)) @ (^ [R3:B]: ((F @ (((sum_Inr @ B) @ A) @ R3))))) @ (top_top @ (set @ B))))))))).
% 9.43/3.46 thf(fact_5947_sum_Osize__gen_I1_J, axiom, (! [B:$tType,A:$tType,Xa:(A > nat),X:(B > nat),X1:A]: (((((((basic_BNF_size_sum @ A) @ B) @ Xa) @ X) @ (((sum_Inl @ A) @ B) @ X1)) = (((plus_plus @ nat) @ (Xa @ X1)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.46 thf(fact_5948_sum_Osize__gen_I2_J, axiom, (! [A:$tType,B:$tType,Xa:(A > nat),X:(B > nat),X22:B]: (((((((basic_BNF_size_sum @ A) @ B) @ Xa) @ X) @ (((sum_Inr @ B) @ A) @ X22)) = (((plus_plus @ nat) @ (X @ X22)) @ (suc @ (zero_zero @ nat))))))).
% 9.43/3.46 thf(fact_5949_Node__def, axiom, (! [A:$tType,B:$tType]: ((((old_Node @ B) @ A) = ((collect @ ((product_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat))) @ (^ [P6:((product_prod @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat))]: ((? [F2:(nat > ((sum_sum @ B) @ nat)),X3:((sum_sum @ A) @ nat),K4:nat]: (((P6 = ((((product_Pair @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ F2) @ X3)) & ((F2 @ K4) = (((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))))))))))).
% 9.43/3.46 thf(fact_5950_Node__K0__I, axiom, (! [B:$tType,A:$tType,A3:((sum_sum @ B) @ nat)]: ((((member @ ((product_prod @ (nat > ((sum_sum @ A) @ nat))) @ ((sum_sum @ B) @ nat))) @ ((((product_Pair @ (nat > ((sum_sum @ A) @ nat))) @ ((sum_sum @ B) @ nat)) @ (^ [K4:nat]: ((((sum_Inr @ nat) @ A) @ (zero_zero @ nat))))) @ A3)) @ ((old_Node @ A) @ B))))).
% 9.43/3.46 thf(fact_5951_nth__item_Opinduct, axiom, (! [A0:nat,P:(nat > $o)]: (((((accp @ nat) @ nth_item_rel) @ A0) => (((((accp @ nat) @ nth_item_rel) @ (zero_zero @ nat)) => (P @ (zero_zero @ nat))) => ((! [N3:nat]: (((((accp @ nat) @ nth_item_rel) @ (suc @ N3)) => ((! [A17:nat,Aa4:nat]: ((((nat_sum_decode @ N3) = (((sum_Inl @ nat) @ nat) @ A17)) => (((nat_sum_decode @ A17) = (((sum_Inl @ nat) @ nat) @ Aa4)) => (P @ Aa4))))) => ((! [A17:nat,B8:nat]: ((((nat_sum_decode @ N3) = (((sum_Inl @ nat) @ nat) @ A17)) => (((nat_sum_decode @ A17) = (((sum_Inr @ nat) @ nat) @ B8)) => (P @ B8))))) => ((! [B8:nat,Ba2:nat,X7:nat,Y6:nat]: ((((nat_sum_decode @ N3) = (((sum_Inr @ nat) @ nat) @ B8)) => (((nat_sum_decode @ B8) = (((sum_Inr @ nat) @ nat) @ Ba2)) => ((((((product_Pair @ nat) @ nat) @ X7) @ Y6) = (nat_prod_decode @ Ba2)) => (P @ X7)))))) => ((! [B8:nat,Ba2:nat,X7:nat,Y6:nat]: ((((nat_sum_decode @ N3) = (((sum_Inr @ nat) @ nat) @ B8)) => (((nat_sum_decode @ B8) = (((sum_Inr @ nat) @ nat) @ Ba2)) => ((((((product_Pair @ nat) @ nat) @ X7) @ Y6) = (nat_prod_decode @ Ba2)) => (P @ Y6)))))) => (P @ (suc @ N3))))))))) => (P @ A0))))))).
% 9.43/3.46 thf(fact_5952_ndepth__K0, axiom, (! [A:$tType,B:$tType,X:((sum_sum @ A) @ nat)]: (((((old_ndepth @ A) @ B) @ (((old_Abs_Node @ B) @ A) @ ((((product_Pair @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ (^ [K4:nat]: ((((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))) @ X))) = (zero_zero @ nat))))).
% 9.43/3.46 thf(fact_5953_Atom__def, axiom, (! [B:$tType,A:$tType]: ((((old_Atom @ A) @ B) = (^ [X3:((sum_sum @ A) @ nat)]: ((((insert2 @ ((old_node @ A) @ B)) @ (((old_Abs_Node @ B) @ A) @ ((((product_Pair @ (nat > ((sum_sum @ B) @ nat))) @ ((sum_sum @ A) @ nat)) @ (^ [K4:nat]: ((((sum_Inr @ nat) @ B) @ (zero_zero @ nat))))) @ X3))) @ (bot_bot @ (set @ ((old_node @ A) @ B)))))))))).
% 9.43/3.46 thf(fact_5954_nth__item_Opsimps_I1_J, axiom, (! [A:$tType]: (((countable @ A) => ((((accp @ nat) @ nth_item_rel) @ (zero_zero @ nat)) => (((nth_item @ A) @ (zero_zero @ nat)) = (undefined @ (set @ ((old_node @ A) @ product_unit))))))))).
% 9.43/3.46 thf(fact_5955_nth__item_Opelims, axiom, (! [A:$tType]: (((countable @ A) => (! [X:nat,Y:(set @ ((old_node @ A) @ product_unit))]: (((((nth_item @ A) @ X) = Y) => ((((accp @ nat) @ nth_item_rel) @ X) => (((X = (zero_zero @ nat)) => ((Y = (undefined @ (set @ ((old_node @ A) @ product_unit)))) => (~ ((((accp @ nat) @ nth_item_rel) @ (zero_zero @ nat)))))) => (~ ((! [N3:nat]: (((X = (suc @ N3)) => ((Y = ((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [I3:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J2:nat]: ((((old_In0 @ A) @ product_unit) @ ((nth_item @ A) @ J2))))) @ (^ [J2:nat]: ((((old_In1 @ A) @ product_unit) @ ((nth_item @ A) @ J2))))) @ (nat_sum_decode @ I3))))) @ (^ [I3:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J2:nat]: ((((old_Leaf @ A) @ product_unit) @ ((from_nat @ A) @ J2))))) @ (^ [J2:nat]: ((((((product_case_prod @ nat) @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ (^ [A7:nat,B5:nat]: (((((old_Scons @ A) @ product_unit) @ ((nth_item @ A) @ A7)) @ ((nth_item @ A) @ B5))))) @ (nat_prod_decode @ J2))))) @ (nat_sum_decode @ I3))))) @ (nat_sum_decode @ N3))) => (~ ((((accp @ nat) @ nth_item_rel) @ (suc @ N3))))))))))))))))))).
% 9.43/3.46 thf(fact_5956_nth__item_Opsimps_I2_J, axiom, (! [A:$tType]: (((countable @ A) => (! [N:nat]: (((((accp @ nat) @ nth_item_rel) @ (suc @ N)) => (((nth_item @ A) @ (suc @ N)) = ((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [I3:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J2:nat]: ((((old_In0 @ A) @ product_unit) @ ((nth_item @ A) @ J2))))) @ (^ [J2:nat]: ((((old_In1 @ A) @ product_unit) @ ((nth_item @ A) @ J2))))) @ (nat_sum_decode @ I3))))) @ (^ [I3:nat]: (((((((sum_case_sum @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ nat) @ (^ [J2:nat]: ((((old_Leaf @ A) @ product_unit) @ ((from_nat @ A) @ J2))))) @ (^ [J2:nat]: ((((((product_case_prod @ nat) @ nat) @ (set @ ((old_node @ A) @ product_unit))) @ (^ [A7:nat,B5:nat]: (((((old_Scons @ A) @ product_unit) @ ((nth_item @ A) @ A7)) @ ((nth_item @ A) @ B5))))) @ (nat_prod_decode @ J2))))) @ (nat_sum_decode @ I3))))) @ (nat_sum_decode @ N)))))))))).
% 9.43/3.46 thf(fact_5957_Scons__def, axiom, (! [B:$tType,A:$tType]: ((((old_Scons @ A) @ B) = (^ [M8:(set @ ((old_node @ A) @ B)),N9:(set @ ((old_node @ A) @ B))]: ((((sup_sup @ (set @ ((old_node @ A) @ B))) @ ((((image2 @ ((old_node @ A) @ B)) @ ((old_node @ A) @ B)) @ (((old_Push_Node @ B) @ A) @ (((sum_Inr @ nat) @ B) @ (one_one @ nat)))) @ M8)) @ ((((image2 @ ((old_node @ A) @ B)) @ ((old_node @ A) @ B)) @ (((old_Push_Node @ B) @ A) @ (((sum_Inr @ nat) @ B) @ (suc @ (one_one @ nat))))) @ N9)))))))).
% 9.43/3.46 thf(fact_5958_In1__def, axiom, (! [A:$tType,B:$tType]: ((((old_In1 @ A) @ B) = (((old_Scons @ A) @ B) @ (((old_Numb @ A) @ B) @ (one_one @ nat))))))).
% 9.43/3.46 thf(fact_5959_dsum__def, axiom, (! [B:$tType,A:$tType]: ((((old_dsum @ A) @ B) = (^ [R3:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),S4:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: ((((sup_sup @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((complete_Sup_Sup @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((image2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((product_case_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ (^ [X3:(set @ ((old_node @ A) @ B)),X9:(set @ ((old_node @ A) @ B))]: ((((insert2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (((old_In0 @ A) @ B) @ X3)) @ (((old_In0 @ A) @ B) @ X9))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))))))))) @ R3))) @ ((complete_Sup_Sup @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((image2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((product_case_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ (^ [Y3:(set @ ((old_node @ A) @ B)),Y8:(set @ ((old_node @ A) @ B))]: ((((insert2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (((old_In1 @ A) @ B) @ Y3)) @ (((old_In1 @ A) @ B) @ Y8))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))))))))) @ S4))))))))).
% 9.43/3.46 thf(fact_5960_dsumE, axiom, (! [B:$tType,A:$tType,W:((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))),R:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),S:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ W) @ ((((old_dsum @ A) @ B) @ R) @ S)) => ((! [X2:(set @ ((old_node @ A) @ B)),X5:(set @ ((old_node @ A) @ B))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ X2) @ X5)) @ R) => (W != ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (((old_In0 @ A) @ B) @ X2)) @ (((old_In0 @ A) @ B) @ X5)))))) => (~ ((! [Y4:(set @ ((old_node @ A) @ B)),Y14:(set @ ((old_node @ A) @ B))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ Y4) @ Y14)) @ S) => (W != ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (((old_In1 @ A) @ B) @ Y4)) @ (((old_In1 @ A) @ B) @ Y14))))))))))))).
% 9.43/3.46 thf(fact_5961_dsum__In0I, axiom, (! [B:$tType,A:$tType,M3:(set @ ((old_node @ A) @ B)),M10:(set @ ((old_node @ A) @ B)),R:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),S:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ M3) @ M10)) @ R) => (((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (((old_In0 @ A) @ B) @ M3)) @ (((old_In0 @ A) @ B) @ M10))) @ ((((old_dsum @ A) @ B) @ R) @ S)))))).
% 9.43/3.46 thf(fact_5962_dsum__In1I, axiom, (! [B:$tType,A:$tType,N6:(set @ ((old_node @ A) @ B)),N11:(set @ ((old_node @ A) @ B)),S:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),R:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ N6) @ N11)) @ S) => (((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (((old_In1 @ A) @ B) @ N6)) @ (((old_In1 @ A) @ B) @ N11))) @ ((((old_dsum @ A) @ B) @ R) @ S)))))).
% 9.43/3.46 thf(fact_5963_dprod__def, axiom, (! [B:$tType,A:$tType]: ((((old_dprod @ A) @ B) = (^ [R3:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),S4:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: (((complete_Sup_Sup @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((image2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((product_case_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ (^ [X3:(set @ ((old_node @ A) @ B)),X9:(set @ ((old_node @ A) @ B))]: (((complete_Sup_Sup @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((image2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ ((((product_case_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))) @ (^ [Y3:(set @ ((old_node @ A) @ B)),Y8:(set @ ((old_node @ A) @ B))]: ((((insert2 @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ ((((old_Scons @ A) @ B) @ X3) @ Y3)) @ ((((old_Scons @ A) @ B) @ X9) @ Y8))) @ (bot_bot @ (set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))))))))) @ S4)))))) @ R3)))))))).
% 9.43/3.46 thf(fact_5964_dprodI, axiom, (! [B:$tType,A:$tType,M3:(set @ ((old_node @ A) @ B)),M10:(set @ ((old_node @ A) @ B)),R:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),N6:(set @ ((old_node @ A) @ B)),N11:(set @ ((old_node @ A) @ B)),S:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ M3) @ M10)) @ R) => ((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ N6) @ N11)) @ S) => (((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ ((((old_Scons @ A) @ B) @ M3) @ N6)) @ ((((old_Scons @ A) @ B) @ M10) @ N11))) @ ((((old_dprod @ A) @ B) @ R) @ S))))))).
% 9.43/3.46 thf(fact_5965_dprodE, axiom, (! [B:$tType,A:$tType,C2:((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))),R:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))),S:(set @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ C2) @ ((((old_dprod @ A) @ B) @ R) @ S)) => (~ ((! [X2:(set @ ((old_node @ A) @ B)),Y4:(set @ ((old_node @ A) @ B)),X5:(set @ ((old_node @ A) @ B))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ X2) @ X5)) @ R) => (! [Y14:(set @ ((old_node @ A) @ B))]: (((((member @ ((product_prod @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B)))) @ ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ Y4) @ Y14)) @ S) => (C2 != ((((product_Pair @ (set @ ((old_node @ A) @ B))) @ (set @ ((old_node @ A) @ B))) @ ((((old_Scons @ A) @ B) @ X2) @ Y4)) @ ((((old_Scons @ A) @ B) @ X5) @ Y14))))))))))))))).
% 9.43/3.46 thf(fact_5966_prop__matchD, axiom, (! [A:$tType,Al:(list @ A),E2:A,Bl:(list @ A),Al2:(list @ A),E6:A,Bl2:(list @ A),P:(A > $o)]: ((((((append @ A) @ Al) @ (((cons @ A) @ E2) @ Bl)) = (((append @ A) @ Al2) @ (((cons @ A) @ E6) @ Bl2))) => ((((list_all @ A) @ P) @ Al) => ((~ ((P @ E2))) => ((~ ((P @ E6))) => ((((list_all @ A) @ P) @ Bl) => ((Al = Al2) & ((E2 = E6) & (Bl = Bl2))))))))))).
% 9.43/3.46 thf(fact_5967_prop__match, axiom, (! [A:$tType,P:(A > $o),Al:(list @ A),E2:A,E6:A,Bl:(list @ A),Al2:(list @ A),Bl2:(list @ A)]: (((((list_all @ A) @ P) @ Al) => ((~ ((P @ E2))) => ((~ ((P @ E6))) => ((((list_all @ A) @ P) @ Bl) => (((((append @ A) @ Al) @ (((cons @ A) @ E2) @ Bl)) = (((append @ A) @ Al2) @ (((cons @ A) @ E6) @ Bl2))) = ((Al = Al2) & ((E2 = E6) & (Bl = Bl2))))))))))).
% 9.43/3.46 thf(fact_5968_Heap__lub__def, axiom, (! [A:$tType]: (((heap_Time_Heap_lub @ A) = (((((((partial_img_lub @ (heap_Time_Heap @ A)) @ ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ (heap_Time_Heap @ A)) @ (heap_Time_execute @ A)) @ (heap_Time_Heap2 @ A)) @ ((((partial_fun_lub @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ ((partial_flat_lub @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 9.43/3.46 thf(fact_5969_heap__step__admissible, axiom, (! [E:$tType,D:$tType,C:$tType,B:$tType,A:$tType,P:(E > (A > (C > (B > (D > $o))))),X:E]: (((((comple1908693960933563346ssible @ (A > (option @ ((product_prod @ B) @ ((product_prod @ C) @ D))))) @ ((((partial_fun_lub @ (option @ ((product_prod @ B) @ ((product_prod @ C) @ D)))) @ (option @ ((product_prod @ B) @ ((product_prod @ C) @ D)))) @ A) @ ((partial_flat_lub @ (option @ ((product_prod @ B) @ ((product_prod @ C) @ D)))) @ (none @ ((product_prod @ B) @ ((product_prod @ C) @ D)))))) @ ((((partial_fun_ord @ (option @ ((product_prod @ B) @ ((product_prod @ C) @ D)))) @ (option @ ((product_prod @ B) @ ((product_prod @ C) @ D)))) @ A) @ ((partial_flat_ord @ (option @ ((product_prod @ B) @ ((product_prod @ C) @ D)))) @ (none @ ((product_prod @ B) @ ((product_prod @ C) @ D)))))) @ (^ [F2:(A > (option @ ((product_prod @ B) @ ((product_prod @ C) @ D))))]: ((! [H:A,H7:C,R3:B,N5:D]: ((((F2 @ H) = ((some @ ((product_prod @ B) @ ((product_prod @ C) @ D))) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D)) @ R3) @ ((((product_Pair @ C) @ D) @ H7) @ N5)))) => (((((P @ X) @ H) @ H7) @ R3) @ N5)))))))))).
% 9.43/3.46 thf(fact_5970_option__admissible, axiom, (! [B:$tType,A:$tType,P:(A > (B > $o))]: (((((comple1908693960933563346ssible @ (A > (option @ B))) @ ((((partial_fun_lub @ (option @ B)) @ (option @ B)) @ A) @ ((partial_flat_lub @ (option @ B)) @ (none @ B)))) @ ((((partial_fun_ord @ (option @ B)) @ (option @ B)) @ A) @ ((partial_flat_ord @ (option @ B)) @ (none @ B)))) @ (^ [F2:(A > (option @ B))]: ((! [X3:A,Y3:B]: ((((F2 @ X3) = ((some @ B) @ Y3)) => ((P @ X3) @ Y3)))))))))).
% 9.43/3.46 thf(fact_5971_option_Olub__upper, axiom, (! [A:$tType,A4:(set @ (option @ A)),X:(option @ A)]: (((((comple1602240252501008431_chain @ (option @ A)) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ A4) => ((((member @ (option @ A)) @ X) @ A4) => ((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ X) @ (((partial_flat_lub @ (option @ A)) @ (none @ A)) @ A4))))))).
% 9.43/3.46 thf(fact_5972_option_Oleq__antisym, axiom, (! [A:$tType,X:(option @ A),Y:(option @ A)]: ((((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ X) @ Y) => (((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ Y) @ X) => (X = Y)))))).
% 9.43/3.46 thf(fact_5973_option_Oleq__trans, axiom, (! [A:$tType,X:(option @ A),Y:(option @ A),Z2:(option @ A)]: ((((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ X) @ Y) => (((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ Y) @ Z2) => ((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ X) @ Z2)))))).
% 9.43/3.46 thf(fact_5974_option_Oleq__refl, axiom, (! [A:$tType,X:(option @ A)]: (((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ X) @ X)))).
% 9.43/3.46 thf(fact_5975_option_Olub__least, axiom, (! [A:$tType,A4:(set @ (option @ A)),Z2:(option @ A)]: (((((comple1602240252501008431_chain @ (option @ A)) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ A4) => ((! [X2:(option @ A)]: (((((member @ (option @ A)) @ X2) @ A4) => ((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ X2) @ Z2)))) => ((((partial_flat_ord @ (option @ A)) @ (none @ A)) @ (((partial_flat_lub @ (option @ A)) @ (none @ A)) @ A4)) @ Z2)))))).
% 9.43/3.46 thf(fact_5976_Heap__ord__def, axiom, (! [A:$tType]: (((heap_Time_Heap_ord @ A) = (((((partial_img_ord @ (heap_Time_Heap @ A)) @ ((heap_ext @ product_unit) > (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))) @ $o) @ (heap_Time_execute @ A)) @ ((((partial_fun_ord @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (heap_ext @ product_unit)) @ ((partial_flat_ord @ (option @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))) @ (none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat)))))))))).
% 9.43/3.46 thf(fact_5977_Partial__Function_Obind__mono, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,B6:((A > (option @ B)) > (option @ C)),C5:(C > ((A > (option @ B)) > (option @ D)))]: (((((((comple7038119648293358887notone @ (A > (option @ B))) @ (option @ C)) @ ((((partial_fun_ord @ (option @ B)) @ (option @ B)) @ A) @ ((partial_flat_ord @ (option @ B)) @ (none @ B)))) @ ((partial_flat_ord @ (option @ C)) @ (none @ C))) @ B6) => ((! [Y4:C]: ((((((comple7038119648293358887notone @ (A > (option @ B))) @ (option @ D)) @ ((((partial_fun_ord @ (option @ B)) @ (option @ B)) @ A) @ ((partial_flat_ord @ (option @ B)) @ (none @ B)))) @ ((partial_flat_ord @ (option @ D)) @ (none @ D))) @ (C5 @ Y4)))) => (((((comple7038119648293358887notone @ (A > (option @ B))) @ (option @ D)) @ ((((partial_fun_ord @ (option @ B)) @ (option @ B)) @ A) @ ((partial_flat_ord @ (option @ B)) @ (none @ B)))) @ ((partial_flat_ord @ (option @ D)) @ (none @ D))) @ (^ [F2:(A > (option @ B))]: (((((bind @ C) @ D) @ (B6 @ F2)) @ (^ [Y3:C]: (((C5 @ Y3) @ F2)))))))))))).
% 9.43/3.46 thf(fact_5978_option_Oconst__mono, axiom, (! [A:$tType,B:$tType,Ord:(B > (B > $o)),C2:(option @ A)]: ((((((comple7038119648293358887notone @ B) @ (option @ A)) @ Ord) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ (^ [F2:B]: (C2)))))).
% 9.43/3.46 thf(fact_5979_option_Ofixp__induct__uc, axiom, (! [B:$tType,A:$tType,C:$tType,U4:(C > (B > (option @ A))),F6:(C > C),C5:((B > (option @ A)) > C),F:C,P:((B > (option @ A)) > $o)]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (option @ A))) @ (option @ A)) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ (^ [F2:(B > (option @ A))]: (((U4 @ (F6 @ (C5 @ F2))) @ X2)))))) => ((F = (C5 @ ((((comple187402453842119260l_fixp @ (B > (option @ A))) @ ((((partial_fun_lub @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ (^ [F2:(B > (option @ A))]: ((U4 @ (F6 @ (C5 @ F2)))))))) => ((! [F4:(B > (option @ A))]: (((U4 @ (C5 @ F4)) = F4))) => (((((comple1908693960933563346ssible @ (B > (option @ A))) @ ((((partial_fun_lub @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ P) => ((P @ (^ [Uu:B]: ((none @ A)))) => ((! [F4:C]: (((P @ (U4 @ F4)) => (P @ (U4 @ (F6 @ F4)))))) => (P @ (U4 @ F))))))))))).
% 9.43/3.46 thf(fact_5980_fixp__induct__option, axiom, (! [C:$tType,B:$tType,A:$tType,U4:(C > (B > (option @ A))),F6:(C > C),C5:((B > (option @ A)) > C),F:C,P:(B > (A > $o)),X:B,Y:A]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (option @ A))) @ (option @ A)) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ (^ [F2:(B > (option @ A))]: (((U4 @ (F6 @ (C5 @ F2))) @ X2)))))) => ((F = (C5 @ ((((comple187402453842119260l_fixp @ (B > (option @ A))) @ ((((partial_fun_lub @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ (^ [F2:(B > (option @ A))]: ((U4 @ (F6 @ (C5 @ F2)))))))) => ((! [F4:(B > (option @ A))]: (((U4 @ (C5 @ F4)) = F4))) => ((! [F4:C,X2:B,Y4:A]: (((! [Xa2:B,Ya3:A]: (((((U4 @ F4) @ Xa2) = ((some @ A) @ Ya3)) => ((P @ Xa2) @ Ya3)))) => ((((U4 @ (F6 @ F4)) @ X2) = ((some @ A) @ Y4)) => ((P @ X2) @ Y4))))) => ((((U4 @ F) @ X) = ((some @ A) @ Y)) => ((P @ X) @ Y))))))))).
% 9.43/3.46 thf(fact_5981_option_Ofixp__rule__uc, axiom, (! [B:$tType,A:$tType,C:$tType,U4:(C > (B > (option @ A))),F6:(C > C),C5:((B > (option @ A)) > C),F:C]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (option @ A))) @ (option @ A)) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ (^ [F2:(B > (option @ A))]: (((U4 @ (F6 @ (C5 @ F2))) @ X2)))))) => ((F = (C5 @ ((((comple187402453842119260l_fixp @ (B > (option @ A))) @ ((((partial_fun_lub @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ (^ [F2:(B > (option @ A))]: ((U4 @ (F6 @ (C5 @ F2)))))))) => ((! [F4:C]: (((C5 @ (U4 @ F4)) = F4))) => (F = (F6 @ F)))))))).
% 9.43/3.46 thf(fact_5982_option_Omono__body__fixp, axiom, (! [A:$tType,B:$tType,F6:((B > (option @ A)) > (B > (option @ A)))]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (option @ A))) @ (option @ A)) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ (^ [F2:(B > (option @ A))]: (((F6 @ F2) @ X2)))))) => (((((comple187402453842119260l_fixp @ (B > (option @ A))) @ ((((partial_fun_lub @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ F6) = (F6 @ ((((comple187402453842119260l_fixp @ (B > (option @ A))) @ ((((partial_fun_lub @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))) @ ((((partial_fun_ord @ (option @ A)) @ (option @ A)) @ B) @ ((partial_flat_ord @ (option @ A)) @ (none @ A)))) @ F6))))))).
% 9.43/3.46 thf(fact_5983_option_Opartial__function__definitions__axioms, axiom, (! [A:$tType]: ((((partia7178651479351089652itions @ (option @ A)) @ ((partial_flat_ord @ (option @ A)) @ (none @ A))) @ ((partial_flat_lub @ (option @ A)) @ (none @ A)))))).
% 9.43/3.46 thf(fact_5984_fixp__mono, axiom, (! [A:$tType]: (((comple9053668089753744459l_ccpo @ A) => (! [F:(A > A),G:(A > A)]: (((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ F) => ((((((comple7038119648293358887notone @ A) @ A) @ (ord_less_eq @ A)) @ (ord_less_eq @ A)) @ G) => ((! [Z11:A]: ((((ord_less_eq @ A) @ (F @ Z11)) @ (G @ Z11)))) => (((ord_less_eq @ A) @ ((comple115746919287870866o_fixp @ A) @ F)) @ ((comple115746919287870866o_fixp @ A) @ G))))))))))).
% 9.43/3.46 thf(fact_5985_heap_Oconst__mono, axiom, (! [A:$tType,B:$tType,Ord:(B > (B > $o)),C2:(heap_Time_Heap @ A)]: ((((((comple7038119648293358887notone @ B) @ (heap_Time_Heap @ A)) @ Ord) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:B]: (C2)))))).
% 9.43/3.46 thf(fact_5986_Heap__Time__Monad_Obind__mono, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,B6:((A > (heap_Time_Heap @ B)) > (heap_Time_Heap @ C)),C5:(C > ((A > (heap_Time_Heap @ B)) > (heap_Time_Heap @ D)))]: (((((((comple7038119648293358887notone @ (A > (heap_Time_Heap @ B))) @ (heap_Time_Heap @ C)) @ ((((partial_fun_ord @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_ord @ B))) @ (heap_Time_Heap_ord @ C)) @ B6) => ((! [Y4:C]: ((((((comple7038119648293358887notone @ (A > (heap_Time_Heap @ B))) @ (heap_Time_Heap @ D)) @ ((((partial_fun_ord @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_ord @ B))) @ (heap_Time_Heap_ord @ D)) @ (C5 @ Y4)))) => (((((comple7038119648293358887notone @ (A > (heap_Time_Heap @ B))) @ (heap_Time_Heap @ D)) @ ((((partial_fun_ord @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_ord @ B))) @ (heap_Time_Heap_ord @ D)) @ (^ [F2:(A > (heap_Time_Heap @ B))]: (((((heap_Time_bind @ C) @ D) @ (B6 @ F2)) @ (^ [Y3:C]: (((C5 @ Y3) @ F2)))))))))))).
% 9.43/3.46 thf(fact_5987_fixp__induct__heap, axiom, (! [C:$tType,A:$tType,B:$tType,U4:(C > (B > (heap_Time_Heap @ A))),F6:(C > C),C5:((B > (heap_Time_Heap @ A)) > C),F:C,P:(B > ((heap_ext @ product_unit) > ((heap_ext @ product_unit) > (A > (nat > $o))))),X:B,H3:(heap_ext @ product_unit),H4:(heap_ext @ product_unit),R:A,N:nat]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (heap_Time_Heap @ A))) @ (heap_Time_Heap @ A)) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:(B > (heap_Time_Heap @ A))]: (((U4 @ (F6 @ (C5 @ F2))) @ X2)))))) => ((F = (C5 @ ((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (^ [F2:(B > (heap_Time_Heap @ A))]: ((U4 @ (F6 @ (C5 @ F2)))))))) => ((! [F4:(B > (heap_Time_Heap @ A))]: (((U4 @ (C5 @ F4)) = F4))) => ((! [F4:C,X2:B,H2:(heap_ext @ product_unit),H6:(heap_ext @ product_unit),R2:A,N3:nat]: (((! [Xa2:B,Ha:(heap_ext @ product_unit),H_a:(heap_ext @ product_unit),Ra3:A,Na:nat]: ((((((((heap_Time_effect @ A) @ ((U4 @ F4) @ Xa2)) @ Ha) @ H_a) @ Ra3) @ Na) => (((((P @ Xa2) @ Ha) @ H_a) @ Ra3) @ Na)))) => (((((((heap_Time_effect @ A) @ ((U4 @ (F6 @ F4)) @ X2)) @ H2) @ H6) @ R2) @ N3) => (((((P @ X2) @ H2) @ H6) @ R2) @ N3))))) => (((((((heap_Time_effect @ A) @ ((U4 @ F) @ X)) @ H3) @ H4) @ R) @ N) => (((((P @ X) @ H3) @ H4) @ R) @ N))))))))).
% 9.43/3.46 thf(fact_5988_refines__adm, axiom, (! [B:$tType,A:$tType,T2:(A > (heap_Time_Heap @ B))]: (((((comple1908693960933563346ssible @ (A > (heap_Time_Heap @ B))) @ ((((partial_fun_lub @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_lub @ B))) @ ((((partial_fun_ord @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_ord @ B))) @ (^ [F2:(A > (heap_Time_Heap @ B))]: ((! [X3:A]: ((((refine_Imp_refines @ B) @ (T2 @ X3)) @ (F2 @ X3)))))))))).
% 9.43/3.46 thf(fact_5989_heap_Ofixp__induct__uc, axiom, (! [B:$tType,A:$tType,C:$tType,U4:(C > (B > (heap_Time_Heap @ A))),F6:(C > C),C5:((B > (heap_Time_Heap @ A)) > C),F:C,P:((B > (heap_Time_Heap @ A)) > $o)]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (heap_Time_Heap @ A))) @ (heap_Time_Heap @ A)) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:(B > (heap_Time_Heap @ A))]: (((U4 @ (F6 @ (C5 @ F2))) @ X2)))))) => ((F = (C5 @ ((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (^ [F2:(B > (heap_Time_Heap @ A))]: ((U4 @ (F6 @ (C5 @ F2)))))))) => ((! [F4:(B > (heap_Time_Heap @ A))]: (((U4 @ (C5 @ F4)) = F4))) => (((((comple1908693960933563346ssible @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ P) => ((P @ (^ [Uu:B]: (((heap_Time_Heap2 @ A) @ (^ [X3:(heap_ext @ product_unit)]: ((none @ ((product_prod @ A) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))))))) => ((! [F4:C]: (((P @ (U4 @ F4)) => (P @ (U4 @ (F6 @ F4)))))) => (P @ (U4 @ F))))))))))).
% 9.43/3.46 thf(fact_5990_heap_Omono__body__fixp, axiom, (! [A:$tType,B:$tType,F6:((B > (heap_Time_Heap @ A)) > (B > (heap_Time_Heap @ A)))]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (heap_Time_Heap @ A))) @ (heap_Time_Heap @ A)) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:(B > (heap_Time_Heap @ A))]: (((F6 @ F2) @ X2)))))) => (((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ F6) = (F6 @ ((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ F6))))))).
% 9.43/3.46 thf(fact_5991_heap_Ofixp__rule__uc, axiom, (! [B:$tType,A:$tType,C:$tType,U4:(C > (B > (heap_Time_Heap @ A))),F6:(C > C),C5:((B > (heap_Time_Heap @ A)) > C),F:C]: (((! [X2:B]: ((((((comple7038119648293358887notone @ (B > (heap_Time_Heap @ A))) @ (heap_Time_Heap @ A)) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (heap_Time_Heap_ord @ A)) @ (^ [F2:(B > (heap_Time_Heap @ A))]: (((U4 @ (F6 @ (C5 @ F2))) @ X2)))))) => ((F = (C5 @ ((((comple187402453842119260l_fixp @ (B > (heap_Time_Heap @ A))) @ ((((partial_fun_lub @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_lub @ A))) @ ((((partial_fun_ord @ (heap_Time_Heap @ A)) @ (heap_Time_Heap @ A)) @ B) @ (heap_Time_Heap_ord @ A))) @ (^ [F2:(B > (heap_Time_Heap @ A))]: ((U4 @ (F6 @ (C5 @ F2)))))))) => ((! [F4:C]: (((C5 @ (U4 @ F4)) = F4))) => (F = (F6 @ F)))))))).
% 9.43/3.46 thf(fact_5992_admissible__heap, axiom, (! [B:$tType,A:$tType,P:(A > ((heap_ext @ product_unit) > ((heap_ext @ product_unit) > (B > (nat > $o)))))]: (((((comple1908693960933563346ssible @ (A > (heap_Time_Heap @ B))) @ ((((partial_fun_lub @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_lub @ B))) @ ((((partial_fun_ord @ (heap_Time_Heap @ B)) @ (heap_Time_Heap @ B)) @ A) @ (heap_Time_Heap_ord @ B))) @ (^ [F2:(A > (heap_Time_Heap @ B))]: ((! [X3:A,H:(heap_ext @ product_unit),H7:(heap_ext @ product_unit),R3:B,N5:nat]: ((((((((heap_Time_effect @ B) @ (F2 @ X3)) @ H) @ H7) @ R3) @ N5) => (((((P @ X3) @ H) @ H7) @ R3) @ N5)))))))))).
% 9.43/3.46 thf(subcl_Heap_Oheap___Typerep_Otyperep, axiom, (! [A:$tType]: (((heap @ A) => (typerep2 @ A))))).
% 9.43/3.46 thf(subcl_Heap_Oheap___HOL_Otype, axiom, (! [A:$tType]: (((heap @ A) => (type @ A))))).
% 9.43/3.46 thf(subcl_Heap_Oheap___Countable_Ocountable, axiom, (! [A:$tType]: (((heap @ A) => (countable @ A))))).
% 9.43/3.46 thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (heap_Time_Heap @ A20)))))).
% 9.43/3.46 thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (heap_Time_Heap @ A20)))))).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1, axiom, (code_term_of @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2, axiom, (typerep2 @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3, axiom, (code_term_of @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4, axiom, (typerep2 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5, axiom, (code_term_of @ code_term)).
% 9.43/3.46 thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6, axiom, (typerep2 @ code_term)).
% 9.43/3.46 thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (heap_ext @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (heap_ext @ A20)))))).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9, axiom, (code_term_of @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Enum_Oenum, axiom, (enum @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Typerep_Otyperep_10, axiom, (typerep2 @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top, axiom, (bounded_lattice_top @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (code_term_of @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Enum_Oenum_12, axiom, (! [A20:$tType,A21:$tType]: ((((enum @ A20) & (enum @ A21)) => (enum @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (typerep2 @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_14, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (typerep2 @ ((old_node @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Typerep_Otyperep_16, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17, axiom, (code_term_of @ typerep)).
% 9.43/3.46 thf(tcon_Typerep_Otyperep___Typerep_Otyperep_18, axiom, (typerep2 @ typerep)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Typerep_Otyperep_19, axiom, (typerep2 @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_20, axiom, (bounded_lattice_top @ assn)).
% 9.43/3.46 thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_21, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (pred @ A20)))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Typerep_Otyperep_22, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (pred @ A20)))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_23, axiom, (! [A20:$tType]: ((bounded_lattice_top @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_24, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (seq @ A20)))))).
% 9.43/3.46 thf(tcon_Predicate_Oseq___Typerep_Otyperep_25, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (seq @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_26, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Enum_Oenum_27, axiom, (! [A20:$tType]: (((enum @ A20) => (enum @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Typerep_Otyperep_28, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_29, axiom, (! [A20:$tType]: (((bounded_lattice_top @ A20) => (bounded_lattice_top @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_30, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (code_term_of @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Enum_Oenum_31, axiom, (! [A20:$tType,A21:$tType]: ((((enum @ A20) & (enum @ A21)) => (enum @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Typerep_Otyperep_32, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (typerep2 @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_33, axiom, (code_term_of @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___Enum_Oenum_34, axiom, (enum @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___Typerep_Otyperep_35, axiom, (typerep2 @ char)).
% 9.43/3.46 thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_36, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (array @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oarray___Typerep_Otyperep_37, axiom, (! [A20:$tType]: ((typerep2 @ (array @ A20))))).
% 9.43/3.46 thf(tcon_List_Olist___Code__Evaluation_Oterm__of_38, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (list @ A20)))))).
% 9.43/3.46 thf(tcon_List_Olist___Typerep_Otyperep_39, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (list @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_40, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (ref @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oref___Typerep_Otyperep_41, axiom, (! [A20:$tType]: ((typerep2 @ (ref @ A20))))).
% 9.43/3.46 thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_42, axiom, (code_term_of @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Enum_Oenum_43, axiom, (enum @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Typerep_Otyperep_44, axiom, (typerep2 @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_45, axiom, (bounded_lattice_top @ $o)).
% 9.43/3.46 thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_46, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Enum_Oenum_47, axiom, (! [A20:$tType]: (((enum @ A20) => (enum @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Typerep_Otyperep_48, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_49, axiom, (! [A20:$tType]: ((bounded_lattice_top @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_50, axiom, (code_term_of @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Typerep_Otyperep_51, axiom, (typerep2 @ rat)).
% 9.43/3.46 thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_52, axiom, (code_term_of @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Typerep_Otyperep_53, axiom, (typerep2 @ num)).
% 9.43/3.46 thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_54, axiom, (code_term_of @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Typerep_Otyperep_55, axiom, (typerep2 @ nat)).
% 9.43/3.46 thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_56, axiom, (code_term_of @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Typerep_Otyperep_57, axiom, (typerep2 @ int)).
% 9.43/3.46 thf(tcon_itself___Code__Evaluation_Oterm__of_58, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (code_term_of @ (itself @ A20)))))).
% 9.43/3.46 thf(tcon_itself___Typerep_Otyperep_59, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (typerep2 @ (itself @ A20)))))).
% 9.43/3.46 thf(tcon_fun___Code__Evaluation_Oterm__of_60, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (code_term_of @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Enum_Oenum_61, axiom, (! [A20:$tType,A21:$tType]: ((((enum @ A20) & (enum @ A21)) => (enum @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Typerep_Otyperep_62, axiom, (! [A20:$tType,A21:$tType]: ((((typerep2 @ A20) & (typerep2 @ A21)) => (typerep2 @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Lattices_Obounded__lattice__top_63, axiom, (! [A20:$tType,A21:$tType]: (((bounded_lattice @ A21) => (bounded_lattice_top @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive, axiom, (! [A20:$tType,A21:$tType]: ((((cl_HOL_Oequal @ A20) & ((quickc3360725361186068524ustive @ A20) & (quickc3360725361186068524ustive @ A21))) => (quickc3360725361186068524ustive @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice, axiom, (! [A20:$tType,A21:$tType]: (((comple6319245703460814977attice @ A21) => (comple6319245703460814977attice @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive, axiom, (! [A20:$tType,A21:$tType]: ((((cl_HOL_Oequal @ A20) & ((quickc658316121487927005ustive @ A20) & (quickc658316121487927005ustive @ A21))) => (quickc658316121487927005ustive @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra, axiom, (! [A20:$tType,A21:$tType]: (((boolea8198339166811842893lgebra @ A21) => (boolea8198339166811842893lgebra @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Complete__Partial__Order_Occpo, axiom, (! [A20:$tType,A21:$tType]: (((comple6319245703460814977attice @ A21) => (comple9053668089753744459l_ccpo @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Quickcheck__Random_Orandom, axiom, (! [A20:$tType,A21:$tType]: ((((code_term_of @ A20) & ((cl_HOL_Oequal @ A20) & (quickcheck_random @ A21))) => (quickcheck_random @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A20:$tType,A21:$tType]: (((semilattice_sup @ A21) => (semilattice_sup @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Lattices_Osemilattice__inf, axiom, (! [A20:$tType,A21:$tType]: (((semilattice_inf @ A21) => (semilattice_inf @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Lattices_Obounded__lattice, axiom, (! [A20:$tType,A21:$tType]: (((bounded_lattice @ A21) => (bounded_lattice @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Complete__Lattices_OSup, axiom, (! [A20:$tType,A21:$tType]: (((complete_Sup @ A21) => (complete_Sup @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Complete__Lattices_OInf, axiom, (! [A20:$tType,A21:$tType]: (((complete_Inf @ A21) => (complete_Inf @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A20:$tType,A21:$tType]: (((order_top @ A21) => (order_top @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A20:$tType,A21:$tType]: (((order_bot @ A21) => (order_bot @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Countable_Ocountable, axiom, (! [A20:$tType,A21:$tType]: ((((finite_finite @ A20) & (countable @ A21)) => (countable @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A20:$tType,A21:$tType]: (((preorder @ A21) => (preorder @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A20:$tType,A21:$tType]: ((((finite_finite @ A20) & (finite_finite @ A21)) => (finite_finite @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Orderings_Oorder, axiom, (! [A20:$tType,A21:$tType]: (((order @ A21) => (order @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Orderings_Oord, axiom, (! [A20:$tType,A21:$tType]: (((ord @ A21) => (ord @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Groups_Ouminus, axiom, (! [A20:$tType,A21:$tType]: (((uminus @ A21) => (uminus @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Lattices_Osup, axiom, (! [A20:$tType,A21:$tType]: (((semilattice_sup @ A21) => (sup @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___Lattices_Oinf, axiom, (! [A20:$tType,A21:$tType]: (((semilattice_inf @ A21) => (inf @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_fun___HOL_Oequal, axiom, (! [A20:$tType,A21:$tType]: ((((enum @ A20) & (cl_HOL_Oequal @ A21)) => (cl_HOL_Oequal @ (A20 > A21)))))).
% 9.43/3.46 thf(tcon_itself___Quickcheck__Random_Orandom_64, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (quickcheck_random @ (itself @ A20)))))).
% 9.43/3.46 thf(tcon_itself___HOL_Oequal_65, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (itself @ A20))))).
% 9.43/3.46 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations, axiom, (bit_un5681908812861735899ations @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct, axiom, (semiri1453513574482234551roduct @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat, axiom, (euclid5411537665997757685th_nat @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat, axiom, (euclid8789492081693882211th_nat @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le, axiom, (ordere1937475149494474687imp_le @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring, axiom, (euclid3128863361964157862miring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel, axiom, (euclid4440199948858584721cancel @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral, axiom, (unique1627219031080169319umeral @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel, axiom, (euclid8851590272496341667cancel @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add, axiom, (strict9044650504122735259up_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add, axiom, (ordere580206878836729694up_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le, axiom, (ordere2412721322843649153imp_le @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations, axiom, (bit_se359711467146920520ations @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_66, axiom, (quickc3360725361186068524ustive @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add, axiom, (strict7427464778891057005id_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add, axiom, (ordere8940638589300402666id_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring, axiom, (euclid3725896446679973847miring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict, axiom, (linord715952674999750819strict @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add, axiom, (linord4140545234300271783up_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations, axiom, (bit_ri3973907225187159222ations @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring, axiom, (linord181362715937106298miring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor, axiom, (semido2269285787275462019factor @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_67, axiom, (quickc658316121487927005ustive @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors, axiom, (semiri3467727345109120633visors @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add, axiom, (ordere6658533253407199908up_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs, axiom, (ordere166539214618696060dd_abs @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add, axiom, (ordere6911136660526730532id_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add, axiom, (linord5086331880401160121up_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add, axiom, (cancel2418104881723323429up_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors, axiom, (ring_15535105094025558882visors @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add, axiom, (cancel1802427076303600483id_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict, axiom, (linord4710134922213307826strict @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel, axiom, (comm_s4317794764714335236cancel @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits, axiom, (bit_semiring_bits @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Onormalization__semidom, axiom, (normal8620421768224518004emidom @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1, axiom, (linord6961819062388156250ring_1 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add, axiom, (ordered_ab_group_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add, axiom, (cancel_semigroup_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__semidom, axiom, (linordered_semidom @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Quickcheck__Random_Orandom_68, axiom, (quickcheck_random @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_69, axiom, (semilattice_sup @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_70, axiom, (semilattice_inf @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult, axiom, (ab_semigroup_mult @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom, axiom, (algebraic_semidom @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult, axiom, (comm_monoid_mult @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oab__semigroup__add, axiom, (ab_semigroup_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Parity_Osemiring__parity, axiom, (semiring_parity @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add, axiom, (comm_monoid_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemiring__modulo, axiom, (semiring_modulo @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__ring, axiom, (linordered_ring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Olinordered__idom, axiom, (linordered_idom @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1, axiom, (comm_semiring_1 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0, axiom, (comm_semiring_0 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Osemigroup__mult, axiom, (semigroup_mult @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Complete__Lattices_OSup_71, axiom, (complete_Sup @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Complete__Lattices_OInf_72, axiom, (complete_Inf @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemidom__modulo, axiom, (semidom_modulo @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemidom__divide, axiom, (semidom_divide @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Num_Osemiring__numeral, axiom, (semiring_numeral @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Osemigroup__add, axiom, (semigroup_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ozero__less__one, axiom, (zero_less_one @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ocomm__semiring, axiom, (comm_semiring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Nat_Osemiring__char__0, axiom, (semiring_char_0 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oab__group__add, axiom, (ab_group_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Countable_Ocountable_73, axiom, (countable @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ozero__neq__one, axiom, (zero_neq_one @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Oordered__ring, axiom, (ordered_ring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn, axiom, (idom_abs_sgn @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Parity_Oring__parity, axiom, (ring_parity @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Orderings_Opreorder_74, axiom, (preorder @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Orderings_Olinorder, axiom, (linorder @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Omonoid__mult, axiom, (monoid_mult @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ocomm__ring__1, axiom, (comm_ring_1 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Omonoid__add, axiom, (monoid_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemiring__1, axiom, (semiring_1 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemiring__0, axiom, (semiring_0 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Orderings_Ono__top, axiom, (no_top @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Orderings_Ono__bot, axiom, (no_bot @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ogroup__add, axiom, (group_add @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___GCD_Osemiring__gcd, axiom, (semiring_gcd @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___GCD_Osemiring__Gcd, axiom, (semiring_Gcd @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Ocomm__ring, axiom, (comm_ring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Orderings_Oorder_75, axiom, (order @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Num_Oneg__numeral, axiom, (neg_numeral @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Nat_Oring__char__0, axiom, (ring_char_0 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Osemiring, axiom, (semiring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Orderings_Oord_76, axiom, (ord @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ouminus_77, axiom, (uminus @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Oring__1, axiom, (ring_1 @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Lattices_Osup_78, axiom, (sup @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Lattices_Oinf_79, axiom, (inf @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___GCD_Oring__gcd, axiom, (ring_gcd @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Power_Opower, axiom, (power @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Num_Onumeral, axiom, (numeral @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Ozero, axiom, (zero @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oplus, axiom, (plus @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Oring, axiom, (ring @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Groups_Oone, axiom, (one @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Rings_Odvd, axiom, (dvd @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___Heap_Oheap, axiom, (heap @ int)).
% 9.43/3.46 thf(tcon_Int_Oint___HOL_Oequal_80, axiom, (cl_HOL_Oequal @ int)).
% 9.43/3.46 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_81, axiom, (bit_un5681908812861735899ations @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_82, axiom, (semiri1453513574482234551roduct @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_83, axiom, (euclid5411537665997757685th_nat @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_84, axiom, (ordere1937475149494474687imp_le @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_85, axiom, (euclid3128863361964157862miring @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_86, axiom, (euclid4440199948858584721cancel @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_87, axiom, (unique1627219031080169319umeral @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_88, axiom, (strict9044650504122735259up_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff, axiom, (ordere1170586879665033532d_diff @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_89, axiom, (ordere580206878836729694up_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_90, axiom, (ordere2412721322843649153imp_le @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_91, axiom, (bit_se359711467146920520ations @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_92, axiom, (quickc3360725361186068524ustive @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_93, axiom, (strict7427464778891057005id_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_94, axiom, (ordere8940638589300402666id_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add, axiom, (canoni5634975068530333245id_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_95, axiom, (euclid3725896446679973847miring @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_96, axiom, (linord4140545234300271783up_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_97, axiom, (linord181362715937106298miring @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_98, axiom, (semido2269285787275462019factor @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_99, axiom, (quickc658316121487927005ustive @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_100, axiom, (semiri3467727345109120633visors @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_101, axiom, (ordere6658533253407199908up_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_102, axiom, (ordere6911136660526730532id_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_103, axiom, (cancel2418104881723323429up_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_104, axiom, (cancel1802427076303600483id_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_105, axiom, (comm_s4317794764714335236cancel @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_106, axiom, (bit_semiring_bits @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Onormalization__semidom_107, axiom, (normal8620421768224518004emidom @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_108, axiom, (cancel_semigroup_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_109, axiom, (linordered_semidom @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_110, axiom, (quickcheck_random @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_111, axiom, (semilattice_sup @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_112, axiom, (semilattice_inf @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_113, axiom, (ab_semigroup_mult @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_114, axiom, (algebraic_semidom @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_115, axiom, (comm_monoid_mult @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff, axiom, (comm_monoid_diff @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_116, axiom, (ab_semigroup_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Parity_Osemiring__parity_117, axiom, (semiring_parity @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_118, axiom, (comm_monoid_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_119, axiom, (semiring_modulo @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_120, axiom, (comm_semiring_1 @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_121, axiom, (comm_semiring_0 @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_122, axiom, (semigroup_mult @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Complete__Lattices_OSup_123, axiom, (complete_Sup @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Complete__Lattices_OInf_124, axiom, (complete_Inf @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_125, axiom, (semidom_modulo @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemidom__divide_126, axiom, (semidom_divide @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Num_Osemiring__numeral_127, axiom, (semiring_numeral @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Osemigroup__add_128, axiom, (semigroup_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Ozero__less__one_129, axiom, (zero_less_one @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_130, axiom, (comm_semiring @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Orderings_Oorder__bot_131, axiom, (order_bot @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_132, axiom, (semiring_char_0 @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Countable_Ocountable_133, axiom, (countable @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_134, axiom, (zero_neq_one @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Orderings_Opreorder_135, axiom, (preorder @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Orderings_Olinorder_136, axiom, (linorder @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Omonoid__mult_137, axiom, (monoid_mult @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Omonoid__add_138, axiom, (monoid_add @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemiring__1_139, axiom, (semiring_1 @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemiring__0_140, axiom, (semiring_0 @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Orderings_Ono__top_141, axiom, (no_top @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_142, axiom, (semiring_gcd @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_143, axiom, (semiring_Gcd @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Orderings_Oorder_144, axiom, (order @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Osemiring_145, axiom, (semiring @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Orderings_Oord_146, axiom, (ord @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Lattices_Osup_147, axiom, (sup @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Lattices_Oinf_148, axiom, (inf @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Power_Opower_149, axiom, (power @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Num_Onumeral_150, axiom, (numeral @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Ozero_151, axiom, (zero @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oplus_152, axiom, (plus @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Groups_Oone_153, axiom, (one @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Rings_Odvd_154, axiom, (dvd @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___Heap_Oheap_155, axiom, (heap @ nat)).
% 9.43/3.46 thf(tcon_Nat_Onat___HOL_Oequal_156, axiom, (cl_HOL_Oequal @ nat)).
% 9.43/3.46 thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_157, axiom, (quickc3360725361186068524ustive @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Quickcheck__Random_Orandom_158, axiom, (quickcheck_random @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Orderings_Opreorder_159, axiom, (preorder @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Orderings_Olinorder_160, axiom, (linorder @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Orderings_Oorder_161, axiom, (order @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Orderings_Oord_162, axiom, (ord @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___Groups_Oplus_163, axiom, (plus @ num)).
% 9.43/3.46 thf(tcon_Num_Onum___HOL_Oequal_164, axiom, (cl_HOL_Oequal @ num)).
% 9.43/3.46 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_165, axiom, (semiri1453513574482234551roduct @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_166, axiom, (ordere1937475149494474687imp_le @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167, axiom, (strict9044650504122735259up_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168, axiom, (ordere580206878836729694up_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169, axiom, (ordere2412721322843649153imp_le @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_170, axiom, (quickc3360725361186068524ustive @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171, axiom, (strict7427464778891057005id_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172, axiom, (ordere8940638589300402666id_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field, axiom, (archim462609752435547400_field @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_173, axiom, (linord715952674999750819strict @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_174, axiom, (linord4140545234300271783up_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_175, axiom, (linord181362715937106298miring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_176, axiom, (quickc658316121487927005ustive @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_177, axiom, (semiri3467727345109120633visors @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_178, axiom, (ordere6658533253407199908up_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_179, axiom, (ordere166539214618696060dd_abs @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling, axiom, (archim2362893244070406136eiling @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_180, axiom, (ordere6911136660526730532id_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_181, axiom, (linord5086331880401160121up_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_182, axiom, (cancel2418104881723323429up_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_183, axiom, (ring_15535105094025558882visors @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_184, axiom, (cancel1802427076303600483id_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_185, axiom, (linord4710134922213307826strict @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_186, axiom, (comm_s4317794764714335236cancel @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_187, axiom, (linord6961819062388156250ring_1 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_188, axiom, (ordered_ab_group_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_189, axiom, (cancel_semigroup_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_190, axiom, (linordered_semidom @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_191, axiom, (quickcheck_random @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Odense__linorder, axiom, (dense_linorder @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_192, axiom, (semilattice_sup @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_193, axiom, (semilattice_inf @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_194, axiom, (ab_semigroup_mult @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_195, axiom, (comm_monoid_mult @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_196, axiom, (ab_semigroup_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Fields_Olinordered__field, axiom, (linordered_field @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_197, axiom, (comm_monoid_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__ring_198, axiom, (linordered_ring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Olinordered__idom_199, axiom, (linordered_idom @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_200, axiom, (comm_semiring_1 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_201, axiom, (comm_semiring_0 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_202, axiom, (semigroup_mult @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Osemidom__divide_203, axiom, (semidom_divide @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Num_Osemiring__numeral_204, axiom, (semiring_numeral @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Osemigroup__add_205, axiom, (semigroup_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Fields_Odivision__ring, axiom, (division_ring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ozero__less__one_206, axiom, (zero_less_one @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_207, axiom, (comm_semiring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_208, axiom, (semiring_char_0 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oab__group__add_209, axiom, (ab_group_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Fields_Ofield__char__0, axiom, (field_char_0 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Countable_Ocountable_210, axiom, (countable @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_211, axiom, (zero_neq_one @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Oordered__ring_212, axiom, (ordered_ring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_213, axiom, (idom_abs_sgn @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Opreorder_214, axiom, (preorder @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Olinorder_215, axiom, (linorder @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Omonoid__mult_216, axiom, (monoid_mult @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_217, axiom, (comm_ring_1 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Omonoid__add_218, axiom, (monoid_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Osemiring__1_219, axiom, (semiring_1 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Osemiring__0_220, axiom, (semiring_0 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Ono__top_221, axiom, (no_top @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Ono__bot_222, axiom, (no_bot @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ogroup__add_223, axiom, (group_add @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Ocomm__ring_224, axiom, (comm_ring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Oorder_225, axiom, (order @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Num_Oneg__numeral_226, axiom, (neg_numeral @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Nat_Oring__char__0_227, axiom, (ring_char_0 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Osemiring_228, axiom, (semiring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Fields_Oinverse, axiom, (inverse @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Orderings_Oord_229, axiom, (ord @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ouminus_230, axiom, (uminus @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Oring__1_231, axiom, (ring_1 @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Lattices_Osup_232, axiom, (sup @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Lattices_Oinf_233, axiom, (inf @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Fields_Ofield, axiom, (field @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Power_Opower_234, axiom, (power @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Num_Onumeral_235, axiom, (numeral @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Ozero_236, axiom, (zero @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oplus_237, axiom, (plus @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Oring_238, axiom, (ring @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Groups_Oone_239, axiom, (one @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___Rings_Odvd_240, axiom, (dvd @ rat)).
% 9.43/3.46 thf(tcon_Rat_Orat___HOL_Oequal_241, axiom, (cl_HOL_Oequal @ rat)).
% 9.43/3.46 thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_242, axiom, (! [A20:$tType]: (((quickc3360725361186068524ustive @ A20) => (quickc3360725361186068524ustive @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243, axiom, (! [A20:$tType]: ((comple6319245703460814977attice @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_244, axiom, (! [A20:$tType]: (((quickc658316121487927005ustive @ A20) => (quickc658316121487927005ustive @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_245, axiom, (! [A20:$tType]: ((boolea8198339166811842893lgebra @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_246, axiom, (! [A20:$tType]: ((comple9053668089753744459l_ccpo @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Quickcheck__Random_Orandom_247, axiom, (! [A20:$tType]: (((quickcheck_random @ A20) => (quickcheck_random @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_248, axiom, (! [A20:$tType]: ((semilattice_sup @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_249, axiom, (! [A20:$tType]: ((semilattice_inf @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Lattices_Obounded__lattice_250, axiom, (! [A20:$tType]: ((bounded_lattice @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Complete__Lattices_OSup_251, axiom, (! [A20:$tType]: ((complete_Sup @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Complete__Lattices_OInf_252, axiom, (! [A20:$tType]: ((complete_Inf @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Orderings_Oorder__top_253, axiom, (! [A20:$tType]: ((order_top @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Orderings_Oorder__bot_254, axiom, (! [A20:$tType]: ((order_bot @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Countable_Ocountable_255, axiom, (! [A20:$tType]: (((finite_finite @ A20) => (countable @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Orderings_Opreorder_256, axiom, (! [A20:$tType]: ((preorder @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_257, axiom, (! [A20:$tType]: (((finite_finite @ A20) => (finite_finite @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_Set_Oset___Orderings_Oorder_258, axiom, (! [A20:$tType]: ((order @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Orderings_Oord_259, axiom, (! [A20:$tType]: ((ord @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Groups_Ouminus_260, axiom, (! [A20:$tType]: ((uminus @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Lattices_Osup_261, axiom, (! [A20:$tType]: ((sup @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___Lattices_Oinf_262, axiom, (! [A20:$tType]: ((inf @ (set @ A20))))).
% 9.43/3.46 thf(tcon_Set_Oset___HOL_Oequal_263, axiom, (! [A20:$tType]: (((cl_HOL_Oequal @ A20) => (cl_HOL_Oequal @ (set @ A20)))))).
% 9.43/3.46 thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_264, axiom, (quickc3360725361186068524ustive @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_265, axiom, (comple6319245703460814977attice @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_266, axiom, (boolea8198339166811842893lgebra @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_267, axiom, (comple9053668089753744459l_ccpo @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_268, axiom, (quickcheck_random @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_269, axiom, (semilattice_sup @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_270, axiom, (semilattice_inf @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_271, axiom, (bounded_lattice @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Complete__Lattices_OSup_272, axiom, (complete_Sup @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Complete__Lattices_OInf_273, axiom, (complete_Inf @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Orderings_Oorder__top_274, axiom, (order_top @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Orderings_Oorder__bot_275, axiom, (order_bot @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Countable_Ocountable_276, axiom, (countable @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Orderings_Opreorder_277, axiom, (preorder @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Orderings_Olinorder_278, axiom, (linorder @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_279, axiom, (finite_finite @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Orderings_Oorder_280, axiom, (order @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Orderings_Oord_281, axiom, (ord @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Groups_Ouminus_282, axiom, (uminus @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Lattices_Osup_283, axiom, (sup @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Lattices_Oinf_284, axiom, (inf @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___Heap_Oheap_285, axiom, (heap @ $o)).
% 9.43/3.46 thf(tcon_HOL_Obool___HOL_Oequal_286, axiom, (cl_HOL_Oequal @ $o)).
% 9.43/3.46 thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_287, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (quickc3360725361186068524ustive @ (ref @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_288, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (quickcheck_random @ (ref @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oref___Countable_Ocountable_289, axiom, (! [A20:$tType]: ((countable @ (ref @ A20))))).
% 9.43/3.46 thf(tcon_Heap_Oref___Heap_Oheap_290, axiom, (! [A20:$tType]: ((heap @ (ref @ A20))))).
% 9.43/3.46 thf(tcon_Heap_Oref___HOL_Oequal_291, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (ref @ A20))))).
% 9.43/3.46 thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_292, axiom, (! [A20:$tType]: (((quickc3360725361186068524ustive @ A20) => (quickc3360725361186068524ustive @ (list @ A20)))))).
% 9.43/3.46 thf(tcon_List_Olist___Quickcheck__Random_Orandom_293, axiom, (! [A20:$tType]: (((quickcheck_random @ A20) => (quickcheck_random @ (list @ A20)))))).
% 9.43/3.46 thf(tcon_List_Olist___Countable_Ocountable_294, axiom, (! [A20:$tType]: (((countable @ A20) => (countable @ (list @ A20)))))).
% 9.43/3.46 thf(tcon_List_Olist___Heap_Oheap_295, axiom, (! [A20:$tType]: (((heap @ A20) => (heap @ (list @ A20)))))).
% 9.43/3.46 thf(tcon_List_Olist___HOL_Oequal_296, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (list @ A20))))).
% 9.43/3.46 thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_297, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (quickc3360725361186068524ustive @ (array @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_298, axiom, (! [A20:$tType]: (((typerep2 @ A20) => (quickcheck_random @ (array @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oarray___Countable_Ocountable_299, axiom, (! [A20:$tType]: ((countable @ (array @ A20))))).
% 9.43/3.46 thf(tcon_Heap_Oarray___Heap_Oheap_300, axiom, (! [A20:$tType]: ((heap @ (array @ A20))))).
% 9.43/3.46 thf(tcon_Heap_Oarray___HOL_Oequal_301, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (array @ A20))))).
% 9.43/3.46 thf(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_302, axiom, (quickc3360725361186068524ustive @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___Quickcheck__Random_Orandom_303, axiom, (quickcheck_random @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___Countable_Ocountable_304, axiom, (countable @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___Finite__Set_Ofinite_305, axiom, (finite_finite @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___Heap_Oheap_306, axiom, (heap @ char)).
% 9.43/3.46 thf(tcon_String_Ochar___HOL_Oequal_307, axiom, (cl_HOL_Oequal @ char)).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_308, axiom, (! [A20:$tType,A21:$tType]: ((((quickc3360725361186068524ustive @ A20) & (quickc3360725361186068524ustive @ A21)) => (quickc3360725361186068524ustive @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_309, axiom, (! [A20:$tType,A21:$tType]: ((((quickcheck_random @ A20) & (quickcheck_random @ A21)) => (quickcheck_random @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Countable_Ocountable_310, axiom, (! [A20:$tType,A21:$tType]: ((((countable @ A20) & (countable @ A21)) => (countable @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_311, axiom, (! [A20:$tType,A21:$tType]: ((((finite_finite @ A20) & (finite_finite @ A21)) => (finite_finite @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___Heap_Oheap_312, axiom, (! [A20:$tType,A21:$tType]: ((((heap @ A20) & (heap @ A21)) => (heap @ ((sum_sum @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Sum__Type_Osum___HOL_Oequal_313, axiom, (! [A20:$tType,A21:$tType]: ((cl_HOL_Oequal @ ((sum_sum @ A20) @ A21))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_314, axiom, (! [A20:$tType]: (((quickc3360725361186068524ustive @ A20) => (quickc3360725361186068524ustive @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_315, axiom, (! [A20:$tType]: (((comple6319245703460814977attice @ A20) => (comple6319245703460814977attice @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_316, axiom, (! [A20:$tType]: (((comple6319245703460814977attice @ A20) => (comple9053668089753744459l_ccpo @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_317, axiom, (! [A20:$tType]: (((quickcheck_random @ A20) => (quickcheck_random @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_318, axiom, (! [A20:$tType]: (((semilattice_sup @ A20) => (semilattice_sup @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_319, axiom, (! [A20:$tType]: (((semilattice_inf @ A20) => (semilattice_inf @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Lattices_Obounded__lattice_320, axiom, (! [A20:$tType]: (((bounded_lattice_top @ A20) => (bounded_lattice @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Complete__Lattices_OSup_321, axiom, (! [A20:$tType]: (((comple6319245703460814977attice @ A20) => (complete_Sup @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Complete__Lattices_OInf_322, axiom, (! [A20:$tType]: (((comple6319245703460814977attice @ A20) => (complete_Inf @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Orderings_Oorder__top_323, axiom, (! [A20:$tType]: (((order_top @ A20) => (order_top @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Orderings_Oorder__bot_324, axiom, (! [A20:$tType]: (((order @ A20) => (order_bot @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Countable_Ocountable_325, axiom, (! [A20:$tType]: (((countable @ A20) => (countable @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Orderings_Opreorder_326, axiom, (! [A20:$tType]: (((preorder @ A20) => (preorder @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Orderings_Olinorder_327, axiom, (! [A20:$tType]: (((linorder @ A20) => (linorder @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Finite__Set_Ofinite_328, axiom, (! [A20:$tType]: (((finite_finite @ A20) => (finite_finite @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Orderings_Oorder_329, axiom, (! [A20:$tType]: (((order @ A20) => (order @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Orderings_Oord_330, axiom, (! [A20:$tType]: (((preorder @ A20) => (ord @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Lattices_Osup_331, axiom, (! [A20:$tType]: (((sup @ A20) => (sup @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Lattices_Oinf_332, axiom, (! [A20:$tType]: (((inf @ A20) => (inf @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___Heap_Oheap_333, axiom, (! [A20:$tType]: (((heap @ A20) => (heap @ (option @ A20)))))).
% 9.43/3.46 thf(tcon_Option_Ooption___HOL_Oequal_334, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (option @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Oseq___HOL_Oequal_335, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (seq @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_336, axiom, (! [A20:$tType]: ((comple6319245703460814977attice @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_337, axiom, (! [A20:$tType]: ((boolea8198339166811842893lgebra @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_338, axiom, (! [A20:$tType]: ((comple9053668089753744459l_ccpo @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_339, axiom, (! [A20:$tType]: ((semilattice_sup @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_340, axiom, (! [A20:$tType]: ((semilattice_inf @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_341, axiom, (! [A20:$tType]: ((bounded_lattice @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Complete__Lattices_OSup_342, axiom, (! [A20:$tType]: ((complete_Sup @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Complete__Lattices_OInf_343, axiom, (! [A20:$tType]: ((complete_Inf @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Orderings_Oorder__top_344, axiom, (! [A20:$tType]: ((order_top @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Orderings_Oorder__bot_345, axiom, (! [A20:$tType]: ((order_bot @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Orderings_Opreorder_346, axiom, (! [A20:$tType]: ((preorder @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Orderings_Oorder_347, axiom, (! [A20:$tType]: ((order @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Orderings_Oord_348, axiom, (! [A20:$tType]: ((ord @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Groups_Ouminus_349, axiom, (! [A20:$tType]: ((uminus @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Lattices_Osup_350, axiom, (! [A20:$tType]: ((sup @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___Lattices_Oinf_351, axiom, (! [A20:$tType]: ((inf @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Predicate_Opred___HOL_Oequal_352, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (pred @ A20))))).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_353, axiom, (boolea8198339166811842893lgebra @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_354, axiom, (semilattice_sup @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_355, axiom, (semilattice_inf @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_356, axiom, (bounded_lattice @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_357, axiom, (ab_semigroup_mult @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_358, axiom, (comm_monoid_mult @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_359, axiom, (semigroup_mult @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Orderings_Oorder__top_360, axiom, (order_top @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_361, axiom, (order_bot @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Orderings_Opreorder_362, axiom, (preorder @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_363, axiom, (monoid_mult @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Orderings_Oorder_364, axiom, (order @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Orderings_Oord_365, axiom, (ord @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Groups_Ouminus_366, axiom, (uminus @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Lattices_Osup_367, axiom, (sup @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Lattices_Oinf_368, axiom, (inf @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Power_Opower_369, axiom, (power @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Groups_Oone_370, axiom, (one @ assn)).
% 9.43/3.46 thf(tcon_Assertions_Oassn___Rings_Odvd_371, axiom, (dvd @ assn)).
% 9.43/3.46 thf(tcon_Typerep_Otyperep___Countable_Ocountable_372, axiom, (countable @ typerep)).
% 9.43/3.46 thf(tcon_Typerep_Otyperep___Heap_Oheap_373, axiom, (heap @ typerep)).
% 9.43/3.46 thf(tcon_Typerep_Otyperep___HOL_Oequal_374, axiom, (cl_HOL_Oequal @ typerep)).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_375, axiom, (! [A20:$tType]: (((quickc3360725361186068524ustive @ A20) => (quickc3360725361186068524ustive @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_376, axiom, (! [A20:$tType]: (((preorder @ A20) => (ordere6658533253407199908up_add @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_377, axiom, (! [A20:$tType]: ((cancel2418104881723323429up_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_378, axiom, (! [A20:$tType]: ((cancel1802427076303600483id_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_379, axiom, (! [A20:$tType]: ((cancel_semigroup_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_380, axiom, (! [A20:$tType]: (((quickcheck_random @ A20) => (quickcheck_random @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_381, axiom, (! [A20:$tType]: ((comm_monoid_diff @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_382, axiom, (! [A20:$tType]: ((ab_semigroup_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_383, axiom, (! [A20:$tType]: ((comm_monoid_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_384, axiom, (! [A20:$tType]: ((complete_Sup @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_385, axiom, (! [A20:$tType]: ((complete_Inf @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_386, axiom, (! [A20:$tType]: ((semigroup_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_387, axiom, (! [A20:$tType]: (((preorder @ A20) => (preorder @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_388, axiom, (! [A20:$tType]: ((monoid_add @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Orderings_Oorder_389, axiom, (! [A20:$tType]: (((preorder @ A20) => (order @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Orderings_Oord_390, axiom, (! [A20:$tType]: (((preorder @ A20) => (ord @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Ozero_391, axiom, (! [A20:$tType]: ((zero @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___Groups_Oplus_392, axiom, (! [A20:$tType]: ((plus @ (multiset @ A20))))).
% 9.43/3.46 thf(tcon_Multiset_Omultiset___HOL_Oequal_393, axiom, (! [A20:$tType]: (((cl_HOL_Oequal @ A20) => (cl_HOL_Oequal @ (multiset @ A20)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_394, axiom, (! [A20:$tType,A21:$tType]: ((((quickc3360725361186068524ustive @ A20) & (quickc3360725361186068524ustive @ A21)) => (quickc3360725361186068524ustive @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_395, axiom, (! [A20:$tType,A21:$tType]: ((((quickc658316121487927005ustive @ A20) & (quickc658316121487927005ustive @ A21)) => (quickc658316121487927005ustive @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_396, axiom, (! [A20:$tType,A21:$tType]: ((((quickcheck_random @ A20) & (quickcheck_random @ A21)) => (quickcheck_random @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Countable_Ocountable_397, axiom, (! [A20:$tType,A21:$tType]: ((((countable @ A20) & (countable @ A21)) => (countable @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_398, axiom, (! [A20:$tType,A21:$tType]: ((((finite_finite @ A20) & (finite_finite @ A21)) => (finite_finite @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___Heap_Oheap_399, axiom, (! [A20:$tType,A21:$tType]: ((((heap @ A20) & (heap @ A21)) => (heap @ ((product_prod @ A20) @ A21)))))).
% 9.43/3.46 thf(tcon_Product__Type_Oprod___HOL_Oequal_400, axiom, (! [A20:$tType,A21:$tType]: ((cl_HOL_Oequal @ ((product_prod @ A20) @ A21))))).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_401, axiom, (quickc3360725361186068524ustive @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_402, axiom, (comple6319245703460814977attice @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_403, axiom, (boolea8198339166811842893lgebra @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_404, axiom, (comple9053668089753744459l_ccpo @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_405, axiom, (quickcheck_random @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_406, axiom, (semilattice_sup @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_407, axiom, (semilattice_inf @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_408, axiom, (bounded_lattice @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_409, axiom, (complete_Sup @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_410, axiom, (complete_Inf @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_411, axiom, (order_top @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_412, axiom, (order_bot @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Countable_Ocountable_413, axiom, (countable @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_414, axiom, (preorder @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_415, axiom, (linorder @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_416, axiom, (finite_finite @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Orderings_Oorder_417, axiom, (order @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Orderings_Oord_418, axiom, (ord @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Groups_Ouminus_419, axiom, (uminus @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Lattices_Osup_420, axiom, (sup @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Lattices_Oinf_421, axiom, (inf @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___Heap_Oheap_422, axiom, (heap @ product_unit)).
% 9.43/3.46 thf(tcon_Product__Type_Ounit___HOL_Oequal_423, axiom, (cl_HOL_Oequal @ product_unit)).
% 9.43/3.46 thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_424, axiom, (! [A20:$tType]: (((quickcheck_random @ A20) => (quickcheck_random @ (heap_ext @ A20)))))).
% 9.43/3.46 thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_425, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (heap_ext @ A20))))).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_426, axiom, (bit_un5681908812861735899ations @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_427, axiom, (semiri1453513574482234551roduct @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_428, axiom, (euclid5411537665997757685th_nat @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_429, axiom, (euclid8789492081693882211th_nat @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_430, axiom, (ordere1937475149494474687imp_le @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_431, axiom, (euclid3128863361964157862miring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_432, axiom, (euclid4440199948858584721cancel @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_433, axiom, (unique1627219031080169319umeral @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_434, axiom, (euclid8851590272496341667cancel @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_435, axiom, (strict9044650504122735259up_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_436, axiom, (ordere580206878836729694up_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_437, axiom, (ordere2412721322843649153imp_le @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_438, axiom, (bit_se359711467146920520ations @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_439, axiom, (quickc3360725361186068524ustive @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_440, axiom, (strict7427464778891057005id_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_441, axiom, (ordere8940638589300402666id_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_442, axiom, (euclid3725896446679973847miring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_443, axiom, (linord715952674999750819strict @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_444, axiom, (linord4140545234300271783up_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_445, axiom, (bit_ri3973907225187159222ations @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_446, axiom, (linord181362715937106298miring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_447, axiom, (quickc658316121487927005ustive @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_448, axiom, (semiri3467727345109120633visors @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_449, axiom, (ordere6658533253407199908up_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_450, axiom, (ordere166539214618696060dd_abs @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_451, axiom, (ordere6911136660526730532id_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_452, axiom, (linord5086331880401160121up_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_453, axiom, (cancel2418104881723323429up_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_454, axiom, (ring_15535105094025558882visors @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_455, axiom, (cancel1802427076303600483id_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_456, axiom, (linord4710134922213307826strict @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_457, axiom, (comm_s4317794764714335236cancel @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_458, axiom, (bit_semiring_bits @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_459, axiom, (linord6961819062388156250ring_1 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_460, axiom, (ordered_ab_group_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_461, axiom, (cancel_semigroup_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_462, axiom, (linordered_semidom @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_463, axiom, (quickcheck_random @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_464, axiom, (ab_semigroup_mult @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_465, axiom, (algebraic_semidom @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_466, axiom, (comm_monoid_mult @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_467, axiom, (ab_semigroup_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_468, axiom, (semiring_parity @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_469, axiom, (comm_monoid_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_470, axiom, (semiring_modulo @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_471, axiom, (linordered_ring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_472, axiom, (linordered_idom @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_473, axiom, (comm_semiring_1 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_474, axiom, (comm_semiring_0 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_475, axiom, (semigroup_mult @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_476, axiom, (semidom_modulo @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_477, axiom, (semidom_divide @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_478, axiom, (semiring_numeral @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_479, axiom, (semigroup_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_480, axiom, (zero_less_one @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_481, axiom, (comm_semiring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_482, axiom, (semiring_char_0 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_483, axiom, (ab_group_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_484, axiom, (zero_neq_one @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_485, axiom, (ordered_ring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_486, axiom, (idom_abs_sgn @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_487, axiom, (ring_parity @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_488, axiom, (preorder @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_489, axiom, (linorder @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_490, axiom, (monoid_mult @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_491, axiom, (comm_ring_1 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_492, axiom, (monoid_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_493, axiom, (semiring_1 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_494, axiom, (semiring_0 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_495, axiom, (group_add @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_496, axiom, (comm_ring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_497, axiom, (order @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_498, axiom, (neg_numeral @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_499, axiom, (ring_char_0 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_500, axiom, (semiring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_501, axiom, (ord @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_502, axiom, (uminus @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_503, axiom, (ring_1 @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Power_Opower_504, axiom, (power @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_505, axiom, (numeral @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_506, axiom, (zero @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_507, axiom, (plus @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_508, axiom, (ring @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_509, axiom, (one @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_510, axiom, (dvd @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_511, axiom, (cl_HOL_Oequal @ code_integer)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_512, axiom, (bit_un5681908812861735899ations @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_513, axiom, (euclid5411537665997757685th_nat @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_514, axiom, (ordere1937475149494474687imp_le @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_515, axiom, (euclid3128863361964157862miring @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_516, axiom, (euclid4440199948858584721cancel @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_517, axiom, (strict9044650504122735259up_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_518, axiom, (ordere580206878836729694up_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_519, axiom, (ordere2412721322843649153imp_le @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_520, axiom, (bit_se359711467146920520ations @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_521, axiom, (quickc3360725361186068524ustive @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_522, axiom, (strict7427464778891057005id_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_523, axiom, (ordere8940638589300402666id_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_524, axiom, (euclid3725896446679973847miring @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_525, axiom, (linord4140545234300271783up_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_526, axiom, (linord181362715937106298miring @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_527, axiom, (quickc658316121487927005ustive @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_528, axiom, (semiri3467727345109120633visors @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_529, axiom, (ordere6658533253407199908up_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_530, axiom, (ordere6911136660526730532id_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_531, axiom, (cancel2418104881723323429up_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_532, axiom, (cancel1802427076303600483id_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_533, axiom, (comm_s4317794764714335236cancel @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_534, axiom, (bit_semiring_bits @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_535, axiom, (cancel_semigroup_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_536, axiom, (linordered_semidom @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_537, axiom, (quickcheck_random @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_538, axiom, (ab_semigroup_mult @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_539, axiom, (algebraic_semidom @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_540, axiom, (comm_monoid_mult @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_541, axiom, (comm_monoid_diff @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_542, axiom, (ab_semigroup_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_543, axiom, (semiring_parity @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_544, axiom, (comm_monoid_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_545, axiom, (semiring_modulo @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_546, axiom, (comm_semiring_1 @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_547, axiom, (comm_semiring_0 @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_548, axiom, (semigroup_mult @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_549, axiom, (semidom_modulo @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_550, axiom, (semidom_divide @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_551, axiom, (semiring_numeral @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_552, axiom, (semigroup_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_553, axiom, (zero_less_one @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_554, axiom, (comm_semiring @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_555, axiom, (semiring_char_0 @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_556, axiom, (zero_neq_one @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_557, axiom, (preorder @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_558, axiom, (linorder @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_559, axiom, (monoid_mult @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_560, axiom, (monoid_add @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_561, axiom, (semiring_1 @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_562, axiom, (semiring_0 @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_563, axiom, (order @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_564, axiom, (semiring @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_565, axiom, (ord @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Power_Opower_566, axiom, (power @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_567, axiom, (numeral @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_568, axiom, (zero @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oplus_569, axiom, (plus @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Groups_Oone_570, axiom, (one @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___Rings_Odvd_571, axiom, (dvd @ code_natural)).
% 9.43/3.46 thf(tcon_Code__Numeral_Onatural___HOL_Oequal_572, axiom, (cl_HOL_Oequal @ code_natural)).
% 9.43/3.46 thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_573, axiom, (! [A20:$tType]: (((quickcheck_random @ A20) => (quickcheck_random @ (heap_Time_Heap @ A20)))))).
% 9.43/3.46 thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_574, axiom, (! [A20:$tType]: ((cl_HOL_Oequal @ (heap_Time_Heap @ A20))))).
% 9.43/3.46 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 9.43/3.46 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 9.43/3.46 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 9.43/3.46 thf(help_fChoice_1_1_T, axiom, (! [A:$tType,P:(A > $o)]: (((P @ ((fChoice @ A) @ P)) = (? [X8:A]: ((P @ X8))))))).
% 9.43/3.46 thf(tfree_0, hypothesis, (heap @ a)).
% 9.43/3.46 thf(conj_0, conjecture, ((((heap_Time_execute @ (array @ a)) @ ((array_of_list @ a) @ xs)) @ h) != (none @ ((product_prod @ (array @ a)) @ ((product_prod @ (heap_ext @ product_unit)) @ nat))))).
% 9.43/3.46 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.EbIqTwV4kM/DTF2DTF24760.p
% 9.43/3.46 ------------------------
% 11.90/5.91 ---- Cleaned THF ---
% 11.90/5.91 thf(ty_t_Heap__Time__Monad_OHeap,type,
% 11.90/5.91 heap_Time_Heap: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Code__Numeral_Onatural,type,
% 11.90/5.91 code_natural: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Code__Numeral_Ointeger,type,
% 11.90/5.91 code_integer: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Code__Evaluation_Oterm,type,
% 11.90/5.91 code_term: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Heap_Oheap_Oheap__ext,type,
% 11.90/5.91 heap_ext: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Product__Type_Ounit,type,
% 11.90/5.91 product_unit: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Product__Type_Oprod,type,
% 11.90/5.91 product_prod: $tType > $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Old__Datatype_Onode,type,
% 11.90/5.91 old_node: $tType > $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Multiset_Omultiset,type,
% 11.90/5.91 multiset: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Typerep_Otyperep,type,
% 11.90/5.91 typerep: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Assertions_Oassn,type,
% 11.90/5.91 assn: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Predicate_Opred,type,
% 11.90/5.91 pred: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Sum__Type_Osum,type,
% 11.90/5.91 sum_sum: $tType > $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Predicate_Oseq,type,
% 11.90/5.91 seq: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Option_Ooption,type,
% 11.90/5.91 option: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_String_Ochar,type,
% 11.90/5.91 char: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Heap_Oarray,type,
% 11.90/5.91 array: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_List_Olist,type,
% 11.90/5.91 list: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Heap_Oref,type,
% 11.90/5.91 ref: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Set_Oset,type,
% 11.90/5.91 set: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Rat_Orat,type,
% 11.90/5.91 rat: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Num_Onum,type,
% 11.90/5.91 num: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Nat_Onat,type,
% 11.90/5.91 nat: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_Int_Oint,type,
% 11.90/5.91 int: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_t_itself,type,
% 11.90/5.91 itself: $tType > $tType ).
% 11.90/5.91
% 11.90/5.91 thf(ty_tf_a,type,
% 11.90/5.91 a: $tType ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 11.90/5.91 bounded_lattice_top:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Typerep_Otyperep,type,
% 11.90/5.91 typerep2:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Enum_Oenum,type,
% 11.90/5.91 enum:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Code__Evaluation_Oterm__of,type,
% 11.90/5.91 code_term_of:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_HOL_Otype,type,
% 11.90/5.91 type:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_HOL_Oequal,type,
% 11.90/5.91 cl_HOL_Oequal:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Heap_Oheap,type,
% 11.90/5.91 heap:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Odvd,type,
% 11.90/5.91 dvd:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oone,type,
% 11.90/5.91 one:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Oring,type,
% 11.90/5.91 ring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oplus,type,
% 11.90/5.91 plus:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ozero,type,
% 11.90/5.91 zero:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Num_Onumeral,type,
% 11.90/5.91 numeral:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Power_Opower,type,
% 11.90/5.91 power:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Fields_Ofield,type,
% 11.90/5.91 field:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_GCD_Oring__gcd,type,
% 11.90/5.91 ring_gcd:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Lattices_Oinf,type,
% 11.90/5.91 inf:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Lattices_Osup,type,
% 11.90/5.91 sup:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Oring__1,type,
% 11.90/5.91 ring_1:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ouminus,type,
% 11.90/5.91 uminus:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Oord,type,
% 11.90/5.91 ord:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Fields_Oinverse,type,
% 11.90/5.91 inverse:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemiring,type,
% 11.90/5.91 semiring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Nat_Oring__char__0,type,
% 11.90/5.91 ring_char_0:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Num_Oneg__numeral,type,
% 11.90/5.91 neg_numeral:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Oorder,type,
% 11.90/5.91 order:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ocomm__ring,type,
% 11.90/5.91 comm_ring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_GCD_Osemiring__Gcd,type,
% 11.90/5.91 semiring_Gcd:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_GCD_Osemiring__gcd,type,
% 11.90/5.91 semiring_gcd:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ogroup__add,type,
% 11.90/5.91 group_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Ono__bot,type,
% 11.90/5.91 no_bot:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Ono__top,type,
% 11.90/5.91 no_top:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemiring__0,type,
% 11.90/5.91 semiring_0:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemiring__1,type,
% 11.90/5.91 semiring_1:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Finite__Set_Ofinite,type,
% 11.90/5.91 finite_finite:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Omonoid__add,type,
% 11.90/5.91 monoid_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ocomm__ring__1,type,
% 11.90/5.91 comm_ring_1:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Omonoid__mult,type,
% 11.90/5.91 monoid_mult:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Olinorder,type,
% 11.90/5.91 linorder:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Opreorder,type,
% 11.90/5.91 preorder:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Parity_Oring__parity,type,
% 11.90/5.91 ring_parity:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Oidom__abs__sgn,type,
% 11.90/5.91 idom_abs_sgn:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Oordered__ring,type,
% 11.90/5.91 ordered_ring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ozero__neq__one,type,
% 11.90/5.91 zero_neq_one:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Countable_Ocountable,type,
% 11.90/5.91 countable:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Fields_Ofield__char__0,type,
% 11.90/5.91 field_char_0:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oab__group__add,type,
% 11.90/5.91 ab_group_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Nat_Osemiring__char__0,type,
% 11.90/5.91 semiring_char_0:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Oorder__bot,type,
% 11.90/5.91 order_bot:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Oorder__top,type,
% 11.90/5.91 order_top:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ocomm__semiring,type,
% 11.90/5.91 comm_semiring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ozero__less__one,type,
% 11.90/5.91 zero_less_one:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Fields_Odivision__ring,type,
% 11.90/5.91 division_ring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Osemigroup__add,type,
% 11.90/5.91 semigroup_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Num_Osemiring__numeral,type,
% 11.90/5.91 semiring_numeral:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemidom__divide,type,
% 11.90/5.91 semidom_divide:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemidom__modulo,type,
% 11.90/5.91 semidom_modulo:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Complete__Lattices_OInf,type,
% 11.90/5.91 complete_Inf:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Complete__Lattices_OSup,type,
% 11.90/5.91 complete_Sup:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Osemigroup__mult,type,
% 11.90/5.91 semigroup_mult:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ocomm__semiring__0,type,
% 11.90/5.91 comm_semiring_0:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ocomm__semiring__1,type,
% 11.90/5.91 comm_semiring_1:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__idom,type,
% 11.90/5.91 linordered_idom:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__ring,type,
% 11.90/5.91 linordered_ring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemiring__modulo,type,
% 11.90/5.91 semiring_modulo:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocomm__monoid__add,type,
% 11.90/5.91 comm_monoid_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Parity_Osemiring__parity,type,
% 11.90/5.91 semiring_parity:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Fields_Olinordered__field,type,
% 11.90/5.91 linordered_field:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oab__semigroup__add,type,
% 11.90/5.91 ab_semigroup_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocomm__monoid__diff,type,
% 11.90/5.91 comm_monoid_diff:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocomm__monoid__mult,type,
% 11.90/5.91 comm_monoid_mult:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Oalgebraic__semidom,type,
% 11.90/5.91 algebraic_semidom:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oab__semigroup__mult,type,
% 11.90/5.91 ab_semigroup_mult:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Lattices_Obounded__lattice,type,
% 11.90/5.91 bounded_lattice:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Lattices_Osemilattice__inf,type,
% 11.90/5.91 semilattice_inf:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 11.90/5.91 semilattice_sup:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Orderings_Odense__linorder,type,
% 11.90/5.91 dense_linorder:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Quickcheck__Random_Orandom,type,
% 11.90/5.91 quickcheck_random:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__semidom,type,
% 11.90/5.91 linordered_semidom:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Complete__Partial__Order_Occpo,type,
% 11.90/5.91 comple9053668089753744459l_ccpo:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocancel__semigroup__add,type,
% 11.90/5.91 cancel_semigroup_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 11.90/5.91 ordered_ab_group_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__semiring__1,type,
% 11.90/5.91 linord6961819062388156250ring_1:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Onormalization__semidom,type,
% 11.90/5.91 normal8620421768224518004emidom:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Bit__Operations_Osemiring__bits,type,
% 11.90/5.91 bit_semiring_bits:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
% 11.90/5.91 comm_s4317794764714335236cancel:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__ring__strict,type,
% 11.90/5.91 linord4710134922213307826strict:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
% 11.90/5.91 cancel1802427076303600483id_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
% 11.90/5.91 ring_15535105094025558882visors:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 11.90/5.91 cancel2418104881723323429up_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Olinordered__ab__group__add,type,
% 11.90/5.91 linord5086331880401160121up_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
% 11.90/5.91 ordere6911136660526730532id_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
% 11.90/5.91 archim2362893244070406136eiling:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
% 11.90/5.91 ordere166539214618696060dd_abs:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
% 11.90/5.91 ordere6658533253407199908up_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
% 11.90/5.91 semiri3467727345109120633visors:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
% 11.90/5.91 boolea8198339166811842893lgebra:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
% 11.90/5.91 quickc658316121487927005ustive:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
% 11.90/5.91 semido2269285787275462019factor:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
% 11.90/5.91 linord181362715937106298miring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
% 11.90/5.91 bit_ri3973907225187159222ations:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
% 11.90/5.91 comple6319245703460814977attice:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
% 11.90/5.91 linord4140545234300271783up_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
% 11.90/5.91 linord715952674999750819strict:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
% 11.90/5.91 archim462609752435547400_field:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
% 11.90/5.91 euclid3725896446679973847miring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
% 11.90/5.91 canoni5634975068530333245id_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
% 11.90/5.91 ordere8940638589300402666id_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
% 11.90/5.91 strict7427464778891057005id_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
% 11.90/5.91 quickc3360725361186068524ustive:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
% 11.90/5.91 bit_se359711467146920520ations:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
% 11.90/5.91 ordere2412721322843649153imp_le:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
% 11.90/5.91 ordere580206878836729694up_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
% 11.90/5.91 ordere1170586879665033532d_diff:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
% 11.90/5.91 strict9044650504122735259up_add:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
% 11.90/5.91 euclid8851590272496341667cancel:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
% 11.90/5.91 unique1627219031080169319umeral:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
% 11.90/5.91 euclid4440199948858584721cancel:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
% 11.90/5.91 euclid3128863361964157862miring:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
% 11.90/5.91 ordere1937475149494474687imp_le:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
% 11.90/5.91 euclid8789492081693882211th_nat:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
% 11.90/5.91 euclid5411537665997757685th_nat:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
% 11.90/5.91 semiri1453513574482234551roduct:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
% 11.90/5.91 bit_un5681908812861735899ations:
% 11.90/5.91 !>[A: $tType] : $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Archimedean__Field_Oceiling,type,
% 11.90/5.91 archimedean_ceiling:
% 11.90/5.91 !>[A: $tType] : ( A > int ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
% 11.90/5.91 archim6421214686448440834_floor:
% 11.90/5.91 !>[A: $tType] : ( A > int ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Archimedean__Field_Ofrac,type,
% 11.90/5.91 archimedean_frac:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Archimedean__Field_Oround,type,
% 11.90/5.91 archimedean_round:
% 11.90/5.91 !>[A: $tType] : ( A > int ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oalloc,type,
% 11.90/5.91 array_alloc:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Ofreeze,type,
% 11.90/5.91 array_freeze:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oget,type,
% 11.90/5.91 array_get:
% 11.90/5.91 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Olen,type,
% 11.90/5.91 array_len:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Olen_H,type,
% 11.90/5.91 array_len2:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( array @ B ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Olength,type,
% 11.90/5.91 array_length:
% 11.90/5.91 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Omake,type,
% 11.90/5.91 array_make:
% 11.90/5.91 !>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Omake_H,type,
% 11.90/5.91 array_make2:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( code_integer > ( B > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Omap__entry,type,
% 11.90/5.91 array_map_entry:
% 11.90/5.91 !>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Onew,type,
% 11.90/5.91 array_new:
% 11.90/5.91 !>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Onew_H,type,
% 11.90/5.91 array_new2:
% 11.90/5.91 !>[A: $tType] : ( code_integer > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Onoteq,type,
% 11.90/5.91 array_noteq:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Onth,type,
% 11.90/5.91 array_nth:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Onth_H,type,
% 11.90/5.91 array_nth2:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > code_integer > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oof__list,type,
% 11.90/5.91 array_of_list:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Opresent,type,
% 11.90/5.91 array_present:
% 11.90/5.91 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oset,type,
% 11.90/5.91 array_set:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oswap,type,
% 11.90/5.91 array_swap:
% 11.90/5.91 !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oupd,type,
% 11.90/5.91 array_upd:
% 11.90/5.91 !>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oupd_H,type,
% 11.90/5.91 array_upd2:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > code_integer > A > ( heap_Time_Heap @ product_unit ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Array__Time_Oupdate,type,
% 11.90/5.91 array_update:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oassn_ORep__assn,type,
% 11.90/5.91 rep_assn: assn > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oex__assn,type,
% 11.90/5.91 ex_assn:
% 11.90/5.91 !>[A: $tType] : ( ( A > assn ) > assn ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oin__range,type,
% 11.90/5.91 in_range: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oin__range__rel,type,
% 11.90/5.91 in_range_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oone__assn__raw,type,
% 11.90/5.91 one_assn_raw: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oone__assn__raw__rel,type,
% 11.90/5.91 one_assn_raw_rel: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Oproper,type,
% 11.90/5.91 proper: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Opure__assn,type,
% 11.90/5.91 pure_assn: $o > assn ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Opure__assn__raw,type,
% 11.90/5.91 pure_assn_raw:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( $o > ( product_prod @ A @ ( set @ B ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Opure__assn__raw__rel,type,
% 11.90/5.91 pure_assn_raw_rel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) > ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_OrelH,type,
% 11.90/5.91 relH: ( set @ nat ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Osnga__assn,type,
% 11.90/5.91 snga_assn:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > assn ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Osnga__assn__raw,type,
% 11.90/5.91 snga_assn_raw:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Osnga__assn__raw__rel,type,
% 11.90/5.91 snga_assn_raw_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Osngr__assn,type,
% 11.90/5.91 sngr_assn:
% 11.90/5.91 !>[A: $tType] : ( ( ref @ A ) > A > assn ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Osngr__assn__raw,type,
% 11.90/5.91 sngr_assn_raw:
% 11.90/5.91 !>[A: $tType] : ( ( ref @ A ) > A > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Osngr__assn__raw__rel,type,
% 11.90/5.91 sngr_assn_raw_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Otimes__assn__raw,type,
% 11.90/5.91 times_assn_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
% 11.90/5.91 times_assn_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Owand__assn,type,
% 11.90/5.91 wand_assn: assn > assn > assn ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Owand__raw,type,
% 11.90/5.91 wand_raw: ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) > ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Assertions_Owand__raw__rel,type,
% 11.90/5.91 wand_raw_rel: ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
% 11.90/5.91 bNF_Cardinal_Csum:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
% 11.90/5.91 bNF_Cardinal_cexp:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
% 11.90/5.91 bNF_Cardinal_cfinite:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
% 11.90/5.91 bNF_Ca4139267488887388095finite:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
% 11.90/5.91 bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
% 11.90/5.91 bNF_Cardinal_cprod:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
% 11.90/5.91 bNF_Cardinal_csum:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
% 11.90/5.91 bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
% 11.90/5.91 bNF_Cardinal_czero:
% 11.90/5.91 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
% 11.90/5.91 bNF_Ca8387033319878233205ardSuc:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
% 11.90/5.91 bNF_Ca6860139660246222851ard_of:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
% 11.90/5.91 bNF_Ca8970107618336181345der_on:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
% 11.90/5.91 bNF_Ca7293521722713021262ofinal:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
% 11.90/5.91 bNF_Ca6246979054910435723ardSuc:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
% 11.90/5.91 bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
% 11.90/5.91 bNF_Ca7133664381575040944arCard:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
% 11.90/5.91 bNF_Ca3754400796208372196lChain:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_OGr,type,
% 11.90/5.91 bNF_Gr:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_OGrp,type,
% 11.90/5.91 bNF_Grp:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_Ocollect,type,
% 11.90/5.91 bNF_collect:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_Oconvol,type,
% 11.90/5.91 bNF_convol:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_Oeq__onp,type,
% 11.90/5.91 bNF_eq_onp:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_OfstOp,type,
% 11.90/5.91 bNF_fstOp:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_Opick__middlep,type,
% 11.90/5.91 bNF_pick_middlep:
% 11.90/5.91 !>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_Orel__fun,type,
% 11.90/5.91 bNF_rel_fun:
% 11.90/5.91 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_OsndOp,type,
% 11.90/5.91 bNF_sndOp:
% 11.90/5.91 !>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Def_Ovimage2p,type,
% 11.90/5.91 bNF_vimage2p:
% 11.90/5.91 !>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Greatest__Fixpoint_OfromCard,type,
% 11.90/5.91 bNF_Gr5436034075474128252omCard:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
% 11.90/5.91 bNF_Greatest_image2:
% 11.90/5.91 !>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
% 11.90/5.91 bNF_Gr4221423524335903396lImage:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
% 11.90/5.91 bNF_Gr7122648621184425601vImage:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard,type,
% 11.90/5.91 bNF_Greatest_toCard:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
% 11.90/5.91 bNF_Gr1419584066657907630d_pred:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
% 11.90/5.91 bNF_Wellorder_Func:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
% 11.90/5.91 bNF_Wellorder_bsqr:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
% 11.90/5.91 bNF_Wellorder_curr:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
% 11.90/5.91 bNF_We2720479622203943262_image:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
% 11.90/5.91 bNF_We413866401316099525erIncl:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
% 11.90/5.91 bNF_Wellorder_ordIso:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
% 11.90/5.91 bNF_Wellorder_ordLeq:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
% 11.90/5.91 bNF_We4044943003108391690rdLess:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
% 11.90/5.91 bNF_Wellorder_compat:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
% 11.90/5.91 bNF_Wellorder_embed:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
% 11.90/5.91 bNF_Wellorder_embedS:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
% 11.90/5.91 bNF_Wellorder_iso:
% 11.90/5.91 !>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
% 11.90/5.91 bNF_Wellorder_wo_rel:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
% 11.90/5.91 bNF_We4791949203932849705sMinim:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
% 11.90/5.91 bNF_We1388413361240627857o_max2:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
% 11.90/5.91 bNF_We6954850376910717587_minim:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
% 11.90/5.91 bNF_Wellorder_wo_suc:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
% 11.90/5.91 basic_BNF_size_prod:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
% 11.90/5.91 basic_BNF_size_sum:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNFs_Ofsts,type,
% 11.90/5.91 basic_fsts:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNFs_Opred__fun,type,
% 11.90/5.91 basic_pred_fun:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNFs_Opred__prod,type,
% 11.90/5.91 basic_pred_prod:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNFs_Orel__prod,type,
% 11.90/5.91 basic_rel_prod:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Basic__BNFs_Osnds,type,
% 11.90/5.91 basic_snds:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Binomial_Obinomial,type,
% 11.90/5.91 binomial: nat > nat > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Binomial_Ogbinomial,type,
% 11.90/5.91 gbinomial:
% 11.90/5.91 !>[A: $tType] : ( A > nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oand__int__rel,type,
% 11.90/5.91 bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oand__not__num,type,
% 11.90/5.91 bit_and_not_num: num > num > ( option @ num ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
% 11.90/5.91 bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oconcat__bit,type,
% 11.90/5.91 bit_concat_bit: nat > int > int > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
% 11.90/5.91 bit_or_not_num_neg: num > num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
% 11.90/5.91 bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
% 11.90/5.91 bit_ri4277139882892585799ns_not:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
% 11.90/5.91 bit_ri4674362597316999326ke_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
% 11.90/5.91 bit_se5824344872417868541ns_and:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
% 11.90/5.91 bit_se4197421643247451524op_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
% 11.90/5.91 bit_se8732182000553998342ip_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
% 11.90/5.91 bit_se2239418461657761734s_mask:
% 11.90/5.91 !>[A: $tType] : ( nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
% 11.90/5.91 bit_se1065995026697491101ons_or:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
% 11.90/5.91 bit_se4730199178511100633sh_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
% 11.90/5.91 bit_se5668285175392031749et_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
% 11.90/5.91 bit_se2584673776208193580ke_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
% 11.90/5.91 bit_se2638667681897837118et_bit:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
% 11.90/5.91 bit_se5824344971392196577ns_xor:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
% 11.90/5.91 bit_se5641148757651400278ts_bit:
% 11.90/5.91 !>[A: $tType] : ( A > nat > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
% 11.90/5.91 bit_se6407376104438227557le_bit:
% 11.90/5.91 !>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Otake__bit__num,type,
% 11.90/5.91 bit_take_bit_num: nat > num > ( option @ num ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
% 11.90/5.91 bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
% 11.90/5.91 bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num,type,
% 11.90/5.91 bit_un6697907153464112080or_num: num > num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oor__num__rel,type,
% 11.90/5.91 bit_un4773296044027857193um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
% 11.90/5.91 bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
% 11.90/5.91 bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
% 11.90/5.91 boolea2506097494486148201lgebra:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
% 11.90/5.91 boolea3799213064322606851m_diff:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Cancellation_Oiterate__add,type,
% 11.90/5.91 iterate_add:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_ONat,type,
% 11.90/5.91 code_Nat: code_integer > code_natural ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_ONeg,type,
% 11.90/5.91 code_Neg: num > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_OPos,type,
% 11.90/5.91 code_Pos: num > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_OSuc,type,
% 11.90/5.91 code_Suc: code_natural > code_natural ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
% 11.90/5.91 code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ocr__integer,type,
% 11.90/5.91 code_cr_integer: int > code_integer > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ocr__natural,type,
% 11.90/5.91 code_cr_natural: nat > code_natural > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Odivmod__abs,type,
% 11.90/5.91 code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Odivmod__integer,type,
% 11.90/5.91 code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Odup,type,
% 11.90/5.91 code_dup: code_integer > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
% 11.90/5.91 code_int_of_integer: code_integer > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
% 11.90/5.91 code_integer_of_int: int > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
% 11.90/5.91 code_integer_of_nat: nat > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
% 11.90/5.91 code_i5400310926305786745atural: code_natural > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Ointeger__of__num,type,
% 11.90/5.91 code_integer_of_num: num > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onat__of__integer,type,
% 11.90/5.91 code_nat_of_integer: code_integer > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
% 11.90/5.91 code_case_natural:
% 11.90/5.91 !>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
% 11.90/5.91 code_nat_of_natural: code_natural > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
% 11.90/5.91 code_natural_of_nat: nat > code_natural ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
% 11.90/5.91 code_rec_natural:
% 11.90/5.91 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
% 11.90/5.91 code_rec_set_natural:
% 11.90/5.91 !>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onatural__of__integer,type,
% 11.90/5.91 code_n4118661773612635043nteger: code_integer > code_natural ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onegative,type,
% 11.90/5.91 code_negative: num > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Onum__of__integer,type,
% 11.90/5.91 code_num_of_integer: code_integer > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Opcr__integer,type,
% 11.90/5.91 code_pcr_integer: int > code_integer > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Opcr__natural,type,
% 11.90/5.91 code_pcr_natural: nat > code_natural > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Opositive,type,
% 11.90/5.91 code_positive: num > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Osize__nat,type,
% 11.90/5.91 code_size_nat: code_natural > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Code__Numeral_Osub,type,
% 11.90/5.91 code_sub: num > num > code_integer ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
% 11.90/5.91 complete_Inf_Inf:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
% 11.90/5.91 complete_Sup_Sup:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
% 11.90/5.91 comple1908693960933563346ssible:
% 11.90/5.91 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
% 11.90/5.91 comple187402453842119260l_fixp:
% 11.90/5.91 !>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
% 11.90/5.91 comple115746919287870866o_fixp:
% 11.90/5.91 !>[A: $tType] : ( ( A > A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Partial__Order_Ochain,type,
% 11.90/5.91 comple1602240252501008431_chain:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Complete__Partial__Order_Omonotone,type,
% 11.90/5.91 comple7038119648293358887notone:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Countable_Ofrom__nat,type,
% 11.90/5.91 from_nat:
% 11.90/5.91 !>[A: $tType] : ( nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Countable_Onth__item,type,
% 11.90/5.91 nth_item:
% 11.90/5.91 !>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Countable_Onth__item__rel,type,
% 11.90/5.91 nth_item_rel: nat > nat > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Countable_Oto__nat,type,
% 11.90/5.91 to_nat:
% 11.90/5.91 !>[A: $tType] : ( A > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Divides_Oadjust__div,type,
% 11.90/5.91 adjust_div: ( product_prod @ int @ int ) > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Divides_Odivmod__nat,type,
% 11.90/5.91 divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Divides_Oeucl__rel__int,type,
% 11.90/5.91 eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
% 11.90/5.91 unique5940410009612947441es_aux:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
% 11.90/5.91 unique8689654367752047608divmod:
% 11.90/5.91 !>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
% 11.90/5.91 unique1321980374590559556d_step:
% 11.90/5.91 !>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Equiv__Relations_Ocongruent,type,
% 11.90/5.91 equiv_congruent:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Equiv__Relations_Ocongruent2,type,
% 11.90/5.91 equiv_congruent2:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Equiv__Relations_Oequiv,type,
% 11.90/5.91 equiv_equiv:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Equiv__Relations_Oequivp,type,
% 11.90/5.91 equiv_equivp:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Equiv__Relations_Oproj,type,
% 11.90/5.91 equiv_proj:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Equiv__Relations_Oquotient,type,
% 11.90/5.91 equiv_quotient:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
% 11.90/5.91 euclid6346220572633701492n_size:
% 11.90/5.91 !>[A: $tType] : ( A > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
% 11.90/5.91 euclid7384307370059645450egment:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
% 11.90/5.91 comm_s3205402744901411588hammer:
% 11.90/5.91 !>[A: $tType] : ( A > nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
% 11.90/5.91 semiring_char_0_fact:
% 11.90/5.91 !>[A: $tType] : ( nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fields_Oinverse__class_Oinverse,type,
% 11.90/5.91 inverse_inverse:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ocard,type,
% 11.90/5.91 finite_card:
% 11.90/5.91 !>[B: $tType] : ( ( set @ B ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
% 11.90/5.91 finite6289374366891150609ommute:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ocomp__fun__idem__axioms,type,
% 11.90/5.91 finite1981172056906289455axioms:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
% 11.90/5.91 finite4980608107308702382axioms:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofinite,type,
% 11.90/5.91 finite_finite2:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofold,type,
% 11.90/5.91 finite_fold:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofolding,type,
% 11.90/5.91 finite_folding:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofolding__idem,type,
% 11.90/5.91 finite_folding_idem:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofolding__idem__axioms,type,
% 11.90/5.91 finite7837460588564673216axioms:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofolding__idem__on,type,
% 11.90/5.91 finite1890593828518410140dem_on:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
% 11.90/5.91 finite6916993218817215295axioms:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Obij__betw,type,
% 11.90/5.91 bij_betw:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Ocomp,type,
% 11.90/5.91 comp:
% 11.90/5.91 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Ofcomp,type,
% 11.90/5.91 fcomp:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( B > C ) > A > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Ofun__upd,type,
% 11.90/5.91 fun_upd:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Oid,type,
% 11.90/5.91 id:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Oinj__on,type,
% 11.90/5.91 inj_on:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Omap__fun,type,
% 11.90/5.91 map_fun:
% 11.90/5.91 !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Omono__on,type,
% 11.90/5.91 mono_on:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Ooverride__on,type,
% 11.90/5.91 override_on:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Ostrict__mono__on,type,
% 11.90/5.91 strict_mono_on:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun_Othe__inv__into,type,
% 11.90/5.91 the_inv_into:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Omax__strict,type,
% 11.90/5.91 fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Omax__weak,type,
% 11.90/5.91 fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Omin__strict,type,
% 11.90/5.91 fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Omin__weak,type,
% 11.90/5.91 fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Opair__leq,type,
% 11.90/5.91 fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Opair__less,type,
% 11.90/5.91 fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Oreduction__pair,type,
% 11.90/5.91 fun_reduction_pair:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Fun__Def_Orp__inv__image,type,
% 11.90/5.91 fun_rp_inv_image:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_OGcd__class_OGcd,type,
% 11.90/5.91 gcd_Gcd:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_OGcd__class_OLcm,type,
% 11.90/5.91 gcd_Lcm:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Obezw,type,
% 11.90/5.91 bezw: nat > nat > ( product_prod @ int @ int ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Obezw__rel,type,
% 11.90/5.91 bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Obounded__quasi__semilattice,type,
% 11.90/5.91 bounde8507323023520639062attice:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
% 11.90/5.91 bounde6485984586167503788ce_set:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
% 11.90/5.91 bounde2362111253966948842tice_F:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Ogcd__class_Ogcd,type,
% 11.90/5.91 gcd_gcd:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Ogcd__class_Olcm,type,
% 11.90/5.91 gcd_lcm:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Ogcd__nat__rel,type,
% 11.90/5.91 gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
% 11.90/5.91 semiring_gcd_Gcd_fin:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
% 11.90/5.91 semiring_gcd_Lcm_fin:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Oabel__semigroup,type,
% 11.90/5.91 abel_semigroup:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Oabel__semigroup__axioms,type,
% 11.90/5.91 abel_s757365448890700780axioms:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Oabs__class_Oabs,type,
% 11.90/5.91 abs_abs:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ocomm__monoid,type,
% 11.90/5.91 comm_monoid:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ocomm__monoid__axioms,type,
% 11.90/5.91 comm_monoid_axioms:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ogroup,type,
% 11.90/5.91 group:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ogroup__axioms,type,
% 11.90/5.91 group_axioms:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 11.90/5.91 minus_minus:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Omonoid,type,
% 11.90/5.91 monoid:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Omonoid__axioms,type,
% 11.90/5.91 monoid_axioms:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Oone__class_Oone,type,
% 11.90/5.91 one_one:
% 11.90/5.91 !>[A: $tType] : A ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Oplus__class_Oplus,type,
% 11.90/5.91 plus_plus:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Osemigroup,type,
% 11.90/5.91 semigroup:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Osgn__class_Osgn,type,
% 11.90/5.91 sgn_sgn:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Otimes__class_Otimes,type,
% 11.90/5.91 times_times:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 11.90/5.91 uminus_uminus:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups_Ozero__class_Ozero,type,
% 11.90/5.91 zero_zero:
% 11.90/5.91 !>[A: $tType] : A ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum,type,
% 11.90/5.91 groups3894954378712506084id_sum:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
% 11.90/5.91 groups7311177749621191930dd_sum:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
% 11.90/5.91 groups1027152243600224163dd_sum:
% 11.90/5.91 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
% 11.90/5.91 groups7121269368397514597t_prod:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
% 11.90/5.91 groups1962203154675924110t_prod:
% 11.90/5.91 !>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
% 11.90/5.91 groups778175481326437816id_set:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
% 11.90/5.91 groups_comm_monoid_F:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
% 11.90/5.91 groups_comm_monoid_G:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Ocomm__monoid__list,type,
% 11.90/5.91 groups1828464146339083142d_list:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
% 11.90/5.91 groups4802862169904069756st_set:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
% 11.90/5.91 groups4207007520872428315er_sum:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Omonoid__add_Osum__list,type,
% 11.90/5.91 groups4543113879258116180m_list:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
% 11.90/5.91 groups8242544230860333062m_list:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Omonoid__list,type,
% 11.90/5.91 groups_monoid_list:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Omonoid__list_OF,type,
% 11.90/5.91 groups_monoid_F:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
% 11.90/5.91 groups5270119922927024881d_list:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_HOL_ONO__MATCH,type,
% 11.90/5.91 nO_MATCH:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( A > B > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_HOL_OThe,type,
% 11.90/5.91 the:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_HOL_OUniq,type,
% 11.90/5.91 uniq:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_HOL_Odefault__class_Odefault,type,
% 11.90/5.91 default_default:
% 11.90/5.91 !>[A: $tType] : A ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_HOL_Oundefined,type,
% 11.90/5.91 undefined:
% 11.90/5.91 !>[A: $tType] : A ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oaddr__of__array,type,
% 11.90/5.91 addr_of_array:
% 11.90/5.91 !>[A: $tType] : ( ( array @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oaddr__of__ref,type,
% 11.90/5.91 addr_of_ref:
% 11.90/5.91 !>[A: $tType] : ( ( ref @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oarray_OArray,type,
% 11.90/5.91 array2:
% 11.90/5.91 !>[A: $tType] : ( nat > ( array @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oheap_Oarrays,type,
% 11.90/5.91 arrays:
% 11.90/5.91 !>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oheap_Oarrays__update,type,
% 11.90/5.91 arrays_update:
% 11.90/5.91 !>[Z: $tType] : ( ( ( typerep > nat > ( list @ nat ) ) > typerep > nat > ( list @ nat ) ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oheap_Olim,type,
% 11.90/5.91 lim:
% 11.90/5.91 !>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oheap_Olim__update,type,
% 11.90/5.91 lim_update:
% 11.90/5.91 !>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oheap_Orefs,type,
% 11.90/5.91 refs:
% 11.90/5.91 !>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap_Oref_ORef,type,
% 11.90/5.91 ref2:
% 11.90/5.91 !>[A: $tType] : ( nat > ( ref @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
% 11.90/5.91 heap_Time_Heap2:
% 11.90/5.91 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Ocase__Heap,type,
% 11.90/5.91 heap_Time_case_Heap:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B ) > ( heap_Time_Heap @ A ) > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
% 11.90/5.91 heap_Time_map_Heap:
% 11.90/5.91 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ Aa ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
% 11.90/5.91 heap_Time_pred_Heap:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( heap_Time_Heap @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
% 11.90/5.91 heap_Time_rec_Heap:
% 11.90/5.91 !>[A: $tType,C: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) > ( heap_Time_Heap @ A ) > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
% 11.90/5.91 heap_Time_rel_Heap:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
% 11.90/5.91 heap_Time_set_Heap:
% 11.90/5.91 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
% 11.90/5.91 heap_Time_size_Heap:
% 11.90/5.91 !>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap__lub,type,
% 11.90/5.91 heap_Time_Heap_lub:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
% 11.90/5.91 heap_Time_Heap_ord:
% 11.90/5.91 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oassert,type,
% 11.90/5.91 heap_Time_assert:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Obind,type,
% 11.90/5.91 heap_Time_bind:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oeffect,type,
% 11.90/5.91 heap_Time_effect:
% 11.90/5.91 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oexecute,type,
% 11.90/5.91 heap_Time_execute:
% 11.90/5.91 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oguard,type,
% 11.90/5.91 heap_Time_guard:
% 11.90/5.91 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oheap,type,
% 11.90/5.91 heap_Time_heap:
% 11.90/5.91 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Olift,type,
% 11.90/5.91 heap_Time_lift:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oraise,type,
% 11.90/5.91 heap_Time_raise:
% 11.90/5.91 !>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oreturn,type,
% 11.90/5.91 heap_Time_return:
% 11.90/5.91 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Osuccess,type,
% 11.90/5.91 heap_Time_success:
% 11.90/5.91 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Otap,type,
% 11.90/5.91 heap_Time_tap:
% 11.90/5.91 !>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
% 11.90/5.91 heap_Time_timeFrame:
% 11.90/5.91 !>[A: $tType] : ( nat > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
% 11.90/5.91 heap_T5500966940807335491me_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Oureturn,type,
% 11.90/5.91 heap_Time_ureturn:
% 11.90/5.91 !>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Heap__Time__Monad_Owait,type,
% 11.90/5.91 heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Hilbert__Choice_Obijection,type,
% 11.90/5.91 hilbert_bijection:
% 11.90/5.91 !>[A: $tType] : ( ( A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Hilbert__Choice_Oinv__into,type,
% 11.90/5.91 hilbert_inv_into:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Hoare__Triple_Ohoare__triple,type,
% 11.90/5.91 hoare_hoare_triple:
% 11.90/5.91 !>[A: $tType] : ( assn > ( heap_Time_Heap @ A ) > ( A > assn ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Hoare__Triple_Onew__addrs,type,
% 11.90/5.91 hoare_new_addrs: ( heap_ext @ product_unit ) > ( set @ nat ) > ( heap_ext @ product_unit ) > ( set @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_If,type,
% 11.90/5.91 if:
% 11.90/5.91 !>[A: $tType] : ( $o > A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
% 11.90/5.91 complete_lattice_lfp:
% 11.90/5.91 !>[A: $tType] : ( ( A > A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_OAbs__Integ,type,
% 11.90/5.91 abs_Integ: ( product_prod @ nat @ nat ) > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_ONeg,type,
% 11.90/5.91 neg: num > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_OPos,type,
% 11.90/5.91 pos: num > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_ORep__Integ,type,
% 11.90/5.91 rep_Integ: int > ( product_prod @ nat @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Odup,type,
% 11.90/5.91 dup: int > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Ointrel,type,
% 11.90/5.91 intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Onat,type,
% 11.90/5.91 nat2: int > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Opcr__int,type,
% 11.90/5.91 pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Opower__int,type,
% 11.90/5.91 power_int:
% 11.90/5.91 !>[A: $tType] : ( A > int > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Oring__1__class_OInts,type,
% 11.90/5.91 ring_1_Ints:
% 11.90/5.91 !>[A: $tType] : ( set @ A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Oring__1__class_Oof__int,type,
% 11.90/5.91 ring_1_of_int:
% 11.90/5.91 !>[A: $tType] : ( int > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Int_Osub,type,
% 11.90/5.91 sub: num > num > int ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices_Oinf__class_Oinf,type,
% 11.90/5.91 inf_inf:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices_Osemilattice__neutr,type,
% 11.90/5.91 semilattice_neutr:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices_Osup__class_Osup,type,
% 11.90/5.91 sup_sup:
% 11.90/5.91 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
% 11.90/5.91 lattic643756798349783984er_Max:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
% 11.90/5.91 lattic643756798350308766er_Min:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
% 11.90/5.91 lattic7752659483105999362nf_fin:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
% 11.90/5.91 lattic5214292709420241887eutr_F:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Osemilattice__set,type,
% 11.90/5.91 lattic149705377957585745ce_set:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
% 11.90/5.91 lattic1715443433743089157tice_F:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 11.90/5.91 lattic5882676163264333800up_fin:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Lifting_OQuotient,type,
% 11.90/5.91 quotient:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oappend,type,
% 11.90/5.91 append:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oarg__min__list,type,
% 11.90/5.91 arg_min_list:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oarg__min__list__rel,type,
% 11.90/5.91 arg_min_list_rel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( list @ A ) ) > ( product_prod @ ( A > B ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Obutlast,type,
% 11.90/5.91 butlast:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oconcat,type,
% 11.90/5.91 concat:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ocount__list,type,
% 11.90/5.91 count_list:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > A > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Odistinct,type,
% 11.90/5.91 distinct:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Odrop,type,
% 11.90/5.91 drop:
% 11.90/5.91 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_OdropWhile,type,
% 11.90/5.91 dropWhile:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oenumerate,type,
% 11.90/5.91 enumerate:
% 11.90/5.91 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oextract,type,
% 11.90/5.91 extract:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ofilter,type,
% 11.90/5.91 filter:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ofind,type,
% 11.90/5.91 find:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ofold,type,
% 11.90/5.91 fold:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ofoldl,type,
% 11.90/5.91 foldl:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ofoldr,type,
% 11.90/5.91 foldr:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ogen__length,type,
% 11.90/5.91 gen_length:
% 11.90/5.91 !>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olast,type,
% 11.90/5.91 last:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olenlex,type,
% 11.90/5.91 lenlex:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olex,type,
% 11.90/5.91 lex:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olexn,type,
% 11.90/5.91 lexn:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olexord,type,
% 11.90/5.91 lexord:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olexordp,type,
% 11.90/5.91 lexordp:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olinorder__class_Osort__key,type,
% 11.90/5.91 linorder_sort_key:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
% 11.90/5.91 linord4507533701916653071of_set:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_OCons,type,
% 11.90/5.91 cons:
% 11.90/5.91 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_ONil,type,
% 11.90/5.91 nil:
% 11.90/5.91 !>[A: $tType] : ( list @ A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Ocase__list,type,
% 11.90/5.91 case_list:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Ohd,type,
% 11.90/5.91 hd:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Olist__all,type,
% 11.90/5.91 list_all:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Olist__all2,type,
% 11.90/5.91 list_all2:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Omap,type,
% 11.90/5.91 map:
% 11.90/5.91 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Oset,type,
% 11.90/5.91 set2:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Osize__list,type,
% 11.90/5.91 size_list:
% 11.90/5.91 !>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist_Otl,type,
% 11.90/5.91 tl:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olist__update,type,
% 11.90/5.91 list_update:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olistrel,type,
% 11.90/5.91 listrel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olistrel1,type,
% 11.90/5.91 listrel1:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olistrel1p,type,
% 11.90/5.91 listrel1p:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olistrelp,type,
% 11.90/5.91 listrelp:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Olists,type,
% 11.90/5.91 lists:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omap__filter,type,
% 11.90/5.91 map_filter:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omap__project,type,
% 11.90/5.91 map_project:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omap__tailrec__rev,type,
% 11.90/5.91 map_tailrec_rev:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omap__tailrec__rev__rel,type,
% 11.90/5.91 map_tailrec_rev_rel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omeasures,type,
% 11.90/5.91 measures:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omin__list,type,
% 11.90/5.91 min_list:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Omin__list__rel,type,
% 11.90/5.91 min_list_rel:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Onth,type,
% 11.90/5.91 nth:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Onths,type,
% 11.90/5.91 nths:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oord__class_Olexordp,type,
% 11.90/5.91 ord_lexordp:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Opartition,type,
% 11.90/5.91 partition:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oproduct,type,
% 11.90/5.91 product:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oproduct__lists,type,
% 11.90/5.91 product_lists:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oremdups,type,
% 11.90/5.91 remdups:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oremdups__adj,type,
% 11.90/5.91 remdups_adj:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oremdups__adj__rel,type,
% 11.90/5.91 remdups_adj_rel:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oremove1,type,
% 11.90/5.91 remove1:
% 11.90/5.91 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oreplicate,type,
% 11.90/5.91 replicate:
% 11.90/5.91 !>[A: $tType] : ( nat > A > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Orev,type,
% 11.90/5.91 rev:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Orotate,type,
% 11.90/5.91 rotate:
% 11.90/5.91 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Orotate1,type,
% 11.90/5.91 rotate1:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oshuffles,type,
% 11.90/5.91 shuffles:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oshuffles__rel,type,
% 11.90/5.91 shuffles_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Osorted__wrt,type,
% 11.90/5.91 sorted_wrt:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Osorted__wrt__rel,type,
% 11.90/5.91 sorted_wrt_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Osplice,type,
% 11.90/5.91 splice:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Osplice__rel,type,
% 11.90/5.91 splice_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Osuccessively,type,
% 11.90/5.91 successively:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Osuccessively__rel,type,
% 11.90/5.91 successively_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Otake,type,
% 11.90/5.91 take:
% 11.90/5.91 !>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_OtakeWhile,type,
% 11.90/5.91 takeWhile:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Othose,type,
% 11.90/5.91 those:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Otranspose,type,
% 11.90/5.91 transpose:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Otranspose__rel,type,
% 11.90/5.91 transpose_rel:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oupt,type,
% 11.90/5.91 upt: nat > nat > ( list @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oupto,type,
% 11.90/5.91 upto: int > int > ( list @ int ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Oupto__rel,type,
% 11.90/5.91 upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_List_Ozip,type,
% 11.90/5.91 zip:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Odom,type,
% 11.90/5.91 dom:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Ograph,type,
% 11.90/5.91 graph:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Omap__add,type,
% 11.90/5.91 map_add:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > A > ( option @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Omap__comp,type,
% 11.90/5.91 map_comp:
% 11.90/5.91 !>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Omap__le,type,
% 11.90/5.91 map_le:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Omap__of,type,
% 11.90/5.91 map_of:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Omap__upds,type,
% 11.90/5.91 map_upds:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Oran,type,
% 11.90/5.91 ran:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Map_Orestrict__map,type,
% 11.90/5.91 restrict_map:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_OCODE__ABORT,type,
% 11.90/5.91 cODE_ABORT:
% 11.90/5.91 !>[A: $tType] : ( ( product_unit > A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_OEps__Opt,type,
% 11.90/5.91 eps_Opt:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( option @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Obijective,type,
% 11.90/5.91 bijective:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Obrk__rel,type,
% 11.90/5.91 brk_rel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Odflt__None__set,type,
% 11.90/5.91 dflt_None_set:
% 11.90/5.91 !>[A: $tType] : ( ( set @ A ) > ( option @ ( set @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Ofilter__rev,type,
% 11.90/5.91 filter_rev:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Ofilter__rev__aux,type,
% 11.90/5.91 filter_rev_aux:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Ofun__of__rel,type,
% 11.90/5.91 fun_of_rel:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Oinv__on,type,
% 11.90/5.91 inv_on:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Olist__all__zip,type,
% 11.90/5.91 list_all_zip:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Olist__all__zip__rel,type,
% 11.90/5.91 list_all_zip_rel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Olist__collect__set,type,
% 11.90/5.91 list_collect_set:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omap__mmupd,type,
% 11.90/5.91 map_mmupd:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > ( option @ A ) ) > ( set @ B ) > A > B > ( option @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omap__to__set,type,
% 11.90/5.91 map_to_set:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omerge,type,
% 11.90/5.91 merge:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omerge__list,type,
% 11.90/5.91 merge_list:
% 11.90/5.91 !>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omerge__list__rel,type,
% 11.90/5.91 merge_list_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omerge__rel,type,
% 11.90/5.91 merge_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort,type,
% 11.90/5.91 mergesort:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__by__rel,type,
% 11.90/5.91 mergesort_by_rel:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__by__rel__merge,type,
% 11.90/5.91 merges9089515139780605204_merge:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
% 11.90/5.91 merges2244889521215249637ge_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__by__rel__rel,type,
% 11.90/5.91 mergesort_by_rel_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__by__rel__split,type,
% 11.90/5.91 merges295452479951948502_split:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__by__rel__split__rel,type,
% 11.90/5.91 merges7066485432131860899it_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Omergesort__remdups,type,
% 11.90/5.91 mergesort_remdups:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Opairself,type,
% 11.90/5.91 pairself:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( product_prod @ A @ A ) > ( product_prod @ B @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Opairself__rel,type,
% 11.90/5.91 pairself_rel:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Opartition__rev,type,
% 11.90/5.91 partition_rev:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Opartition__rev__rel,type,
% 11.90/5.91 partition_rev_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Oquicksort__by__rel,type,
% 11.90/5.91 quicksort_by_rel:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Oquicksort__by__rel__rel,type,
% 11.90/5.91 quicksort_by_rel_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Orel__of,type,
% 11.90/5.91 rel_of:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( ( product_prod @ A @ B ) > $o ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Orel__restrict,type,
% 11.90/5.91 rel_restrict:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Oremove__rev,type,
% 11.90/5.91 remove_rev:
% 11.90/5.91 !>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Orevg,type,
% 11.90/5.91 revg:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Orevg__rel,type,
% 11.90/5.91 revg_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Oset__to__map,type,
% 11.90/5.91 set_to_map:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( option @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Oslice,type,
% 11.90/5.91 slice:
% 11.90/5.91 !>[A: $tType] : ( nat > nat > ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Osu__rel__fun,type,
% 11.90/5.91 su_rel_fun:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( A > B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Othe__default,type,
% 11.90/5.91 the_default:
% 11.90/5.91 !>[A: $tType] : ( A > ( option @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Ouncurry,type,
% 11.90/5.91 uncurry:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Ozipf,type,
% 11.90/5.91 zipf:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( list @ A ) > ( list @ B ) > ( list @ C ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Misc_Ozipf__rel,type,
% 11.90/5.91 zipf_rel:
% 11.90/5.91 !>[A: $tType,B: $tType,C: $tType] : ( ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Oadd__mset,type,
% 11.90/5.91 add_mset:
% 11.90/5.91 !>[A: $tType] : ( A > ( multiset @ A ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset,type,
% 11.90/5.91 comm_monoid_sum_mset:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ocomm__monoid__add__class_Osum__mset,type,
% 11.90/5.91 comm_m7189776963980413722m_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ocomm__monoid__mset,type,
% 11.90/5.91 comm_monoid_mset:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ocomm__monoid__mset_OF,type,
% 11.90/5.91 comm_monoid_F:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ocomm__monoid__mult__class_Oprod__mset,type,
% 11.90/5.91 comm_m9189036328036947845d_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ofilter__mset,type,
% 11.90/5.91 filter_mset:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( multiset @ A ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ofold__mset,type,
% 11.90/5.91 fold_mset:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( multiset @ A ) > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Oimage__mset,type,
% 11.90/5.91 image_mset:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B ) > ( multiset @ A ) > ( multiset @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ointer__mset,type,
% 11.90/5.91 inter_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Olinorder__class_Opart,type,
% 11.90/5.91 linorder_part:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Oms__strict,type,
% 11.90/5.91 ms_strict: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Oms__weak,type,
% 11.90/5.91 ms_weak: set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omset,type,
% 11.90/5.91 mset:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omset__set,type,
% 11.90/5.91 mset_set:
% 11.90/5.91 !>[B: $tType] : ( ( set @ B ) > ( multiset @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omult,type,
% 11.90/5.91 mult:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omult1,type,
% 11.90/5.91 mult1:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omulteqp__code,type,
% 11.90/5.91 multeqp_code:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omultiset_OAbs__multiset,type,
% 11.90/5.91 abs_multiset:
% 11.90/5.91 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omultiset_Ocount,type,
% 11.90/5.91 count:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > A > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omultp,type,
% 11.90/5.91 multp:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Omultp__code,type,
% 11.90/5.91 multp_code:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > $o ) > ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Opcr__multiset,type,
% 11.90/5.91 pcr_multiset:
% 11.90/5.91 !>[C: $tType,B: $tType] : ( ( C > B > $o ) > ( C > nat ) > ( multiset @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Opw__leq,type,
% 11.90/5.91 pw_leq: ( multiset @ ( product_prod @ nat @ nat ) ) > ( multiset @ ( product_prod @ nat @ nat ) ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Orepeat__mset,type,
% 11.90/5.91 repeat_mset:
% 11.90/5.91 !>[A: $tType] : ( nat > ( multiset @ A ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Oreplicate__mset,type,
% 11.90/5.91 replicate_mset:
% 11.90/5.91 !>[A: $tType] : ( nat > A > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Oset__mset,type,
% 11.90/5.91 set_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Osize__multiset,type,
% 11.90/5.91 size_multiset:
% 11.90/5.91 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Osubset__eq__mset__impl,type,
% 11.90/5.91 subset_eq_mset_impl:
% 11.90/5.91 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( option @ $o ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
% 11.90/5.91 subset751672762298770561pl_rel:
% 11.90/5.91 !>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Osubset__mset,type,
% 11.90/5.91 subset_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Osubseteq__mset,type,
% 11.90/5.91 subseteq_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Ounion__mset,type,
% 11.90/5.91 union_mset:
% 11.90/5.91 !>[A: $tType] : ( ( multiset @ A ) > ( multiset @ A ) > ( multiset @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Multiset_Owcount,type,
% 11.90/5.91 wcount:
% 11.90/5.91 !>[A: $tType] : ( ( A > nat ) > ( multiset @ A ) > A > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_OSuc,type,
% 11.90/5.91 suc: nat > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_Ocompow,type,
% 11.90/5.91 compow:
% 11.90/5.91 !>[A: $tType] : ( nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_Onat_Ocase__nat,type,
% 11.90/5.91 case_nat:
% 11.90/5.91 !>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
% 11.90/5.91 rec_nat:
% 11.90/5.91 !>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
% 11.90/5.91 semiring_1_of_nat:
% 11.90/5.91 !>[A: $tType] : ( nat > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
% 11.90/5.91 semiri8178284476397505188at_aux:
% 11.90/5.91 !>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat_Osize__class_Osize,type,
% 11.90/5.91 size_size:
% 11.90/5.91 !>[A: $tType] : ( A > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 11.90/5.91 nat_list_decode: nat > ( list @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 11.90/5.91 nat_list_decode_rel: nat > nat > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 11.90/5.91 nat_list_encode: ( list @ nat ) > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 11.90/5.91 nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 11.90/5.91 nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 11.90/5.91 nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 11.90/5.91 nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 11.90/5.91 nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 11.90/5.91 nat_set_decode: nat > ( set @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 11.90/5.91 nat_set_encode: ( set @ nat ) > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Osum__decode,type,
% 11.90/5.91 nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Nat__Bijection_Otriangle,type,
% 11.90/5.91 nat_triangle: nat > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_OBitM,type,
% 11.90/5.91 bitM: num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oinc,type,
% 11.90/5.91 inc: num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onat__of__num,type,
% 11.90/5.91 nat_of_num: num > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
% 11.90/5.91 neg_numeral_dbl:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
% 11.90/5.91 neg_numeral_dbl_dec:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
% 11.90/5.91 neg_numeral_dbl_inc:
% 11.90/5.91 !>[A: $tType] : ( A > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
% 11.90/5.91 neg_numeral_is_num:
% 11.90/5.91 !>[A: $tType] : ( A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oneg__numeral__class_Osub,type,
% 11.90/5.91 neg_numeral_sub:
% 11.90/5.91 !>[A: $tType] : ( num > num > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onum_OBit0,type,
% 11.90/5.91 bit0: num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onum_OBit1,type,
% 11.90/5.91 bit1: num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onum_OOne,type,
% 11.90/5.91 one2: num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onum_Ocase__num,type,
% 11.90/5.91 case_num:
% 11.90/5.91 !>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onum_Osize__num,type,
% 11.90/5.91 size_num: num > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onum__of__nat,type,
% 11.90/5.91 num_of_nat: nat > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Onumeral__class_Onumeral,type,
% 11.90/5.91 numeral_numeral:
% 11.90/5.91 !>[A: $tType] : ( num > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Opred__numeral,type,
% 11.90/5.91 pred_numeral: num > nat ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Oring__1__class_Oiszero,type,
% 11.90/5.91 ring_1_iszero:
% 11.90/5.91 !>[A: $tType] : ( A > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Num_Osqr,type,
% 11.90/5.91 sqr: num > num ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_OAtom,type,
% 11.90/5.91 old_Atom:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_OIn0,type,
% 11.90/5.91 old_In0:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_OIn1,type,
% 11.90/5.91 old_In1:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_OLeaf,type,
% 11.90/5.91 old_Leaf:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_ONode,type,
% 11.90/5.91 old_Node:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_ONumb,type,
% 11.90/5.91 old_Numb:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_OPush__Node,type,
% 11.90/5.91 old_Push_Node:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_OScons,type,
% 11.90/5.91 old_Scons:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_Odprod,type,
% 11.90/5.91 old_dprod:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_Odsum,type,
% 11.90/5.91 old_dsum:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_Ondepth,type,
% 11.90/5.91 old_ndepth:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
% 11.90/5.91 old_Abs_Node:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Obind,type,
% 11.90/5.91 bind:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ocombine__options,type,
% 11.90/5.91 combine_options:
% 11.90/5.91 !>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ois__none,type,
% 11.90/5.91 is_none:
% 11.90/5.91 !>[A: $tType] : ( ( option @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_ONone,type,
% 11.90/5.91 none:
% 11.90/5.91 !>[A: $tType] : ( option @ A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_OSome,type,
% 11.90/5.91 some:
% 11.90/5.91 !>[A: $tType] : ( A > ( option @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Ocase__option,type,
% 11.90/5.91 case_option:
% 11.90/5.91 !>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Omap__option,type,
% 11.90/5.91 map_option:
% 11.90/5.91 !>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Opred__option,type,
% 11.90/5.91 pred_option:
% 11.90/5.91 !>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Orec__option,type,
% 11.90/5.91 rec_option:
% 11.90/5.91 !>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Orel__option,type,
% 11.90/5.91 rel_option:
% 11.90/5.91 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Oset__option,type,
% 11.90/5.91 set_option:
% 11.90/5.91 !>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Osize__option,type,
% 11.90/5.91 size_option:
% 11.90/5.91 !>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Ooption_Othe,type,
% 11.90/5.91 the2:
% 11.90/5.91 !>[A: $tType] : ( ( option @ A ) > A ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Option_Othese,type,
% 11.90/5.91 these:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Order__Relation_OAbove,type,
% 11.90/5.91 order_Above:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Order__Relation_OAboveS,type,
% 11.90/5.91 order_AboveS:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 11.90/5.91
% 11.90/5.91 thf(sy_c_Order__Relation_OUnder,type,
% 11.90/5.91 order_Under:
% 11.90/5.91 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_OUnderS,type,
% 11.90/5.92 order_UnderS:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Oabove,type,
% 11.90/5.92 order_above:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_OaboveS,type,
% 11.90/5.92 order_aboveS:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Olinear__order__on,type,
% 11.90/5.92 order_679001287576687338der_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Oofilter,type,
% 11.90/5.92 order_ofilter:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Opartial__order__on,type,
% 11.90/5.92 order_7125193373082350890der_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Opreorder__on,type,
% 11.90/5.92 order_preorder_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Orelation__of,type,
% 11.90/5.92 order_relation_of:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Ounder,type,
% 11.90/5.92 order_under:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_OunderS,type,
% 11.90/5.92 order_underS:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Order__Relation_Owell__order__on,type,
% 11.90/5.92 order_well_order_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Obot__class_Obot,type,
% 11.90/5.92 bot_bot:
% 11.90/5.92 !>[A: $tType] : A ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oord_Omax,type,
% 11.90/5.92 max:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oord_Omin,type,
% 11.90/5.92 min:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oord__class_Oless,type,
% 11.90/5.92 ord_less:
% 11.90/5.92 !>[A: $tType] : ( A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 11.90/5.92 ord_less_eq:
% 11.90/5.92 !>[A: $tType] : ( A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oord__class_Omax,type,
% 11.90/5.92 ord_max:
% 11.90/5.92 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oord__class_Omin,type,
% 11.90/5.92 ord_min:
% 11.90/5.92 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oorder__class_Omono,type,
% 11.90/5.92 order_mono:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
% 11.90/5.92 order_strict_mono:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Orderings_Otop__class_Otop,type,
% 11.90/5.92 top_top:
% 11.90/5.92 !>[A: $tType] : A ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Oflat__lub,type,
% 11.90/5.92 partial_flat_lub:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Oflat__ord,type,
% 11.90/5.92 partial_flat_ord:
% 11.90/5.92 !>[A: $tType] : ( A > A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Ofun__lub,type,
% 11.90/5.92 partial_fun_lub:
% 11.90/5.92 !>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Ofun__ord,type,
% 11.90/5.92 partial_fun_ord:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Oimg__lub,type,
% 11.90/5.92 partial_img_lub:
% 11.90/5.92 !>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( A > C ) > ( D > B ) > ( ( set @ C ) > D ) > ( set @ A ) > B ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Oimg__ord,type,
% 11.90/5.92 partial_img_ord:
% 11.90/5.92 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Partial__Function_Opartial__function__definitions,type,
% 11.90/5.92 partia7178651479351089652itions:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Power_Opower_Opower,type,
% 11.90/5.92 power2:
% 11.90/5.92 !>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Power_Opower__class_Opower,type,
% 11.90/5.92 power_power:
% 11.90/5.92 !>[A: $tType] : ( A > nat > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Predicate_OSeq,type,
% 11.90/5.92 seq2:
% 11.90/5.92 !>[A: $tType] : ( ( product_unit > ( seq @ A ) ) > ( pred @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Predicate_Obind,type,
% 11.90/5.92 bind2:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Predicate_Oiterate__upto,type,
% 11.90/5.92 iterate_upto:
% 11.90/5.92 !>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( pred @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Predicate_Oiterate__upto__rel,type,
% 11.90/5.92 iterate_upto_rel:
% 11.90/5.92 !>[A: $tType] : ( ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Predicate_Oseq_OEmpty,type,
% 11.90/5.92 empty:
% 11.90/5.92 !>[A: $tType] : ( seq @ A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Predicate_Oseq_OInsert,type,
% 11.90/5.92 insert:
% 11.90/5.92 !>[A: $tType] : ( A > ( pred @ A ) > ( seq @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_OPair,type,
% 11.90/5.92 product_Pair:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_OSigma,type,
% 11.90/5.92 product_Sigma:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_OUnity,type,
% 11.90/5.92 product_Unity: product_unit ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oapfst,type,
% 11.90/5.92 product_apfst:
% 11.90/5.92 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oapsnd,type,
% 11.90/5.92 product_apsnd:
% 11.90/5.92 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Ocurry,type,
% 11.90/5.92 product_curry:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 11.90/5.92 produc5280177257484947105e_prod:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Omap__prod,type,
% 11.90/5.92 product_map_prod:
% 11.90/5.92 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
% 11.90/5.92 product_rec_bool:
% 11.90/5.92 !>[T: $tType] : ( T > T > $o > T ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
% 11.90/5.92 product_rec_set_bool:
% 11.90/5.92 !>[T: $tType] : ( T > T > $o > T > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 11.90/5.92 product_rec_prod:
% 11.90/5.92 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
% 11.90/5.92 product_rec_set_prod:
% 11.90/5.92 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
% 11.90/5.92 product_rec_set_unit:
% 11.90/5.92 !>[T: $tType] : ( T > product_unit > T > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
% 11.90/5.92 product_rec_unit:
% 11.90/5.92 !>[T: $tType] : ( T > product_unit > T ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
% 11.90/5.92 product_case_prod:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oprod_Ofst,type,
% 11.90/5.92 product_fst:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oprod_Osnd,type,
% 11.90/5.92 product_snd:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oprod_Oswap,type,
% 11.90/5.92 product_swap:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oproduct,type,
% 11.90/5.92 product_product:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Oscomp,type,
% 11.90/5.92 product_scomp:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
% 11.90/5.92 product_Abs_unit: $o > product_unit ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Ounit_ORep__unit,type,
% 11.90/5.92 product_Rep_unit: product_unit > $o ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Product__Type_Ounit_Ocase__unit,type,
% 11.90/5.92 product_case_unit:
% 11.90/5.92 !>[A: $tType] : ( A > product_unit > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Pure_Otype,type,
% 11.90/5.92 type2:
% 11.90/5.92 !>[A: $tType] : ( itself @ A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
% 11.90/5.92 linorder_quicksort:
% 11.90/5.92 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
% 11.90/5.92 linord6200660962353139674rt_rel:
% 11.90/5.92 !>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Quotient_OQuotient3,type,
% 11.90/5.92 quotient3:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Oinc__shift,type,
% 11.90/5.92 inc_shift: code_natural > code_natural > code_natural ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Oiterate,type,
% 11.90/5.92 iterate:
% 11.90/5.92 !>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Oiterate__rel,type,
% 11.90/5.92 iterate_rel:
% 11.90/5.92 !>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Olog,type,
% 11.90/5.92 log: code_natural > code_natural > code_natural ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Olog__rel,type,
% 11.90/5.92 log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Ominus__shift,type,
% 11.90/5.92 minus_shift: code_natural > code_natural > code_natural > code_natural ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Onext,type,
% 11.90/5.92 next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Opick,type,
% 11.90/5.92 pick:
% 11.90/5.92 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Orange,type,
% 11.90/5.92 range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Oselect,type,
% 11.90/5.92 select:
% 11.90/5.92 !>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Oselect__weight,type,
% 11.90/5.92 select_weight:
% 11.90/5.92 !>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random_Osplit__seed,type,
% 11.90/5.92 split_seed: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random__Pred_Obind,type,
% 11.90/5.92 random_bind:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random__Pred_Oempty,type,
% 11.90/5.92 random_empty:
% 11.90/5.92 !>[A: $tType] : ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random__Pred_Oiterate__upto,type,
% 11.90/5.92 random_iterate_upto:
% 11.90/5.92 !>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Random__Pred_Ounion,type,
% 11.90/5.92 random_union:
% 11.90/5.92 !>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_OAbs__Rat,type,
% 11.90/5.92 abs_Rat: ( product_prod @ int @ int ) > rat ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_OFract,type,
% 11.90/5.92 fract: int > int > rat ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_OFrct,type,
% 11.90/5.92 frct: ( product_prod @ int @ int ) > rat ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_ORep__Rat,type,
% 11.90/5.92 rep_Rat: rat > ( product_prod @ int @ int ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
% 11.90/5.92 field_char_0_of_rat:
% 11.90/5.92 !>[A: $tType] : ( rat > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Onormalize,type,
% 11.90/5.92 normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Oof__int,type,
% 11.90/5.92 of_int: int > rat ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Opcr__rat,type,
% 11.90/5.92 pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Opositive,type,
% 11.90/5.92 positive: rat > $o ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Oquotient__of,type,
% 11.90/5.92 quotient_of: rat > ( product_prod @ int @ int ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rat_Oratrel,type,
% 11.90/5.92 ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Oalloc,type,
% 11.90/5.92 ref_alloc:
% 11.90/5.92 !>[A: $tType] : ( A > ( heap_ext @ product_unit ) > ( product_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Ochange,type,
% 11.90/5.92 ref_change:
% 11.90/5.92 !>[A: $tType] : ( ( A > A ) > ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Oget,type,
% 11.90/5.92 ref_get:
% 11.90/5.92 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Olookup,type,
% 11.90/5.92 ref_lookup:
% 11.90/5.92 !>[A: $tType] : ( ( ref @ A ) > ( heap_Time_Heap @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Opresent,type,
% 11.90/5.92 ref_present:
% 11.90/5.92 !>[A: $tType] : ( ( heap_ext @ product_unit ) > ( ref @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Oref,type,
% 11.90/5.92 ref_ref:
% 11.90/5.92 !>[A: $tType] : ( A > ( heap_Time_Heap @ ( ref @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Oset,type,
% 11.90/5.92 ref_set:
% 11.90/5.92 !>[A: $tType] : ( ( ref @ A ) > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Ref__Time_Oupdate,type,
% 11.90/5.92 ref_update:
% 11.90/5.92 !>[A: $tType] : ( ( ref @ A ) > A > ( heap_Time_Heap @ product_unit ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Refine__Imp__Hol_Oassert_H,type,
% 11.90/5.92 refine_Imp_assert: $o > ( heap_Time_Heap @ product_unit ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Refine__Imp__Hol_Orefines,type,
% 11.90/5.92 refine_Imp_refines:
% 11.90/5.92 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_ODomain,type,
% 11.90/5.92 domain:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_ODomainp,type,
% 11.90/5.92 domainp:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_OField,type,
% 11.90/5.92 field2:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_OId,type,
% 11.90/5.92 id2:
% 11.90/5.92 !>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_OId__on,type,
% 11.90/5.92 id_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_OImage,type,
% 11.90/5.92 image:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_ORange,type,
% 11.90/5.92 range2:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_ORangep,type,
% 11.90/5.92 rangep:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oantisym,type,
% 11.90/5.92 antisym:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oantisymp,type,
% 11.90/5.92 antisymp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oasym,type,
% 11.90/5.92 asym:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oasymp,type,
% 11.90/5.92 asymp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oconverse,type,
% 11.90/5.92 converse:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oconversep,type,
% 11.90/5.92 conversep:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oinv__image,type,
% 11.90/5.92 inv_image:
% 11.90/5.92 !>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oirrefl,type,
% 11.90/5.92 irrefl:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oirreflp,type,
% 11.90/5.92 irreflp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Orefl__on,type,
% 11.90/5.92 refl_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Oreflp,type,
% 11.90/5.92 reflp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Orelcomp,type,
% 11.90/5.92 relcomp:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Orelcompp,type,
% 11.90/5.92 relcompp:
% 11.90/5.92 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Osingle__valued,type,
% 11.90/5.92 single_valued:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Osingle__valuedp,type,
% 11.90/5.92 single_valuedp:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Ototal__on,type,
% 11.90/5.92 total_on:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Otrans,type,
% 11.90/5.92 trans:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Relation_Otransp,type,
% 11.90/5.92 transp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
% 11.90/5.92 algebr8660921524188924756oprime:
% 11.90/5.92 !>[A: $tType] : ( A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Odivide__class_Odivide,type,
% 11.90/5.92 divide_divide:
% 11.90/5.92 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Odvd__class_Odvd,type,
% 11.90/5.92 dvd_dvd:
% 11.90/5.92 !>[A: $tType] : ( A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Omodulo__class_Omodulo,type,
% 11.90/5.92 modulo_modulo:
% 11.90/5.92 !>[A: $tType] : ( A > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
% 11.90/5.92 normal6383669964737779283malize:
% 11.90/5.92 !>[A: $tType] : ( A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
% 11.90/5.92 unit_f5069060285200089521factor:
% 11.90/5.92 !>[A: $tType] : ( A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
% 11.90/5.92 zero_neq_one_of_bool:
% 11.90/5.92 !>[A: $tType] : ( $o > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_OCollect,type,
% 11.90/5.92 collect:
% 11.90/5.92 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_OPow,type,
% 11.90/5.92 pow:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_Obind,type,
% 11.90/5.92 bind3:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_Odisjnt,type,
% 11.90/5.92 disjnt:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_Oimage,type,
% 11.90/5.92 image2:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_Oinsert,type,
% 11.90/5.92 insert2:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_Ois__singleton,type,
% 11.90/5.92 is_singleton:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set_Ovimage,type,
% 11.90/5.92 vimage:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
% 11.90/5.92 set_fo6178422350223883121st_nat:
% 11.90/5.92 !>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
% 11.90/5.92 set_fo1817059534552279752at_rel:
% 11.90/5.92 !>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
% 11.90/5.92 set_ord_atLeast:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
% 11.90/5.92 set_or1337092689740270186AtMost:
% 11.90/5.92 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
% 11.90/5.92 set_or7035219750837199246ssThan:
% 11.90/5.92 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OatMost,type,
% 11.90/5.92 set_ord_atMost:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
% 11.90/5.92 set_ord_greaterThan:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
% 11.90/5.92 set_or3652927894154168847AtMost:
% 11.90/5.92 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
% 11.90/5.92 set_or5935395276787703475ssThan:
% 11.90/5.92 !>[A: $tType] : ( A > A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
% 11.90/5.92 set_ord_lessThan:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_String_Ochar_OChar,type,
% 11.90/5.92 char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_String_Ointeger__of__char,type,
% 11.90/5.92 integer_of_char: char > code_integer ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
% 11.90/5.92 unique5772411509450598832har_of:
% 11.90/5.92 !>[A: $tType] : ( A > char ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Sum__Type_OInl,type,
% 11.90/5.92 sum_Inl:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Sum__Type_OInr,type,
% 11.90/5.92 sum_Inr:
% 11.90/5.92 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Sum__Type_OPlus,type,
% 11.90/5.92 sum_Plus:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Sum__Type_Omap__sum,type,
% 11.90/5.92 sum_map_sum:
% 11.90/5.92 !>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 11.90/5.92 sum_case_sum:
% 11.90/5.92 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
% 11.90/5.92 syntax7388354845996824322omatch:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( A > B > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Time__Reasoning_Ofails,type,
% 11.90/5.92 time_fails:
% 11.90/5.92 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Time__Reasoning_Othe__heap,type,
% 11.90/5.92 time_the_heap:
% 11.90/5.92 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Time__Reasoning_Othe__res,type,
% 11.90/5.92 time_the_res:
% 11.90/5.92 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Time__Reasoning_Otime,type,
% 11.90/5.92 time_time:
% 11.90/5.92 !>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > nat ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Transitive__Closure_Oacyclic,type,
% 11.90/5.92 transitive_acyclic:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Transitive__Closure_Ontrancl,type,
% 11.90/5.92 transitive_ntrancl:
% 11.90/5.92 !>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Transitive__Closure_Ortrancl,type,
% 11.90/5.92 transitive_rtrancl:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Transitive__Closure_Ortranclp,type,
% 11.90/5.92 transitive_rtranclp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Transitive__Closure_Otrancl,type,
% 11.90/5.92 transitive_trancl:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Transitive__Closure_Otranclp,type,
% 11.90/5.92 transitive_tranclp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Typedef_Otype__definition,type,
% 11.90/5.92 type_definition:
% 11.90/5.92 !>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Typerep_Otyperep__class_Otyperep,type,
% 11.90/5.92 typerep_typerep:
% 11.90/5.92 !>[A: $tType] : ( ( itself @ A ) > typerep ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Oacc,type,
% 11.90/5.92 acc:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Oaccp,type,
% 11.90/5.92 accp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Ofinite__psubset,type,
% 11.90/5.92 finite_psubset:
% 11.90/5.92 !>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Oless__than,type,
% 11.90/5.92 less_than: set @ ( product_prod @ nat @ nat ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Olex__prod,type,
% 11.90/5.92 lex_prod:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Omax__ext,type,
% 11.90/5.92 max_ext:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Omax__extp,type,
% 11.90/5.92 max_extp:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Omeasure,type,
% 11.90/5.92 measure:
% 11.90/5.92 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Omin__ext,type,
% 11.90/5.92 min_ext:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Omlex__prod,type,
% 11.90/5.92 mlex_prod:
% 11.90/5.92 !>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Opred__nat,type,
% 11.90/5.92 pred_nat: set @ ( product_prod @ nat @ nat ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_Owf,type,
% 11.90/5.92 wf:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wellfounded_OwfP,type,
% 11.90/5.92 wfP:
% 11.90/5.92 !>[A: $tType] : ( ( A > A > $o ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wfrec_Oadm__wf,type,
% 11.90/5.92 adm_wf:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wfrec_Ocut,type,
% 11.90/5.92 cut:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Wfrec_Osame__fst,type,
% 11.90/5.92 same_fst:
% 11.90/5.92 !>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Zorn_OChains,type,
% 11.90/5.92 chains:
% 11.90/5.92 !>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Zorn_Oinit__seg__of,type,
% 11.90/5.92 init_seg_of:
% 11.90/5.92 !>[A: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_Zorn_Opred__on_Ochain,type,
% 11.90/5.92 pred_chain:
% 11.90/5.92 !>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_fChoice,type,
% 11.90/5.92 fChoice:
% 11.90/5.92 !>[A: $tType] : ( ( A > $o ) > A ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_c_member,type,
% 11.90/5.92 member:
% 11.90/5.92 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 11.90/5.92
% 11.90/5.92 thf(sy_v_h,type,
% 11.90/5.92 h: heap_ext @ product_unit ).
% 11.90/5.92
% 11.90/5.92 thf(sy_v_xs,type,
% 11.90/5.92 xs: list @ a ).
% 11.90/5.92
% 11.90/5.92 thf(fact_0_fails__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( time_fails @ A )
% 11.90/5.92 = ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ M @ H )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_1_Heap__eqI,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
% 11.90/5.92 ( ! [H2: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ F @ H2 )
% 11.90/5.92 = ( heap_Time_execute @ A @ G @ H2 ) )
% 11.90/5.92 => ( F = G ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_2_execute__raise,axiom,
% 11.90/5.92 ! [A: $tType,S: list @ char] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S ) )
% 11.90/5.92 = ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_3_execute__guard_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ~ ( P @ H3 )
% 11.90/5.92 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_4_execute__assert_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ~ ( P @ X )
% 11.90/5.92 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_5_successI,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 => ( heap_Time_success @ A @ F @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_6_success__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_success @ A )
% 11.90/5.92 = ( ^ [F2: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ F2 @ H )
% 11.90/5.92 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_7_Heap__ordE,axiom,
% 11.90/5.92 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_Heap_ord @ A @ X @ Y )
% 11.90/5.92 => ( ( ( heap_Time_execute @ A @ X @ H3 )
% 11.90/5.92 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 => ( ( heap_Time_execute @ A @ X @ H3 )
% 11.90/5.92 = ( heap_Time_execute @ A @ Y @ H3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_8_Heap__ordI,axiom,
% 11.90/5.92 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A] :
% 11.90/5.92 ( ! [H2: heap_ext @ product_unit] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ X @ H2 )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 | ( ( heap_Time_execute @ A @ X @ H2 )
% 11.90/5.92 = ( heap_Time_execute @ A @ Y @ H2 ) ) )
% 11.90/5.92 => ( heap_Time_Heap_ord @ A @ X @ Y ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_9_execute__bind_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 11.90/5.92 = ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_10_the__default_Osimps_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,X: A] :
% 11.90/5.92 ( ( the_default @ A @ X @ ( none @ A ) )
% 11.90/5.92 = X ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_11_raise__bind,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,E2: list @ char,F: B > ( heap_Time_Heap @ A )] :
% 11.90/5.92 ( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E2 ) @ F )
% 11.90/5.92 = ( heap_Time_raise @ A @ E2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_12_heap_Oleq__refl,axiom,
% 11.90/5.92 ! [A: $tType,X: heap_Time_Heap @ A] : ( heap_Time_Heap_ord @ A @ X @ X ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_13_heap_Oleq__trans,axiom,
% 11.90/5.92 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( heap_Time_Heap_ord @ A @ X @ Y )
% 11.90/5.92 => ( ( heap_Time_Heap_ord @ A @ Y @ Z2 )
% 11.90/5.92 => ( heap_Time_Heap_ord @ A @ X @ Z2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_14_heap_Oleq__antisym,axiom,
% 11.90/5.92 ! [A: $tType,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( heap_Time_Heap_ord @ A @ X @ Y )
% 11.90/5.92 => ( ( heap_Time_Heap_ord @ A @ Y @ X )
% 11.90/5.92 => ( X = Y ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_15_assert__cong,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B ),X: A] :
% 11.90/5.92 ( ( P = P2 )
% 11.90/5.92 => ( ! [X2: A] :
% 11.90/5.92 ( ( P2 @ X2 )
% 11.90/5.92 => ( ( F @ X2 )
% 11.90/5.92 = ( F3 @ X2 ) ) )
% 11.90/5.92 => ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X ) @ F )
% 11.90/5.92 = ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X ) @ F3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_16_success__ifI,axiom,
% 11.90/5.92 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,E2: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( C2
% 11.90/5.92 => ( heap_Time_success @ A @ T2 @ H3 ) )
% 11.90/5.92 => ( ( ~ C2
% 11.90/5.92 => ( heap_Time_success @ A @ E2 @ H3 ) )
% 11.90/5.92 => ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_17_success__guardI,axiom,
% 11.90/5.92 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( P @ H3 )
% 11.90/5.92 => ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_18_success__assertI,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( P @ X )
% 11.90/5.92 => ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_19_fails__bind,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,M2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( time_fails @ A @ ( heap_Time_bind @ B @ A @ M2 @ F ) @ H3 )
% 11.90/5.92 = ( ~ ( time_fails @ B @ M2 @ H3 )
% 11.90/5.92 => ( time_fails @ A @ ( F @ ( time_the_res @ B @ M2 @ H3 ) ) @ ( time_the_heap @ B @ M2 @ H3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_20_success__of__listI,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [Xs: list @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_21_success__tapI,axiom,
% 11.90/5.92 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_22_success__heapI,axiom,
% 11.90/5.92 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_23_success__ureturnI,axiom,
% 11.90/5.92 ! [A: $tType,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_24_bind__ureturn,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
% 11.90/5.92 = F ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_25_ureturn__bind,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 11.90/5.92 ( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X ) @ F )
% 11.90/5.92 = ( F @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_26_option_Osimps_I6_J,axiom,
% 11.90/5.92 ! [A: $tType,C: $tType,F1: C,F22: A > C] :
% 11.90/5.92 ( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
% 11.90/5.92 = F1 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_27_success__bind__effectI,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ X @ N )
% 11.90/5.92 => ( ( heap_Time_success @ B @ ( G @ X ) @ H4 )
% 11.90/5.92 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_28_execute__guard_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( P @ H3 )
% 11.90/5.92 => ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_29_combine__options__simps_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,F: A > A > A,Y: option @ A] :
% 11.90/5.92 ( ( combine_options @ A @ F @ ( none @ A ) @ Y )
% 11.90/5.92 = Y ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_30_combine__options__simps_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,F: A > A > A,X: option @ A] :
% 11.90/5.92 ( ( combine_options @ A @ F @ X @ ( none @ A ) )
% 11.90/5.92 = X ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_31_option_Oinject,axiom,
% 11.90/5.92 ! [A: $tType,X22: A,Y2: A] :
% 11.90/5.92 ( ( ( some @ A @ X22 )
% 11.90/5.92 = ( some @ A @ Y2 ) )
% 11.90/5.92 = ( X22 = Y2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_32_not__None__eq,axiom,
% 11.90/5.92 ! [A: $tType,X: option @ A] :
% 11.90/5.92 ( ( X
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 = ( ? [Y3: A] :
% 11.90/5.92 ( X
% 11.90/5.92 = ( some @ A @ Y3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_33_not__Some__eq,axiom,
% 11.90/5.92 ! [A: $tType,X: option @ A] :
% 11.90/5.92 ( ( ! [Y3: A] :
% 11.90/5.92 ( X
% 11.90/5.92 != ( some @ A @ Y3 ) ) )
% 11.90/5.92 = ( X
% 11.90/5.92 = ( none @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_34_combine__options__simps_I3_J,axiom,
% 11.90/5.92 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 11.90/5.92 ( ( combine_options @ A @ F @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
% 11.90/5.92 = ( some @ A @ ( F @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_35_option_Osimps_I7_J,axiom,
% 11.90/5.92 ! [C: $tType,A: $tType,F1: C,F22: A > C,X22: A] :
% 11.90/5.92 ( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
% 11.90/5.92 = ( F22 @ X22 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_36_effect__deterministic_I3_J,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
% 11.90/5.92 => ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
% 11.90/5.92 => ( N = N2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_37_effect__deterministic_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
% 11.90/5.92 => ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
% 11.90/5.92 => ( H4 = H5 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_38_effect__deterministic_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N2: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ A3 @ N )
% 11.90/5.92 => ( ( heap_Time_effect @ A @ F @ H3 @ H5 @ B2 @ N2 )
% 11.90/5.92 => ( A3 = B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_39_effect__ifE,axiom,
% 11.90/5.92 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ( ( C2
% 11.90/5.92 => ~ ( heap_Time_effect @ A @ T2 @ H3 @ H4 @ R @ N ) )
% 11.90/5.92 => ~ ( ~ C2
% 11.90/5.92 => ~ ( heap_Time_effect @ A @ E2 @ H3 @ H4 @ R @ N ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_40_effect__ifI,axiom,
% 11.90/5.92 ! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,E2: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( C2
% 11.90/5.92 => ( heap_Time_effect @ A @ T2 @ H3 @ H4 @ R @ N ) )
% 11.90/5.92 => ( ( ~ C2
% 11.90/5.92 => ( heap_Time_effect @ A @ E2 @ H3 @ H4 @ R @ N ) )
% 11.90/5.92 => ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H3 @ H4 @ R @ N ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_41_combine__options__assoc,axiom,
% 11.90/5.92 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A,Z2: option @ A] :
% 11.90/5.92 ( ! [X2: A,Y4: A,Z3: A] :
% 11.90/5.92 ( ( F @ ( F @ X2 @ Y4 ) @ Z3 )
% 11.90/5.92 = ( F @ X2 @ ( F @ Y4 @ Z3 ) ) )
% 11.90/5.92 => ( ( combine_options @ A @ F @ ( combine_options @ A @ F @ X @ Y ) @ Z2 )
% 11.90/5.92 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_42_combine__options__commute,axiom,
% 11.90/5.92 ! [A: $tType,F: A > A > A,X: option @ A,Y: option @ A] :
% 11.90/5.92 ( ! [X2: A,Y4: A] :
% 11.90/5.92 ( ( F @ X2 @ Y4 )
% 11.90/5.92 = ( F @ Y4 @ X2 ) )
% 11.90/5.92 => ( ( combine_options @ A @ F @ X @ Y )
% 11.90/5.92 = ( combine_options @ A @ F @ Y @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_43_combine__options__left__commute,axiom,
% 11.90/5.92 ! [A: $tType,F: A > A > A,Y: option @ A,X: option @ A,Z2: option @ A] :
% 11.90/5.92 ( ! [X2: A,Y4: A] :
% 11.90/5.92 ( ( F @ X2 @ Y4 )
% 11.90/5.92 = ( F @ Y4 @ X2 ) )
% 11.90/5.92 => ( ! [X2: A,Y4: A,Z3: A] :
% 11.90/5.92 ( ( F @ ( F @ X2 @ Y4 ) @ Z3 )
% 11.90/5.92 = ( F @ X2 @ ( F @ Y4 @ Z3 ) ) )
% 11.90/5.92 => ( ( combine_options @ A @ F @ Y @ ( combine_options @ A @ F @ X @ Z2 ) )
% 11.90/5.92 = ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y @ Z2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_44_mem__Collect__eq,axiom,
% 11.90/5.92 ! [A: $tType,A3: A,P: A > $o] :
% 11.90/5.92 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 11.90/5.92 = ( P @ A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_45_Collect__mem__eq,axiom,
% 11.90/5.92 ! [A: $tType,A4: set @ A] :
% 11.90/5.92 ( ( collect @ A
% 11.90/5.92 @ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
% 11.90/5.92 = A4 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_46_Collect__cong,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,Q: A > $o] :
% 11.90/5.92 ( ! [X2: A] :
% 11.90/5.92 ( ( P @ X2 )
% 11.90/5.92 = ( Q @ X2 ) )
% 11.90/5.92 => ( ( collect @ A @ P )
% 11.90/5.92 = ( collect @ A @ Q ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_47_ext,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 11.90/5.92 ( ! [X2: A] :
% 11.90/5.92 ( ( F @ X2 )
% 11.90/5.92 = ( G @ X2 ) )
% 11.90/5.92 => ( F = G ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_48_option_Odistinct_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,X22: A] :
% 11.90/5.92 ( ( none @ A )
% 11.90/5.92 != ( some @ A @ X22 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_49_option_OdiscI,axiom,
% 11.90/5.92 ! [A: $tType,Option: option @ A,X22: A] :
% 11.90/5.92 ( ( Option
% 11.90/5.92 = ( some @ A @ X22 ) )
% 11.90/5.92 => ( Option
% 11.90/5.92 != ( none @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_50_option_Oexhaust,axiom,
% 11.90/5.92 ! [A: $tType,Y: option @ A] :
% 11.90/5.92 ( ( Y
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 => ~ ! [X23: A] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( some @ A @ X23 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_51_split__option__ex,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ^ [P3: ( option @ A ) > $o] :
% 11.90/5.92 ? [X4: option @ A] : ( P3 @ X4 ) )
% 11.90/5.92 = ( ^ [P4: ( option @ A ) > $o] :
% 11.90/5.92 ( ( P4 @ ( none @ A ) )
% 11.90/5.92 | ? [X3: A] : ( P4 @ ( some @ A @ X3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_52_split__option__all,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ^ [P3: ( option @ A ) > $o] :
% 11.90/5.92 ! [X4: option @ A] : ( P3 @ X4 ) )
% 11.90/5.92 = ( ^ [P4: ( option @ A ) > $o] :
% 11.90/5.92 ( ( P4 @ ( none @ A ) )
% 11.90/5.92 & ! [X3: A] : ( P4 @ ( some @ A @ X3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_53_combine__options__cases,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
% 11.90/5.92 ( ( ( X
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 => ( P @ X @ Y ) )
% 11.90/5.92 => ( ( ( Y
% 11.90/5.92 = ( none @ B ) )
% 11.90/5.92 => ( P @ X @ Y ) )
% 11.90/5.92 => ( ! [A5: A,B3: B] :
% 11.90/5.92 ( ( X
% 11.90/5.92 = ( some @ A @ A5 ) )
% 11.90/5.92 => ( ( Y
% 11.90/5.92 = ( some @ B @ B3 ) )
% 11.90/5.92 => ( P @ X @ Y ) ) )
% 11.90/5.92 => ( P @ X @ Y ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_54_success__effectE,axiom,
% 11.90/5.92 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_success @ A @ C2 @ H3 )
% 11.90/5.92 => ~ ! [R2: A,H6: heap_ext @ product_unit,N3: nat] :
% 11.90/5.92 ~ ( heap_Time_effect @ A @ C2 @ H3 @ H6 @ R2 @ N3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_55_effect__success,axiom,
% 11.90/5.92 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ( heap_Time_success @ A @ C2 @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_56_effect__raiseE,axiom,
% 11.90/5.92 ! [A: $tType,X: list @ char,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H3 @ H4 @ R @ N ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_57_the__default_Osimps_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,Uu2: A,X: A] :
% 11.90/5.92 ( ( the_default @ A @ Uu2 @ ( some @ A @ X ) )
% 11.90/5.92 = X ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_58_Heap__cases,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ! [X2: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ H6 ) ) )
% 11.90/5.92 => ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_59_successE,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_success @ A @ F @ H3 )
% 11.90/5.92 => ~ ! [R2: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ H6 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_60_execute__heap,axiom,
% 11.90/5.92 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
% 11.90/5.92 = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ F ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_61_some__opt__sym__eq__trivial,axiom,
% 11.90/5.92 ! [A: $tType,X: A] :
% 11.90/5.92 ( ( eps_Opt @ A
% 11.90/5.92 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.90/5.92 @ X ) )
% 11.90/5.92 = ( some @ A @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_62_success__newI,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [N: nat,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_63_success__makeI,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [N: nat,F: nat > A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_64_mmupd__in__upd,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,K: A,K2: set @ A,M2: A > ( option @ B ),V: B] :
% 11.90/5.92 ( ( member @ A @ K @ K2 )
% 11.90/5.92 => ( ( map_mmupd @ A @ B @ M2 @ K2 @ V @ K )
% 11.90/5.92 = ( some @ B @ V ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_65_option_Ocollapse,axiom,
% 11.90/5.92 ! [A: $tType,Option: option @ A] :
% 11.90/5.92 ( ( Option
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 => ( ( some @ A @ ( the2 @ A @ Option ) )
% 11.90/5.92 = Option ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_66_not__Some__eq2,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,V: option @ ( product_prod @ A @ B )] :
% 11.90/5.92 ( ( ! [X3: A,Y3: B] :
% 11.90/5.92 ( V
% 11.90/5.92 != ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) )
% 11.90/5.92 = ( V
% 11.90/5.92 = ( none @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_67_effect__tapE,axiom,
% 11.90/5.92 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ~ ( ( H4 = H3 )
% 11.90/5.92 => ( ( R
% 11.90/5.92 = ( F @ H3 ) )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_68_effect__tapI,axiom,
% 11.90/5.92 ! [A: $tType,H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
% 11.90/5.92 ( ( H4 = H3 )
% 11.90/5.92 => ( ( R
% 11.90/5.92 = ( F @ H3 ) )
% 11.90/5.92 => ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H3 @ H4 @ R @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_69_effect__assertI,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A,H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( P @ X )
% 11.90/5.92 => ( ( H4 = H3 )
% 11.90/5.92 => ( ( R = X )
% 11.90/5.92 => ( ( N
% 11.90/5.92 = ( one_one @ nat ) )
% 11.90/5.92 => ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_70_mmupd__notin__upd,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,K: A,K2: set @ A,M2: A > ( option @ B ),V: B] :
% 11.90/5.92 ( ~ ( member @ A @ K @ K2 )
% 11.90/5.92 => ( ( map_mmupd @ A @ B @ M2 @ K2 @ V @ K )
% 11.90/5.92 = ( M2 @ K ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_71_fun__comp__eq__conv,axiom,
% 11.90/5.92 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,Fg: A > B] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ F @ G )
% 11.90/5.92 = Fg )
% 11.90/5.92 = ( ! [X3: A] :
% 11.90/5.92 ( ( F @ ( G @ X3 ) )
% 11.90/5.92 = ( Fg @ X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_72_comp__cong__right,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,X: A > B,Y: A > B,F: B > C] :
% 11.90/5.92 ( ( X = Y )
% 11.90/5.92 => ( ( comp @ B @ C @ A @ F @ X )
% 11.90/5.92 = ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_73_comp__cong__left,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,C: $tType,X: A > B,Y: A > B,F: C > A] :
% 11.90/5.92 ( ( X = Y )
% 11.90/5.92 => ( ( comp @ A @ B @ C @ X @ F )
% 11.90/5.92 = ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_74_bex2I,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B ),P: A > B > $o] :
% 11.90/5.92 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 11.90/5.92 => ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 11.90/5.92 => ( P @ A3 @ B2 ) )
% 11.90/5.92 => ? [A5: A,B3: B] :
% 11.90/5.92 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ S2 )
% 11.90/5.92 & ( P @ A5 @ B3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_75_pairself_Ocases,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
% 11.90/5.92 ~ ! [F4: A > B,A5: A,B3: A] :
% 11.90/5.92 ( X
% 11.90/5.92 != ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F4 @ ( product_Pair @ A @ A @ A5 @ B3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_76_fails__array__new,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [N: nat,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ~ ( time_fails @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_77_timeFrame_Ocases,axiom,
% 11.90/5.92 ! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.90/5.92 ( ! [N3: nat,R2: A,H2: heap_ext @ product_unit,N4: nat] :
% 11.90/5.92 ( X
% 11.90/5.92 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N3 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) ) )
% 11.90/5.92 => ~ ! [N3: nat] :
% 11.90/5.92 ( X
% 11.90/5.92 != ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N3 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_78_execute__assert_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( P @ X )
% 11.90/5.92 => ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_79_execute__tap,axiom,
% 11.90/5.92 ! [A: $tType,F: ( heap_ext @ product_unit ) > A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_80_option_Osel,axiom,
% 11.90/5.92 ! [A: $tType,X22: A] :
% 11.90/5.92 ( ( the2 @ A @ ( some @ A @ X22 ) )
% 11.90/5.92 = X22 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_81_option_Oexpand,axiom,
% 11.90/5.92 ! [A: $tType,Option: option @ A,Option2: option @ A] :
% 11.90/5.92 ( ( ( Option
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 = ( Option2
% 11.90/5.92 = ( none @ A ) ) )
% 11.90/5.92 => ( ( ( Option
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 => ( ( Option2
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 => ( ( the2 @ A @ Option )
% 11.90/5.92 = ( the2 @ A @ Option2 ) ) ) )
% 11.90/5.92 => ( Option = Option2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_82_effect__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_effect @ A )
% 11.90/5.92 = ( ^ [C3: heap_Time_Heap @ A,H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R3: A,N5: nat] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ C3 @ H )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ N5 ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_83_effectI,axiom,
% 11.90/5.92 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,R: A,H4: heap_ext @ product_unit,N: nat] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ C2 @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 11.90/5.92 => ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_84_map__mmupd__def,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( map_mmupd @ B @ A )
% 11.90/5.92 = ( ^ [M: B > ( option @ A ),K3: set @ B,V2: A,K4: B] : ( if @ ( option @ A ) @ ( member @ B @ K4 @ K3 ) @ ( some @ A @ V2 ) @ ( M @ K4 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_85_map__mmupdE,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,M2: B > ( option @ A ),K2: set @ B,V: A,K: B,X: A] :
% 11.90/5.92 ( ( ( map_mmupd @ B @ A @ M2 @ K2 @ V @ K )
% 11.90/5.92 = ( some @ A @ X ) )
% 11.90/5.92 => ( ( ~ ( member @ B @ K @ K2 )
% 11.90/5.92 => ( ( M2 @ K )
% 11.90/5.92 != ( some @ A @ X ) ) )
% 11.90/5.92 => ~ ( ( member @ B @ K @ K2 )
% 11.90/5.92 => ( X != V ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_86_success__bind__executeI,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 11.90/5.92 => ( ( heap_Time_success @ B @ ( G @ X ) @ H4 )
% 11.90/5.92 => ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_87_Eps__Opt__eq__Some__implies,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A] :
% 11.90/5.92 ( ( ( eps_Opt @ A @ P )
% 11.90/5.92 = ( some @ A @ X ) )
% 11.90/5.92 => ( P @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_88_Eps__Opt__eq__Some,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A] :
% 11.90/5.92 ( ! [X5: A] :
% 11.90/5.92 ( ( P @ X )
% 11.90/5.92 => ( ( P @ X5 )
% 11.90/5.92 => ( X5 = X ) ) )
% 11.90/5.92 => ( ( ( eps_Opt @ A @ P )
% 11.90/5.92 = ( some @ A @ X ) )
% 11.90/5.92 = ( P @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_89_option_Oexhaust__sel,axiom,
% 11.90/5.92 ! [A: $tType,Option: option @ A] :
% 11.90/5.92 ( ( Option
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 => ( Option
% 11.90/5.92 = ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_90_effect__assertE,axiom,
% 11.90/5.92 ! [A: $tType,P: A > $o,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ~ ( ( P @ X )
% 11.90/5.92 => ( ( R = X )
% 11.90/5.92 => ( ( H4 = H3 )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_91_comp__apply,axiom,
% 11.90/5.92 ! [C: $tType,A: $tType,B: $tType] :
% 11.90/5.92 ( ( comp @ B @ A @ C )
% 11.90/5.92 = ( ^ [F2: B > A,G2: C > B,X3: C] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_92_prod_Oinject,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
% 11.90/5.92 ( ( ( product_Pair @ A @ B @ X1 @ X22 )
% 11.90/5.92 = ( product_Pair @ A @ B @ Y1 @ Y2 ) )
% 11.90/5.92 = ( ( X1 = Y1 )
% 11.90/5.92 & ( X22 = Y2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_93_old_Oprod_Oinject,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
% 11.90/5.92 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 11.90/5.92 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 11.90/5.92 = ( ( A3 = A6 )
% 11.90/5.92 & ( B2 = B4 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_94_execute__bind_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 11.90/5.92 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 11.90/5.92 = ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H4 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_95_prod__cases3,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
% 11.90/5.92 ~ ! [A5: A,B3: B,C4: C] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_96_prod__cases4,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 11.90/5.92 ~ ! [A5: A,B3: B,C4: C,D2: D] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_97_prod__cases5,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 11.90/5.92 ~ ! [A5: A,B3: B,C4: C,D2: D,E3: E] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_98_prod__cases6,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
% 11.90/5.92 ~ ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E3 @ F4 ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_99_prod__cases7,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F5: $tType,G3: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 11.90/5.92 ~ ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5,G4: G3] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_100_prod__induct3,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
% 11.90/5.92 ( ! [A5: A,B3: B,C4: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B3 @ C4 ) ) )
% 11.90/5.92 => ( P @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_101_prod__induct4,axiom,
% 11.90/5.92 ! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
% 11.90/5.92 ( ! [A5: A,B3: B,C4: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C4 @ D2 ) ) ) )
% 11.90/5.92 => ( P @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_102_timeFrame_Osimps_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,N: nat] :
% 11.90/5.92 ( ( heap_Time_timeFrame @ A @ N @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_103_Pair__inject,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B] :
% 11.90/5.92 ( ( ( product_Pair @ A @ B @ A3 @ B2 )
% 11.90/5.92 = ( product_Pair @ A @ B @ A6 @ B4 ) )
% 11.90/5.92 => ~ ( ( A3 = A6 )
% 11.90/5.92 => ( B2 != B4 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_104_prod__cases,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P5: product_prod @ A @ B] :
% 11.90/5.92 ( ! [A5: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.90/5.92 => ( P @ P5 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_105_surj__pair,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,P5: product_prod @ A @ B] :
% 11.90/5.92 ? [X2: A,Y4: B] :
% 11.90/5.92 ( P5
% 11.90/5.92 = ( product_Pair @ A @ B @ X2 @ Y4 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_106_old_Oprod_Oexhaust,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
% 11.90/5.92 ~ ! [A5: A,B3: B] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( product_Pair @ A @ B @ A5 @ B3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_107_one__reorient,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( one @ A )
% 11.90/5.92 => ! [X: A] :
% 11.90/5.92 ( ( ( one_one @ A )
% 11.90/5.92 = X )
% 11.90/5.92 = ( X
% 11.90/5.92 = ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_108_comp__eq__dest__lhs,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 11.90/5.92 = C2 )
% 11.90/5.92 => ( ( A3 @ ( B2 @ V ) )
% 11.90/5.92 = ( C2 @ V ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_109_comp__eq__elim,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 11.90/5.92 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 11.90/5.92 => ! [V3: A] :
% 11.90/5.92 ( ( A3 @ ( B2 @ V3 ) )
% 11.90/5.92 = ( C2 @ ( D3 @ V3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_110_comp__eq__dest,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 11.90/5.92 = ( comp @ D @ B @ A @ C2 @ D3 ) )
% 11.90/5.92 => ( ( A3 @ ( B2 @ V ) )
% 11.90/5.92 = ( C2 @ ( D3 @ V ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_111_comp__assoc,axiom,
% 11.90/5.92 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H3: A > C] :
% 11.90/5.92 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H3 )
% 11.90/5.92 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_112_comp__def,axiom,
% 11.90/5.92 ! [A: $tType,C: $tType,B: $tType] :
% 11.90/5.92 ( ( comp @ B @ C @ A )
% 11.90/5.92 = ( ^ [F2: B > C,G2: A > B,X3: A] : ( F2 @ ( G2 @ X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_113_prod__induct7,axiom,
% 11.90/5.92 ! [G3: $tType,F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) )] :
% 11.90/5.92 ( ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F5 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F5 @ G3 ) @ E3 @ ( product_Pair @ F5 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 11.90/5.92 => ( P @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_114_prod__induct6,axiom,
% 11.90/5.92 ! [F5: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) )] :
% 11.90/5.92 ( ! [A5: A,B3: B,C4: C,D2: D,E3: E,F4: F5] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F5 ) ) @ C4 @ ( product_Pair @ D @ ( product_prod @ E @ F5 ) @ D2 @ ( product_Pair @ E @ F5 @ E3 @ F4 ) ) ) ) ) )
% 11.90/5.92 => ( P @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_115_prod__induct5,axiom,
% 11.90/5.92 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
% 11.90/5.92 ( ! [A5: A,B3: B,C4: C,D2: D,E3: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C4 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) )
% 11.90/5.92 => ( P @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_116_old_Oprod_Orec,axiom,
% 11.90/5.92 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
% 11.90/5.92 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 11.90/5.92 = ( F1 @ A3 @ B2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_117_timeFrame_Oelims,axiom,
% 11.90/5.92 ! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( ( heap_Time_timeFrame @ A @ X @ Xa )
% 11.90/5.92 = Y )
% 11.90/5.92 => ( ! [R2: A,H2: heap_ext @ product_unit,N4: nat] :
% 11.90/5.92 ( ( Xa
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) )
% 11.90/5.92 => ( Y
% 11.90/5.92 != ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) ) )
% 11.90/5.92 => ~ ( ( Xa
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 => ( Y
% 11.90/5.92 != ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_118_execute__bind__eq__SomeI,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,X: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y: B,H5: heap_ext @ product_unit,N2: nat] :
% 11.90/5.92 ( ( ( heap_Time_execute @ A @ F @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
% 11.90/5.92 => ( ( ( heap_Time_execute @ B @ ( G @ X ) @ H4 )
% 11.90/5.92 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N2 ) ) ) )
% 11.90/5.92 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_119_uncurry__apply,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
% 11.90/5.92 ( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 11.90/5.92 = ( F @ A3 @ B2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_120_timeFrame_Osimps_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,N: nat,R: A,H3: heap_ext @ product_unit,N2: nat] :
% 11.90/5.92 ( ( heap_Time_timeFrame @ A @ N @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N2 ) ) ) )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_121_heap__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_heap @ A )
% 11.90/5.92 = ( ^ [F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( heap_Time_Heap2 @ A @ ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ F2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_122_internal__case__prod__conv,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
% 11.90/5.92 ( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 11.90/5.92 = ( C2 @ A3 @ B2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_123_option_Osplit__sel,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 11.90/5.92 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 11.90/5.92 = ( ( ( Option
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 => ( P @ F1 ) )
% 11.90/5.92 & ( ( Option
% 11.90/5.92 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 11.90/5.92 => ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_124_option_Osplit__sel__asm,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
% 11.90/5.92 ( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 11.90/5.92 = ( ~ ( ( ( Option
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 & ~ ( P @ F1 ) )
% 11.90/5.92 | ( ( Option
% 11.90/5.92 = ( some @ A @ ( the2 @ A @ Option ) ) )
% 11.90/5.92 & ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_125_fold__atLeastAtMost__nat_Ocases,axiom,
% 11.90/5.92 ! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
% 11.90/5.92 ~ ! [F4: nat > A > A,A5: nat,B3: nat,Acc: A] :
% 11.90/5.92 ( X
% 11.90/5.92 != ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_126_one__natural_Orsp,axiom,
% 11.90/5.92 ( ( one_one @ nat )
% 11.90/5.92 = ( one_one @ nat ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_127_timeFrame_Opelims,axiom,
% 11.90/5.92 ! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( ( heap_Time_timeFrame @ A @ X @ Xa )
% 11.90/5.92 = Y )
% 11.90/5.92 => ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ Xa ) )
% 11.90/5.92 => ( ! [R2: A,H2: heap_ext @ product_unit,N4: nat] :
% 11.90/5.92 ( ( Xa
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) )
% 11.90/5.92 => ( ( Y
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) )
% 11.90/5.92 => ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N4 ) ) ) ) ) ) )
% 11.90/5.92 => ~ ( ( Xa
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 => ( ( Y
% 11.90/5.92 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
% 11.90/5.92 => ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_128_add__right__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel_semigroup_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ B2 @ A3 )
% 11.90/5.92 = ( plus_plus @ A @ C2 @ A3 ) )
% 11.90/5.92 = ( B2 = C2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_129_add__left__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel_semigroup_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ C2 ) )
% 11.90/5.92 = ( B2 = C2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_130_Heap_Oinject,axiom,
% 11.90/5.92 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.90/5.92 ( ( ( heap_Time_Heap2 @ A @ X )
% 11.90/5.92 = ( heap_Time_Heap2 @ A @ Ya ) )
% 11.90/5.92 = ( X = Ya ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_131_Heap__execute,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
% 11.90/5.92 = F ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_132_timeFrame__assoc,axiom,
% 11.90/5.92 ! [A: $tType,N: nat,N2: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N2 @ F ) )
% 11.90/5.92 = ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N2 ) @ F ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_133_add__right__imp__eq,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel_semigroup_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ B2 @ A3 )
% 11.90/5.92 = ( plus_plus @ A @ C2 @ A3 ) )
% 11.90/5.92 => ( B2 = C2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_134_add__left__imp__eq,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel_semigroup_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ C2 ) )
% 11.90/5.92 => ( B2 = C2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_135_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_semigroup_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_136_ab__semigroup__add__class_Oadd_Ocommute,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_semigroup_add @ A )
% 11.90/5.92 => ( ( plus_plus @ A )
% 11.90/5.92 = ( ^ [A7: A,B5: A] : ( plus_plus @ A @ B5 @ A7 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_137_add_Oright__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ B2 @ A3 )
% 11.90/5.92 = ( plus_plus @ A @ C2 @ A3 ) )
% 11.90/5.92 = ( B2 = C2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_138_add_Oleft__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ C2 ) )
% 11.90/5.92 = ( B2 = C2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_139_add_Oassoc,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( semigroup_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_140_group__cancel_Oadd2,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( comm_monoid_add @ A )
% 11.90/5.92 => ! [B6: A,K: A,B2: A,A3: A] :
% 11.90/5.92 ( ( B6
% 11.90/5.92 = ( plus_plus @ A @ K @ B2 ) )
% 11.90/5.92 => ( ( plus_plus @ A @ A3 @ B6 )
% 11.90/5.92 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_141_group__cancel_Oadd1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( comm_monoid_add @ A )
% 11.90/5.92 => ! [A4: A,K: A,A3: A,B2: A] :
% 11.90/5.92 ( ( A4
% 11.90/5.92 = ( plus_plus @ A @ K @ A3 ) )
% 11.90/5.92 => ( ( plus_plus @ A @ A4 @ B2 )
% 11.90/5.92 = ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_142_add__mono__thms__linordered__semiring_I4_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.92 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.92 ( ( ( I = J )
% 11.90/5.92 & ( K = L ) )
% 11.90/5.92 => ( ( plus_plus @ A @ I @ K )
% 11.90/5.92 = ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_143_Heap_Oexhaust,axiom,
% 11.90/5.92 ! [A: $tType,Y: heap_Time_Heap @ A] :
% 11.90/5.92 ~ ! [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.90/5.92 ( Y
% 11.90/5.92 != ( heap_Time_Heap2 @ A @ X2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_144_execute_Osimps,axiom,
% 11.90/5.92 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( heap_Time_Heap2 @ A @ F ) )
% 11.90/5.92 = F ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_145_option_Osimps_I5_J,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F1: B,F22: A > B,X22: A] :
% 11.90/5.92 ( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X22 ) )
% 11.90/5.92 = ( F22 @ X22 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_146_option_Osimps_I4_J,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F1: B,F22: A > B] :
% 11.90/5.92 ( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
% 11.90/5.92 = F1 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_147_effect__bindE,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit,H5: heap_ext @ product_unit,R4: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H3 @ H5 @ R4 @ N )
% 11.90/5.92 => ~ ! [H6: heap_ext @ product_unit,R2: B,N1: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ B @ F @ H3 @ H6 @ R2 @ N1 )
% 11.90/5.92 => ! [N22: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( G @ R2 ) @ H6 @ H5 @ R4 @ N22 )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_148_effect__bindI,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H5: heap_ext @ product_unit,R4: B,N2: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ F @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ( ( heap_Time_effect @ B @ ( G @ R ) @ H4 @ H5 @ R4 @ N2 )
% 11.90/5.92 => ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 @ H5 @ R4 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_149_option_Ocase__eq__if,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( case_option @ B @ A )
% 11.90/5.92 = ( ^ [F12: B,F23: A > B,Option3: option @ A] :
% 11.90/5.92 ( if @ B
% 11.90/5.92 @ ( Option3
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 @ F12
% 11.90/5.92 @ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_150_uncurry__curry__id,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 11.90/5.92 ( ( uncurry @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 11.90/5.92 = F ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_151_curry__uncurry__id,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 11.90/5.92 ( ( product_curry @ A @ B @ C @ ( uncurry @ A @ B @ C @ F ) )
% 11.90/5.92 = F ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_152_accp_Ocases,axiom,
% 11.90/5.92 ! [A: $tType,R: A > A > $o,A3: A] :
% 11.90/5.92 ( ( accp @ A @ R @ A3 )
% 11.90/5.92 => ! [Y6: A] :
% 11.90/5.92 ( ( R @ Y6 @ A3 )
% 11.90/5.92 => ( accp @ A @ R @ Y6 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_153_accp_Osimps,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( accp @ A )
% 11.90/5.92 = ( ^ [R3: A > A > $o,A7: A] :
% 11.90/5.92 ? [X3: A] :
% 11.90/5.92 ( ( A7 = X3 )
% 11.90/5.92 & ! [Y3: A] :
% 11.90/5.92 ( ( R3 @ Y3 @ X3 )
% 11.90/5.92 => ( accp @ A @ R3 @ Y3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_154_accpI,axiom,
% 11.90/5.92 ! [A: $tType,R: A > A > $o,X: A] :
% 11.90/5.92 ( ! [Y4: A] :
% 11.90/5.92 ( ( R @ Y4 @ X )
% 11.90/5.92 => ( accp @ A @ R @ Y4 ) )
% 11.90/5.92 => ( accp @ A @ R @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_155_accp__induct,axiom,
% 11.90/5.92 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 11.90/5.92 ( ( accp @ A @ R @ A3 )
% 11.90/5.92 => ( ! [X2: A] :
% 11.90/5.92 ( ( accp @ A @ R @ X2 )
% 11.90/5.92 => ( ! [Y6: A] :
% 11.90/5.92 ( ( R @ Y6 @ X2 )
% 11.90/5.92 => ( P @ Y6 ) )
% 11.90/5.92 => ( P @ X2 ) ) )
% 11.90/5.92 => ( P @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_156_accp__downward,axiom,
% 11.90/5.92 ! [A: $tType,R: A > A > $o,B2: A,A3: A] :
% 11.90/5.92 ( ( accp @ A @ R @ B2 )
% 11.90/5.92 => ( ( R @ A3 @ B2 )
% 11.90/5.92 => ( accp @ A @ R @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_157_not__accp__down,axiom,
% 11.90/5.92 ! [A: $tType,R5: A > A > $o,X: A] :
% 11.90/5.92 ( ~ ( accp @ A @ R5 @ X )
% 11.90/5.92 => ~ ! [Z3: A] :
% 11.90/5.92 ( ( R5 @ Z3 @ X )
% 11.90/5.92 => ( accp @ A @ R5 @ Z3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_158_accp__induct__rule,axiom,
% 11.90/5.92 ! [A: $tType,R: A > A > $o,A3: A,P: A > $o] :
% 11.90/5.92 ( ( accp @ A @ R @ A3 )
% 11.90/5.92 => ( ! [X2: A] :
% 11.90/5.92 ( ( accp @ A @ R @ X2 )
% 11.90/5.92 => ( ! [Y6: A] :
% 11.90/5.92 ( ( R @ Y6 @ X2 )
% 11.90/5.92 => ( P @ Y6 ) )
% 11.90/5.92 => ( P @ X2 ) ) )
% 11.90/5.92 => ( P @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_159_fun_Omap__comp,axiom,
% 11.90/5.92 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 11.90/5.92 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 11.90/5.92 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_160_comp__cong,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F3: D > A,G5: E > D,X6: E] :
% 11.90/5.92 ( ( ( F @ ( G @ X ) )
% 11.90/5.92 = ( F3 @ ( G5 @ X6 ) ) )
% 11.90/5.92 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 11.90/5.92 = ( comp @ D @ A @ E @ F3 @ G5 @ X6 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_161_curryI,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
% 11.90/5.92 ( ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 11.90/5.92 => ( product_curry @ A @ B @ $o @ F @ A3 @ B2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_162_curry__conv,axiom,
% 11.90/5.92 ! [A: $tType,C: $tType,B: $tType] :
% 11.90/5.92 ( ( product_curry @ B @ C @ A )
% 11.90/5.92 = ( ^ [F2: ( product_prod @ B @ C ) > A,A7: B,B5: C] : ( F2 @ ( product_Pair @ B @ C @ A7 @ B5 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_163_curryE,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
% 11.90/5.92 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
% 11.90/5.92 => ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_164_curryD,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
% 11.90/5.92 ( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
% 11.90/5.92 => ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_165_fold__atLeastAtMost__nat_Opinduct,axiom,
% 11.90/5.92 ! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
% 11.90/5.92 ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
% 11.90/5.92 => ( ! [F4: nat > A > A,A5: nat,B3: nat,Acc: A] :
% 11.90/5.92 ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A5 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) )
% 11.90/5.92 => ( ( ~ ( ord_less @ nat @ B3 @ A5 )
% 11.90/5.92 => ( P @ F4 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F4 @ A5 @ Acc ) ) )
% 11.90/5.92 => ( P @ F4 @ A5 @ B3 @ Acc ) ) )
% 11.90/5.92 => ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_166_in__lex__prod,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,A3: A,B2: B,A6: A,B4: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 11.90/5.92 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A6 @ B4 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
% 11.90/5.92 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A6 ) @ R )
% 11.90/5.92 | ( ( A3 = A6 )
% 11.90/5.92 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B4 ) @ S ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_167_rewriteR__comp__comp2,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H3: A > C,R1: D > B,R22: A > D,F: B > E,L: D > E] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ G @ H3 )
% 11.90/5.92 = ( comp @ D @ B @ A @ R1 @ R22 ) )
% 11.90/5.92 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 11.90/5.92 = L )
% 11.90/5.92 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H3 )
% 11.90/5.92 = ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_168_rewriteL__comp__comp2,axiom,
% 11.90/5.92 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H3: E > A,R: E > D] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ F @ G )
% 11.90/5.92 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 11.90/5.92 => ( ( ( comp @ A @ D @ E @ L2 @ H3 )
% 11.90/5.92 = R )
% 11.90/5.92 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H3 ) )
% 11.90/5.92 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_169_rewriteR__comp__comp,axiom,
% 11.90/5.92 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H3: A > C,R: A > B,F: B > D] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ G @ H3 )
% 11.90/5.92 = R )
% 11.90/5.92 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H3 )
% 11.90/5.92 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_170_rewriteL__comp__comp,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H3: D > A] :
% 11.90/5.92 ( ( ( comp @ C @ B @ A @ F @ G )
% 11.90/5.92 = L )
% 11.90/5.92 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H3 ) )
% 11.90/5.92 = ( comp @ A @ B @ D @ L @ H3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_171_type__copy__map__cong0,axiom,
% 11.90/5.92 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M3: B > A,G: C > B,X: C,N6: D > A,H3: C > D,F: A > E] :
% 11.90/5.92 ( ( ( M3 @ ( G @ X ) )
% 11.90/5.92 = ( N6 @ ( H3 @ X ) ) )
% 11.90/5.92 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M3 ) @ G @ X )
% 11.90/5.92 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N6 ) @ H3 @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_172_comp__apply__eq,axiom,
% 11.90/5.92 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H3: D > A,K: C > D] :
% 11.90/5.92 ( ( ( F @ ( G @ X ) )
% 11.90/5.92 = ( H3 @ ( K @ X ) ) )
% 11.90/5.92 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 11.90/5.92 = ( comp @ D @ A @ C @ H3 @ K @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_173_effect__returnE,axiom,
% 11.90/5.92 ! [A: $tType,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ~ ( ( R = X )
% 11.90/5.92 => ( ( H4 = H3 )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_174_effect__returnI,axiom,
% 11.90/5.92 ! [A: $tType,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A] :
% 11.90/5.92 ( ( H3 = H4 )
% 11.90/5.92 => ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H3 @ H4 @ X @ ( one_one @ nat ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_175_add_Oright__commute,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_semigroup_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.92 = ( plus_plus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_176_add__less__cancel__right,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.92 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_177_add__less__cancel__left,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.92 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_178_fold__if__return,axiom,
% 11.90/5.92 ! [A: $tType,B2: $o,C2: A,D3: A] :
% 11.90/5.92 ( ( B2
% 11.90/5.92 => ( ( heap_Time_return @ A @ C2 )
% 11.90/5.92 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D3 ) ) ) )
% 11.90/5.92 & ( ~ B2
% 11.90/5.92 => ( ( heap_Time_return @ A @ D3 )
% 11.90/5.92 = ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_179_add__less__imp__less__right,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.92 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.92 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_180_add__less__imp__less__left,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.92 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 11.90/5.92 => ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_181_add__strict__right__mono,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_182_add__strict__left__mono,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_183_add__strict__mono,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( strict9044650504122735259up_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.92 => ( ( ord_less @ A @ C2 @ D3 )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_184_add__mono__thms__linordered__field_I1_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.92 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.92 ( ( ( ord_less @ A @ I @ J )
% 11.90/5.92 & ( K = L ) )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_185_add__mono__thms__linordered__field_I2_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.92 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.92 ( ( ( I = J )
% 11.90/5.92 & ( ord_less @ A @ K @ L ) )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_186_add__mono__thms__linordered__field_I5_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.92 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.92 ( ( ( ord_less @ A @ I @ J )
% 11.90/5.92 & ( ord_less @ A @ K @ L ) )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_187_success__returnI,axiom,
% 11.90/5.92 ! [A: $tType,X: A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_188_fails__return,axiom,
% 11.90/5.92 ! [A: $tType,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ~ ( time_fails @ A @ ( heap_Time_return @ A @ X ) @ H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_189_add_Oright__assoc,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_semigroup_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_190_nat__add__left__cancel__less,axiom,
% 11.90/5.92 ! [K: nat,M2: nat,N: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 11.90/5.92 = ( ord_less @ nat @ M2 @ N ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_191_less__option__Some,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: A,Y: A] :
% 11.90/5.92 ( ( ord_less @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 11.90/5.92 = ( ord_less @ A @ X @ Y ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_192_fold__atLeastAtMost__nat_Opelims,axiom,
% 11.90/5.92 ! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
% 11.90/5.92 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
% 11.90/5.92 = Y )
% 11.90/5.92 => ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
% 11.90/5.92 => ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
% 11.90/5.92 => ( Y = Xc ) )
% 11.90/5.92 & ( ~ ( ord_less @ nat @ Xb @ Xa )
% 11.90/5.92 => ( Y
% 11.90/5.92 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
% 11.90/5.92 => ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_193_fold__atLeastAtMost__nat_Opsimps,axiom,
% 11.90/5.92 ! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
% 11.90/5.92 ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
% 11.90/5.92 => ( ( ( ord_less @ nat @ B2 @ A3 )
% 11.90/5.92 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 11.90/5.92 = Acc2 ) )
% 11.90/5.92 & ( ~ ( ord_less @ nat @ B2 @ A3 )
% 11.90/5.92 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 11.90/5.92 = ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_194_in__measure,axiom,
% 11.90/5.92 ! [A: $tType,X: A,Y: A,F: A > nat] :
% 11.90/5.92 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F ) )
% 11.90/5.92 = ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_195_less__add__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_semidom @ A )
% 11.90/5.92 => ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_196_add__mono1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord181362715937106298miring @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_197_lift__collapse,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( heap_Time_lift @ B @ A )
% 11.90/5.92 = ( ^ [F2: B > A,X3: B] : ( heap_Time_return @ A @ ( F2 @ X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_198_add__lessD1,axiom,
% 11.90/5.92 ! [I: nat,J: nat,K: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 11.90/5.92 => ( ord_less @ nat @ I @ K ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_199_add__less__mono,axiom,
% 11.90/5.92 ! [I: nat,J: nat,K: nat,L: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ I @ J )
% 11.90/5.92 => ( ( ord_less @ nat @ K @ L )
% 11.90/5.92 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_200_not__add__less1,axiom,
% 11.90/5.92 ! [I: nat,J: nat] :
% 11.90/5.92 ~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_201_less__option__None,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: option @ A] :
% 11.90/5.92 ~ ( ord_less @ ( option @ A ) @ X @ ( none @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_202_less__option__None__Some__code,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_203_less__option__None__Some,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: A] : ( ord_less @ ( option @ A ) @ ( none @ A ) @ ( some @ A @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_204_less__option__None__is__Some,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: option @ A] :
% 11.90/5.92 ( ( ord_less @ ( option @ A ) @ ( none @ A ) @ X )
% 11.90/5.92 => ? [Z3: A] :
% 11.90/5.92 ( X
% 11.90/5.92 = ( some @ A @ Z3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_205_lift__def,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( heap_Time_lift @ A @ B )
% 11.90/5.92 = ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_206_fold__atLeastAtMost__nat_Osimps,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( set_fo6178422350223883121st_nat @ A )
% 11.90/5.92 = ( ^ [F2: nat > A > A,A7: nat,B5: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A7 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A7 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A7 @ Acc3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_207_fold__atLeastAtMost__nat_Oelims,axiom,
% 11.90/5.92 ! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
% 11.90/5.92 ( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
% 11.90/5.92 = Y )
% 11.90/5.92 => ( ( ( ord_less @ nat @ Xb @ Xa )
% 11.90/5.92 => ( Y = Xc ) )
% 11.90/5.92 & ( ~ ( ord_less @ nat @ Xb @ Xa )
% 11.90/5.92 => ( Y
% 11.90/5.92 = ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_208_less__add__eq__less,axiom,
% 11.90/5.92 ! [K: nat,L: nat,M2: nat,N: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ K @ L )
% 11.90/5.92 => ( ( ( plus_plus @ nat @ M2 @ L )
% 11.90/5.92 = ( plus_plus @ nat @ K @ N ) )
% 11.90/5.92 => ( ord_less @ nat @ M2 @ N ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_209_trans__less__add2,axiom,
% 11.90/5.92 ! [I: nat,J: nat,M2: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ I @ J )
% 11.90/5.92 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_210_trans__less__add1,axiom,
% 11.90/5.92 ! [I: nat,J: nat,M2: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ I @ J )
% 11.90/5.92 => ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_211_add__less__mono1,axiom,
% 11.90/5.92 ! [I: nat,J: nat,K: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ I @ J )
% 11.90/5.92 => ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_212_not__add__less2,axiom,
% 11.90/5.92 ! [J: nat,I: nat] :
% 11.90/5.92 ~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_213_less__numeral__extra_I4_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord181362715937106298miring @ A )
% 11.90/5.92 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_214_mlex__iff,axiom,
% 11.90/5.92 ! [A: $tType,X: A,Y: A,F: A > nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.92 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) )
% 11.90/5.92 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 11.90/5.92 | ( ( ( F @ X )
% 11.90/5.92 = ( F @ Y ) )
% 11.90/5.92 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_215_mlex__less,axiom,
% 11.90/5.92 ! [A: $tType,F: A > nat,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.92 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 11.90/5.92 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_216_pairself_Opelims,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
% 11.90/5.92 ( ( ( pairself @ A @ B @ X @ Xa )
% 11.90/5.92 = Y )
% 11.90/5.92 => ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ Xa ) )
% 11.90/5.92 => ~ ! [A5: A,B3: A] :
% 11.90/5.92 ( ( Xa
% 11.90/5.92 = ( product_Pair @ A @ A @ A5 @ B3 ) )
% 11.90/5.92 => ( ( Y
% 11.90/5.92 = ( product_Pair @ B @ B @ ( X @ A5 ) @ ( X @ B3 ) ) )
% 11.90/5.92 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X @ ( product_Pair @ A @ A @ A5 @ B3 ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_217_execute__return,axiom,
% 11.90/5.92 ! [A: $tType,X: A] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
% 11.90/5.92 = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_218_same__fstI,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,P: A > $o,X: A,Y7: B,Y: B,R5: A > ( set @ ( product_prod @ B @ B ) )] :
% 11.90/5.92 ( ( P @ X )
% 11.90/5.92 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R5 @ X ) )
% 11.90/5.92 => ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P @ R5 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_219_zero__less__two,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord181362715937106298miring @ A )
% 11.90/5.92 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_220_is__num__normalize_I1_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_221_add_Osafe__commute,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_semigroup_add @ A )
% 11.90/5.92 => ! [X: A,Y: A,A3: A,B2: A] :
% 11.90/5.92 ( ( syntax7388354845996824322omatch @ A @ A @ ( plus_plus @ A @ X @ Y ) @ A3 )
% 11.90/5.92 => ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( plus_plus @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_222_bijective__def,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType] :
% 11.90/5.92 ( ( bijective @ A @ B )
% 11.90/5.92 = ( ^ [R6: set @ ( product_prod @ A @ B )] :
% 11.90/5.92 ( ! [X3: A,Y3: B,Z5: B] :
% 11.90/5.92 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 )
% 11.90/5.92 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R6 ) )
% 11.90/5.92 => ( Y3 = Z5 ) )
% 11.90/5.92 & ! [X3: A,Y3: A,Z5: B] :
% 11.90/5.92 ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R6 )
% 11.90/5.92 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R6 ) )
% 11.90/5.92 => ( X3 = Y3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_223_ssubst__Pair__rhs,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,R: A,S: B,R5: set @ ( product_prod @ A @ B ),S3: B] :
% 11.90/5.92 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R5 )
% 11.90/5.92 => ( ( S3 = S )
% 11.90/5.92 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R5 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_224_add__is__0,axiom,
% 11.90/5.92 ! [M2: nat,N: nat] :
% 11.90/5.92 ( ( ( plus_plus @ nat @ M2 @ N )
% 11.90/5.92 = ( zero_zero @ nat ) )
% 11.90/5.92 = ( ( M2
% 11.90/5.92 = ( zero_zero @ nat ) )
% 11.90/5.92 & ( N
% 11.90/5.92 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_225_Nat_Oadd__0__right,axiom,
% 11.90/5.92 ! [M2: nat] :
% 11.90/5.92 ( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
% 11.90/5.92 = M2 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_226_timeFrame__zero,axiom,
% 11.90/5.92 ! [A: $tType,H3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H3 )
% 11.90/5.92 = H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_227_Heap__Time__Monad_Obind__bind,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
% 11.90/5.92 ( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
% 11.90/5.92 = ( heap_Time_bind @ A @ B @ F
% 11.90/5.92 @ ^ [X3: A] : ( heap_Time_bind @ C @ B @ ( G @ X3 ) @ K ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_228_not__gr__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [N: A] :
% 11.90/5.92 ( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
% 11.90/5.92 = ( N
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_229_add__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( monoid_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_230_zero__eq__add__iff__both__eq__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [X: A,Y: A] :
% 11.90/5.92 ( ( ( zero_zero @ A )
% 11.90/5.92 = ( plus_plus @ A @ X @ Y ) )
% 11.90/5.92 = ( ( X
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 & ( Y
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_231_add__eq__0__iff__both__eq__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [X: A,Y: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ X @ Y )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 = ( ( X
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 & ( Y
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_232_add__cancel__right__right,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 = ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( B2
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_233_add__cancel__right__left,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 = ( plus_plus @ A @ B2 @ A3 ) )
% 11.90/5.92 = ( B2
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_234_add__cancel__left__right,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = A3 )
% 11.90/5.92 = ( B2
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_235_add__cancel__left__left,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ B2 @ A3 )
% 11.90/5.92 = A3 )
% 11.90/5.92 = ( B2
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_236_double__zero__sym,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( zero_zero @ A )
% 11.90/5.92 = ( plus_plus @ A @ A3 @ A3 ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_237_add_Oright__neutral,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( monoid_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_238_add__gr__0,axiom,
% 11.90/5.92 ! [M2: nat,N: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.92 = ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 11.90/5.92 | ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_239_less__one,axiom,
% 11.90/5.92 ! [N: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ N @ ( one_one @ nat ) )
% 11.90/5.92 = ( N
% 11.90/5.92 = ( zero_zero @ nat ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_240_some__opt__eq__trivial,axiom,
% 11.90/5.92 ! [A: $tType,X: A] :
% 11.90/5.92 ( ( eps_Opt @ A
% 11.90/5.92 @ ^ [Y3: A] : ( Y3 = X ) )
% 11.90/5.92 = ( some @ A @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_241_zero__less__double__add__iff__zero__less__single__add,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 11.90/5.92 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_242_double__add__less__zero__iff__single__add__less__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_243_less__add__same__cancel2,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 11.90/5.92 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_244_less__add__same__cancel1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_245_add__less__same__cancel2,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_246_add__less__same__cancel1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_247_some__opt__false__trivial,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( eps_Opt @ A
% 11.90/5.92 @ ^ [Uu: A] : $false )
% 11.90/5.92 = ( none @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_248_zero__reorient,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( zero @ A )
% 11.90/5.92 => ! [X: A] :
% 11.90/5.92 ( ( ( zero_zero @ A )
% 11.90/5.92 = X )
% 11.90/5.92 = ( X
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_249_fun_Omap__ident,axiom,
% 11.90/5.92 ! [A: $tType,D: $tType,T2: D > A] :
% 11.90/5.92 ( ( comp @ A @ A @ D
% 11.90/5.92 @ ^ [X3: A] : X3
% 11.90/5.92 @ T2 )
% 11.90/5.92 = T2 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_250_option_Odisc__eq__case_I2_J,axiom,
% 11.90/5.92 ! [A: $tType,Option: option @ A] :
% 11.90/5.92 ( ( Option
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 = ( case_option @ $o @ A @ $false
% 11.90/5.92 @ ^ [Uu: A] : $true
% 11.90/5.92 @ Option ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_251_option_Odisc__eq__case_I1_J,axiom,
% 11.90/5.92 ! [A: $tType,Option: option @ A] :
% 11.90/5.92 ( ( Option
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 = ( case_option @ $o @ A @ $true
% 11.90/5.92 @ ^ [Uu: A] : $false
% 11.90/5.92 @ Option ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_252_option_Ocase__distrib,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,H3: B > C,F1: B,F22: A > B,Option: option @ A] :
% 11.90/5.92 ( ( H3 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
% 11.90/5.92 = ( case_option @ C @ A @ ( H3 @ F1 )
% 11.90/5.92 @ ^ [X3: A] : ( H3 @ ( F22 @ X3 ) )
% 11.90/5.92 @ Option ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_253_effect__LetI,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: B,N: nat] :
% 11.90/5.92 ( ( X = T2 )
% 11.90/5.92 => ( ( heap_Time_effect @ B @ ( F @ X ) @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ( heap_Time_effect @ B @ ( F @ T2 ) @ H3 @ H4 @ R @ N ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_254_distrib__if__bind,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D3: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
% 11.90/5.92 ( ( B2
% 11.90/5.92 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F )
% 11.90/5.92 = ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
% 11.90/5.92 & ( ~ B2
% 11.90/5.92 => ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D3 ) @ F )
% 11.90/5.92 = ( heap_Time_bind @ B @ A @ D3 @ F ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_255_success__LetI,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( X = T2 )
% 11.90/5.92 => ( ( heap_Time_success @ B @ ( F @ X ) @ H3 )
% 11.90/5.92 => ( heap_Time_success @ B @ ( F @ T2 ) @ H3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_256_curry__K,axiom,
% 11.90/5.92 ! [B: $tType,C: $tType,A: $tType,C2: C] :
% 11.90/5.92 ( ( product_curry @ A @ B @ C
% 11.90/5.92 @ ^ [X3: product_prod @ A @ B] : C2 )
% 11.90/5.92 = ( ^ [X3: A,Y3: B] : C2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_257_case__optionE,axiom,
% 11.90/5.92 ! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
% 11.90/5.92 ( ( case_option @ $o @ A @ P @ Q @ X )
% 11.90/5.92 => ( ( ( X
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 => ~ P )
% 11.90/5.92 => ~ ! [Y4: A] :
% 11.90/5.92 ( ( X
% 11.90/5.92 = ( some @ A @ Y4 ) )
% 11.90/5.92 => ~ ( Q @ Y4 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_258_curry__def,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/5.92 ( ( product_curry @ A @ B @ C )
% 11.90/5.92 = ( ^ [C3: ( product_prod @ A @ B ) > C,X3: A,Y3: B] : ( C3 @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_259_combine__options__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( combine_options @ A )
% 11.90/5.92 = ( ^ [F2: A > A > A,X3: option @ A,Y3: option @ A] :
% 11.90/5.92 ( case_option @ ( option @ A ) @ A @ Y3
% 11.90/5.92 @ ^ [Z5: A] :
% 11.90/5.92 ( case_option @ ( option @ A ) @ A @ ( some @ A @ Z5 )
% 11.90/5.92 @ ^ [Aa2: A] : ( some @ A @ ( F2 @ Z5 @ Aa2 ) )
% 11.90/5.92 @ Y3 )
% 11.90/5.92 @ X3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_260_less__option__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ( ( ord_less @ ( option @ A ) )
% 11.90/5.92 = ( ^ [X3: option @ A] :
% 11.90/5.92 ( case_option @ $o @ A @ $false
% 11.90/5.92 @ ^ [Y3: A] :
% 11.90/5.92 ( case_option @ $o @ A @ $true
% 11.90/5.92 @ ^ [Z5: A] : ( ord_less @ A @ Z5 @ Y3 )
% 11.90/5.92 @ X3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_261_zero__less__iff__neq__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [N: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ N )
% 11.90/5.92 = ( N
% 11.90/5.92 != ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_262_gr__implies__not__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [M2: A,N: A] :
% 11.90/5.92 ( ( ord_less @ A @ M2 @ N )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_263_not__less__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [N: A] :
% 11.90/5.92 ~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_264_gr__zeroI,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [N: A] :
% 11.90/5.92 ( ( N
% 11.90/5.92 != ( zero_zero @ A ) )
% 11.90/5.92 => ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_265_add_Ogroup__left__neutral,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_266_add_Ocomm__neutral,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( comm_monoid_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_267_comm__monoid__add__class_Oadd__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( comm_monoid_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_268_zero__neq__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( zero_neq_one @ A )
% 11.90/5.92 => ( ( zero_zero @ A )
% 11.90/5.92 != ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_269_array__make,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ( ( array_new @ A )
% 11.90/5.92 = ( ^ [N5: nat,X3: A] :
% 11.90/5.92 ( array_make @ A @ N5
% 11.90/5.92 @ ^ [Uu: nat] : X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_270_plus__nat_Oadd__0,axiom,
% 11.90/5.92 ! [N: nat] :
% 11.90/5.92 ( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.92 = N ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_271_add__eq__self__zero,axiom,
% 11.90/5.92 ! [M2: nat,N: nat] :
% 11.90/5.92 ( ( ( plus_plus @ nat @ M2 @ N )
% 11.90/5.92 = M2 )
% 11.90/5.92 => ( N
% 11.90/5.92 = ( zero_zero @ nat ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_272_less__numeral__extra_I1_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord181362715937106298miring @ A )
% 11.90/5.92 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_273_ureturn__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_ureturn @ A )
% 11.90/5.92 = ( ^ [X3: A] :
% 11.90/5.92 ( heap_Time_heap @ A
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_274_raise__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_raise @ A )
% 11.90/5.92 = ( ^ [S4: list @ char] :
% 11.90/5.92 ( heap_Time_Heap2 @ A
% 11.90/5.92 @ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_275_bind__lift,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
% 11.90/5.92 ( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
% 11.90/5.92 = ( heap_Time_bind @ B @ A @ F
% 11.90/5.92 @ ^ [X3: B] : ( heap_Time_return @ A @ ( G @ X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_276_guard__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_guard @ A )
% 11.90/5.92 = ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.90/5.92 ( heap_Time_Heap2 @ A
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_277_pos__add__strict,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( strict7427464778891057005id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 => ( ( ord_less @ A @ B2 @ C2 )
% 11.90/5.92 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_278_canonically__ordered__monoid__add__class_OlessE,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.92 => ~ ! [C4: A] :
% 11.90/5.92 ( ( B2
% 11.90/5.92 = ( plus_plus @ A @ A3 @ C4 ) )
% 11.90/5.92 => ( C4
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_279_add__pos__pos,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.92 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_280_add__neg__neg,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.92 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 11.90/5.92 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_281_add__less__zeroD,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [X: A,Y: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
% 11.90/5.92 => ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
% 11.90/5.92 | ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_282_zero__less__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( zero_less_one @ A )
% 11.90/5.92 => ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_283_not__one__less__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord181362715937106298miring @ A )
% 11.90/5.92 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_284_execute__ureturn,axiom,
% 11.90/5.92 ! [A: $tType,X: A] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
% 11.90/5.92 = ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_285_less__imp__add__positive,axiom,
% 11.90/5.92 ! [I: nat,J: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ I @ J )
% 11.90/5.92 => ? [K5: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K5 )
% 11.90/5.92 & ( ( plus_plus @ nat @ I @ K5 )
% 11.90/5.92 = J ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_286_pairself_Osimps,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F: A > B,A3: A,B2: A] :
% 11.90/5.92 ( ( pairself @ A @ B @ F @ ( product_Pair @ A @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_287_pairself_Oelims,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
% 11.90/5.92 ( ( ( pairself @ A @ B @ X @ Xa )
% 11.90/5.92 = Y )
% 11.90/5.92 => ~ ! [A5: A,B3: A] :
% 11.90/5.92 ( ( Xa
% 11.90/5.92 = ( product_Pair @ A @ A @ A5 @ B3 ) )
% 11.90/5.92 => ( Y
% 11.90/5.92 != ( product_Pair @ B @ B @ ( X @ A5 ) @ ( X @ B3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_288_return__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_return @ A )
% 11.90/5.92 = ( ^ [X3: A] :
% 11.90/5.92 ( heap_Time_heap @ A
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_289_effect__ureturnI,axiom,
% 11.90/5.92 ! [A: $tType,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A] :
% 11.90/5.92 ( ( H3 = H4 )
% 11.90/5.92 => ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 @ H4 @ X @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_290_effect__ureturnE,axiom,
% 11.90/5.92 ! [A: $tType,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H3 @ H4 @ R @ N )
% 11.90/5.92 => ~ ( ( R = X )
% 11.90/5.92 => ( ( H4 = H3 )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_291_tap__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap_Time_tap @ A )
% 11.90/5.92 = ( ^ [F2: ( heap_ext @ product_unit ) > A] :
% 11.90/5.92 ( heap_Time_Heap2 @ A
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F2 @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_292_double__eq__0__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ A3 )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_293_return__bind,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
% 11.90/5.92 ( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F )
% 11.90/5.92 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 11.90/5.92 @ ^ [Uu: product_unit] : ( F @ X ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_294_bind__return,axiom,
% 11.90/5.92 ! [A: $tType,F: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
% 11.90/5.92 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 11.90/5.92 @ ^ [Uu: product_unit] : F ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_295_add__0__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( semiri1453513574482234551roduct @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( B2
% 11.90/5.92 = ( plus_plus @ A @ B2 @ A3 ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_296_verit__sum__simplify,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_297_dbl__inc__simps_I2_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_298_disjE__realizer2,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,P: $o,Q: A > $o,X: option @ A,R5: B > $o,F: B,G: A > B] :
% 11.90/5.92 ( ( case_option @ $o @ A @ P @ Q @ X )
% 11.90/5.92 => ( ( P
% 11.90/5.92 => ( R5 @ F ) )
% 11.90/5.92 => ( ! [Q2: A] :
% 11.90/5.92 ( ( Q @ Q2 )
% 11.90/5.92 => ( R5 @ ( G @ Q2 ) ) )
% 11.90/5.92 => ( R5 @ ( case_option @ B @ A @ F @ G @ X ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_299_set__to__map__def,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( set_to_map @ B @ A )
% 11.90/5.92 = ( ^ [S5: set @ ( product_prod @ B @ A ),K4: B] :
% 11.90/5.92 ( eps_Opt @ A
% 11.90/5.92 @ ^ [V2: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K4 @ V2 ) @ S5 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_300_K__record__comp,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
% 11.90/5.92 ( ( comp @ C @ B @ A
% 11.90/5.92 @ ^ [X3: C] : C2
% 11.90/5.92 @ F )
% 11.90/5.92 = ( ^ [X3: A] : C2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_301_pred__equals__eq2,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.92 ( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 ) )
% 11.90/5.92 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) ) )
% 11.90/5.92 = ( R5 = S2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_302_dbl__inc__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( neg_numeral_dbl_inc @ A )
% 11.90/5.92 = ( ^ [X3: A] : ( plus_plus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_303_zero__natural_Orsp,axiom,
% 11.90/5.92 ( ( zero_zero @ nat )
% 11.90/5.92 = ( zero_zero @ nat ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_304_fails__wait,axiom,
% 11.90/5.92 ! [X: nat,H3: heap_ext @ product_unit] :
% 11.90/5.92 ~ ( time_fails @ product_unit @ ( heap_Time_wait @ X ) @ H3 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_305_Euclid__induct,axiom,
% 11.90/5.92 ! [P: nat > nat > $o,A3: nat,B2: nat] :
% 11.90/5.92 ( ! [A5: nat,B3: nat] :
% 11.90/5.92 ( ( P @ A5 @ B3 )
% 11.90/5.92 = ( P @ B3 @ A5 ) )
% 11.90/5.92 => ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
% 11.90/5.92 => ( ! [A5: nat,B3: nat] :
% 11.90/5.92 ( ( P @ A5 @ B3 )
% 11.90/5.92 => ( P @ A5 @ ( plus_plus @ nat @ A5 @ B3 ) ) )
% 11.90/5.92 => ( P @ A3 @ B2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_306_lookup__chain,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType] :
% 11.90/5.92 ( ( heap @ B )
% 11.90/5.92 => ! [R: ref @ B,F: heap_Time_Heap @ A] :
% 11.90/5.92 ( ( heap_Time_bind @ B @ A @ ( ref_lookup @ B @ R )
% 11.90/5.92 @ ^ [Uu: B] : F )
% 11.90/5.92 = ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
% 11.90/5.92 @ ^ [Uu: product_unit] : F ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_307_Ints__odd__less__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_308_num__of__nat__plus__distrib,axiom,
% 11.90/5.92 ! [M2: nat,N: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 11.90/5.92 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.92 => ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.92 = ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_309_divides__aux__eq,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.92 => ! [Q3: A,R: A] :
% 11.90/5.92 ( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q3 @ R ) )
% 11.90/5.92 = ( R
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_310_sgn__pos,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 => ( ( sgn_sgn @ A @ A3 )
% 11.90/5.92 = ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_311_abs__add__one__gt__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_312_map__option__case,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( map_option @ B @ A )
% 11.90/5.92 = ( ^ [F2: B > A] :
% 11.90/5.92 ( case_option @ ( option @ A ) @ B @ ( none @ A )
% 11.90/5.92 @ ^ [X3: B] : ( some @ A @ ( F2 @ X3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_313_add__neg__numeral__special_I7_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_314_add__neg__numeral__special_I8_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_315_dbl__dec__simps_I3_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_316_neg__equal__iff__equal,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ( uminus_uminus @ A @ A3 )
% 11.90/5.92 = ( uminus_uminus @ A @ B2 ) )
% 11.90/5.92 = ( A3 = B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_317_add_Oinverse__inverse,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = A3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_318_abs__idempotent,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
% 11.90/5.92 = ( abs_abs @ A @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_319_neg__equal__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( uminus_uminus @ A @ A3 )
% 11.90/5.92 = A3 )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_320_equal__neg__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 = ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_321_neg__equal__0__iff__equal,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( uminus_uminus @ A @ A3 )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_322_neg__0__equal__iff__equal,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( zero_zero @ A )
% 11.90/5.92 = ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( ( zero_zero @ A )
% 11.90/5.92 = A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_323_add_Oinverse__neutral,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_324_neg__less__iff__less,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_325_minus__add__distrib,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_326_minus__add__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = B2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_327_add__minus__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
% 11.90/5.92 = B2 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_328_abs__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ( ( abs_abs @ A @ ( zero_zero @ A ) )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_329_abs__eq__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( abs_abs @ A @ A3 )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_330_abs__0__eq,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( zero_zero @ A )
% 11.90/5.92 = ( abs_abs @ A @ A3 ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_331_abs__add__abs,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
% 11.90/5.92 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_332_abs__1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( idom_abs_sgn @ A )
% 11.90/5.92 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_333_abs__minus__cancel,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( abs_abs @ A @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_334_sgn__1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( idom_abs_sgn @ A )
% 11.90/5.92 => ( ( sgn_sgn @ A @ ( one_one @ A ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_335_map__option__eq__Some,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: B > A,Xo: option @ B,Y: A] :
% 11.90/5.92 ( ( ( map_option @ B @ A @ F @ Xo )
% 11.90/5.92 = ( some @ A @ Y ) )
% 11.90/5.92 = ( ? [Z5: B] :
% 11.90/5.92 ( ( Xo
% 11.90/5.92 = ( some @ B @ Z5 ) )
% 11.90/5.92 & ( ( F @ Z5 )
% 11.90/5.92 = Y ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_336_None__eq__map__option__iff,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: B > A,X: option @ B] :
% 11.90/5.92 ( ( ( none @ A )
% 11.90/5.92 = ( map_option @ B @ A @ F @ X ) )
% 11.90/5.92 = ( X
% 11.90/5.92 = ( none @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_337_map__option__is__None,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: B > A,Opt: option @ B] :
% 11.90/5.92 ( ( ( map_option @ B @ A @ F @ Opt )
% 11.90/5.92 = ( none @ A ) )
% 11.90/5.92 = ( Opt
% 11.90/5.92 = ( none @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_338_option_Omap__disc__iff,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F: A > B,A3: option @ A] :
% 11.90/5.92 ( ( ( map_option @ A @ B @ F @ A3 )
% 11.90/5.92 = ( none @ B ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( none @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_339_neg__less__0__iff__less,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.92 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_340_neg__0__less__iff__less,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_341_neg__less__pos,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 11.90/5.92 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_342_less__neg__neg,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_343_ab__left__minus,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_344_add_Oright__inverse,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_345_zero__less__abs__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 != ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_346_abs__neg__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_347_case__map__option,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,C: $tType,G: A,H3: B > A,F: C > B,X: option @ C] :
% 11.90/5.92 ( ( case_option @ A @ B @ G @ H3 @ ( map_option @ C @ B @ F @ X ) )
% 11.90/5.92 = ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H3 @ F ) @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_348_dbl__inc__simps_I4_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_349_abs__sgn__eq__1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 != ( zero_zero @ A ) )
% 11.90/5.92 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_350_dbl__dec__simps_I2_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
% 11.90/5.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_351_sgn__neg,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.92 => ( ( sgn_sgn @ A @ A3 )
% 11.90/5.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_352_minus__equation__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ( uminus_uminus @ A @ A3 )
% 11.90/5.92 = B2 )
% 11.90/5.92 = ( ( uminus_uminus @ A @ B2 )
% 11.90/5.92 = A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_353_equation__minus__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 = ( uminus_uminus @ A @ B2 ) )
% 11.90/5.92 = ( B2
% 11.90/5.92 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_354_same__sgn__abs__add,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ( sgn_sgn @ A @ B2 )
% 11.90/5.92 = ( sgn_sgn @ A @ A3 ) )
% 11.90/5.92 => ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_355_sgn__minus__1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( idom_abs_sgn @ A )
% 11.90/5.92 => ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_356_option_Omap__ident,axiom,
% 11.90/5.92 ! [A: $tType,T2: option @ A] :
% 11.90/5.92 ( ( map_option @ A @ A
% 11.90/5.92 @ ^ [X3: A] : X3
% 11.90/5.92 @ T2 )
% 11.90/5.92 = T2 ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_357_abs__of__neg,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.92 => ( ( abs_abs @ A @ A3 )
% 11.90/5.92 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_358_abs__sgn__eq,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( A3
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 11.90/5.92 = ( zero_zero @ A ) ) )
% 11.90/5.92 & ( ( A3
% 11.90/5.92 != ( zero_zero @ A ) )
% 11.90/5.92 => ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_359_abs__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ( ( abs_abs @ A @ ( one_one @ A ) )
% 11.90/5.92 = ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_360_map__option_Ocompositionality,axiom,
% 11.90/5.92 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Option: option @ A] :
% 11.90/5.92 ( ( map_option @ B @ C @ F @ ( map_option @ A @ B @ G @ Option ) )
% 11.90/5.92 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Option ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_361_option_Omap__comp,axiom,
% 11.90/5.92 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: option @ A] :
% 11.90/5.92 ( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F @ V ) )
% 11.90/5.92 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_362_map__option_Ocomp,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 11.90/5.92 ( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F ) @ ( map_option @ A @ B @ G ) )
% 11.90/5.92 = ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_363_minus__less__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 11.90/5.92 = ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_364_less__minus__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 11.90/5.92 = ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_365_is__num__normalize_I8_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( neg_numeral @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_366_add_Oinverse__distrib__swap,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_367_group__cancel_Oneg1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_group_add @ A )
% 11.90/5.92 => ! [A4: A,K: A,A3: A] :
% 11.90/5.92 ( ( A4
% 11.90/5.92 = ( plus_plus @ A @ K @ A3 ) )
% 11.90/5.92 => ( ( uminus_uminus @ A @ A4 )
% 11.90/5.92 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_368_one__neq__neg__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ring_char_0 @ A )
% 11.90/5.92 => ( ( one_one @ A )
% 11.90/5.92 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_369_map__option__cong,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > B,G: A > B] :
% 11.90/5.92 ( ( X = Y )
% 11.90/5.92 => ( ! [A5: A] :
% 11.90/5.92 ( ( Y
% 11.90/5.92 = ( some @ A @ A5 ) )
% 11.90/5.92 => ( ( F @ A5 )
% 11.90/5.92 = ( G @ A5 ) ) )
% 11.90/5.92 => ( ( map_option @ A @ B @ F @ X )
% 11.90/5.92 = ( map_option @ A @ B @ G @ Y ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_370_option_Osimps_I9_J,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,F: A > B,X22: A] :
% 11.90/5.92 ( ( map_option @ A @ B @ F @ ( some @ A @ X22 ) )
% 11.90/5.92 = ( some @ B @ ( F @ X22 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_371_option_Osimps_I8_J,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType,F: A > B] :
% 11.90/5.92 ( ( map_option @ A @ B @ F @ ( none @ A ) )
% 11.90/5.92 = ( none @ B ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_372_same__sgn__sgn__add,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ( sgn_sgn @ A @ B2 )
% 11.90/5.92 = ( sgn_sgn @ A @ A3 ) )
% 11.90/5.92 => ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( sgn_sgn @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_373_Ints__add,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ring_1 @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_374_success__lookupI,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [R: ref @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( ref_lookup @ A @ R ) @ H3 ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_375_Ints__1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ring_1 @ A )
% 11.90/5.92 => ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_376_Ints__nonzero__abs__less1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [X: A] :
% 11.90/5.92 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
% 11.90/5.92 => ( X
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_377_sgn__if,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ( ( sgn_sgn @ A )
% 11.90/5.92 = ( ^ [X3: A] :
% 11.90/5.92 ( if @ A
% 11.90/5.92 @ ( X3
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 @ ( zero_zero @ A )
% 11.90/5.92 @ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X3 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_378_sgn__1__neg,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( sgn_sgn @ A @ A3 )
% 11.90/5.92 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.92 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_379_abs__not__less__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_380_abs__of__pos,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.92 => ( ( abs_abs @ A @ A3 )
% 11.90/5.92 = A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_381_option_Orec__o__map,axiom,
% 11.90/5.92 ! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F: A > B] :
% 11.90/5.92 ( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F ) )
% 11.90/5.92 = ( rec_option @ C @ A @ G
% 11.90/5.92 @ ^ [X3: A] : ( Ga @ ( F @ X3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_382_neg__eq__iff__add__eq__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ( uminus_uminus @ A @ A3 )
% 11.90/5.92 = B2 )
% 11.90/5.92 = ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_383_eq__neg__iff__add__eq__0,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 = ( uminus_uminus @ A @ B2 ) )
% 11.90/5.92 = ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_384_add_Oinverse__unique,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 => ( ( uminus_uminus @ A @ A3 )
% 11.90/5.92 = B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_385_ab__group__add__class_Oab__left__minus,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ab_group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 11.90/5.92 = ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_386_add__eq__0__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 = ( B2
% 11.90/5.92 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_387_zero__neq__neg__one,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ring_char_0 @ A )
% 11.90/5.92 => ( ( zero_zero @ A )
% 11.90/5.92 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_388_less__minus__one__simps_I4_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_389_less__minus__one__simps_I2_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_390_option_Omap__sel,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,A3: option @ A,F: A > B] :
% 11.90/5.92 ( ( A3
% 11.90/5.92 != ( none @ A ) )
% 11.90/5.92 => ( ( the2 @ B @ ( map_option @ A @ B @ F @ A3 ) )
% 11.90/5.92 = ( F @ ( the2 @ A @ A3 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_391_Ints__double__eq__0__iff,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ring_char_0 @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( ( ( plus_plus @ A @ A3 @ A3 )
% 11.90/5.92 = ( zero_zero @ A ) )
% 11.90/5.92 = ( A3
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_392_less__minus__one__simps_I3_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_393_less__minus__one__simps_I1_J,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_394_sgn__1__pos,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ( sgn_sgn @ A @ A3 )
% 11.90/5.92 = ( one_one @ A ) )
% 11.90/5.92 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_395_Ints__odd__nonzero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ring_char_0 @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
% 11.90/5.92 != ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_396_map__option__o__empty,axiom,
% 11.90/5.92 ! [C: $tType,B: $tType,A: $tType,F: C > B] :
% 11.90/5.92 ( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F )
% 11.90/5.92 @ ^ [X3: A] : ( none @ C ) )
% 11.90/5.92 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_397_execute__lookup,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [R: ref @ A,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R ) @ H3 )
% 11.90/5.92 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( ref_get @ A @ H3 @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_398_change__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ( ( ref_change @ A )
% 11.90/5.92 = ( ^ [F2: A > A,R3: ref @ A] :
% 11.90/5.92 ( heap_Time_bind @ A @ A @ ( ref_lookup @ A @ R3 )
% 11.90/5.92 @ ^ [X3: A] :
% 11.90/5.92 ( heap_Time_bind @ product_unit @ A @ ( ref_update @ A @ R3 @ ( F2 @ X3 ) )
% 11.90/5.92 @ ^ [Uu: product_unit] : ( heap_Time_return @ A @ ( F2 @ X3 ) ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_399_fails__assert_H,axiom,
% 11.90/5.92 ! [P: $o,H3: heap_ext @ product_unit] :
% 11.90/5.92 ( ( time_fails @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
% 11.90/5.92 = ~ P ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_400_effect__lookupI,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,X: A,R: ref @ A,N: nat] :
% 11.90/5.92 ( ( H4 = H3 )
% 11.90/5.92 => ( ( X
% 11.90/5.92 = ( ref_get @ A @ H3 @ R ) )
% 11.90/5.92 => ( ( N
% 11.90/5.92 = ( one_one @ nat ) )
% 11.90/5.92 => ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H3 @ H4 @ X @ N ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_401_effect__lookupE,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ! [R: ref @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,X: A,N: nat] :
% 11.90/5.92 ( ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H3 @ H4 @ X @ N )
% 11.90/5.92 => ~ ( ( H4 = H3 )
% 11.90/5.92 => ( ( X
% 11.90/5.92 = ( ref_get @ A @ H3 @ R ) )
% 11.90/5.92 => ( N
% 11.90/5.92 != ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_402_add_Ogroup__axioms,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( group_add @ A )
% 11.90/5.92 => ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_403_num__of__nat__double,axiom,
% 11.90/5.92 ! [N: nat] :
% 11.90/5.92 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.92 => ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
% 11.90/5.92 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_404_Ints__nonzero__abs__ge1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linordered_idom @ A )
% 11.90/5.92 => ! [X: A] :
% 11.90/5.92 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 11.90/5.92 => ( ( X
% 11.90/5.92 != ( zero_zero @ A ) )
% 11.90/5.92 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_405_lookup__def,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( heap @ A )
% 11.90/5.92 => ( ( ref_lookup @ A )
% 11.90/5.92 = ( ^ [R3: ref @ A] :
% 11.90/5.92 ( heap_Time_tap @ A
% 11.90/5.92 @ ^ [H: heap_ext @ product_unit] : ( ref_get @ A @ H @ R3 ) ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_406_map__conv__bind__option,axiom,
% 11.90/5.92 ! [A: $tType,B: $tType] :
% 11.90/5.92 ( ( map_option @ B @ A )
% 11.90/5.92 = ( ^ [F2: B > A,X3: option @ B] : ( bind @ B @ A @ X3 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F2 ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_407_nat__add__left__cancel__le,axiom,
% 11.90/5.92 ! [K: nat,M2: nat,N: nat] :
% 11.90/5.92 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 11.90/5.92 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_408_less__eq__option__None__code,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_409_bind__assoc,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,C: $tType,X: option @ C,F: C > ( option @ B ),G: B > ( option @ A )] :
% 11.90/5.92 ( ( bind @ B @ A @ ( bind @ C @ B @ X @ F ) @ G )
% 11.90/5.92 = ( bind @ C @ A @ X
% 11.90/5.92 @ ^ [Y3: C] : ( bind @ B @ A @ ( F @ Y3 ) @ G ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_410_le__zero__eq,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.92 => ! [N: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
% 11.90/5.92 = ( N
% 11.90/5.92 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_411_add__le__cancel__left,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.92 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_412_add__le__cancel__right,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.92 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_413_neg__le__iff__le,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_414_less__eq__option__Some,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: A,Y: A] :
% 11.90/5.92 ( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 11.90/5.92 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_415_less__eq__option__Some__None,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( preorder @ A )
% 11.90/5.92 => ! [X: A] :
% 11.90/5.92 ~ ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X ) @ ( none @ A ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_416_bind__runit,axiom,
% 11.90/5.92 ! [A: $tType,X: option @ A] :
% 11.90/5.92 ( ( bind @ A @ A @ X @ ( some @ A ) )
% 11.90/5.92 = X ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_417_bind__rzero,axiom,
% 11.90/5.92 ! [B: $tType,A: $tType,X: option @ B] :
% 11.90/5.92 ( ( bind @ B @ A @ X
% 11.90/5.92 @ ^ [X3: B] : ( none @ A ) )
% 11.90/5.92 = ( none @ A ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_418_add__le__same__cancel1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [B2: A,A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_419_add__le__same__cancel2,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_420_le__add__same__cancel1,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_421_le__add__same__cancel2,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordere1937475149494474687imp_le @ A )
% 11.90/5.92 => ! [A3: A,B2: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_422_double__add__le__zero__iff__single__add__le__zero,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_423_zero__le__double__add__iff__zero__le__single__add,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_424_neg__0__le__iff__le,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.92 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_425_neg__le__0__iff__le,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( ordered_ab_group_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.92 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.92
% 11.90/5.92 thf(fact_426_less__eq__neg__nonpos,axiom,
% 11.90/5.92 ! [A: $tType] :
% 11.90/5.92 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.92 => ! [A3: A] :
% 11.90/5.92 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_427_neg__less__eq__nonneg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
% 11.90/5.93 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_428_abs__of__nonneg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( abs_abs @ A @ A3 )
% 11.90/5.93 = A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_429_abs__le__self__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
% 11.90/5.93 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_430_abs__le__zero__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.93 = ( A3
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_431_abs__of__nonpos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( abs_abs @ A @ A3 )
% 11.90/5.93 = ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_432_ord__eq__le__eq__trans,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ord @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( A3 = B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 11.90/5.93 => ( ( C2 = D3 )
% 11.90/5.93 => ( ord_less_eq @ A @ A3 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_433_bind__option__cong__code,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B )] :
% 11.90/5.93 ( ( X = Y )
% 11.90/5.93 => ( ( bind @ A @ B @ X @ F )
% 11.90/5.93 = ( bind @ A @ B @ Y @ F ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_434_group_Oleft__cancel,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( ( F @ A3 @ B2 )
% 11.90/5.93 = ( F @ A3 @ C2 ) )
% 11.90/5.93 = ( B2 = C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_435_group_Oleft__inverse,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( F @ ( Inverse @ A3 ) @ A3 )
% 11.90/5.93 = Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_436_group_Oright__cancel,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( ( F @ B2 @ A3 )
% 11.90/5.93 = ( F @ C2 @ A3 ) )
% 11.90/5.93 = ( B2 = C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_437_group_Oright__inverse,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( F @ A3 @ ( Inverse @ A3 ) )
% 11.90/5.93 = Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_438_group_Oinverse__unique,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( ( F @ A3 @ B2 )
% 11.90/5.93 = Z2 )
% 11.90/5.93 => ( ( Inverse @ A3 )
% 11.90/5.93 = B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_439_group_Oinverse__inverse,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( Inverse @ ( Inverse @ A3 ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_440_group_Oinverse__neutral,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( Inverse @ Z2 )
% 11.90/5.93 = Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_441_group_Ogroup__left__neutral,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( F @ Z2 @ A3 )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_442_group_Oinverse__distrib__swap,axiom,
% 11.90/5.93 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
% 11.90/5.93 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.90/5.93 => ( ( Inverse @ ( F @ A3 @ B2 ) )
% 11.90/5.93 = ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_443_accp__subset,axiom,
% 11.90/5.93 ! [A: $tType,R12: A > A > $o,R23: A > A > $o] :
% 11.90/5.93 ( ( ord_less_eq @ ( A > A > $o ) @ R12 @ R23 )
% 11.90/5.93 => ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R23 ) @ ( accp @ A @ R12 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_444_le__some__optE,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( preorder @ A )
% 11.90/5.93 => ! [M2: A,X: option @ A] :
% 11.90/5.93 ( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M2 ) @ X )
% 11.90/5.93 => ~ ! [M4: A] :
% 11.90/5.93 ( ( X
% 11.90/5.93 = ( some @ A @ M4 ) )
% 11.90/5.93 => ~ ( ord_less_eq @ A @ M2 @ M4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_445_pred__subset__eq2,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.93 ( ( ord_less_eq @ ( A > B > $o )
% 11.90/5.93 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
% 11.90/5.93 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
% 11.90/5.93 = ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_446_less__eq__option__def,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( preorder @ A )
% 11.90/5.93 => ( ( ord_less_eq @ ( option @ A ) )
% 11.90/5.93 = ( ^ [X3: option @ A,Y3: option @ A] :
% 11.90/5.93 ( case_option @ $o @ A @ $true
% 11.90/5.93 @ ^ [Z5: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z5 ) @ Y3 )
% 11.90/5.93 @ X3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_447_zero__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.93 => ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_448_add__mono__thms__linordered__semiring_I3_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.93 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.93 ( ( ( ord_less_eq @ A @ I @ J )
% 11.90/5.93 & ( K = L ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_449_add__mono__thms__linordered__semiring_I2_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.93 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.93 ( ( ( I = J )
% 11.90/5.93 & ( ord_less_eq @ A @ K @ L ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_450_add__mono__thms__linordered__semiring_I1_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.93 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.93 ( ( ( ord_less_eq @ A @ I @ J )
% 11.90/5.93 & ( ord_less_eq @ A @ K @ L ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_451_add__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ C2 @ D3 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_452_add__left__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_453_less__eqE,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ~ ! [C4: A] :
% 11.90/5.93 ( B2
% 11.90/5.93 != ( plus_plus @ A @ A3 @ C4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_454_add__right__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6658533253407199908up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_455_le__iff__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( canoni5634975068530333245id_add @ A )
% 11.90/5.93 => ( ( ord_less_eq @ A )
% 11.90/5.93 = ( ^ [A7: A,B5: A] :
% 11.90/5.93 ? [C3: A] :
% 11.90/5.93 ( B5
% 11.90/5.93 = ( plus_plus @ A @ A7 @ C3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_456_add__le__imp__le__left,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.93 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 11.90/5.93 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_457_add__le__imp__le__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.93 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_458_accp__subset__induct,axiom,
% 11.90/5.93 ! [A: $tType,D4: A > $o,R5: A > A > $o,X: A,P: A > $o] :
% 11.90/5.93 ( ( ord_less_eq @ ( A > $o ) @ D4 @ ( accp @ A @ R5 ) )
% 11.90/5.93 => ( ! [X2: A,Z3: A] :
% 11.90/5.93 ( ( D4 @ X2 )
% 11.90/5.93 => ( ( R5 @ Z3 @ X2 )
% 11.90/5.93 => ( D4 @ Z3 ) ) )
% 11.90/5.93 => ( ( D4 @ X )
% 11.90/5.93 => ( ! [X2: A] :
% 11.90/5.93 ( ( D4 @ X2 )
% 11.90/5.93 => ( ! [Z6: A] :
% 11.90/5.93 ( ( R5 @ Z6 @ X2 )
% 11.90/5.93 => ( P @ Z6 ) )
% 11.90/5.93 => ( P @ X2 ) ) )
% 11.90/5.93 => ( P @ X ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_459_le__numeral__extra_I4_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord181362715937106298miring @ A )
% 11.90/5.93 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_460_subrelI,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
% 11.90/5.93 ( ! [X2: A,Y4: B] :
% 11.90/5.93 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 11.90/5.93 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ S ) )
% 11.90/5.93 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_461_le__minus__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_462_minus__le__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 11.90/5.93 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_463_le__imp__neg__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_464_exists__leI,axiom,
% 11.90/5.93 ! [N: nat,P: nat > $o] :
% 11.90/5.93 ( ( ! [N7: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ N7 @ N )
% 11.90/5.93 => ~ ( P @ N7 ) )
% 11.90/5.93 => ( P @ N ) )
% 11.90/5.93 => ? [N4: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ N4 @ N )
% 11.90/5.93 & ( P @ N4 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_465_nat__le__iff__add,axiom,
% 11.90/5.93 ( ( ord_less_eq @ nat )
% 11.90/5.93 = ( ^ [M: nat,N5: nat] :
% 11.90/5.93 ? [K4: nat] :
% 11.90/5.93 ( N5
% 11.90/5.93 = ( plus_plus @ nat @ M @ K4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_466_trans__le__add2,axiom,
% 11.90/5.93 ! [I: nat,J: nat,M2: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_467_trans__le__add1,axiom,
% 11.90/5.93 ! [I: nat,J: nat,M2: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_468_add__le__mono1,axiom,
% 11.90/5.93 ! [I: nat,J: nat,K: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_469_add__le__mono,axiom,
% 11.90/5.93 ! [I: nat,J: nat,K: nat,L: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ K @ L )
% 11.90/5.93 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_470_le__Suc__ex,axiom,
% 11.90/5.93 ! [K: nat,L: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ L )
% 11.90/5.93 => ? [N3: nat] :
% 11.90/5.93 ( L
% 11.90/5.93 = ( plus_plus @ nat @ K @ N3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_471_add__leD2,axiom,
% 11.90/5.93 ! [M2: nat,K: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 11.90/5.93 => ( ord_less_eq @ nat @ K @ N ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_472_add__leD1,axiom,
% 11.90/5.93 ! [M2: nat,K: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 11.90/5.93 => ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_473_le__add2,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_474_le__add1,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_475_add__leE,axiom,
% 11.90/5.93 ! [M2: nat,K: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
% 11.90/5.93 => ~ ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.93 => ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_476_abs__le__D1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_477_abs__ge__self,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_478_less__eq__option__None,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( preorder @ A )
% 11.90/5.93 => ! [X: option @ A] : ( ord_less_eq @ ( option @ A ) @ ( none @ A ) @ X ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_479_less__eq__option__None__is__None,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( preorder @ A )
% 11.90/5.93 => ! [X: option @ A] :
% 11.90/5.93 ( ( ord_less_eq @ ( option @ A ) @ X @ ( none @ A ) )
% 11.90/5.93 => ( X
% 11.90/5.93 = ( none @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_480_bind_Obind__lunit,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,X: A,F: A > ( option @ B )] :
% 11.90/5.93 ( ( bind @ A @ B @ ( some @ A @ X ) @ F )
% 11.90/5.93 = ( F @ X ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_481_Option_Obind__cong,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 11.90/5.93 ( ( X = Y )
% 11.90/5.93 => ( ! [A5: A] :
% 11.90/5.93 ( ( Y
% 11.90/5.93 = ( some @ A @ A5 ) )
% 11.90/5.93 => ( ( F @ A5 )
% 11.90/5.93 = ( G @ A5 ) ) )
% 11.90/5.93 => ( ( bind @ A @ B @ X @ F )
% 11.90/5.93 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_482_bind__eq__Some__conv,axiom,
% 11.90/5.93 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A ),X: A] :
% 11.90/5.93 ( ( ( bind @ B @ A @ F @ G )
% 11.90/5.93 = ( some @ A @ X ) )
% 11.90/5.93 = ( ? [Y3: B] :
% 11.90/5.93 ( ( F
% 11.90/5.93 = ( some @ B @ Y3 ) )
% 11.90/5.93 & ( ( G @ Y3 )
% 11.90/5.93 = ( some @ A @ X ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_483_bind_Obind__lzero,axiom,
% 11.90/5.93 ! [A: $tType,B: $tType,F: A > ( option @ B )] :
% 11.90/5.93 ( ( bind @ A @ B @ ( none @ A ) @ F )
% 11.90/5.93 = ( none @ B ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_484_map__option__bind,axiom,
% 11.90/5.93 ! [A: $tType,B: $tType,C: $tType,F: B > A,X: option @ C,G: C > ( option @ B )] :
% 11.90/5.93 ( ( map_option @ B @ A @ F @ ( bind @ C @ B @ X @ G ) )
% 11.90/5.93 = ( bind @ C @ A @ X @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F ) @ G ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_485_add__decreasing,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ C2 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_486_add__increasing,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 11.90/5.93 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_487_add__decreasing2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_488_add__increasing2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [C2: A,B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.93 => ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_489_add__nonneg__nonneg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_490_add__nonpos__nonpos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_491_add__nonneg__eq__0__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 11.90/5.93 => ( ( ( plus_plus @ A @ X @ Y )
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 = ( ( X
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 & ( Y
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_492_add__nonpos__eq__0__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ( plus_plus @ A @ X @ Y )
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 = ( ( X
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 & ( Y
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_493_not__one__le__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord181362715937106298miring @ A )
% 11.90/5.93 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_494_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord181362715937106298miring @ A )
% 11.90/5.93 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_495_zero__less__one__class_Ozero__le__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( zero_less_one @ A )
% 11.90/5.93 => ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_496_add__mono__thms__linordered__field_I4_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.93 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.93 ( ( ( ord_less_eq @ A @ I @ J )
% 11.90/5.93 & ( ord_less @ A @ K @ L ) )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_497_add__mono__thms__linordered__field_I3_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.93 => ! [I: A,J: A,K: A,L: A] :
% 11.90/5.93 ( ( ( ord_less @ A @ I @ J )
% 11.90/5.93 & ( ord_less_eq @ A @ K @ L ) )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_498_add__le__less__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less @ A @ C2 @ D3 )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_499_add__less__le__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere580206878836729694up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ C2 @ D3 )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_500_le__minus__one__simps_I2_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_501_le__minus__one__simps_I4_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_502_abs__ge__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_503_abs__triangle__ineq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_504_abs__leI,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_505_abs__le__D2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_506_abs__le__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
% 11.90/5.93 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 & ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_507_abs__ge__minus__self,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_508_mono__nat__linear__lb,axiom,
% 11.90/5.93 ! [F: nat > nat,M2: nat,K: nat] :
% 11.90/5.93 ( ! [M5: nat,N3: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ M5 @ N3 )
% 11.90/5.93 => ( ord_less @ nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
% 11.90/5.93 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_509_nat__geq__1__eq__neqz,axiom,
% 11.90/5.93 ! [X: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X )
% 11.90/5.93 = ( X
% 11.90/5.93 != ( zero_zero @ nat ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_510_bind__split__asm,axiom,
% 11.90/5.93 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F: B > ( option @ A )] :
% 11.90/5.93 ( ( P @ ( bind @ B @ A @ M2 @ F ) )
% 11.90/5.93 = ( ~ ( ( ( M2
% 11.90/5.93 = ( none @ B ) )
% 11.90/5.93 & ~ ( P @ ( none @ A ) ) )
% 11.90/5.93 | ? [X3: B] :
% 11.90/5.93 ( ( M2
% 11.90/5.93 = ( some @ B @ X3 ) )
% 11.90/5.93 & ~ ( P @ ( F @ X3 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_511_bind__split,axiom,
% 11.90/5.93 ! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F: B > ( option @ A )] :
% 11.90/5.93 ( ( P @ ( bind @ B @ A @ M2 @ F ) )
% 11.90/5.93 = ( ( ( M2
% 11.90/5.93 = ( none @ B ) )
% 11.90/5.93 => ( P @ ( none @ A ) ) )
% 11.90/5.93 & ! [V2: B] :
% 11.90/5.93 ( ( M2
% 11.90/5.93 = ( some @ B @ V2 ) )
% 11.90/5.93 => ( P @ ( F @ V2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_512_mlex__leq,axiom,
% 11.90/5.93 ! [A: $tType,F: A > nat,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
% 11.90/5.93 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
% 11.90/5.93 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_513_bind__eq__None__conv,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,A3: option @ B,F: B > ( option @ A )] :
% 11.90/5.93 ( ( ( bind @ B @ A @ A3 @ F )
% 11.90/5.93 = ( none @ A ) )
% 11.90/5.93 = ( ( A3
% 11.90/5.93 = ( none @ B ) )
% 11.90/5.93 | ( ( F @ ( the2 @ B @ A3 ) )
% 11.90/5.93 = ( none @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_514_bind__map__option,axiom,
% 11.90/5.93 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: option @ C,G: B > ( option @ A )] :
% 11.90/5.93 ( ( bind @ B @ A @ ( map_option @ C @ B @ F @ X ) @ G )
% 11.90/5.93 = ( bind @ C @ A @ X @ ( comp @ B @ ( option @ A ) @ C @ G @ F ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_515_add__neg__nonpos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_516_add__nonneg__pos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.93 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_517_add__nonpos__neg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_518_add__pos__nonneg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere6911136660526730532id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.93 => ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_519_add__strict__increasing,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere8940638589300402666id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ C2 )
% 11.90/5.93 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_520_add__strict__increasing2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere8940638589300402666id_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less @ A @ B2 @ C2 )
% 11.90/5.93 => ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_521_discrete,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.93 => ( ( ord_less @ A )
% 11.90/5.93 = ( ^ [A7: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_522_le__minus__one__simps_I1_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_523_le__minus__one__simps_I3_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_524_dense__eq0__I,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 & ( dense_linorder @ A ) )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ! [E3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E3 ) )
% 11.90/5.93 => ( X
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_525_abs__minus__le__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_526_success__changeI,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( heap @ A )
% 11.90/5.93 => ! [F: A > A,R: ref @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( ref_change @ A @ F @ R ) @ H3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_527_semiring__norm_I6_J,axiom,
% 11.90/5.93 ! [M2: num,N: num] :
% 11.90/5.93 ( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.93 = ( bit0 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_528_field__le__epsilon,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ! [E3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
% 11.90/5.93 => ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E3 ) ) )
% 11.90/5.93 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_529_ex__has__greatest__nat__lemma,axiom,
% 11.90/5.93 ! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
% 11.90/5.93 ( ( P @ K )
% 11.90/5.93 => ( ! [X2: A] :
% 11.90/5.93 ( ( P @ X2 )
% 11.90/5.93 => ? [Y6: A] :
% 11.90/5.93 ( ( P @ Y6 )
% 11.90/5.93 & ~ ( ord_less_eq @ nat @ ( F @ Y6 ) @ ( F @ X2 ) ) ) )
% 11.90/5.93 => ? [Y4: A] :
% 11.90/5.93 ( ( P @ Y4 )
% 11.90/5.93 & ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_530_map__comp__def,axiom,
% 11.90/5.93 ! [A: $tType,C: $tType,B: $tType] :
% 11.90/5.93 ( ( map_comp @ B @ C @ A )
% 11.90/5.93 = ( ^ [F2: B > ( option @ C ),G2: A > ( option @ B ),K4: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F2 @ ( G2 @ K4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_531_convex__bound__lt,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord715952674999750819strict @ A )
% 11.90/5.93 => ! [X: A,A3: A,Y: A,U: A,V: A] :
% 11.90/5.93 ( ( ord_less @ A @ X @ A3 )
% 11.90/5.93 => ( ( ord_less @ A @ Y @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 11.90/5.93 => ( ( ( plus_plus @ A @ U @ V )
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 => ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_532_map__comp__empty_I1_J,axiom,
% 11.90/5.93 ! [C: $tType,B: $tType,A: $tType,M2: C > ( option @ B )] :
% 11.90/5.93 ( ( map_comp @ C @ B @ A @ M2
% 11.90/5.93 @ ^ [X3: A] : ( none @ C ) )
% 11.90/5.93 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_533_map__comp__empty_I2_J,axiom,
% 11.90/5.93 ! [B: $tType,D: $tType,C: $tType,M2: C > ( option @ B )] :
% 11.90/5.93 ( ( map_comp @ B @ D @ C
% 11.90/5.93 @ ^ [X3: B] : ( none @ D )
% 11.90/5.93 @ M2 )
% 11.90/5.93 = ( ^ [X3: C] : ( none @ D ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_534_map__comp__None__iff,axiom,
% 11.90/5.93 ! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K: C] :
% 11.90/5.93 ( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K )
% 11.90/5.93 = ( none @ A ) )
% 11.90/5.93 = ( ( ( M22 @ K )
% 11.90/5.93 = ( none @ B ) )
% 11.90/5.93 | ? [K6: B] :
% 11.90/5.93 ( ( ( M22 @ K )
% 11.90/5.93 = ( some @ B @ K6 ) )
% 11.90/5.93 & ( ( M1 @ K6 )
% 11.90/5.93 = ( none @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_535_Ints__eq__abs__less1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 11.90/5.93 => ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
% 11.90/5.93 => ( ( X = Y )
% 11.90/5.93 = ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_536_nat0__intermed__int__val,axiom,
% 11.90/5.93 ! [N: nat,F: nat > int,K: int] :
% 11.90/5.93 ( ! [I2: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ I2 @ N )
% 11.90/5.93 => ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) @ ( F @ I2 ) ) ) @ ( one_one @ int ) ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
% 11.90/5.93 => ( ( ord_less_eq @ int @ K @ ( F @ N ) )
% 11.90/5.93 => ? [I2: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I2 @ N )
% 11.90/5.93 & ( ( F @ I2 )
% 11.90/5.93 = K ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_537_diff__diff__left,axiom,
% 11.90/5.93 ! [I: nat,J: nat,K: nat] :
% 11.90/5.93 ( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
% 11.90/5.93 = ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_538_nat__1__eq__mult__iff,axiom,
% 11.90/5.93 ! [M2: nat,N: nat] :
% 11.90/5.93 ( ( ( one_one @ nat )
% 11.90/5.93 = ( times_times @ nat @ M2 @ N ) )
% 11.90/5.93 = ( ( M2
% 11.90/5.93 = ( one_one @ nat ) )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_539_nat__mult__eq__1__iff,axiom,
% 11.90/5.93 ! [M2: nat,N: nat] :
% 11.90/5.93 ( ( ( times_times @ nat @ M2 @ N )
% 11.90/5.93 = ( one_one @ nat ) )
% 11.90/5.93 = ( ( M2
% 11.90/5.93 = ( one_one @ nat ) )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_540_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ A3 )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_541_diff__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_542_zero__diff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( comm_monoid_diff @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_543_diff__0__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_544_diff__self,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ A3 )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_545_mult_Oright__neutral,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( monoid_mult @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_546_mult__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( monoid_mult @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_547_add__diff__cancel__right_H,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel2418104881723323429up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_548_add__diff__cancel__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel2418104881723323429up_add @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_549_add__diff__cancel__left_H,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel2418104881723323429up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
% 11.90/5.93 = B2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_550_add__diff__cancel__left,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel2418104881723323429up_add @ A )
% 11.90/5.93 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_551_diff__add__cancel,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_552_add__diff__cancel,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_553_minus__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_554_Nat_Oadd__diff__assoc,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 11.90/5.93 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_555_Nat_Oadd__diff__assoc2,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 11.90/5.93 = ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_556_Nat_Odiff__diff__right,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 11.90/5.93 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_557_zle__add1__eq__le,axiom,
% 11.90/5.93 ! [W: int,Z2: int] :
% 11.90/5.93 ( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
% 11.90/5.93 = ( ord_less_eq @ int @ W @ Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_558_map__comp__simps_I2_J,axiom,
% 11.90/5.93 ! [B: $tType,C: $tType,A: $tType,M22: B > ( option @ A ),K: B,K7: A,M1: A > ( option @ C )] :
% 11.90/5.93 ( ( ( M22 @ K )
% 11.90/5.93 = ( some @ A @ K7 ) )
% 11.90/5.93 => ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K )
% 11.90/5.93 = ( M1 @ K7 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_559_map__comp__simps_I1_J,axiom,
% 11.90/5.93 ! [B: $tType,A: $tType,C: $tType,M22: B > ( option @ A ),K: B,M1: A > ( option @ C )] :
% 11.90/5.93 ( ( ( M22 @ K )
% 11.90/5.93 = ( none @ A ) )
% 11.90/5.93 => ( ( map_comp @ A @ C @ B @ M1 @ M22 @ K )
% 11.90/5.93 = ( none @ C ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_560_diff__ge__0__iff__ge,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_561_zero__comp__diff__simps_I1_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_562_diff__gt__0__iff__gt,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_563_zero__comp__diff__simps_I2_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_564_le__add__diff__inverse,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_semidom @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.93 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_565_le__add__diff__inverse2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_semidom @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.93 => ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.93 = A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_566_mult__cancel__left1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_15535105094025558882visors @ A )
% 11.90/5.93 => ! [C2: A,B2: A] :
% 11.90/5.93 ( ( C2
% 11.90/5.93 = ( times_times @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( ( C2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 | ( B2
% 11.90/5.93 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_567_mult__cancel__left2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_15535105094025558882visors @ A )
% 11.90/5.93 => ! [C2: A,A3: A] :
% 11.90/5.93 ( ( ( times_times @ A @ C2 @ A3 )
% 11.90/5.93 = C2 )
% 11.90/5.93 = ( ( C2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 | ( A3
% 11.90/5.93 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_568_mult__cancel__right1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_15535105094025558882visors @ A )
% 11.90/5.93 => ! [C2: A,B2: A] :
% 11.90/5.93 ( ( C2
% 11.90/5.93 = ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( ( C2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 | ( B2
% 11.90/5.93 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_569_mult__cancel__right2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_15535105094025558882visors @ A )
% 11.90/5.93 => ! [A3: A,C2: A] :
% 11.90/5.93 ( ( ( times_times @ A @ A3 @ C2 )
% 11.90/5.93 = C2 )
% 11.90/5.93 = ( ( C2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 | ( A3
% 11.90/5.93 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_570_diff__add__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( comm_monoid_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_571_diff__numeral__special_I9_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_572_diff__0,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_573_mult__minus1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_1 @ A )
% 11.90/5.93 => ! [Z2: A] :
% 11.90/5.93 ( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z2 )
% 11.90/5.93 = ( uminus_uminus @ A @ Z2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_574_mult__minus1__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_1 @ A )
% 11.90/5.93 => ! [Z2: A] :
% 11.90/5.93 ( ( times_times @ A @ Z2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.93 = ( uminus_uminus @ A @ Z2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_575_uminus__add__conv__diff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
% 11.90/5.93 = ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_576_diff__minus__eq__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
% 11.90/5.93 = ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_577_diff__numeral__special_I12_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_578_le__imp__0__less,axiom,
% 11.90/5.93 ! [Z2: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 11.90/5.93 => ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_579_q__pos__lemma,axiom,
% 11.90/5.93 ! [B4: int,Q4: int,R4: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 11.90/5.93 => ( ( ord_less @ int @ R4 @ B4 )
% 11.90/5.93 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 11.90/5.93 => ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_580_zdiv__mono2__lemma,axiom,
% 11.90/5.93 ! [B2: int,Q3: int,R: int,B4: int,Q4: int,R4: int] :
% 11.90/5.93 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R )
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 11.90/5.93 => ( ( ord_less @ int @ R4 @ B4 )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 11.90/5.93 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 11.90/5.93 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_581_zdiv__mono2__neg__lemma,axiom,
% 11.90/5.93 ! [B2: int,Q3: int,R: int,B4: int,Q4: int,R4: int] :
% 11.90/5.93 ( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R )
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) )
% 11.90/5.93 => ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q4 ) @ R4 ) @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( ord_less @ int @ R @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 11.90/5.93 => ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
% 11.90/5.93 => ( ( ord_less_eq @ int @ B4 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_582_unique__quotient__lemma,axiom,
% 11.90/5.93 ! [B2: int,Q4: int,R4: int,Q3: int,R: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
% 11.90/5.93 => ( ( ord_less @ int @ R4 @ B2 )
% 11.90/5.93 => ( ( ord_less @ int @ R @ B2 )
% 11.90/5.93 => ( ord_less_eq @ int @ Q4 @ Q3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_583_unique__quotient__lemma__neg,axiom,
% 11.90/5.93 ! [B2: int,Q4: int,R4: int,Q3: int,R: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( ord_less @ int @ B2 @ R )
% 11.90/5.93 => ( ( ord_less @ int @ B2 @ R4 )
% 11.90/5.93 => ( ord_less_eq @ int @ Q3 @ Q4 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_584_int__ge__induct,axiom,
% 11.90/5.93 ! [K: int,I: int,P: int > $o] :
% 11.90/5.93 ( ( ord_less_eq @ int @ K @ I )
% 11.90/5.93 => ( ( P @ K )
% 11.90/5.93 => ( ! [I2: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ K @ I2 )
% 11.90/5.93 => ( ( P @ I2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 11.90/5.93 => ( P @ I ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_585_int__induct,axiom,
% 11.90/5.93 ! [P: int > $o,K: int,I: int] :
% 11.90/5.93 ( ( P @ K )
% 11.90/5.93 => ( ! [I2: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ K @ I2 )
% 11.90/5.93 => ( ( P @ I2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 11.90/5.93 => ( ! [I2: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ I2 @ K )
% 11.90/5.93 => ( ( P @ I2 )
% 11.90/5.93 => ( P @ ( minus_minus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 11.90/5.93 => ( P @ I ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_586_zless__imp__add1__zle,axiom,
% 11.90/5.93 ! [W: int,Z2: int] :
% 11.90/5.93 ( ( ord_less @ int @ W @ Z2 )
% 11.90/5.93 => ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_587_add1__zle__eq,axiom,
% 11.90/5.93 ! [W: int,Z2: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 )
% 11.90/5.93 = ( ord_less @ int @ W @ Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_588_nat__eq__add__iff1,axiom,
% 11.90/5.93 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ J @ I )
% 11.90/5.93 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 )
% 11.90/5.93 = N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_589_nat__eq__add__iff2,axiom,
% 11.90/5.93 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( M2
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_590_nat__le__add__iff1,axiom,
% 11.90/5.93 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ J @ I )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_591_nat__le__add__iff2,axiom,
% 11.90/5.93 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_592_nat__diff__add__eq1,axiom,
% 11.90/5.93 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ J @ I )
% 11.90/5.93 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_593_nat__diff__add__eq2,axiom,
% 11.90/5.93 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_594_bounded__Max__nat,axiom,
% 11.90/5.93 ! [P: nat > $o,X: nat,M3: nat] :
% 11.90/5.93 ( ( P @ X )
% 11.90/5.93 => ( ! [X2: nat] :
% 11.90/5.93 ( ( P @ X2 )
% 11.90/5.93 => ( ord_less_eq @ nat @ X2 @ M3 ) )
% 11.90/5.93 => ~ ! [M5: nat] :
% 11.90/5.93 ( ( P @ M5 )
% 11.90/5.93 => ~ ! [X7: nat] :
% 11.90/5.93 ( ( P @ X7 )
% 11.90/5.93 => ( ord_less_eq @ nat @ X7 @ M5 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_595_subset__Collect__conv,axiom,
% 11.90/5.93 ! [A: $tType,S2: set @ A,P: A > $o] :
% 11.90/5.93 ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( collect @ A @ P ) )
% 11.90/5.93 = ( ! [X3: A] :
% 11.90/5.93 ( ( member @ A @ X3 @ S2 )
% 11.90/5.93 => ( P @ X3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_596_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel2418104881723323429up_add @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
% 11.90/5.93 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_597_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_semigroup_mult @ A )
% 11.90/5.93 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
% 11.90/5.93 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_598_ab__semigroup__mult__class_Omult_Ocommute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_semigroup_mult @ A )
% 11.90/5.93 => ( ( times_times @ A )
% 11.90/5.93 = ( ^ [A7: A,B5: A] : ( times_times @ A @ B5 @ A7 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_599_mult_Oassoc,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semigroup_mult @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_600_diff__eq__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ( minus_minus @ A @ A3 @ B2 )
% 11.90/5.93 = ( minus_minus @ A @ C2 @ D3 ) )
% 11.90/5.93 => ( ( A3 = B2 )
% 11.90/5.93 = ( C2 = D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_601_one__integer_Orsp,axiom,
% 11.90/5.93 ( ( one_one @ int )
% 11.90/5.93 = ( one_one @ int ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_602_int__gr__induct,axiom,
% 11.90/5.93 ! [K: int,I: int,P: int > $o] :
% 11.90/5.93 ( ( ord_less @ int @ K @ I )
% 11.90/5.93 => ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 11.90/5.93 => ( ! [I2: int] :
% 11.90/5.93 ( ( ord_less @ int @ K @ I2 )
% 11.90/5.93 => ( ( P @ I2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) ) ) )
% 11.90/5.93 => ( P @ I ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_603_zless__add1__eq,axiom,
% 11.90/5.93 ! [W: int,Z2: int] :
% 11.90/5.93 ( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
% 11.90/5.93 = ( ( ord_less @ int @ W @ Z2 )
% 11.90/5.93 | ( W = Z2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_604_odd__nonzero,axiom,
% 11.90/5.93 ! [Z2: int] :
% 11.90/5.93 ( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
% 11.90/5.93 != ( zero_zero @ int ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_605_odd__less__0__iff,axiom,
% 11.90/5.93 ! [Z2: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
% 11.90/5.93 = ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_606_eq__add__iff1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 11.90/5.93 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 11.90/5.93 = ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 )
% 11.90/5.93 = D3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_607_eq__add__iff2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 11.90/5.93 ( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 11.90/5.93 = ( C2
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_608_square__diff__square__factored,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( comm_ring @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
% 11.90/5.93 = ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_609_left__add__mult__distrib,axiom,
% 11.90/5.93 ! [I: nat,U: nat,J: nat,K: nat] :
% 11.90/5.93 ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_610_nat__less__add__iff1,axiom,
% 11.90/5.93 ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ J @ I )
% 11.90/5.93 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_611_nat__less__add__iff2,axiom,
% 11.90/5.93 ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
% 11.90/5.93 = ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_612_le__add__iff1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_613_le__add__iff2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_614_less__add__iff1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 11.90/5.93 = ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_615_less__add__iff2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
% 11.90/5.93 = ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_616_square__diff__one__factored,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_1 @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
% 11.90/5.93 = ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_617_diff__eq__diff__less__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ( minus_minus @ A @ A3 @ B2 )
% 11.90/5.93 = ( minus_minus @ A @ C2 @ D3 ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 = ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_618_diff__right__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_619_diff__left__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_620_diff__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,D3: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ D3 @ C2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_621_eq__iff__diff__eq__0,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ( ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 11.90/5.93 = ( ^ [A7: A,B5: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A7 @ B5 )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_622_diff__strict__right__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_623_diff__strict__left__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( ord_less @ A @ B2 @ A3 )
% 11.90/5.93 => ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_624_diff__eq__diff__less,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ( minus_minus @ A @ A3 @ B2 )
% 11.90/5.93 = ( minus_minus @ A @ C2 @ D3 ) )
% 11.90/5.93 => ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 = ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_625_diff__strict__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,D3: A,C2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less @ A @ D3 @ C2 )
% 11.90/5.93 => ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_626_diff__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel2418104881723323429up_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_627_add__implies__diff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( cancel1802427076303600483id_add @ A )
% 11.90/5.93 => ! [C2: A,B2: A,A3: A] :
% 11.90/5.93 ( ( ( plus_plus @ A @ C2 @ B2 )
% 11.90/5.93 = A3 )
% 11.90/5.93 => ( C2
% 11.90/5.93 = ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_628_diff__add__eq__diff__diff__swap,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_629_diff__add__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_630_diff__diff__eq2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_631_add__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_632_eq__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( minus_minus @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.93 = C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_633_diff__eq__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ( minus_minus @ A @ A3 @ B2 )
% 11.90/5.93 = C2 )
% 11.90/5.93 = ( A3
% 11.90/5.93 = ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_634_group__cancel_Osub1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_group_add @ A )
% 11.90/5.93 => ! [A4: A,K: A,A3: A,B2: A] :
% 11.90/5.93 ( ( A4
% 11.90/5.93 = ( plus_plus @ A @ K @ A3 ) )
% 11.90/5.93 => ( ( minus_minus @ A @ A4 @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_635_minus__diff__commute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_group_add @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
% 11.90/5.93 = ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_636_ring__class_Oring__distribs_I2_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_637_ring__class_Oring__distribs_I1_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_638_comm__semiring__class_Odistrib,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( comm_semiring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_639_distrib__left,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_640_distrib__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_641_combine__common__factor,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring @ A )
% 11.90/5.93 => ! [A3: A,E2: A,B2: A,C2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_642_crossproduct__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiri1453513574482234551roduct @ A )
% 11.90/5.93 => ! [W: A,Y: A,X: A,Z2: A] :
% 11.90/5.93 ( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y ) @ ( times_times @ A @ X @ Z2 ) )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ W @ Z2 ) @ ( times_times @ A @ X @ Y ) ) )
% 11.90/5.93 = ( ( W = X )
% 11.90/5.93 | ( Y = Z2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_643_crossproduct__noteq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiri1453513574482234551roduct @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ( A3 != B2 )
% 11.90/5.93 & ( C2 != D3 ) )
% 11.90/5.93 = ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) )
% 11.90/5.93 != ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_644_comm__monoid__mult__class_Omult__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( comm_monoid_mult @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( times_times @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_645_mult_Ocomm__neutral,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( comm_monoid_mult @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( times_times @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_646_abs__minus__commute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_647_Nat_Odiff__cancel,axiom,
% 11.90/5.93 ! [K: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
% 11.90/5.93 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_648_diff__cancel2,axiom,
% 11.90/5.93 ! [M2: nat,K: nat,N: nat] :
% 11.90/5.93 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) )
% 11.90/5.93 = ( minus_minus @ nat @ M2 @ N ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_649_diff__add__inverse,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] :
% 11.90/5.93 ( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
% 11.90/5.93 = M2 ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_650_diff__add__inverse2,axiom,
% 11.90/5.93 ! [M2: nat,N: nat] :
% 11.90/5.93 ( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
% 11.90/5.93 = M2 ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_651_le__diff__conv,axiom,
% 11.90/5.93 ! [J: nat,K: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 11.90/5.93 = ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_652_Nat_Ole__diff__conv2,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 11.90/5.93 = ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_653_Nat_Odiff__add__assoc,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
% 11.90/5.93 = ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_654_Nat_Odiff__add__assoc2,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
% 11.90/5.93 = ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_655_Nat_Ole__imp__diff__is__add,axiom,
% 11.90/5.93 ! [I: nat,J: nat,K: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.93 => ( ( ( minus_minus @ nat @ J @ I )
% 11.90/5.93 = K )
% 11.90/5.93 = ( J
% 11.90/5.93 = ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_656_add__mult__distrib,axiom,
% 11.90/5.93 ! [M2: nat,N: nat,K: nat] :
% 11.90/5.93 ( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N ) @ K )
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_657_add__mult__distrib2,axiom,
% 11.90/5.93 ! [K: nat,M2: nat,N: nat] :
% 11.90/5.93 ( ( times_times @ nat @ K @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_658_mlex__leI,axiom,
% 11.90/5.93 ! [A3: nat,A6: nat,B2: nat,B4: nat,N6: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ A3 @ A6 )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ B2 @ B4 )
% 11.90/5.93 => ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_659_nat__mult__1,axiom,
% 11.90/5.93 ! [N: nat] :
% 11.90/5.93 ( ( times_times @ nat @ ( one_one @ nat ) @ N )
% 11.90/5.93 = N ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_660_nat__mult__1__right,axiom,
% 11.90/5.93 ! [N: nat] :
% 11.90/5.93 ( ( times_times @ nat @ N @ ( one_one @ nat ) )
% 11.90/5.93 = N ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_661_mult__eq__if,axiom,
% 11.90/5.93 ( ( times_times @ nat )
% 11.90/5.93 = ( ^ [M: nat,N5: nat] :
% 11.90/5.93 ( if @ nat
% 11.90/5.93 @ ( M
% 11.90/5.93 = ( zero_zero @ nat ) )
% 11.90/5.93 @ ( zero_zero @ nat )
% 11.90/5.93 @ ( plus_plus @ nat @ N5 @ ( times_times @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_662_lambda__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( monoid_mult @ A )
% 11.90/5.93 => ( ( ^ [X3: A] : X3 )
% 11.90/5.93 = ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_663_is__num__normalize_I6_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( neg_numeral_is_num @ A @ X )
% 11.90/5.93 => ( ( neg_numeral_is_num @ A @ Y )
% 11.90/5.93 => ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_664_is__num__add__commute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( neg_numeral_is_num @ A @ X )
% 11.90/5.93 => ( ( neg_numeral_is_num @ A @ Y )
% 11.90/5.93 => ( ( plus_plus @ A @ X @ Y )
% 11.90/5.93 = ( plus_plus @ A @ Y @ X ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_665_is__num__add__left__commute,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.93 ( ( neg_numeral_is_num @ A @ X )
% 11.90/5.93 => ( ( neg_numeral_is_num @ A @ Y )
% 11.90/5.93 => ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y @ Z2 ) )
% 11.90/5.93 = ( plus_plus @ A @ Y @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_666_is__num__normalize_I4_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_667_le__iff__diff__le__0,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ( ( ord_less_eq @ A )
% 11.90/5.93 = ( ^ [A7: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_668_less__iff__diff__less__0,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ( ( ord_less @ A )
% 11.90/5.93 = ( ^ [A7: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A7 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_669_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ( minus_minus @ A @ B2 @ A3 )
% 11.90/5.93 = C2 )
% 11.90/5.93 = ( B2
% 11.90/5.93 = ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_670_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
% 11.90/5.93 = B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_671_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_672_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
% 11.90/5.93 = ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_673_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
% 11.90/5.93 = ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_676_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_677_le__add__diff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_678_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
% 11.90/5.93 = B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_679_le__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_680_diff__le__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_681_add__le__imp__le__diff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_semidom @ A )
% 11.90/5.93 => ! [I: A,K: A,N: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 11.90/5.93 => ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_682_add__le__add__imp__diff__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_semidom @ A )
% 11.90/5.93 => ! [I: A,K: A,N: A,J: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 11.90/5.93 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
% 11.90/5.93 => ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_683_less__diff__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_684_diff__less__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordered_ab_group_add @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_685_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_semidom @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ~ ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 => ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.93 = A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_686_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_group_add @ A )
% 11.90/5.93 => ( ( minus_minus @ A )
% 11.90/5.93 = ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_687_diff__conv__add__uminus,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( group_add @ A )
% 11.90/5.93 => ( ( minus_minus @ A )
% 11.90/5.93 = ( ^ [A7: A,B5: A] : ( plus_plus @ A @ A7 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_688_group__cancel_Osub2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ab_group_add @ A )
% 11.90/5.93 => ! [B6: A,K: A,B2: A,A3: A] :
% 11.90/5.93 ( ( B6
% 11.90/5.93 = ( plus_plus @ A @ K @ B2 ) )
% 11.90/5.93 => ( ( minus_minus @ A @ A3 @ B6 )
% 11.90/5.93 = ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_689_abs__triangle__ineq2__sym,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_690_abs__triangle__ineq3,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_691_abs__triangle__ineq2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_692_add__scale__eq__noteq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiri1453513574482234551roduct @ A )
% 11.90/5.93 => ! [R: A,A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( R
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ( A3 = B2 )
% 11.90/5.93 & ( C2 != D3 ) )
% 11.90/5.93 => ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
% 11.90/5.93 != ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D3 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_693_less__1__mult,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_semidom @ A )
% 11.90/5.93 => ! [M2: A,N: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( one_one @ A ) @ M2 )
% 11.90/5.93 => ( ( ord_less @ A @ ( one_one @ A ) @ N )
% 11.90/5.93 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_694_square__eq__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ring_15535105094025558882visors @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ( times_times @ A @ X @ X )
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 = ( ( X
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 | ( X
% 11.90/5.93 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_695_diff__add__0,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] :
% 11.90/5.93 ( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) )
% 11.90/5.93 = ( zero_zero @ nat ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_696_less__diff__conv,axiom,
% 11.90/5.93 ! [I: nat,J: nat,K: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
% 11.90/5.93 = ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_697_add__diff__inverse__nat,axiom,
% 11.90/5.93 ! [M2: nat,N: nat] :
% 11.90/5.93 ( ~ ( ord_less @ nat @ M2 @ N )
% 11.90/5.93 => ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M2 @ N ) )
% 11.90/5.93 = M2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_698_less__diff__conv2,axiom,
% 11.90/5.93 ! [K: nat,J: nat,I: nat] :
% 11.90/5.93 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.93 => ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
% 11.90/5.93 = ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_699_mlex__snd__decrI,axiom,
% 11.90/5.93 ! [A3: nat,A6: nat,B2: nat,B4: nat,N6: nat] :
% 11.90/5.93 ( ( A3 = A6 )
% 11.90/5.93 => ( ( ord_less @ nat @ B2 @ B4 )
% 11.90/5.93 => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_700_mlex__fst__decrI,axiom,
% 11.90/5.93 ! [A3: nat,A6: nat,B2: nat,N6: nat,B4: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ A3 @ A6 )
% 11.90/5.93 => ( ( ord_less @ nat @ B2 @ N6 )
% 11.90/5.93 => ( ( ord_less @ nat @ B4 @ N6 )
% 11.90/5.93 => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N6 ) @ B4 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_701_mlex__bound,axiom,
% 11.90/5.93 ! [A3: nat,A4: nat,B2: nat,N6: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ A3 @ A4 )
% 11.90/5.93 => ( ( ord_less @ nat @ B2 @ N6 )
% 11.90/5.93 => ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A3 @ N6 ) @ B2 ) @ ( times_times @ nat @ A4 @ N6 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_702_mult__eq__self__implies__10,axiom,
% 11.90/5.93 ! [M2: nat,N: nat] :
% 11.90/5.93 ( ( M2
% 11.90/5.93 = ( times_times @ nat @ M2 @ N ) )
% 11.90/5.93 => ( ( N
% 11.90/5.93 = ( one_one @ nat ) )
% 11.90/5.93 | ( M2
% 11.90/5.93 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_703_map__comp__Some__iff,axiom,
% 11.90/5.93 ! [A: $tType,C: $tType,B: $tType,M1: B > ( option @ A ),M22: C > ( option @ B ),K: C,V: A] :
% 11.90/5.93 ( ( ( map_comp @ B @ A @ C @ M1 @ M22 @ K )
% 11.90/5.93 = ( some @ A @ V ) )
% 11.90/5.93 = ( ? [K6: B] :
% 11.90/5.93 ( ( ( M22 @ K )
% 11.90/5.93 = ( some @ B @ K6 ) )
% 11.90/5.93 & ( ( M1 @ K6 )
% 11.90/5.93 = ( some @ A @ V ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_704_field__le__mult__one__interval,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ! [Z3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
% 11.90/5.93 => ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
% 11.90/5.93 => ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_705_abs__diff__le__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [X: A,A3: A,R: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 11.90/5.93 = ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 11.90/5.93 & ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_706_abs__diff__triangle__ineq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_707_abs__triangle__ineq4,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ordere166539214618696060dd_abs @ A )
% 11.90/5.93 => ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_708_abs__diff__less__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [X: A,A3: A,R: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
% 11.90/5.93 = ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
% 11.90/5.93 & ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_709_sum__squares__ge__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_ring @ A )
% 11.90/5.93 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_710_mult__left__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord181362715937106298miring @ A )
% 11.90/5.93 => ! [C2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_711_mult__le__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord181362715937106298miring @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_712_mult__right__le__one__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 11.90/5.93 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_713_mult__left__le__one__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
% 11.90/5.93 => ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_714_not__sum__squares__lt__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_ring @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_715_nat__diff__split,axiom,
% 11.90/5.93 ! [P: nat > $o,A3: nat,B2: nat] :
% 11.90/5.93 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 11.90/5.93 = ( ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.93 => ( P @ ( zero_zero @ nat ) ) )
% 11.90/5.93 & ! [D5: nat] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ nat @ B2 @ D5 ) )
% 11.90/5.93 => ( P @ D5 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_716_nat__diff__split__asm,axiom,
% 11.90/5.93 ! [P: nat > $o,A3: nat,B2: nat] :
% 11.90/5.93 ( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
% 11.90/5.93 = ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.93 & ~ ( P @ ( zero_zero @ nat ) ) )
% 11.90/5.93 | ? [D5: nat] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ nat @ B2 @ D5 ) )
% 11.90/5.93 & ~ ( P @ D5 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_717_dbl__dec__def,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ( ( neg_numeral_dbl_dec @ A )
% 11.90/5.93 = ( ^ [X3: A] : ( minus_minus @ A @ ( plus_plus @ A @ X3 @ X3 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_718_mult__le__cancel__left1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 11.90/5.93 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_719_mult__le__cancel__left2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [C2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 11.90/5.93 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_720_mult__le__cancel__right1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
% 11.90/5.93 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_721_mult__le__cancel__right2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [A3: A,C2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
% 11.90/5.93 & ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_722_mult__less__cancel__left1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ C2 @ ( times_times @ A @ C2 @ B2 ) )
% 11.90/5.93 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 11.90/5.93 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_723_mult__less__cancel__left2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [C2: A,A3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
% 11.90/5.93 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 11.90/5.93 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_724_mult__less__cancel__right1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [C2: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ C2 @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
% 11.90/5.93 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_725_mult__less__cancel__right2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_idom @ A )
% 11.90/5.93 => ! [A3: A,C2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
% 11.90/5.93 = ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
% 11.90/5.93 & ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_726_convex__bound__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord6961819062388156250ring_1 @ A )
% 11.90/5.93 => ! [X: A,A3: A,Y: A,U: A,V: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ X @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ Y @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
% 11.90/5.93 => ( ( ( plus_plus @ A @ U @ V )
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A3 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_727_is__num_Ocases,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( neg_numeral_is_num @ A @ A3 )
% 11.90/5.93 => ( ( A3
% 11.90/5.93 != ( one_one @ A ) )
% 11.90/5.93 => ( ! [X2: A] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( uminus_uminus @ A @ X2 ) )
% 11.90/5.93 => ~ ( neg_numeral_is_num @ A @ X2 ) )
% 11.90/5.93 => ~ ! [X2: A,Y4: A] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ A @ X2 @ Y4 ) )
% 11.90/5.93 => ( ( neg_numeral_is_num @ A @ X2 )
% 11.90/5.93 => ~ ( neg_numeral_is_num @ A @ Y4 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_728_is__num_Osimps,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( neg_numeral @ A )
% 11.90/5.93 => ( ( neg_numeral_is_num @ A )
% 11.90/5.93 = ( ^ [A7: A] :
% 11.90/5.93 ( ( A7
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 | ? [X3: A] :
% 11.90/5.93 ( ( A7
% 11.90/5.93 = ( uminus_uminus @ A @ X3 ) )
% 11.90/5.93 & ( neg_numeral_is_num @ A @ X3 ) )
% 11.90/5.93 | ? [X3: A,Y3: A] :
% 11.90/5.93 ( ( A7
% 11.90/5.93 = ( plus_plus @ A @ X3 @ Y3 ) )
% 11.90/5.93 & ( neg_numeral_is_num @ A @ X3 )
% 11.90/5.93 & ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_729_sum__squares__gt__zero__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord4710134922213307826strict @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
% 11.90/5.93 = ( ( X
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 | ( Y
% 11.90/5.93 != ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_730_sum__squares__le__zero__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord4710134922213307826strict @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
% 11.90/5.93 = ( ( X
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 & ( Y
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_731_incr__lemma,axiom,
% 11.90/5.93 ! [D3: int,Z2: int,X: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
% 11.90/5.93 => ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_732_decr__lemma,axiom,
% 11.90/5.93 ! [D3: int,X: int,Z2: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
% 11.90/5.93 => ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_733_sum__squares__eq__zero__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linord4710134922213307826strict @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 = ( ( X
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 & ( Y
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_734_frac__unique__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A,A3: A] :
% 11.90/5.93 ( ( ( archimedean_frac @ A @ X )
% 11.90/5.93 = A3 )
% 11.90/5.93 = ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
% 11.90/5.93 & ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 & ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_735_scaling__mono,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [U: A,V: A,R: A,S: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ U @ V )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
% 11.90/5.93 => ( ( ord_less_eq @ A @ R @ S )
% 11.90/5.93 => ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_736_divide__le__eq__1__neg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 11.90/5.93 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_737_divide__le__eq__1__pos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 11.90/5.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_738_div__by__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semidom_divide @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_739_div__self,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semidom_divide @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ A3 @ A3 )
% 11.90/5.93 = ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_740_zero__eq__1__divide__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ( zero_zero @ A )
% 11.90/5.93 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 11.90/5.93 = ( A3
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_741_one__divide__eq__0__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 = ( A3
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_742_eq__divide__eq__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ( one_one @ A )
% 11.90/5.93 = ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ( A3
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 & ( A3 = B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_743_divide__eq__eq__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ( divide_divide @ A @ B2 @ A3 )
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 = ( ( A3
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 & ( A3 = B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_744_divide__self__if,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ( A3
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ A3 @ A3 )
% 11.90/5.93 = ( zero_zero @ A ) ) )
% 11.90/5.93 & ( ( A3
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ A3 @ A3 )
% 11.90/5.93 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_745_divide__self,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ A3 @ A3 )
% 11.90/5.93 = ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_746_one__eq__divide__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ( one_one @ A )
% 11.90/5.93 = ( divide_divide @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 & ( A3 = B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_747_divide__eq__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 = ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 & ( A3 = B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_748_divide__minus1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.93 = ( uminus_uminus @ A @ X ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_749_zero__le__divide__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_750_divide__le__0__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.93 = ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_751_zero__less__divide__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 11.90/5.93 = ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_752_less__divide__eq__1__pos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_753_less__divide__eq__1__neg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_754_divide__less__eq__1__pos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 11.90/5.93 = ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_755_divide__less__eq__1__neg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 11.90/5.93 = ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_756_divide__less__0__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
% 11.90/5.93 = ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_757_nonzero__divide__mult__cancel__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_758_nonzero__divide__mult__cancel__left,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 11.90/5.93 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_759_le__divide__eq__1__pos,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_760_le__divide__eq__1__neg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_761_int__distrib_I2_J,axiom,
% 11.90/5.93 ! [W: int,Z1: int,Z22: int] :
% 11.90/5.93 ( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_762_int__distrib_I1_J,axiom,
% 11.90/5.93 ! [Z1: int,Z22: int,W: int] :
% 11.90/5.93 ( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_763_plus__int__code_I2_J,axiom,
% 11.90/5.93 ! [L: int] :
% 11.90/5.93 ( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
% 11.90/5.93 = L ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_764_plus__int__code_I1_J,axiom,
% 11.90/5.93 ! [K: int] :
% 11.90/5.93 ( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
% 11.90/5.93 = K ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_765_zero__integer_Orsp,axiom,
% 11.90/5.93 ( ( zero_zero @ int )
% 11.90/5.93 = ( zero_zero @ int ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_766_add__divide__distrib,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_767_int__div__pos__eq,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 11.90/5.93 => ( ( ord_less @ int @ R @ B2 )
% 11.90/5.93 => ( ( divide_divide @ int @ A3 @ B2 )
% 11.90/5.93 = Q3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_768_int__div__neg__eq,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( ord_less @ int @ B2 @ R )
% 11.90/5.93 => ( ( divide_divide @ int @ A3 @ B2 )
% 11.90/5.93 = Q3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_769_split__zdiv,axiom,
% 11.90/5.93 ! [P: int > $o,N: int,K: int] :
% 11.90/5.93 ( ( P @ ( divide_divide @ int @ N @ K ) )
% 11.90/5.93 = ( ( ( K
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 => ( P @ ( zero_zero @ int ) ) )
% 11.90/5.93 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 11.90/5.93 => ! [I3: int,J2: int] :
% 11.90/5.93 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
% 11.90/5.93 & ( ord_less @ int @ J2 @ K )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 11.90/5.93 => ( P @ I3 ) ) )
% 11.90/5.93 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 11.90/5.93 => ! [I3: int,J2: int] :
% 11.90/5.93 ( ( ( ord_less @ int @ K @ J2 )
% 11.90/5.93 & ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 11.90/5.93 => ( P @ I3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_770_frac__lt__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_771_frac__1__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 11.90/5.93 = ( archimedean_frac @ A @ X ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_772_right__inverse__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.93 = ( one_one @ A ) )
% 11.90/5.93 = ( A3 = B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_773_less__divide__eq__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 & ( ord_less @ A @ A3 @ B2 ) )
% 11.90/5.93 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 & ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_774_divide__less__eq__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 & ( ord_less @ A @ B2 @ A3 ) )
% 11.90/5.93 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 & ( ord_less @ A @ A3 @ B2 ) )
% 11.90/5.93 | ( A3
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_775_divide__add__eq__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [Z2: A,X: A,Y: A] :
% 11.90/5.93 ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_776_add__divide__eq__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [Z2: A,X: A,Y: A] :
% 11.90/5.93 ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z2 ) )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_777_add__num__frac,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [Y: A,Z2: A,X: A] :
% 11.90/5.93 ( ( Y
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X @ Y ) )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_778_add__frac__num,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [Y: A,X: A,Z2: A] :
% 11.90/5.93 ( ( Y
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z2 )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_779_add__frac__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [Y: A,Z2: A,X: A,W: A] :
% 11.90/5.93 ( ( Y
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W @ Z2 ) )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_780_add__divide__eq__if__simps_I1_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [Z2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( ( Z2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
% 11.90/5.93 = A3 ) )
% 11.90/5.93 & ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_781_add__divide__eq__if__simps_I2_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [Z2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( ( Z2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
% 11.90/5.93 = B2 ) )
% 11.90/5.93 & ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_782_less__half__sum,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_783_gt__half__sum,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.93 => ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_784_divide__eq__minus__1__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( field @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.93 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.93 = ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 & ( A3
% 11.90/5.93 = ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_785_div__pos__geq,axiom,
% 11.90/5.93 ! [L: int,K: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 11.90/5.93 => ( ( ord_less_eq @ int @ L @ K )
% 11.90/5.93 => ( ( divide_divide @ int @ K @ L )
% 11.90/5.93 = ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_786_frac__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ( archimedean_frac @ A @ X )
% 11.90/5.93 = X )
% 11.90/5.93 = ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.93 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_787_frac__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 11.90/5.93 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
% 11.90/5.93 = ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
% 11.90/5.93 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 11.90/5.93 => ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
% 11.90/5.93 = ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_788_incr__mult__lemma,axiom,
% 11.90/5.93 ! [D3: int,P: int > $o,K: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ X2 @ D3 ) ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ( P @ X7 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ X7 @ ( times_times @ int @ K @ D3 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_789_divide__le__eq__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 & ( ord_less_eq @ A @ B2 @ A3 ) )
% 11.90/5.93 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 11.90/5.93 | ( A3
% 11.90/5.93 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_790_le__divide__eq__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( linordered_field @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
% 11.90/5.93 = ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.93 & ( ord_less_eq @ A @ A3 @ B2 ) )
% 11.90/5.93 | ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/5.93 & ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_791_minus__divide__add__eq__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [Z2: A,X: A,Y: A] :
% 11.90/5.93 ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_792_add__divide__eq__if__simps_I3_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( division_ring @ A )
% 11.90/5.93 => ! [Z2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( ( Z2
% 11.90/5.93 = ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
% 11.90/5.93 = B2 ) )
% 11.90/5.93 & ( ( Z2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
% 11.90/5.93 = ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_793_frac__neg,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 11.90/5.93 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 11.90/5.93 = ( zero_zero @ A ) ) )
% 11.90/5.93 & ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 11.90/5.93 => ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
% 11.90/5.93 = ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_794_div__mult__self4,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,C2: A,A3: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_795_div__mult__self3,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,C2: A,A3: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_796_div__mult__self2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_797_div__mult__self1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_798_div__minus1__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid8851590272496341667cancel @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.93 = ( uminus_uminus @ A @ A3 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_799_div__pos__neg__trivial,axiom,
% 11.90/5.93 ! [K: int,L: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( divide_divide @ int @ K @ L )
% 11.90/5.93 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_800_bits__div__by__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( bit_semiring_bits @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_801_dividend__less__times__div,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.93 => ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_802_dividend__less__div__times,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.93 => ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_803_split__div,axiom,
% 11.90/5.93 ! [P: nat > $o,M2: nat,N: nat] :
% 11.90/5.93 ( ( P @ ( divide_divide @ nat @ M2 @ N ) )
% 11.90/5.93 = ( ( ( N
% 11.90/5.93 = ( zero_zero @ nat ) )
% 11.90/5.93 => ( P @ ( zero_zero @ nat ) ) )
% 11.90/5.93 & ( ( N
% 11.90/5.93 != ( zero_zero @ nat ) )
% 11.90/5.93 => ! [I3: nat,J2: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ J2 @ N )
% 11.90/5.93 => ( ( M2
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J2 ) )
% 11.90/5.93 => ( P @ I3 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_804_div__less__dividend,axiom,
% 11.90/5.93 ! [N: nat,M2: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ ( one_one @ nat ) @ N )
% 11.90/5.93 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 11.90/5.93 => ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_805_div__eq__dividend__iff,axiom,
% 11.90/5.93 ! [M2: nat,N: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 11.90/5.93 => ( ( ( divide_divide @ nat @ M2 @ N )
% 11.90/5.93 = M2 )
% 11.90/5.93 = ( N
% 11.90/5.93 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_806_div__add__self1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_807_div__add__self2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( B2
% 11.90/5.93 != ( zero_zero @ A ) )
% 11.90/5.93 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.93 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_808_upto_Opinduct,axiom,
% 11.90/5.93 ! [A0: int,A1: int,P: int > int > $o] :
% 11.90/5.93 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 11.90/5.93 => ( ! [I2: int,J3: int] :
% 11.90/5.93 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J3 ) )
% 11.90/5.93 => ( ( ( ord_less_eq @ int @ I2 @ J3 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) )
% 11.90/5.93 => ( P @ I2 @ J3 ) ) )
% 11.90/5.93 => ( P @ A0 @ A1 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_809_eucl__rel__int_Osimps,axiom,
% 11.90/5.93 ( eucl_rel_int
% 11.90/5.93 = ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
% 11.90/5.93 ( ? [K4: int] :
% 11.90/5.93 ( ( A12 = K4 )
% 11.90/5.93 & ( A23
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 & ( A33
% 11.90/5.93 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K4 ) ) )
% 11.90/5.93 | ? [L3: int,K4: int,Q5: int] :
% 11.90/5.93 ( ( A12 = K4 )
% 11.90/5.93 & ( A23 = L3 )
% 11.90/5.93 & ( A33
% 11.90/5.93 = ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
% 11.90/5.93 & ( L3
% 11.90/5.93 != ( zero_zero @ int ) )
% 11.90/5.93 & ( K4
% 11.90/5.93 = ( times_times @ int @ Q5 @ L3 ) ) )
% 11.90/5.93 | ? [R3: int,L3: int,K4: int,Q5: int] :
% 11.90/5.93 ( ( A12 = K4 )
% 11.90/5.93 & ( A23 = L3 )
% 11.90/5.93 & ( A33
% 11.90/5.93 = ( product_Pair @ int @ int @ Q5 @ R3 ) )
% 11.90/5.93 & ( ( sgn_sgn @ int @ R3 )
% 11.90/5.93 = ( sgn_sgn @ int @ L3 ) )
% 11.90/5.93 & ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ L3 ) )
% 11.90/5.93 & ( K4
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ Q5 @ L3 ) @ R3 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_810_eucl__rel__int_Ocases,axiom,
% 11.90/5.93 ! [A1: int,A22: int,A32: product_prod @ int @ int] :
% 11.90/5.93 ( ( eucl_rel_int @ A1 @ A22 @ A32 )
% 11.90/5.93 => ( ( ( A22
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 => ( A32
% 11.90/5.93 != ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
% 11.90/5.93 => ( ! [Q2: int] :
% 11.90/5.93 ( ( A32
% 11.90/5.93 = ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) )
% 11.90/5.93 => ( ( A22
% 11.90/5.93 != ( zero_zero @ int ) )
% 11.90/5.93 => ( A1
% 11.90/5.93 != ( times_times @ int @ Q2 @ A22 ) ) ) )
% 11.90/5.93 => ~ ! [R2: int,Q2: int] :
% 11.90/5.93 ( ( A32
% 11.90/5.93 = ( product_Pair @ int @ int @ Q2 @ R2 ) )
% 11.90/5.93 => ( ( ( sgn_sgn @ int @ R2 )
% 11.90/5.93 = ( sgn_sgn @ int @ A22 ) )
% 11.90/5.93 => ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ A22 ) )
% 11.90/5.93 => ( A1
% 11.90/5.93 != ( plus_plus @ int @ ( times_times @ int @ Q2 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_811_eucl__rel__int__iff,axiom,
% 11.90/5.93 ! [K: int,L: int,Q3: int,R: int] :
% 11.90/5.93 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.93 = ( ( K
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ L @ Q3 ) @ R ) )
% 11.90/5.93 & ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 11.90/5.93 & ( ord_less @ int @ R @ L ) ) )
% 11.90/5.93 & ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
% 11.90/5.93 => ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( ord_less @ int @ L @ R )
% 11.90/5.93 & ( ord_less_eq @ int @ R @ ( zero_zero @ int ) ) ) )
% 11.90/5.93 & ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
% 11.90/5.93 => ( Q3
% 11.90/5.93 = ( zero_zero @ int ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_812_power_Opower__eq__if,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( power2 @ A )
% 11.90/5.93 = ( ^ [One: A,Times: A > A > A,P6: A,M: nat] :
% 11.90/5.93 ( if @ A
% 11.90/5.93 @ ( M
% 11.90/5.93 = ( zero_zero @ nat ) )
% 11.90/5.93 @ One
% 11.90/5.93 @ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_813_split__neg__lemma,axiom,
% 11.90/5.93 ! [K: int,P: int > int > $o,N: int] :
% 11.90/5.93 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 11.90/5.93 = ( ! [I3: int,J2: int] :
% 11.90/5.93 ( ( ( ord_less @ int @ K @ J2 )
% 11.90/5.93 & ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 11.90/5.93 => ( P @ I3 @ J2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_814_split__pos__lemma,axiom,
% 11.90/5.93 ! [K: int,P: int > int > $o,N: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 11.90/5.93 => ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
% 11.90/5.93 = ( ! [I3: int,J2: int] :
% 11.90/5.93 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
% 11.90/5.93 & ( ord_less @ int @ J2 @ K )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 11.90/5.93 => ( P @ I3 @ J2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_815_mod__add__self2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_816_mod__add__self1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_817_mod__by__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semidom_modulo @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_818_bits__mod__by__1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( bit_semiring_bits @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_819_mod__mult__self1,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_820_mod__mult__self2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_821_mod__mult__self3,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [C2: A,B2: A,A3: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_822_mod__mult__self4,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [B2: A,C2: A,A3: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_823_mod__minus1__right,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid8851590272496341667cancel @ A )
% 11.90/5.93 => ! [A3: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.93 = ( zero_zero @ A ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_824_mod__int__unique,axiom,
% 11.90/5.93 ! [K: int,L: int,Q3: int,R: int] :
% 11.90/5.93 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.93 => ( ( modulo_modulo @ int @ K @ L )
% 11.90/5.93 = R ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_825_unique__quotient,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int,Q4: int,R4: int] :
% 11.90/5.93 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.93 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R4 ) )
% 11.90/5.93 => ( Q3 = Q4 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_826_unique__remainder,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int,Q4: int,R4: int] :
% 11.90/5.93 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.93 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ R4 ) )
% 11.90/5.93 => ( R = R4 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_827_full__exhaustive__int_H_Ocases,axiom,
% 11.90/5.93 ! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
% 11.90/5.93 ~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
% 11.90/5.93 ( X
% 11.90/5.93 != ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_828_exhaustive__int_H_Ocases,axiom,
% 11.90/5.93 ! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
% 11.90/5.93 ~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I2: int] :
% 11.90/5.93 ( X
% 11.90/5.93 != ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_829_small__lazy_H_Ocases,axiom,
% 11.90/5.93 ! [X: product_prod @ int @ int] :
% 11.90/5.93 ~ ! [D2: int,I2: int] :
% 11.90/5.93 ( X
% 11.90/5.93 != ( product_Pair @ int @ int @ D2 @ I2 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_830_eucl__rel__int,axiom,
% 11.90/5.93 ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_831_mod__add__right__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_832_mod__add__left__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_833_mod__add__cong,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,C2: A,A6: A,B2: A,B4: A] :
% 11.90/5.93 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ A6 @ C2 ) )
% 11.90/5.93 => ( ( ( modulo_modulo @ A @ B2 @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ B4 @ C2 ) )
% 11.90/5.93 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ ( plus_plus @ A @ A6 @ B4 ) @ C2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_834_mod__add__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_835_eucl__rel__int__by0,axiom,
% 11.90/5.93 ! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_836_div__int__unique,axiom,
% 11.90/5.93 ! [K: int,L: int,Q3: int,R: int] :
% 11.90/5.93 ( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.93 => ( ( divide_divide @ int @ K @ L )
% 11.90/5.93 = Q3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_837_mod__eqE,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid8851590272496341667cancel @ A )
% 11.90/5.93 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.93 ( ( ( modulo_modulo @ A @ A3 @ C2 )
% 11.90/5.93 = ( modulo_modulo @ A @ B2 @ C2 ) )
% 11.90/5.93 => ~ ! [D2: A] :
% 11.90/5.93 ( B2
% 11.90/5.93 != ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_838_div__add1__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_839_mult__div__mod__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring_modulo @ A )
% 11.90/5.93 => ! [B2: A,A3: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_840_mod__mult__div__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring_modulo @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_841_mod__div__mult__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring_modulo @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_842_div__mult__mod__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring_modulo @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
% 11.90/5.93 = A3 ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_843_mod__div__decomp,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semiring_modulo @ A )
% 11.90/5.93 => ! [A3: A,B2: A] :
% 11.90/5.93 ( A3
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_844_cancel__div__mod__rules_I1_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semidom_modulo @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_845_cancel__div__mod__rules_I2_J,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( semidom_modulo @ A )
% 11.90/5.93 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.93 ( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_846_div__mult1__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.93 ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_847_eucl__rel__int__dividesI,axiom,
% 11.90/5.93 ! [L: int,K: int,Q3: int] :
% 11.90/5.93 ( ( L
% 11.90/5.93 != ( zero_zero @ int ) )
% 11.90/5.93 => ( ( K
% 11.90/5.93 = ( times_times @ int @ Q3 @ L ) )
% 11.90/5.93 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_848_div__mod__decomp__int,axiom,
% 11.90/5.93 ! [A4: int,N: int] :
% 11.90/5.93 ( A4
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A4 @ N ) @ N ) @ ( modulo_modulo @ int @ A4 @ N ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_849_mod__pos__neg__trivial,axiom,
% 11.90/5.93 ! [K: int,L: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( modulo_modulo @ int @ K @ L )
% 11.90/5.93 = ( plus_plus @ int @ K @ L ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_850_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.93 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.93 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.93 => ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.93 = ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_851_int__mod__pos__eq,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
% 11.90/5.93 => ( ( ord_less @ int @ R @ B2 )
% 11.90/5.93 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 11.90/5.93 = R ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_852_int__mod__neg__eq,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int] :
% 11.90/5.93 ( ( A3
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B2 @ Q3 ) @ R ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
% 11.90/5.93 => ( ( ord_less @ int @ B2 @ R )
% 11.90/5.93 => ( ( modulo_modulo @ int @ A3 @ B2 )
% 11.90/5.93 = R ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_853_split__zmod,axiom,
% 11.90/5.93 ! [P: int > $o,N: int,K: int] :
% 11.90/5.93 ( ( P @ ( modulo_modulo @ int @ N @ K ) )
% 11.90/5.93 = ( ( ( K
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 => ( P @ N ) )
% 11.90/5.93 & ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
% 11.90/5.93 => ! [I3: int,J2: int] :
% 11.90/5.93 ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J2 )
% 11.90/5.93 & ( ord_less @ int @ J2 @ K )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 11.90/5.93 => ( P @ J2 ) ) )
% 11.90/5.93 & ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 11.90/5.93 => ! [I3: int,J2: int] :
% 11.90/5.93 ( ( ( ord_less @ int @ K @ J2 )
% 11.90/5.93 & ( ord_less_eq @ int @ J2 @ ( zero_zero @ int ) )
% 11.90/5.93 & ( N
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ K @ I3 ) @ J2 ) ) )
% 11.90/5.93 => ( P @ J2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_854_zmod__zmult2__eq,axiom,
% 11.90/5.93 ! [C2: int,A3: int,B2: int] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
% 11.90/5.93 => ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_855_zminus1__lemma,axiom,
% 11.90/5.93 ! [A3: int,B2: int,Q3: int,R: int] :
% 11.90/5.93 ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.93 => ( ( B2
% 11.90/5.93 != ( zero_zero @ int ) )
% 11.90/5.93 => ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
% 11.90/5.93 @ ( product_Pair @ int @ int
% 11.90/5.93 @ ( if @ int
% 11.90/5.93 @ ( R
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 @ ( uminus_uminus @ int @ Q3 )
% 11.90/5.93 @ ( minus_minus @ int @ ( uminus_uminus @ int @ Q3 ) @ ( one_one @ int ) ) )
% 11.90/5.93 @ ( if @ int
% 11.90/5.93 @ ( R
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 @ ( zero_zero @ int )
% 11.90/5.93 @ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_856_eucl__rel__int__remainderI,axiom,
% 11.90/5.93 ! [R: int,L: int,K: int,Q3: int] :
% 11.90/5.93 ( ( ( sgn_sgn @ int @ R )
% 11.90/5.93 = ( sgn_sgn @ int @ L ) )
% 11.90/5.93 => ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
% 11.90/5.93 => ( ( K
% 11.90/5.93 = ( plus_plus @ int @ ( times_times @ int @ Q3 @ L ) @ R ) )
% 11.90/5.93 => ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q3 @ R ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_857_verit__le__mono__div__int,axiom,
% 11.90/5.93 ! [A4: int,B6: int,N: int] :
% 11.90/5.93 ( ( ord_less @ int @ A4 @ B6 )
% 11.90/5.93 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 11.90/5.93 => ( ord_less_eq @ int
% 11.90/5.93 @ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
% 11.90/5.93 @ ( if @ int
% 11.90/5.93 @ ( ( modulo_modulo @ int @ B6 @ N )
% 11.90/5.93 = ( zero_zero @ int ) )
% 11.90/5.93 @ ( one_one @ int )
% 11.90/5.93 @ ( zero_zero @ int ) ) )
% 11.90/5.93 @ ( divide_divide @ int @ B6 @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_858_bezw__0,axiom,
% 11.90/5.93 ! [X: nat] :
% 11.90/5.93 ( ( bezw @ X @ ( zero_zero @ nat ) )
% 11.90/5.93 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_859_verit__le__mono__div,axiom,
% 11.90/5.93 ! [A4: nat,B6: nat,N: nat] :
% 11.90/5.93 ( ( ord_less @ nat @ A4 @ B6 )
% 11.90/5.93 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.93 => ( ord_less_eq @ nat
% 11.90/5.93 @ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
% 11.90/5.93 @ ( if @ nat
% 11.90/5.93 @ ( ( modulo_modulo @ nat @ B6 @ N )
% 11.90/5.93 = ( zero_zero @ nat ) )
% 11.90/5.93 @ ( one_one @ nat )
% 11.90/5.93 @ ( zero_zero @ nat ) ) )
% 11.90/5.93 @ ( divide_divide @ nat @ B6 @ N ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_860_zero__le__ceiling,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 11.90/5.93 = ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_861_ceiling__less__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
% 11.90/5.93 = ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_862_floor__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 11.90/5.93 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
% 11.90/5.93 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
% 11.90/5.93 & ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
% 11.90/5.93 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
% 11.90/5.93 = ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_863_cpmi,axiom,
% 11.90/5.93 ! [D4: int,P: int > $o,P2: int > $o,B6: set @ int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.93 => ( ? [Z6: int] :
% 11.90/5.93 ! [X2: int] :
% 11.90/5.93 ( ( ord_less @ int @ X2 @ Z6 )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 = ( P2 @ X2 ) ) )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ B6 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ( ! [X2: int,K5: int] :
% 11.90/5.93 ( ( P2 @ X2 )
% 11.90/5.93 = ( P2 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K5 @ D4 ) ) ) )
% 11.90/5.93 => ( ( ? [X8: int] : ( P @ X8 ) )
% 11.90/5.93 = ( ? [X3: int] :
% 11.90/5.93 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 & ( P2 @ X3 ) )
% 11.90/5.93 | ? [X3: int] :
% 11.90/5.93 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 & ? [Y3: int] :
% 11.90/5.93 ( ( member @ int @ Y3 @ B6 )
% 11.90/5.93 & ( P @ ( plus_plus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_864_cppi,axiom,
% 11.90/5.93 ! [D4: int,P: int > $o,P2: int > $o,A4: set @ int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.93 => ( ? [Z6: int] :
% 11.90/5.93 ! [X2: int] :
% 11.90/5.93 ( ( ord_less @ int @ Z6 @ X2 )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 = ( P2 @ X2 ) ) )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ A4 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ( ! [X2: int,K5: int] :
% 11.90/5.93 ( ( P2 @ X2 )
% 11.90/5.93 = ( P2 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K5 @ D4 ) ) ) )
% 11.90/5.93 => ( ( ? [X8: int] : ( P @ X8 ) )
% 11.90/5.93 = ( ? [X3: int] :
% 11.90/5.93 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 & ( P2 @ X3 ) )
% 11.90/5.93 | ? [X3: int] :
% 11.90/5.93 ( ( member @ int @ X3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 & ? [Y3: int] :
% 11.90/5.93 ( ( member @ int @ Y3 @ A4 )
% 11.90/5.93 & ( P @ ( minus_minus @ int @ Y3 @ X3 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_865_bset_I6_J,axiom,
% 11.90/5.93 ! [D4: int,B6: set @ int,T2: int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ X7 @ T2 )
% 11.90/5.93 => ( ord_less_eq @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_866_bset_I8_J,axiom,
% 11.90/5.93 ! [D4: int,T2: int,B6: set @ int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.93 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ T2 @ X7 )
% 11.90/5.93 => ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_867_aset_I6_J,axiom,
% 11.90/5.93 ! [D4: int,T2: int,A4: set @ int] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.93 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ord_less_eq @ int @ X7 @ T2 )
% 11.90/5.93 => ( ord_less_eq @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_868_atLeastAtMost__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( ord @ A )
% 11.90/5.93 => ! [I: A,L: A,U: A] :
% 11.90/5.93 ( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 11.90/5.93 = ( ( ord_less_eq @ A @ L @ I )
% 11.90/5.93 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_869_Icc__eq__Icc,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( order @ A )
% 11.90/5.93 => ! [L: A,H3: A,L4: A,H4: A] :
% 11.90/5.93 ( ( ( set_or1337092689740270186AtMost @ A @ L @ H3 )
% 11.90/5.93 = ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) )
% 11.90/5.93 = ( ( ( L = L4 )
% 11.90/5.93 & ( H3 = H4 ) )
% 11.90/5.93 | ( ~ ( ord_less_eq @ A @ L @ H3 )
% 11.90/5.93 & ~ ( ord_less_eq @ A @ L4 @ H4 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_870_atLeastatMost__subset__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( preorder @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.93 = ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.93 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_871_floor__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
% 11.90/5.93 = ( one_one @ int ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_872_ceiling__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
% 11.90/5.93 = ( one_one @ int ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_873_floor__add2,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A,Y: A] :
% 11.90/5.93 ( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
% 11.90/5.93 | ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
% 11.90/5.93 => ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
% 11.90/5.93 = ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_874_zero__less__floor,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 11.90/5.93 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_875_floor__le__zero,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
% 11.90/5.93 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_876_one__le__floor,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 11.90/5.93 = ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_877_floor__less__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 11.90/5.93 = ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_878_ceiling__le__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
% 11.90/5.93 = ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_879_one__less__ceiling,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
% 11.90/5.93 = ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_880_ceiling__add__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
% 11.90/5.93 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_881_floor__diff__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 11.90/5.93 = ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_882_ceiling__diff__one,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A] :
% 11.90/5.93 ( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
% 11.90/5.93 = ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_883_atLeastatMost__psubset__iff,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( preorder @ A )
% 11.90/5.93 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.93 ( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.93 = ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.93 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.93 & ( ord_less_eq @ A @ B2 @ D3 )
% 11.90/5.93 & ( ( ord_less @ A @ C2 @ A3 )
% 11.90/5.93 | ( ord_less @ A @ B2 @ D3 ) ) ) )
% 11.90/5.93 & ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_884_nat__mod__eq__iff,axiom,
% 11.90/5.93 ! [X: nat,N: nat,Y: nat] :
% 11.90/5.93 ( ( ( modulo_modulo @ nat @ X @ N )
% 11.90/5.93 = ( modulo_modulo @ nat @ Y @ N ) )
% 11.90/5.93 = ( ? [Q1: nat,Q22: nat] :
% 11.90/5.93 ( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
% 11.90/5.93 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_885_le__floor__add,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_886_bset_I1_J,axiom,
% 11.90/5.93 ! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
% 11.90/5.93 ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ B6 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ B6 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( Q @ X2 )
% 11.90/5.93 => ( Q @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ( P @ X7 )
% 11.90/5.93 & ( Q @ X7 ) )
% 11.90/5.93 => ( ( P @ ( minus_minus @ int @ X7 @ D4 ) )
% 11.90/5.93 & ( Q @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_887_bset_I2_J,axiom,
% 11.90/5.93 ! [D4: int,B6: set @ int,P: int > $o,Q: int > $o] :
% 11.90/5.93 ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ B6 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ B6 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( Q @ X2 )
% 11.90/5.93 => ( Q @ ( minus_minus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ( P @ X7 )
% 11.90/5.93 | ( Q @ X7 ) )
% 11.90/5.93 => ( ( P @ ( minus_minus @ int @ X7 @ D4 ) )
% 11.90/5.93 | ( Q @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_888_aset_I1_J,axiom,
% 11.90/5.93 ! [D4: int,A4: set @ int,P: int > $o,Q: int > $o] :
% 11.90/5.93 ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ A4 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ A4 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( Q @ X2 )
% 11.90/5.93 => ( Q @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ( P @ X7 )
% 11.90/5.93 & ( Q @ X7 ) )
% 11.90/5.93 => ( ( P @ ( plus_plus @ int @ X7 @ D4 ) )
% 11.90/5.93 & ( Q @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_889_aset_I2_J,axiom,
% 11.90/5.93 ! [D4: int,A4: set @ int,P: int > $o,Q: int > $o] :
% 11.90/5.93 ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ A4 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( P @ X2 )
% 11.90/5.93 => ( P @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ( ! [X2: int] :
% 11.90/5.93 ( ! [Xa2: int] :
% 11.90/5.93 ( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb2: int] :
% 11.90/5.93 ( ( member @ int @ Xb2 @ A4 )
% 11.90/5.93 => ( X2
% 11.90/5.93 != ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
% 11.90/5.93 => ( ( Q @ X2 )
% 11.90/5.93 => ( Q @ ( plus_plus @ int @ X2 @ D4 ) ) ) )
% 11.90/5.93 => ! [X7: int] :
% 11.90/5.93 ( ! [Xa3: int] :
% 11.90/5.93 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.93 => ! [Xb3: int] :
% 11.90/5.93 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.93 => ( X7
% 11.90/5.93 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.93 => ( ( ( P @ X7 )
% 11.90/5.93 | ( Q @ X7 ) )
% 11.90/5.93 => ( ( P @ ( plus_plus @ int @ X7 @ D4 ) )
% 11.90/5.93 | ( Q @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_890_nat__mod__eq__lemma,axiom,
% 11.90/5.93 ! [X: nat,N: nat,Y: nat] :
% 11.90/5.93 ( ( ( modulo_modulo @ nat @ X @ N )
% 11.90/5.93 = ( modulo_modulo @ nat @ Y @ N ) )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ Y @ X )
% 11.90/5.93 => ? [Q2: nat] :
% 11.90/5.93 ( X
% 11.90/5.93 = ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q2 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_891_mod__eq__nat2E,axiom,
% 11.90/5.93 ! [M2: nat,Q3: nat,N: nat] :
% 11.90/5.93 ( ( ( modulo_modulo @ nat @ M2 @ Q3 )
% 11.90/5.93 = ( modulo_modulo @ nat @ N @ Q3 ) )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.93 => ~ ! [S6: nat] :
% 11.90/5.93 ( N
% 11.90/5.93 != ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q3 @ S6 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_892_mod__eq__nat1E,axiom,
% 11.90/5.93 ! [M2: nat,Q3: nat,N: nat] :
% 11.90/5.93 ( ( ( modulo_modulo @ nat @ M2 @ Q3 )
% 11.90/5.93 = ( modulo_modulo @ nat @ N @ Q3 ) )
% 11.90/5.93 => ( ( ord_less_eq @ nat @ N @ M2 )
% 11.90/5.93 => ~ ! [S6: nat] :
% 11.90/5.93 ( M2
% 11.90/5.93 != ( plus_plus @ nat @ N @ ( times_times @ nat @ Q3 @ S6 ) ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_893_ceiling__add__le,axiom,
% 11.90/5.93 ! [A: $tType] :
% 11.90/5.93 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.93 => ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% 11.90/5.93
% 11.90/5.93 thf(fact_894_mod__mult2__eq,axiom,
% 11.90/5.93 ! [M2: nat,N: nat,Q3: nat] :
% 11.90/5.93 ( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q3 ) )
% 11.90/5.93 = ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_895_div__mod__decomp,axiom,
% 11.90/5.94 ! [A4: nat,N: nat] :
% 11.90/5.94 ( A4
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A4 @ N ) @ N ) @ ( modulo_modulo @ nat @ A4 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_896_one__add__floor,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
% 11.90/5.94 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_897_split__mod,axiom,
% 11.90/5.94 ! [P: nat > $o,M2: nat,N: nat] :
% 11.90/5.94 ( ( P @ ( modulo_modulo @ nat @ M2 @ N ) )
% 11.90/5.94 = ( ( ( N
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 => ( P @ M2 ) )
% 11.90/5.94 & ( ( N
% 11.90/5.94 != ( zero_zero @ nat ) )
% 11.90/5.94 => ! [I3: nat,J2: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ J2 @ N )
% 11.90/5.94 => ( ( M2
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ N @ I3 ) @ J2 ) )
% 11.90/5.94 => ( P @ J2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_898_bset_I3_J,axiom,
% 11.90/5.94 ! [D4: int,T2: int,B6: set @ int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( X7 = T2 )
% 11.90/5.94 => ( ( minus_minus @ int @ X7 @ D4 )
% 11.90/5.94 = T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_899_bset_I4_J,axiom,
% 11.90/5.94 ! [D4: int,T2: int,B6: set @ int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ( ( member @ int @ T2 @ B6 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( X7 != T2 )
% 11.90/5.94 => ( ( minus_minus @ int @ X7 @ D4 )
% 11.90/5.94 != T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_900_bset_I5_J,axiom,
% 11.90/5.94 ! [D4: int,B6: set @ int,T2: int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( ord_less @ int @ X7 @ T2 )
% 11.90/5.94 => ( ord_less @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_901_bset_I7_J,axiom,
% 11.90/5.94 ! [D4: int,T2: int,B6: set @ int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ( ( member @ int @ T2 @ B6 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( ord_less @ int @ T2 @ X7 )
% 11.90/5.94 => ( ord_less @ int @ T2 @ ( minus_minus @ int @ X7 @ D4 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_902_aset_I3_J,axiom,
% 11.90/5.94 ! [D4: int,T2: int,A4: set @ int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( X7 = T2 )
% 11.90/5.94 => ( ( plus_plus @ int @ X7 @ D4 )
% 11.90/5.94 = T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_903_aset_I4_J,axiom,
% 11.90/5.94 ! [D4: int,T2: int,A4: set @ int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ( ( member @ int @ T2 @ A4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( X7 != T2 )
% 11.90/5.94 => ( ( plus_plus @ int @ X7 @ D4 )
% 11.90/5.94 != T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_904_aset_I5_J,axiom,
% 11.90/5.94 ! [D4: int,T2: int,A4: set @ int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ( ( member @ int @ T2 @ A4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( ord_less @ int @ X7 @ T2 )
% 11.90/5.94 => ( ord_less @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_905_aset_I7_J,axiom,
% 11.90/5.94 ! [D4: int,A4: set @ int,T2: int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( ord_less @ int @ T2 @ X7 )
% 11.90/5.94 => ( ord_less @ int @ T2 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_906_aset_I8_J,axiom,
% 11.90/5.94 ! [D4: int,A4: set @ int,T2: int] :
% 11.90/5.94 ( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( ord_less_eq @ int @ T2 @ X7 )
% 11.90/5.94 => ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X7 @ D4 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_907_ceiling__divide__lower,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Q3: A,P5: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
% 11.90/5.94 => ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) @ P5 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_908_floor__divide__upper,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Q3: A,P5: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ Q3 )
% 11.90/5.94 => ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q3 ) ) ) @ ( one_one @ A ) ) @ Q3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_909_power__decreasing__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [B2: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.94 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
% 11.90/5.94 = ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_910_divmod__nat__def,axiom,
% 11.90/5.94 ( divmod_nat
% 11.90/5.94 = ( ^ [M: nat,N5: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M @ N5 ) @ ( modulo_modulo @ nat @ M @ N5 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_911_distrib__left__NO__MATCH,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( semiring @ A )
% 11.90/5.94 => ! [X: B,Y: B,A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A3 )
% 11.90/5.94 => ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_912_distrib__right__NO__MATCH,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( semiring @ A )
% 11.90/5.94 => ! [X: B,Y: B,C2: A,A3: A,B2: A] :
% 11.90/5.94 ( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C2 )
% 11.90/5.94 => ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_913_Suc__times__mod__eq,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
% 11.90/5.94 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N ) ) @ M2 )
% 11.90/5.94 = ( one_one @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_914_div__add__self1__no__field,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.94 & ( field @ B ) )
% 11.90/5.94 => ! [X: B,B2: A,A3: A] :
% 11.90/5.94 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 11.90/5.94 => ( ( B2
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
% 11.90/5.94 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_915_div__add__self2__no__field,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.94 & ( field @ B ) )
% 11.90/5.94 => ! [X: B,B2: A,A3: A] :
% 11.90/5.94 ( ( nO_MATCH @ B @ A @ X @ B2 )
% 11.90/5.94 => ( ( B2
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.94 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_916_power__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( power_power @ A @ ( one_one @ A ) @ N )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_917_add__Suc__right,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.94 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_918_power__one__right,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( power_power @ A @ A3 @ ( one_one @ nat ) )
% 11.90/5.94 = A3 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_919_power__inject__exp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ( ( power_power @ A @ A3 @ M2 )
% 11.90/5.94 = ( power_power @ A @ A3 @ N ) )
% 11.90/5.94 = ( M2 = N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_920_mult__Suc__right,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( times_times @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.94 = ( plus_plus @ nat @ M2 @ ( times_times @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_921_diff__Suc__1,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
% 11.90/5.94 = N ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_922_of__int__eq__1__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_char_0 @ A )
% 11.90/5.94 => ! [Z2: int] :
% 11.90/5.94 ( ( ( ring_1_of_int @ A @ Z2 )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 = ( Z2
% 11.90/5.94 = ( one_one @ int ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_923_of__int__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ( ( ring_1_of_int @ A @ ( one_one @ int ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_924_of__int__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [W: int,Z2: int] :
% 11.90/5.94 ( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_925_power__strict__increasing__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [B2: A,X: nat,Y: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 11.90/5.94 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
% 11.90/5.94 = ( ord_less @ nat @ X @ Y ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_926_left__minus__one__mult__self,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_ring_1 @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
% 11.90/5.94 = A3 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_927_minus__one__mult__self,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_ring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_928_diff__Suc__diff__eq2,axiom,
% 11.90/5.94 ! [K: nat,J: nat,I: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.94 => ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
% 11.90/5.94 = ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_929_diff__Suc__diff__eq1,axiom,
% 11.90/5.94 ! [K: nat,J: nat,I: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ K @ J )
% 11.90/5.94 => ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
% 11.90/5.94 = ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_930_Suc__diff,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
% 11.90/5.94 => ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
% 11.90/5.94 = ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_931_Suc__mod__mult__self4,axiom,
% 11.90/5.94 ! [N: nat,K: nat,M2: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M2 ) ) @ N )
% 11.90/5.94 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_932_Suc__mod__mult__self3,axiom,
% 11.90/5.94 ! [K: nat,N: nat,M2: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M2 ) ) @ N )
% 11.90/5.94 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_933_Suc__mod__mult__self2,axiom,
% 11.90/5.94 ! [M2: nat,N: nat,K: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N @ K ) ) ) @ N )
% 11.90/5.94 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_934_Suc__mod__mult__self1,axiom,
% 11.90/5.94 ! [M2: nat,K: nat,N: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K @ N ) ) ) @ N )
% 11.90/5.94 = ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_935_ceiling__add__of__int,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,Z2: int] :
% 11.90/5.94 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_936_power__strict__decreasing__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [B2: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.94 => ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ord_less @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
% 11.90/5.94 = ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_937_power__increasing__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [B2: A,X: nat,Y: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ B2 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
% 11.90/5.94 = ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_938_Suc__diff__1,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 11.90/5.94 = N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_939_of__int__1__le__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [Z2: int] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
% 11.90/5.94 = ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_940_of__int__le__1__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [Z2: int] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_941_of__int__less__1__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [Z2: int] :
% 11.90/5.94 ( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_942_of__int__1__less__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [Z2: int] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
% 11.90/5.94 = ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_943_power__gt1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_944_add__Suc__shift,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 11.90/5.94 = ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_945_add__Suc,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
% 11.90/5.94 = ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_946_nat__arith_Osuc1,axiom,
% 11.90/5.94 ! [A4: nat,K: nat,A3: nat] :
% 11.90/5.94 ( ( A4
% 11.90/5.94 = ( plus_plus @ nat @ K @ A3 ) )
% 11.90/5.94 => ( ( suc @ A4 )
% 11.90/5.94 = ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_947_power__Suc__le__self,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_948_power__Suc__less__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_949_power__minus_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [X: A,N: nat] :
% 11.90/5.94 ( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
% 11.90/5.94 => ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
% 11.90/5.94 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_950_one__le__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_951_left__right__inverse__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [X: A,Y: A,N: nat] :
% 11.90/5.94 ( ( ( times_times @ A @ X @ Y )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 => ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_952_power__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( power @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_953_power__one__over,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( division_ring @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
% 11.90/5.94 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_954_power__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [A3: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.94 = ( times_times @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_955_nat__compl__induct_H,axiom,
% 11.90/5.94 ! [P: nat > $o,N: nat] :
% 11.90/5.94 ( ( P @ ( zero_zero @ nat ) )
% 11.90/5.94 => ( ! [N3: nat] :
% 11.90/5.94 ( ! [Nn: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ Nn @ N3 )
% 11.90/5.94 => ( P @ Nn ) )
% 11.90/5.94 => ( P @ ( suc @ N3 ) ) )
% 11.90/5.94 => ( P @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_956_nat__compl__induct,axiom,
% 11.90/5.94 ! [P: nat > $o,N: nat] :
% 11.90/5.94 ( ( P @ ( zero_zero @ nat ) )
% 11.90/5.94 => ( ! [N3: nat] :
% 11.90/5.94 ( ! [Nn: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ Nn @ N3 )
% 11.90/5.94 => ( P @ Nn ) )
% 11.90/5.94 => ( P @ ( suc @ N3 ) ) )
% 11.90/5.94 => ( P @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_957_one__is__add,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ( suc @ ( zero_zero @ nat ) )
% 11.90/5.94 = ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.94 = ( ( ( M2
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.94 & ( N
% 11.90/5.94 = ( zero_zero @ nat ) ) )
% 11.90/5.94 | ( ( M2
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 & ( N
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_958_add__is__1,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ( plus_plus @ nat @ M2 @ N )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.94 = ( ( ( M2
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.94 & ( N
% 11.90/5.94 = ( zero_zero @ nat ) ) )
% 11.90/5.94 | ( ( M2
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 & ( N
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_959_nat__in__between__eq_I2_J,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat] :
% 11.90/5.94 ( ( ( ord_less_eq @ nat @ A3 @ B2 )
% 11.90/5.94 & ( ord_less @ nat @ B2 @ ( suc @ A3 ) ) )
% 11.90/5.94 = ( B2 = A3 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_960_nat__in__between__eq_I1_J,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat] :
% 11.90/5.94 ( ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.94 & ( ord_less_eq @ nat @ B2 @ ( suc @ A3 ) ) )
% 11.90/5.94 = ( B2
% 11.90/5.94 = ( suc @ A3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_961_less__imp__Suc__add,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ M2 @ N )
% 11.90/5.94 => ? [K5: nat] :
% 11.90/5.94 ( N
% 11.90/5.94 = ( suc @ ( plus_plus @ nat @ M2 @ K5 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_962_less__iff__Suc__add,axiom,
% 11.90/5.94 ( ( ord_less @ nat )
% 11.90/5.94 = ( ^ [M: nat,N5: nat] :
% 11.90/5.94 ? [K4: nat] :
% 11.90/5.94 ( N5
% 11.90/5.94 = ( suc @ ( plus_plus @ nat @ M @ K4 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_963_less__add__Suc2,axiom,
% 11.90/5.94 ! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_964_less__add__Suc1,axiom,
% 11.90/5.94 ! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_965_less__natE,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ M2 @ N )
% 11.90/5.94 => ~ ! [Q2: nat] :
% 11.90/5.94 ( N
% 11.90/5.94 != ( suc @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_966_Suc__to__right,axiom,
% 11.90/5.94 ! [N: nat,M2: nat] :
% 11.90/5.94 ( ( ( suc @ N )
% 11.90/5.94 = M2 )
% 11.90/5.94 => ( N
% 11.90/5.94 = ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_967_One__nat__def,axiom,
% 11.90/5.94 ( ( one_one @ nat )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_968_mult__Suc,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( times_times @ nat @ ( suc @ M2 ) @ N )
% 11.90/5.94 = ( plus_plus @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_969_Suc__eq__plus1__left,axiom,
% 11.90/5.94 ( suc
% 11.90/5.94 = ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_970_plus__1__eq__Suc,axiom,
% 11.90/5.94 ( ( plus_plus @ nat @ ( one_one @ nat ) )
% 11.90/5.94 = suc ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_971_Suc__eq__plus1,axiom,
% 11.90/5.94 ( suc
% 11.90/5.94 = ( ^ [N5: nat] : ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_972_diff__Suc__eq__diff__pred,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.94 = ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_973_power__le__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_974_power__less__power__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_975_power__gt1__lemma,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_976_power__0__left,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( ( N
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 11.90/5.94 = ( one_one @ A ) ) )
% 11.90/5.94 & ( ( N
% 11.90/5.94 != ( zero_zero @ nat ) )
% 11.90/5.94 => ( ( power_power @ A @ ( zero_zero @ A ) @ N )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_977_power__strict__increasing,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [N: nat,N6: nat,A3: A] :
% 11.90/5.94 ( ( ord_less @ nat @ N @ N6 )
% 11.90/5.94 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_978_power__less__imp__less__exp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 11.90/5.94 => ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_979_power__minus,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 11.90/5.94 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_980_power__increasing,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [N: nat,N6: nat,A3: A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ N @ N6 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_981_floor__add__int,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,Z2: int] :
% 11.90/5.94 ( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
% 11.90/5.94 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_982_int__add__floor,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Z2: int,X: A] :
% 11.90/5.94 ( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
% 11.90/5.94 = ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_983_nat__induct__non__zero,axiom,
% 11.90/5.94 ! [N: nat,P: nat > $o] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ( P @ ( one_one @ nat ) )
% 11.90/5.94 => ( ! [N3: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 11.90/5.94 => ( ( P @ N3 )
% 11.90/5.94 => ( P @ ( suc @ N3 ) ) ) )
% 11.90/5.94 => ( P @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_984_of__int__leD,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [N: int,X: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 11.90/5.94 => ( ( N
% 11.90/5.94 = ( zero_zero @ int ) )
% 11.90/5.94 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_985_of__int__lessD,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [N: int,X: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
% 11.90/5.94 => ( ( N
% 11.90/5.94 = ( zero_zero @ int ) )
% 11.90/5.94 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_986_floor__exists,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim462609752435547400_field @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ? [Z3: int] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
% 11.90/5.94 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_987_floor__exists1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim462609752435547400_field @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ? [X2: int] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ X )
% 11.90/5.94 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) ) )
% 11.90/5.94 & ! [Y6: int] :
% 11.90/5.94 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y6 ) @ X )
% 11.90/5.94 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y6 @ ( one_one @ int ) ) ) ) )
% 11.90/5.94 => ( Y6 = X2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_988_power__Suc__less,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_989_power__strict__decreasing,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [N: nat,N6: nat,A3: A] :
% 11.90/5.94 ( ( ord_less @ nat @ N @ N6 )
% 11.90/5.94 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_990_power__decreasing,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [N: nat,N6: nat,A3: A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ N @ N6 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_991_power__le__imp__le__exp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 11.90/5.94 => ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_992_self__le__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_993_one__less__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_994_nz__le__conv__less,axiom,
% 11.90/5.94 ! [K: nat,M2: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 11.90/5.94 => ( ( ord_less_eq @ nat @ K @ M2 )
% 11.90/5.94 => ( ord_less @ nat @ ( minus_minus @ nat @ K @ ( suc @ ( zero_zero @ nat ) ) ) @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_995_ceiling__altdef,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ( ( archimedean_ceiling @ A )
% 11.90/5.94 = ( ^ [X3: A] :
% 11.90/5.94 ( if @ int
% 11.90/5.94 @ ( X3
% 11.90/5.94 = ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X3 ) ) )
% 11.90/5.94 @ ( archim6421214686448440834_floor @ A @ X3 )
% 11.90/5.94 @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X3 ) @ ( one_one @ int ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_996_Suc__diff__eq__diff__pred,axiom,
% 11.90/5.94 ! [N: nat,M2: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
% 11.90/5.94 = ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_997_Suc__pred_H,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( N
% 11.90/5.94 = ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_998_add__eq__if,axiom,
% 11.90/5.94 ( ( plus_plus @ nat )
% 11.90/5.94 = ( ^ [M: nat,N5: nat] :
% 11.90/5.94 ( if @ nat
% 11.90/5.94 @ ( M
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 @ N5
% 11.90/5.94 @ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_999_Suc__n__minus__m__eq,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( ord_less @ nat @ ( one_one @ nat ) @ M2 )
% 11.90/5.94 => ( ( suc @ ( minus_minus @ nat @ N @ M2 ) )
% 11.90/5.94 = ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1000_floor__unique,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Z2: int,X: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
% 11.90/5.94 => ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
% 11.90/5.94 => ( ( archim6421214686448440834_floor @ A @ X )
% 11.90/5.94 = Z2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1001_floor__eq__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,A3: int] :
% 11.90/5.94 ( ( ( archim6421214686448440834_floor @ A @ X )
% 11.90/5.94 = A3 )
% 11.90/5.94 = ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
% 11.90/5.94 & ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1002_floor__split,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [P: int > $o,T2: A] :
% 11.90/5.94 ( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
% 11.90/5.94 = ( ! [I3: int] :
% 11.90/5.94 ( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I3 ) @ T2 )
% 11.90/5.94 & ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) ) )
% 11.90/5.94 => ( P @ I3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1003_ceiling__correct,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
% 11.90/5.94 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1004_ceiling__unique,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Z2: int,X: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X )
% 11.90/5.94 => ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) )
% 11.90/5.94 => ( ( archimedean_ceiling @ A @ X )
% 11.90/5.94 = Z2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1005_ceiling__eq__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,A3: int] :
% 11.90/5.94 ( ( ( archimedean_ceiling @ A @ X )
% 11.90/5.94 = A3 )
% 11.90/5.94 = ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
% 11.90/5.94 & ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1006_ceiling__split,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [P: int > $o,T2: A] :
% 11.90/5.94 ( ( P @ ( archimedean_ceiling @ A @ T2 ) )
% 11.90/5.94 = ( ! [I3: int] :
% 11.90/5.94 ( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I3 ) @ ( one_one @ A ) ) @ T2 )
% 11.90/5.94 & ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I3 ) ) )
% 11.90/5.94 => ( P @ I3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1007_less__floor__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Z2: int,X: A] :
% 11.90/5.94 ( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
% 11.90/5.94 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1008_floor__le__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,Z2: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
% 11.90/5.94 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1009_ceiling__less__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,Z2: int] :
% 11.90/5.94 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
% 11.90/5.94 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1010_le__ceiling__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [Z2: int,X: A] :
% 11.90/5.94 ( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
% 11.90/5.94 = ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1011_floor__correct,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
% 11.90/5.94 & ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1012_power__diff__power__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.94 => ! [A3: A,N: nat,M2: nat] :
% 11.90/5.94 ( ( A3
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( ( ord_less_eq @ nat @ N @ M2 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 11.90/5.94 = ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
% 11.90/5.94 & ( ~ ( ord_less_eq @ nat @ N @ M2 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1013_power__eq__if,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( power @ A )
% 11.90/5.94 => ( ( power_power @ A )
% 11.90/5.94 = ( ^ [P6: A,M: nat] :
% 11.90/5.94 ( if @ A
% 11.90/5.94 @ ( M
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 @ ( one_one @ A )
% 11.90/5.94 @ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1014_power__minus__mult,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
% 11.90/5.94 = ( power_power @ A @ A3 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1015_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [K: nat,N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ K @ N )
% 11.90/5.94 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
% 11.90/5.94 = ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1016_option_Osize__gen_I2_J,axiom,
% 11.90/5.94 ! [A: $tType,X: A > nat,X22: A] :
% 11.90/5.94 ( ( size_option @ A @ X @ ( some @ A @ X22 ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1017_Heap_Osize__gen,axiom,
% 11.90/5.94 ! [A: $tType,Xa: A > nat,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.90/5.94 ( ( heap_Time_size_Heap @ A @ Xa @ ( heap_Time_Heap2 @ A @ X ) )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1018_num_Osize__gen_I2_J,axiom,
% 11.90/5.94 ! [X22: num] :
% 11.90/5.94 ( ( size_num @ ( bit0 @ X22 ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1019_normalize__negative,axiom,
% 11.90/5.94 ! [Q3: int,P5: int] :
% 11.90/5.94 ( ( ord_less @ int @ Q3 @ ( zero_zero @ int ) )
% 11.90/5.94 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/5.94 = ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1020_option_Osize__gen_I1_J,axiom,
% 11.90/5.94 ! [A: $tType,X: A > nat] :
% 11.90/5.94 ( ( size_option @ A @ X @ ( none @ A ) )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1021_Heap_Osize_I2_J,axiom,
% 11.90/5.94 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.90/5.94 ( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X ) )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1022_num_Osize_I5_J,axiom,
% 11.90/5.94 ! [X22: num] :
% 11.90/5.94 ( ( size_size @ num @ ( bit0 @ X22 ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1023_option_Osize_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( size_size @ ( option @ A ) @ ( none @ A ) )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1024_normalize__denom__zero,axiom,
% 11.90/5.94 ! [P5: int] :
% 11.90/5.94 ( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
% 11.90/5.94 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1025_option_Osize__neq,axiom,
% 11.90/5.94 ! [A: $tType,X: option @ A] :
% 11.90/5.94 ( ( size_size @ ( option @ A ) @ X )
% 11.90/5.94 != ( zero_zero @ nat ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1026_Heap_Osize__neq,axiom,
% 11.90/5.94 ! [A: $tType,X: heap_Time_Heap @ A] :
% 11.90/5.94 ( ( size_size @ ( heap_Time_Heap @ A ) @ X )
% 11.90/5.94 != ( zero_zero @ nat ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1027_option_Osize__gen__o__map,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,F: B > nat,G: A > B] :
% 11.90/5.94 ( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F ) @ ( map_option @ A @ B @ G ) )
% 11.90/5.94 = ( size_option @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1028_normalize__denom__pos,axiom,
% 11.90/5.94 ! [R: product_prod @ int @ int,P5: int,Q3: int] :
% 11.90/5.94 ( ( ( normalize @ R )
% 11.90/5.94 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/5.94 => ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1029_normalize__crossproduct,axiom,
% 11.90/5.94 ! [Q3: int,S: int,P5: int,R: int] :
% 11.90/5.94 ( ( Q3
% 11.90/5.94 != ( zero_zero @ int ) )
% 11.90/5.94 => ( ( S
% 11.90/5.94 != ( zero_zero @ int ) )
% 11.90/5.94 => ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/5.94 = ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
% 11.90/5.94 => ( ( times_times @ int @ P5 @ S )
% 11.90/5.94 = ( times_times @ int @ R @ Q3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1030_option_Osize_I4_J,axiom,
% 11.90/5.94 ! [A: $tType,X22: A] :
% 11.90/5.94 ( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
% 11.90/5.94 = ( suc @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1031_divmod__nat__if,axiom,
% 11.90/5.94 ( divmod_nat
% 11.90/5.94 = ( ^ [M: nat,N5: nat] :
% 11.90/5.94 ( if @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ( ( N5
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 | ( ord_less @ nat @ M @ N5 ) )
% 11.90/5.94 @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M )
% 11.90/5.94 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
% 11.90/5.94 @ ( divmod_nat @ ( minus_minus @ nat @ M @ N5 ) @ N5 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1032_sum__gp__multiplied,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( monoid_mult @ A )
% 11.90/5.94 & ( comm_ring @ A ) )
% 11.90/5.94 => ! [M2: nat,N: nat,X: A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
% 11.90/5.94 = ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1033_pair__lessI2,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 11.90/5.94 => ( ( ord_less @ nat @ S @ T2 )
% 11.90/5.94 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1034_pair__less__iff1,axiom,
% 11.90/5.94 ! [X: nat,Y: nat,Z2: nat] :
% 11.90/5.94 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z2 ) ) @ fun_pair_less )
% 11.90/5.94 = ( ord_less @ nat @ Y @ Z2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1035_Suc__times__numeral__mod__eq,axiom,
% 11.90/5.94 ! [K: num,N: nat] :
% 11.90/5.94 ( ( ( numeral_numeral @ nat @ K )
% 11.90/5.94 != ( one_one @ nat ) )
% 11.90/5.94 => ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
% 11.90/5.94 = ( one_one @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1036_one__int_Otransfer,axiom,
% 11.90/5.94 pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1037_less__than__iff,axiom,
% 11.90/5.94 ! [X: nat,Y: nat] :
% 11.90/5.94 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
% 11.90/5.94 = ( ord_less @ nat @ X @ Y ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1038_one__int__def,axiom,
% 11.90/5.94 ( ( one_one @ int )
% 11.90/5.94 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1039_sum__telescope_H_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ab_group_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,F: nat > A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) ) ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
% 11.90/5.94 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1040_case__prod__conv,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
% 11.90/5.94 ( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
% 11.90/5.94 = ( F @ A3 @ B2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1041_case__prod__curry,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
% 11.90/5.94 = F ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1042_curry__case__prod,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
% 11.90/5.94 ( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
% 11.90/5.94 = F ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1043_distrib__right__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( numeral @ A )
% 11.90/5.94 & ( semiring @ A ) )
% 11.90/5.94 => ! [A3: A,B2: A,V: num] :
% 11.90/5.94 ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1044_distrib__left__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( numeral @ A )
% 11.90/5.94 & ( semiring @ A ) )
% 11.90/5.94 => ! [V: num,B2: A,C2: A] :
% 11.90/5.94 ( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1045_add__neg__numeral__simps_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1046_add__numeral__left,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [V: num,W: num,Z2: A] :
% 11.90/5.94 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1047_numeral__plus__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1048_semiring__norm_I168_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [V: num,W: num,Y: A] :
% 11.90/5.94 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
% 11.90/5.94 = ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1049_diff__numeral__simps_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1050_diff__numeral__simps_I2_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1051_ceiling__add__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,V: num] :
% 11.90/5.94 ( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1052_power__add__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [A3: A,M2: num,N: num] :
% 11.90/5.94 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
% 11.90/5.94 = ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1053_power__add__numeral2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( monoid_mult @ A )
% 11.90/5.94 => ! [A3: A,M2: num,N: num,B2: A] :
% 11.90/5.94 ( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M2 ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
% 11.90/5.94 = ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1054_numeral__less__floor,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [V: num,X: A] :
% 11.90/5.94 ( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 11.90/5.94 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1055_floor__le__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,V: num] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 11.90/5.94 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1056_sum_Ocl__ivl__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [N: nat,M2: nat,G: nat > A] :
% 11.90/5.94 ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.94 = ( zero_zero @ A ) ) )
% 11.90/5.94 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1057_ceiling__less__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,V: num] :
% 11.90/5.94 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
% 11.90/5.94 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1058_numeral__le__ceiling,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [V: num,X: A] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
% 11.90/5.94 = ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1059_neg__numeral__less__floor,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [V: num,X: A] :
% 11.90/5.94 ( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
% 11.90/5.94 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1060_floor__le__neg__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,V: num] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 11.90/5.94 = ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1061_ceiling__less__neg__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,V: num] :
% 11.90/5.94 ( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
% 11.90/5.94 = ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1062_neg__numeral__le__ceiling,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [V: num,X: A] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
% 11.90/5.94 = ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1063_nested__case__prod__simp,axiom,
% 11.90/5.94 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 11.90/5.94 ( ( product_case_prod @ B @ C @ ( D > A ) )
% 11.90/5.94 = ( ^ [F2: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
% 11.90/5.94 ( product_case_prod @ B @ C @ A
% 11.90/5.94 @ ^ [A7: B,B5: C] : ( F2 @ A7 @ B5 @ Y3 )
% 11.90/5.94 @ X3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1064_prod_Ocase__distrib,axiom,
% 11.90/5.94 ! [C: $tType,D: $tType,B: $tType,A: $tType,H3: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
% 11.90/5.94 ( ( H3 @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 11.90/5.94 = ( product_case_prod @ A @ B @ D
% 11.90/5.94 @ ^ [X12: A,X24: B] : ( H3 @ ( F @ X12 @ X24 ) )
% 11.90/5.94 @ Prod ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1065_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1066_sum_OatLeastAtMost__shift__bounds,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1067_split__cong,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType,Q3: product_prod @ A @ B,F: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
% 11.90/5.94 ( ! [X2: A,Y4: B] :
% 11.90/5.94 ( ( ( product_Pair @ A @ B @ X2 @ Y4 )
% 11.90/5.94 = Q3 )
% 11.90/5.94 => ( ( F @ X2 @ Y4 )
% 11.90/5.94 = ( G @ X2 @ Y4 ) ) )
% 11.90/5.94 => ( ( P5 = Q3 )
% 11.90/5.94 => ( ( product_case_prod @ A @ B @ C @ F @ P5 )
% 11.90/5.94 = ( product_case_prod @ A @ B @ C @ G @ Q3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1068_old_Oprod_Ocase,axiom,
% 11.90/5.94 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 11.90/5.94 = ( F @ X1 @ X22 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1069_uminus__int_Oabs__eq,axiom,
% 11.90/5.94 ! [X: product_prod @ nat @ nat] :
% 11.90/5.94 ( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
% 11.90/5.94 = ( abs_Integ
% 11.90/5.94 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 )
% 11.90/5.94 @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1070_case__prod__Pair__iden,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
% 11.90/5.94 = P5 ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1071_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1072_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1073_cond__case__prod__eta,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
% 11.90/5.94 ( ! [X2: A,Y4: B] :
% 11.90/5.94 ( ( F @ X2 @ Y4 )
% 11.90/5.94 = ( G @ ( product_Pair @ A @ B @ X2 @ Y4 ) ) )
% 11.90/5.94 => ( ( product_case_prod @ A @ B @ C @ F )
% 11.90/5.94 = G ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1074_case__prod__eta,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ C
% 11.90/5.94 @ ^ [X3: A,Y3: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
% 11.90/5.94 = F ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1075_case__prodE2,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
% 11.90/5.94 ( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
% 11.90/5.94 => ~ ! [X2: B,Y4: C] :
% 11.90/5.94 ( ( Z2
% 11.90/5.94 = ( product_Pair @ B @ C @ X2 @ Y4 ) )
% 11.90/5.94 => ~ ( Q @ ( P @ X2 @ Y4 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1076_eq__Abs__Integ,axiom,
% 11.90/5.94 ! [Z2: int] :
% 11.90/5.94 ~ ! [X2: nat,Y4: nat] :
% 11.90/5.94 ( Z2
% 11.90/5.94 != ( abs_Integ @ ( product_Pair @ nat @ nat @ X2 @ Y4 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1077_one__le__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linord181362715937106298miring @ A )
% 11.90/5.94 => ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1078_not__numeral__less__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linord181362715937106298miring @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1079_one__plus__numeral__commute,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [X: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 11.90/5.94 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1080_numeral__Bit0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1081_one__neq__neg__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_char_0 @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( one_one @ A )
% 11.90/5.94 != ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1082_numeral__neq__neg__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_char_0 @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( numeral_numeral @ A @ N )
% 11.90/5.94 != ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1083_uncurry__def,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/5.94 ( ( uncurry @ A @ B @ C )
% 11.90/5.94 = ( product_case_prod @ A @ B @ C ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1084_internal__case__prod__def,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/5.94 ( ( produc5280177257484947105e_prod @ A @ B @ C )
% 11.90/5.94 = ( product_case_prod @ A @ B @ C ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1085_sum_OatLeastAtMost__rev,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,N: nat,M2: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1086_numeral__code_I2_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( numeral_numeral @ A @ ( bit0 @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1087_sum__shift__lb__Suc0__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [F: nat > A,K: nat] :
% 11.90/5.94 ( ( ( F @ ( zero_zero @ nat ) )
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1088_sum_OatLeast0__atMost__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1089_sum_OatLeast0__atMost__Suc__shift,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1090_sum_OatLeast__Suc__atMost,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1091_sum_Onat__ivl__Suc_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1092_neg__numeral__le__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1093_neg__one__le__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1094_neg__numeral__le__neg__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1095_not__numeral__le__neg__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1096_not__one__le__neg__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1097_not__neg__one__less__neg__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1098_not__one__less__neg__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1099_not__numeral__less__neg__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1100_neg__one__less__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1101_neg__numeral__less__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1102_sum_OSuc__reindex__ivl,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( G @ M2 )
% 11.90/5.94 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1103_sum__Suc__diff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ab_group_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,F: nat > A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( minus_minus @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1104_sum__power__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( monoid_mult @ A )
% 11.90/5.94 & ( comm_ring @ A ) )
% 11.90/5.94 => ! [X: A,M2: nat,I4: set @ nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I3 ) )
% 11.90/5.94 @ I4 )
% 11.90/5.94 = ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I4 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1105_sum_OatLeast__atMost__pred__shift,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ( comp @ nat @ A @ nat @ G
% 11.90/5.94 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1106_sum__atLeastAtMost__code,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [F: nat > A,A3: nat,B2: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 11.90/5.94 = ( set_fo6178422350223883121st_nat @ A
% 11.90/5.94 @ ^ [A7: nat] : ( plus_plus @ A @ ( F @ A7 ) )
% 11.90/5.94 @ A3
% 11.90/5.94 @ B2
% 11.90/5.94 @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1107_sum_Oub__add__nat,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,G: nat > A,P5: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1108_sum_OatLeastAtMost__shift__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1109_zero__int__def,axiom,
% 11.90/5.94 ( ( zero_zero @ int )
% 11.90/5.94 = ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1110_pair__lessI1,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.94 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1111_zero__int_Otransfer,axiom,
% 11.90/5.94 pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1112_sum__natinterval__diff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ab_group_add @ A )
% 11.90/5.94 => ! [M2: nat,N: nat,F: nat > A] :
% 11.90/5.94 ( ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( minus_minus @ A @ ( F @ M2 ) @ ( F @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
% 11.90/5.94 & ( ~ ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [K4: nat] : ( minus_minus @ A @ ( F @ K4 ) @ ( F @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1113_convex__sum__bound__le,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType] :
% 11.90/5.94 ( ( linordered_idom @ B )
% 11.90/5.94 => ! [I4: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
% 11.90/5.94 ( ! [I2: A] :
% 11.90/5.94 ( ( member @ A @ I2 @ I4 )
% 11.90/5.94 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I2 ) ) )
% 11.90/5.94 => ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I4 )
% 11.90/5.94 = ( one_one @ B ) )
% 11.90/5.94 => ( ! [I2: A] :
% 11.90/5.94 ( ( member @ A @ I2 @ I4 )
% 11.90/5.94 => ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I2 ) @ B2 ) ) @ Delta ) )
% 11.90/5.94 => ( ord_less_eq @ B
% 11.90/5.94 @ ( abs_abs @ B
% 11.90/5.94 @ ( minus_minus @ B
% 11.90/5.94 @ ( groups7311177749621191930dd_sum @ A @ B
% 11.90/5.94 @ ^ [I3: A] : ( times_times @ B @ ( A3 @ I3 ) @ ( X @ I3 ) )
% 11.90/5.94 @ I4 )
% 11.90/5.94 @ B2 ) )
% 11.90/5.94 @ Delta ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1114_sum__comp__morphism,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,C: $tType] :
% 11.90/5.94 ( ( ( comm_monoid_add @ B )
% 11.90/5.94 & ( comm_monoid_add @ A ) )
% 11.90/5.94 => ! [H3: B > A,G: C > B,A4: set @ C] :
% 11.90/5.94 ( ( ( H3 @ ( zero_zero @ B ) )
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 => ( ! [X2: B,Y4: B] :
% 11.90/5.94 ( ( H3 @ ( plus_plus @ B @ X2 @ Y4 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( H3 @ X2 ) @ ( H3 @ Y4 ) ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H3 @ G ) @ A4 )
% 11.90/5.94 = ( H3 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A4 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1115_sum__gp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( division_ring @ A )
% 11.90/5.94 & ( comm_ring @ A ) )
% 11.90/5.94 => ! [N: nat,M2: nat,X: A] :
% 11.90/5.94 ( ( ( ord_less @ nat @ N @ M2 )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( zero_zero @ A ) ) )
% 11.90/5.94 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.90/5.94 => ( ( ( X
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
% 11.90/5.94 & ( ( X
% 11.90/5.94 != ( one_one @ A ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1116_divmod__digit__1_I1_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
% 11.90/5.94 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
% 11.90/5.94 = ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1117_sum__gp__offset,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( division_ring @ A )
% 11.90/5.94 & ( comm_ring @ A ) )
% 11.90/5.94 => ! [X: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( ( X
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
% 11.90/5.94 & ( ( X
% 11.90/5.94 != ( one_one @ A ) )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
% 11.90/5.94 = ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1118_Sum__Icc__nat,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.94 @ ^ [X3: nat] : X3
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.94 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1119_arith__series__nat,axiom,
% 11.90/5.94 ! [A3: nat,D3: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.94 @ ^ [I3: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I3 @ D3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.94 = ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1120_mod__double__modulus,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [M2: A,X: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.94 => ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 11.90/5.94 = ( modulo_modulo @ A @ X @ M2 ) )
% 11.90/5.94 | ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1121_sum_Odistrib,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: B > A,H3: B > A,A4: set @ B] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ B @ A
% 11.90/5.94 @ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H3 @ X3 ) )
% 11.90/5.94 @ A4 )
% 11.90/5.94 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ A4 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1122_case__prodI,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
% 11.90/5.94 ( ( F @ A3 @ B2 )
% 11.90/5.94 => ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1123_case__prodI2,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
% 11.90/5.94 ( ! [A5: A,B3: B] :
% 11.90/5.94 ( ( P5
% 11.90/5.94 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.90/5.94 => ( C2 @ A5 @ B3 ) )
% 11.90/5.94 => ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1124_mem__case__prodI,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B2: C] :
% 11.90/5.94 ( ( member @ A @ Z2 @ ( C2 @ A3 @ B2 ) )
% 11.90/5.94 => ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1125_mem__case__prodI2,axiom,
% 11.90/5.94 ! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
% 11.90/5.94 ( ! [A5: A,B3: B] :
% 11.90/5.94 ( ( P5
% 11.90/5.94 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.90/5.94 => ( member @ C @ Z2 @ ( C2 @ A5 @ B3 ) ) )
% 11.90/5.94 => ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1126_case__prodI2_H,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
% 11.90/5.94 ( ! [A5: A,B3: B] :
% 11.90/5.94 ( ( ( product_Pair @ A @ B @ A5 @ B3 )
% 11.90/5.94 = P5 )
% 11.90/5.94 => ( C2 @ A5 @ B3 @ X ) )
% 11.90/5.94 => ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1127_one__eq__numeral__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_char_0 @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( ( one_one @ A )
% 11.90/5.94 = ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( one2 = N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1128_numeral__eq__one__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_char_0 @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( ( numeral_numeral @ A @ N )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 = ( N = one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1129_of__nat__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1130_of__nat__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ( ( semiring_1_of_nat @ A @ ( one_one @ nat ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1131_of__nat__1__eq__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_char_0 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( ( one_one @ A )
% 11.90/5.94 = ( semiring_1_of_nat @ A @ N ) )
% 11.90/5.94 = ( N
% 11.90/5.94 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1132_of__nat__eq__1__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_char_0 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( ( semiring_1_of_nat @ A @ N )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 = ( N
% 11.90/5.94 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1133_semiring__norm_I2_J,axiom,
% 11.90/5.94 ( ( plus_plus @ num @ one2 @ one2 )
% 11.90/5.94 = ( bit0 @ one2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1134_numeral__le__one__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linord181362715937106298miring @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1135_one__less__numeral__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linord181362715937106298miring @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( ord_less @ num @ one2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1136_numeral__eq__neg__one__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_char_0 @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( N = one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1137_neg__one__eq__numeral__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_char_0 @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( ( uminus_uminus @ A @ ( one_one @ A ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.94 = ( N = one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1138_of__nat__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ! [M2: nat] :
% 11.90/5.94 ( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1139_Suc__numeral,axiom,
% 11.90/5.94 ! [N: num] :
% 11.90/5.94 ( ( suc @ ( numeral_numeral @ nat @ N ) )
% 11.90/5.94 = ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1140_not__neg__one__le__neg__numeral__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
% 11.90/5.94 = ( M2 != one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1141_one__add__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1142_neg__numeral__less__neg__one__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( M2 != one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1143_bits__one__mod__two__eq__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1144_one__mod__two__eq__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1145_add__2__eq__Suc,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 = ( suc @ ( suc @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1146_add__2__eq__Suc_H,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( suc @ ( suc @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1147_Suc__1,axiom,
% 11.90/5.94 ( ( suc @ ( one_one @ nat ) )
% 11.90/5.94 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1148_numeral__plus__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1149_one__plus__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1150_add__self__div__2,axiom,
% 11.90/5.94 ! [M2: nat] :
% 11.90/5.94 ( ( divide_divide @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = M2 ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1151_bits__1__div__2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1152_one__div__two__eq__zero,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1153_add__neg__numeral__special_I9_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1154_diff__numeral__special_I11_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1155_diff__numeral__special_I10_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1156_not__mod__2__eq__0__eq__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1157_not__mod__2__eq__1__eq__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 != ( one_one @ A ) )
% 11.90/5.94 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1158_minus__1__div__2__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid8789492081693882211th_nat @ A )
% 11.90/5.94 => ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1159_bits__minus__1__mod__2__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1160_minus__1__mod__2__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid8789492081693882211th_nat @ A )
% 11.90/5.94 => ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1161_diff__numeral__special_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1162_diff__numeral__special_I4_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1163_add__self__mod__2,axiom,
% 11.90/5.94 ! [M2: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( zero_zero @ nat ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1164_power__minus1__even,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1165_mod2__gr__0,axiom,
% 11.90/5.94 ! [M2: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 11.90/5.94 = ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1166_lex__prod__def,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( lex_prod @ A @ B )
% 11.90/5.94 = ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
% 11.90/5.94 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 11.90/5.94 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 11.90/5.94 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 11.90/5.94 @ ^ [A7: A,B5: B] :
% 11.90/5.94 ( product_case_prod @ A @ B @ $o
% 11.90/5.94 @ ^ [A8: A,B7: B] :
% 11.90/5.94 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A8 ) @ Ra )
% 11.90/5.94 | ( ( A7 = A8 )
% 11.90/5.94 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1167_same__fst__def,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( same_fst @ A @ B )
% 11.90/5.94 = ( ^ [P4: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
% 11.90/5.94 ( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
% 11.90/5.94 @ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
% 11.90/5.94 @ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
% 11.90/5.94 @ ^ [X9: A,Y8: B] :
% 11.90/5.94 ( product_case_prod @ A @ B @ $o
% 11.90/5.94 @ ^ [X3: A,Y3: B] :
% 11.90/5.94 ( ( X9 = X3 )
% 11.90/5.94 & ( P4 @ X3 )
% 11.90/5.94 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y3 ) @ ( R6 @ X3 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1168_mem__case__prodE,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),P5: product_prod @ B @ C] :
% 11.90/5.94 ( ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P5 ) )
% 11.90/5.94 => ~ ! [X2: B,Y4: C] :
% 11.90/5.94 ( ( P5
% 11.90/5.94 = ( product_Pair @ B @ C @ X2 @ Y4 ) )
% 11.90/5.94 => ~ ( member @ A @ Z2 @ ( C2 @ X2 @ Y4 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1169_case__prodD,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 11.90/5.94 => ( F @ A3 @ B2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1170_case__prodE,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
% 11.90/5.94 => ~ ! [X2: A,Y4: B] :
% 11.90/5.94 ( ( P5
% 11.90/5.94 = ( product_Pair @ A @ B @ X2 @ Y4 ) )
% 11.90/5.94 => ~ ( C2 @ X2 @ Y4 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1171_add__One__commute,axiom,
% 11.90/5.94 ! [N: num] :
% 11.90/5.94 ( ( plus_plus @ num @ one2 @ N )
% 11.90/5.94 = ( plus_plus @ num @ N @ one2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1172_case__prodD_H,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,C: $tType,R5: A > B > C > $o,A3: A,B2: B,C2: C] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ ( C > $o ) @ R5 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C2 )
% 11.90/5.94 => ( R5 @ A3 @ B2 @ C2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1173_case__prodE_H,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z2 )
% 11.90/5.94 => ~ ! [X2: A,Y4: B] :
% 11.90/5.94 ( ( P5
% 11.90/5.94 = ( product_Pair @ A @ B @ X2 @ Y4 ) )
% 11.90/5.94 => ~ ( C2 @ X2 @ Y4 @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1174_sum_OatLeast__int__atMost__int__shift,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: int > A,M2: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1175_less__int_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 11.90/5.94 ( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 11.90/5.94 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.94 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.94 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.94 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
% 11.90/5.94 @ Xa
% 11.90/5.94 @ X ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1176_less__eq__int_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 11.90/5.94 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.94 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.94 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.94 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) )
% 11.90/5.94 @ Xa
% 11.90/5.94 @ X ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1177_Collect__case__prod__mono,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
% 11.90/5.94 ( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B6 )
% 11.90/5.94 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1178_zadd__int__left,axiom,
% 11.90/5.94 ! [M2: nat,N: nat,Z2: int] :
% 11.90/5.94 ( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z2 ) )
% 11.90/5.94 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N ) ) @ Z2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1179_int__plus,axiom,
% 11.90/5.94 ! [N: nat,M2: nat] :
% 11.90/5.94 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M2 ) )
% 11.90/5.94 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1180_int__ops_I5_J,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat] :
% 11.90/5.94 ( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
% 11.90/5.94 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1181_int__ops_I2_J,axiom,
% 11.90/5.94 ( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
% 11.90/5.94 = ( one_one @ int ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1182_zle__iff__zadd,axiom,
% 11.90/5.94 ( ( ord_less_eq @ int )
% 11.90/5.94 = ( ^ [W2: int,Z5: int] :
% 11.90/5.94 ? [N5: nat] :
% 11.90/5.94 ( Z5
% 11.90/5.94 = ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1183_numeral__One,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ( ( numeral_numeral @ A @ one2 )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1184_numerals_I1_J,axiom,
% 11.90/5.94 ( ( numeral_numeral @ nat @ one2 )
% 11.90/5.94 = ( one_one @ nat ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1185_execute__bind__case,axiom,
% 11.90/5.94 ! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H3: heap_ext @ product_unit] :
% 11.90/5.94 ( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H3 )
% 11.90/5.94 = ( case_option @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.94 @ ( product_case_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.94 @ ^ [X3: B] :
% 11.90/5.94 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.94 @ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ A @ N5 @ ( heap_Time_execute @ A @ ( G @ X3 ) @ H7 ) ) ) )
% 11.90/5.94 @ ( heap_Time_execute @ B @ F @ H3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1186_Sum__Icc__int,axiom,
% 11.90/5.94 ! [M2: int,N: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ M2 @ N )
% 11.90/5.94 => ( ( groups7311177749621191930dd_sum @ int @ int
% 11.90/5.94 @ ^ [X3: int] : X3
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) )
% 11.90/5.94 = ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M2 @ ( minus_minus @ int @ M2 @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1187_Heap__Time__Monad_Obind__def,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( heap_Time_bind @ A @ B )
% 11.90/5.94 = ( ^ [F2: heap_Time_Heap @ A,G2: A > ( heap_Time_Heap @ B )] :
% 11.90/5.94 ( heap_Time_Heap2 @ B
% 11.90/5.94 @ ^ [H: heap_ext @ product_unit] :
% 11.90/5.94 ( case_option @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.94 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.94 @ ^ [R3: A] :
% 11.90/5.94 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.90/5.94 @ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ B @ N5 @ ( heap_Time_execute @ B @ ( G2 @ R3 ) @ H7 ) ) ) )
% 11.90/5.94 @ ( heap_Time_execute @ A @ F2 @ H ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1188_zless__iff__Suc__zadd,axiom,
% 11.90/5.94 ( ( ord_less @ int )
% 11.90/5.94 = ( ^ [W2: int,Z5: int] :
% 11.90/5.94 ? [N5: nat] :
% 11.90/5.94 ( Z5
% 11.90/5.94 = ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1189_int__ops_I4_J,axiom,
% 11.90/5.94 ! [A3: nat] :
% 11.90/5.94 ( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
% 11.90/5.94 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1190_int__Suc,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( semiring_1_of_nat @ int @ ( suc @ N ) )
% 11.90/5.94 = ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1191_int__bit__induct,axiom,
% 11.90/5.94 ! [P: int > $o,K: int] :
% 11.90/5.94 ( ( P @ ( zero_zero @ int ) )
% 11.90/5.94 => ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.94 => ( ! [K5: int] :
% 11.90/5.94 ( ( P @ K5 )
% 11.90/5.94 => ( ( K5
% 11.90/5.94 != ( zero_zero @ int ) )
% 11.90/5.94 => ( P @ ( times_times @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
% 11.90/5.94 => ( ! [K5: int] :
% 11.90/5.94 ( ( P @ K5 )
% 11.90/5.94 => ( ( K5
% 11.90/5.94 != ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.94 => ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
% 11.90/5.94 => ( P @ K ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1192_uminus__numeral__One,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1193_int__def,axiom,
% 11.90/5.94 ( ( semiring_1_of_nat @ int )
% 11.90/5.94 = ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1194_num__of__nat__One,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
% 11.90/5.94 => ( ( num_of_nat @ N )
% 11.90/5.94 = one2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1195_neg__zdiv__mult__2,axiom,
% 11.90/5.94 ! [A3: int,B2: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 11.90/5.94 => ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.94 = ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1196_pos__zdiv__mult__2,axiom,
% 11.90/5.94 ! [A3: int,B2: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 11.90/5.94 => ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.94 = ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1197_pos__zmod__mult__2,axiom,
% 11.90/5.94 ! [A3: int,B2: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
% 11.90/5.94 => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.94 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1198_mod__mult2__eq_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [A3: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1199_numeral__num__of__nat__unfold,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( ( N
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 11.90/5.94 = ( one_one @ A ) ) )
% 11.90/5.94 & ( ( N
% 11.90/5.94 != ( zero_zero @ nat ) )
% 11.90/5.94 => ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
% 11.90/5.94 = ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1200_double__arith__series,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_semiring_1 @ A )
% 11.90/5.94 => ! [A3: A,D3: A,N: nat] :
% 11.90/5.94 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
% 11.90/5.94 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 11.90/5.94 = ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1201_double__gauss__sum,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_semiring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
% 11.90/5.94 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1202_neg__zmod__mult__2,axiom,
% 11.90/5.94 ! [A3: int,B2: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
% 11.90/5.94 => ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.94 = ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1203_mult__2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_numeral @ A )
% 11.90/5.94 => ! [Z2: A] :
% 11.90/5.94 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
% 11.90/5.94 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1204_mult__2__right,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_numeral @ A )
% 11.90/5.94 => ! [Z2: A] :
% 11.90/5.94 ( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1205_left__add__twice,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_numeral @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1206_pos__eucl__rel__int__mult__2,axiom,
% 11.90/5.94 ! [B2: int,A3: int,Q3: int,R: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
% 11.90/5.94 => ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.94 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1207_one__power2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1208_nat__1__add__1,axiom,
% 11.90/5.94 ( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
% 11.90/5.94 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1209_Suc__nat__number__of__add,axiom,
% 11.90/5.94 ! [V: num,N: nat] :
% 11.90/5.94 ( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1210_double__gauss__sum__from__Suc__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_semiring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
% 11.90/5.94 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1211_arith__series,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [A3: A,D3: A,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I3 ) @ D3 ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1212_gauss__sum,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1213_nat__approx__posE,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [E2: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
% 11.90/5.94 => ~ ! [N3: nat] :
% 11.90/5.94 ~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1214_neg__eucl__rel__int__mult__2,axiom,
% 11.90/5.94 ! [B2: int,A3: int,Q3: int,R: int] :
% 11.90/5.94 ( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
% 11.90/5.94 => ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.94 => ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q3 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1215_gauss__sum__from__Suc__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1216_exp__add__not__zero__imp__right,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 != ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1217_exp__add__not__zero__imp__left,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
% 11.90/5.94 != ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1218_sum__power2__eq__zero__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A,Y: A] :
% 11.90/5.94 ( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 = ( ( X
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 & ( Y
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1219_div__exp__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [A3: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1220_power2__eq__1__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_15535105094025558882visors @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 = ( ( A3
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 | ( A3
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1221_plus__int_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 11.90/5.94 ( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 11.90/5.94 = ( abs_Integ
% 11.90/5.94 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.94 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.94 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) )
% 11.90/5.94 @ Xa
% 11.90/5.94 @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1222_abs__square__eq__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 = ( ( abs_abs @ A @ X )
% 11.90/5.94 = ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1223_minus__int_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 11.90/5.94 ( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 11.90/5.94 = ( abs_Integ
% 11.90/5.94 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.94 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.94 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) )
% 11.90/5.94 @ Xa
% 11.90/5.94 @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1224_nat__induct2,axiom,
% 11.90/5.94 ! [P: nat > $o,N: nat] :
% 11.90/5.94 ( ( P @ ( zero_zero @ nat ) )
% 11.90/5.94 => ( ( P @ ( one_one @ nat ) )
% 11.90/5.94 => ( ! [N3: nat] :
% 11.90/5.94 ( ( P @ N3 )
% 11.90/5.94 => ( P @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 11.90/5.94 => ( P @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1225_times__int_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
% 11.90/5.94 ( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
% 11.90/5.94 = ( abs_Integ
% 11.90/5.94 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.94 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.94 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) )
% 11.90/5.94 @ Xa
% 11.90/5.94 @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1226_sum__power2__le__zero__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A,Y: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
% 11.90/5.94 = ( ( X
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 & ( Y
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1227_sum__power2__ge__zero,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1228_bits__stable__imp__add__self,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = A3 )
% 11.90/5.94 => ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1229_not__sum__power2__lt__zero,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A,Y: A] :
% 11.90/5.94 ~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1230_sum__power2__gt__zero__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A,Y: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 11.90/5.94 = ( ( X
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 | ( Y
% 11.90/5.94 != ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1231_power2__sum,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_semiring_1 @ A )
% 11.90/5.94 => ! [X: A,Y: A] :
% 11.90/5.94 ( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1232_square__le__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 11.90/5.94 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 11.90/5.94 => ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1233_abs__square__le__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1234_abs__square__less__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_idom @ A )
% 11.90/5.94 => ! [X: A] :
% 11.90/5.94 ( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1235_div__exp__mod__exp__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [A3: A,N: nat,M2: nat] :
% 11.90/5.94 ( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 11.90/5.94 = ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1236_ex__power__ivl2,axiom,
% 11.90/5.94 ! [B2: nat,K: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 11.90/5.94 => ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
% 11.90/5.94 => ? [N3: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
% 11.90/5.94 & ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1237_ex__power__ivl1,axiom,
% 11.90/5.94 ! [B2: nat,K: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
% 11.90/5.94 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
% 11.90/5.94 => ? [N3: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K )
% 11.90/5.94 & ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1238_power2__diff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_ring_1 @ A )
% 11.90/5.94 => ! [X: A,Y: A] :
% 11.90/5.94 ( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1239_power__minus1__odd,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1240_mask__eq__sum__exp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 @ ( collect @ nat
% 11.90/5.94 @ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1241_sum_Oin__pairs,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_add @ A )
% 11.90/5.94 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 11.90/5.94 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.94 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1242_gauss__sum__nat,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.94 @ ^ [X3: nat] : X3
% 11.90/5.94 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.94 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1243_divmod__step__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [L: num,R: A,Q3: A] :
% 11.90/5.94 ( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 11.90/5.94 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R ) )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L ) ) ) ) )
% 11.90/5.94 & ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
% 11.90/5.94 => ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q3 @ R ) )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1244_divmod__step__def,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ( ( unique1321980374590559556d_step @ A )
% 11.90/5.94 = ( ^ [L3: num] :
% 11.90/5.94 ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 11.90/5.94 @ ^ [Q5: A,R3: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R3 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R3 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1245_divmod__step__nat__def,axiom,
% 11.90/5.94 ( ( unique1321980374590559556d_step @ nat )
% 11.90/5.94 = ( ^ [L3: num] :
% 11.90/5.94 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.94 @ ^ [Q5: nat,R3: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R3 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R3 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1246_divmod__step__int__def,axiom,
% 11.90/5.94 ( ( unique1321980374590559556d_step @ int )
% 11.90/5.94 = ( ^ [L3: num] :
% 11.90/5.94 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.94 @ ^ [Q5: int,R3: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R3 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R3 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1247_set__bit__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1248_mlex__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( mlex_prod @ A )
% 11.90/5.94 = ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
% 11.90/5.94 ( collect @ ( product_prod @ A @ A )
% 11.90/5.94 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.94 @ ^ [X3: A,Y3: A] :
% 11.90/5.94 ( ( ord_less @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
% 11.90/5.94 | ( ( ord_less_eq @ nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) )
% 11.90/5.94 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1249_unset__bit__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1250_flip__bit__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1251_set__bit__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1252_split__part,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
% 11.90/5.94 ( ( product_case_prod @ A @ B @ $o
% 11.90/5.94 @ ^ [A7: A,B5: B] :
% 11.90/5.94 ( P
% 11.90/5.94 & ( Q @ A7 @ B5 ) ) )
% 11.90/5.94 = ( ^ [Ab: product_prod @ A @ B] :
% 11.90/5.94 ( P
% 11.90/5.94 & ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1253_prod_Odisc__eq__case,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 11.90/5.94 ( product_case_prod @ A @ B @ $o
% 11.90/5.94 @ ^ [Uu: A,Uv: B] : $true
% 11.90/5.94 @ Prod ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1254_divmod__step__integer__def,axiom,
% 11.90/5.94 ( ( unique1321980374590559556d_step @ code_integer )
% 11.90/5.94 = ( ^ [L3: num] :
% 11.90/5.94 ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.94 @ ^ [Q5: code_integer,R3: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R3 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R3 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q5 ) @ R3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1255_signed__take__bit__rec,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 11.90/5.94 = ( ^ [N5: nat,A7: A] :
% 11.90/5.94 ( if @ A
% 11.90/5.94 @ ( N5
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 11.90/5.94 @ ( plus_plus @ A @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1256_round__unique,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,Y: int] :
% 11.90/5.94 ( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
% 11.90/5.94 => ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
% 11.90/5.94 => ( ( archimedean_round @ A @ X )
% 11.90/5.94 = Y ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1257_rel__of__def,axiom,
% 11.90/5.94 ! [B: $tType,A: $tType] :
% 11.90/5.94 ( ( rel_of @ A @ B )
% 11.90/5.94 = ( ^ [M: A > ( option @ B ),P4: ( product_prod @ A @ B ) > $o] :
% 11.90/5.94 ( collect @ ( product_prod @ A @ B )
% 11.90/5.94 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.94 @ ^ [K4: A,V2: B] :
% 11.90/5.94 ( ( ( M @ K4 )
% 11.90/5.94 = ( some @ B @ V2 ) )
% 11.90/5.94 & ( P4 @ ( product_Pair @ A @ B @ K4 @ V2 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1258_dbl__simps_I4_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1259_round__altdef,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ( ( archimedean_round @ A )
% 11.90/5.94 = ( ^ [X3: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X3 ) ) @ ( archimedean_ceiling @ A @ X3 ) @ ( archim6421214686448440834_floor @ A @ X3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1260_round__unique_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A,N: int] :
% 11.90/5.94 ( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 11.90/5.94 => ( ( archimedean_round @ A @ X )
% 11.90/5.94 = N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1261_of__int__round__abs__le,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1262_divmod__algorithm__code_I5_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.94 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 11.90/5.94 @ ^ [Q5: A,R3: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R3 ) )
% 11.90/5.94 @ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1263_signed__take__bit__of__minus__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1264_signed__take__bit__Suc__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1265_signed__take__bit__numeral__of__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ! [K: num] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1266_round__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ( ( archimedean_round @ A @ ( one_one @ A ) )
% 11.90/5.94 = ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1267_divmod__algorithm__code_I2_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1268_dbl__simps_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1269_divmod__algorithm__code_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1270_less__eq__integer__code_I1_J,axiom,
% 11.90/5.94 ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1271_sgn__integer__code,axiom,
% 11.90/5.94 ( ( sgn_sgn @ code_integer )
% 11.90/5.94 = ( ^ [K4: code_integer] :
% 11.90/5.94 ( if @ code_integer
% 11.90/5.94 @ ( K4
% 11.90/5.94 = ( zero_zero @ code_integer ) )
% 11.90/5.94 @ ( zero_zero @ code_integer )
% 11.90/5.94 @ ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1272_exhaustive__integer_H_Ocases,axiom,
% 11.90/5.94 ! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
% 11.90/5.94 ~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1273_full__exhaustive__integer_H_Ocases,axiom,
% 11.90/5.94 ! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
% 11.90/5.94 ~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I2: code_integer] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1274_times__integer__code_I1_J,axiom,
% 11.90/5.94 ! [K: code_integer] :
% 11.90/5.94 ( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 11.90/5.94 = ( zero_zero @ code_integer ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1275_times__integer__code_I2_J,axiom,
% 11.90/5.94 ! [L: code_integer] :
% 11.90/5.94 ( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 11.90/5.94 = ( zero_zero @ code_integer ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1276_minus__integer__code_I1_J,axiom,
% 11.90/5.94 ! [K: code_integer] :
% 11.90/5.94 ( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 11.90/5.94 = K ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1277_plus__integer__code_I2_J,axiom,
% 11.90/5.94 ! [L: code_integer] :
% 11.90/5.94 ( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 11.90/5.94 = L ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1278_plus__integer__code_I1_J,axiom,
% 11.90/5.94 ! [K: code_integer] :
% 11.90/5.94 ( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 11.90/5.94 = K ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1279_minus__integer__code_I2_J,axiom,
% 11.90/5.94 ! [L: code_integer] :
% 11.90/5.94 ( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
% 11.90/5.94 = ( uminus_uminus @ code_integer @ L ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1280_divmod__integer_H__def,axiom,
% 11.90/5.94 ( ( unique8689654367752047608divmod @ code_integer )
% 11.90/5.94 = ( ^ [M: num,N5: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N5 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M ) @ ( numeral_numeral @ code_integer @ N5 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1281_signed__take__bit__add,axiom,
% 11.90/5.94 ! [N: nat,K: int,L: int] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
% 11.90/5.94 = ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1282_dbl__def,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( neg_numeral_dbl @ A )
% 11.90/5.94 = ( ^ [X3: A] : ( plus_plus @ A @ X3 @ X3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1283_divmod__int__def,axiom,
% 11.90/5.94 ( ( unique8689654367752047608divmod @ int )
% 11.90/5.94 = ( ^ [M: num,N5: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1284_divmod__def,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ( ( unique8689654367752047608divmod @ A )
% 11.90/5.94 = ( ^ [M: num,N5: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N5 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1285_divmod_H__nat__def,axiom,
% 11.90/5.94 ( ( unique8689654367752047608divmod @ nat )
% 11.90/5.94 = ( ^ [M: num,N5: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N5 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N5 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1286_signed__take__bit__int__greater__eq,axiom,
% 11.90/5.94 ! [K: int,N: nat] :
% 11.90/5.94 ( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
% 11.90/5.94 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1287_divmod__divmod__step,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ( ( unique8689654367752047608divmod @ A )
% 11.90/5.94 = ( ^ [M: num,N5: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M @ N5 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M ) ) @ ( unique1321980374590559556d_step @ A @ N5 @ ( unique8689654367752047608divmod @ A @ M @ ( bit0 @ N5 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1288_round__def,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ( ( archimedean_round @ A )
% 11.90/5.94 = ( ^ [X3: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1289_signed__take__bit__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1290_of__int__round__le,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1291_of__int__round__ge,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1292_of__int__round__gt,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( archim2362893244070406136eiling @ A )
% 11.90/5.94 => ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1293_integer__of__int__code,axiom,
% 11.90/5.94 ( code_integer_of_int
% 11.90/5.94 = ( ^ [K4: int] :
% 11.90/5.94 ( if @ code_integer @ ( ord_less @ int @ K4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K4 ) ) )
% 11.90/5.94 @ ( if @ code_integer
% 11.90/5.94 @ ( K4
% 11.90/5.94 = ( zero_zero @ int ) )
% 11.90/5.94 @ ( zero_zero @ code_integer )
% 11.90/5.94 @ ( if @ code_integer
% 11.90/5.94 @ ( ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( zero_zero @ int ) )
% 11.90/5.94 @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 11.90/5.94 @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1294_divmod__algorithm__code_I6_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.94 = ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
% 11.90/5.94 @ ^ [Q5: A,R3: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R3 ) @ ( one_one @ A ) ) )
% 11.90/5.94 @ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1295_of__nat__code,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_1 @ A )
% 11.90/5.94 => ( ( semiring_1_of_nat @ A )
% 11.90/5.94 = ( ^ [N5: nat] :
% 11.90/5.94 ( semiri8178284476397505188at_aux @ A
% 11.90/5.94 @ ^ [I3: A] : ( plus_plus @ A @ I3 @ ( one_one @ A ) )
% 11.90/5.94 @ N5
% 11.90/5.94 @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1296_signed__take__bit__Suc__minus__bit1,axiom,
% 11.90/5.94 ! [N: nat,K: num] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1297_concat__bit__Suc,axiom,
% 11.90/5.94 ! [N: nat,K: int,L: int] :
% 11.90/5.94 ( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
% 11.90/5.94 = ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1298_even__succ__mod__exp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1299_even__succ__div__exp,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.94 = ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1300_take__bit__rec,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ( ( bit_se2584673776208193580ke_bit @ A )
% 11.90/5.94 = ( ^ [N5: nat,A7: A] :
% 11.90/5.94 ( if @ A
% 11.90/5.94 @ ( N5
% 11.90/5.94 = ( zero_zero @ nat ) )
% 11.90/5.94 @ ( zero_zero @ A )
% 11.90/5.94 @ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1301_nat__dvd__1__iff__1,axiom,
% 11.90/5.94 ! [M2: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ M2 @ ( one_one @ nat ) )
% 11.90/5.94 = ( M2
% 11.90/5.94 = ( one_one @ nat ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1302_integer__of__int__eq__of__int,axiom,
% 11.90/5.94 ( code_integer_of_int
% 11.90/5.94 = ( ring_1_of_int @ code_integer ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1303_dvd__add__triv__right__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_s4317794764714335236cancel @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1304_dvd__add__triv__left__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_s4317794764714335236cancel @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1305_dvd__add__times__triv__right__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_s4317794764714335236cancel @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1306_dvd__add__times__triv__left__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_s4317794764714335236cancel @ A )
% 11.90/5.94 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1307_unit__prod,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1308_div__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ C2 @ A3 )
% 11.90/5.94 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1309_unit__div__1__div__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 11.90/5.94 = A3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1310_unit__div__1__unit,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1311_unit__div,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1312_take__bit__Suc__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1313_take__bit__numeral__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.94 => ! [L: num] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1314_semiring__norm_I9_J,axiom,
% 11.90/5.94 ! [M2: num,N: num] :
% 11.90/5.94 ( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.94 = ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1315_semiring__norm_I7_J,axiom,
% 11.90/5.94 ! [M2: num,N: num] :
% 11.90/5.94 ( ( plus_plus @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( bit1 @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1316_semiring__norm_I16_J,axiom,
% 11.90/5.94 ! [M2: num,N: num] :
% 11.90/5.94 ( ( times_times @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ ( bit0 @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1317_unit__mult__div__div,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
% 11.90/5.94 = ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1318_unit__div__mult__self,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
% 11.90/5.94 = B2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1319_take__bit__of__1__eq__0__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 = ( N
% 11.90/5.94 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1320_semiring__norm_I10_J,axiom,
% 11.90/5.94 ! [M2: num,N: num] :
% 11.90/5.94 ( ( plus_plus @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M2 @ N ) @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1321_semiring__norm_I8_J,axiom,
% 11.90/5.94 ! [M2: num] :
% 11.90/5.94 ( ( plus_plus @ num @ ( bit1 @ M2 ) @ one2 )
% 11.90/5.94 = ( bit0 @ ( plus_plus @ num @ M2 @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1322_semiring__norm_I5_J,axiom,
% 11.90/5.94 ! [M2: num] :
% 11.90/5.94 ( ( plus_plus @ num @ ( bit0 @ M2 ) @ one2 )
% 11.90/5.94 = ( bit1 @ M2 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1323_semiring__norm_I4_J,axiom,
% 11.90/5.94 ! [N: num] :
% 11.90/5.94 ( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
% 11.90/5.94 = ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1324_semiring__norm_I3_J,axiom,
% 11.90/5.94 ! [N: num] :
% 11.90/5.94 ( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
% 11.90/5.94 = ( bit1 @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1325_odd__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) )
% 11.90/5.94 = ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.94 != ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1326_even__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.94 = ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1327_dbl__inc__simps_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1328_even__plus__one__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
% 11.90/5.94 = ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1329_even__diff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_parity @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1330_neg__one__even__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 11.90/5.94 = ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1331_neg__one__odd__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1332_even__diff__nat,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) )
% 11.90/5.94 = ( ( ord_less @ nat @ M2 @ N )
% 11.90/5.94 | ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1333_divmod__algorithm__code_I4_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1334_mod__Suc__eq__mod__add3,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 11.90/5.94 = ( modulo_modulo @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1335_Suc__mod__eq__add3__mod__numeral,axiom,
% 11.90/5.94 ! [M2: nat,V: num] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 11.90/5.94 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1336_div__Suc__eq__div__add3,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( divide_divide @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
% 11.90/5.94 = ( divide_divide @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1337_Suc__div__eq__add3__div__numeral,axiom,
% 11.90/5.94 ! [M2: nat,V: num] :
% 11.90/5.94 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
% 11.90/5.94 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1338_dbl__dec__simps_I4_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1339_even__succ__div__2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1340_odd__succ__div__two,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1341_even__succ__div__two,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1342_odd__two__times__div__two__nat,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 => ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
% 11.90/5.94 = ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1343_odd__two__times__div__two__succ,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
% 11.90/5.94 = A3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1344_semiring__parity__class_Oeven__mask__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
% 11.90/5.94 = ( N
% 11.90/5.94 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1345_zmod__numeral__Bit1,axiom,
% 11.90/5.94 ! [V: num,W: num] :
% 11.90/5.94 ( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1346_divmod__algorithm__code_I8_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( ( ord_less @ num @ M2 @ N )
% 11.90/5.94 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
% 11.90/5.94 & ( ~ ( ord_less @ num @ M2 @ N )
% 11.90/5.94 => ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1347_divmod__algorithm__code_I7_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( ( ord_less_eq @ num @ M2 @ N )
% 11.90/5.94 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
% 11.90/5.94 & ( ~ ( ord_less_eq @ num @ M2 @ N )
% 11.90/5.94 => ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.94 = ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1348_signed__take__bit__Suc__bit1,axiom,
% 11.90/5.94 ! [N: nat,K: num] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1349_uminus__integer__code_I1_J,axiom,
% 11.90/5.94 ( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
% 11.90/5.94 = ( zero_zero @ code_integer ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1350_take__bit__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: nat,X: int] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1351_abs__integer__code,axiom,
% 11.90/5.94 ( ( abs_abs @ code_integer )
% 11.90/5.94 = ( ^ [K4: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K4 ) @ K4 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1352_less__integer__code_I1_J,axiom,
% 11.90/5.94 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1353_unset__bit__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: nat,X: int] :
% 11.90/5.94 ( ( bit_se2638667681897837118et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1354_flip__bit__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: nat,X: int] :
% 11.90/5.94 ( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1355_set__bit__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: nat,X: int] :
% 11.90/5.94 ( ( bit_se5668285175392031749et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1356_take__bit__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.94 => ! [N: nat,A3: A,B2: A] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
% 11.90/5.94 = ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1357_zero__integer__def,axiom,
% 11.90/5.94 ( ( zero_zero @ code_integer )
% 11.90/5.94 = ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1358_less__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( ord_less @ int @ Xa @ X ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1359_uminus__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [X: int] :
% 11.90/5.94 ( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1360_abs__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [X: int] :
% 11.90/5.94 ( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1361_dvd__add__right__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_s4317794764714335236cancel @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ B2 )
% 11.90/5.94 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1362_dvd__add__left__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_s4317794764714335236cancel @ A )
% 11.90/5.94 => ! [A3: A,C2: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ C2 )
% 11.90/5.94 => ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1363_dvd__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_semiring_1 @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ B2 )
% 11.90/5.94 => ( ( dvd_dvd @ A @ A3 @ C2 )
% 11.90/5.94 => ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1364_divide__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1365_dvd__unit__imp__unit,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ B2 )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1366_unit__imp__dvd,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1367_one__dvd,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_monoid_mult @ A )
% 11.90/5.94 => ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1368_modulo__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( modulo_modulo @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1369_sgn__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [X: int] :
% 11.90/5.94 ( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1370_concat__bit__assoc,axiom,
% 11.90/5.94 ! [N: nat,K: int,M2: nat,L: int,R: int] :
% 11.90/5.94 ( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M2 @ L @ R ) )
% 11.90/5.94 = ( bit_concat_bit @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1371_not__is__unit__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1372_pinf_I9_J,axiom,
% 11.90/5.94 ! [B: $tType] :
% 11.90/5.94 ( ( ( plus @ B )
% 11.90/5.94 & ( linorder @ B )
% 11.90/5.94 & ( dvd @ B ) )
% 11.90/5.94 => ! [D3: B,S: B] :
% 11.90/5.94 ? [Z3: B] :
% 11.90/5.94 ! [X7: B] :
% 11.90/5.94 ( ( ord_less @ B @ Z3 @ X7 )
% 11.90/5.94 => ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) )
% 11.90/5.94 = ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1373_pinf_I10_J,axiom,
% 11.90/5.94 ! [B: $tType] :
% 11.90/5.94 ( ( ( plus @ B )
% 11.90/5.94 & ( linorder @ B )
% 11.90/5.94 & ( dvd @ B ) )
% 11.90/5.94 => ! [D3: B,S: B] :
% 11.90/5.94 ? [Z3: B] :
% 11.90/5.94 ! [X7: B] :
% 11.90/5.94 ( ( ord_less @ B @ Z3 @ X7 )
% 11.90/5.94 => ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) )
% 11.90/5.94 = ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1374_minf_I9_J,axiom,
% 11.90/5.94 ! [B: $tType] :
% 11.90/5.94 ( ( ( plus @ B )
% 11.90/5.94 & ( linorder @ B )
% 11.90/5.94 & ( dvd @ B ) )
% 11.90/5.94 => ! [D3: B,S: B] :
% 11.90/5.94 ? [Z3: B] :
% 11.90/5.94 ! [X7: B] :
% 11.90/5.94 ( ( ord_less @ B @ X7 @ Z3 )
% 11.90/5.94 => ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) )
% 11.90/5.94 = ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1375_minf_I10_J,axiom,
% 11.90/5.94 ! [B: $tType] :
% 11.90/5.94 ( ( ( plus @ B )
% 11.90/5.94 & ( linorder @ B )
% 11.90/5.94 & ( dvd @ B ) )
% 11.90/5.94 => ! [D3: B,S: B] :
% 11.90/5.94 ? [Z3: B] :
% 11.90/5.94 ! [X7: B] :
% 11.90/5.94 ( ( ord_less @ B @ X7 @ Z3 )
% 11.90/5.94 => ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) )
% 11.90/5.94 = ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X7 @ S ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1376_unit__mult__right__cancel,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ( times_times @ A @ B2 @ A3 )
% 11.90/5.94 = ( times_times @ A @ C2 @ A3 ) )
% 11.90/5.94 = ( B2 = C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1377_unit__mult__left__cancel,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ( times_times @ A @ A3 @ B2 )
% 11.90/5.94 = ( times_times @ A @ A3 @ C2 ) )
% 11.90/5.94 = ( B2 = C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1378_mult__unit__dvd__iff_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1379_dvd__mult__unit__iff_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1380_mult__unit__dvd__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1381_dvd__mult__unit__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1382_is__unit__mult__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1383_div__plus__div__distrib__dvd__left,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.94 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ C2 @ A3 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1384_div__plus__div__distrib__dvd__right,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.94 => ! [C2: A,B2: A,A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ C2 @ B2 )
% 11.90/5.94 => ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1385_unit__div__cancel,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ( divide_divide @ A @ B2 @ A3 )
% 11.90/5.94 = ( divide_divide @ A @ C2 @ A3 ) )
% 11.90/5.94 = ( B2 = C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1386_div__unit__dvd__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1387_dvd__div__unit__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1388_xor__num_Ocases,axiom,
% 11.90/5.94 ! [X: product_prod @ num @ num] :
% 11.90/5.94 ( ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ one2 @ one2 ) )
% 11.90/5.94 => ( ! [N3: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
% 11.90/5.94 => ( ! [N3: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
% 11.90/5.94 => ( ! [M5: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) )
% 11.90/5.94 => ( ! [M5: num,N3: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
% 11.90/5.94 => ( ! [M5: num,N3: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
% 11.90/5.94 => ( ! [M5: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) )
% 11.90/5.94 => ( ! [M5: num,N3: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
% 11.90/5.94 => ~ ! [M5: num,N3: num] :
% 11.90/5.94 ( X
% 11.90/5.94 != ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1389_plus__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1390_bezout__lemma__nat,axiom,
% 11.90/5.94 ! [D3: nat,A3: nat,B2: nat,X: nat,Y: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ D3 @ A3 )
% 11.90/5.94 => ( ( dvd_dvd @ nat @ D3 @ B2 )
% 11.90/5.94 => ( ( ( ( times_times @ nat @ A3 @ X )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D3 ) )
% 11.90/5.94 | ( ( times_times @ nat @ B2 @ X )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y ) @ D3 ) ) )
% 11.90/5.94 => ? [X2: nat,Y4: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ D3 @ A3 )
% 11.90/5.94 & ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A3 @ B2 ) )
% 11.90/5.94 & ( ( ( times_times @ nat @ A3 @ X2 )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y4 ) @ D3 ) )
% 11.90/5.94 | ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X2 )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D3 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1391_bezout__add__nat,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat] :
% 11.90/5.94 ? [D2: nat,X2: nat,Y4: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 11.90/5.94 & ( dvd_dvd @ nat @ D2 @ B2 )
% 11.90/5.94 & ( ( ( times_times @ nat @ A3 @ X2 )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) )
% 11.90/5.94 | ( ( times_times @ nat @ B2 @ X2 )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1392_one__integer__def,axiom,
% 11.90/5.94 ( ( one_one @ code_integer )
% 11.90/5.94 = ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1393_less__eq__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( ord_less_eq @ int @ Xa @ X ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1394_times__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( times_times @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1395_zdvd__period,axiom,
% 11.90/5.94 ! [A3: int,D3: int,X: int,T2: int,C2: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ A3 @ D3 )
% 11.90/5.94 => ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
% 11.90/5.94 = ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D3 ) ) @ T2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1396_zdvd__reduce,axiom,
% 11.90/5.94 ! [K: int,N: int,M2: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M2 ) ) )
% 11.90/5.94 = ( dvd_dvd @ int @ K @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1397_minus__integer_Oabs__eq,axiom,
% 11.90/5.94 ! [Xa: int,X: int] :
% 11.90/5.94 ( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.94 = ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1398_unity__coeff__ex,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( dvd @ A )
% 11.90/5.94 & ( semiring_0 @ A ) )
% 11.90/5.94 => ! [P: A > $o,L: A] :
% 11.90/5.94 ( ( ? [X3: A] : ( P @ ( times_times @ A @ L @ X3 ) ) )
% 11.90/5.94 = ( ? [X3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X3 @ ( zero_zero @ A ) ) )
% 11.90/5.94 & ( P @ X3 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1399_unit__dvdE,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ~ ( ( A3
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ! [C4: A] :
% 11.90/5.94 ( B2
% 11.90/5.94 != ( times_times @ A @ A3 @ C4 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1400_inf__period_I4_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( comm_ring @ A )
% 11.90/5.94 & ( dvd @ A ) )
% 11.90/5.94 => ! [D3: A,D4: A,T2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ D3 @ D4 )
% 11.90/5.94 => ! [X7: A,K8: A] :
% 11.90/5.94 ( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T2 ) ) )
% 11.90/5.94 = ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K8 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1401_inf__period_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ( comm_ring @ A )
% 11.90/5.94 & ( dvd @ A ) )
% 11.90/5.94 => ! [D3: A,D4: A,T2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ D3 @ D4 )
% 11.90/5.94 => ! [X7: A,K8: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X7 @ T2 ) )
% 11.90/5.94 = ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X7 @ ( times_times @ A @ K8 @ D4 ) ) @ T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1402_unit__div__eq__0__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 = ( A3
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1403_numeral__Bit1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1404_unit__eq__div1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.94 = C2 )
% 11.90/5.94 = ( A3
% 11.90/5.94 = ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1405_unit__eq__div2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( A3
% 11.90/5.94 = ( divide_divide @ A @ C2 @ B2 ) )
% 11.90/5.94 = ( ( times_times @ A @ A3 @ B2 )
% 11.90/5.94 = C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1406_div__mult__unit2,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [C2: A,B2: A,A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B2 @ A3 )
% 11.90/5.94 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1407_unit__div__commute,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
% 11.90/5.94 = ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1408_unit__div__mult__swap,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1409_is__unit__div__mult2__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [B2: A,C2: A,A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/5.94 = ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1410_unit__imp__mod__eq__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid3725896446679973847miring @ A )
% 11.90/5.94 => ! [B2: A,A3: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 11.90/5.94 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1411_is__unit__power__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
% 11.90/5.94 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 | ( N
% 11.90/5.94 = ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1412_bezout__add__strong__nat,axiom,
% 11.90/5.94 ! [A3: nat,B2: nat] :
% 11.90/5.94 ( ( A3
% 11.90/5.94 != ( zero_zero @ nat ) )
% 11.90/5.94 => ? [D2: nat,X2: nat,Y4: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ D2 @ A3 )
% 11.90/5.94 & ( dvd_dvd @ nat @ D2 @ B2 )
% 11.90/5.94 & ( ( times_times @ nat @ A3 @ X2 )
% 11.90/5.94 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1413_take__bit__Suc__bit1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.94 => ! [N: nat,K: num] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1414_numeral__code_I3_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( numeral_numeral @ A @ ( bit1 @ N ) )
% 11.90/5.94 = ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1415_stable__imp__take__bit__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [A3: A,N: nat] :
% 11.90/5.94 ( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = A3 )
% 11.90/5.94 => ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 11.90/5.94 = ( zero_zero @ A ) ) )
% 11.90/5.94 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
% 11.90/5.94 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1416_is__unitE,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,C2: A] :
% 11.90/5.94 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.94 => ~ ( ( A3
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ! [B3: A] :
% 11.90/5.94 ( ( B3
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.94 = B3 )
% 11.90/5.94 => ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
% 11.90/5.94 = A3 )
% 11.90/5.94 => ( ( ( times_times @ A @ A3 @ B3 )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ C2 @ A3 )
% 11.90/5.94 != ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1417_is__unit__div__mult__cancel__left,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( A3
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
% 11.90/5.94 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1418_is__unit__div__mult__cancel__right,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ( A3
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/5.94 => ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
% 11.90/5.94 = ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1419_odd__even__add,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A,B2: A] :
% 11.90/5.94 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
% 11.90/5.94 => ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1420_odd__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1421_dvd__power__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( algebraic_semidom @ A )
% 11.90/5.94 => ! [X: A,M2: nat,N: nat] :
% 11.90/5.94 ( ( X
% 11.90/5.94 != ( zero_zero @ A ) )
% 11.90/5.94 => ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
% 11.90/5.94 = ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
% 11.90/5.94 | ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1422_dvd__power,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( comm_semiring_1 @ A )
% 11.90/5.94 => ! [N: nat,X: A] :
% 11.90/5.94 ( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.94 | ( X
% 11.90/5.94 = ( one_one @ A ) ) )
% 11.90/5.94 => ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1423_Suc3__eq__add__3,axiom,
% 11.90/5.94 ! [N: nat] :
% 11.90/5.94 ( ( suc @ ( suc @ ( suc @ N ) ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1424_dvd__mult__cancel2,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 11.90/5.94 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M2 ) @ M2 )
% 11.90/5.94 = ( N
% 11.90/5.94 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1425_dvd__mult__cancel1,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
% 11.90/5.94 => ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N ) @ M2 )
% 11.90/5.94 = ( N
% 11.90/5.94 = ( one_one @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1426_dvd__minus__add,axiom,
% 11.90/5.94 ! [Q3: nat,N: nat,R: nat,M2: nat] :
% 11.90/5.94 ( ( ord_less_eq @ nat @ Q3 @ N )
% 11.90/5.94 => ( ( ord_less_eq @ nat @ Q3 @ ( times_times @ nat @ R @ M2 ) )
% 11.90/5.94 => ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q3 ) )
% 11.90/5.94 = ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M2 ) @ Q3 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1427_power__dvd__imp__le,axiom,
% 11.90/5.94 ! [I: nat,M2: nat,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
% 11.90/5.94 => ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
% 11.90/5.94 => ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1428_aset_I10_J,axiom,
% 11.90/5.94 ! [D3: int,D4: int,A4: set @ int,T2: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ D3 @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 11.90/5.94 => ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1429_aset_I9_J,axiom,
% 11.90/5.94 ! [D3: int,D4: int,A4: set @ int,T2: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ D3 @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ A4 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 11.90/5.94 => ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( plus_plus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1430_bset_I10_J,axiom,
% 11.90/5.94 ! [D3: int,D4: int,B6: set @ int,T2: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ D3 @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 11.90/5.94 => ~ ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1431_bset_I9_J,axiom,
% 11.90/5.94 ! [D3: int,D4: int,B6: set @ int,T2: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ D3 @ D4 )
% 11.90/5.94 => ! [X7: int] :
% 11.90/5.94 ( ! [Xa3: int] :
% 11.90/5.94 ( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
% 11.90/5.94 => ! [Xb3: int] :
% 11.90/5.94 ( ( member @ int @ Xb3 @ B6 )
% 11.90/5.94 => ( X7
% 11.90/5.94 != ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
% 11.90/5.94 => ( ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ X7 @ T2 ) )
% 11.90/5.94 => ( dvd_dvd @ int @ D3 @ ( plus_plus @ int @ ( minus_minus @ int @ X7 @ D4 ) @ T2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1432_num_Osize__gen_I3_J,axiom,
% 11.90/5.94 ! [X32: num] :
% 11.90/5.94 ( ( size_num @ ( bit1 @ X32 ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1433_num_Osize_I6_J,axiom,
% 11.90/5.94 ! [X32: num] :
% 11.90/5.94 ( ( size_size @ num @ ( bit1 @ X32 ) )
% 11.90/5.94 = ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1434_odd__iff__mod__2__eq__one,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.94 = ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1435_Suc__mod__eq__add3__mod,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
% 11.90/5.94 = ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1436_Suc__div__eq__add3__div,axiom,
% 11.90/5.94 ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
% 11.90/5.94 = ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1437_even__diff__iff,axiom,
% 11.90/5.94 ! [K: int,L: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
% 11.90/5.94 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1438_even__abs__add__iff,axiom,
% 11.90/5.94 ! [K: int,L: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
% 11.90/5.94 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1439_even__add__abs__iff,axiom,
% 11.90/5.94 ! [K: int,L: int] :
% 11.90/5.94 ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
% 11.90/5.94 = ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1440_signed__take__bit__eq__take__bit__shift,axiom,
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ int )
% 11.90/5.94 = ( ^ [N5: nat,K4: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ ( plus_plus @ int @ K4 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1441_take__bit__incr__eq,axiom,
% 11.90/5.94 ! [N: nat,K: int] :
% 11.90/5.94 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 11.90/5.94 != ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
% 11.90/5.94 => ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1442_oddE,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ~ ! [B3: A] :
% 11.90/5.94 ( A3
% 11.90/5.94 != ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1443_parity__cases,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 != ( zero_zero @ A ) ) )
% 11.90/5.94 => ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 != ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1444_mod2__eq__if,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( semiring_parity @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( zero_zero @ A ) ) )
% 11.90/5.94 & ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.94 => ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.94 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1445_minus__one__power__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 11.90/5.94 = ( one_one @ A ) ) )
% 11.90/5.94 & ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 => ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 11.90/5.94 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1446_take__bit__int__greater__eq,axiom,
% 11.90/5.94 ! [K: int,N: nat] :
% 11.90/5.94 ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
% 11.90/5.94 => ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1447_take__bit__Suc__minus__1__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1448_take__bit__Suc,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [N: nat,A3: A] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1449_take__bit__numeral__minus__1__eq,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.94 => ! [K: num] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1450_even__mask__div__iff_H,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.94 => ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 11.90/5.94 = ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1451_even__mask__div__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_semiring_bits @ A )
% 11.90/5.94 => ! [M2: nat,N: nat] :
% 11.90/5.94 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
% 11.90/5.94 = ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 11.90/5.94 = ( zero_zero @ A ) )
% 11.90/5.94 | ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1452_Code__Numeral_Onegative__def,axiom,
% 11.90/5.94 ( code_negative
% 11.90/5.94 = ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1453_take__bit__numeral__minus__numeral__int,axiom,
% 11.90/5.94 ! [M2: num,N: num] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/5.94 = ( case_option @ int @ num @ ( zero_zero @ int )
% 11.90/5.94 @ ^ [Q5: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ int @ Q5 ) ) )
% 11.90/5.94 @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1454_signed__take__bit__numeral__minus__bit1,axiom,
% 11.90/5.94 ! [L: num,K: num] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1455_execute__change,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( heap @ A )
% 11.90/5.94 => ! [F: A > A,R: ref @ A,H3: heap_ext @ product_unit] :
% 11.90/5.94 ( ( heap_Time_execute @ A @ ( ref_change @ A @ F @ R ) @ H3 )
% 11.90/5.94 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1456_signed__take__bit__numeral__bit1,axiom,
% 11.90/5.94 ! [L: num,K: num] :
% 11.90/5.94 ( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1457_take__bit__Suc__minus__bit1,axiom,
% 11.90/5.94 ! [N: nat,K: num] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 11.90/5.94 = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1458_divmod__BitM__2__eq,axiom,
% 11.90/5.94 ! [M2: num] :
% 11.90/5.94 ( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
% 11.90/5.94 = ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1459_flip__bit__0,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [A3: A] :
% 11.90/5.94 ( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
% 11.90/5.94 = ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1460_of__bool__eq_I2_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( zero_neq_one @ A )
% 11.90/5.94 => ( ( zero_neq_one_of_bool @ A @ $true )
% 11.90/5.94 = ( one_one @ A ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1461_of__bool__eq__1__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( zero_neq_one @ A )
% 11.90/5.94 => ! [P: $o] :
% 11.90/5.94 ( ( ( zero_neq_one_of_bool @ A @ P )
% 11.90/5.94 = ( one_one @ A ) )
% 11.90/5.94 = P ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1462_of__bool__less__one__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( linordered_semidom @ A )
% 11.90/5.94 => ! [P: $o] :
% 11.90/5.94 ( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
% 11.90/5.94 = ~ P ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1463_of__bool__not__iff,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( ring_1 @ A )
% 11.90/5.94 => ! [P: $o] :
% 11.90/5.94 ( ( zero_neq_one_of_bool @ A @ ~ P )
% 11.90/5.94 = ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1464_take__bit__num__simps_I3_J,axiom,
% 11.90/5.94 ! [N: nat,M2: num] :
% 11.90/5.94 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
% 11.90/5.94 = ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 11.90/5.94 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
% 11.90/5.94 @ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1465_take__bit__num__simps_I4_J,axiom,
% 11.90/5.94 ! [N: nat,M2: num] :
% 11.90/5.94 ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
% 11.90/5.94 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1466_take__bit__num__simps_I6_J,axiom,
% 11.90/5.94 ! [R: num,M2: num] :
% 11.90/5.94 ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit0 @ M2 ) )
% 11.90/5.94 = ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 11.90/5.94 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
% 11.90/5.94 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M2 ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1467_take__bit__of__1,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [N: nat] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
% 11.90/5.94 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1468_add__neg__numeral__special_I6_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1469_add__neg__numeral__special_I5_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1470_diff__numeral__special_I5_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [N: num] :
% 11.90/5.94 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1471_diff__numeral__special_I6_J,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( neg_numeral @ A )
% 11.90/5.94 => ! [M2: num] :
% 11.90/5.94 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.94 = ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1472_take__bit__num__simps_I7_J,axiom,
% 11.90/5.94 ! [R: num,M2: num] :
% 11.90/5.94 ( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit1 @ M2 ) )
% 11.90/5.94 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M2 ) ) ) ) ).
% 11.90/5.94
% 11.90/5.94 thf(fact_1473_take__bit__numeral__numeral,axiom,
% 11.90/5.94 ! [A: $tType] :
% 11.90/5.94 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.94 => ! [M2: num,N: num] :
% 11.90/5.94 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.94 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1474_bits__1__div__exp,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.95 = ( zero_neq_one_of_bool @ A
% 11.90/5.95 @ ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1475_one__div__2__pow__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.95 = ( zero_neq_one_of_bool @ A
% 11.90/5.95 @ ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1476_one__mod__2__pow__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.95 = ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1477_mult__inc,axiom,
% 11.90/5.95 ! [X: num,Y: num] :
% 11.90/5.95 ( ( times_times @ num @ X @ ( inc @ Y ) )
% 11.90/5.95 = ( plus_plus @ num @ ( times_times @ num @ X @ Y ) @ X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1478_add__inc,axiom,
% 11.90/5.95 ! [X: num,Y: num] :
% 11.90/5.95 ( ( plus_plus @ num @ X @ ( inc @ Y ) )
% 11.90/5.95 = ( inc @ ( plus_plus @ num @ X @ Y ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1479_of__bool__less__eq__one,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linordered_semidom @ A )
% 11.90/5.95 => ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1480_of__bool__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( zero_neq_one @ A )
% 11.90/5.95 => ( ( zero_neq_one_of_bool @ A )
% 11.90/5.95 = ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1481_split__of__bool,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( zero_neq_one @ A )
% 11.90/5.95 => ! [P: A > $o,P5: $o] :
% 11.90/5.95 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 11.90/5.95 = ( ( P5
% 11.90/5.95 => ( P @ ( one_one @ A ) ) )
% 11.90/5.95 & ( ~ P5
% 11.90/5.95 => ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1482_split__of__bool__asm,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( zero_neq_one @ A )
% 11.90/5.95 => ! [P: A > $o,P5: $o] :
% 11.90/5.95 ( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
% 11.90/5.95 = ( ~ ( ( P5
% 11.90/5.95 & ~ ( P @ ( one_one @ A ) ) )
% 11.90/5.95 | ( ~ P5
% 11.90/5.95 & ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1483_add__One,axiom,
% 11.90/5.95 ! [X: num] :
% 11.90/5.95 ( ( plus_plus @ num @ X @ one2 )
% 11.90/5.95 = ( inc @ X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1484_effect__updateI,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( heap @ A )
% 11.90/5.95 => ! [H4: heap_ext @ product_unit,R: ref @ A,V: A,H3: heap_ext @ product_unit,N: nat,X: product_unit] :
% 11.90/5.95 ( ( H4
% 11.90/5.95 = ( ref_set @ A @ R @ V @ H3 ) )
% 11.90/5.95 => ( ( N
% 11.90/5.95 = ( one_one @ nat ) )
% 11.90/5.95 => ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R @ V ) @ H3 @ H4 @ X @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1485_effect__updateE,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( heap @ A )
% 11.90/5.95 => ! [R4: ref @ A,V: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: product_unit,N: nat] :
% 11.90/5.95 ( ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R4 @ V ) @ H3 @ H4 @ R @ N )
% 11.90/5.95 => ~ ( ( H4
% 11.90/5.95 = ( ref_set @ A @ R4 @ V @ H3 ) )
% 11.90/5.95 => ( N
% 11.90/5.95 != ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1486_pred__numeral__def,axiom,
% 11.90/5.95 ( pred_numeral
% 11.90/5.95 = ( ^ [K4: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K4 ) @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1487_BitM__plus__one,axiom,
% 11.90/5.95 ! [N: num] :
% 11.90/5.95 ( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
% 11.90/5.95 = ( bit0 @ N ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1488_one__plus__BitM,axiom,
% 11.90/5.95 ! [N: num] :
% 11.90/5.95 ( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
% 11.90/5.95 = ( bit0 @ N ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1489_numeral__inc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( numeral @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( numeral_numeral @ A @ ( inc @ X ) )
% 11.90/5.95 = ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1490_numeral__BitM,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( neg_numeral @ A )
% 11.90/5.95 => ! [N: num] :
% 11.90/5.95 ( ( numeral_numeral @ A @ ( bitM @ N ) )
% 11.90/5.95 = ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1491_take__bit__numeral__minus__bit1,axiom,
% 11.90/5.95 ! [L: num,K: num] :
% 11.90/5.95 ( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
% 11.90/5.95 = ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1492_bits__induct,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ! [P: A > $o,A3: A] :
% 11.90/5.95 ( ! [A5: A] :
% 11.90/5.95 ( ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.95 = A5 )
% 11.90/5.95 => ( P @ A5 ) )
% 11.90/5.95 => ( ! [A5: A,B3: $o] :
% 11.90/5.95 ( ( P @ A5 )
% 11.90/5.95 => ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
% 11.90/5.95 = A5 )
% 11.90/5.95 => ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) ) ) ) )
% 11.90/5.95 => ( P @ A3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1493_effect__changeE,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( heap @ A )
% 11.90/5.95 => ! [F: A > A,R4: ref @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.95 ( ( heap_Time_effect @ A @ ( ref_change @ A @ F @ R4 ) @ H3 @ H4 @ R @ N )
% 11.90/5.95 => ~ ( ( H4
% 11.90/5.95 = ( ref_set @ A @ R4 @ ( F @ ( ref_get @ A @ H3 @ R4 ) ) @ H3 ) )
% 11.90/5.95 => ( ( R
% 11.90/5.95 = ( F @ ( ref_get @ A @ H3 @ R4 ) ) )
% 11.90/5.95 => ( N
% 11.90/5.95 != ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1494_effect__changeI,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( heap @ A )
% 11.90/5.95 => ! [H4: heap_ext @ product_unit,R: ref @ A,F: A > A,H3: heap_ext @ product_unit,X: A,N: nat] :
% 11.90/5.95 ( ( H4
% 11.90/5.95 = ( ref_set @ A @ R @ ( F @ ( ref_get @ A @ H3 @ R ) ) @ H3 ) )
% 11.90/5.95 => ( ( X
% 11.90/5.95 = ( F @ ( ref_get @ A @ H3 @ R ) ) )
% 11.90/5.95 => ( ( N
% 11.90/5.95 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 11.90/5.95 => ( heap_Time_effect @ A @ ( ref_change @ A @ F @ R ) @ H3 @ H4 @ X @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1495_divide__int__unfold,axiom,
% 11.90/5.95 ! [L: int,K: int,N: nat,M2: nat] :
% 11.90/5.95 ( ( ( ( ( sgn_sgn @ int @ L )
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 | ( ( sgn_sgn @ int @ K )
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 | ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) )
% 11.90/5.95 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.95 = ( zero_zero @ int ) ) )
% 11.90/5.95 & ( ~ ( ( ( sgn_sgn @ int @ L )
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 | ( ( sgn_sgn @ int @ K )
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 | ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) )
% 11.90/5.95 => ( ( ( ( sgn_sgn @ int @ K )
% 11.90/5.95 = ( sgn_sgn @ int @ L ) )
% 11.90/5.95 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.95 = ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) )
% 11.90/5.95 & ( ( ( sgn_sgn @ int @ K )
% 11.90/5.95 != ( sgn_sgn @ int @ L ) )
% 11.90/5.95 => ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.95 = ( uminus_uminus @ int
% 11.90/5.95 @ ( semiring_1_of_nat @ int
% 11.90/5.95 @ ( plus_plus @ nat @ ( divide_divide @ nat @ M2 @ N )
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1496_take__bit__numeral__bit1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [L: num,K: num] :
% 11.90/5.95 ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1497_Divides_Oadjust__div__eq,axiom,
% 11.90/5.95 ! [Q3: int,R: int] :
% 11.90/5.95 ( ( adjust_div @ ( product_Pair @ int @ int @ Q3 @ R ) )
% 11.90/5.95 = ( plus_plus @ int @ Q3
% 11.90/5.95 @ ( zero_neq_one_of_bool @ int
% 11.90/5.95 @ ( R
% 11.90/5.95 != ( zero_zero @ int ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1498_mask__numeral,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [N: num] :
% 11.90/5.95 ( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1499_and__int__unfold,axiom,
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int )
% 11.90/5.95 = ( ^ [K4: int,L3: int] :
% 11.90/5.95 ( if @ int
% 11.90/5.95 @ ( ( K4
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 | ( L3
% 11.90/5.95 = ( zero_zero @ int ) ) )
% 11.90/5.95 @ ( zero_zero @ int )
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.95 @ L3
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( L3
% 11.90/5.95 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.95 @ K4
% 11.90/5.95 @ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1500_the__res__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( time_the_res @ A )
% 11.90/5.95 = ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 11.90/5.95 ( case_option @ A @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ A )
% 11.90/5.95 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A
% 11.90/5.95 @ ^ [R3: A] :
% 11.90/5.95 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ A
% 11.90/5.95 @ ^ [X3: heap_ext @ product_unit,Xa4: nat] : R3 ) )
% 11.90/5.95 @ ( heap_Time_execute @ A @ M @ H ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1501_take__bit__eq__mask__iff__exp__dvd,axiom,
% 11.90/5.95 ! [N: nat,K: int] :
% 11.90/5.95 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 11.90/5.95 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 11.90/5.95 = ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1502_Code__Numeral_Opositive__def,axiom,
% 11.90/5.95 ( code_positive
% 11.90/5.95 = ( numeral_numeral @ code_integer ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1503_divide__int__def,axiom,
% 11.90/5.95 ( ( divide_divide @ int )
% 11.90/5.95 = ( ^ [K4: int,L3: int] :
% 11.90/5.95 ( if @ int
% 11.90/5.95 @ ( L3
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 @ ( zero_zero @ int )
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( ( sgn_sgn @ int @ K4 )
% 11.90/5.95 = ( sgn_sgn @ int @ L3 ) )
% 11.90/5.95 @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
% 11.90/5.95 @ ( uminus_uminus @ int
% 11.90/5.95 @ ( semiring_1_of_nat @ int
% 11.90/5.95 @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K4 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ~ ( dvd_dvd @ int @ L3 @ K4 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1504_sum__gp0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( division_ring @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( ( X
% 11.90/5.95 = ( one_one @ A ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
% 11.90/5.95 & ( ( X
% 11.90/5.95 != ( one_one @ A ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1505_atMost__eq__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( order @ A )
% 11.90/5.95 => ! [X: A,Y: A] :
% 11.90/5.95 ( ( ( set_ord_atMost @ A @ X )
% 11.90/5.95 = ( set_ord_atMost @ A @ Y ) )
% 11.90/5.95 = ( X = Y ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1506_atMost__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ord @ A )
% 11.90/5.95 => ! [I: A,K: A] :
% 11.90/5.95 ( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
% 11.90/5.95 = ( ord_less_eq @ A @ I @ K ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1507_atMost__subset__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( preorder @ A )
% 11.90/5.95 => ! [X: A,Y: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
% 11.90/5.95 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1508_bit_Oconj__one__right,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1509_and_Oright__neutral,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = A3 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1510_and_Oleft__neutral,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
% 11.90/5.95 = A3 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1511_and__numerals_I2_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [Y: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1512_and__numerals_I8_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1513_Icc__subset__Iic__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( preorder @ A )
% 11.90/5.95 => ! [L: A,H3: A,H4: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) @ ( set_ord_atMost @ A @ H4 ) )
% 11.90/5.95 = ( ~ ( ord_less_eq @ A @ L @ H3 )
% 11.90/5.95 | ( ord_less_eq @ A @ H3 @ H4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1514_mask__Suc__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1515_sum_OatMost__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1516_take__bit__minus__one__eq__mask,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1517_and__numerals_I1_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [Y: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 11.90/5.95 = ( zero_zero @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1518_and__numerals_I5_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 11.90/5.95 = ( zero_zero @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1519_nat__numeral__diff__1,axiom,
% 11.90/5.95 ! [V: num] :
% 11.90/5.95 ( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
% 11.90/5.95 = ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1520_and__numerals_I7_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num,Y: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1521_mask__integer_Oabs__eq,axiom,
% 11.90/5.95 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 11.90/5.95 = ( ^ [X3: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1522_not__Iic__eq__Icc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( no_bot @ A )
% 11.90/5.95 => ! [H4: A,L: A,H3: A] :
% 11.90/5.95 ( ( set_ord_atMost @ A @ H4 )
% 11.90/5.95 != ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1523_atMost__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ord @ A )
% 11.90/5.95 => ( ( set_ord_atMost @ A )
% 11.90/5.95 = ( ^ [U2: A] :
% 11.90/5.95 ( collect @ A
% 11.90/5.95 @ ^ [X3: A] : ( ord_less_eq @ A @ X3 @ U2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1524_and__eq__minus__1__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [A3: A,B2: A] :
% 11.90/5.95 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( ( A3
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 & ( B2
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1525_atMost__atLeast0,axiom,
% 11.90/5.95 ( ( set_ord_atMost @ nat )
% 11.90/5.95 = ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1526_nat__one__as__int,axiom,
% 11.90/5.95 ( ( one_one @ nat )
% 11.90/5.95 = ( nat2 @ ( one_one @ int ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1527_not__Iic__le__Icc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( no_bot @ A )
% 11.90/5.95 => ! [H3: A,L4: A,H4: A] :
% 11.90/5.95 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H3 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1528_nat__int__add,axiom,
% 11.90/5.95 ! [A3: nat,B2: nat] :
% 11.90/5.95 ( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
% 11.90/5.95 = ( plus_plus @ nat @ A3 @ B2 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1529_option_Othe__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( the2 @ A )
% 11.90/5.95 = ( case_option @ A @ A @ ( undefined @ A )
% 11.90/5.95 @ ^ [X24: A] : X24 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1530_nat__plus__as__int,axiom,
% 11.90/5.95 ( ( plus_plus @ nat )
% 11.90/5.95 = ( ^ [A7: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1531_nat__add__distrib,axiom,
% 11.90/5.95 ! [Z2: int,Z7: int] :
% 11.90/5.95 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 11.90/5.95 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
% 11.90/5.95 => ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z7 ) )
% 11.90/5.95 = ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1532_Suc__as__int,axiom,
% 11.90/5.95 ( suc
% 11.90/5.95 = ( ^ [A7: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A7 ) @ ( one_one @ int ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1533_nat__abs__triangle__ineq,axiom,
% 11.90/5.95 ! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1534_sum_OatMost__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 11.90/5.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1535_sum__telescope,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ab_group_add @ A )
% 11.90/5.95 => ! [F: nat > A,I: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( minus_minus @ A @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ I ) )
% 11.90/5.95 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1536_the__heap__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( time_the_heap @ A )
% 11.90/5.95 = ( ^ [M: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 11.90/5.95 ( case_option @ ( heap_ext @ product_unit ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ ( heap_ext @ product_unit ) )
% 11.90/5.95 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( heap_ext @ product_unit )
% 11.90/5.95 @ ^ [X3: A] :
% 11.90/5.95 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( heap_ext @ product_unit )
% 11.90/5.95 @ ^ [H7: heap_ext @ product_unit,Xa4: nat] : H7 ) )
% 11.90/5.95 @ ( heap_Time_execute @ A @ M @ H ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1537_and__one__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.95 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1538_one__and__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.95 = ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1539_take__bit__eq__mask__iff,axiom,
% 11.90/5.95 ! [N: nat,K: int] :
% 11.90/5.95 ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
% 11.90/5.95 = ( bit_se2239418461657761734s_mask @ int @ N ) )
% 11.90/5.95 = ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
% 11.90/5.95 = ( zero_zero @ int ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1540_Suc__nat__eq__nat__zadd1,axiom,
% 11.90/5.95 ! [Z2: int] :
% 11.90/5.95 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
% 11.90/5.95 => ( ( suc @ ( nat2 @ Z2 ) )
% 11.90/5.95 = ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1541_Divides_Oadjust__div__def,axiom,
% 11.90/5.95 ( adjust_div
% 11.90/5.95 = ( product_case_prod @ int @ int @ int
% 11.90/5.95 @ ^ [Q5: int,R3: int] :
% 11.90/5.95 ( plus_plus @ int @ Q5
% 11.90/5.95 @ ( zero_neq_one_of_bool @ int
% 11.90/5.95 @ ( R3
% 11.90/5.95 != ( zero_zero @ int ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1542_sum__gp__basic,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
% 11.90/5.95 = ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1543_sum__power__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [M2: nat,N: nat,X: A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1544_sum_Otriangle__reindex__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 11.90/5.95 @ ( collect @ ( product_prod @ nat @ nat )
% 11.90/5.95 @ ( product_case_prod @ nat @ nat @ $o
% 11.90/5.95 @ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] :
% 11.90/5.95 ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ K4 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1545_and__int__rec,axiom,
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int )
% 11.90/5.95 = ( ^ [K4: int,L3: int] :
% 11.90/5.95 ( plus_plus @ int
% 11.90/5.95 @ ( zero_neq_one_of_bool @ int
% 11.90/5.95 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 11.90/5.95 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 11.90/5.95 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1546_mask__half__int,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
% 11.90/5.95 = ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1547_sum_Oin__pairs__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1548_mask__nat__def,axiom,
% 11.90/5.95 ( ( bit_se2239418461657761734s_mask @ nat )
% 11.90/5.95 = ( ^ [N5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1549_mask__eq__exp__minus__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ( ( bit_se2239418461657761734s_mask @ A )
% 11.90/5.95 = ( ^ [N5: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1550_sum_Ozero__middle,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [P5: nat,K: nat,G: nat > A,H3: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 11.90/5.95 => ( ( ord_less_eq @ nat @ K @ P5 )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( zero_zero @ A ) @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ P5 ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1551_and__minus__numerals_I3_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 11.90/5.95 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1552_and__minus__numerals_I7_J,axiom,
% 11.90/5.95 ! [N: num,M2: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
% 11.90/5.95 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1553_and__minus__numerals_I4_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 11.90/5.95 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1554_and__minus__numerals_I8_J,axiom,
% 11.90/5.95 ! [N: num,M2: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
% 11.90/5.95 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1555_gbinomial__partial__row__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1556_gbinomial__r__part__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1557_and__num_Oelims,axiom,
% 11.90/5.95 ! [X: num,Xa: num,Y: option @ num] :
% 11.90/5.95 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 11.90/5.95 = Y )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ one2 ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ? [N3: num] :
% 11.90/5.95 ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( none @ num ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ? [N3: num] :
% 11.90/5.95 ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ one2 ) ) ) )
% 11.90/5.95 => ( ( ? [M5: num] :
% 11.90/5.95 ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( none @ num ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ( ? [M5: num] :
% 11.90/5.95 ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ one2 ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ~ ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 11.90/5.95 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1558_take__bit__Suc__from__most,axiom,
% 11.90/5.95 ! [N: nat,K: int] :
% 11.90/5.95 ( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
% 11.90/5.95 = ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1559_gbinomial__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( semiring_char_0 @ A )
% 11.90/5.95 & ( semidom_divide @ A ) )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
% 11.90/5.95 = A3 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1560_gbinomial__0_I1_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( semiring_char_0 @ A )
% 11.90/5.95 & ( semidom_divide @ A ) )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1561_and__nat__numerals_I2_J,axiom,
% 11.90/5.95 ! [Y: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( one_one @ nat ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1562_and__nat__numerals_I4_J,axiom,
% 11.90/5.95 ! [X: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.95 = ( one_one @ nat ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1563_bit__disjunctive__add__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ! [A3: A,B2: A,N: nat] :
% 11.90/5.95 ( ! [N3: nat] :
% 11.90/5.95 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
% 11.90/5.95 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 11.90/5.95 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 11.90/5.95 | ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1564_bit__1__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
% 11.90/5.95 = ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1565_not__bit__1__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1566_bit__numeral__simps_I1_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [N: num] :
% 11.90/5.95 ~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1567_gbinomial__Suc__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 11.90/5.95 = ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1568_and__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [Xa: int,X: int] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1569_gbinomial__addition__formula,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( gbinomial @ A @ A3 @ ( suc @ K ) )
% 11.90/5.95 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1570_gbinomial__absorb__comp,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
% 11.90/5.95 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1571_gbinomial__mult__1_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
% 11.90/5.95 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1572_gbinomial__mult__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
% 11.90/5.95 = ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1573_Suc__times__gbinomial,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [K: nat,A3: A] :
% 11.90/5.95 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
% 11.90/5.95 = ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1574_gbinomial__absorption,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [K: nat,A3: A] :
% 11.90/5.95 ( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
% 11.90/5.95 = ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1575_gbinomial__parallel__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K4 ) ) @ K4 )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1576_gbinomial__rec,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 11.90/5.95 = ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1577_gbinomial__factors,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
% 11.90/5.95 = ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1578_gbinomial__negated__upper,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ( ( gbinomial @ A )
% 11.90/5.95 = ( ^ [A7: A,K4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A7 ) @ ( one_one @ A ) ) @ K4 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1579_gbinomial__index__swap,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [K: nat,N: nat] :
% 11.90/5.95 ( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1580_int__bit__bound,axiom,
% 11.90/5.95 ! [K: int] :
% 11.90/5.95 ~ ! [N3: nat] :
% 11.90/5.95 ( ! [M6: nat] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ N3 @ M6 )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ M6 )
% 11.90/5.95 = ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) )
% 11.90/5.95 => ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
% 11.90/5.95 = ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1581_gbinomial__minus,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,K: nat] :
% 11.90/5.95 ( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1582_gbinomial__reduce__nat,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [K: nat,A3: A] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 11.90/5.95 => ( ( gbinomial @ A @ A3 @ K )
% 11.90/5.95 = ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1583_gbinomial__sum__lower__neg,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [A3: A,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1584_numeral__and__num,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.95 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1585_gbinomial__sum__up__index,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [K: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J2 ) @ K )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.95 = ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1586_gbinomial__partial__sum__poly,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [M2: nat,A3: A,X: A,Y: A] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K4 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1587_even__bit__succ__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
% 11.90/5.95 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 11.90/5.95 | ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1588_odd__bit__iff__bit__pred,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
% 11.90/5.95 = ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
% 11.90/5.95 | ( N
% 11.90/5.95 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1589_and__num_Osimps_I9_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.95 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 11.90/5.95 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1590_gbinomial__absorption_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [K: nat,A3: A] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 11.90/5.95 => ( ( gbinomial @ A @ A3 @ K )
% 11.90/5.95 = ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1591_gbinomial__partial__sum__poly__xpos,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [M2: nat,A3: A,X: A,Y: A] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A3 ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K4 ) @ A3 ) @ ( one_one @ A ) ) @ K4 ) @ ( power_power @ A @ X @ K4 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1592_gbinomial__sum__nat__pow2,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ K4 ) ) @ K4 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K4 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1593_bit__sum__mult__2__cases,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [A3: A,B2: A,N: nat] :
% 11.90/5.95 ( ! [J3: nat] :
% 11.90/5.95 ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J3 ) )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
% 11.90/5.95 = ( ( ( N
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
% 11.90/5.95 & ( ( N
% 11.90/5.95 != ( zero_zero @ nat ) )
% 11.90/5.95 => ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1594_and__not__num_Osimps_I8_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.95 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 11.90/5.95 @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1595_bit__rec,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_semiring_bits @ A )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ A )
% 11.90/5.95 = ( ^ [A7: A,N5: nat] :
% 11.90/5.95 ( ( ( N5
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 => ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A7 ) )
% 11.90/5.95 & ( ( N5
% 11.90/5.95 != ( zero_zero @ nat ) )
% 11.90/5.95 => ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1596_and__nat__unfold,axiom,
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] :
% 11.90/5.95 ( if @ nat
% 11.90/5.95 @ ( ( M
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 | ( N5
% 11.90/5.95 = ( zero_zero @ nat ) ) )
% 11.90/5.95 @ ( zero_zero @ nat )
% 11.90/5.95 @ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1597_and__nat__rec,axiom,
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] :
% 11.90/5.95 ( plus_plus @ nat
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
% 11.90/5.95 & ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
% 11.90/5.95 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1598_set__bit__eq,axiom,
% 11.90/5.95 ( ( bit_se5668285175392031749et_bit @ int )
% 11.90/5.95 = ( ^ [N5: nat,K4: int] :
% 11.90/5.95 ( plus_plus @ int @ K4
% 11.90/5.95 @ ( times_times @ int
% 11.90/5.95 @ ( zero_neq_one_of_bool @ int
% 11.90/5.95 @ ~ ( bit_se5641148757651400278ts_bit @ int @ K4 @ N5 ) )
% 11.90/5.95 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1599_and__not__num_Oelims,axiom,
% 11.90/5.95 ! [X: num,Xa: num,Y: option @ num] :
% 11.90/5.95 ( ( ( bit_and_not_num @ X @ Xa )
% 11.90/5.95 = Y )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( none @ num ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ? [N3: num] :
% 11.90/5.95 ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ one2 ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ? [N3: num] :
% 11.90/5.95 ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( none @ num ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( bit0 @ M5 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( bit0 @ M5 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 11.90/5.95 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ~ ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1600_and__not__num_Opelims,axiom,
% 11.90/5.95 ! [X: num,Xa: num,Y: option @ num] :
% 11.90/5.95 ( ( ( bit_and_not_num @ X @ Xa )
% 11.90/5.95 = Y )
% 11.90/5.95 => ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( none @ num ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ one2 ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( none @ num ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( bit0 @ M5 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( bit0 @ M5 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 11.90/5.95 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ~ ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1601_and__num_Opelims,axiom,
% 11.90/5.95 ! [X: num,Xa: num,Y: option @ num] :
% 11.90/5.95 ( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
% 11.90/5.95 = Y )
% 11.90/5.95 => ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ one2 ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( none @ num ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ one2 ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( none @ num ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ one2 ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ~ ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
% 11.90/5.95 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1602_gbinomial__code,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ( ( gbinomial @ A )
% 11.90/5.95 = ( ^ [A7: A,K4: nat] :
% 11.90/5.95 ( if @ A
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 @ ( one_one @ A )
% 11.90/5.95 @ ( divide_divide @ A
% 11.90/5.95 @ ( set_fo6178422350223883121st_nat @ A
% 11.90/5.95 @ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ L3 ) ) )
% 11.90/5.95 @ ( zero_zero @ nat )
% 11.90/5.95 @ ( minus_minus @ nat @ K4 @ ( one_one @ nat ) )
% 11.90/5.95 @ ( one_one @ A ) )
% 11.90/5.95 @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1603_xor__num_Oelims,axiom,
% 11.90/5.95 ! [X: num,Xa: num,Y: option @ num] :
% 11.90/5.95 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 11.90/5.95 = Y )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( none @ num ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( bit1 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( bit0 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( bit1 @ M5 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( bit0 @ M5 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ~ ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( Y
% 11.90/5.95 != ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1604_choose__alternating__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_ring_1 @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1605_choose__linear__sum,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [I3: nat] : ( times_times @ nat @ I3 @ ( binomial @ N @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1606_binomial__r__part__sum,axiom,
% 11.90/5.95 ! [M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1607_sum_Otriangle__reindex,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 11.90/5.95 @ ( collect @ ( product_prod @ nat @ nat )
% 11.90/5.95 @ ( product_case_prod @ nat @ nat @ $o
% 11.90/5.95 @ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] :
% 11.90/5.95 ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ K4 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1608_lessThan__eq__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [X: A,Y: A] :
% 11.90/5.95 ( ( ( set_ord_lessThan @ A @ X )
% 11.90/5.95 = ( set_ord_lessThan @ A @ Y ) )
% 11.90/5.95 = ( X = Y ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1609_lessThan__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ord @ A )
% 11.90/5.95 => ! [I: A,K: A] :
% 11.90/5.95 ( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
% 11.90/5.95 = ( ord_less @ A @ I @ K ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1610_binomial__n__n,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( binomial @ N @ N )
% 11.90/5.95 = ( one_one @ nat ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1611_lessThan__subset__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [X: A,Y: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
% 11.90/5.95 = ( ord_less_eq @ A @ X @ Y ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1612_binomial__Suc__Suc,axiom,
% 11.90/5.95 ! [N: nat,K: nat] :
% 11.90/5.95 ( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
% 11.90/5.95 = ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1613_binomial__n__0,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( binomial @ N @ ( zero_zero @ nat ) )
% 11.90/5.95 = ( one_one @ nat ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1614_sum_OlessThan__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1615_choose__one,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( binomial @ N @ ( one_one @ nat ) )
% 11.90/5.95 = N ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1616_lessThan__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ord @ A )
% 11.90/5.95 => ( ( set_ord_lessThan @ A )
% 11.90/5.95 = ( ^ [U2: A] :
% 11.90/5.95 ( collect @ A
% 11.90/5.95 @ ^ [X3: A] : ( ord_less @ A @ X3 @ U2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1617_choose__mult__lemma,axiom,
% 11.90/5.95 ! [M2: nat,R: nat,K: nat] :
% 11.90/5.95 ( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R ) @ K ) @ ( plus_plus @ nat @ M2 @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K ) @ K ) )
% 11.90/5.95 = ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R ) @ M2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1618_lessThan__Suc__atMost,axiom,
% 11.90/5.95 ! [K: nat] :
% 11.90/5.95 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 11.90/5.95 = ( set_ord_atMost @ nat @ K ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1619_bit__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [X: int] :
% 11.90/5.95 ( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1620_lessThan__strict__subset__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [M2: A,N: A] :
% 11.90/5.95 ( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N ) )
% 11.90/5.95 = ( ord_less @ A @ M2 @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1621_Suc__times__binomial__add,axiom,
% 11.90/5.95 ! [A3: nat,B2: nat] :
% 11.90/5.95 ( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
% 11.90/5.95 = ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1622_binomial__absorb__comp,axiom,
% 11.90/5.95 ! [N: nat,K: nat] :
% 11.90/5.95 ( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
% 11.90/5.95 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1623_fact__fact__dvd__fact,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linordered_semidom @ A )
% 11.90/5.95 => ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1624_sum_Onat__diff__reindex,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1625_Iic__subset__Iio__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
% 11.90/5.95 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1626_binomial__absorption,axiom,
% 11.90/5.95 ! [K: nat,N: nat] :
% 11.90/5.95 ( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
% 11.90/5.95 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1627_sum__diff__distrib,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ord @ A )
% 11.90/5.95 => ! [Q: A > nat,P: A > nat,N: A] :
% 11.90/5.95 ( ! [X2: A] : ( ord_less_eq @ nat @ ( Q @ X2 ) @ ( P @ X2 ) )
% 11.90/5.95 => ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ A @ nat
% 11.90/5.95 @ ^ [X3: A] : ( minus_minus @ nat @ ( P @ X3 ) @ ( Q @ X3 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1628_sum_OlessThan__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 11.90/5.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1629_sum__lessThan__telescope,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ab_group_add @ A )
% 11.90/5.95 => ! [F: nat > A,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [N5: nat] : ( minus_minus @ A @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ M2 ) )
% 11.90/5.95 = ( minus_minus @ A @ ( F @ M2 ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1630_sum__lessThan__telescope_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ab_group_add @ A )
% 11.90/5.95 => ! [F: nat > A,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [N5: nat] : ( minus_minus @ A @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ M2 ) )
% 11.90/5.95 = ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1631_sum_OatLeast1__atMost__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1632_sum__bounds__lt__plus1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [F: nat > A,Mm: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( F @ ( suc @ K4 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ Mm ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1633_sum_Onested__swap_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [A3: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] :
% 11.90/5.95 ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1634_sum__choose__lower,axiom,
% 11.90/5.95 ! [R: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [K4: nat] : ( binomial @ ( plus_plus @ nat @ R @ K4 ) @ K4 )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1635_choose__rising__sum_I2_J,axiom,
% 11.90/5.95 ! [N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ M2 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1636_choose__rising__sum_I1_J,axiom,
% 11.90/5.95 ! [N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [J2: nat] : ( binomial @ ( plus_plus @ nat @ N @ J2 ) @ N )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ M2 ) )
% 11.90/5.95 = ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1637_binomial__code,axiom,
% 11.90/5.95 ( binomial
% 11.90/5.95 = ( ^ [N5: nat,K4: nat] : ( if @ nat @ ( ord_less @ nat @ N5 @ K4 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( binomial @ N5 @ ( minus_minus @ nat @ N5 @ K4 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N5 @ K4 ) @ ( one_one @ nat ) ) @ N5 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K4 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1638_choose__reduce__nat,axiom,
% 11.90/5.95 ! [N: nat,K: nat] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.95 => ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 11.90/5.95 => ( ( binomial @ N @ K )
% 11.90/5.95 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1639_times__binomial__minus1__eq,axiom,
% 11.90/5.95 ! [K: nat,N: nat] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
% 11.90/5.95 => ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
% 11.90/5.95 = ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1640_one__diff__power__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1641_power__diff__1__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
% 11.90/5.95 = ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1642_geometric__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field @ A )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( X
% 11.90/5.95 != ( one_one @ A ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1643_sum_OatMost__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
% 11.90/5.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1644_vandermonde,axiom,
% 11.90/5.95 ! [M2: nat,N: nat,R: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ nat @ ( binomial @ M2 @ K4 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ R ) )
% 11.90/5.95 = ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1645_binomial__addition__formula,axiom,
% 11.90/5.95 ! [N: nat,K: nat] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.95 => ( ( binomial @ N @ ( suc @ K ) )
% 11.90/5.95 = ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1646_power__diff__sumr2,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat,Y: A] :
% 11.90/5.95 ( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( minus_minus @ A @ X @ Y )
% 11.90/5.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) ) @ ( power_power @ A @ X @ I3 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1647_diff__power__eq__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat,Y: A] :
% 11.90/5.95 ( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( minus_minus @ A @ X @ Y )
% 11.90/5.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1648_sum__gp__strict,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( division_ring @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( ( X
% 11.90/5.95 = ( one_one @ A ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( semiring_1_of_nat @ A @ N ) ) )
% 11.90/5.95 & ( ( X
% 11.90/5.95 != ( one_one @ A ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1649_binomial,axiom,
% 11.90/5.95 ! [A3: nat,B2: nat,N: nat] :
% 11.90/5.95 ( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K4 ) ) @ ( power_power @ nat @ A3 @ K4 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1650_choose__two,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.95 = ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1651_one__diff__power__eq_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( monoid_mult @ A )
% 11.90/5.95 & ( comm_ring @ A ) )
% 11.90/5.95 => ! [X: A,N: nat] :
% 11.90/5.95 ( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
% 11.90/5.95 @ ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1652_binomial__ring,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A,B2: A,N: nat] :
% 11.90/5.95 ( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( power_power @ A @ A3 @ K4 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1653_xor__num_Osimps_I6_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.95 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1654_xor__num_Osimps_I8_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.95 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1655_choose__alternating__linear__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_ring_1 @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( N
% 11.90/5.95 != ( one_one @ nat ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I3 ) @ ( semiring_1_of_nat @ A @ I3 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I3 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( zero_zero @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1656_fact__Suc__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1657_fact__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1658_fact__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1659_fact__code,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A )
% 11.90/5.95 = ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1660_fact__reduce,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A @ N )
% 11.90/5.95 = ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1661_fact__num__eq__if,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A )
% 11.90/5.95 = ( ^ [M: nat] :
% 11.90/5.95 ( if @ A
% 11.90/5.95 @ ( M
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 @ ( one_one @ A )
% 11.90/5.95 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1662_xor__num_Opelims,axiom,
% 11.90/5.95 ! [X: num,Xa: num,Y: option @ num] :
% 11.90/5.95 ( ( ( bit_un2480387367778600638or_num @ X @ Xa )
% 11.90/5.95 = Y )
% 11.90/5.95 => ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( none @ num ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( bit1 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ( ( X = one2 )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( bit0 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( bit1 @ M5 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit0 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ( ( Xa = one2 )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( bit0 @ M5 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 11.90/5.95 => ( ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit0 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.90/5.95 => ~ ! [M5: num] :
% 11.90/5.95 ( ( X
% 11.90/5.95 = ( bit1 @ M5 ) )
% 11.90/5.95 => ! [N3: num] :
% 11.90/5.95 ( ( Xa
% 11.90/5.95 = ( bit1 @ N3 ) )
% 11.90/5.95 => ( ( Y
% 11.90/5.95 = ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
% 11.90/5.95 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1663_dvd__fact,axiom,
% 11.90/5.95 ! [M2: nat,N: nat] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
% 11.90/5.95 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( dvd_dvd @ nat @ M2 @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1664_fact__ge__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linordered_semidom @ A )
% 11.90/5.95 => ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1665_pochhammer__double,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [Z2: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1666_fact__double,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1667_pochhammer__code,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ( ( comm_s3205402744901411588hammer @ A )
% 11.90/5.95 = ( ^ [A7: A,N5: nat] :
% 11.90/5.95 ( if @ A
% 11.90/5.95 @ ( N5
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 @ ( one_one @ A )
% 11.90/5.95 @ ( set_fo6178422350223883121st_nat @ A
% 11.90/5.95 @ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ O ) ) )
% 11.90/5.95 @ ( zero_zero @ nat )
% 11.90/5.95 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) )
% 11.90/5.95 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1668_xor__Suc__0__eq,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.95 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1669_Suc__0__xor__eq,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 11.90/5.95 = ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1670_push__bit__numeral__minus__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [N: num] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1671_num__of__integer__code,axiom,
% 11.90/5.95 ( code_num_of_integer
% 11.90/5.95 = ( ^ [K4: code_integer] :
% 11.90/5.95 ( if @ num @ ( ord_less_eq @ code_integer @ K4 @ ( one_one @ code_integer ) ) @ one2
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ num
% 11.90/5.95 @ ^ [L3: code_integer,J2: code_integer] :
% 11.90/5.95 ( if @ num
% 11.90/5.95 @ ( J2
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) )
% 11.90/5.95 @ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one2 ) )
% 11.90/5.95 @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1672_Sum__Ico__nat,axiom,
% 11.90/5.95 ! [M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.95 @ ^ [X3: nat] : X3
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.90/5.95 = ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1673_ivl__diff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [I: A,N: A,M2: A] :
% 11.90/5.95 ( ( ord_less_eq @ A @ I @ N )
% 11.90/5.95 => ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
% 11.90/5.95 = ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1674_push__bit__push__bit,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,A3: A] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
% 11.90/5.95 = ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1675_lessThan__minus__lessThan,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [N: A,M2: A] :
% 11.90/5.95 ( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M2 ) )
% 11.90/5.95 = ( set_or7035219750837199246ssThan @ A @ M2 @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1676_pochhammer__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
% 11.90/5.95 = A3 ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1677_atLeastLessThan__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ord @ A )
% 11.90/5.95 => ! [I: A,L: A,U: A] :
% 11.90/5.95 ( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 11.90/5.95 = ( ( ord_less_eq @ A @ L @ I )
% 11.90/5.95 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1678_ivl__subset,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [I: A,J: A,M2: A,N: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
% 11.90/5.95 = ( ( ord_less_eq @ A @ J @ I )
% 11.90/5.95 | ( ( ord_less_eq @ A @ M2 @ I )
% 11.90/5.95 & ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1679_pochhammer__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1680_xor__numerals_I1_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [Y: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1681_xor__numerals_I2_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [Y: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1682_xor__numerals_I5_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1683_xor__numerals_I8_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1684_push__bit__of__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
% 11.90/5.95 = ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1685_sum_Oop__ivl__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [N: nat,M2: nat,G: nat > A] :
% 11.90/5.95 ( ( ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( zero_zero @ A ) ) )
% 11.90/5.95 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1686_xor__numerals_I4_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num,Y: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1687_xor__numerals_I6_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num,Y: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1688_flip__bit__nat__def,axiom,
% 11.90/5.95 ( ( bit_se8732182000553998342ip_bit @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1689_push__bit__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [Xa: nat,X: int] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1690_atLeastLessThan__inj_I2_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.95 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 11.90/5.95 = ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.95 => ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.95 => ( ( ord_less @ A @ C2 @ D3 )
% 11.90/5.95 => ( B2 = D3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1691_atLeastLessThan__inj_I1_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.95 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 11.90/5.95 = ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.95 => ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.95 => ( ( ord_less @ A @ C2 @ D3 )
% 11.90/5.95 => ( A3 = C2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1692_atLeastLessThan__eq__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.95 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.95 => ( ( ord_less @ A @ C2 @ D3 )
% 11.90/5.95 => ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 11.90/5.95 = ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.95 = ( ( A3 = C2 )
% 11.90/5.95 & ( B2 = D3 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1693_flip__bit__eq__xor,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ( ( bit_se8732182000553998342ip_bit @ A )
% 11.90/5.95 = ( ^ [N5: nat,A7: A] : ( bit_se5824344971392196577ns_xor @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1694_push__bit__add,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [N: nat,A3: A,B2: A] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.95 = ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1695_atLeastLessThan__subset__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( linorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.95 => ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.95 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.95 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1696_atLeastLessThanSuc__atLeastAtMost,axiom,
% 11.90/5.95 ! [L: nat,U: nat] :
% 11.90/5.95 ( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
% 11.90/5.95 = ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1697_lessThan__atLeast0,axiom,
% 11.90/5.95 ( ( set_ord_lessThan @ nat )
% 11.90/5.95 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1698_push__bit__take__bit,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,A3: A] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 11.90/5.95 = ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1699_pochhammer__fact,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( semiring_char_0 @ A )
% 11.90/5.95 & ( comm_semiring_1 @ A ) )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A )
% 11.90/5.95 = ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1700_sum_Oshift__bounds__Suc__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1701_sum_Oshift__bounds__nat__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1702_sum_Oivl__cong,axiom,
% 11.90/5.95 ! [A: $tType,B: $tType] :
% 11.90/5.95 ( ( ( ord @ B )
% 11.90/5.95 & ( comm_monoid_add @ A ) )
% 11.90/5.95 => ! [A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
% 11.90/5.95 ( ( A3 = C2 )
% 11.90/5.95 => ( ( B2 = D3 )
% 11.90/5.95 => ( ! [X2: B] :
% 11.90/5.95 ( ( ord_less_eq @ B @ C2 @ X2 )
% 11.90/5.95 => ( ( ord_less @ B @ X2 @ D3 )
% 11.90/5.95 => ( ( G @ X2 )
% 11.90/5.95 = ( H3 @ X2 ) ) ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1703_sum_OatLeastLessThan__concat,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,P5: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( ord_less_eq @ nat @ N @ P5 )
% 11.90/5.95 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1704_sum__diff__nat__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ab_group_add @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,P5: nat,F: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( ord_less_eq @ nat @ N @ P5 )
% 11.90/5.95 => ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1705_pochhammer__0__left,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( ( N
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 11.90/5.95 = ( one_one @ A ) ) )
% 11.90/5.95 & ( ( N
% 11.90/5.95 != ( zero_zero @ nat ) )
% 11.90/5.95 => ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
% 11.90/5.95 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1706_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1707_sum_OatLeastLessThan__shift__bounds,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1708_concat__bit__eq,axiom,
% 11.90/5.95 ( bit_concat_bit
% 11.90/5.95 = ( ^ [N5: nat,K4: int,L3: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N5 @ K4 ) @ ( bit_se4730199178511100633sh_bit @ int @ N5 @ L3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1709_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( preorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.95 = ( ( ord_less_eq @ A @ A3 @ B2 )
% 11.90/5.95 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.95 & ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1710_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( dense_linorder @ A )
% 11.90/5.95 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.95 ( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.95 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.95 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.95 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1711_sum__shift__lb__Suc0__0__upt,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [F: nat > A,K: nat] :
% 11.90/5.95 ( ( ( F @ ( zero_zero @ nat ) )
% 11.90/5.95 = ( zero_zero @ A ) )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1712_sum_OatLeast0__lessThan__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1713_sum_OatLeast__Suc__lessThan,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.90/5.95 = ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1714_sum_OatLeastLessThan__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [A3: nat,B2: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1715_sum_Olast__plus,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1716_pochhammer__rec,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 11.90/5.95 = ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1717_pochhammer__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1718_pochhammer__rec_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [Z2: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1719_pochhammer__product_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [Z2: A,N: nat,M2: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
% 11.90/5.95 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1720_bit__iff__and__push__bit__not__eq__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ( ( bit_se5641148757651400278ts_bit @ A )
% 11.90/5.95 = ( ^ [A7: A,N5: nat] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) )
% 11.90/5.95 != ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1721_sum__Suc__diff_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ab_group_add @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,F: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( minus_minus @ A @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.90/5.95 = ( minus_minus @ A @ ( F @ N ) @ ( F @ M2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1722_sum_OatLeastLessThan__rev,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1723_sum_Onested__swap,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [A3: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] :
% 11.90/5.95 ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1724_sum_Onat__group,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [M: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1725_numeral__xor__num,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.95 = ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1726_sum_Ohead__if,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [N: nat,M2: nat,G: nat > A] :
% 11.90/5.95 ( ( ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( zero_zero @ A ) ) )
% 11.90/5.95 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1727_sum_OatLeast0__lessThan__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1728_sum_OatLeastLessThan__shift__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1729_pochhammer__product,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,Z2: A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
% 11.90/5.95 = ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1730_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1731_sum_OatLeast__lessThan__pred__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ( comp @ nat @ A @ nat @ G
% 11.90/5.95 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1732_pochhammer__absorb__comp,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_ring_1 @ A )
% 11.90/5.95 => ! [R: A,K: nat] :
% 11.90/5.95 ( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K ) )
% 11.90/5.95 = ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1733_pochhammer__same,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( ( semiring_char_0 @ A )
% 11.90/5.95 & ( comm_ring_1 @ A )
% 11.90/5.95 & ( semiri3467727345109120633visors @ A ) )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1734_pochhammer__minus,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_ring_1 @ A )
% 11.90/5.95 => ! [B2: A,K: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1735_pochhammer__minus_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_ring_1 @ A )
% 11.90/5.95 => ! [B2: A,K: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
% 11.90/5.95 = ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1736_gbinomial__pochhammer_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ( ( gbinomial @ A )
% 11.90/5.95 = ( ^ [A7: A,K4: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A7 @ ( semiring_1_of_nat @ A @ K4 ) ) @ ( one_one @ A ) ) @ K4 ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1737_gbinomial__pochhammer,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ( ( gbinomial @ A )
% 11.90/5.95 = ( ^ [A7: A,K4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K4 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A7 ) @ K4 ) ) @ ( semiring_char_0_fact @ A @ K4 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1738_xor__nat__unfold,axiom,
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] :
% 11.90/5.95 ( if @ nat
% 11.90/5.95 @ ( M
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 @ N5
% 11.90/5.95 @ ( if @ nat
% 11.90/5.95 @ ( N5
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 @ M
% 11.90/5.95 @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1739_num__of__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [X: int] :
% 11.90/5.95 ( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( num_of_nat @ ( nat2 @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1740_signed__take__bit__code,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( bit_ri4674362597316999326ke_bit @ A )
% 11.90/5.95 = ( ^ [N5: nat,A7: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ N5 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N5 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A7 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1741_xor__nat__rec,axiom,
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] :
% 11.90/5.95 ( plus_plus @ nat
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
% 11.90/5.95 != ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
% 11.90/5.95 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1742_xor__one__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.95 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 11.90/5.95 @ ( zero_neq_one_of_bool @ A
% 11.90/5.95 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1743_one__xor__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [A3: A] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.95 = ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
% 11.90/5.95 @ ( zero_neq_one_of_bool @ A
% 11.90/5.95 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1744_pochhammer__binomial__sum,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_ring_1 @ A )
% 11.90/5.95 => ! [A3: A,B2: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K4 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K4 ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1745_divmod__integer__def,axiom,
% 11.90/5.95 ( code_divmod_integer
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K4 @ L3 ) @ ( modulo_modulo @ code_integer @ K4 @ L3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1746_sum__power2,axiom,
% 11.90/5.95 ! [K: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
% 11.90/5.95 = ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1747_nat__of__integer__code,axiom,
% 11.90/5.95 ( code_nat_of_integer
% 11.90/5.95 = ( ^ [K4: code_integer] :
% 11.90/5.95 ( if @ nat @ ( ord_less_eq @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ nat
% 11.90/5.95 @ ^ [L3: code_integer,J2: code_integer] :
% 11.90/5.95 ( if @ nat
% 11.90/5.95 @ ( J2
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) )
% 11.90/5.95 @ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( one_one @ nat ) ) )
% 11.90/5.95 @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1748_pochhammer__times__pochhammer__half,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( field_char_0 @ A )
% 11.90/5.95 => ! [Z2: A,N: nat] :
% 11.90/5.95 ( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K4 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1749_int__of__integer__code,axiom,
% 11.90/5.95 ( code_int_of_integer
% 11.90/5.95 = ( ^ [K4: code_integer] :
% 11.90/5.95 ( if @ int @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K4 ) ) )
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( zero_zero @ int )
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ int
% 11.90/5.95 @ ^ [L3: code_integer,J2: code_integer] :
% 11.90/5.95 ( if @ int
% 11.90/5.95 @ ( J2
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) )
% 11.90/5.95 @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
% 11.90/5.95 @ ( code_divmod_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1750_divmod__integer__eq__cases,axiom,
% 11.90/5.95 ( code_divmod_integer
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] :
% 11.90/5.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( L3
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
% 11.90/5.95 @ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L3
% 11.90/5.95 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( ( sgn_sgn @ code_integer @ K4 )
% 11.90/5.95 = ( sgn_sgn @ code_integer @ L3 ) )
% 11.90/5.95 @ ( code_divmod_abs @ K4 @ L3 )
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ^ [R3: code_integer,S4: code_integer] :
% 11.90/5.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( S4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S4 ) ) )
% 11.90/5.95 @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1751_time__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( time_time @ A )
% 11.90/5.95 = ( ^ [C3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
% 11.90/5.95 ( case_option @ nat @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( zero_zero @ nat )
% 11.90/5.95 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ nat
% 11.90/5.95 @ ^ [X3: A] :
% 11.90/5.95 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ nat
% 11.90/5.95 @ ^ [Xa4: heap_ext @ product_unit,T3: nat] : T3 ) )
% 11.90/5.95 @ ( heap_Time_execute @ A @ C3 @ H ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1752_bit__cut__integer__def,axiom,
% 11.90/5.95 ( code_bit_cut_integer
% 11.90/5.95 = ( ^ [K4: code_integer] :
% 11.90/5.95 ( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
% 11.90/5.95 @ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1753_integer__of__num_I3_J,axiom,
% 11.90/5.95 ! [N: num] :
% 11.90/5.95 ( ( code_integer_of_num @ ( bit1 @ N ) )
% 11.90/5.95 = ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1754_less__eq__int_Orep__eq,axiom,
% 11.90/5.95 ( ( ord_less_eq @ int )
% 11.90/5.95 = ( ^ [X3: int,Xa4: int] :
% 11.90/5.95 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.95 @ ^ [Y3: nat,Z5: nat] :
% 11.90/5.95 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.95 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
% 11.90/5.95 @ ( rep_Integ @ X3 )
% 11.90/5.95 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1755_nat__of__integer__of__nat,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 11.90/5.95 = N ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1756_int__of__integer__of__nat,axiom,
% 11.90/5.95 ! [N: nat] :
% 11.90/5.95 ( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
% 11.90/5.95 = ( semiring_1_of_nat @ int @ N ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1757_integer__of__int__int__of__integer,axiom,
% 11.90/5.95 ! [K: code_integer] :
% 11.90/5.95 ( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
% 11.90/5.95 = K ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1758_int__of__integer__integer__of__int,axiom,
% 11.90/5.95 ! [K: int] :
% 11.90/5.95 ( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
% 11.90/5.95 = K ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1759_int__of__integer__inverse,axiom,
% 11.90/5.95 ! [X: code_integer] :
% 11.90/5.95 ( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
% 11.90/5.95 = X ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1760_apsnd__conv,axiom,
% 11.90/5.95 ! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y: C] :
% 11.90/5.95 ( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y ) )
% 11.90/5.95 = ( product_Pair @ A @ B @ X @ ( F @ Y ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1761_prod_Oneutral__const,axiom,
% 11.90/5.95 ! [B: $tType,A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [A4: set @ B] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.95 @ ^ [Uu: B] : ( one_one @ A )
% 11.90/5.95 @ A4 )
% 11.90/5.95 = ( one_one @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1762_of__int__integer__of,axiom,
% 11.90/5.95 ! [K: code_integer] :
% 11.90/5.95 ( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
% 11.90/5.95 = K ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1763_int__of__integer__of__int,axiom,
% 11.90/5.95 ! [K: int] :
% 11.90/5.95 ( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
% 11.90/5.95 = K ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1764_zero__integer_Orep__eq,axiom,
% 11.90/5.95 ( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
% 11.90/5.95 = ( zero_zero @ int ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1765_int__of__integer__numeral,axiom,
% 11.90/5.95 ! [K: num] :
% 11.90/5.95 ( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
% 11.90/5.95 = ( numeral_numeral @ int @ K ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1766_plus__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1767_one__integer_Orep__eq,axiom,
% 11.90/5.95 ( ( code_int_of_integer @ ( one_one @ code_integer ) )
% 11.90/5.95 = ( one_one @ int ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1768_uminus__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
% 11.90/5.95 = ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1769_times__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1770_minus__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1771_abs__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
% 11.90/5.95 = ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1772_divide__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1773_modulo__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1774_sgn__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
% 11.90/5.95 = ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1775_prod_OatMost__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1776_prod_OlessThan__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1777_nat__of__integer__non__positive,axiom,
% 11.90/5.95 ! [K: code_integer] :
% 11.90/5.95 ( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
% 11.90/5.95 => ( ( code_nat_of_integer @ K )
% 11.90/5.95 = ( zero_zero @ nat ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1778_prod_Ocl__ivl__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [N: nat,M2: nat,G: nat > A] :
% 11.90/5.95 ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( one_one @ A ) ) )
% 11.90/5.95 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1779_prod_Oop__ivl__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [N: nat,M2: nat,G: nat > A] :
% 11.90/5.95 ( ( ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( one_one @ A ) ) )
% 11.90/5.95 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1780_xor__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1781_push__bit__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: nat,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1782_xor__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [Xa: int,X: int] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1783_nat__of__integer_Orep__eq,axiom,
% 11.90/5.95 ( code_nat_of_integer
% 11.90/5.95 = ( ^ [X3: code_integer] : ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1784_integer__eq__iff,axiom,
% 11.90/5.95 ( ( ^ [Y5: code_integer,Z4: code_integer] : ( Y5 = Z4 ) )
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ K4 )
% 11.90/5.95 = ( code_int_of_integer @ L3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1785_integer__eqI,axiom,
% 11.90/5.95 ! [K: code_integer,L: code_integer] :
% 11.90/5.95 ( ( ( code_int_of_integer @ K )
% 11.90/5.95 = ( code_int_of_integer @ L ) )
% 11.90/5.95 => ( K = L ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1786_int__of__integer__inject,axiom,
% 11.90/5.95 ! [X: code_integer,Y: code_integer] :
% 11.90/5.95 ( ( ( code_int_of_integer @ X )
% 11.90/5.95 = ( code_int_of_integer @ Y ) )
% 11.90/5.95 = ( X = Y ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1787_prod_Onot__neutral__contains__not__neutral,axiom,
% 11.90/5.95 ! [B: $tType,A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: B > A,A4: set @ B] :
% 11.90/5.95 ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 11.90/5.95 != ( one_one @ A ) )
% 11.90/5.95 => ~ ! [A5: B] :
% 11.90/5.95 ( ( member @ B @ A5 @ A4 )
% 11.90/5.95 => ( ( G @ A5 )
% 11.90/5.95 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1788_prod_Oneutral,axiom,
% 11.90/5.95 ! [B: $tType,A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [A4: set @ B,G: B > A] :
% 11.90/5.95 ( ! [X2: B] :
% 11.90/5.95 ( ( member @ B @ X2 @ A4 )
% 11.90/5.95 => ( ( G @ X2 )
% 11.90/5.95 = ( one_one @ A ) ) )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 11.90/5.95 = ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1789_apsnd__compose,axiom,
% 11.90/5.95 ! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
% 11.90/5.95 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
% 11.90/5.95 = ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1790_prod__ge__1,axiom,
% 11.90/5.95 ! [A: $tType,B: $tType] :
% 11.90/5.95 ( ( linord181362715937106298miring @ A )
% 11.90/5.95 => ! [A4: set @ B,F: B > A] :
% 11.90/5.95 ( ! [X2: B] :
% 11.90/5.95 ( ( member @ B @ X2 @ A4 )
% 11.90/5.95 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X2 ) ) )
% 11.90/5.95 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1791_int__of__integer__less__iff,axiom,
% 11.90/5.95 ! [X: code_integer,Y: code_integer] :
% 11.90/5.95 ( ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Y ) )
% 11.90/5.95 = ( ord_less @ code_integer @ X @ Y ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1792_integer__less__iff,axiom,
% 11.90/5.95 ( ( ord_less @ code_integer )
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1793_less__integer_Orep__eq,axiom,
% 11.90/5.95 ( ( ord_less @ code_integer )
% 11.90/5.95 = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1794_less__eq__integer_Orep__eq,axiom,
% 11.90/5.95 ( ( ord_less_eq @ code_integer )
% 11.90/5.95 = ( ^ [X3: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1795_integer__less__eq__iff,axiom,
% 11.90/5.95 ( ( ord_less_eq @ code_integer )
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K4 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1796_prod_Oshift__bounds__cl__Suc__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1797_prod_Oshift__bounds__Suc__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1798_prod_Oshift__bounds__cl__nat__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1799_prod_Oshift__bounds__nat__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1800_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
% 11.90/5.95 ! [L: code_integer,U: code_integer] :
% 11.90/5.95 ( ( set_or7035219750837199246ssThan @ code_integer @ L @ ( plus_plus @ code_integer @ U @ ( one_one @ code_integer ) ) )
% 11.90/5.95 = ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1801_prod__le__1,axiom,
% 11.90/5.95 ! [B: $tType,A: $tType] :
% 11.90/5.95 ( ( linord181362715937106298miring @ A )
% 11.90/5.95 => ! [A4: set @ B,F: B > A] :
% 11.90/5.95 ( ! [X2: B] :
% 11.90/5.95 ( ( member @ B @ X2 @ A4 )
% 11.90/5.95 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X2 ) )
% 11.90/5.95 & ( ord_less_eq @ A @ ( F @ X2 ) @ ( one_one @ A ) ) ) )
% 11.90/5.95 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1802_prod_Oivl__cong,axiom,
% 11.90/5.95 ! [A: $tType,B: $tType] :
% 11.90/5.95 ( ( ( ord @ B )
% 11.90/5.95 & ( comm_monoid_mult @ A ) )
% 11.90/5.95 => ! [A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
% 11.90/5.95 ( ( A3 = C2 )
% 11.90/5.95 => ( ( B2 = D3 )
% 11.90/5.95 => ( ! [X2: B] :
% 11.90/5.95 ( ( ord_less_eq @ B @ C2 @ X2 )
% 11.90/5.95 => ( ( ord_less @ B @ X2 @ D3 )
% 11.90/5.95 => ( ( G @ X2 )
% 11.90/5.95 = ( H3 @ X2 ) ) ) )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1803_take__bit__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: nat,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1804_and__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1805_bit__integer_Orep__eq,axiom,
% 11.90/5.95 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 11.90/5.95 = ( ^ [X3: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1806_prod_OatLeastLessThan__concat,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,P5: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( ord_less_eq @ nat @ N @ P5 )
% 11.90/5.95 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1807_mask__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: nat] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
% 11.90/5.95 = ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1808_integer__of__num__def,axiom,
% 11.90/5.95 ( code_integer_of_num
% 11.90/5.95 = ( numeral_numeral @ code_integer ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1809_unset__bit__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: nat,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1810_set__bit__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: nat,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1811_flip__bit__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: nat,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1812_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1813_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1814_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 11.90/5.95 ! [L: int,U: int] :
% 11.90/5.95 ( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
% 11.90/5.95 = ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1815_prod_OatLeastAtMost__shift__bounds,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1816_prod_OatLeastLessThan__shift__bounds,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1817_prod_OatLeastAtMost__rev,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1818_prod_Onat__diff__reindex,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1819_nat__of__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [X: int] :
% 11.90/5.95 ( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( nat2 @ X ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1820_num__of__integer_Orep__eq,axiom,
% 11.90/5.95 ( code_num_of_integer
% 11.90/5.95 = ( ^ [X3: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1821_prod_OatLeast0__atMost__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1822_prod_OatLeast0__lessThan__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1823_prod_OatLeast__Suc__lessThan,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1824_prod_Onat__ivl__Suc_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1825_prod_OatLeast__Suc__atMost,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1826_prod_OatLeastLessThan__Suc,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [A3: nat,B2: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1827_divmod__abs__code_I6_J,axiom,
% 11.90/5.95 ! [J: code_integer] :
% 11.90/5.95 ( ( code_divmod_abs @ ( zero_zero @ code_integer ) @ J )
% 11.90/5.95 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1828_prod_Olast__plus,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1829_prod_OSuc__reindex__ivl,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ M2 )
% 11.90/5.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1830_prod_OatMost__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 11.90/5.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1831_prod_OlessThan__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 11.90/5.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1832_prod_OatLeastLessThan__rev,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1833_prod_Onested__swap,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [A3: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] :
% 11.90/5.95 ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1834_prod_OatLeast1__atMost__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1835_prod_Onat__group,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,K: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [M: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1836_fact__prod,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( semiring_char_0 @ A )
% 11.90/5.95 => ( ( semiring_char_0_fact @ A )
% 11.90/5.95 = ( ^ [N5: nat] :
% 11.90/5.95 ( semiring_1_of_nat @ A
% 11.90/5.95 @ ( groups7121269368397514597t_prod @ nat @ nat
% 11.90/5.95 @ ^ [X3: nat] : X3
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1837_prod_Onested__swap_H,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [A3: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] :
% 11.90/5.95 ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1838_prod__atLeastAtMost__code,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [F: nat > A,A3: nat,B2: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 11.90/5.95 = ( set_fo6178422350223883121st_nat @ A
% 11.90/5.95 @ ^ [A7: nat] : ( times_times @ A @ ( F @ A7 ) )
% 11.90/5.95 @ A3
% 11.90/5.95 @ B2
% 11.90/5.95 @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1839_prod_Ohead__if,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [N: nat,M2: nat,G: nat > A] :
% 11.90/5.95 ( ( ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( one_one @ A ) ) )
% 11.90/5.95 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1840_prod_Oub__add__nat,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A,P5: nat] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
% 11.90/5.95 = ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1841_prod_OatLeast0__atMost__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1842_prod_OatLeast0__lessThan__Suc__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1843_prod_OatLeastLessThan__shift__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1844_integer__of__num__triv_I1_J,axiom,
% 11.90/5.95 ( ( code_integer_of_num @ one2 )
% 11.90/5.95 = ( one_one @ code_integer ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1845_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat,M2: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1846_prod_OatMost__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.95 = ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
% 11.90/5.95 @ ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1847_pochhammer__prod,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ( ( comm_s3205402744901411588hammer @ A )
% 11.90/5.95 = ( ^ [A7: A,N5: nat] :
% 11.90/5.95 ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ I3 ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1848_prod_OatLeast__atMost__pred__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ( comp @ nat @ A @ nat @ G
% 11.90/5.95 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1849_prod_OatLeast__lessThan__pred__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ( comp @ nat @ A @ nat @ G
% 11.90/5.95 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 11.90/5.95 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1850_integer__of__num_I2_J,axiom,
% 11.90/5.95 ! [N: num] :
% 11.90/5.95 ( ( code_integer_of_num @ ( bit0 @ N ) )
% 11.90/5.95 = ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1851_divmod__abs__code_I5_J,axiom,
% 11.90/5.95 ! [J: code_integer] :
% 11.90/5.95 ( ( code_divmod_abs @ J @ ( zero_zero @ code_integer ) )
% 11.90/5.95 = ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( abs_abs @ code_integer @ J ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1852_prod_OatLeastAtMost__shift__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1853_sum_OatLeast__int__lessThan__int__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_add @ A )
% 11.90/5.95 => ! [G: int > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.95 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1854_pochhammer__Suc__prod,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I3 ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1855_pochhammer__prod__rev,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ( ( comm_s3205402744901411588hammer @ A )
% 11.90/5.95 = ( ^ [A7: A,N5: nat] :
% 11.90/5.95 ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( plus_plus @ A @ A7 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N5 @ I3 ) ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1856_fact__div__fact,axiom,
% 11.90/5.95 ! [N: nat,M2: nat] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ N @ M2 )
% 11.90/5.95 => ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ nat
% 11.90/5.95 @ ^ [X3: nat] : X3
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1857_integer__of__num__triv_I2_J,axiom,
% 11.90/5.95 ( ( code_integer_of_num @ ( bit0 @ one2 ) )
% 11.90/5.95 = ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1858_prod_Oin__pairs,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1859_prod_Oin__pairs__0,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1860_nat__of__integer__less__iff,axiom,
% 11.90/5.95 ! [X: code_integer,Y: code_integer] :
% 11.90/5.95 ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X )
% 11.90/5.95 => ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
% 11.90/5.95 => ( ( ord_less @ nat @ ( code_nat_of_integer @ X ) @ ( code_nat_of_integer @ Y ) )
% 11.90/5.95 = ( ord_less @ code_integer @ X @ Y ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1861_pochhammer__Suc__prod__rev,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_semiring_1 @ A )
% 11.90/5.95 => ! [A3: A,N: nat] :
% 11.90/5.95 ( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I3 ) ) )
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1862_divmod__integer__code,axiom,
% 11.90/5.95 ( code_divmod_integer
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] :
% 11.90/5.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L3 )
% 11.90/5.95 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ ( code_divmod_abs @ K4 @ L3 )
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ^ [R3: code_integer,S4: code_integer] :
% 11.90/5.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( S4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S4 ) ) )
% 11.90/5.95 @ ( code_divmod_abs @ K4 @ L3 ) ) )
% 11.90/5.95 @ ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( L3
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K4 )
% 11.90/5.95 @ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
% 11.90/5.95 @ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K4 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K4 @ L3 )
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ^ [R3: code_integer,S4: code_integer] :
% 11.90/5.95 ( if @ ( product_prod @ code_integer @ code_integer )
% 11.90/5.95 @ ( S4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S4 ) ) )
% 11.90/5.95 @ ( code_divmod_abs @ K4 @ L3 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1863_bit__cut__integer__code,axiom,
% 11.90/5.95 ( code_bit_cut_integer
% 11.90/5.95 = ( ^ [K4: code_integer] :
% 11.90/5.95 ( if @ ( product_prod @ code_integer @ $o )
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( zero_zero @ code_integer ) )
% 11.90/5.95 @ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
% 11.90/5.95 @ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
% 11.90/5.95 @ ^ [R3: code_integer,S4: code_integer] :
% 11.90/5.95 ( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K4 ) @ R3 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R3 ) @ S4 ) )
% 11.90/5.95 @ ( S4
% 11.90/5.95 = ( one_one @ code_integer ) ) )
% 11.90/5.95 @ ( code_divmod_abs @ K4 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1864_prod_Ozero__middle,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [P5: nat,K: nat,G: nat > A,H3: nat > A] :
% 11.90/5.95 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 11.90/5.95 => ( ( ord_less_eq @ nat @ K @ P5 )
% 11.90/5.95 => ( ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ ( one_one @ A ) @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ P5 ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1865_xor__int__rec,axiom,
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ int )
% 11.90/5.95 = ( ^ [K4: int,L3: int] :
% 11.90/5.95 ( plus_plus @ int
% 11.90/5.95 @ ( zero_neq_one_of_bool @ int
% 11.90/5.95 @ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) )
% 11.90/5.95 != ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 11.90/5.95 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1866_divmod__abs__def,axiom,
% 11.90/5.95 ( code_divmod_abs
% 11.90/5.95 = ( ^ [K4: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L3 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K4 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1867_less__int_Orep__eq,axiom,
% 11.90/5.95 ( ( ord_less @ int )
% 11.90/5.95 = ( ^ [X3: int,Xa4: int] :
% 11.90/5.95 ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.95 @ ^ [Y3: nat,Z5: nat] :
% 11.90/5.95 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.95 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
% 11.90/5.95 @ ( rep_Integ @ X3 )
% 11.90/5.95 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1868_make_H__def,axiom,
% 11.90/5.95 ! [B: $tType,A: $tType] :
% 11.90/5.95 ( ( ( heap @ A )
% 11.90/5.95 & ( semiring_1 @ B ) )
% 11.90/5.95 => ( ( array_make2 @ B @ A )
% 11.90/5.95 = ( ^ [I3: code_integer,F2: B > A] : ( array_make @ A @ ( code_nat_of_integer @ I3 ) @ ( comp @ B @ A @ nat @ F2 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1869_new_H__def,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( heap @ A )
% 11.90/5.95 => ( ( array_new2 @ A )
% 11.90/5.95 = ( comp @ nat @ ( A > ( heap_Time_Heap @ ( array @ A ) ) ) @ code_integer @ ( array_new @ A ) @ code_nat_of_integer ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1870_xor__int__unfold,axiom,
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ int )
% 11.90/5.95 = ( ^ [K4: int,L3: int] :
% 11.90/5.95 ( if @ int
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.95 @ ( bit_ri4277139882892585799ns_not @ int @ L3 )
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( L3
% 11.90/5.95 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.95 @ ( bit_ri4277139882892585799ns_not @ int @ K4 )
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( K4
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 @ L3
% 11.90/5.95 @ ( if @ int
% 11.90/5.95 @ ( L3
% 11.90/5.95 = ( zero_zero @ int ) )
% 11.90/5.95 @ K4
% 11.90/5.95 @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1871_take__bit__num__code,axiom,
% 11.90/5.95 ( bit_take_bit_num
% 11.90/5.95 = ( ^ [N5: nat,M: num] :
% 11.90/5.95 ( product_case_prod @ nat @ num @ ( option @ num )
% 11.90/5.95 @ ^ [A7: nat,X3: num] :
% 11.90/5.95 ( case_nat @ ( option @ num ) @ ( none @ num )
% 11.90/5.95 @ ^ [O: nat] :
% 11.90/5.95 ( case_num @ ( option @ num ) @ ( some @ num @ one2 )
% 11.90/5.95 @ ^ [P6: num] :
% 11.90/5.95 ( case_option @ ( option @ num ) @ num @ ( none @ num )
% 11.90/5.95 @ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
% 11.90/5.95 @ ( bit_take_bit_num @ O @ P6 ) )
% 11.90/5.95 @ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 11.90/5.95 @ X3 )
% 11.90/5.95 @ A7 )
% 11.90/5.95 @ ( product_Pair @ nat @ num @ N5 @ M ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1872_prod_Otriangle__reindex,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 11.90/5.95 @ ( collect @ ( product_prod @ nat @ nat )
% 11.90/5.95 @ ( product_case_prod @ nat @ nat @ $o
% 11.90/5.95 @ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] :
% 11.90/5.95 ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ K4 ) )
% 11.90/5.95 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1873_case__nat__add__eq__if,axiom,
% 11.90/5.95 ! [A: $tType,A3: A,F: nat > A,V: num,N: nat] :
% 11.90/5.95 ( ( case_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 11.90/5.95 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1874_rec__nat__add__eq__if,axiom,
% 11.90/5.95 ! [A: $tType,A3: A,F: nat > A > A,V: num,N: nat] :
% 11.90/5.95 ( ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
% 11.90/5.95 = ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1875_bit_Ocompl__zero,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1876_bit_Ocompl__one,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( zero_zero @ A ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1877_bit_Oxor__cancel__right,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1878_bit_Oxor__cancel__left,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1879_bit_Oxor__one__right,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1880_bit_Oxor__one__left,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 11.90/5.95 = ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1881_push__bit__minus__one__eq__not__mask,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [N: nat] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1882_not__one__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
% 11.90/5.95 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1883_not__add__distrib,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [A3: A,B2: A] :
% 11.90/5.95 ( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B2 ) )
% 11.90/5.95 = ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1884_not__diff__distrib,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [A3: A,B2: A] :
% 11.90/5.95 ( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B2 ) )
% 11.90/5.95 = ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1885_minus__eq__not__plus__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( uminus_uminus @ A )
% 11.90/5.95 = ( ^ [A7: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1886_minus__eq__not__minus__1,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( uminus_uminus @ A )
% 11.90/5.95 = ( ^ [A7: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1887_not__eq__complement,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( bit_ri4277139882892585799ns_not @ A )
% 11.90/5.95 = ( ^ [A7: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1888_prod__int__eq,axiom,
% 11.90/5.95 ! [I: nat,J: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ int @ int
% 11.90/5.95 @ ^ [X3: int] : X3
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1889_prod__int__plus__eq,axiom,
% 11.90/5.95 ! [I: nat,J: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ int @ int
% 11.90/5.95 @ ^ [X3: int] : X3
% 11.90/5.95 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1890_push__bit__mask__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [M2: nat,N: nat] :
% 11.90/5.95 ( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2239418461657761734s_mask @ A @ N ) )
% 11.90/5.95 = ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1891_prod_OatLeast__int__lessThan__int__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: int > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1892_unset__bit__eq__and__not,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ( ( bit_se2638667681897837118et_bit @ A )
% 11.90/5.95 = ( ^ [N5: nat,A7: A] : ( bit_se5824344872417868541ns_and @ A @ A7 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1893_prod_OatLeast__int__atMost__int__shift,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: int > A,M2: nat,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1894_int__numeral__not__and__num,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 11.90/5.95 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M2 ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1895_int__numeral__and__not__num,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/5.95 = ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1896_prod_Otriangle__reindex__eq,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( comm_monoid_mult @ A )
% 11.90/5.95 => ! [G: nat > nat > A,N: nat] :
% 11.90/5.95 ( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
% 11.90/5.95 @ ( collect @ ( product_prod @ nat @ nat )
% 11.90/5.95 @ ( product_case_prod @ nat @ nat @ $o
% 11.90/5.95 @ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 11.90/5.95 = ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [K4: nat] :
% 11.90/5.95 ( groups7121269368397514597t_prod @ nat @ A
% 11.90/5.95 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ K4 ) )
% 11.90/5.95 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1897_and__not__numerals_I8_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 11.90/5.95 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1898_not__int__rec,axiom,
% 11.90/5.95 ( ( bit_ri4277139882892585799ns_not @ int )
% 11.90/5.95 = ( ^ [K4: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1899_Nitpick_Ocase__nat__unfold,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( case_nat @ A )
% 11.90/5.95 = ( ^ [X3: A,F2: nat > A,N5: nat] :
% 11.90/5.95 ( if @ A
% 11.90/5.95 @ ( N5
% 11.90/5.95 = ( zero_zero @ nat ) )
% 11.90/5.95 @ X3
% 11.90/5.95 @ ( F2 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1900_or__int__rec,axiom,
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ int )
% 11.90/5.95 = ( ^ [K4: int,L3: int] :
% 11.90/5.95 ( plus_plus @ int
% 11.90/5.95 @ ( zero_neq_one_of_bool @ int
% 11.90/5.95 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 11.90/5.95 | ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 11.90/5.95 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1901_or__numerals_I4_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num,Y: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1902_or__numerals_I6_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num,Y: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1903_or__numerals_I7_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num,Y: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1904_or__nat__rec,axiom,
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] :
% 11.90/5.95 ( plus_plus @ nat
% 11.90/5.95 @ ( zero_neq_one_of_bool @ nat
% 11.90/5.95 @ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
% 11.90/5.95 | ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
% 11.90/5.95 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1905_or__not__numerals_I9_J,axiom,
% 11.90/5.95 ! [M2: num,N: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
% 11.90/5.95 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1906_bit_Odisj__one__left,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1907_bit_Odisj__one__right,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1908_or__numerals_I8_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1909_or__numerals_I2_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [Y: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1910_bit_Odisj__cancel__left,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1911_bit_Odisj__cancel__right,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.95 => ! [X: A] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
% 11.90/5.95 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1912_or__numerals_I5_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [X: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1913_or__numerals_I1_J,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_un5681908812861735899ations @ A )
% 11.90/5.95 => ! [Y: num] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
% 11.90/5.95 = ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1914_or__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [Xa: int,X: int] :
% 11.90/5.95 ( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1915_or__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer,Xa: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa ) )
% 11.90/5.95 = ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1916_disjunctive__add,axiom,
% 11.90/5.95 ! [A: $tType] :
% 11.90/5.95 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.95 => ! [A3: A,B2: A] :
% 11.90/5.95 ( ! [N3: nat] :
% 11.90/5.95 ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N3 )
% 11.90/5.95 | ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
% 11.90/5.95 => ( ( plus_plus @ A @ A3 @ B2 )
% 11.90/5.95 = ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1917_plus__and__or,axiom,
% 11.90/5.95 ! [X: int,Y: int] :
% 11.90/5.95 ( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) )
% 11.90/5.95 = ( plus_plus @ int @ X @ Y ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1918_not__integer_Orep__eq,axiom,
% 11.90/5.95 ! [X: code_integer] :
% 11.90/5.95 ( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
% 11.90/5.95 = ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1919_set__bit__nat__def,axiom,
% 11.90/5.95 ( ( bit_se5668285175392031749et_bit @ nat )
% 11.90/5.95 = ( ^ [M: nat,N5: nat] : ( bit_se1065995026697491101ons_or @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.95
% 11.90/5.95 thf(fact_1920_not__integer_Oabs__eq,axiom,
% 11.90/5.95 ! [X: int] :
% 11.90/5.95 ( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.95 = ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1921_set__bit__eq__or,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ( ( bit_se5668285175392031749et_bit @ A )
% 11.90/5.96 = ( ^ [N5: nat,A7: A] : ( bit_se1065995026697491101ons_or @ A @ A7 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1922_bit_Ocomplement__unique,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.96 => ! [A3: A,X: A,Y: A] :
% 11.90/5.96 ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
% 11.90/5.96 = ( zero_zero @ A ) )
% 11.90/5.96 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
% 11.90/5.96 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.96 => ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y )
% 11.90/5.96 = ( zero_zero @ A ) )
% 11.90/5.96 => ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y )
% 11.90/5.96 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.96 => ( X = Y ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1923_bit_Ocompl__unique,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.96 => ! [X: A,Y: A] :
% 11.90/5.96 ( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
% 11.90/5.96 = ( zero_zero @ A ) )
% 11.90/5.96 => ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
% 11.90/5.96 = ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.96 => ( ( bit_ri4277139882892585799ns_not @ A @ X )
% 11.90/5.96 = Y ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1924_or__one__eq,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.96 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1925_one__or__eq,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.96 = ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1926_mask__Suc__double,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [N: nat] :
% 11.90/5.96 ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
% 11.90/5.96 = ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1927_or__not__numerals_I5_J,axiom,
% 11.90/5.96 ! [M2: num,N: num] :
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 11.90/5.96 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1928_Suc__0__or__eq,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 11.90/5.96 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1929_or__Suc__0__eq,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.96 = ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1930_or__not__numerals_I8_J,axiom,
% 11.90/5.96 ! [M2: num,N: num] :
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
% 11.90/5.96 = ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1931_or__int__unfold,axiom,
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ int )
% 11.90/5.96 = ( ^ [K4: int,L3: int] :
% 11.90/5.96 ( if @ int
% 11.90/5.96 @ ( ( K4
% 11.90/5.96 = ( uminus_uminus @ int @ ( one_one @ int ) ) )
% 11.90/5.96 | ( L3
% 11.90/5.96 = ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 11.90/5.96 @ ( uminus_uminus @ int @ ( one_one @ int ) )
% 11.90/5.96 @ ( if @ int
% 11.90/5.96 @ ( K4
% 11.90/5.96 = ( zero_zero @ int ) )
% 11.90/5.96 @ L3
% 11.90/5.96 @ ( if @ int
% 11.90/5.96 @ ( L3
% 11.90/5.96 = ( zero_zero @ int ) )
% 11.90/5.96 @ K4
% 11.90/5.96 @ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1932_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.96 => ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1933_or__nat__unfold,axiom,
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ nat )
% 11.90/5.96 = ( ^ [M: nat,N5: nat] :
% 11.90/5.96 ( if @ nat
% 11.90/5.96 @ ( M
% 11.90/5.96 = ( zero_zero @ nat ) )
% 11.90/5.96 @ N5
% 11.90/5.96 @ ( if @ nat
% 11.90/5.96 @ ( N5
% 11.90/5.96 = ( zero_zero @ nat ) )
% 11.90/5.96 @ M
% 11.90/5.96 @ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1934_drop__bit__rec,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ( ( bit_se4197421643247451524op_bit @ A )
% 11.90/5.96 = ( ^ [N5: nat,A7: A] :
% 11.90/5.96 ( if @ A
% 11.90/5.96 @ ( N5
% 11.90/5.96 = ( zero_zero @ nat ) )
% 11.90/5.96 @ A7
% 11.90/5.96 @ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A7 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1935_image__atLeastZeroLessThan__integer,axiom,
% 11.90/5.96 ! [U: code_integer] :
% 11.90/5.96 ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ U )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U )
% 11.90/5.96 = ( image2 @ nat @ code_integer @ ( semiring_1_of_nat @ code_integer ) @ ( set_ord_lessThan @ nat @ ( code_nat_of_integer @ U ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1936_uminus__int__def,axiom,
% 11.90/5.96 ( ( uminus_uminus @ int )
% 11.90/5.96 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
% 11.90/5.96 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.96 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1937_bit_Oabstract__boolean__algebra__axioms,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.96 => ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1938_map__fun__apply,axiom,
% 11.90/5.96 ! [D: $tType,A: $tType,C: $tType,B: $tType] :
% 11.90/5.96 ( ( map_fun @ B @ C @ D @ A )
% 11.90/5.96 = ( ^ [F2: B > C,G2: D > A,H: C > D,X3: B] : ( G2 @ ( H @ ( F2 @ X3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1939_drop__bit__drop__bit,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [M2: nat,N: nat,A3: A] :
% 11.90/5.96 ( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
% 11.90/5.96 = ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1940_image__add__atLeastAtMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_semidom @ A )
% 11.90/5.96 => ! [K: A,I: A,J: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1941_image__diff__atLeastAtMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_idom @ A )
% 11.90/5.96 => ! [D3: A,A3: A,B2: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( minus_minus @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D3 @ B2 ) @ ( minus_minus @ A @ D3 @ A3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1942_image__add__atLeastLessThan,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_semidom @ A )
% 11.90/5.96 => ! [K: A,I: A,J: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1943_max__0__1_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1944_max__0__1_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1945_image__uminus__atLeastAtMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordered_ab_group_add @ A )
% 11.90/5.96 => ! [X: A,Y: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1946_max__0__1_I6_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ! [X: num] :
% 11.90/5.96 ( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 11.90/5.96 = ( numeral_numeral @ A @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1947_max__0__1_I5_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ! [X: num] :
% 11.90/5.96 ( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 11.90/5.96 = ( numeral_numeral @ A @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1948_image__add__atMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordered_ab_group_add @ A )
% 11.90/5.96 => ! [C2: A,A3: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A3 ) )
% 11.90/5.96 = ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1949_image__add__atLeastAtMost_H,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_semidom @ A )
% 11.90/5.96 => ! [K: A,I: A,J: A] :
% 11.90/5.96 ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1950_image__minus__const__atLeastAtMost_H,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_idom @ A )
% 11.90/5.96 => ! [D3: A,A3: A,B2: A] :
% 11.90/5.96 ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1951_image__add__atLeastLessThan_H,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_semidom @ A )
% 11.90/5.96 => ! [K: A,I: A,J: A] :
% 11.90/5.96 ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1952_drop__bit__of__1,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [N: nat] :
% 11.90/5.96 ( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
% 11.90/5.96 = ( zero_neq_one_of_bool @ A
% 11.90/5.96 @ ( N
% 11.90/5.96 = ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1953_image__mult__atLeastAtMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [D3: A,A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
% 11.90/5.96 => ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1954_image__divide__atLeastAtMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [D3: A,A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [C3: A] : ( divide_divide @ A @ C3 @ D3 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1955_drop__bit__integer_Orep__eq,axiom,
% 11.90/5.96 ! [X: nat,Xa: code_integer] :
% 11.90/5.96 ( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa ) )
% 11.90/5.96 = ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1956_drop__bit__integer_Oabs__eq,axiom,
% 11.90/5.96 ! [Xa: nat,X: int] :
% 11.90/5.96 ( ( bit_se4197421643247451524op_bit @ code_integer @ Xa @ ( code_integer_of_int @ X ) )
% 11.90/5.96 = ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1957_max__add__distrib__left,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.96 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.96 ( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
% 11.90/5.96 = ( ord_max @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1958_max__add__distrib__right,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.96 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.96 ( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z2 ) )
% 11.90/5.96 = ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1959_max__diff__distrib__left,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordered_ab_group_add @ A )
% 11.90/5.96 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.96 ( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z2 )
% 11.90/5.96 = ( ord_max @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1960_translation__Compl,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ab_group_add @ A )
% 11.90/5.96 => ! [A3: A,T2: set @ A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 11.90/5.96 = ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1961_nat__add__max__right,axiom,
% 11.90/5.96 ! [M2: nat,N: nat,Q3: nat] :
% 11.90/5.96 ( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q3 ) )
% 11.90/5.96 = ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1962_nat__add__max__left,axiom,
% 11.90/5.96 ! [M2: nat,N: nat,Q3: nat] :
% 11.90/5.96 ( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q3 )
% 11.90/5.96 = ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q3 ) @ ( plus_plus @ nat @ N @ Q3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1963_image__eq__imp__comp,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A4: set @ B,G: C > A,B6: set @ C,H3: A > D] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ A4 )
% 11.90/5.96 = ( image2 @ C @ A @ G @ B6 ) )
% 11.90/5.96 => ( ( image2 @ B @ D @ ( comp @ A @ D @ B @ H3 @ F ) @ A4 )
% 11.90/5.96 = ( image2 @ C @ D @ ( comp @ A @ D @ C @ H3 @ G ) @ B6 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1964_image__comp,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
% 11.90/5.96 ( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ R ) )
% 11.90/5.96 = ( image2 @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1965_map__fun__def,axiom,
% 11.90/5.96 ! [B: $tType,D: $tType,A: $tType,C: $tType] :
% 11.90/5.96 ( ( map_fun @ C @ A @ B @ D )
% 11.90/5.96 = ( ^ [F2: C > A,G2: B > D,H: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G2 @ H ) @ F2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1966_map__fun_Ocompositionality,axiom,
% 11.90/5.96 ! [D: $tType,F5: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F5,H3: C > A,I: B > D,Fun: A > B] :
% 11.90/5.96 ( ( map_fun @ E @ C @ D @ F5 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H3 @ I @ Fun ) )
% 11.90/5.96 = ( map_fun @ E @ A @ B @ F5 @ ( comp @ C @ A @ E @ H3 @ F ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Fun ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1967_map__fun_Ocomp,axiom,
% 11.90/5.96 ! [E: $tType,C: $tType,A: $tType,F5: $tType,D: $tType,B: $tType,F: E > C,G: D > F5,H3: C > A,I: B > D] :
% 11.90/5.96 ( ( comp @ ( C > D ) @ ( E > F5 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F5 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H3 @ I ) )
% 11.90/5.96 = ( map_fun @ E @ A @ B @ F5 @ ( comp @ C @ A @ E @ H3 @ F ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1968_translation__subtract__Compl,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ab_group_add @ A )
% 11.90/5.96 => ! [A3: A,T2: set @ A] :
% 11.90/5.96 ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.96 @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
% 11.90/5.96 = ( uminus_uminus @ ( set @ A )
% 11.90/5.96 @ ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.96 @ T2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1969_nat__minus__add__max,axiom,
% 11.90/5.96 ! [N: nat,M2: nat] :
% 11.90/5.96 ( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M2 ) @ M2 )
% 11.90/5.96 = ( ord_max @ nat @ N @ M2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1970_translation__diff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ab_group_add @ A )
% 11.90/5.96 => ! [A3: A,S: set @ A,T2: set @ A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
% 11.90/5.96 = ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1971_translation__subtract__diff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ab_group_add @ A )
% 11.90/5.96 => ! [A3: A,S: set @ A,T2: set @ A] :
% 11.90/5.96 ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.96 @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
% 11.90/5.96 = ( minus_minus @ ( set @ A )
% 11.90/5.96 @ ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.96 @ S )
% 11.90/5.96 @ ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.96 @ T2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1972_take__bit__drop__bit,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [M2: nat,N: nat,A3: A] :
% 11.90/5.96 ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
% 11.90/5.96 = ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1973_bit__drop__bit__eq,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [N: nat,A3: A] :
% 11.90/5.96 ( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
% 11.90/5.96 = ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1974_div__push__bit__of__1__eq__drop__bit,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [A3: A,N: nat] :
% 11.90/5.96 ( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
% 11.90/5.96 = ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1975_bit__iff__and__drop__bit__eq__1,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ( ( bit_se5641148757651400278ts_bit @ A )
% 11.90/5.96 = ( ^ [A7: A,N5: nat] :
% 11.90/5.96 ( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N5 @ A7 ) @ ( one_one @ A ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1976_bits__ident,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_se359711467146920520ations @ A )
% 11.90/5.96 => ! [N: nat,A3: A] :
% 11.90/5.96 ( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
% 11.90/5.96 = A3 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1977_slice__eq__mask,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/5.96 => ! [N: nat,M2: nat,A3: A] :
% 11.90/5.96 ( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) )
% 11.90/5.96 = ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M2 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1978_image__add__0,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [S2: set @ A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
% 11.90/5.96 = S2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1979_times__int__def,axiom,
% 11.90/5.96 ( ( times_times @ int )
% 11.90/5.96 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
% 11.90/5.96 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.96 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.96 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.96 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1980_minus__int__def,axiom,
% 11.90/5.96 ( ( minus_minus @ int )
% 11.90/5.96 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
% 11.90/5.96 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.96 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.96 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.96 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1981_plus__int__def,axiom,
% 11.90/5.96 ( ( plus_plus @ int )
% 11.90/5.96 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
% 11.90/5.96 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.96 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.96 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.96 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1982_mlex__prod__def,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( mlex_prod @ A )
% 11.90/5.96 = ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
% 11.90/5.96 ( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
% 11.90/5.96 @ ^ [X3: A] : ( product_Pair @ nat @ A @ ( F2 @ X3 ) @ X3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1983_rat__minus__code,axiom,
% 11.90/5.96 ! [P5: rat,Q3: rat] :
% 11.90/5.96 ( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q3 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int,C3: int] :
% 11.90/5.96 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
% 11.90/5.96 @ ( quotient_of @ Q3 ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1984_rat__plus__code,axiom,
% 11.90/5.96 ! [P5: rat,Q3: rat] :
% 11.90/5.96 ( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q3 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int,C3: int] :
% 11.90/5.96 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ B5 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
% 11.90/5.96 @ ( quotient_of @ Q3 ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1985_image__Suc__atLeastAtMost,axiom,
% 11.90/5.96 ! [I: nat,J: nat] :
% 11.90/5.96 ( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1986_image__Suc__atLeastLessThan,axiom,
% 11.90/5.96 ! [I: nat,J: nat] :
% 11.90/5.96 ( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1987_in__inv__image,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
% 11.90/5.96 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R @ F ) )
% 11.90/5.96 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1988_pair__imageI,axiom,
% 11.90/5.96 ! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A4: set @ ( product_prod @ A @ B ),F: A > B > C] :
% 11.90/5.96 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A4 )
% 11.90/5.96 => ( member @ C @ ( F @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1989_int__of__integer__max,axiom,
% 11.90/5.96 ! [K: code_integer,L: code_integer] :
% 11.90/5.96 ( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
% 11.90/5.96 = ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1990_rat__one__code,axiom,
% 11.90/5.96 ( ( quotient_of @ ( one_one @ rat ) )
% 11.90/5.96 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1991_of__nat__of__integer,axiom,
% 11.90/5.96 ! [K: code_integer] :
% 11.90/5.96 ( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
% 11.90/5.96 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1992_rat__zero__code,axiom,
% 11.90/5.96 ( ( quotient_of @ ( zero_zero @ rat ) )
% 11.90/5.96 = ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1993_quotient__of__number_I3_J,axiom,
% 11.90/5.96 ! [K: num] :
% 11.90/5.96 ( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
% 11.90/5.96 = ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1994_quotient__of__number_I4_J,axiom,
% 11.90/5.96 ( ( quotient_of @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 11.90/5.96 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1995_quotient__of__number_I5_J,axiom,
% 11.90/5.96 ! [K: num] :
% 11.90/5.96 ( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
% 11.90/5.96 = ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1996_diff__rat__def,axiom,
% 11.90/5.96 ( ( minus_minus @ rat )
% 11.90/5.96 = ( ^ [Q5: rat,R3: rat] : ( plus_plus @ rat @ Q5 @ ( uminus_uminus @ rat @ R3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1997_quotient__of__div,axiom,
% 11.90/5.96 ! [R: rat,N: int,D3: int] :
% 11.90/5.96 ( ( ( quotient_of @ R )
% 11.90/5.96 = ( product_Pair @ int @ int @ N @ D3 ) )
% 11.90/5.96 => ( R
% 11.90/5.96 = ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1998_zero__notin__Suc__image,axiom,
% 11.90/5.96 ! [A4: set @ nat] :
% 11.90/5.96 ~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A4 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_1999_None__notin__image__Some,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A] :
% 11.90/5.96 ~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2000_plus__integer__def,axiom,
% 11.90/5.96 ( ( plus_plus @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( plus_plus @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2001_times__integer__def,axiom,
% 11.90/5.96 ( ( times_times @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( times_times @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2002_minus__integer__def,axiom,
% 11.90/5.96 ( ( minus_minus @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( minus_minus @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2003_divide__integer__def,axiom,
% 11.90/5.96 ( ( divide_divide @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( divide_divide @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2004_modulo__integer__def,axiom,
% 11.90/5.96 ( ( modulo_modulo @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( modulo_modulo @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2005_and__integer__def,axiom,
% 11.90/5.96 ( ( bit_se5824344872417868541ns_and @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344872417868541ns_and @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2006_or__integer__def,axiom,
% 11.90/5.96 ( ( bit_se1065995026697491101ons_or @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se1065995026697491101ons_or @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2007_xor__integer__def,axiom,
% 11.90/5.96 ( ( bit_se5824344971392196577ns_xor @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344971392196577ns_xor @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2008_image__int__atLeastAtMost,axiom,
% 11.90/5.96 ! [A3: nat,B2: nat] :
% 11.90/5.96 ( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2009_image__int__atLeastLessThan,axiom,
% 11.90/5.96 ! [A3: nat,B2: nat] :
% 11.90/5.96 ( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2010_quotient__of__denom__pos,axiom,
% 11.90/5.96 ! [R: rat,P5: int,Q3: int] :
% 11.90/5.96 ( ( ( quotient_of @ R )
% 11.90/5.96 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/5.96 => ( ord_less @ int @ ( zero_zero @ int ) @ Q3 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2011_image__Suc__atMost,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2012_image__Suc__lessThan,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2013_inv__image__def,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( inv_image @ B @ A )
% 11.90/5.96 = ( ^ [R3: set @ ( product_prod @ B @ B ),F2: A > B] :
% 11.90/5.96 ( collect @ ( product_prod @ A @ A )
% 11.90/5.96 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.96 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2014_rat__uminus__code,axiom,
% 11.90/5.96 ! [P5: rat] :
% 11.90/5.96 ( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A7 ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2015_rat__abs__code,axiom,
% 11.90/5.96 ! [P5: rat] :
% 11.90/5.96 ( ( quotient_of @ ( abs_abs @ rat @ P5 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A7 ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2016_uminus__integer__def,axiom,
% 11.90/5.96 ( ( uminus_uminus @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2017_abs__integer__def,axiom,
% 11.90/5.96 ( ( abs_abs @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2018_image__add__int__atLeastLessThan,axiom,
% 11.90/5.96 ! [L: int,U: int] :
% 11.90/5.96 ( ( image2 @ int @ int
% 11.90/5.96 @ ^ [X3: int] : ( plus_plus @ int @ X3 @ L )
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2019_sgn__integer__def,axiom,
% 11.90/5.96 ( ( sgn_sgn @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2020_not__integer__def,axiom,
% 11.90/5.96 ( ( bit_ri4277139882892585799ns_not @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2021_image__add__integer__atLeastLessThan,axiom,
% 11.90/5.96 ! [L: code_integer,U: code_integer] :
% 11.90/5.96 ( ( image2 @ code_integer @ code_integer
% 11.90/5.96 @ ^ [X3: code_integer] : ( plus_plus @ code_integer @ X3 @ L )
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ ( minus_minus @ code_integer @ U @ L ) ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2022_rat__times__code,axiom,
% 11.90/5.96 ! [P5: rat,Q3: rat] :
% 11.90/5.96 ( ( quotient_of @ ( times_times @ rat @ P5 @ Q3 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int,C3: int] :
% 11.90/5.96 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ B5 ) @ ( times_times @ int @ C3 @ D5 ) ) )
% 11.90/5.96 @ ( quotient_of @ Q3 ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2023_rat__divide__code,axiom,
% 11.90/5.96 ! [P5: rat,Q3: rat] :
% 11.90/5.96 ( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q3 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int,C3: int] :
% 11.90/5.96 ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [B5: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A7 @ D5 ) @ ( times_times @ int @ C3 @ B5 ) ) )
% 11.90/5.96 @ ( quotient_of @ Q3 ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2024_image__atLeastZeroLessThan__int,axiom,
% 11.90/5.96 ! [U: int] :
% 11.90/5.96 ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
% 11.90/5.96 = ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2025_quotient__of__int,axiom,
% 11.90/5.96 ! [A3: int] :
% 11.90/5.96 ( ( quotient_of @ ( of_int @ A3 ) )
% 11.90/5.96 = ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2026_rp__inv__image__def,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( fun_rp_inv_image @ A @ B )
% 11.90/5.96 = ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
% 11.90/5.96 @ ^ [R6: set @ ( product_prod @ A @ A ),S5: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F2 ) @ ( inv_image @ A @ B @ S5 @ F2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2027_rat__inverse__code,axiom,
% 11.90/5.96 ! [P5: rat] :
% 11.90/5.96 ( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
% 11.90/5.96 = ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
% 11.90/5.96 @ ^ [A7: int,B5: int] :
% 11.90/5.96 ( if @ ( product_prod @ int @ int )
% 11.90/5.96 @ ( A7
% 11.90/5.96 = ( zero_zero @ int ) )
% 11.90/5.96 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.96 @ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A7 ) @ B5 ) @ ( abs_abs @ int @ A7 ) ) )
% 11.90/5.96 @ ( quotient_of @ P5 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2028_diff__numeral__special_I8_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num] :
% 11.90/5.96 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2029_diff__numeral__special_I7_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [N: num] :
% 11.90/5.96 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2030_minus__sub__one__diff__one,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [M2: num] :
% 11.90/5.96 ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
% 11.90/5.96 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2031_add__neg__numeral__special_I4_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [N: num] :
% 11.90/5.96 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2032_int__of__integer__sub,axiom,
% 11.90/5.96 ! [K: num,L: num] :
% 11.90/5.96 ( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
% 11.90/5.96 = ( neg_numeral_sub @ int @ K @ L ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2033_inverse__1,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ( ( inverse_inverse @ A @ ( one_one @ A ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2034_inverse__eq__1__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( field @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( ( inverse_inverse @ A @ X )
% 11.90/5.96 = ( one_one @ A ) )
% 11.90/5.96 = ( X
% 11.90/5.96 = ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2035_right__inverse,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( A3
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2036_left__inverse,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( A3
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2037_inverse__eq__divide__numeral,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [W: num] :
% 11.90/5.96 ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
% 11.90/5.96 = ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2038_add__neg__numeral__simps_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num,N: num] :
% 11.90/5.96 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2039_add__neg__numeral__simps_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num,N: num] :
% 11.90/5.96 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2040_semiring__norm_I166_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [V: num,W: num,Y: A] :
% 11.90/5.96 ( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y ) )
% 11.90/5.96 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W ) @ Y ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2041_semiring__norm_I167_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [V: num,W: num,Y: A] :
% 11.90/5.96 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y ) )
% 11.90/5.96 = ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V ) @ Y ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2042_inverse__eq__divide__neg__numeral,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [W: num] :
% 11.90/5.96 ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
% 11.90/5.96 = ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2043_diff__numeral__special_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num] :
% 11.90/5.96 ( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2044_diff__numeral__special_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [N: num] :
% 11.90/5.96 ( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2045_add__neg__numeral__special_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num] :
% 11.90/5.96 ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2046_add__neg__numeral__special_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num] :
% 11.90/5.96 ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2047_add__neg__numeral__special_I3_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( neg_numeral @ A )
% 11.90/5.96 => ! [M2: num] :
% 11.90/5.96 ( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
% 11.90/5.96 = ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2048_inverse__unique,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( times_times @ A @ A3 @ B2 )
% 11.90/5.96 = ( one_one @ A ) )
% 11.90/5.96 => ( ( inverse_inverse @ A @ A3 )
% 11.90/5.96 = B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2049_inverse__eq__divide,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ( ( inverse_inverse @ A )
% 11.90/5.96 = ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2050_prod__inversef,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( field @ A )
% 11.90/5.96 => ! [F: B > A,A4: set @ B] :
% 11.90/5.96 ( ( groups7121269368397514597t_prod @ B @ A @ ( comp @ A @ A @ B @ ( inverse_inverse @ A ) @ F ) @ A4 )
% 11.90/5.96 = ( inverse_inverse @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2051_inverse__le__1__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 11.90/5.96 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 11.90/5.96 | ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2052_one__less__inverse__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 11.90/5.96 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.96 & ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2053_one__less__inverse,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.96 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.96 => ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2054_division__ring__inverse__add,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( A3
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( B2
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 11.90/5.96 = ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2055_inverse__add,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( field @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( A3
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( B2
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
% 11.90/5.96 = ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2056_field__class_Ofield__inverse,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( field @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( A3
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2057_nonzero__inverse__eq__divide,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( division_ring @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( A3
% 11.90/5.96 != ( zero_zero @ A ) )
% 11.90/5.96 => ( ( inverse_inverse @ A @ A3 )
% 11.90/5.96 = ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2058_one__le__inverse__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
% 11.90/5.96 = ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
% 11.90/5.96 & ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2059_inverse__less__1__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
% 11.90/5.96 = ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
% 11.90/5.96 | ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2060_one__le__inverse,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/5.96 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.96 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2061_Frct__code__post_I5_J,axiom,
% 11.90/5.96 ! [K: num] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
% 11.90/5.96 = ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2062_Code__Numeral_Odup__def,axiom,
% 11.90/5.96 ( code_dup
% 11.90/5.96 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
% 11.90/5.96 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2063_Frct__code__post_I6_J,axiom,
% 11.90/5.96 ! [K: num,L: num] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
% 11.90/5.96 = ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2064_Frct__code__post_I4_J,axiom,
% 11.90/5.96 ! [K: num] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
% 11.90/5.96 = ( numeral_numeral @ rat @ K ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2065_eq__numeral__iff__iszero_I8_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [Y: num] :
% 11.90/5.96 ( ( ( one_one @ A )
% 11.90/5.96 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
% 11.90/5.96 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2066_eq__numeral__iff__iszero_I7_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [X: num] :
% 11.90/5.96 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 11.90/5.96 = ( one_one @ A ) )
% 11.90/5.96 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2067_Inf_OINF__image,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
% 11.90/5.96 ( ( Inf @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 11.90/5.96 = ( Inf @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2068_not__iszero__1,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2069_Code__Numeral_Odup__code_I1_J,axiom,
% 11.90/5.96 ( ( code_dup @ ( zero_zero @ code_integer ) )
% 11.90/5.96 = ( zero_zero @ code_integer ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2070_not__iszero__neg__1,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2071_dup_Orep__eq,axiom,
% 11.90/5.96 ! [X: code_integer] :
% 11.90/5.96 ( ( code_int_of_integer @ ( code_dup @ X ) )
% 11.90/5.96 = ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2072_dup_Oabs__eq,axiom,
% 11.90/5.96 ! [X: int] :
% 11.90/5.96 ( ( code_dup @ ( code_integer_of_int @ X ) )
% 11.90/5.96 = ( code_integer_of_int @ ( plus_plus @ int @ X @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2073_eq__numeral__iff__iszero_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [X: num,Y: num] :
% 11.90/5.96 ( ( ( numeral_numeral @ A @ X )
% 11.90/5.96 = ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
% 11.90/5.96 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2074_eq__numeral__iff__iszero_I3_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [X: num,Y: num] :
% 11.90/5.96 ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
% 11.90/5.96 = ( numeral_numeral @ A @ Y ) )
% 11.90/5.96 = ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2075_Frct__code__post_I2_J,axiom,
% 11.90/5.96 ! [A3: int] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
% 11.90/5.96 = ( zero_zero @ rat ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2076_Frct__code__post_I1_J,axiom,
% 11.90/5.96 ! [A3: int] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
% 11.90/5.96 = ( zero_zero @ rat ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2077_Frct__code__post_I3_J,axiom,
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) )
% 11.90/5.96 = ( one_one @ rat ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2078_eq__numeral__iff__iszero_I6_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [Y: num] :
% 11.90/5.96 ( ( ( one_one @ A )
% 11.90/5.96 = ( numeral_numeral @ A @ Y ) )
% 11.90/5.96 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2079_eq__numeral__iff__iszero_I5_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ring_1 @ A )
% 11.90/5.96 => ! [X: num] :
% 11.90/5.96 ( ( ( numeral_numeral @ A @ X )
% 11.90/5.96 = ( one_one @ A ) )
% 11.90/5.96 = ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2080_Frct__code__post_I7_J,axiom,
% 11.90/5.96 ! [A3: int,B2: int] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
% 11.90/5.96 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2081_Frct__code__post_I8_J,axiom,
% 11.90/5.96 ! [A3: int,B2: int] :
% 11.90/5.96 ( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
% 11.90/5.96 = ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2082_Sup_OSUP__image,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A4: set @ C] :
% 11.90/5.96 ( ( Sup @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 11.90/5.96 = ( Sup @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2083_Code__Numeral_Osub__code_I9_J,axiom,
% 11.90/5.96 ! [M2: num,N: num] :
% 11.90/5.96 ( ( code_sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.96 = ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2084_Code__Numeral_Osub__code_I8_J,axiom,
% 11.90/5.96 ! [M2: num,N: num] :
% 11.90/5.96 ( ( code_sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.96 = ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2085_less__eq__int__def,axiom,
% 11.90/5.96 ( ( ord_less_eq @ int )
% 11.90/5.96 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
% 11.90/5.96 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.96 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.96 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.96 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2086_less__int__def,axiom,
% 11.90/5.96 ( ( ord_less @ int )
% 11.90/5.96 = ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
% 11.90/5.96 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.96 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.96 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.96 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2087_image__affinity__atLeastAtMost__div__diff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [A3: A,B2: A,M2: A,C2: A] :
% 11.90/5.96 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2088_image__affinity__atLeastAtMost__div,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [A3: A,B2: A,M2: A,C2: A] :
% 11.90/5.96 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( plus_plus @ A @ ( divide_divide @ A @ X3 @ M2 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2089_image__affinity__atLeastAtMost__diff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [A3: A,B2: A,M2: A,C2: A] :
% 11.90/5.96 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2090_id__apply,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( id @ A )
% 11.90/5.96 = ( ^ [X3: A] : X3 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2091_bijective__Empty,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2092_image__id,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( id @ A ) )
% 11.90/5.96 = ( id @ ( set @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2093_fun_Omap__id,axiom,
% 11.90/5.96 ! [A: $tType,D: $tType,T2: D > A] :
% 11.90/5.96 ( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
% 11.90/5.96 = T2 ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2094_id__comp,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,G: A > B] :
% 11.90/5.96 ( ( comp @ B @ B @ A @ ( id @ B ) @ G )
% 11.90/5.96 = G ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2095_comp__id,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/5.96 ( ( comp @ A @ B @ A @ F @ ( id @ A ) )
% 11.90/5.96 = F ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2096_lessThan__0,axiom,
% 11.90/5.96 ( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
% 11.90/5.96 = ( bot_bot @ ( set @ nat ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2097_apsnd__id,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
% 11.90/5.96 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2098_map__mmupd__empty,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,M2: A > ( option @ B ),V: B] :
% 11.90/5.96 ( ( map_mmupd @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) @ V )
% 11.90/5.96 = M2 ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2099_set__to__map__empty,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.96 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2100_atLeastatMost__empty__iff2,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( preorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( bot_bot @ ( set @ A ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2101_atLeastatMost__empty__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( preorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2102_atLeastatMost__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [B2: A,A3: A] :
% 11.90/5.96 ( ( ord_less @ A @ B2 @ A3 )
% 11.90/5.96 => ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2103_atLeastLessThan__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [B2: A,A3: A] :
% 11.90/5.96 ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2104_atLeastLessThan__empty__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( preorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2105_atLeastLessThan__empty__iff2,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( preorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( bot_bot @ ( set @ A ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2106_prod_Oempty,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [G: B > A] :
% 11.90/5.96 ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2107_group__add__class_Ominus__comp__minus,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( group_add @ A )
% 11.90/5.96 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 11.90/5.96 = ( id @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2108_boolean__algebra__class_Ominus__comp__minus,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( boolea8198339166811842893lgebra @ A )
% 11.90/5.96 => ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
% 11.90/5.96 = ( id @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2109_rel__of__empty,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
% 11.90/5.96 ( ( rel_of @ A @ B
% 11.90/5.96 @ ^ [X3: A] : ( none @ B )
% 11.90/5.96 @ P )
% 11.90/5.96 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2110_comp__the__Some,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
% 11.90/5.96 = ( id @ A ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2111_fun_Omap__id0,axiom,
% 11.90/5.96 ! [A: $tType,D: $tType] :
% 11.90/5.96 ( ( comp @ A @ A @ D @ ( id @ A ) )
% 11.90/5.96 = ( id @ ( D > A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2112_Iio__eq__empty__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ( linorder @ A )
% 11.90/5.96 & ( order_bot @ A ) )
% 11.90/5.96 => ! [N: A] :
% 11.90/5.96 ( ( ( set_ord_lessThan @ A @ N )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( N
% 11.90/5.96 = ( bot_bot @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2113_option_Omap__id0,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( map_option @ A @ A @ ( id @ A ) )
% 11.90/5.96 = ( id @ ( option @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2114_set__notEmptyE,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A] :
% 11.90/5.96 ( ( S2
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ~ ! [X2: A] :
% 11.90/5.96 ~ ( member @ A @ X2 @ S2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2115_memb__imp__not__empty,axiom,
% 11.90/5.96 ! [A: $tType,X: A,S2: set @ A] :
% 11.90/5.96 ( ( member @ A @ X @ S2 )
% 11.90/5.96 => ( S2
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2116_id__def,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( id @ A )
% 11.90/5.96 = ( ^ [X3: A] : X3 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2117_eq__id__iff,axiom,
% 11.90/5.96 ! [A: $tType,F: A > A] :
% 11.90/5.96 ( ( ! [X3: A] :
% 11.90/5.96 ( ( F @ X3 )
% 11.90/5.96 = X3 ) )
% 11.90/5.96 = ( F
% 11.90/5.96 = ( id @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2118_comp__eq__id__dest,axiom,
% 11.90/5.96 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
% 11.90/5.96 ( ( ( comp @ C @ B @ A @ A3 @ B2 )
% 11.90/5.96 = ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
% 11.90/5.96 => ( ( A3 @ ( B2 @ V ) )
% 11.90/5.96 = ( C2 @ V ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2119_pointfree__idE,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
% 11.90/5.96 ( ( ( comp @ B @ A @ A @ F @ G )
% 11.90/5.96 = ( id @ A ) )
% 11.90/5.96 => ( ( F @ ( G @ X ) )
% 11.90/5.96 = X ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2120_option_Omap__id,axiom,
% 11.90/5.96 ! [A: $tType,T2: option @ A] :
% 11.90/5.96 ( ( map_option @ A @ A @ ( id @ A ) @ T2 )
% 11.90/5.96 = T2 ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2121_not__empty__eq__Iic__eq__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( preorder @ A )
% 11.90/5.96 => ! [H3: A] :
% 11.90/5.96 ( ( bot_bot @ ( set @ A ) )
% 11.90/5.96 != ( set_ord_atMost @ A @ H3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2122_lessThan__non__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( no_bot @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( set_ord_lessThan @ A @ X )
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2123_subset__minus__empty,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.96 => ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2124_atLeastLessThan0,axiom,
% 11.90/5.96 ! [M2: nat] :
% 11.90/5.96 ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
% 11.90/5.96 = ( bot_bot @ ( set @ nat ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2125_lessThan__empty__iff,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( ( set_ord_lessThan @ nat @ N )
% 11.90/5.96 = ( bot_bot @ ( set @ nat ) ) )
% 11.90/5.96 = ( N
% 11.90/5.96 = ( zero_zero @ nat ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2126_set__to__map__empty__iff_I1_J,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.96 ( ( ( set_to_map @ A @ B @ S2 )
% 11.90/5.96 = ( ^ [X3: A] : ( none @ B ) ) )
% 11.90/5.96 = ( S2
% 11.90/5.96 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2127_Code__Numeral_Osub__code_I1_J,axiom,
% 11.90/5.96 ( ( code_sub @ one2 @ one2 )
% 11.90/5.96 = ( zero_zero @ code_integer ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2128_set__to__map__empty__iff_I2_J,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.96 ( ( ( ^ [X3: A] : ( none @ B ) )
% 11.90/5.96 = ( set_to_map @ A @ B @ S2 ) )
% 11.90/5.96 = ( S2
% 11.90/5.96 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2129_less__integer__def,axiom,
% 11.90/5.96 ( ( ord_less @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2130_less__eq__integer__def,axiom,
% 11.90/5.96 ( ( ord_less_eq @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( int > $o ) @ ( code_integer > $o ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ $o @ $o @ code_int_of_integer @ ( id @ $o ) ) @ ( ord_less_eq @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2131_Code__Numeral_Osub__code_I6_J,axiom,
% 11.90/5.96 ! [M2: num,N: num] :
% 11.90/5.96 ( ( code_sub @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
% 11.90/5.96 = ( code_dup @ ( code_sub @ M2 @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2132_Code__Numeral_Osub__code_I7_J,axiom,
% 11.90/5.96 ! [M2: num,N: num] :
% 11.90/5.96 ( ( code_sub @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
% 11.90/5.96 = ( code_dup @ ( code_sub @ M2 @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2133_sub_Orep__eq,axiom,
% 11.90/5.96 ! [X: num,Xa: num] :
% 11.90/5.96 ( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
% 11.90/5.96 = ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2134_sub_Oabs__eq,axiom,
% 11.90/5.96 ( code_sub
% 11.90/5.96 = ( ^ [Xa4: num,X3: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2135_image__mult__atLeastAtMost__if,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [C2: A,X: A,Y: A] :
% 11.90/5.96 ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.96 => ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.96 => ( ( ( ord_less_eq @ A @ X @ Y )
% 11.90/5.96 => ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ A @ X @ Y )
% 11.90/5.96 => ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2136_image__mult__atLeastAtMost__if_H,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [X: A,Y: A,C2: A] :
% 11.90/5.96 ( ( ( ord_less_eq @ A @ X @ Y )
% 11.90/5.96 => ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ A @ X @ Y )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( times_times @ A @ X3 @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2137_image__affinity__atLeastAtMost,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linordered_field @ A )
% 11.90/5.96 => ! [A3: A,B2: A,M2: A,C2: A] :
% 11.90/5.96 ( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 & ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/5.96 => ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X3 ) @ C2 )
% 11.90/5.96 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2138_and__int_Osimps,axiom,
% 11.90/5.96 ( ( bit_se5824344872417868541ns_and @ int )
% 11.90/5.96 = ( ^ [K4: int,L3: int] :
% 11.90/5.96 ( if @ int
% 11.90/5.96 @ ( ( member @ int @ K4 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ L3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 @ ( uminus_uminus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
% 11.90/5.96 @ ( plus_plus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K4 )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
% 11.90/5.96 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2139_and__int_Oelims,axiom,
% 11.90/5.96 ! [X: int,Xa: int,Y: int] :
% 11.90/5.96 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
% 11.90/5.96 = Y )
% 11.90/5.96 => ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( Y
% 11.90/5.96 = ( uminus_uminus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
% 11.90/5.96 & ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( Y
% 11.90/5.96 = ( plus_plus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
% 11.90/5.96 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2140_and__int_Opelims,axiom,
% 11.90/5.96 ! [X: int,Xa: int,Y: int] :
% 11.90/5.96 ( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
% 11.90/5.96 = Y )
% 11.90/5.96 => ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
% 11.90/5.96 => ~ ( ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( Y
% 11.90/5.96 = ( uminus_uminus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
% 11.90/5.96 & ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( Y
% 11.90/5.96 = ( plus_plus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
% 11.90/5.96 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
% 11.90/5.96 => ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2141_and__int_Opsimps,axiom,
% 11.90/5.96 ! [K: int,L: int] :
% 11.90/5.96 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
% 11.90/5.96 => ( ( ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 11.90/5.96 = ( uminus_uminus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
% 11.90/5.96 & ( ~ ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
% 11.90/5.96 = ( plus_plus @ int
% 11.90/5.96 @ ( zero_neq_one_of_bool @ int
% 11.90/5.96 @ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
% 11.90/5.96 & ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
% 11.90/5.96 @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2142_and__int_Opinduct,axiom,
% 11.90/5.96 ! [A0: int,A1: int,P: int > int > $o] :
% 11.90/5.96 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
% 11.90/5.96 => ( ! [K5: int,L5: int] :
% 11.90/5.96 ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K5 @ L5 ) )
% 11.90/5.96 => ( ( ~ ( ( member @ int @ K5 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
% 11.90/5.96 & ( member @ int @ L5 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
% 11.90/5.96 => ( P @ ( divide_divide @ int @ K5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L5 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
% 11.90/5.96 => ( P @ K5 @ L5 ) ) )
% 11.90/5.96 => ( P @ A0 @ A1 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2143_mask__mod__exp,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.96 => ! [N: nat,M2: nat] :
% 11.90/5.96 ( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
% 11.90/5.96 = ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2144_pair__leqI2,axiom,
% 11.90/5.96 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 11.90/5.96 ( ( ord_less_eq @ nat @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_less_eq @ nat @ S @ T2 )
% 11.90/5.96 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2145_min__eq__arg_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [M2: A,N: A] :
% 11.90/5.96 ( ( ( ord_min @ A @ M2 @ N )
% 11.90/5.96 = M2 )
% 11.90/5.96 = ( ord_less_eq @ A @ M2 @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2146_min__eq__arg_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [M2: A,N: A] :
% 11.90/5.96 ( ( ( ord_min @ A @ M2 @ N )
% 11.90/5.96 = N )
% 11.90/5.96 = ( ord_less_eq @ A @ N @ M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2147_min__arg__le_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [N: A,M2: A] :
% 11.90/5.96 ( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M2 @ N ) )
% 11.90/5.96 = ( ( ord_min @ A @ M2 @ N )
% 11.90/5.96 = N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2148_min__arg__le_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [M2: A,N: A] :
% 11.90/5.96 ( ( ord_less_eq @ A @ M2 @ ( ord_min @ A @ M2 @ N ) )
% 11.90/5.96 = ( ( ord_min @ A @ M2 @ N )
% 11.90/5.96 = M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2149_min__arg__not__ge_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [M2: A,N: A] :
% 11.90/5.96 ( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ M2 ) )
% 11.90/5.96 = ( ( ord_min @ A @ M2 @ N )
% 11.90/5.96 = M2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2150_min__arg__not__ge_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [M2: A,N: A] :
% 11.90/5.96 ( ( ~ ( ord_less @ A @ ( ord_min @ A @ M2 @ N ) @ N ) )
% 11.90/5.96 = ( ( ord_min @ A @ M2 @ N )
% 11.90/5.96 = N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2151_min__less__self__conv_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 )
% 11.90/5.96 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2152_min__less__self__conv_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
% 11.90/5.96 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2153_min__simps_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_min @ A @ A3 @ B2 )
% 11.90/5.96 = A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2154_min__simps_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [B2: A,A3: A] :
% 11.90/5.96 ( ( ord_less @ A @ B2 @ A3 )
% 11.90/5.96 => ( ( ord_min @ A @ A3 @ B2 )
% 11.90/5.96 = B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2155_min__0__1_I2_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
% 11.90/5.96 = ( zero_zero @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2156_min__0__1_I1_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
% 11.90/5.96 = ( zero_zero @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2157_min__0__1_I5_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ! [X: num] :
% 11.90/5.96 ( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2158_min__0__1_I6_J,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord181362715937106298miring @ A )
% 11.90/5.96 => ! [X: num] :
% 11.90/5.96 ( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2159_atLeastAtMost__singleton__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.96 ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 = ( ( A3 = B2 )
% 11.90/5.96 & ( B2 = C2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2160_atLeastAtMost__singleton,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( set_or1337092689740270186AtMost @ A @ A3 @ A3 )
% 11.90/5.96 = ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2161_min__Suc__gt_I2_J,axiom,
% 11.90/5.96 ! [A3: nat,B2: nat] :
% 11.90/5.96 ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_min @ nat @ B2 @ ( suc @ A3 ) )
% 11.90/5.96 = ( suc @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2162_min__Suc__gt_I1_J,axiom,
% 11.90/5.96 ! [A3: nat,B2: nat] :
% 11.90/5.96 ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_min @ nat @ ( suc @ A3 ) @ B2 )
% 11.90/5.96 = ( suc @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2163_case__prod__Pair,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
% 11.90/5.96 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2164_single__Diff__lessThan,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( preorder @ A )
% 11.90/5.96 => ! [K: A] :
% 11.90/5.96 ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
% 11.90/5.96 = ( insert2 @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2165_min__add__distrib__right,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.96 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.96 ( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z2 ) )
% 11.90/5.96 = ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2166_min__add__distrib__left,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordere2412721322843649153imp_le @ A )
% 11.90/5.96 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.96 ( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
% 11.90/5.96 = ( ord_min @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2167_min__diff__distrib__left,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordered_ab_group_add @ A )
% 11.90/5.96 => ! [X: A,Y: A,Z2: A] :
% 11.90/5.96 ( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z2 )
% 11.90/5.96 = ( ord_min @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2168_bot__option__def,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ( ( bot_bot @ ( option @ A ) )
% 11.90/5.96 = ( none @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2169_bot__empty__eq2,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( bot_bot @ ( A > B > $o ) )
% 11.90/5.96 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2170_map__fun_Oidentity,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( map_fun @ A @ A @ B @ B
% 11.90/5.96 @ ^ [X3: A] : X3
% 11.90/5.96 @ ^ [X3: B] : X3 )
% 11.90/5.96 = ( id @ ( A > B ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2171_map__option_Oidentity,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( map_option @ A @ A
% 11.90/5.96 @ ^ [X3: A] : X3 )
% 11.90/5.96 = ( id @ ( option @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2172_minus__max__eq__min,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.96 => ! [X: A,Y: A] :
% 11.90/5.96 ( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
% 11.90/5.96 = ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2173_minus__min__eq__max,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linord5086331880401160121up_add @ A )
% 11.90/5.96 => ! [X: A,Y: A] :
% 11.90/5.96 ( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
% 11.90/5.96 = ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2174_insert__minus__eq,axiom,
% 11.90/5.96 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 11.90/5.96 ( ( X != Y )
% 11.90/5.96 => ( ( minus_minus @ ( set @ A ) @ ( insert2 @ A @ X @ A4 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 = ( insert2 @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2175_set__minus__singleton__eq,axiom,
% 11.90/5.96 ! [A: $tType,X: A,X10: set @ A] :
% 11.90/5.96 ( ~ ( member @ A @ X @ X10 )
% 11.90/5.96 => ( ( minus_minus @ ( set @ A ) @ X10 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.96 = X10 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2176_atLeastAtMost__singleton_H,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( A3 = B2 )
% 11.90/5.96 => ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
% 11.90/5.96 = ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2177_remove__subset,axiom,
% 11.90/5.96 ! [A: $tType,X: A,S2: set @ A] :
% 11.90/5.96 ( ( member @ A @ X @ S2 )
% 11.90/5.96 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ S2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2178_nat__of__integer__def,axiom,
% 11.90/5.96 ( code_nat_of_integer
% 11.90/5.96 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2179_bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se5641148757651400278ts_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2180_mask__integer__def,axiom,
% 11.90/5.96 ( ( bit_se2239418461657761734s_mask @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2181_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ A )
% 11.90/5.96 = ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2182_atLeastAtMostPlus1__int__conv,axiom,
% 11.90/5.96 ! [M2: int,N: int] :
% 11.90/5.96 ( ( ord_less_eq @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 11.90/5.96 => ( ( set_or1337092689740270186AtMost @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
% 11.90/5.96 = ( insert2 @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2183_simp__from__to,axiom,
% 11.90/5.96 ( ( set_or1337092689740270186AtMost @ int )
% 11.90/5.96 = ( ^ [I3: int,J2: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J2 @ I3 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert2 @ int @ I3 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2184_Code__Numeral_Osub__def,axiom,
% 11.90/5.96 ( code_sub
% 11.90/5.96 = ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
% 11.90/5.96 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2185_num__of__integer__def,axiom,
% 11.90/5.96 ( code_num_of_integer
% 11.90/5.96 = ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2186_take__bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se2584673776208193580ke_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2584673776208193580ke_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2187_drop__bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se4197421643247451524op_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4197421643247451524op_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2188_push__bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se4730199178511100633sh_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4730199178511100633sh_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2189_unset__bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se2638667681897837118et_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2638667681897837118et_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2190_set__bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se5668285175392031749et_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5668285175392031749et_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2191_flip__bit__integer__def,axiom,
% 11.90/5.96 ( ( bit_se8732182000553998342ip_bit @ code_integer )
% 11.90/5.96 = ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se8732182000553998342ip_bit @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2192_pair__leqI1,axiom,
% 11.90/5.96 ! [A3: nat,B2: nat,S: nat,T2: nat] :
% 11.90/5.96 ( ( ord_less @ nat @ A3 @ B2 )
% 11.90/5.96 => ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2193_image__minus__const__atLeastLessThan__nat,axiom,
% 11.90/5.96 ! [C2: nat,Y: nat,X: nat] :
% 11.90/5.96 ( ( ( ord_less @ nat @ C2 @ Y )
% 11.90/5.96 => ( ( image2 @ nat @ nat
% 11.90/5.96 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less @ nat @ C2 @ Y )
% 11.90/5.96 => ( ( ( ord_less @ nat @ X @ Y )
% 11.90/5.96 => ( ( image2 @ nat @ nat
% 11.90/5.96 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
% 11.90/5.96 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less @ nat @ X @ Y )
% 11.90/5.96 => ( ( image2 @ nat @ nat
% 11.90/5.96 @ ^ [I3: nat] : ( minus_minus @ nat @ I3 @ C2 )
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
% 11.90/5.96 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2194_card__atLeastAtMost__int,axiom,
% 11.90/5.96 ! [L: int,U: int] :
% 11.90/5.96 ( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
% 11.90/5.96 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2195_range__mod,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.96 => ( ( image2 @ nat @ nat
% 11.90/5.96 @ ^ [M: nat] : ( modulo_modulo @ nat @ M @ N )
% 11.90/5.96 @ ( top_top @ ( set @ nat ) ) )
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2196_Some__image__these__eq,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ ( option @ A )] :
% 11.90/5.96 ( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
% 11.90/5.96 = ( collect @ ( option @ A )
% 11.90/5.96 @ ^ [X3: option @ A] :
% 11.90/5.96 ( ( member @ ( option @ A ) @ X3 @ A4 )
% 11.90/5.96 & ( X3
% 11.90/5.96 != ( none @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2197_prod__mono2,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( linordered_idom @ B )
% 11.90/5.96 => ! [B6: set @ A,A4: set @ A,F: A > B] :
% 11.90/5.96 ( ( finite_finite2 @ A @ B6 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.96 => ( ! [B3: A] :
% 11.90/5.96 ( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) )
% 11.90/5.96 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B3 ) ) )
% 11.90/5.96 => ( ! [A5: A] :
% 11.90/5.96 ( ( member @ A @ A5 @ A4 )
% 11.90/5.96 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A5 ) ) )
% 11.90/5.96 => ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B6 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2198_the__dflt__None__nonempty,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A] :
% 11.90/5.96 ( ( S2
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ( dflt_None_set @ A @ S2 )
% 11.90/5.96 = ( some @ ( set @ A ) @ S2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2199_the__dflt__None__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( none @ ( set @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2200_atMost__UNIV__triv,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2201_finite__atLeastAtMost,axiom,
% 11.90/5.96 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2202_finite__atLeastLessThan,axiom,
% 11.90/5.96 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2203_int__of__integer__min,axiom,
% 11.90/5.96 ! [K: code_integer,L: code_integer] :
% 11.90/5.96 ( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
% 11.90/5.96 = ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2204_finite__atMost,axiom,
% 11.90/5.96 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2205_finite__lessThan,axiom,
% 11.90/5.96 ! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2206_finite__atLeastAtMost__int,axiom,
% 11.90/5.96 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2207_finite__atLeastLessThan__int,axiom,
% 11.90/5.96 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2208_finite__atLeastLessThan__integer,axiom,
% 11.90/5.96 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2209_finite__atLeastAtMost__integer,axiom,
% 11.90/5.96 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or1337092689740270186AtMost @ code_integer @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2210_finite__option__UNIV,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 11.90/5.96 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2211_card__eq__UNIV,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( finite_finite @ A )
% 11.90/5.96 => ! [S2: set @ A] :
% 11.90/5.96 ( ( ( finite_card @ A @ S2 )
% 11.90/5.96 = ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
% 11.90/5.96 = ( S2
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2212_card__eq__UNIV2,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( finite_finite @ A )
% 11.90/5.96 => ! [S2: set @ A] :
% 11.90/5.96 ( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( finite_card @ A @ S2 ) )
% 11.90/5.96 = ( S2
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2213_these__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2214_these__insert__None,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ ( option @ A )] :
% 11.90/5.96 ( ( these @ A @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ A4 ) )
% 11.90/5.96 = ( these @ A @ A4 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2215_the__dflt__None__set,axiom,
% 11.90/5.96 ! [A: $tType,X: set @ A] :
% 11.90/5.96 ( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X ) )
% 11.90/5.96 = X ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2216_surj__plus,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ab_group_add @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2217_infinite__Icc__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
% 11.90/5.96 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2218_infinite__Ico__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
% 11.90/5.96 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2219_prod_Oinfinite,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [A4: set @ B,G: B > A] :
% 11.90/5.96 ( ~ ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2220_card__ge__UNIV,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( finite_finite @ A )
% 11.90/5.96 => ! [S2: set @ A] :
% 11.90/5.96 ( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S2 ) )
% 11.90/5.96 = ( S2
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2221_surj__diff__right,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ab_group_add @ A )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ( ( image2 @ A @ A
% 11.90/5.96 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.96 @ ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2222_prod__eq__1__iff,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A,F: A > nat] :
% 11.90/5.96 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.96 => ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A4 )
% 11.90/5.96 = ( one_one @ nat ) )
% 11.90/5.96 = ( ! [X3: A] :
% 11.90/5.96 ( ( member @ A @ X3 @ A4 )
% 11.90/5.96 => ( ( F @ X3 )
% 11.90/5.96 = ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2223_prod_Odelta,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,A3: B,B2: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ S2 )
% 11.90/5.96 => ( ( ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( B2 @ A3 ) ) )
% 11.90/5.96 & ( ~ ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2224_prod_Odelta_H,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,A3: B,B2: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ S2 )
% 11.90/5.96 => ( ( ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( B2 @ A3 ) ) )
% 11.90/5.96 & ( ~ ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( A3 = K4 ) @ ( B2 @ K4 ) @ ( one_one @ A ) )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2225_these__insert__Some,axiom,
% 11.90/5.96 ! [A: $tType,X: A,A4: set @ ( option @ A )] :
% 11.90/5.96 ( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) )
% 11.90/5.96 = ( insert2 @ A @ X @ ( these @ A @ A4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2226_these__image__Some__eq,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A] :
% 11.90/5.96 ( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
% 11.90/5.96 = A4 ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2227_sum_Oinsert,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [A4: set @ B,X: B,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ~ ( member @ B @ X @ A4 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 11.90/5.96 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2228_atLeastLessThan__singleton,axiom,
% 11.90/5.96 ! [M2: nat] :
% 11.90/5.96 ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
% 11.90/5.96 = ( insert2 @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2229_atMost__0,axiom,
% 11.90/5.96 ( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
% 11.90/5.96 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2230_card__atLeastLessThan__int,axiom,
% 11.90/5.96 ! [L: int,U: int] :
% 11.90/5.96 ( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
% 11.90/5.96 = ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2231_card__doubleton__eq__2__iff,axiom,
% 11.90/5.96 ! [A: $tType,A3: A,B2: A] :
% 11.90/5.96 ( ( ( finite_card @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.96 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.96 = ( A3 != B2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2232_atMost__eq__UNIV__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order_top @ A )
% 11.90/5.96 => ! [X: A] :
% 11.90/5.96 ( ( ( set_ord_atMost @ A @ X )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( X
% 11.90/5.96 = ( top_top @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2233_card__le__if__inj__on__rel,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,B6: set @ A,A4: set @ B,R: B > A > $o] :
% 11.90/5.96 ( ( finite_finite2 @ A @ B6 )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ A4 )
% 11.90/5.96 => ? [B8: A] :
% 11.90/5.96 ( ( member @ A @ B8 @ B6 )
% 11.90/5.96 & ( R @ A5 @ B8 ) ) )
% 11.90/5.96 => ( ! [A13: B,A24: B,B3: A] :
% 11.90/5.96 ( ( member @ B @ A13 @ A4 )
% 11.90/5.96 => ( ( member @ B @ A24 @ A4 )
% 11.90/5.96 => ( ( member @ A @ B3 @ B6 )
% 11.90/5.96 => ( ( R @ A13 @ B3 )
% 11.90/5.96 => ( ( R @ A24 @ B3 )
% 11.90/5.96 => ( A13 = A24 ) ) ) ) ) )
% 11.90/5.96 => ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2234_top__option__def,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order_top @ A )
% 11.90/5.96 => ( ( top_top @ ( option @ A ) )
% 11.90/5.96 = ( some @ A @ ( top_top @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2235_finite__if__eq__beyond__finite,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A,S3: set @ A] :
% 11.90/5.96 ( ( finite_finite2 @ A @ S2 )
% 11.90/5.96 => ( finite_finite2 @ ( set @ A )
% 11.90/5.96 @ ( collect @ ( set @ A )
% 11.90/5.96 @ ^ [S4: set @ A] :
% 11.90/5.96 ( ( minus_minus @ ( set @ A ) @ S4 @ S2 )
% 11.90/5.96 = ( minus_minus @ ( set @ A ) @ S3 @ S2 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2236_finite__range__Some,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
% 11.90/5.96 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2237_obtain__subset__with__card__n,axiom,
% 11.90/5.96 ! [A: $tType,N: nat,S2: set @ A] :
% 11.90/5.96 ( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
% 11.90/5.96 => ~ ! [T4: set @ A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ A ) @ T4 @ S2 )
% 11.90/5.96 => ( ( ( finite_card @ A @ T4 )
% 11.90/5.96 = N )
% 11.90/5.96 => ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2238_atLeastAtMost__eq__UNIV__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( bounded_lattice @ A )
% 11.90/5.96 => ! [X: A,Y: A] :
% 11.90/5.96 ( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( ( X
% 11.90/5.96 = ( bot_bot @ A ) )
% 11.90/5.96 & ( Y
% 11.90/5.96 = ( top_top @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2239_UNIV__option__conv,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( top_top @ ( set @ ( option @ A ) ) )
% 11.90/5.96 = ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2240_bounded__nat__set__is__finite,axiom,
% 11.90/5.96 ! [N6: set @ nat,N: nat] :
% 11.90/5.96 ( ! [X2: nat] :
% 11.90/5.96 ( ( member @ nat @ X2 @ N6 )
% 11.90/5.96 => ( ord_less @ nat @ X2 @ N ) )
% 11.90/5.96 => ( finite_finite2 @ nat @ N6 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2241_finite__nat__set__iff__bounded,axiom,
% 11.90/5.96 ( ( finite_finite2 @ nat )
% 11.90/5.96 = ( ^ [N9: set @ nat] :
% 11.90/5.96 ? [M: nat] :
% 11.90/5.96 ! [X3: nat] :
% 11.90/5.96 ( ( member @ nat @ X3 @ N9 )
% 11.90/5.96 => ( ord_less @ nat @ X3 @ M ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2242_finite__nat__set__iff__bounded__le,axiom,
% 11.90/5.96 ( ( finite_finite2 @ nat )
% 11.90/5.96 = ( ^ [N9: set @ nat] :
% 11.90/5.96 ? [M: nat] :
% 11.90/5.96 ! [X3: nat] :
% 11.90/5.96 ( ( member @ nat @ X3 @ N9 )
% 11.90/5.96 => ( ord_less_eq @ nat @ X3 @ M ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2243_surjD,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 => ? [X2: B] :
% 11.90/5.96 ( Y
% 11.90/5.96 = ( F @ X2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2244_surjE,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType,F: B > A,Y: A] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 => ~ ! [X2: B] :
% 11.90/5.96 ( Y
% 11.90/5.96 != ( F @ X2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2245_surjI,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 11.90/5.96 ( ! [X2: A] :
% 11.90/5.96 ( ( G @ ( F @ X2 ) )
% 11.90/5.96 = X2 )
% 11.90/5.96 => ( ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2246_surj__def,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,F: B > A] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( ! [Y3: A] :
% 11.90/5.96 ? [X3: B] :
% 11.90/5.96 ( Y3
% 11.90/5.96 = ( F @ X3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2247_these__empty__eq,axiom,
% 11.90/5.96 ! [A: $tType,B6: set @ ( option @ A )] :
% 11.90/5.96 ( ( ( these @ A @ B6 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( ( B6
% 11.90/5.96 = ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 11.90/5.96 | ( B6
% 11.90/5.96 = ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2248_these__not__empty__eq,axiom,
% 11.90/5.96 ! [A: $tType,B6: set @ ( option @ A )] :
% 11.90/5.96 ( ( ( these @ A @ B6 )
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( ( B6
% 11.90/5.96 != ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 11.90/5.96 & ( B6
% 11.90/5.96 != ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2249_not__UNIV__eq__Icc,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( no_top @ A )
% 11.90/5.96 => ! [L4: A,H4: A] :
% 11.90/5.96 ( ( top_top @ ( set @ A ) )
% 11.90/5.96 != ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2250_infinite__Iic,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ( linorder @ A )
% 11.90/5.96 & ( no_bot @ A ) )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2251_infinite__Iio,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ( linorder @ A )
% 11.90/5.96 & ( no_bot @ A ) )
% 11.90/5.96 => ! [A3: A] :
% 11.90/5.96 ~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2252_not__UNIV__eq__Iic,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( no_top @ A )
% 11.90/5.96 => ! [H4: A] :
% 11.90/5.96 ( ( top_top @ ( set @ A ) )
% 11.90/5.96 != ( set_ord_atMost @ A @ H4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2253_card__1__singletonI,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A,X: A] :
% 11.90/5.96 ( ( finite_finite2 @ A @ S2 )
% 11.90/5.96 => ( ( ( finite_card @ A @ S2 )
% 11.90/5.96 = ( one_one @ nat ) )
% 11.90/5.96 => ( ( member @ A @ X @ S2 )
% 11.90/5.96 => ( S2
% 11.90/5.96 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2254_finite__M__bounded__by__nat,axiom,
% 11.90/5.96 ! [P: nat > $o,I: nat] :
% 11.90/5.96 ( finite_finite2 @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [K4: nat] :
% 11.90/5.96 ( ( P @ K4 )
% 11.90/5.96 & ( ord_less @ nat @ K4 @ I ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2255_finite__less__ub,axiom,
% 11.90/5.96 ! [F: nat > nat,U: nat] :
% 11.90/5.96 ( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F @ N3 ) )
% 11.90/5.96 => ( finite_finite2 @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [N5: nat] : ( ord_less_eq @ nat @ ( F @ N5 ) @ U ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2256_card__insert__disjoint_H,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A,X: A] :
% 11.90/5.96 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.96 => ( ~ ( member @ A @ X @ A4 )
% 11.90/5.96 => ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ A4 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/5.96 = ( finite_card @ A @ A4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2257_in__these__eq,axiom,
% 11.90/5.96 ! [A: $tType,X: A,A4: set @ ( option @ A )] :
% 11.90/5.96 ( ( member @ A @ X @ ( these @ A @ A4 ) )
% 11.90/5.96 = ( member @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2258_infinite__Icc,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2259_infinite__Ico,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2260_subset__eq__atLeast0__atMost__finite,axiom,
% 11.90/5.96 ! [N6: set @ nat,N: nat] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.96 => ( finite_finite2 @ nat @ N6 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2261_subset__eq__atLeast0__lessThan__finite,axiom,
% 11.90/5.96 ! [N6: set @ nat,N: nat] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.96 => ( finite_finite2 @ nat @ N6 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2262_atMost__Suc,axiom,
% 11.90/5.96 ! [K: nat] :
% 11.90/5.96 ( ( set_ord_atMost @ nat @ ( suc @ K ) )
% 11.90/5.96 = ( insert2 @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2263_surj__fun__eq,axiom,
% 11.90/5.96 ! [B: $tType,C: $tType,A: $tType,F: B > A,X10: set @ B,G1: A > C,G22: A > C] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ X10 )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ X10 )
% 11.90/5.96 => ( ( comp @ A @ C @ B @ G1 @ F @ X2 )
% 11.90/5.96 = ( comp @ A @ C @ B @ G22 @ F @ X2 ) ) )
% 11.90/5.96 => ( G1 = G22 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2264_comp__surj,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 => ( ( ( image2 @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( top_top @ ( set @ C ) ) )
% 11.90/5.96 => ( ( image2 @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2265_fun_Oset__map,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
% 11.90/5.96 ( ( image2 @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
% 11.90/5.96 = ( image2 @ A @ B @ F @ ( image2 @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2266_fun_Omap__cong,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
% 11.90/5.96 ( ( X = Ya )
% 11.90/5.96 => ( ! [Z3: A] :
% 11.90/5.96 ( ( member @ A @ Z3 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 11.90/5.96 => ( ( F @ Z3 )
% 11.90/5.96 = ( G @ Z3 ) ) )
% 11.90/5.96 => ( ( comp @ A @ B @ D @ F @ X )
% 11.90/5.96 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2267_fun_Omap__cong0,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,D: $tType,X: D > A,F: A > B,G: A > B] :
% 11.90/5.96 ( ! [Z3: A] :
% 11.90/5.96 ( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 11.90/5.96 => ( ( F @ Z3 )
% 11.90/5.96 = ( G @ Z3 ) ) )
% 11.90/5.96 => ( ( comp @ A @ B @ D @ F @ X )
% 11.90/5.96 = ( comp @ A @ B @ D @ G @ X ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2268_fun_Oinj__map__strong,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F: A > B,Fa: A > B] :
% 11.90/5.96 ( ! [Z3: A,Za: A] :
% 11.90/5.96 ( ( member @ A @ Z3 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
% 11.90/5.96 => ( ( member @ A @ Za @ ( image2 @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
% 11.90/5.96 => ( ( ( F @ Z3 )
% 11.90/5.96 = ( Fa @ Za ) )
% 11.90/5.96 => ( Z3 = Za ) ) ) )
% 11.90/5.96 => ( ( ( comp @ A @ B @ D @ F @ X )
% 11.90/5.96 = ( comp @ A @ B @ D @ Fa @ Xa ) )
% 11.90/5.96 => ( X = Xa ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2269_lessThan__Suc,axiom,
% 11.90/5.96 ! [K: nat] :
% 11.90/5.96 ( ( set_ord_lessThan @ nat @ ( suc @ K ) )
% 11.90/5.96 = ( insert2 @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2270_not__UNIV__le__Icc,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( no_top @ A )
% 11.90/5.96 => ! [L: A,H3: A] :
% 11.90/5.96 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2271_surj__id,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2272_not__UNIV__le__Iic,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( no_top @ A )
% 11.90/5.96 => ! [H3: A] :
% 11.90/5.96 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2273_sum_Ofinite__Collect__op,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [I4: set @ B,X: B > A,Y: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [I3: B] :
% 11.90/5.96 ( ( member @ B @ I3 @ I4 )
% 11.90/5.96 & ( ( X @ I3 )
% 11.90/5.96 != ( zero_zero @ A ) ) ) ) )
% 11.90/5.96 => ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [I3: B] :
% 11.90/5.96 ( ( member @ B @ I3 @ I4 )
% 11.90/5.96 & ( ( Y @ I3 )
% 11.90/5.96 != ( zero_zero @ A ) ) ) ) )
% 11.90/5.96 => ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [I3: B] :
% 11.90/5.96 ( ( member @ B @ I3 @ I4 )
% 11.90/5.96 & ( ( plus_plus @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
% 11.90/5.96 != ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2274_prod_Ofinite__Collect__op,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [I4: set @ B,X: B > A,Y: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [I3: B] :
% 11.90/5.96 ( ( member @ B @ I3 @ I4 )
% 11.90/5.96 & ( ( X @ I3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 => ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [I3: B] :
% 11.90/5.96 ( ( member @ B @ I3 @ I4 )
% 11.90/5.96 & ( ( Y @ I3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 => ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [I3: B] :
% 11.90/5.96 ( ( member @ B @ I3 @ I4 )
% 11.90/5.96 & ( ( times_times @ A @ ( X @ I3 ) @ ( Y @ I3 ) )
% 11.90/5.96 != ( one_one @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2275_finite__atLeastZeroLessThan__int,axiom,
% 11.90/5.96 ! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2276_card__eq__sum,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( finite_card @ A )
% 11.90/5.96 = ( groups7311177749621191930dd_sum @ A @ nat
% 11.90/5.96 @ ^ [X3: A] : ( one_one @ nat ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2277_prod_Ointer__filter,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [A4: set @ B,G: B > A,P: B > $o] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ A4 )
% 11.90/5.96 & ( P @ X3 ) ) ) )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( G @ X3 ) @ ( one_one @ A ) )
% 11.90/5.96 @ A4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2278_prod__gen__delta,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,A3: B,B2: B > A,C2: A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ S2 )
% 11.90/5.96 => ( ( ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
% 11.90/5.96 & ( ~ ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ C2 )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( power_power @ A @ C2 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2279_finite__atLeastZeroLessThan__integer,axiom,
% 11.90/5.96 ! [U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or7035219750837199246ssThan @ code_integer @ ( zero_zero @ code_integer ) @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2280_card__2__iff_H,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A] :
% 11.90/5.96 ( ( ( finite_card @ A @ S2 )
% 11.90/5.96 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.96 = ( ? [X3: A] :
% 11.90/5.96 ( ( member @ A @ X3 @ S2 )
% 11.90/5.96 & ? [Y3: A] :
% 11.90/5.96 ( ( member @ A @ Y3 @ S2 )
% 11.90/5.96 & ( X3 != Y3 )
% 11.90/5.96 & ! [Z5: A] :
% 11.90/5.96 ( ( member @ A @ Z5 @ S2 )
% 11.90/5.96 => ( ( Z5 = X3 )
% 11.90/5.96 | ( Z5 = Y3 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2281_sum_Orelated,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [R5: A > A > $o,S2: set @ B,H3: B > A,G: B > A] :
% 11.90/5.96 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
% 11.90/5.96 => ( ! [X13: A,Y12: A,X23: A,Y22: A] :
% 11.90/5.96 ( ( ( R5 @ X13 @ X23 )
% 11.90/5.96 & ( R5 @ Y12 @ Y22 ) )
% 11.90/5.96 => ( R5 @ ( plus_plus @ A @ X13 @ Y12 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
% 11.90/5.96 => ( ( finite_finite2 @ B @ S2 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ S2 )
% 11.90/5.96 => ( R5 @ ( H3 @ X2 ) @ ( G @ X2 ) ) )
% 11.90/5.96 => ( R5 @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2282_prod_Orelated,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [R5: A > A > $o,S2: set @ B,H3: B > A,G: B > A] :
% 11.90/5.96 ( ( R5 @ ( one_one @ A ) @ ( one_one @ A ) )
% 11.90/5.96 => ( ! [X13: A,Y12: A,X23: A,Y22: A] :
% 11.90/5.96 ( ( ( R5 @ X13 @ X23 )
% 11.90/5.96 & ( R5 @ Y12 @ Y22 ) )
% 11.90/5.96 => ( R5 @ ( times_times @ A @ X13 @ Y12 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
% 11.90/5.96 => ( ( finite_finite2 @ B @ S2 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ S2 )
% 11.90/5.96 => ( R5 @ ( H3 @ X2 ) @ ( G @ X2 ) ) )
% 11.90/5.96 => ( R5 @ ( groups7121269368397514597t_prod @ B @ A @ H3 @ S2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2283_sum_Oinsert__if,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [A4: set @ B,X: B,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( ( member @ B @ X @ A4 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 11.90/5.96 = ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) )
% 11.90/5.96 & ( ~ ( member @ B @ X @ A4 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 11.90/5.96 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2284_atLeast0__atMost__Suc,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2285_atLeast0__lessThan__Suc,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2286_Icc__eq__insert__lb__nat,axiom,
% 11.90/5.96 ! [M2: nat,N: nat] :
% 11.90/5.96 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.96 => ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
% 11.90/5.96 = ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2287_atLeastAtMostSuc__conv,axiom,
% 11.90/5.96 ! [M2: nat,N: nat] :
% 11.90/5.96 ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.96 => ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2288_atLeastAtMost__insertL,axiom,
% 11.90/5.96 ! [M2: nat,N: nat] :
% 11.90/5.96 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.96 => ( ( insert2 @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
% 11.90/5.96 = ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2289_prod_Oreindex__bij__witness__not__neutral,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,C: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S7: set @ B,T5: set @ C,S2: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H3: C > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ S7 )
% 11.90/5.96 => ( ( finite_finite2 @ C @ T5 )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
% 11.90/5.96 => ( ( I @ ( J @ A5 ) )
% 11.90/5.96 = A5 ) )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
% 11.90/5.96 => ( member @ C @ ( J @ A5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
% 11.90/5.96 => ( ! [B3: C] :
% 11.90/5.96 ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 11.90/5.96 => ( ( J @ ( I @ B3 ) )
% 11.90/5.96 = B3 ) )
% 11.90/5.96 => ( ! [B3: C] :
% 11.90/5.96 ( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 11.90/5.96 => ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) ) )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ S7 )
% 11.90/5.96 => ( ( G @ A5 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [B3: C] :
% 11.90/5.96 ( ( member @ C @ B3 @ T5 )
% 11.90/5.96 => ( ( H3 @ B3 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ S2 )
% 11.90/5.96 => ( ( H3 @ ( J @ A5 ) )
% 11.90/5.96 = ( G @ A5 ) ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ C @ A @ H3 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2290_dflt__None__set__def,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dflt_None_set @ A )
% 11.90/5.96 = ( ^ [S5: set @ A] :
% 11.90/5.96 ( if @ ( option @ ( set @ A ) )
% 11.90/5.96 @ ( S5
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 @ ( none @ ( set @ A ) )
% 11.90/5.96 @ ( some @ ( set @ A ) @ S5 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2291_sum__eq__1__iff,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A,F: A > nat] :
% 11.90/5.96 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.96 => ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 )
% 11.90/5.96 = ( one_one @ nat ) )
% 11.90/5.96 = ( ? [X3: A] :
% 11.90/5.96 ( ( member @ A @ X3 @ A4 )
% 11.90/5.96 & ( ( F @ X3 )
% 11.90/5.96 = ( one_one @ nat ) )
% 11.90/5.96 & ! [Y3: A] :
% 11.90/5.96 ( ( member @ A @ Y3 @ A4 )
% 11.90/5.96 => ( ( X3 != Y3 )
% 11.90/5.96 => ( ( F @ Y3 )
% 11.90/5.96 = ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2292_surj__Compl__image__subset,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType,F: B > A,A4: set @ B] :
% 11.90/5.96 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.96 = ( top_top @ ( set @ A ) ) )
% 11.90/5.96 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F @ A4 ) ) @ ( image2 @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2293_card__atLeastZeroLessThan__int,axiom,
% 11.90/5.96 ! [U: int] :
% 11.90/5.96 ( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
% 11.90/5.96 = ( nat2 @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2294_atMost__nat__numeral,axiom,
% 11.90/5.96 ! [K: num] :
% 11.90/5.96 ( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
% 11.90/5.96 = ( insert2 @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2295_lessThan__nat__numeral,axiom,
% 11.90/5.96 ! [K: num] :
% 11.90/5.96 ( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
% 11.90/5.96 = ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2296_sum__Suc,axiom,
% 11.90/5.96 ! [A: $tType,F: A > nat,A4: set @ A] :
% 11.90/5.96 ( ( groups7311177749621191930dd_sum @ A @ nat
% 11.90/5.96 @ ^ [X3: A] : ( suc @ ( F @ X3 ) )
% 11.90/5.96 @ A4 )
% 11.90/5.96 = ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 ) @ ( finite_card @ A @ A4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2297_prod_Osetdiff__irrelevant,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [A4: set @ B,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G
% 11.90/5.96 @ ( minus_minus @ ( set @ B ) @ A4
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( G @ X3 )
% 11.90/5.96 = ( one_one @ A ) ) ) ) )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2298_notin__range__Some,axiom,
% 11.90/5.96 ! [A: $tType,X: option @ A] :
% 11.90/5.96 ( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
% 11.90/5.96 = ( X
% 11.90/5.96 = ( none @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2299_less__1__prod2,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( linordered_idom @ B )
% 11.90/5.96 => ! [I4: set @ A,I: A,F: A > B] :
% 11.90/5.96 ( ( finite_finite2 @ A @ I4 )
% 11.90/5.96 => ( ( member @ A @ I @ I4 )
% 11.90/5.96 => ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
% 11.90/5.96 => ( ! [I2: A] :
% 11.90/5.96 ( ( member @ A @ I2 @ I4 )
% 11.90/5.96 => ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I2 ) ) )
% 11.90/5.96 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I4 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2300_less__1__prod,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( linordered_idom @ B )
% 11.90/5.96 => ! [I4: set @ A,F: A > B] :
% 11.90/5.96 ( ( finite_finite2 @ A @ I4 )
% 11.90/5.96 => ( ( I4
% 11.90/5.96 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 => ( ! [I2: A] :
% 11.90/5.96 ( ( member @ A @ I2 @ I4 )
% 11.90/5.96 => ( ord_less @ B @ ( one_one @ B ) @ ( F @ I2 ) ) )
% 11.90/5.96 => ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I4 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2301_sum_Oreindex__nontrivial,axiom,
% 11.90/5.96 ! [C: $tType,A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [A4: set @ B,H3: B > C,G: C > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ! [X2: B,Y4: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ A4 )
% 11.90/5.96 => ( ( member @ B @ Y4 @ A4 )
% 11.90/5.96 => ( ( X2 != Y4 )
% 11.90/5.96 => ( ( ( H3 @ X2 )
% 11.90/5.96 = ( H3 @ Y4 ) )
% 11.90/5.96 => ( ( G @ ( H3 @ X2 ) )
% 11.90/5.96 = ( zero_zero @ A ) ) ) ) ) )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 11.90/5.96 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2302_prod_Oreindex__nontrivial,axiom,
% 11.90/5.96 ! [C: $tType,A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [A4: set @ B,H3: B > C,G: C > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ! [X2: B,Y4: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ A4 )
% 11.90/5.96 => ( ( member @ B @ Y4 @ A4 )
% 11.90/5.96 => ( ( X2 != Y4 )
% 11.90/5.96 => ( ( ( H3 @ X2 )
% 11.90/5.96 = ( H3 @ Y4 ) )
% 11.90/5.96 => ( ( G @ ( H3 @ X2 ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ) ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2303_sum_Osubset__diff,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [B6: set @ B,A4: set @ B,G: B > A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ B ) @ B6 @ A4 )
% 11.90/5.96 => ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
% 11.90/5.96 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2304_prod_Osame__carrier,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H3: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ C5 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
% 11.90/5.96 => ( ( G @ A5 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [B3: B] :
% 11.90/5.96 ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
% 11.90/5.96 => ( ( H3 @ B3 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ B6 ) )
% 11.90/5.96 = ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ C5 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2305_prod_Osame__carrierI,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [C5: set @ B,A4: set @ B,B6: set @ B,G: B > A,H3: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ C5 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
% 11.90/5.96 => ( ! [A5: B] :
% 11.90/5.96 ( ( member @ B @ A5 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
% 11.90/5.96 => ( ( G @ A5 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [B3: B] :
% 11.90/5.96 ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
% 11.90/5.96 => ( ( H3 @ B3 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ C5 ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ B6 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2306_prod_Omono__neutral__left,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [T6: set @ B,S2: set @ B,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ T6 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2307_prod_Omono__neutral__right,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [T6: set @ B,S2: set @ B,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ T6 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2308_prod_Omono__neutral__cong__left,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [T6: set @ B,S2: set @ B,H3: B > A,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ T6 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( H3 @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ S2 )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( H3 @ X2 ) ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2309_prod_Omono__neutral__cong__right,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [T6: set @ B,S2: set @ B,G: B > A,H3: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ T6 )
% 11.90/5.96 => ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ S2 )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( H3 @ X2 ) ) )
% 11.90/5.96 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
% 11.90/5.96 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ S2 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2310_atLeastLessThanSuc,axiom,
% 11.90/5.96 ! [M2: nat,N: nat] :
% 11.90/5.96 ( ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
% 11.90/5.96 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2311_Option_Othese__def,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( these @ A )
% 11.90/5.96 = ( ^ [A9: set @ ( option @ A )] :
% 11.90/5.96 ( image2 @ ( option @ A ) @ A @ ( the2 @ A )
% 11.90/5.96 @ ( collect @ ( option @ A )
% 11.90/5.96 @ ^ [X3: option @ A] :
% 11.90/5.96 ( ( member @ ( option @ A ) @ X3 @ A9 )
% 11.90/5.96 & ( X3
% 11.90/5.96 != ( none @ A ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2312_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2313_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2314_atMost__Suc__eq__insert__0,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_ord_atMost @ nat @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2315_lessThan__Suc__eq__insert__0,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_ord_lessThan @ nat @ ( suc @ N ) )
% 11.90/5.96 = ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2316_card__2__iff,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A] :
% 11.90/5.96 ( ( ( finite_card @ A @ S2 )
% 11.90/5.96 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.96 = ( ? [X3: A,Y3: A] :
% 11.90/5.96 ( ( S2
% 11.90/5.96 = ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.96 & ( X3 != Y3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2317_card__3__iff,axiom,
% 11.90/5.96 ! [A: $tType,S2: set @ A] :
% 11.90/5.96 ( ( ( finite_card @ A @ S2 )
% 11.90/5.96 = ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 11.90/5.96 = ( ? [X3: A,Y3: A,Z5: A] :
% 11.90/5.96 ( ( S2
% 11.90/5.96 = ( insert2 @ A @ X3 @ ( insert2 @ A @ Y3 @ ( insert2 @ A @ Z5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 11.90/5.96 & ( X3 != Y3 )
% 11.90/5.96 & ( Y3 != Z5 )
% 11.90/5.96 & ( X3 != Z5 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2318_sum__image__le,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType,C: $tType] :
% 11.90/5.96 ( ( ordere6911136660526730532id_add @ B )
% 11.90/5.96 => ! [I4: set @ C,G: A > B,F: C > A] :
% 11.90/5.96 ( ( finite_finite2 @ C @ I4 )
% 11.90/5.96 => ( ! [I2: C] :
% 11.90/5.96 ( ( member @ C @ I2 @ I4 )
% 11.90/5.96 => ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I2 ) ) ) )
% 11.90/5.96 => ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image2 @ C @ A @ F @ I4 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I4 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2319_sum_Oremove,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [A4: set @ B,X: B,G: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( member @ B @ X @ A4 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
% 11.90/5.96 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2320_sum_Oinsert__remove,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [A4: set @ B,G: B > A,X: B] :
% 11.90/5.96 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A4 ) )
% 11.90/5.96 = ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2321_atLeast1__atMost__eq__remove0,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 11.90/5.96 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2322_atLeast1__lessThan__eq__remove0,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
% 11.90/5.96 = ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2323_atLeastLessThan__nat__numeral,axiom,
% 11.90/5.96 ! [M2: nat,K: num] :
% 11.90/5.96 ( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
% 11.90/5.96 = ( insert2 @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K ) ) ) ) )
% 11.90/5.96 & ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K ) )
% 11.90/5.96 => ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K ) )
% 11.90/5.96 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2324_sum_Odelta__remove,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_add @ A )
% 11.90/5.96 => ! [S2: set @ B,A3: B,B2: B > A,C2: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B @ S2 )
% 11.90/5.96 => ( ( ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 11.90/5.96 & ( ~ ( member @ B @ A3 @ S2 )
% 11.90/5.96 => ( ( groups7311177749621191930dd_sum @ B @ A
% 11.90/5.96 @ ^ [K4: B] : ( if @ A @ ( K4 = A3 ) @ ( B2 @ K4 ) @ ( C2 @ K4 ) )
% 11.90/5.96 @ S2 )
% 11.90/5.96 = ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2325_prod__le__power,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( linordered_semidom @ A )
% 11.90/5.96 => ! [A4: set @ B,F: B > A,N: A,K: nat] :
% 11.90/5.96 ( ! [I2: B] :
% 11.90/5.96 ( ( member @ B @ I2 @ A4 )
% 11.90/5.96 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I2 ) )
% 11.90/5.96 & ( ord_less_eq @ A @ ( F @ I2 ) @ N ) ) )
% 11.90/5.96 => ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
% 11.90/5.96 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
% 11.90/5.96 => ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2326_card__Diff__insert,axiom,
% 11.90/5.96 ! [A: $tType,A3: A,A4: set @ A,B6: set @ A] :
% 11.90/5.96 ( ( member @ A @ A3 @ A4 )
% 11.90/5.96 => ( ~ ( member @ A @ A3 @ B6 )
% 11.90/5.96 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ B6 ) ) )
% 11.90/5.96 = ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2327_card__insert__le__m1,axiom,
% 11.90/5.96 ! [A: $tType,N: nat,Y: set @ A,X: A] :
% 11.90/5.96 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.96 => ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 11.90/5.96 => ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y ) ) @ N ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2328_card__Diff__singleton__if,axiom,
% 11.90/5.96 ! [A: $tType,X: A,A4: set @ A] :
% 11.90/5.96 ( ( ( member @ A @ X @ A4 )
% 11.90/5.96 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.96 = ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
% 11.90/5.96 & ( ~ ( member @ A @ X @ A4 )
% 11.90/5.96 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.96 = ( finite_card @ A @ A4 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2329_card__Diff__singleton,axiom,
% 11.90/5.96 ! [A: $tType,X: A,A4: set @ A] :
% 11.90/5.96 ( ( member @ A @ X @ A4 )
% 11.90/5.96 => ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.96 = ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2330_in__finite__psubset,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.96 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B6 ) @ ( finite_psubset @ A ) )
% 11.90/5.96 = ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.96 & ( finite_finite2 @ A @ B6 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2331_card__1__singletonE,axiom,
% 11.90/5.96 ! [A: $tType,A4: set @ A] :
% 11.90/5.96 ( ( ( finite_card @ A @ A4 )
% 11.90/5.96 = ( one_one @ nat ) )
% 11.90/5.96 => ~ ! [X2: A] :
% 11.90/5.96 ( A4
% 11.90/5.96 != ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2332_integer__of__nat__def,axiom,
% 11.90/5.96 ( code_integer_of_nat
% 11.90/5.96 = ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2333_card__UNIV__unit,axiom,
% 11.90/5.96 ( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
% 11.90/5.96 = ( one_one @ nat ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2334_card__lessThan,axiom,
% 11.90/5.96 ! [U: nat] :
% 11.90/5.96 ( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
% 11.90/5.96 = U ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2335_card__atLeastLessThan,axiom,
% 11.90/5.96 ! [L: nat,U: nat] :
% 11.90/5.96 ( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
% 11.90/5.96 = ( minus_minus @ nat @ U @ L ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2336_card__atMost,axiom,
% 11.90/5.96 ! [U: nat] :
% 11.90/5.96 ( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
% 11.90/5.96 = ( suc @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2337_Collect__const__case__prod,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType,P: $o] :
% 11.90/5.96 ( ( P
% 11.90/5.96 => ( ( collect @ ( product_prod @ A @ B )
% 11.90/5.96 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.96 @ ^ [A7: A,B5: B] : P ) )
% 11.90/5.96 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
% 11.90/5.96 & ( ~ P
% 11.90/5.96 => ( ( collect @ ( product_prod @ A @ B )
% 11.90/5.96 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.96 @ ^ [A7: A,B5: B] : P ) )
% 11.90/5.96 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2338_nat__of__integer__integer__of__nat,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
% 11.90/5.96 = N ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2339_card__atLeastAtMost,axiom,
% 11.90/5.96 ! [L: nat,U: nat] :
% 11.90/5.96 ( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
% 11.90/5.96 = ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2340_integer__of__nat_Orep__eq,axiom,
% 11.90/5.96 ! [X: nat] :
% 11.90/5.96 ( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
% 11.90/5.96 = ( semiring_1_of_nat @ int @ X ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2341_int__of__integer__integer__of__nat,axiom,
% 11.90/5.96 ! [N: nat] :
% 11.90/5.96 ( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
% 11.90/5.96 = ( semiring_1_of_nat @ int @ N ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2342_top__empty__eq2,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( top_top @ ( A > B > $o ) )
% 11.90/5.96 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2343_int__of__integer__induct,axiom,
% 11.90/5.96 ! [Y: int,P: int > $o] :
% 11.90/5.96 ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 11.90/5.96 => ( ! [X2: code_integer] : ( P @ ( code_int_of_integer @ X2 ) )
% 11.90/5.96 => ( P @ Y ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2344_int__of__integer__cases,axiom,
% 11.90/5.96 ! [Y: int] :
% 11.90/5.96 ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 11.90/5.96 => ~ ! [X2: code_integer] :
% 11.90/5.96 ( Y
% 11.90/5.96 != ( code_int_of_integer @ X2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2345_int__of__integer,axiom,
% 11.90/5.96 ! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2346_integer__of__int__inject,axiom,
% 11.90/5.96 ! [X: int,Y: int] :
% 11.90/5.96 ( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
% 11.90/5.96 => ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 11.90/5.96 => ( ( ( code_integer_of_int @ X )
% 11.90/5.96 = ( code_integer_of_int @ Y ) )
% 11.90/5.96 = ( X = Y ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2347_integer__of__int__induct,axiom,
% 11.90/5.96 ! [P: code_integer > $o,X: code_integer] :
% 11.90/5.96 ( ! [Y4: int] :
% 11.90/5.96 ( ( member @ int @ Y4 @ ( top_top @ ( set @ int ) ) )
% 11.90/5.96 => ( P @ ( code_integer_of_int @ Y4 ) ) )
% 11.90/5.96 => ( P @ X ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2348_integer__of__int__cases,axiom,
% 11.90/5.96 ! [X: code_integer] :
% 11.90/5.96 ~ ! [Y4: int] :
% 11.90/5.96 ( ( X
% 11.90/5.96 = ( code_integer_of_int @ Y4 ) )
% 11.90/5.96 => ~ ( member @ int @ Y4 @ ( top_top @ ( set @ int ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2349_subset__card__intvl__is__intvl,axiom,
% 11.90/5.96 ! [A4: set @ nat,K: nat] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ nat ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) )
% 11.90/5.96 => ( A4
% 11.90/5.96 = ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ K @ ( finite_card @ nat @ A4 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2350_integer__of__int__inverse,axiom,
% 11.90/5.96 ! [Y: int] :
% 11.90/5.96 ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
% 11.90/5.96 => ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
% 11.90/5.96 = Y ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2351_subset__eq__atLeast0__lessThan__card,axiom,
% 11.90/5.96 ! [N6: set @ nat,N: nat] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.90/5.96 => ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2352_integer__of__nat__eq__of__nat,axiom,
% 11.90/5.96 ( code_integer_of_nat
% 11.90/5.96 = ( semiring_1_of_nat @ code_integer ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2353_card__less__Suc2,axiom,
% 11.90/5.96 ! [M3: set @ nat,I: nat] :
% 11.90/5.96 ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M3 )
% 11.90/5.96 => ( ( finite_card @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [K4: nat] :
% 11.90/5.96 ( ( member @ nat @ ( suc @ K4 ) @ M3 )
% 11.90/5.96 & ( ord_less @ nat @ K4 @ I ) ) ) )
% 11.90/5.96 = ( finite_card @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [K4: nat] :
% 11.90/5.96 ( ( member @ nat @ K4 @ M3 )
% 11.90/5.96 & ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2354_card__less__Suc,axiom,
% 11.90/5.96 ! [M3: set @ nat,I: nat] :
% 11.90/5.96 ( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
% 11.90/5.96 => ( ( suc
% 11.90/5.96 @ ( finite_card @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [K4: nat] :
% 11.90/5.96 ( ( member @ nat @ ( suc @ K4 ) @ M3 )
% 11.90/5.96 & ( ord_less @ nat @ K4 @ I ) ) ) ) )
% 11.90/5.96 = ( finite_card @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [K4: nat] :
% 11.90/5.96 ( ( member @ nat @ K4 @ M3 )
% 11.90/5.96 & ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2355_card__less,axiom,
% 11.90/5.96 ! [M3: set @ nat,I: nat] :
% 11.90/5.96 ( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
% 11.90/5.96 => ( ( finite_card @ nat
% 11.90/5.96 @ ( collect @ nat
% 11.90/5.96 @ ^ [K4: nat] :
% 11.90/5.96 ( ( member @ nat @ K4 @ M3 )
% 11.90/5.96 & ( ord_less @ nat @ K4 @ ( suc @ I ) ) ) ) )
% 11.90/5.96 != ( zero_zero @ nat ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2356_integer__of__nat__0,axiom,
% 11.90/5.96 ( ( code_integer_of_nat @ ( zero_zero @ nat ) )
% 11.90/5.96 = ( zero_zero @ code_integer ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2357_integer__of__nat_Oabs__eq,axiom,
% 11.90/5.96 ( code_integer_of_nat
% 11.90/5.96 = ( ^ [X3: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X3 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2358_integer__of__nat__numeral,axiom,
% 11.90/5.96 ! [N: num] :
% 11.90/5.96 ( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
% 11.90/5.96 = ( numeral_numeral @ code_integer @ N ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2359_integer__of__nat__1,axiom,
% 11.90/5.96 ( ( code_integer_of_nat @ ( one_one @ nat ) )
% 11.90/5.96 = ( one_one @ code_integer ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2360_card__sum__le__nat__sum,axiom,
% 11.90/5.96 ! [S2: set @ nat] :
% 11.90/5.96 ( ord_less_eq @ nat
% 11.90/5.96 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.96 @ ^ [X3: nat] : X3
% 11.90/5.96 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
% 11.90/5.96 @ ( groups7311177749621191930dd_sum @ nat @ nat
% 11.90/5.96 @ ^ [X3: nat] : X3
% 11.90/5.96 @ S2 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2361_set__encode__insert,axiom,
% 11.90/5.96 ! [A4: set @ nat,N: nat] :
% 11.90/5.96 ( ( finite_finite2 @ nat @ A4 )
% 11.90/5.96 => ( ~ ( member @ nat @ N @ A4 )
% 11.90/5.96 => ( ( nat_set_encode @ ( insert2 @ nat @ N @ A4 ) )
% 11.90/5.96 = ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2362_card__greaterThanLessThan__int,axiom,
% 11.90/5.96 ! [L: int,U: int] :
% 11.90/5.96 ( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
% 11.90/5.96 = ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2363_wmax__insertI,axiom,
% 11.90/5.96 ! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.96 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak )
% 11.90/5.96 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2364_wmin__insertI,axiom,
% 11.90/5.96 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.96 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak )
% 11.90/5.96 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2365_smin__insertI,axiom,
% 11.90/5.96 ! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.96 ( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict )
% 11.90/5.96 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2366_prod_Oinsert_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [I4: set @ B,P5: B > A,I: B] :
% 11.90/5.96 ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ I4 )
% 11.90/5.96 & ( ( P5 @ X3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 => ( ( ( member @ B @ I @ I4 )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 11.90/5.96 = ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) )
% 11.90/5.96 & ( ~ ( member @ B @ I @ I4 )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 11.90/5.96 = ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2367_greaterThanLessThan__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ord @ A )
% 11.90/5.96 => ! [I: A,L: A,U: A] :
% 11.90/5.96 ( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 11.90/5.96 = ( ( ord_less @ A @ L @ I )
% 11.90/5.96 & ( ord_less @ A @ I @ U ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2368_finite__greaterThanLessThan__int,axiom,
% 11.90/5.96 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2369_greaterThanLessThan__empty__iff2,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( bot_bot @ ( set @ A ) )
% 11.90/5.96 = ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
% 11.90/5.96 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2370_greaterThanLessThan__empty__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.96 = ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2371_greaterThanLessThan__empty,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( order @ A )
% 11.90/5.96 => ! [L: A,K: A] :
% 11.90/5.96 ( ( ord_less_eq @ A @ L @ K )
% 11.90/5.96 => ( ( set_or5935395276787703475ssThan @ A @ K @ L )
% 11.90/5.96 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2372_infinite__Ioo__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
% 11.90/5.96 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2373_image__uminus__greaterThanLessThan,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( ordered_ab_group_add @ A )
% 11.90/5.96 => ! [X: A,Y: A] :
% 11.90/5.96 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
% 11.90/5.96 = ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2374_prod_Oempty_H,axiom,
% 11.90/5.96 ! [B: $tType,A: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [P5: B > A] :
% 11.90/5.96 ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
% 11.90/5.96 = ( one_one @ A ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2375_smin__emptyI,axiom,
% 11.90/5.96 ! [X10: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.96 ( ( X10
% 11.90/5.96 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 11.90/5.96 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2376_wmax__emptyI,axiom,
% 11.90/5.96 ! [X10: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.96 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X10 )
% 11.90/5.96 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X10 ) @ fun_max_weak ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2377_wmin__emptyI,axiom,
% 11.90/5.96 ! [X10: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2378_prod_Onon__neutral_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [G: B > A,I4: set @ B] :
% 11.90/5.96 ( ( groups1962203154675924110t_prod @ B @ A @ G
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ I4 )
% 11.90/5.96 & ( ( G @ X3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 = ( groups1962203154675924110t_prod @ B @ A @ G @ I4 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2379_infinite__Ioo,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2380_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
% 11.90/5.96 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.96 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2381_prod__decode__aux_Ocases,axiom,
% 11.90/5.96 ! [X: product_prod @ nat @ nat] :
% 11.90/5.96 ~ ! [K5: nat,M5: nat] :
% 11.90/5.96 ( X
% 11.90/5.96 != ( product_Pair @ nat @ nat @ K5 @ M5 ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2382_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 11.90/5.96 ! [L: int,U: int] :
% 11.90/5.96 ( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 11.90/5.96 = ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2383_prod_Omono__neutral__left_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,T6: set @ B,G: B > A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
% 11.90/5.96 = ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2384_prod_Omono__neutral__right_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,T6: set @ B,G: B > A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 11.90/5.96 = ( groups1962203154675924110t_prod @ B @ A @ G @ S2 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2385_prod_Omono__neutral__cong__left_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,T6: set @ B,H3: B > A,G: B > A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [I2: B] :
% 11.90/5.96 ( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( H3 @ I2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ S2 )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( H3 @ X2 ) ) )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
% 11.90/5.96 = ( groups1962203154675924110t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2386_prod_Omono__neutral__cong__right_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [S2: set @ B,T6: set @ B,G: B > A,H3: B > A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( one_one @ A ) ) )
% 11.90/5.96 => ( ! [X2: B] :
% 11.90/5.96 ( ( member @ B @ X2 @ S2 )
% 11.90/5.96 => ( ( G @ X2 )
% 11.90/5.96 = ( H3 @ X2 ) ) )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
% 11.90/5.96 = ( groups1962203154675924110t_prod @ B @ A @ H3 @ S2 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2387_prod_Odistrib_H,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ! [I4: set @ B,G: B > A,H3: B > A] :
% 11.90/5.96 ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ I4 )
% 11.90/5.96 & ( ( G @ X3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 => ( ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ I4 )
% 11.90/5.96 & ( ( H3 @ X3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A
% 11.90/5.96 @ ^ [I3: B] : ( times_times @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
% 11.90/5.96 @ I4 )
% 11.90/5.96 = ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I4 ) @ ( groups1962203154675924110t_prod @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2388_prod_OG__def,axiom,
% 11.90/5.96 ! [A: $tType,B: $tType] :
% 11.90/5.96 ( ( comm_monoid_mult @ A )
% 11.90/5.96 => ( ( groups1962203154675924110t_prod @ B @ A )
% 11.90/5.96 = ( ^ [P6: B > A,I5: set @ B] :
% 11.90/5.96 ( if @ A
% 11.90/5.96 @ ( finite_finite2 @ B
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ I5 )
% 11.90/5.96 & ( ( P6 @ X3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 @ ( groups7121269368397514597t_prod @ B @ A @ P6
% 11.90/5.96 @ ( collect @ B
% 11.90/5.96 @ ^ [X3: B] :
% 11.90/5.96 ( ( member @ B @ X3 @ I5 )
% 11.90/5.96 & ( ( P6 @ X3 )
% 11.90/5.96 != ( one_one @ A ) ) ) ) )
% 11.90/5.96 @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2389_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.96 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.96 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2390_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( dense_linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.96 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.96 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.96 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.96 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2391_atLeastAtMost__diff__ends,axiom,
% 11.90/5.96 ! [A: $tType] :
% 11.90/5.96 ( ( linorder @ A )
% 11.90/5.96 => ! [A3: A,B2: A] :
% 11.90/5.96 ( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.96 = ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2392_smax__insertI,axiom,
% 11.90/5.96 ! [Y: product_prod @ nat @ nat,Y9: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X10: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.96 ( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y9 )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
% 11.90/5.96 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X10 @ Y9 ) @ fun_max_strict )
% 11.90/5.96 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ X10 ) @ Y9 ) @ fun_max_strict ) ) ) ) ).
% 11.90/5.96
% 11.90/5.96 thf(fact_2393_set__decode__plus__power__2,axiom,
% 11.90/5.96 ! [N: nat,Z2: nat] :
% 11.90/5.96 ( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
% 11.90/5.96 => ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
% 11.90/5.96 = ( insert2 @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2394_prod__decode__aux_Osimps,axiom,
% 11.90/5.97 ( nat_prod_decode_aux
% 11.90/5.97 = ( ^ [K4: nat,M: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M @ K4 ) @ ( product_Pair @ nat @ nat @ M @ ( minus_minus @ nat @ K4 @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K4 ) @ ( minus_minus @ nat @ M @ ( suc @ K4 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2395_prod__decode__aux_Oelims,axiom,
% 11.90/5.97 ! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
% 11.90/5.97 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 11.90/5.97 = Y )
% 11.90/5.97 => ( ( ( ord_less_eq @ nat @ Xa @ X )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
% 11.90/5.97 & ( ~ ( ord_less_eq @ nat @ Xa @ X )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2396_min__rpair__set,axiom,
% 11.90/5.97 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2397_triangle__Suc,axiom,
% 11.90/5.97 ! [N: nat] :
% 11.90/5.97 ( ( nat_triangle @ ( suc @ N ) )
% 11.90/5.97 = ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2398_finite__greaterThanLessThan,axiom,
% 11.90/5.97 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2399_finite__greaterThanLessThan__integer,axiom,
% 11.90/5.97 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2400_card__greaterThanLessThan,axiom,
% 11.90/5.97 ! [L: nat,U: nat] :
% 11.90/5.97 ( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
% 11.90/5.97 = ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2401_max__rpair__set,axiom,
% 11.90/5.97 fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2402_atLeastSucLessThan__greaterThanLessThan,axiom,
% 11.90/5.97 ! [L: nat,U: nat] :
% 11.90/5.97 ( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
% 11.90/5.97 = ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2403_smax__emptyI,axiom,
% 11.90/5.97 ! [Y9: set @ ( product_prod @ nat @ nat )] :
% 11.90/5.97 ( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y9 )
% 11.90/5.97 => ( ( Y9
% 11.90/5.97 != ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
% 11.90/5.97 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y9 ) @ fun_max_strict ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2404_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
% 11.90/5.97 ! [L: code_integer,U: code_integer] :
% 11.90/5.97 ( ( set_or7035219750837199246ssThan @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
% 11.90/5.97 = ( set_or5935395276787703475ssThan @ code_integer @ L @ U ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2405_prod__decode__aux_Opelims,axiom,
% 11.90/5.97 ! [X: nat,Xa: nat,Y: product_prod @ nat @ nat] :
% 11.90/5.97 ( ( ( nat_prod_decode_aux @ X @ Xa )
% 11.90/5.97 = Y )
% 11.90/5.97 => ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
% 11.90/5.97 => ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
% 11.90/5.97 & ( ~ ( ord_less_eq @ nat @ Xa @ X )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
% 11.90/5.97 => ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2406_total__on__singleton,axiom,
% 11.90/5.97 ! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2407_prod__encode__prod__decode__aux,axiom,
% 11.90/5.97 ! [K: nat,M2: nat] :
% 11.90/5.97 ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
% 11.90/5.97 = ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2408_sum_Oinsert_H,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comm_monoid_add @ A )
% 11.90/5.97 => ! [I4: set @ B,P5: B > A,I: B] :
% 11.90/5.97 ( ( finite_finite2 @ B
% 11.90/5.97 @ ( collect @ B
% 11.90/5.97 @ ^ [X3: B] :
% 11.90/5.97 ( ( member @ B @ X3 @ I4 )
% 11.90/5.97 & ( ( P5 @ X3 )
% 11.90/5.97 != ( zero_zero @ A ) ) ) ) )
% 11.90/5.97 => ( ( ( member @ B @ I @ I4 )
% 11.90/5.97 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 11.90/5.97 = ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I4 ) ) )
% 11.90/5.97 & ( ~ ( member @ B @ I @ I4 )
% 11.90/5.97 => ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert2 @ B @ I @ I4 ) )
% 11.90/5.97 = ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I4 ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2409_max__ext_Omax__extI,axiom,
% 11.90/5.97 ! [A: $tType,X10: set @ A,Y9: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( finite_finite2 @ A @ X10 )
% 11.90/5.97 => ( ( finite_finite2 @ A @ Y9 )
% 11.90/5.97 => ( ( Y9
% 11.90/5.97 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 => ( ! [X2: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ X10 )
% 11.90/5.97 => ? [Xa2: A] :
% 11.90/5.97 ( ( member @ A @ Xa2 @ Y9 )
% 11.90/5.97 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R5 ) ) )
% 11.90/5.97 => ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X10 @ Y9 ) @ ( max_ext @ A @ R5 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2410_total__on__def,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( total_on @ A )
% 11.90/5.97 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ! [X3: A] :
% 11.90/5.97 ( ( member @ A @ X3 @ A9 )
% 11.90/5.97 => ! [Y3: A] :
% 11.90/5.97 ( ( member @ A @ Y3 @ A9 )
% 11.90/5.97 => ( ( X3 != Y3 )
% 11.90/5.97 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 11.90/5.97 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2411_total__onI,axiom,
% 11.90/5.97 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ! [X2: A,Y4: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ A4 )
% 11.90/5.97 => ( ( member @ A @ Y4 @ A4 )
% 11.90/5.97 => ( ( X2 != Y4 )
% 11.90/5.97 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 11.90/5.97 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R ) ) ) ) )
% 11.90/5.97 => ( total_on @ A @ A4 @ R ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2412_sum_Odistrib__triv_H,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comm_monoid_add @ A )
% 11.90/5.97 => ! [I4: set @ B,G: B > A,H3: B > A] :
% 11.90/5.97 ( ( finite_finite2 @ B @ I4 )
% 11.90/5.97 => ( ( groups1027152243600224163dd_sum @ B @ A
% 11.90/5.97 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
% 11.90/5.97 @ I4 )
% 11.90/5.97 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2413_le__prod__encode__1,axiom,
% 11.90/5.97 ! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2414_le__prod__encode__2,axiom,
% 11.90/5.97 ! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2415_sum_Odistrib_H,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comm_monoid_add @ A )
% 11.90/5.97 => ! [I4: set @ B,G: B > A,H3: B > A] :
% 11.90/5.97 ( ( finite_finite2 @ B
% 11.90/5.97 @ ( collect @ B
% 11.90/5.97 @ ^ [X3: B] :
% 11.90/5.97 ( ( member @ B @ X3 @ I4 )
% 11.90/5.97 & ( ( G @ X3 )
% 11.90/5.97 != ( zero_zero @ A ) ) ) ) )
% 11.90/5.97 => ( ( finite_finite2 @ B
% 11.90/5.97 @ ( collect @ B
% 11.90/5.97 @ ^ [X3: B] :
% 11.90/5.97 ( ( member @ B @ X3 @ I4 )
% 11.90/5.97 & ( ( H3 @ X3 )
% 11.90/5.97 != ( zero_zero @ A ) ) ) ) )
% 11.90/5.97 => ( ( groups1027152243600224163dd_sum @ B @ A
% 11.90/5.97 @ ^ [I3: B] : ( plus_plus @ A @ ( G @ I3 ) @ ( H3 @ I3 ) )
% 11.90/5.97 @ I4 )
% 11.90/5.97 = ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I4 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H3 @ I4 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2416_prod__encode__def,axiom,
% 11.90/5.97 ( nat_prod_encode
% 11.90/5.97 = ( product_case_prod @ nat @ nat @ nat
% 11.90/5.97 @ ^ [M: nat,N5: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M @ N5 ) ) @ M ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2417_max__ext_Ocases,axiom,
% 11.90/5.97 ! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
% 11.90/5.97 => ~ ( ( finite_finite2 @ A @ A1 )
% 11.90/5.97 => ( ( finite_finite2 @ A @ A22 )
% 11.90/5.97 => ( ( A22
% 11.90/5.97 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 => ~ ! [X7: A] :
% 11.90/5.97 ( ( member @ A @ X7 @ A1 )
% 11.90/5.97 => ? [Xa3: A] :
% 11.90/5.97 ( ( member @ A @ Xa3 @ A22 )
% 11.90/5.97 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R5 ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2418_max__ext_Osimps,axiom,
% 11.90/5.97 ! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
% 11.90/5.97 = ( ( finite_finite2 @ A @ A1 )
% 11.90/5.97 & ( finite_finite2 @ A @ A22 )
% 11.90/5.97 & ( A22
% 11.90/5.97 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 & ! [X3: A] :
% 11.90/5.97 ( ( member @ A @ X3 @ A1 )
% 11.90/5.97 => ? [Y3: A] :
% 11.90/5.97 ( ( member @ A @ Y3 @ A22 )
% 11.90/5.97 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2419_max__extp__eq,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( max_extp @ A )
% 11.90/5.97 = ( ^ [R3: A > A > $o,X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2420_max__extp__max__ext__eq,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( max_extp @ A
% 11.90/5.97 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) )
% 11.90/5.97 = ( ^ [X3: set @ A,Y3: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X3 @ Y3 ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2421_one__div__numeral,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.97 => ! [N: num] :
% 11.90/5.97 ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.97 = ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2422_sqr_Osimps_I3_J,axiom,
% 11.90/5.97 ! [N: num] :
% 11.90/5.97 ( ( sqr @ ( bit1 @ N ) )
% 11.90/5.97 = ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2423_one__mod__numeral,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( unique1627219031080169319umeral @ A )
% 11.90/5.97 => ! [N: num] :
% 11.90/5.97 ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
% 11.90/5.97 = ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2424_card__UNION,axiom,
% 11.90/5.97 ! [A: $tType,A4: set @ ( set @ A )] :
% 11.90/5.97 ( ( finite_finite2 @ ( set @ A ) @ A4 )
% 11.90/5.97 => ( ! [X2: set @ A] :
% 11.90/5.97 ( ( member @ ( set @ A ) @ X2 @ A4 )
% 11.90/5.97 => ( finite_finite2 @ A @ X2 ) )
% 11.90/5.97 => ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
% 11.90/5.97 = ( nat2
% 11.90/5.97 @ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
% 11.90/5.97 @ ^ [I5: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I5 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I5 ) ) ) )
% 11.90/5.97 @ ( collect @ ( set @ ( set @ A ) )
% 11.90/5.97 @ ^ [I5: set @ ( set @ A )] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I5 @ A4 )
% 11.90/5.97 & ( I5
% 11.90/5.97 != ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2425_Sup__atMost,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [Y: A] :
% 11.90/5.97 ( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
% 11.90/5.97 = Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2426_Sup__lessThan,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 & ( dense_linorder @ A ) )
% 11.90/5.97 => ! [Y: A] :
% 11.90/5.97 ( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
% 11.90/5.97 = Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2427_fst__apsnd,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
% 11.90/5.97 ( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
% 11.90/5.97 = ( product_fst @ A @ C @ X ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2428_fst__comp__apsnd,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 11.90/5.97 ( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 11.90/5.97 = ( product_fst @ A @ B ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2429_apsnd__eq__conv,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
% 11.90/5.97 ( ( ( product_apsnd @ C @ B @ A @ F @ X )
% 11.90/5.97 = ( product_apsnd @ C @ B @ A @ G @ X ) )
% 11.90/5.97 = ( ( F @ ( product_snd @ A @ C @ X ) )
% 11.90/5.97 = ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2430_snd__apsnd,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
% 11.90/5.97 ( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
% 11.90/5.97 = ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2431_snd__comp__apsnd,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,F: B > C] :
% 11.90/5.97 ( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
% 11.90/5.97 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2432_fst__divmod__integer,axiom,
% 11.90/5.97 ! [K: code_integer,L: code_integer] :
% 11.90/5.97 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 11.90/5.97 = ( divide_divide @ code_integer @ K @ L ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2433_snd__divmod__integer,axiom,
% 11.90/5.97 ! [K: code_integer,L: code_integer] :
% 11.90/5.97 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
% 11.90/5.97 = ( modulo_modulo @ code_integer @ K @ L ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2434_Sup__atLeastAtMost,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ X @ Y )
% 11.90/5.97 => ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.97 = Y ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2435_Inf__atLeastAtMost,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ X @ Y )
% 11.90/5.97 => ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
% 11.90/5.97 = X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2436_Sup__atLeastLessThan,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 & ( dense_linorder @ A ) )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( ord_less @ A @ X @ Y )
% 11.90/5.97 => ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
% 11.90/5.97 = Y ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2437_Inf__atLeastLessThan,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( ord_less @ A @ X @ Y )
% 11.90/5.97 => ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
% 11.90/5.97 = X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2438_prod_Ocollapse,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 11.90/5.97 ( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
% 11.90/5.97 = Prod ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2439_Inf__atMost,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [X: A] :
% 11.90/5.97 ( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
% 11.90/5.97 = ( bot_bot @ A ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2440_Sup__greaterThanLessThan,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 & ( dense_linorder @ A ) )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( ord_less @ A @ X @ Y )
% 11.90/5.97 => ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
% 11.90/5.97 = Y ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2441_img__fst,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 11.90/5.97 => ( member @ A @ A3 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2442_Inf__greaterThanLessThan,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 & ( dense_linorder @ A ) )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( ord_less @ A @ X @ Y )
% 11.90/5.97 => ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
% 11.90/5.97 = X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2443_img__snd,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ S2 )
% 11.90/5.97 => ( member @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2444_fst__divmod__abs,axiom,
% 11.90/5.97 ! [K: code_integer,L: code_integer] :
% 11.90/5.97 ( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 11.90/5.97 = ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2445_snd__divmod__abs,axiom,
% 11.90/5.97 ! [K: code_integer,L: code_integer] :
% 11.90/5.97 ( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
% 11.90/5.97 = ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2446_Inf__atMostLessThan,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [X: A] :
% 11.90/5.97 ( ( ord_less @ A @ ( top_top @ A ) @ X )
% 11.90/5.97 => ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
% 11.90/5.97 = ( bot_bot @ A ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2447_split__comp__eq,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
% 11.90/5.97 ( ( ^ [U2: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
% 11.90/5.97 = ( product_case_prod @ D @ B @ C
% 11.90/5.97 @ ^ [X3: D] : ( F @ ( G @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2448_case__prod__beta_H,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_case_prod @ A @ B @ C )
% 11.90/5.97 = ( ^ [F2: A > B > C,X3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X3 ) @ ( product_snd @ A @ B @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2449_case__prod__unfold,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_case_prod @ A @ B @ C )
% 11.90/5.97 = ( ^ [C3: A > B > C,P6: product_prod @ A @ B] : ( C3 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2450_Product__Type_OCollect__case__prodD,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: A > B > $o] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) )
% 11.90/5.97 => ( A4 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2451_case__prod__beta,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType] :
% 11.90/5.97 ( ( product_case_prod @ B @ C @ A )
% 11.90/5.97 = ( ^ [F2: B > C > A,P6: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2452_split__beta,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_case_prod @ A @ B @ C )
% 11.90/5.97 = ( ^ [F2: A > B > C,Prod2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2453_effect__heapI,axiom,
% 11.90/5.97 ! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A] :
% 11.90/5.97 ( ( N
% 11.90/5.97 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( H4
% 11.90/5.97 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( R
% 11.90/5.97 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
% 11.90/5.97 => ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H3 @ H4 @ R @ N ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2454_effect__heapE,axiom,
% 11.90/5.97 ! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.97 ( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H3 @ H4 @ R @ N )
% 11.90/5.97 => ~ ( ( H4
% 11.90/5.97 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( N
% 11.90/5.97 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( R
% 11.90/5.97 != ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2455_effect__guardI,axiom,
% 11.90/5.97 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
% 11.90/5.97 ( ( P @ H3 )
% 11.90/5.97 => ( ( H4
% 11.90/5.97 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( N
% 11.90/5.97 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( R
% 11.90/5.97 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
% 11.90/5.97 => ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2456_effect__guardE,axiom,
% 11.90/5.97 ! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.97 ( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H3 @ H4 @ R @ N )
% 11.90/5.97 => ~ ( ( H4
% 11.90/5.97 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( N
% 11.90/5.97 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) ) )
% 11.90/5.97 => ( ( R
% 11.90/5.97 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H3 ) ) )
% 11.90/5.97 => ~ ( P @ H3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2457_effectE,axiom,
% 11.90/5.97 ! [A: $tType,C2: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/5.97 ( ( heap_Time_effect @ A @ C2 @ H3 @ H4 @ R @ N )
% 11.90/5.97 => ~ ( ( R
% 11.90/5.97 = ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) )
% 11.90/5.97 => ( ( H4
% 11.90/5.97 = ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) ) )
% 11.90/5.97 => ( ( N
% 11.90/5.97 = ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ C2 @ H3 ) ) ) ) )
% 11.90/5.97 => ~ ( heap_Time_success @ A @ C2 @ H3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2458_prod__eq__iff,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( ^ [Y5: product_prod @ A @ B,Z4: product_prod @ A @ B] : ( Y5 = Z4 ) )
% 11.90/5.97 = ( ^ [S4: product_prod @ A @ B,T3: product_prod @ A @ B] :
% 11.90/5.97 ( ( ( product_fst @ A @ B @ S4 )
% 11.90/5.97 = ( product_fst @ A @ B @ T3 ) )
% 11.90/5.97 & ( ( product_snd @ A @ B @ S4 )
% 11.90/5.97 = ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2459_prod__eqI,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,Q3: product_prod @ A @ B] :
% 11.90/5.97 ( ( ( product_fst @ A @ B @ P5 )
% 11.90/5.97 = ( product_fst @ A @ B @ Q3 ) )
% 11.90/5.97 => ( ( ( product_snd @ A @ B @ P5 )
% 11.90/5.97 = ( product_snd @ A @ B @ Q3 ) )
% 11.90/5.97 => ( P5 = Q3 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2460_prod_Oexpand,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
% 11.90/5.97 ( ( ( ( product_fst @ A @ B @ Prod )
% 11.90/5.97 = ( product_fst @ A @ B @ Prod3 ) )
% 11.90/5.97 & ( ( product_snd @ A @ B @ Prod )
% 11.90/5.97 = ( product_snd @ A @ B @ Prod3 ) ) )
% 11.90/5.97 => ( Prod = Prod3 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2461_All__prod__contract,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P: A > B > $o] :
% 11.90/5.97 ( ( ! [A7: A,X8: B] : ( P @ A7 @ X8 ) )
% 11.90/5.97 = ( ! [Z5: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z5 ) @ ( product_snd @ A @ B @ Z5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2462_Ex__prod__contract,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P: A > B > $o] :
% 11.90/5.97 ( ( ? [A7: A,X8: B] : ( P @ A7 @ X8 ) )
% 11.90/5.97 = ( ? [Z5: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z5 ) @ ( product_snd @ A @ B @ Z5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2463_prod_Osplit__sel,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 11.90/5.97 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 11.90/5.97 = ( ( Prod
% 11.90/5.97 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 11.90/5.97 => ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2464_prod_Osplit__sel__asm,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
% 11.90/5.97 ( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
% 11.90/5.97 = ( ~ ( ( Prod
% 11.90/5.97 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
% 11.90/5.97 & ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2465_surjective__pairing,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
% 11.90/5.97 ( T2
% 11.90/5.97 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2466_prod_Oexhaust__sel,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
% 11.90/5.97 ( Prod
% 11.90/5.97 = ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2467_exI__realizer,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P: A > B > $o,Y: A,X: B] :
% 11.90/5.97 ( ( P @ Y @ X )
% 11.90/5.97 => ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2468_conjI__realizer,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,P: A > $o,P5: A,Q: B > $o,Q3: B] :
% 11.90/5.97 ( ( P @ P5 )
% 11.90/5.97 => ( ( Q @ Q3 )
% 11.90/5.97 => ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q3 ) ) )
% 11.90/5.97 & ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P5 @ Q3 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2469_execute__bind__success,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H3: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
% 11.90/5.97 ( ( heap_Time_success @ A @ F @ H3 )
% 11.90/5.97 => ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H3 )
% 11.90/5.97 = ( heap_Time_timeFrame @ B @ ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) @ ( heap_Time_execute @ B @ ( G @ ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) @ ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( the2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( heap_Time_execute @ A @ F @ H3 ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2470_sndI,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 11.90/5.97 ( ( X
% 11.90/5.97 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 11.90/5.97 => ( ( product_snd @ A @ B @ X )
% 11.90/5.97 = Z2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2471_sndE,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: B > $o] :
% 11.90/5.97 ( ( X
% 11.90/5.97 = ( product_Pair @ A @ B @ A3 @ B2 ) )
% 11.90/5.97 => ( ( P @ ( product_snd @ A @ B @ X ) )
% 11.90/5.97 => ( P @ B2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2472_snd__eqD,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: B,Y: A,A3: A] :
% 11.90/5.97 ( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 11.90/5.97 = A3 )
% 11.90/5.97 => ( Y = A3 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2473_snd__conv,axiom,
% 11.90/5.97 ! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
% 11.90/5.97 ( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
% 11.90/5.97 = X22 ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2474_Some__Inf,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [A4: set @ A] :
% 11.90/5.97 ( ( some @ A @ ( complete_Inf_Inf @ A @ A4 ) )
% 11.90/5.97 = ( complete_Inf_Inf @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2475_fstI,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z2: B] :
% 11.90/5.97 ( ( X
% 11.90/5.97 = ( product_Pair @ A @ B @ Y @ Z2 ) )
% 11.90/5.97 => ( ( product_fst @ A @ B @ X )
% 11.90/5.97 = Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2476_fstE,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A3: A,B2: B,P: A > $o] :
% 11.90/5.97 ( ( X
% 11.90/5.97 = ( product_Pair @ A @ B @ A3 @ B2 ) )
% 11.90/5.97 => ( ( P @ ( product_fst @ A @ B @ X ) )
% 11.90/5.97 => ( P @ A3 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2477_fst__eqD,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: A,Y: B,A3: A] :
% 11.90/5.97 ( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 11.90/5.97 = A3 )
% 11.90/5.97 => ( X = A3 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2478_fst__conv,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X1: A,X22: B] :
% 11.90/5.97 ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
% 11.90/5.97 = X1 ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2479_snd__fst__flip,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( product_snd @ B @ A )
% 11.90/5.97 = ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
% 11.90/5.97 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2480_fst__snd__flip,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_fst @ A @ B )
% 11.90/5.97 = ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 11.90/5.97 @ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2481_finite__UNION__then__finite,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,B6: B > ( set @ A ),A4: set @ B,A3: B] :
% 11.90/5.97 ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A4 ) ) )
% 11.90/5.97 => ( ( member @ B @ A3 @ A4 )
% 11.90/5.97 => ( finite_finite2 @ A @ ( B6 @ A3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2482_case__prod__comp,axiom,
% 11.90/5.97 ! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
% 11.90/5.97 ( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
% 11.90/5.97 = ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2483_fn__snd__conv,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,F: B > C] :
% 11.90/5.97 ( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_snd @ A @ B @ X3 ) ) )
% 11.90/5.97 = ( product_case_prod @ A @ B @ C
% 11.90/5.97 @ ^ [Uu: A] : F ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2484_snd__def,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_snd @ A @ B )
% 11.90/5.97 = ( product_case_prod @ A @ B @ B
% 11.90/5.97 @ ^ [X12: A,X24: B] : X24 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2485_fn__fst__conv,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 11.90/5.97 ( ( ^ [X3: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X3 ) ) )
% 11.90/5.97 = ( product_case_prod @ A @ B @ C
% 11.90/5.97 @ ^ [A7: A,Uu: B] : ( F @ A7 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2486_fst__def,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_fst @ A @ B )
% 11.90/5.97 = ( product_case_prod @ A @ B @ A
% 11.90/5.97 @ ^ [X12: A,X24: B] : X12 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2487_Collect__split__mono__strong,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X10: set @ A,A4: set @ ( product_prod @ A @ B ),Y9: set @ B,P: A > B > $o,Q: A > B > $o] :
% 11.90/5.97 ( ( X10
% 11.90/5.97 = ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) )
% 11.90/5.97 => ( ( Y9
% 11.90/5.97 = ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) )
% 11.90/5.97 => ( ! [X2: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ X10 )
% 11.90/5.97 => ! [Xa3: B] :
% 11.90/5.97 ( ( member @ B @ Xa3 @ Y9 )
% 11.90/5.97 => ( ( P @ X2 @ Xa3 )
% 11.90/5.97 => ( Q @ X2 @ Xa3 ) ) ) )
% 11.90/5.97 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
% 11.90/5.97 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2488_Some__Sup,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [A4: set @ A] :
% 11.90/5.97 ( ( A4
% 11.90/5.97 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 => ( ( some @ A @ ( complete_Sup_Sup @ A @ A4 ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2489_Inf__option__def,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ( ( complete_Inf_Inf @ ( option @ A ) )
% 11.90/5.97 = ( ^ [A9: set @ ( option @ A )] : ( if @ ( option @ A ) @ ( member @ ( option @ A ) @ ( none @ A ) @ A9 ) @ ( none @ A ) @ ( some @ A @ ( complete_Inf_Inf @ A @ ( these @ A @ A9 ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2490_Some__INF,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [F: B > A,A4: set @ B] :
% 11.90/5.97 ( ( some @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A4 ) ) )
% 11.90/5.97 = ( complete_Inf_Inf @ ( option @ A )
% 11.90/5.97 @ ( image2 @ B @ ( option @ A )
% 11.90/5.97 @ ^ [X3: B] : ( some @ A @ ( F @ X3 ) )
% 11.90/5.97 @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2491_INF__image,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,C: $tType] :
% 11.90/5.97 ( ( complete_Inf @ A )
% 11.90/5.97 => ! [G: B > A,F: C > B,A4: set @ C] :
% 11.90/5.97 ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 11.90/5.97 = ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2492_SUP__image,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,C: $tType] :
% 11.90/5.97 ( ( complete_Sup @ A )
% 11.90/5.97 => ! [G: B > A,F: C > B,A4: set @ C] :
% 11.90/5.97 ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A4 ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2493_in__snd__imageE,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,Y: A,S2: set @ ( product_prod @ B @ A )] :
% 11.90/5.97 ( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) )
% 11.90/5.97 => ~ ! [X2: B] :
% 11.90/5.97 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ S2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2494_in__fst__imageE,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: A,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ( member @ A @ X @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
% 11.90/5.97 => ~ ! [Y4: B] :
% 11.90/5.97 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2495_Some__SUP,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ B )
% 11.90/5.97 => ! [A4: set @ A,F: A > B] :
% 11.90/5.97 ( ( A4
% 11.90/5.97 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 => ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A4 ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( option @ B )
% 11.90/5.97 @ ( image2 @ A @ ( option @ B )
% 11.90/5.97 @ ^ [X3: A] : ( some @ B @ ( F @ X3 ) )
% 11.90/5.97 @ A4 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2496_finite__range__prod,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,F: B > ( product_prod @ A @ C )] :
% 11.90/5.97 ( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
% 11.90/5.97 => ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
% 11.90/5.97 => ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2497_UN__atMost__UNIV,axiom,
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 11.90/5.97 = ( top_top @ ( set @ nat ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2498_UN__lessThan__UNIV,axiom,
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 11.90/5.97 = ( top_top @ ( set @ nat ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2499_Sup__option__def,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ( ( complete_Sup_Sup @ ( option @ A ) )
% 11.90/5.97 = ( ^ [A9: set @ ( option @ A )] :
% 11.90/5.97 ( if @ ( option @ A )
% 11.90/5.97 @ ( ( A9
% 11.90/5.97 = ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 11.90/5.97 | ( A9
% 11.90/5.97 = ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) )
% 11.90/5.97 @ ( none @ A )
% 11.90/5.97 @ ( some @ A @ ( complete_Sup_Sup @ A @ ( these @ A @ A9 ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2500_snd__image__mp,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ A,X: B,Y: A] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) @ B6 )
% 11.90/5.97 => ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 )
% 11.90/5.97 => ( member @ A @ Y @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2501_fst__image__mp,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B6: set @ A,X: A,Y: B] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 ) @ B6 )
% 11.90/5.97 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ A4 )
% 11.90/5.97 => ( member @ A @ X @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2502_prod_OUnion__comp,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comm_monoid_mult @ A )
% 11.90/5.97 => ! [B6: set @ ( set @ B ),G: B > A] :
% 11.90/5.97 ( ! [X2: set @ B] :
% 11.90/5.97 ( ( member @ ( set @ B ) @ X2 @ B6 )
% 11.90/5.97 => ( finite_finite2 @ B @ X2 ) )
% 11.90/5.97 => ( ! [A14: set @ B] :
% 11.90/5.97 ( ( member @ ( set @ B ) @ A14 @ B6 )
% 11.90/5.97 => ! [A25: set @ B] :
% 11.90/5.97 ( ( member @ ( set @ B ) @ A25 @ B6 )
% 11.90/5.97 => ( ( A14 != A25 )
% 11.90/5.97 => ! [X2: B] :
% 11.90/5.97 ( ( member @ B @ X2 @ A14 )
% 11.90/5.97 => ( ( member @ B @ X2 @ A25 )
% 11.90/5.97 => ( ( G @ X2 )
% 11.90/5.97 = ( one_one @ A ) ) ) ) ) ) )
% 11.90/5.97 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
% 11.90/5.97 = ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B6 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2503_UN__UN__finite__eq,axiom,
% 11.90/5.97 ! [A: $tType,A4: nat > ( set @ A )] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ A )
% 11.90/5.97 @ ( image2 @ nat @ ( set @ A )
% 11.90/5.97 @ ^ [N5: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
% 11.90/5.97 @ ( top_top @ ( set @ nat ) ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2504_UN__le__add__shift,axiom,
% 11.90/5.97 ! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ A )
% 11.90/5.97 @ ( image2 @ nat @ ( set @ A )
% 11.90/5.97 @ ^ [I3: nat] : ( M3 @ ( plus_plus @ nat @ I3 @ K ) )
% 11.90/5.97 @ ( set_ord_atMost @ nat @ N ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2505_UN__le__add__shift__strict,axiom,
% 11.90/5.97 ! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ A )
% 11.90/5.97 @ ( image2 @ nat @ ( set @ A )
% 11.90/5.97 @ ^ [I3: nat] : ( M3 @ ( plus_plus @ nat @ I3 @ K ) )
% 11.90/5.97 @ ( set_ord_lessThan @ nat @ N ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2506_rat__sgn__code,axiom,
% 11.90/5.97 ! [P5: rat] :
% 11.90/5.97 ( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
% 11.90/5.97 = ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2507_bezw__non__0,axiom,
% 11.90/5.97 ! [Y: nat,X: nat] :
% 11.90/5.97 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
% 11.90/5.97 => ( ( bezw @ X @ Y )
% 11.90/5.97 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2508_UN__finite__subset,axiom,
% 11.90/5.97 ! [A: $tType,A4: nat > ( set @ A ),C5: set @ A] :
% 11.90/5.97 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
% 11.90/5.97 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2509_UN__finite2__eq,axiom,
% 11.90/5.97 ! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
% 11.90/5.97 ( ! [N3: nat] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
% 11.90/5.97 => ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2510_bezw_Oelims,axiom,
% 11.90/5.97 ! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
% 11.90/5.97 ( ( ( bezw @ X @ Xa )
% 11.90/5.97 = Y )
% 11.90/5.97 => ( ( ( Xa
% 11.90/5.97 = ( zero_zero @ nat ) )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 11.90/5.97 & ( ( Xa
% 11.90/5.97 != ( zero_zero @ nat ) )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2511_bezw_Osimps,axiom,
% 11.90/5.97 ( bezw
% 11.90/5.97 = ( ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( Y3
% 11.90/5.97 = ( zero_zero @ nat ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X3 @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2512_max__ext__def,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( max_ext @ A )
% 11.90/5.97 = ( ^ [R6: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 11.90/5.97 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 11.90/5.97 @ ( max_extp @ A
% 11.90/5.97 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2513_UN__finite2__subset,axiom,
% 11.90/5.97 ! [A: $tType,A4: nat > ( set @ A ),B6: nat > ( set @ A ),K: nat] :
% 11.90/5.97 ( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K ) ) ) ) )
% 11.90/5.97 => ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2514_bezw_Opelims,axiom,
% 11.90/5.97 ! [X: nat,Xa: nat,Y: product_prod @ int @ int] :
% 11.90/5.97 ( ( ( bezw @ X @ Xa )
% 11.90/5.97 = Y )
% 11.90/5.97 => ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
% 11.90/5.97 => ~ ( ( ( ( Xa
% 11.90/5.97 = ( zero_zero @ nat ) )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
% 11.90/5.97 & ( ( Xa
% 11.90/5.97 != ( zero_zero @ nat ) )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
% 11.90/5.97 => ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2515_snd__diag__id,axiom,
% 11.90/5.97 ! [A: $tType,Z2: A] :
% 11.90/5.97 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
% 11.90/5.97 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 11.90/5.97 @ Z2 )
% 11.90/5.97 = ( id @ A @ Z2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2516_fst__diag__id,axiom,
% 11.90/5.97 ! [A: $tType,Z2: A] :
% 11.90/5.97 ( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
% 11.90/5.97 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 11.90/5.97 @ Z2 )
% 11.90/5.97 = ( id @ A @ Z2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2517_size__prod__simp,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( basic_BNF_size_prod @ A @ B )
% 11.90/5.97 = ( ^ [F2: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2518_fst__diag__snd,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
% 11.90/5.97 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
% 11.90/5.97 @ ( product_snd @ A @ B ) ) )
% 11.90/5.97 = ( product_snd @ A @ B ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2519_snd__diag__fst,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
% 11.90/5.97 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 11.90/5.97 @ ( product_fst @ A @ B ) ) )
% 11.90/5.97 = ( product_fst @ A @ B ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2520_None__in__Inf,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ! [A4: set @ ( option @ A )] :
% 11.90/5.97 ( ( member @ ( option @ A ) @ ( none @ A ) @ A4 )
% 11.90/5.97 => ( ( complete_Inf_Inf @ ( option @ A ) @ A4 )
% 11.90/5.97 = ( none @ A ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2521_empty__Sup,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ( ( complete_Sup_Sup @ ( option @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
% 11.90/5.97 = ( none @ A ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2522_singleton__None__Sup,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( comple6319245703460814977attice @ A )
% 11.90/5.97 => ( ( complete_Sup_Sup @ ( option @ A ) @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) )
% 11.90/5.97 = ( none @ A ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2523_INF__INT__eq2,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
% 11.90/5.97 ( ( complete_Inf_Inf @ ( A > B > $o )
% 11.90/5.97 @ ( image2 @ C @ ( A > B > $o )
% 11.90/5.97 @ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R @ I3 ) )
% 11.90/5.97 @ S2 ) )
% 11.90/5.97 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2524_INF__Int__eq2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
% 11.90/5.97 ( ( complete_Inf_Inf @ ( A > B > $o )
% 11.90/5.97 @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
% 11.90/5.97 @ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
% 11.90/5.97 @ S2 ) )
% 11.90/5.97 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2525_SUP__UN__eq2,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( A > B > $o )
% 11.90/5.97 @ ( image2 @ C @ ( A > B > $o )
% 11.90/5.97 @ ^ [I3: C,X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( R @ I3 ) )
% 11.90/5.97 @ S2 ) )
% 11.90/5.97 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2526_SUP__Sup__eq2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( A > B > $o )
% 11.90/5.97 @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
% 11.90/5.97 @ ^ [I3: set @ ( product_prod @ A @ B ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ I3 )
% 11.90/5.97 @ S2 ) )
% 11.90/5.97 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2527_Inf__INT__eq2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( complete_Inf_Inf @ ( A > B > $o ) )
% 11.90/5.97 = ( ^ [S5: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S5 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2528_Sup__SUP__eq2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( A > B > $o ) )
% 11.90/5.97 = ( ^ [S5: set @ ( A > B > $o ),X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S5 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2529_fst__diag__fst,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
% 11.90/5.97 @ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 11.90/5.97 @ ( product_fst @ A @ B ) ) )
% 11.90/5.97 = ( product_fst @ A @ B ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2530_snd__diag__snd,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
% 11.90/5.97 @ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [X3: B] : ( product_Pair @ B @ B @ X3 @ X3 )
% 11.90/5.97 @ ( product_snd @ A @ B ) ) )
% 11.90/5.97 = ( product_snd @ A @ B ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2531_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P: A > B > $o,X: A,Y: B,A3: product_prod @ A @ B] :
% 11.90/5.97 ( ( P @ X @ Y )
% 11.90/5.97 => ( ( A3
% 11.90/5.97 = ( product_Pair @ A @ B @ X @ Y ) )
% 11.90/5.97 => ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2532_normalize__def,axiom,
% 11.90/5.97 ( normalize
% 11.90/5.97 = ( ^ [P6: product_prod @ int @ int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
% 11.90/5.97 @ ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( ( product_snd @ int @ int @ P6 )
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2533_range__prod,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,F: C > ( product_prod @ A @ B )] :
% 11.90/5.97 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F @ ( top_top @ ( set @ C ) ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) )
% 11.90/5.97 @ ^ [Uu: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2534_fun_Oin__rel,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,D: $tType,R5: A > B > $o,A3: D > A,B2: D > B] :
% 11.90/5.97 ( ( bNF_rel_fun @ D @ D @ A @ B
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ R5
% 11.90/5.97 @ A3
% 11.90/5.97 @ B2 )
% 11.90/5.97 = ( ? [Z5: D > ( product_prod @ A @ B )] :
% 11.90/5.97 ( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z5
% 11.90/5.97 @ ( collect @ ( D > ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) ) )
% 11.90/5.97 & ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z5 )
% 11.90/5.97 = A3 )
% 11.90/5.97 & ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z5 )
% 11.90/5.97 = B2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2535_apfst__apsnd,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 11.90/5.97 ( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
% 11.90/5.97 = ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2536_apsnd__apfst,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
% 11.90/5.97 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
% 11.90/5.97 = ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2537_eq__snd__iff,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,B2: A,P5: product_prod @ B @ A] :
% 11.90/5.97 ( ( B2
% 11.90/5.97 = ( product_snd @ B @ A @ P5 ) )
% 11.90/5.97 = ( ? [A7: B] :
% 11.90/5.97 ( P5
% 11.90/5.97 = ( product_Pair @ B @ A @ A7 @ B2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2538_SigmaI,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( member @ A @ A3 @ A4 )
% 11.90/5.97 => ( ( member @ B @ B2 @ ( B6 @ A3 ) )
% 11.90/5.97 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2539_mem__Sigma__iff,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 11.90/5.97 = ( ( member @ A @ A3 @ A4 )
% 11.90/5.97 & ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2540_gcd__add1,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [M2: A,N: A] :
% 11.90/5.97 ( ( gcd_gcd @ A @ ( plus_plus @ A @ M2 @ N ) @ N )
% 11.90/5.97 = ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2541_gcd__add2,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [M2: A,N: A] :
% 11.90/5.97 ( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ M2 @ N ) )
% 11.90/5.97 = ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2542_gcd_Obottom__left__bottom,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A] :
% 11.90/5.97 ( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
% 11.90/5.97 = ( one_one @ A ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2543_gcd_Obottom__right__bottom,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A] :
% 11.90/5.97 ( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.97 = ( one_one @ A ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2544_apfst__conv,axiom,
% 11.90/5.97 ! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y: B] :
% 11.90/5.97 ( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y ) )
% 11.90/5.97 = ( product_Pair @ A @ B @ ( F @ X ) @ Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2545_apfst__eq__conv,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
% 11.90/5.97 ( ( ( product_apfst @ C @ A @ B @ F @ X )
% 11.90/5.97 = ( product_apfst @ C @ A @ B @ G @ X ) )
% 11.90/5.97 = ( ( F @ ( product_fst @ C @ B @ X ) )
% 11.90/5.97 = ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2546_fst__apfst,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
% 11.90/5.97 ( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
% 11.90/5.97 = ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2547_snd__apfst,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
% 11.90/5.97 ( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
% 11.90/5.97 = ( product_snd @ C @ A @ X ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2548_Collect__case__prod,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P: A > $o,Q: B > $o] :
% 11.90/5.97 ( ( collect @ ( product_prod @ A @ B )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.97 @ ^ [A7: A,B5: B] :
% 11.90/5.97 ( ( P @ A7 )
% 11.90/5.97 & ( Q @ B5 ) ) ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 11.90/5.97 @ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2549_is__unit__gcd__iff,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A,B2: A] :
% 11.90/5.97 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 11.90/5.97 = ( ( gcd_gcd @ A @ A3 @ B2 )
% 11.90/5.97 = ( one_one @ A ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2550_Sigma__empty1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
% 11.90/5.97 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2551_Sigma__empty2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
% 11.90/5.97 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2552_Times__empty,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.90/5.97 ( ( ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 )
% 11.90/5.97 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.97 = ( ( A4
% 11.90/5.97 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 | ( B6
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2553_Sigma__UNIV__cancel,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,X10: set @ B] :
% 11.90/5.97 ( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : X10 )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 11.90/5.97 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2554_Compl__Times__UNIV2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A] :
% 11.90/5.97 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A4 )
% 11.90/5.97 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2555_Compl__Times__UNIV1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ B] :
% 11.90/5.97 ( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 11.90/5.97 @ ^ [Uu: A] : A4 ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 11.90/5.97 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2556_snd__comp__apfst,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 11.90/5.97 ( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 11.90/5.97 = ( product_snd @ A @ B ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2557_UNIV__Times__UNIV,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 11.90/5.97 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
% 11.90/5.97 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2558_apfst__id,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_apfst @ A @ A @ B @ ( id @ A ) )
% 11.90/5.97 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2559_pairself__image__cart,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,F: B > A,A4: set @ B,B6: set @ B] :
% 11.90/5.97 ( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F )
% 11.90/5.97 @ ( product_Sigma @ B @ B @ A4
% 11.90/5.97 @ ^ [Uu: B] : B6 ) )
% 11.90/5.97 = ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A4 )
% 11.90/5.97 @ ^ [Uu: A] : ( image2 @ B @ A @ F @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2560_fst__image__times,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,B6: set @ B,A4: set @ A] :
% 11.90/5.97 ( ( ( B6
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) )
% 11.90/5.97 => ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 ) )
% 11.90/5.97 = ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.97 & ( ( B6
% 11.90/5.97 != ( bot_bot @ ( set @ B ) ) )
% 11.90/5.97 => ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 ) )
% 11.90/5.97 = A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2561_snd__image__times,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
% 11.90/5.97 ( ( ( A4
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) )
% 11.90/5.97 => ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 11.90/5.97 @ ( product_Sigma @ B @ A @ A4
% 11.90/5.97 @ ^ [Uu: B] : B6 ) )
% 11.90/5.97 = ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.97 & ( ( A4
% 11.90/5.97 != ( bot_bot @ ( set @ B ) ) )
% 11.90/5.97 => ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 11.90/5.97 @ ( product_Sigma @ B @ A @ A4
% 11.90/5.97 @ ^ [Uu: B] : B6 ) )
% 11.90/5.97 = B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2562_fst__comp__apfst,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,F: A > C] :
% 11.90/5.97 ( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
% 11.90/5.97 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2563_fun_Omap__transfer,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,D: $tType,G3: $tType,F5: $tType,Rb2: A > F5 > $o,Sd: B > G3 > $o] :
% 11.90/5.97 ( bNF_rel_fun @ ( A > B ) @ ( F5 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F5 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F5 @ B @ G3 @ Rb2 @ Sd )
% 11.90/5.97 @ ( bNF_rel_fun @ ( D > A ) @ ( D > F5 ) @ ( D > B ) @ ( D > G3 )
% 11.90/5.97 @ ( bNF_rel_fun @ D @ D @ A @ F5
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ Rb2 )
% 11.90/5.97 @ ( bNF_rel_fun @ D @ D @ B @ G3
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ Sd ) )
% 11.90/5.97 @ ( comp @ A @ B @ D )
% 11.90/5.97 @ ( comp @ F5 @ G3 @ D ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2564_comp__transfer,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,E: $tType,F5: $tType,D: $tType,C: $tType,B6: A > C > $o,C5: B > D > $o,A4: E > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F5 > C ) > F5 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B6 @ C5 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F5 > C ) @ ( E > B ) @ ( F5 > D ) @ ( bNF_rel_fun @ E @ F5 @ A @ C @ A4 @ B6 ) @ ( bNF_rel_fun @ E @ F5 @ B @ D @ A4 @ C5 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2565_power__transfer,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( ( power @ B )
% 11.90/5.97 & ( power @ A ) )
% 11.90/5.97 => ! [R5: A > B > $o] :
% 11.90/5.97 ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 11.90/5.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 11.90/5.97 => ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R5
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ A @ B
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ R5 )
% 11.90/5.97 @ ( power_power @ A )
% 11.90/5.97 @ ( power_power @ B ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2566_times__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( times_times @ int )
% 11.90/5.97 @ ( times_times @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2567_times__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( times_times @ nat )
% 11.90/5.97 @ ( times_times @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2568_Times__eq__cancel2,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B6: set @ B] :
% 11.90/5.97 ( ( member @ A @ X @ C5 )
% 11.90/5.97 => ( ( ( product_Sigma @ B @ A @ A4
% 11.90/5.97 @ ^ [Uu: B] : C5 )
% 11.90/5.97 = ( product_Sigma @ B @ A @ B6
% 11.90/5.97 @ ^ [Uu: B] : C5 ) )
% 11.90/5.97 = ( A4 = B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2569_Sigma__cong,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: A > ( set @ B ),D4: A > ( set @ B )] :
% 11.90/5.97 ( ( A4 = B6 )
% 11.90/5.97 => ( ! [X2: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ B6 )
% 11.90/5.97 => ( ( C5 @ X2 )
% 11.90/5.97 = ( D4 @ X2 ) ) )
% 11.90/5.97 => ( ( product_Sigma @ A @ B @ A4 @ C5 )
% 11.90/5.97 = ( product_Sigma @ A @ B @ B6 @ D4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2570_divide__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( divide_divide @ nat )
% 11.90/5.97 @ ( divide_divide @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2571_divide__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( divide_divide @ int )
% 11.90/5.97 @ ( divide_divide @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2572_or__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se1065995026697491101ons_or @ nat )
% 11.90/5.97 @ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2573_or__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se1065995026697491101ons_or @ int )
% 11.90/5.97 @ ( bit_se1065995026697491101ons_or @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2574_set__bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se5668285175392031749et_bit @ int )
% 11.90/5.97 @ ( bit_se5668285175392031749et_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2575_set__bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se5668285175392031749et_bit @ nat )
% 11.90/5.97 @ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2576_xor__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se5824344971392196577ns_xor @ int )
% 11.90/5.97 @ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2577_xor__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se5824344971392196577ns_xor @ nat )
% 11.90/5.97 @ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2578_and__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se5824344872417868541ns_and @ int )
% 11.90/5.97 @ ( bit_se5824344872417868541ns_and @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2579_and__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se5824344872417868541ns_and @ nat )
% 11.90/5.97 @ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2580_modulo__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( modulo_modulo @ int )
% 11.90/5.97 @ ( modulo_modulo @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2581_modulo__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( modulo_modulo @ nat )
% 11.90/5.97 @ ( modulo_modulo @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2582_less__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( ord_less @ int )
% 11.90/5.97 @ ( ord_less @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2583_less__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( ord_less @ nat )
% 11.90/5.97 @ ( ord_less @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2584_unset__bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se2638667681897837118et_bit @ int )
% 11.90/5.97 @ ( bit_se2638667681897837118et_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2585_take__bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se2584673776208193580ke_bit @ int )
% 11.90/5.97 @ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2586_flip__bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se8732182000553998342ip_bit @ int )
% 11.90/5.97 @ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2587_flip__bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se8732182000553998342ip_bit @ nat )
% 11.90/5.97 @ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2588_push__bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se4730199178511100633sh_bit @ nat )
% 11.90/5.97 @ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2589_push__bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se4730199178511100633sh_bit @ int )
% 11.90/5.97 @ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2590_plus__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( plus_plus @ int )
% 11.90/5.97 @ ( plus_plus @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2591_plus__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( plus_plus @ nat )
% 11.90/5.97 @ ( plus_plus @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2592_drop__bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se4197421643247451524op_bit @ int )
% 11.90/5.97 @ ( bit_se4197421643247451524op_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2593_minus__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( minus_minus @ nat )
% 11.90/5.97 @ ( minus_minus @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2594_minus__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( minus_minus @ int )
% 11.90/5.97 @ ( minus_minus @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2595_less__eq__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ $o @ $o
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( ord_less_eq @ int )
% 11.90/5.97 @ ( ord_less_eq @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2596_less__eq__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ $o @ $o
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( ord_less_eq @ nat )
% 11.90/5.97 @ ( ord_less_eq @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2597_SigmaE,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C2: product_prod @ A @ B,A4: set @ A,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 11.90/5.97 => ~ ! [X2: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ A4 )
% 11.90/5.97 => ! [Y4: B] :
% 11.90/5.97 ( ( member @ B @ Y4 @ ( B6 @ X2 ) )
% 11.90/5.97 => ( C2
% 11.90/5.97 != ( product_Pair @ A @ B @ X2 @ Y4 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2598_SigmaD1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 11.90/5.97 => ( member @ A @ A3 @ A4 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2599_SigmaD2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 11.90/5.97 => ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2600_SigmaE2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,B2: B,A4: set @ A,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 11.90/5.97 => ~ ( ( member @ A @ A3 @ A4 )
% 11.90/5.97 => ~ ( member @ B @ B2 @ ( B6 @ A3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2601_Suc_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ suc
% 11.90/5.97 @ suc ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2602_sub_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
% 11.90/5.97 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ num @ num @ int @ int
% 11.90/5.97 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) )
% 11.90/5.97 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2603_transfer__rule__numeral,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( ( monoid_add @ B )
% 11.90/5.97 & ( semiring_numeral @ B )
% 11.90/5.97 & ( monoid_add @ A )
% 11.90/5.97 & ( semiring_numeral @ A ) )
% 11.90/5.97 => ! [R5: A > B > $o] :
% 11.90/5.97 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 11.90/5.97 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 11.90/5.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 11.90/5.97 => ( bNF_rel_fun @ num @ num @ A @ B
% 11.90/5.97 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.97 @ R5
% 11.90/5.97 @ ( numeral_numeral @ A )
% 11.90/5.97 @ ( numeral_numeral @ B ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2604_uminus__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( uminus_uminus @ int )
% 11.90/5.97 @ ( uminus_uminus @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2605_integer__of__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ int @ int
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( semiring_1_of_nat @ int )
% 11.90/5.97 @ ( semiring_1_of_nat @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2606_Sigma__Diff__distrib1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I4 @ J4 ) @ C5 )
% 11.90/5.97 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2607_natural__of__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ nat @ nat
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ nat2
% 11.90/5.97 @ nat2 ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2608_abs__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( abs_abs @ int )
% 11.90/5.97 @ ( abs_abs @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2609_bit__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bit_se5641148757651400278ts_bit @ int )
% 11.90/5.97 @ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2610_not__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bit_ri4277139882892585799ns_not @ int )
% 11.90/5.97 @ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2611_sgn__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( sgn_sgn @ int )
% 11.90/5.97 @ ( sgn_sgn @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2612_mask__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ int @ int
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bit_se2239418461657761734s_mask @ int )
% 11.90/5.97 @ ( bit_se2239418461657761734s_mask @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2613_mask__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bit_se2239418461657761734s_mask @ nat )
% 11.90/5.97 @ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2614_transfer__rule__of__int,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( ( ring_1 @ B )
% 11.90/5.97 & ( ring_1 @ A ) )
% 11.90/5.97 => ! [R5: A > B > $o] :
% 11.90/5.97 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 11.90/5.97 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 11.90/5.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 11.90/5.97 => ( ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
% 11.90/5.97 => ( bNF_rel_fun @ int @ int @ A @ B
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ R5
% 11.90/5.97 @ ( ring_1_of_int @ A )
% 11.90/5.97 @ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2615_fun_Orel__map_I1_J,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y: D > B] :
% 11.90/5.97 ( ( bNF_rel_fun @ D @ D @ C @ B
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ Sb
% 11.90/5.97 @ ( comp @ A @ C @ D @ I @ X )
% 11.90/5.97 @ Y )
% 11.90/5.97 = ( bNF_rel_fun @ D @ D @ A @ B
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
% 11.90/5.97 @ X
% 11.90/5.97 @ Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2616_fun_Orel__map_I2_J,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G: B > C,Y: D > B] :
% 11.90/5.97 ( ( bNF_rel_fun @ D @ D @ A @ C
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ Sa
% 11.90/5.97 @ X
% 11.90/5.97 @ ( comp @ B @ C @ D @ G @ Y ) )
% 11.90/5.97 = ( bNF_rel_fun @ D @ D @ A @ B
% 11.90/5.97 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
% 11.90/5.97 @ X
% 11.90/5.97 @ Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2617_times__eq__iff,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
% 11.90/5.97 ( ( ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 )
% 11.90/5.97 = ( product_Sigma @ A @ B @ C5
% 11.90/5.97 @ ^ [Uu: A] : D4 ) )
% 11.90/5.97 = ( ( ( A4 = C5 )
% 11.90/5.97 & ( B6 = D4 ) )
% 11.90/5.97 | ( ( ( A4
% 11.90/5.97 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 | ( B6
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) ) )
% 11.90/5.97 & ( ( C5
% 11.90/5.97 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 | ( D4
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2618_dup_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
% 11.90/5.97 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2619_Sigma__Diff__distrib2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ I4
% 11.90/5.97 @ ^ [I3: A] : ( minus_minus @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
% 11.90/5.97 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2620_Times__Diff__distrib1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.97 @ ^ [Uu: A] : C5 )
% 11.90/5.97 = ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : C5 )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ B6
% 11.90/5.97 @ ^ [Uu: A] : C5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2621_Sigma__Union,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X10: set @ ( set @ A ),B6: A > ( set @ B )] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X10 ) @ B6 )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [A9: set @ A] : ( product_Sigma @ A @ B @ A9 @ B6 )
% 11.90/5.97 @ X10 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2622_Collect__case__prod__Sigma,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,P: A > $o,Q: A > B > $o] :
% 11.90/5.97 ( ( collect @ ( product_prod @ A @ B )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.97 @ ^ [X3: A,Y3: B] :
% 11.90/5.97 ( ( P @ X3 )
% 11.90/5.97 & ( Q @ X3 @ Y3 ) ) ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ ( collect @ A @ P )
% 11.90/5.97 @ ^ [X3: A] : ( collect @ B @ ( Q @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2623_apfst__compose,axiom,
% 11.90/5.97 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
% 11.90/5.97 ( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
% 11.90/5.97 = ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2624_apsnd__apfst__commute,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P5: product_prod @ D @ C] :
% 11.90/5.97 ( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P5 ) )
% 11.90/5.97 = ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2625_gcd__add__mult,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [M2: A,K: A,N: A] :
% 11.90/5.97 ( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K @ M2 ) @ N ) )
% 11.90/5.97 = ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2626_Sigma__mono,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,C5: set @ A,B6: A > ( set @ B ),D4: A > ( set @ B )] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
% 11.90/5.97 => ( ! [X2: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ A4 )
% 11.90/5.97 => ( ord_less_eq @ ( set @ B ) @ ( B6 @ X2 ) @ ( D4 @ X2 ) ) )
% 11.90/5.97 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) @ ( product_Sigma @ A @ B @ C5 @ D4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2627_bezout__int,axiom,
% 11.90/5.97 ! [X: int,Y: int] :
% 11.90/5.97 ? [U3: int,V4: int] :
% 11.90/5.97 ( ( plus_plus @ int @ ( times_times @ int @ U3 @ X ) @ ( times_times @ int @ V4 @ Y ) )
% 11.90/5.97 = ( gcd_gcd @ int @ X @ Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2628_Times__subset__cancel2,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: A,C5: set @ A,A4: set @ B,B6: set @ B] :
% 11.90/5.97 ( ( member @ A @ X @ C5 )
% 11.90/5.97 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.97 @ ( product_Sigma @ B @ A @ A4
% 11.90/5.97 @ ^ [Uu: B] : C5 )
% 11.90/5.97 @ ( product_Sigma @ B @ A @ B6
% 11.90/5.97 @ ^ [Uu: B] : C5 ) )
% 11.90/5.97 = ( ord_less_eq @ ( set @ B ) @ A4 @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2629_UN__Times__distrib,axiom,
% 11.90/5.97 ! [C: $tType,D: $tType,B: $tType,A: $tType,E4: C > ( set @ A ),F6: D > ( set @ B ),A4: set @ C,B6: set @ D] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [A7: C,B5: D] :
% 11.90/5.97 ( product_Sigma @ A @ B @ ( E4 @ A7 )
% 11.90/5.97 @ ^ [Uu: A] : ( F6 @ B5 ) ) )
% 11.90/5.97 @ ( product_Sigma @ C @ D @ A4
% 11.90/5.97 @ ^ [Uu: C] : B6 ) ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E4 @ A4 ) )
% 11.90/5.97 @ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F6 @ B6 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2630_mem__Times__iff,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ X
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 ) )
% 11.90/5.97 = ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
% 11.90/5.97 & ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2631_in__prod__fst__sndI,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
% 11.90/5.97 ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A4 )
% 11.90/5.97 => ( ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 )
% 11.90/5.97 => ( member @ ( product_prod @ A @ B ) @ X
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2632_swap__product,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
% 11.90/5.97 ( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 11.90/5.97 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [I3: B,J2: A] : ( product_Pair @ A @ B @ J2 @ I3 ) )
% 11.90/5.97 @ ( product_Sigma @ B @ A @ A4
% 11.90/5.97 @ ^ [Uu: B] : B6 ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ B6
% 11.90/5.97 @ ^ [Uu: A] : A4 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2633_Sigma__empty__iff,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,I4: set @ A,X10: A > ( set @ B )] :
% 11.90/5.97 ( ( ( product_Sigma @ A @ B @ I4 @ X10 )
% 11.90/5.97 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.97 = ( ! [X3: A] :
% 11.90/5.97 ( ( member @ A @ X3 @ I4 )
% 11.90/5.97 => ( ( X10 @ X3 )
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2634_gcd__mult__unit2,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.97 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.97 => ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 11.90/5.97 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2635_gcd__mult__unit1,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.97 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.97 => ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 11.90/5.97 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2636_gcd__div__unit1,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.97 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.97 => ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 11.90/5.97 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2637_gcd__div__unit2,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( semiring_gcd @ A )
% 11.90/5.97 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.97 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.97 => ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 11.90/5.97 = ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2638_less__int_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 11.90/5.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 11.90/5.97 @ ( ord_less @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2639_less__eq__int_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
% 11.90/5.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 11.90/5.97 @ ( ord_less_eq @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2640_plus__int_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
% 11.90/5.97 @ ( plus_plus @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2641_minus__int_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
% 11.90/5.97 @ ( minus_minus @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2642_times__subset__iff,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,A4: set @ A,C5: set @ B,B6: set @ A,D4: set @ B] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : C5 )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ B6
% 11.90/5.97 @ ^ [Uu: A] : D4 ) )
% 11.90/5.97 = ( ( A4
% 11.90/5.97 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 | ( C5
% 11.90/5.97 = ( bot_bot @ ( set @ B ) ) )
% 11.90/5.97 | ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.97 & ( ord_less_eq @ ( set @ B ) @ C5 @ D4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2643_image__paired__Times,axiom,
% 11.90/5.97 ! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A4: set @ C,B6: set @ D] :
% 11.90/5.97 ( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 11.90/5.97 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 11.90/5.97 @ ( product_Sigma @ C @ D @ A4
% 11.90/5.97 @ ^ [Uu: C] : B6 ) )
% 11.90/5.97 = ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A4 )
% 11.90/5.97 @ ^ [Uu: A] : ( image2 @ D @ B @ G @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2644_fst__image__Sigma,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,B6: A > ( set @ B )] :
% 11.90/5.97 ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ B6 ) )
% 11.90/5.97 = ( collect @ A
% 11.90/5.97 @ ^ [X3: A] :
% 11.90/5.97 ( ( member @ A @ X3 @ A4 )
% 11.90/5.97 & ( ( B6 @ X3 )
% 11.90/5.97 != ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2645_Sigma__def,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( product_Sigma @ A @ B )
% 11.90/5.97 = ( ^ [A9: set @ A,B9: A > ( set @ B )] :
% 11.90/5.97 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [X3: A] :
% 11.90/5.97 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.97 @ ( B9 @ X3 ) ) )
% 11.90/5.97 @ A9 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2646_times__int_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
% 11.90/5.97 @ ( times_times @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2647_num__of__integer_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ num @ num
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.97 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 11.90/5.97 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2648_snd__image__Sigma,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,A4: set @ B,B6: B > ( set @ A )] :
% 11.90/5.97 ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A4 @ B6 ) )
% 11.90/5.97 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2649_subset__fst__snd,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4
% 11.90/5.97 @ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
% 11.90/5.97 @ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2650_int__transfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ pcr_int
% 11.90/5.97 @ ^ [N5: nat] : ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) )
% 11.90/5.97 @ ( semiring_1_of_nat @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2651_uminus__int_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
% 11.90/5.97 @ ( uminus_uminus @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2652_eq__fst__iff,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,A3: A,P5: product_prod @ A @ B] :
% 11.90/5.97 ( ( A3
% 11.90/5.97 = ( product_fst @ A @ B @ P5 ) )
% 11.90/5.97 = ( ? [B5: B] :
% 11.90/5.97 ( P5
% 11.90/5.97 = ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2653_transfer__rule__of__nat,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( ( semiring_1 @ B )
% 11.90/5.97 & ( semiring_1 @ A ) )
% 11.90/5.97 => ! [R5: A > B > $o] :
% 11.90/5.97 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 11.90/5.97 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 11.90/5.97 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 11.90/5.97 => ( bNF_rel_fun @ nat @ nat @ A @ B
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ R5
% 11.90/5.97 @ ( semiring_1_of_nat @ A )
% 11.90/5.97 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2654_transfer__rule__of__bool,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( ( zero_neq_one @ B )
% 11.90/5.97 & ( zero_neq_one @ A ) )
% 11.90/5.97 => ! [R5: A > B > $o] :
% 11.90/5.97 ( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 11.90/5.97 => ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
% 11.90/5.97 => ( bNF_rel_fun @ $o @ $o @ A @ B
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 11.90/5.97 @ R5
% 11.90/5.97 @ ( zero_neq_one_of_bool @ A )
% 11.90/5.97 @ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2655_plus__rat_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 11.90/5.97 @ ( plus_plus @ rat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2656_inverse__rat_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( ( product_fst @ int @ int @ X3 )
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
% 11.90/5.97 @ ( inverse_inverse @ rat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2657_Id__on__def,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( id_on @ A )
% 11.90/5.97 = ( ^ [A9: set @ A] :
% 11.90/5.97 ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
% 11.90/5.97 @ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
% 11.90/5.97 @ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 11.90/5.97 @ A9 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2658_times__rat_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 11.90/5.97 @ ( times_times @ rat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2659_gcd__1__nat,axiom,
% 11.90/5.97 ! [M2: nat] :
% 11.90/5.97 ( ( gcd_gcd @ nat @ M2 @ ( one_one @ nat ) )
% 11.90/5.97 = ( one_one @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2660_Id__onI,axiom,
% 11.90/5.97 ! [A: $tType,A3: A,A4: set @ A] :
% 11.90/5.97 ( ( member @ A @ A3 @ A4 )
% 11.90/5.97 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2661_unset__bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se2638667681897837118et_bit @ nat )
% 11.90/5.97 @ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2662_take__bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se2584673776208193580ke_bit @ nat )
% 11.90/5.97 @ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2663_bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bit_se5641148757651400278ts_bit @ nat )
% 11.90/5.97 @ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2664_drop__bit__natural_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.97 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bit_se4197421643247451524op_bit @ nat )
% 11.90/5.97 @ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2665_Id__on__iff,axiom,
% 11.90/5.97 ! [A: $tType,X: A,Y: A,A4: set @ A] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A4 ) )
% 11.90/5.97 = ( ( X = Y )
% 11.90/5.97 & ( member @ A @ X @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2666_Id__on__eqI,axiom,
% 11.90/5.97 ! [A: $tType,A3: A,B2: A,A4: set @ A] :
% 11.90/5.97 ( ( A3 = B2 )
% 11.90/5.97 => ( ( member @ A @ A3 @ A4 )
% 11.90/5.97 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2667_Id__onE,axiom,
% 11.90/5.97 ! [A: $tType,C2: product_prod @ A @ A,A4: set @ A] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A4 ) )
% 11.90/5.97 => ~ ! [X2: A] :
% 11.90/5.97 ( ( member @ A @ X2 @ A4 )
% 11.90/5.97 => ( C2
% 11.90/5.97 != ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2668_bezout__nat,axiom,
% 11.90/5.97 ! [A3: nat,B2: nat] :
% 11.90/5.97 ( ( A3
% 11.90/5.97 != ( zero_zero @ nat ) )
% 11.90/5.97 => ? [X2: nat,Y4: nat] :
% 11.90/5.97 ( ( times_times @ nat @ A3 @ X2 )
% 11.90/5.97 = ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2669_one__rat_Otransfer,axiom,
% 11.90/5.97 pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2670_zero__rat_Otransfer,axiom,
% 11.90/5.97 pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2671_bezw__aux,axiom,
% 11.90/5.97 ! [X: nat,Y: nat] :
% 11.90/5.97 ( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y ) )
% 11.90/5.97 = ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2672_uminus__rat_Otransfer,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
% 11.90/5.97 @ ( uminus_uminus @ rat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2673_gcd__nat_Opelims,axiom,
% 11.90/5.97 ! [X: nat,Xa: nat,Y: nat] :
% 11.90/5.97 ( ( ( gcd_gcd @ nat @ X @ Xa )
% 11.90/5.97 = Y )
% 11.90/5.97 => ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
% 11.90/5.97 => ~ ( ( ( ( Xa
% 11.90/5.97 = ( zero_zero @ nat ) )
% 11.90/5.97 => ( Y = X ) )
% 11.90/5.97 & ( ( Xa
% 11.90/5.97 != ( zero_zero @ nat ) )
% 11.90/5.97 => ( Y
% 11.90/5.97 = ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
% 11.90/5.97 => ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2674_o__rsp_I1_J,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,E: $tType,F5: $tType,D: $tType,C: $tType,R23: A > C > $o,R32: B > D > $o,R12: E > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F5 > C ) > F5 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R23 @ R32 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F5 > C ) @ ( E > B ) @ ( F5 > D ) @ ( bNF_rel_fun @ E @ F5 @ A @ C @ R12 @ R23 ) @ ( bNF_rel_fun @ E @ F5 @ B @ D @ R12 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2675_o__rsp_I2_J,axiom,
% 11.90/5.97 ! [E: $tType,F5: $tType,H8: $tType,G3: $tType,R12: E > F5 > $o] :
% 11.90/5.97 ( bNF_rel_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( E > G3 ) > E > H8 ) @ ( ( F5 > G3 ) > F5 > H8 )
% 11.90/5.97 @ ^ [Y5: G3 > H8,Z4: G3 > H8] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ ( E > G3 ) @ ( F5 > G3 ) @ ( E > H8 ) @ ( F5 > H8 )
% 11.90/5.97 @ ( bNF_rel_fun @ E @ F5 @ G3 @ G3 @ R12
% 11.90/5.97 @ ^ [Y5: G3,Z4: G3] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( bNF_rel_fun @ E @ F5 @ H8 @ H8 @ R12
% 11.90/5.97 @ ^ [Y5: H8,Z4: H8] : ( Y5 = Z4 ) ) )
% 11.90/5.97 @ ( comp @ G3 @ H8 @ E )
% 11.90/5.97 @ ( comp @ G3 @ H8 @ F5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2676_apfst__convE,axiom,
% 11.90/5.97 ! [C: $tType,A: $tType,B: $tType,Q3: product_prod @ A @ B,F: C > A,P5: product_prod @ C @ B] :
% 11.90/5.97 ( ( Q3
% 11.90/5.97 = ( product_apfst @ C @ A @ B @ F @ P5 ) )
% 11.90/5.97 => ~ ! [X2: C,Y4: B] :
% 11.90/5.97 ( ( P5
% 11.90/5.97 = ( product_Pair @ C @ B @ X2 @ Y4 ) )
% 11.90/5.97 => ( Q3
% 11.90/5.97 != ( product_Pair @ A @ B @ ( F @ X2 ) @ Y4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2677_UN__le__eq__Un0,axiom,
% 11.90/5.97 ! [A: $tType,M3: nat > ( set @ A ),N: nat] :
% 11.90/5.97 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_ord_atMost @ nat @ N ) ) )
% 11.90/5.97 = ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M3 @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2678_times__int_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ U2 ) @ ( times_times @ nat @ Y3 @ V2 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X3 @ V2 ) @ ( times_times @ nat @ Y3 @ U2 ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2679_sup__Some,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( sup @ A )
% 11.90/5.97 => ! [X: A,Y: A] :
% 11.90/5.97 ( ( sup_sup @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 11.90/5.97 = ( some @ A @ ( sup_sup @ A @ X @ Y ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2680_intrel__iff,axiom,
% 11.90/5.97 ! [X: nat,Y: nat,U: nat,V: nat] :
% 11.90/5.97 ( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ U @ V ) )
% 11.90/5.97 = ( ( plus_plus @ nat @ X @ V )
% 11.90/5.97 = ( plus_plus @ nat @ U @ Y ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2681_insert__Times__insert,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,B6: set @ B] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
% 11.90/5.97 @ ^ [Uu: A] : ( insert2 @ B @ B2 @ B6 ) )
% 11.90/5.97 = ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
% 11.90/5.97 @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : ( insert2 @ B @ B2 @ B6 ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A4 )
% 11.90/5.97 @ ^ [Uu: A] : B6 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2682_Sigma__Un__distrib1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I4 @ J4 ) @ C5 )
% 11.90/5.97 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2683_Sigma__Un__distrib2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ I4
% 11.90/5.97 @ ^ [I3: A] : ( sup_sup @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
% 11.90/5.97 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2684_Times__Un__distrib1,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
% 11.90/5.97 ( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.97 @ ^ [Uu: A] : C5 )
% 11.90/5.97 = ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : C5 )
% 11.90/5.97 @ ( product_Sigma @ A @ B @ B6
% 11.90/5.97 @ ^ [Uu: A] : C5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2685_set__diff__diff__left,axiom,
% 11.90/5.97 ! [A: $tType,A4: set @ A,B6: set @ A,C5: set @ A] :
% 11.90/5.97 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/5.97 = ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B6 @ C5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2686_sup__option__def,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( sup @ A )
% 11.90/5.97 => ( ( sup_sup @ ( option @ A ) )
% 11.90/5.97 = ( ^ [X3: option @ A,Y3: option @ A] :
% 11.90/5.97 ( case_option @ ( option @ A ) @ A @ Y3
% 11.90/5.97 @ ^ [X9: A] :
% 11.90/5.97 ( case_option @ ( option @ A ) @ A @ X3
% 11.90/5.97 @ ^ [Z5: A] : ( some @ A @ ( sup_sup @ A @ X9 @ Z5 ) )
% 11.90/5.97 @ Y3 )
% 11.90/5.97 @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2687_ivl__disj__un__two__touch_I4_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,M2: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.97 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 11.90/5.97 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2688_ivl__disj__un__two_I3_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,M2: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.97 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 11.90/5.97 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2689_max__ext__additive,axiom,
% 11.90/5.97 ! [A: $tType,A4: set @ A,B6: set @ A,R5: set @ ( product_prod @ A @ A ),C5: set @ A,D4: set @ A] :
% 11.90/5.97 ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A4 @ B6 ) @ ( max_ext @ A @ R5 ) )
% 11.90/5.97 => ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C5 @ D4 ) @ ( max_ext @ A @ R5 ) )
% 11.90/5.97 => ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ A ) @ B6 @ D4 ) ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2690_zero__int_Orsp,axiom,
% 11.90/5.97 intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2691_ivl__disj__un__two_I7_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,M2: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.97 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 11.90/5.97 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2692_card__Un__le,axiom,
% 11.90/5.97 ! [A: $tType,A4: set @ A,B6: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2693_ivl__disj__un__one_I2_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 11.90/5.97 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2694_atLeastLessThan__add__Un,axiom,
% 11.90/5.97 ! [I: nat,J: nat,K: nat] :
% 11.90/5.97 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/5.97 => ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
% 11.90/5.97 = ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2695_uminus__int_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] : ( product_Pair @ nat @ nat @ Y3 @ X3 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2696_one__int_Orsp,axiom,
% 11.90/5.97 intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2697_ivl__disj__un__two__touch_I2_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,M2: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.97 => ( ( ord_less @ A @ M2 @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 11.90/5.97 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2698_ivl__disj__un__two_I1_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,M2: A,U: A] :
% 11.90/5.97 ( ( ord_less @ A @ L @ M2 )
% 11.90/5.97 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 11.90/5.97 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2699_ivl__disj__un__singleton_I2_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [U: A] :
% 11.90/5.97 ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.97 = ( set_ord_atMost @ A @ U ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2700_ivl__disj__un__one_I4_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 11.90/5.97 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2701_ivl__disj__un__one_I1_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,U: A] :
% 11.90/5.97 ( ( ord_less @ A @ L @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 11.90/5.97 = ( set_ord_lessThan @ A @ U ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2702_intrel__def,axiom,
% 11.90/5.97 ( intrel
% 11.90/5.97 = ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] :
% 11.90/5.97 ( ( plus_plus @ nat @ X3 @ V2 )
% 11.90/5.97 = ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2703_ivl__disj__un__singleton_I6_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.97 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2704_ivl__disj__un__singleton_I3_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,U: A] :
% 11.90/5.97 ( ( ord_less @ A @ L @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 11.90/5.97 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2705_ivl__disj__un__two_I4_J,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( linorder @ A )
% 11.90/5.97 => ! [L: A,M2: A,U: A] :
% 11.90/5.97 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.97 => ( ( ord_less @ A @ M2 @ U )
% 11.90/5.97 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 11.90/5.97 = ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2706_less__int_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 11.90/5.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2707_less__eq__int_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
% 11.90/5.97 @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
% 11.90/5.97 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ $o
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ U2 @ Y3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2708_plus__int_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ U2 ) @ ( plus_plus @ nat @ Y3 @ V2 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2709_minus__int_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) )
% 11.90/5.97 @ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
% 11.90/5.97 @ ^ [X3: nat,Y3: nat] :
% 11.90/5.97 ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
% 11.90/5.97 @ ^ [U2: nat,V2: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X3 @ V2 ) @ ( plus_plus @ nat @ Y3 @ U2 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2710_max__weak__def,axiom,
% 11.90/5.97 ( fun_max_weak
% 11.90/5.97 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2711_min__weak__def,axiom,
% 11.90/5.97 ( fun_min_weak
% 11.90/5.97 = ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2712_fold__union__pair,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,B6: set @ A,X: B,A4: set @ ( product_prod @ B @ A )] :
% 11.90/5.97 ( ( finite_finite2 @ A @ B6 )
% 11.90/5.97 => ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.97 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.97 @ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.97 @ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
% 11.90/5.97 @ B6 ) )
% 11.90/5.97 @ A4 )
% 11.90/5.97 = ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.97 @ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) )
% 11.90/5.97 @ A4
% 11.90/5.97 @ B6 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2713_Field__insert,axiom,
% 11.90/5.97 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
% 11.90/5.97 = ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2714_set__to__map__insert,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
% 11.90/5.97 => ( ( set_to_map @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ Kv @ S2 ) )
% 11.90/5.97 = ( fun_upd @ A @ ( option @ B ) @ ( set_to_map @ A @ B @ S2 ) @ ( product_fst @ A @ B @ Kv ) @ ( some @ B @ ( product_snd @ A @ B @ Kv ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2715_plus__rat_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2716_inverse__rat_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( ( product_fst @ int @ int @ X3 )
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( ( product_fst @ int @ int @ X3 )
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2717_fun__upd__upd,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z2: B] :
% 11.90/5.97 ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z2 )
% 11.90/5.97 = ( fun_upd @ A @ B @ F @ X @ Z2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2718_fun__upd__triv,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 11.90/5.97 ( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
% 11.90/5.97 = F ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2719_fun__upd__apply,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( fun_upd @ B @ A )
% 11.90/5.97 = ( ^ [F2: B > A,X3: B,Y3: A,Z5: B] : ( if @ A @ ( Z5 = X3 ) @ Y3 @ ( F2 @ Z5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2720_image__update,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: A,A4: set @ A,F: A > B,N: B] :
% 11.90/5.97 ( ~ ( member @ A @ X @ A4 )
% 11.90/5.97 => ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F @ X @ N ) @ A4 )
% 11.90/5.97 = ( image2 @ A @ B @ F @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2721_empty__upd__none,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: A] :
% 11.90/5.97 ( ( fun_upd @ A @ ( option @ B )
% 11.90/5.97 @ ^ [X3: A] : ( none @ B )
% 11.90/5.97 @ X
% 11.90/5.97 @ ( none @ B ) )
% 11.90/5.97 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2722_sup__None__1,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( sup @ A )
% 11.90/5.97 => ! [Y: option @ A] :
% 11.90/5.97 ( ( sup_sup @ ( option @ A ) @ ( none @ A ) @ Y )
% 11.90/5.97 = Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2723_sup__None__2,axiom,
% 11.90/5.97 ! [A: $tType] :
% 11.90/5.97 ( ( sup @ A )
% 11.90/5.97 => ! [X: option @ A] :
% 11.90/5.97 ( ( sup_sup @ ( option @ A ) @ X @ ( none @ A ) )
% 11.90/5.97 = X ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2724_finite__Field__eq__finite,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( finite_finite2 @ A @ ( field2 @ A @ R5 ) )
% 11.90/5.97 = ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2725_image__map__upd,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: A,A4: set @ A,M2: A > ( option @ B ),Y: B] :
% 11.90/5.97 ( ~ ( member @ A @ X @ A4 )
% 11.90/5.97 => ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ A4 )
% 11.90/5.97 = ( image2 @ A @ ( option @ B ) @ M2 @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2726_map__option__o__map__upd,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,F: C > B,M2: A > ( option @ C ),A3: A,B2: C] :
% 11.90/5.97 ( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( fun_upd @ A @ ( option @ C ) @ M2 @ A3 @ ( some @ C @ B2 ) ) )
% 11.90/5.97 = ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ M2 ) @ A3 @ ( some @ B @ ( F @ B2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2727_fun__upd__def,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( fun_upd @ A @ B )
% 11.90/5.97 = ( ^ [F2: A > B,A7: A,B5: B,X3: A] : ( if @ B @ ( X3 = A7 ) @ B5 @ ( F2 @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2728_fun__upd__eqD,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z2: B] :
% 11.90/5.97 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 11.90/5.97 = ( fun_upd @ A @ B @ G @ X @ Z2 ) )
% 11.90/5.97 => ( Y = Z2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2729_fun__upd__idem,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
% 11.90/5.97 ( ( ( F @ X )
% 11.90/5.97 = Y )
% 11.90/5.97 => ( ( fun_upd @ B @ A @ F @ X @ Y )
% 11.90/5.97 = F ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2730_fun__upd__same,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
% 11.90/5.97 ( ( fun_upd @ B @ A @ F @ X @ Y @ X )
% 11.90/5.97 = Y ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2731_fun__upd__other,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,Z2: A,X: A,F: A > B,Y: B] :
% 11.90/5.97 ( ( Z2 != X )
% 11.90/5.97 => ( ( fun_upd @ A @ B @ F @ X @ Y @ Z2 )
% 11.90/5.97 = ( F @ Z2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2732_fun__upd__twist,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,A3: A,C2: A,M2: A > B,B2: B,D3: B] :
% 11.90/5.97 ( ( A3 != C2 )
% 11.90/5.97 => ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ A3 @ B2 ) @ C2 @ D3 )
% 11.90/5.97 = ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M2 @ C2 @ D3 ) @ A3 @ B2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2733_fun__upd__idem__iff,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
% 11.90/5.97 ( ( ( fun_upd @ A @ B @ F @ X @ Y )
% 11.90/5.97 = F )
% 11.90/5.97 = ( ( F @ X )
% 11.90/5.97 = Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2734_map__upd__Some__unfold,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,M2: B > ( option @ A ),A3: B,B2: A,X: B,Y: A] :
% 11.90/5.97 ( ( ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) @ X )
% 11.90/5.97 = ( some @ A @ Y ) )
% 11.90/5.97 = ( ( ( X = A3 )
% 11.90/5.97 & ( B2 = Y ) )
% 11.90/5.97 | ( ( X != A3 )
% 11.90/5.97 & ( ( M2 @ X )
% 11.90/5.97 = ( some @ A @ Y ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2735_map__upd__triv,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X: A] :
% 11.90/5.97 ( ( ( T2 @ K )
% 11.90/5.97 = ( some @ A @ X ) )
% 11.90/5.97 => ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X ) )
% 11.90/5.97 = T2 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2736_map__upd__eqD1,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,X: B,N: A > ( option @ B ),Y: B] :
% 11.90/5.97 ( ( ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ X ) )
% 11.90/5.97 = ( fun_upd @ A @ ( option @ B ) @ N @ A3 @ ( some @ B @ Y ) ) )
% 11.90/5.97 => ( X = Y ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2737_FieldI2,axiom,
% 11.90/5.97 ! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
% 11.90/5.97 => ( member @ A @ J @ ( field2 @ A @ R5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2738_FieldI1,axiom,
% 11.90/5.97 ! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
% 11.90/5.97 => ( member @ A @ I @ ( field2 @ A @ R5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2739_sup__Un__eq2,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ( sup_sup @ ( A > B > $o )
% 11.90/5.97 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
% 11.90/5.97 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
% 11.90/5.97 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2740_fun__upd__comp,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,F: C > B,G: A > C,X: A,Y: C] :
% 11.90/5.97 ( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G @ X @ Y ) )
% 11.90/5.97 = ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G ) @ X @ ( F @ Y ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2741_map__upd__nonempty,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X: B] :
% 11.90/5.97 ( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X ) )
% 11.90/5.97 != ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2742_one__rat_Orsp,axiom,
% 11.90/5.97 ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2743_R__subset__Field,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.97 @ ( product_Sigma @ A @ A @ ( field2 @ A @ R5 )
% 11.90/5.97 @ ^ [Uu: A] : ( field2 @ A @ R5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2744_fst__in__Field,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_fst @ A @ A ) @ R5 ) @ ( field2 @ A @ R5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2745_snd__in__Field,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ A ) @ A @ ( product_snd @ A @ A ) @ R5 ) @ ( field2 @ A @ R5 ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2746_zero__rat_Orsp,axiom,
% 11.90/5.97 ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2747_Fract_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
% 11.90/5.97 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.97 @ ratrel )
% 11.90/5.97 @ ^ [A7: int,B5: int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( B5
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ A7 @ B5 ) )
% 11.90/5.97 @ ^ [A7: int,B5: int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( B5
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ A7 @ B5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2748_sum_Oeq__fold,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comm_monoid_add @ A )
% 11.90/5.97 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 11.90/5.97 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2749_prod_Oeq__fold,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType] :
% 11.90/5.97 ( ( comm_monoid_mult @ A )
% 11.90/5.97 => ( ( groups7121269368397514597t_prod @ B @ A )
% 11.90/5.97 = ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2750_finite__range__updI,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,F: B > ( option @ A ),A3: B,B2: A] :
% 11.90/5.97 ( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F @ ( top_top @ ( set @ B ) ) ) )
% 11.90/5.97 => ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2751_fun__upd__image,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: B,A4: set @ B,F: B > A,Y: A] :
% 11.90/5.97 ( ( ( member @ B @ X @ A4 )
% 11.90/5.97 => ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A4 )
% 11.90/5.97 = ( insert2 @ A @ Y @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
% 11.90/5.97 & ( ~ ( member @ B @ X @ A4 )
% 11.90/5.97 => ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y ) @ A4 )
% 11.90/5.97 = ( image2 @ B @ A @ F @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2752_times__rat_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2753_uminus__rat_Orsp,axiom,
% 11.90/5.97 ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2754_product__fold,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.90/5.97 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.97 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.97 => ( ( product_Sigma @ A @ B @ A4
% 11.90/5.97 @ ^ [Uu: A] : B6 )
% 11.90/5.97 = ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [X3: A,Z5: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.97 @ ^ [Y3: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 11.90/5.97 @ Z5
% 11.90/5.97 @ B6 )
% 11.90/5.97 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
% 11.90/5.97 @ A4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2755_Id__on__fold,axiom,
% 11.90/5.97 ! [A: $tType,A4: set @ A] :
% 11.90/5.97 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.97 => ( ( id_on @ A @ A4 )
% 11.90/5.97 = ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
% 11.90/5.97 @ ^ [X3: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) )
% 11.90/5.97 @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/5.97 @ A4 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2756_plus__rat_Oabs__eq,axiom,
% 11.90/5.97 ! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
% 11.90/5.97 ( ( ratrel @ Xa @ Xa )
% 11.90/5.97 => ( ( ratrel @ X @ X )
% 11.90/5.97 => ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 11.90/5.97 = ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2757_inverse__rat_Oabs__eq,axiom,
% 11.90/5.97 ! [X: product_prod @ int @ int] :
% 11.90/5.97 ( ( ratrel @ X @ X )
% 11.90/5.97 => ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
% 11.90/5.97 = ( abs_Rat
% 11.90/5.97 @ ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( ( product_fst @ int @ int @ X )
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2758_times__rat_Oabs__eq,axiom,
% 11.90/5.97 ! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
% 11.90/5.97 ( ( ratrel @ Xa @ Xa )
% 11.90/5.97 => ( ( ratrel @ X @ X )
% 11.90/5.97 => ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
% 11.90/5.97 = ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2759_uminus__rat_Oabs__eq,axiom,
% 11.90/5.97 ! [X: product_prod @ int @ int] :
% 11.90/5.97 ( ( ratrel @ X @ X )
% 11.90/5.97 => ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
% 11.90/5.97 = ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2760_graph__map__upd,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,M2: A > ( option @ B ),K: A,V: B] :
% 11.90/5.97 ( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
% 11.90/5.97 = ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2761_insert__relcomp__union__fold,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X10: set @ ( product_prod @ C @ B )] :
% 11.90/5.97 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
% 11.90/5.97 => ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S2 ) @ X10 )
% 11.90/5.97 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 11.90/5.97 @ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] :
% 11.90/5.97 ( if @ ( set @ ( product_prod @ C @ B ) )
% 11.90/5.97 @ ( ( product_snd @ C @ A @ X )
% 11.90/5.97 = W2 )
% 11.90/5.97 @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A10 )
% 11.90/5.97 @ A10 ) )
% 11.90/5.97 @ X10
% 11.90/5.97 @ S2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2762_graph__empty,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( graph @ A @ B
% 11.90/5.97 @ ^ [X3: A] : ( none @ B ) )
% 11.90/5.97 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2763_relcomp_Ocases,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
% 11.90/5.97 => ~ ! [B3: B] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B3 ) @ R )
% 11.90/5.97 => ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ A22 ) @ S ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2764_relcomp_Osimps,axiom,
% 11.90/5.97 ! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
% 11.90/5.97 = ( ? [A7: A,B5: B,C3: C] :
% 11.90/5.97 ( ( A1 = A7 )
% 11.90/5.97 & ( A22 = C3 )
% 11.90/5.97 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R )
% 11.90/5.97 & ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C3 ) @ S ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2765_relcomp_OrelcompI,axiom,
% 11.90/5.97 ! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),C2: C,S: set @ ( product_prod @ B @ C )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 11.90/5.97 => ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S )
% 11.90/5.97 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2766_relcompE,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S ) )
% 11.90/5.97 => ~ ! [X2: A,Y4: C,Z3: B] :
% 11.90/5.97 ( ( Xz
% 11.90/5.97 = ( product_Pair @ A @ B @ X2 @ Z3 ) )
% 11.90/5.97 => ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y4 ) @ R )
% 11.90/5.97 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2767_relcompEpair,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R @ S ) )
% 11.90/5.97 => ~ ! [B3: C] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B3 ) @ R )
% 11.90/5.97 => ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B3 @ C2 ) @ S ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2768_in__graphI,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,M2: B > ( option @ A ),K: B,V: A] :
% 11.90/5.97 ( ( ( M2 @ K )
% 11.90/5.97 = ( some @ A @ V ) )
% 11.90/5.97 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2769_in__graphD,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B )] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M2 ) )
% 11.90/5.97 => ( ( M2 @ K )
% 11.90/5.97 = ( some @ B @ V ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2770_max__ext__compat,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
% 11.90/5.97 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S2 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2771_min__ext__compat,axiom,
% 11.90/5.97 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 11.90/5.97 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
% 11.90/5.97 => ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S2 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R5 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2772_relcomp__fold,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
% 11.90/5.97 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ R5 )
% 11.90/5.97 => ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
% 11.90/5.97 => ( ( relcomp @ A @ B @ C @ R5 @ S2 )
% 11.90/5.97 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
% 11.90/5.97 @ ^ [X3: A,Y3: B,A9: set @ ( product_prod @ A @ C )] :
% 11.90/5.97 ( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
% 11.90/5.97 @ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
% 11.90/5.97 @ ^ [W2: B,Z5: C,A10: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y3 = W2 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Z5 ) @ A10 ) @ A10 ) )
% 11.90/5.97 @ A9
% 11.90/5.97 @ S2 ) )
% 11.90/5.97 @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
% 11.90/5.97 @ R5 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2773_one__rat__def,axiom,
% 11.90/5.97 ( ( one_one @ rat )
% 11.90/5.97 = ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2774_graph__fun__upd__None,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A] :
% 11.90/5.97 ( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) )
% 11.90/5.97 = ( collect @ ( product_prod @ A @ B )
% 11.90/5.97 @ ^ [E5: product_prod @ A @ B] :
% 11.90/5.97 ( ( member @ ( product_prod @ A @ B ) @ E5 @ ( graph @ A @ B @ M2 ) )
% 11.90/5.97 & ( ( product_fst @ A @ B @ E5 )
% 11.90/5.97 != K ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2775_insert__relcomp__fold,axiom,
% 11.90/5.97 ! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R5: set @ ( product_prod @ C @ A )] :
% 11.90/5.97 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
% 11.90/5.97 => ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R5 ) @ S2 )
% 11.90/5.97 = ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 11.90/5.97 @ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] :
% 11.90/5.97 ( if @ ( set @ ( product_prod @ C @ B ) )
% 11.90/5.97 @ ( ( product_snd @ C @ A @ X )
% 11.90/5.97 = W2 )
% 11.90/5.97 @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A10 )
% 11.90/5.97 @ A10 ) )
% 11.90/5.97 @ ( relcomp @ C @ A @ B @ R5 @ S2 )
% 11.90/5.97 @ S2 ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2776_zero__rat__def,axiom,
% 11.90/5.97 ( ( zero_zero @ rat )
% 11.90/5.97 = ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2777_plus__rat__def,axiom,
% 11.90/5.97 ( ( plus_plus @ rat )
% 11.90/5.97 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y3 ) @ ( product_snd @ int @ int @ X3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2778_inverse__rat__def,axiom,
% 11.90/5.97 ( ( inverse_inverse @ rat )
% 11.90/5.97 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] :
% 11.90/5.97 ( if @ ( product_prod @ int @ int )
% 11.90/5.97 @ ( ( product_fst @ int @ int @ X3 )
% 11.90/5.97 = ( zero_zero @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.97 @ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_fst @ int @ int @ X3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2779_times__rat__def,axiom,
% 11.90/5.97 ( ( times_times @ rat )
% 11.90/5.97 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int,Y3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_fst @ int @ int @ Y3 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X3 ) @ ( product_snd @ int @ int @ Y3 ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2780_uminus__rat__def,axiom,
% 11.90/5.97 ( ( uminus_uminus @ rat )
% 11.90/5.97 = ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
% 11.90/5.97 @ ^ [X3: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X3 ) ) @ ( product_snd @ int @ int @ X3 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2781_comp__fun__commute__relcomp__fold,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,C: $tType,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.97 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
% 11.90/5.97 => ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
% 11.90/5.97 @ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 11.90/5.97 @ ^ [X3: C,Y3: A,A9: set @ ( product_prod @ C @ B )] :
% 11.90/5.97 ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
% 11.90/5.97 @ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
% 11.90/5.97 @ ^ [W2: A,Z5: B,A10: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y3 = W2 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X3 @ Z5 ) @ A10 ) @ A10 ) )
% 11.90/5.97 @ A9
% 11.90/5.97 @ S2 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2782_restrict__upd__same,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
% 11.90/5.97 ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.97 = ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2783_restrict__map__UNIV,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,F: A > ( option @ B )] :
% 11.90/5.97 ( ( restrict_map @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.97 = F ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2784_restrict__out,axiom,
% 11.90/5.97 ! [A: $tType,B: $tType,X: A,A4: set @ A,M2: A > ( option @ B )] :
% 11.90/5.97 ( ~ ( member @ A @ X @ A4 )
% 11.90/5.97 => ( ( restrict_map @ A @ B @ M2 @ A4 @ X )
% 11.90/5.97 = ( none @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2785_restrict__map__empty,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,D4: set @ A] :
% 11.90/5.97 ( ( restrict_map @ A @ B
% 11.90/5.97 @ ^ [X3: A] : ( none @ B )
% 11.90/5.97 @ D4 )
% 11.90/5.97 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2786_restrict__map__to__empty,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.97 ( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
% 11.90/5.97 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2787_fun__upd__None__restrict,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType,X: A,D4: set @ A,M2: A > ( option @ B )] :
% 11.90/5.97 ( ( ( member @ A @ X @ D4 )
% 11.90/5.97 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
% 11.90/5.97 = ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
% 11.90/5.97 & ( ~ ( member @ A @ X @ D4 )
% 11.90/5.97 => ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D4 ) @ X @ ( none @ B ) )
% 11.90/5.97 = ( restrict_map @ A @ B @ M2 @ D4 ) ) ) ) ).
% 11.90/5.97
% 11.90/5.97 thf(fact_2788_le__map__restrict,axiom,
% 11.90/5.97 ! [B: $tType,A: $tType] :
% 11.90/5.97 ( ( order @ B )
% 11.90/5.97 => ! [M2: A > ( option @ B ),X10: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M2 @ X10 ) @ M2 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2789_restrict__map__subset__eq,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B ),R5: set @ A,M7: A > ( option @ B ),R7: set @ A] :
% 11.90/5.98 ( ( ( restrict_map @ A @ B @ M2 @ R5 )
% 11.90/5.98 = M7 )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ A ) @ R7 @ R5 )
% 11.90/5.98 => ( ( restrict_map @ A @ B @ M2 @ R7 )
% 11.90/5.98 = ( restrict_map @ A @ B @ M7 @ R7 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2790_restrict__map__eq_I2_J,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: set @ B,K: B,V: A] :
% 11.90/5.98 ( ( ( restrict_map @ B @ A @ M2 @ A4 @ K )
% 11.90/5.98 = ( some @ A @ V ) )
% 11.90/5.98 = ( ( ( M2 @ K )
% 11.90/5.98 = ( some @ A @ V ) )
% 11.90/5.98 & ( member @ B @ K @ A4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2791_restrict__map__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( restrict_map @ A @ B )
% 11.90/5.98 = ( ^ [M: A > ( option @ B ),A9: set @ A,X3: A] : ( if @ ( option @ B ) @ ( member @ A @ X3 @ A9 ) @ ( M @ X3 ) @ ( none @ B ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2792_comp__fun__commute_Ointro,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: A > B > B] :
% 11.90/5.98 ( ! [Y4: A,X2: A] :
% 11.90/5.98 ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 11.90/5.98 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 11.90/5.98 => ( finite6289374366891150609ommute @ A @ B @ F ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2793_comp__fun__commute_Ocomp__fun__commute,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 11.90/5.98 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 11.90/5.98 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 11.90/5.98 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2794_comp__fun__commute__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( finite6289374366891150609ommute @ A @ B )
% 11.90/5.98 = ( ^ [F2: A > B > B] :
% 11.90/5.98 ! [Y3: A,X3: A] :
% 11.90/5.98 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 11.90/5.98 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2795_graph__restrictD_I1_J,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B ),A4: set @ A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A4 ) ) )
% 11.90/5.98 => ( member @ A @ K @ A4 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2796_map__restrict__insert__none__simp,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,M2: B > ( option @ A ),X: B,S: set @ B] :
% 11.90/5.98 ( ( ( M2 @ X )
% 11.90/5.98 = ( none @ A ) )
% 11.90/5.98 => ( ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ ( insert2 @ B @ X @ S ) ) )
% 11.90/5.98 = ( restrict_map @ B @ A @ M2 @ ( uminus_uminus @ ( set @ B ) @ S ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2797_restrict__map__upd,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: A > ( option @ B ),S2: set @ A,K: A,V: B] :
% 11.90/5.98 ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F @ S2 ) @ K @ ( some @ B @ V ) )
% 11.90/5.98 = ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ K @ ( some @ B @ V ) ) @ ( insert2 @ A @ K @ S2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2798_graph__restrictD_I2_J,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,K: A,V: B,M2: A > ( option @ B ),A4: set @ A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A4 ) ) )
% 11.90/5.98 => ( ( M2 @ K )
% 11.90/5.98 = ( some @ B @ V ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2799_restrict__complement__singleton__eq,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
% 11.90/5.98 ( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.98 = ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2800_map__upd__eq__restrict,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A] :
% 11.90/5.98 ( ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( none @ B ) )
% 11.90/5.98 = ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2801_comp__fun__commute__product__fold,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,B6: set @ A] :
% 11.90/5.98 ( ( finite_finite2 @ A @ B6 )
% 11.90/5.98 => ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.98 @ ^ [X3: B,Z5: set @ ( product_prod @ B @ A )] :
% 11.90/5.98 ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
% 11.90/5.98 @ ^ [Y3: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) )
% 11.90/5.98 @ Z5
% 11.90/5.98 @ B6 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2802_Sup__fin_Oeq__fold_H,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semilattice_sup @ A )
% 11.90/5.98 => ( ( lattic5882676163264333800up_fin @ A )
% 11.90/5.98 = ( ^ [A9: set @ A] :
% 11.90/5.98 ( the2 @ A
% 11.90/5.98 @ ( finite_fold @ A @ ( option @ A )
% 11.90/5.98 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( sup_sup @ A @ X3 ) @ Y3 ) )
% 11.90/5.98 @ ( none @ A )
% 11.90/5.98 @ A9 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2803_ran__map__upd,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
% 11.90/5.98 ( ( ( M2 @ A3 )
% 11.90/5.98 = ( none @ A ) )
% 11.90/5.98 => ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) ) )
% 11.90/5.98 = ( insert2 @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2804_Gcd__fin_Oeq__fold,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semiring_gcd @ A )
% 11.90/5.98 => ( ( semiring_gcd_Gcd_fin @ A )
% 11.90/5.98 = ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A9 ) @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2805_Max_Oeq__fold_H,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ( ( lattic643756798349783984er_Max @ A )
% 11.90/5.98 = ( ^ [A9: set @ A] :
% 11.90/5.98 ( the2 @ A
% 11.90/5.98 @ ( finite_fold @ A @ ( option @ A )
% 11.90/5.98 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_max @ A @ X3 ) @ Y3 ) )
% 11.90/5.98 @ ( none @ A )
% 11.90/5.98 @ A9 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2806_Min_Oeq__fold_H,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ( ( lattic643756798350308766er_Min @ A )
% 11.90/5.98 = ( ^ [A9: set @ A] :
% 11.90/5.98 ( the2 @ A
% 11.90/5.98 @ ( finite_fold @ A @ ( option @ A )
% 11.90/5.98 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( ord_min @ A @ X3 ) @ Y3 ) )
% 11.90/5.98 @ ( none @ A )
% 11.90/5.98 @ A9 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2807_pairself__image__eq,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: B > A,P: B > B > $o] :
% 11.90/5.98 ( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
% 11.90/5.98 = ( collect @ ( product_prod @ A @ A )
% 11.90/5.98 @ ^ [Uu: product_prod @ A @ A] :
% 11.90/5.98 ? [A7: B,B5: B] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( product_Pair @ A @ A @ ( F @ A7 ) @ ( F @ B5 ) ) )
% 11.90/5.98 & ( P @ A7 @ B5 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2808_eq__or__mem__image__simp,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: B > A,A3: B,B6: set @ B] :
% 11.90/5.98 ( ( collect @ A
% 11.90/5.98 @ ^ [Uu: A] :
% 11.90/5.98 ? [L3: B] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( F @ L3 ) )
% 11.90/5.98 & ( ( L3 = A3 )
% 11.90/5.98 | ( member @ B @ L3 @ B6 ) ) ) )
% 11.90/5.98 = ( insert2 @ A @ ( F @ A3 )
% 11.90/5.98 @ ( collect @ A
% 11.90/5.98 @ ^ [Uu: A] :
% 11.90/5.98 ? [L3: B] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( F @ L3 ) )
% 11.90/5.98 & ( member @ B @ L3 @ B6 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2809_Gcd__fin_Oinfinite,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semiring_gcd @ A )
% 11.90/5.98 => ! [A4: set @ A] :
% 11.90/5.98 ( ~ ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( semiring_gcd_Gcd_fin @ A @ A4 )
% 11.90/5.98 = ( one_one @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2810_is__unit__Gcd__fin__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semiring_gcd @ A )
% 11.90/5.98 => ! [A4: set @ A] :
% 11.90/5.98 ( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( one_one @ A ) )
% 11.90/5.98 = ( ( semiring_gcd_Gcd_fin @ A @ A4 )
% 11.90/5.98 = ( one_one @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2811_Eps__Opt__eq__None,axiom,
% 11.90/5.98 ! [A: $tType,P: A > $o] :
% 11.90/5.98 ( ( ( eps_Opt @ A @ P )
% 11.90/5.98 = ( none @ A ) )
% 11.90/5.98 = ( ~ ? [X8: A] : ( P @ X8 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2812_ran__empty,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( ran @ B @ A
% 11.90/5.98 @ ^ [X3: B] : ( none @ A ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2813_map__update__eta__repair_I2_J,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
% 11.90/5.98 ( ( ( M2 @ K )
% 11.90/5.98 = ( none @ B ) )
% 11.90/5.98 => ( ( ran @ A @ B
% 11.90/5.98 @ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
% 11.90/5.98 = ( insert2 @ B @ V @ ( ran @ A @ B @ M2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2814_ran__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( ran @ A @ B )
% 11.90/5.98 = ( ^ [M: A > ( option @ B )] :
% 11.90/5.98 ( collect @ B
% 11.90/5.98 @ ^ [B5: B] :
% 11.90/5.98 ? [A7: A] :
% 11.90/5.98 ( ( M @ A7 )
% 11.90/5.98 = ( some @ B @ B5 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2815_ranI,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
% 11.90/5.98 ( ( ( M2 @ A3 )
% 11.90/5.98 = ( some @ A @ B2 ) )
% 11.90/5.98 => ( member @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2816_fs__contract,axiom,
% 11.90/5.98 ! [B: $tType,C: $tType,A: $tType,F: A > B > C,S2: set @ C] :
% 11.90/5.98 ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 11.90/5.98 @ ( collect @ ( product_prod @ A @ B )
% 11.90/5.98 @ ^ [Uu: product_prod @ A @ B] :
% 11.90/5.98 ? [P6: product_prod @ A @ B] :
% 11.90/5.98 ( ( Uu = P6 )
% 11.90/5.98 & ( member @ C @ ( F @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) @ S2 ) ) ) )
% 11.90/5.98 = ( collect @ A
% 11.90/5.98 @ ^ [A7: A] :
% 11.90/5.98 ? [B5: B] : ( member @ C @ ( F @ A7 @ B5 ) @ S2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2817_ran__restrictD,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,Y: A,M2: B > ( option @ A ),A4: set @ B] :
% 11.90/5.98 ( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M2 @ A4 ) ) )
% 11.90/5.98 => ? [X2: B] :
% 11.90/5.98 ( ( member @ B @ X2 @ A4 )
% 11.90/5.98 & ( ( M2 @ X2 )
% 11.90/5.98 = ( some @ A @ Y ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2818_Un__interval,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [B1: A,B22: A,B32: A,F: A > B] :
% 11.90/5.98 ( ( ord_less_eq @ A @ B1 @ B22 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ B22 @ B32 )
% 11.90/5.98 => ( ( sup_sup @ ( set @ B )
% 11.90/5.98 @ ( collect @ B
% 11.90/5.98 @ ^ [Uu: B] :
% 11.90/5.98 ? [I3: A] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( F @ I3 ) )
% 11.90/5.98 & ( ord_less_eq @ A @ B1 @ I3 )
% 11.90/5.98 & ( ord_less @ A @ I3 @ B22 ) ) )
% 11.90/5.98 @ ( collect @ B
% 11.90/5.98 @ ^ [Uu: B] :
% 11.90/5.98 ? [I3: A] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( F @ I3 ) )
% 11.90/5.98 & ( ord_less_eq @ A @ B22 @ I3 )
% 11.90/5.98 & ( ord_less @ A @ I3 @ B32 ) ) ) )
% 11.90/5.98 = ( collect @ B
% 11.90/5.98 @ ^ [Uu: B] :
% 11.90/5.98 ? [I3: A] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( F @ I3 ) )
% 11.90/5.98 & ( ord_less_eq @ A @ B1 @ I3 )
% 11.90/5.98 & ( ord_less @ A @ I3 @ B32 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2819_fold__atLeastAtMost__nat,axiom,
% 11.90/5.98 ! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: A] :
% 11.90/5.98 ( ( finite6289374366891150609ommute @ nat @ A @ F )
% 11.90/5.98 => ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
% 11.90/5.98 = ( finite_fold @ nat @ A @ F @ Acc2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2820_Min_Oinfinite,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A4: set @ A] :
% 11.90/5.98 ( ~ ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( lattic643756798350308766er_Min @ A @ A4 )
% 11.90/5.98 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2821_relcomp__unfold,axiom,
% 11.90/5.98 ! [B: $tType,C: $tType,A: $tType] :
% 11.90/5.98 ( ( relcomp @ A @ C @ B )
% 11.90/5.98 = ( ^ [R3: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
% 11.90/5.98 ( collect @ ( product_prod @ A @ B )
% 11.90/5.98 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.98 @ ^ [X3: A,Z5: B] :
% 11.90/5.98 ? [Y3: C] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R3 )
% 11.90/5.98 & ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z5 ) @ S4 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2822_Max_Oinfinite,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A4: set @ A] :
% 11.90/5.98 ( ~ ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( lattic643756798349783984er_Max @ A @ A4 )
% 11.90/5.98 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2823_graph__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( graph @ A @ B )
% 11.90/5.98 = ( ^ [M: A > ( option @ B )] :
% 11.90/5.98 ( collect @ ( product_prod @ A @ B )
% 11.90/5.98 @ ^ [Uu: product_prod @ A @ B] :
% 11.90/5.98 ? [A7: A,B5: B] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( product_Pair @ A @ B @ A7 @ B5 ) )
% 11.90/5.98 & ( ( M @ A7 )
% 11.90/5.98 = ( some @ B @ B5 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2824_Sup__fin_Oinfinite,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semilattice_sup @ A )
% 11.90/5.98 => ! [A4: set @ A] :
% 11.90/5.98 ( ~ ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( lattic5882676163264333800up_fin @ A @ A4 )
% 11.90/5.98 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2825_Min__add__commute,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( linord4140545234300271783up_add @ A )
% 11.90/5.98 => ! [S2: set @ B,F: B > A,K: A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ S2 )
% 11.90/5.98 => ( ( S2
% 11.90/5.98 != ( bot_bot @ ( set @ B ) ) )
% 11.90/5.98 => ( ( lattic643756798350308766er_Min @ A
% 11.90/5.98 @ ( image2 @ B @ A
% 11.90/5.98 @ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
% 11.90/5.98 @ S2 ) )
% 11.90/5.98 = ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2826_card__le__Suc__Max,axiom,
% 11.90/5.98 ! [S2: set @ nat] :
% 11.90/5.98 ( ( finite_finite2 @ nat @ S2 )
% 11.90/5.98 => ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2827_Max__add__commute,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( linord4140545234300271783up_add @ A )
% 11.90/5.98 => ! [S2: set @ B,F: B > A,K: A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ S2 )
% 11.90/5.98 => ( ( S2
% 11.90/5.98 != ( bot_bot @ ( set @ B ) ) )
% 11.90/5.98 => ( ( lattic643756798349783984er_Max @ A
% 11.90/5.98 @ ( image2 @ B @ A
% 11.90/5.98 @ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ K )
% 11.90/5.98 @ S2 ) )
% 11.90/5.98 = ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2828_set__to__map__ran,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,S2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( set_to_map @ B @ A @ S2 ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2829_map__to__set__upd,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A,V: B] :
% 11.90/5.98 ( ( map_to_set @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V ) ) )
% 11.90/5.98 = ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V )
% 11.90/5.98 @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( map_to_set @ A @ B @ M2 )
% 11.90/5.98 @ ( collect @ ( product_prod @ A @ B )
% 11.90/5.98 @ ^ [Uu: product_prod @ A @ B] :
% 11.90/5.98 ? [V5: B] :
% 11.90/5.98 ( Uu
% 11.90/5.98 = ( product_Pair @ A @ B @ K @ V5 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2830_brk__rel__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( brk_rel @ A @ B )
% 11.90/5.98 = ( ^ [R6: set @ ( product_prod @ A @ B )] :
% 11.90/5.98 ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
% 11.90/5.98 @ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
% 11.90/5.98 @ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
% 11.90/5.98 ? [X3: A,Y3: B] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) )
% 11.90/5.98 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R6 ) ) )
% 11.90/5.98 @ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
% 11.90/5.98 @ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
% 11.90/5.98 ? [X3: A,Y3: B] :
% 11.90/5.98 ( Uu
% 11.90/5.98 = ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X3 ) @ ( product_Pair @ $o @ B @ $false @ Y3 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2831_image2__def,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,C: $tType] :
% 11.90/5.98 ( ( bNF_Greatest_image2 @ C @ A @ B )
% 11.90/5.98 = ( ^ [A9: set @ C,F2: C > A,G2: C > B] :
% 11.90/5.98 ( collect @ ( product_prod @ A @ B )
% 11.90/5.98 @ ^ [Uu: product_prod @ A @ B] :
% 11.90/5.98 ? [A7: C] :
% 11.90/5.98 ( ( Uu
% 11.90/5.98 = ( product_Pair @ A @ B @ ( F2 @ A7 ) @ ( G2 @ A7 ) ) )
% 11.90/5.98 & ( member @ C @ A7 @ A9 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2832_greaterThan__Suc,axiom,
% 11.90/5.98 ! [K: nat] :
% 11.90/5.98 ( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
% 11.90/5.98 = ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert2 @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2833_refl__on__singleton,axiom,
% 11.90/5.98 ! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2834_time__refines,axiom,
% 11.90/5.98 ! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.98 ( ( refine_Imp_refines @ A @ C2 @ C6 )
% 11.90/5.98 => ( ~ ( time_fails @ A @ C6 @ H3 )
% 11.90/5.98 => ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H3 ) @ ( time_time @ A @ C6 @ H3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2835_greaterThan__eq__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( ( set_ord_greaterThan @ A @ X )
% 11.90/5.98 = ( set_ord_greaterThan @ A @ Y ) )
% 11.90/5.98 = ( X = Y ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2836_greaterThan__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ! [I: A,K: A] :
% 11.90/5.98 ( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
% 11.90/5.98 = ( ord_less @ A @ K @ I ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2837_Inf__greaterThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ( comple6319245703460814977attice @ A )
% 11.90/5.98 & ( dense_linorder @ A ) )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
% 11.90/5.98 = X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2838_greaterThan__subset__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
% 11.90/5.98 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2839_Compl__greaterThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [K: A] :
% 11.90/5.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
% 11.90/5.98 = ( set_ord_atMost @ A @ K ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2840_Compl__atMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [K: A] :
% 11.90/5.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
% 11.90/5.98 = ( set_ord_greaterThan @ A @ K ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2841_map__to__set__empty,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( map_to_set @ A @ B
% 11.90/5.98 @ ^ [X3: A] : ( none @ B ) )
% 11.90/5.98 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2842_Sup__greaterThanAtLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( comple6319245703460814977attice @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( ord_less @ A @ X @ ( top_top @ A ) )
% 11.90/5.98 => ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
% 11.90/5.98 = ( top_top @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2843_image__uminus__greaterThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ordered_ab_group_add @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
% 11.90/5.98 = ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2844_image__uminus__lessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ordered_ab_group_add @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
% 11.90/5.98 = ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2845_refl__onD2,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.90/5.98 ( ( refl_on @ A @ A4 @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/5.98 => ( member @ A @ Y @ A4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2846_refl__onD1,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.90/5.98 ( ( refl_on @ A @ A4 @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/5.98 => ( member @ A @ X @ A4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2847_refl__onD,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.90/5.98 ( ( refl_on @ A @ A4 @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ A4 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2848_greaterThan__non__empty,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_top @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( set_ord_greaterThan @ A @ X )
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2849_infinite__Ioi,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ( linorder @ A )
% 11.90/5.98 & ( no_top @ A ) )
% 11.90/5.98 => ! [A3: A] :
% 11.90/5.98 ~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2850_greaterThan__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_ord_greaterThan @ A )
% 11.90/5.98 = ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2851_fails__refines,axiom,
% 11.90/5.98 ! [A: $tType,C2: heap_Time_Heap @ A,C6: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
% 11.90/5.98 ( ( refine_Imp_refines @ A @ C2 @ C6 )
% 11.90/5.98 => ( ( time_fails @ A @ C2 @ H3 )
% 11.90/5.98 => ( time_fails @ A @ C6 @ H3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2852_map__to__set__inverse,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.98 ( ( set_to_map @ A @ B @ ( map_to_set @ A @ B @ M2 ) )
% 11.90/5.98 = M2 ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2853_map__to__set__empty__iff_I1_J,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.98 ( ( ( map_to_set @ A @ B @ M2 )
% 11.90/5.98 = ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.98 = ( M2
% 11.90/5.98 = ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2854_map__to__set__empty__iff_I2_J,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.98 ( ( ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
% 11.90/5.98 = ( map_to_set @ A @ B @ M2 ) )
% 11.90/5.98 = ( M2
% 11.90/5.98 = ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2855_image2__eqI,axiom,
% 11.90/5.98 ! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A4: set @ B] :
% 11.90/5.98 ( ( B2
% 11.90/5.98 = ( F @ X ) )
% 11.90/5.98 => ( ( C2
% 11.90/5.98 = ( G @ X ) )
% 11.90/5.98 => ( ( member @ B @ X @ A4 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A4 @ F @ G ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2856_refl__on__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( refl_on @ A )
% 11.90/5.98 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
% 11.90/5.98 @ ( product_Sigma @ A @ A @ A9
% 11.90/5.98 @ ^ [Uu: A] : A9 ) )
% 11.90/5.98 & ! [X3: A] :
% 11.90/5.98 ( ( member @ A @ X3 @ A9 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2857_refl__onI,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/5.98 @ ( product_Sigma @ A @ A @ A4
% 11.90/5.98 @ ^ [Uu: A] : A4 ) )
% 11.90/5.98 => ( ! [X2: A] :
% 11.90/5.98 ( ( member @ A @ X2 @ A4 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R ) )
% 11.90/5.98 => ( refl_on @ A @ A4 @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2858_map__to__set__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( map_to_set @ A @ B )
% 11.90/5.98 = ( ^ [M: A > ( option @ B )] :
% 11.90/5.98 ( collect @ ( product_prod @ A @ B )
% 11.90/5.98 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.98 @ ^ [K4: A,V2: B] :
% 11.90/5.98 ( ( M @ K4 )
% 11.90/5.98 = ( some @ B @ V2 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2859_INT__greaterThan__UNIV,axiom,
% 11.90/5.98 ( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 11.90/5.98 = ( bot_bot @ ( set @ nat ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2860_map__to__set__ran,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( ran @ B @ A )
% 11.90/5.98 = ( ^ [M: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2861_greaterThan__0,axiom,
% 11.90/5.98 ( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
% 11.90/5.98 = ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2862_refines__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( refine_Imp_refines @ A )
% 11.90/5.98 = ( ^ [P6: heap_Time_Heap @ A,Q5: heap_Time_Heap @ A] :
% 11.90/5.98 ! [H: heap_ext @ product_unit] :
% 11.90/5.98 ( case_option @ $o @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ $true
% 11.90/5.98 @ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ $o
% 11.90/5.98 @ ^ [R3: A] :
% 11.90/5.98 ( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ $o
% 11.90/5.98 @ ^ [H7: heap_ext @ product_unit,T3: nat] :
% 11.90/5.98 ( ( heap_Time_execute @ A @ P6 @ H )
% 11.90/5.98 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T3 ) ) ) ) ) )
% 11.90/5.98 @ ( heap_Time_execute @ A @ Q5 @ H ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2863_refines__empty,axiom,
% 11.90/5.98 ! [A: $tType,M2: heap_Time_Heap @ A] :
% 11.90/5.98 ( refine_Imp_refines @ A @ M2
% 11.90/5.98 @ ( heap_Time_Heap2 @ A
% 11.90/5.98 @ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2864_refines__bind,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,M2: heap_Time_Heap @ A,M7: heap_Time_Heap @ A,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B )] :
% 11.90/5.98 ( ( refine_Imp_refines @ A @ M2 @ M7 )
% 11.90/5.98 => ( ! [X2: A] : ( refine_Imp_refines @ B @ ( F @ X2 ) @ ( F3 @ X2 ) )
% 11.90/5.98 => ( refine_Imp_refines @ B @ ( heap_Time_bind @ A @ B @ M2 @ F ) @ ( heap_Time_bind @ A @ B @ M7 @ F3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2865_refl__on__domain,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( refl_on @ A @ A4 @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ A4 )
% 11.90/5.98 & ( member @ A @ B2 @ A4 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2866_refines__option,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,A3: option @ A,A6: option @ A,M1: heap_Time_Heap @ B,M12: heap_Time_Heap @ B,M22: A > ( heap_Time_Heap @ B ),M23: A > ( heap_Time_Heap @ B )] :
% 11.90/5.98 ( ( A3 = A6 )
% 11.90/5.98 => ( ( refine_Imp_refines @ B @ M1 @ M12 )
% 11.90/5.98 => ( ! [X2: A] : ( refine_Imp_refines @ B @ ( M22 @ X2 ) @ ( M23 @ X2 ) )
% 11.90/5.98 => ( refine_Imp_refines @ B @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M1 @ M22 @ A3 ) @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M12 @ M23 @ A6 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2867_linear__order__on__singleton,axiom,
% 11.90/5.98 ! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2868_Total__subset__Id,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( total_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) )
% 11.90/5.98 => ( ( R
% 11.90/5.98 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 11.90/5.98 | ? [A5: A] :
% 11.90/5.98 ( R
% 11.90/5.98 = ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2869_relation__of__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order_relation_of @ A )
% 11.90/5.98 = ( ^ [P4: A > A > $o,A9: set @ A] :
% 11.90/5.98 ( collect @ ( product_prod @ A @ A )
% 11.90/5.98 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.98 @ ^ [A7: A,B5: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 )
% 11.90/5.98 @ ( product_Sigma @ A @ A @ A9
% 11.90/5.98 @ ^ [Uu: A] : A9 ) )
% 11.90/5.98 & ( P4 @ A7 @ B5 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2870_euclidean__size__times__nonunit,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3725896446679973847miring @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( A3
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ( B2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.98 => ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2871_surj__swap,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
% 11.90/5.98 = ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2872_euclidean__size__integer_Orep__eq,axiom,
% 11.90/5.98 ( ( euclid6346220572633701492n_size @ code_integer )
% 11.90/5.98 = ( ^ [X3: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2873_swap__swap,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
% 11.90/5.98 ( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P5 ) )
% 11.90/5.98 = P5 ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2874_pair__in__Id__conv,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
% 11.90/5.98 = ( A3 = B2 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2875_IdI,axiom,
% 11.90/5.98 ! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2876_swap__simp,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,X: B,Y: A] :
% 11.90/5.98 ( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
% 11.90/5.98 = ( product_Pair @ A @ B @ Y @ X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2877_case__swap,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,C: $tType,F: C > B > A,P5: product_prod @ C @ B] :
% 11.90/5.98 ( ( product_case_prod @ B @ C @ A
% 11.90/5.98 @ ^ [Y3: B,X3: C] : ( F @ X3 @ Y3 )
% 11.90/5.98 @ ( product_swap @ C @ B @ P5 ) )
% 11.90/5.98 = ( product_case_prod @ C @ B @ A @ F @ P5 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2878_swap__comp__swap,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
% 11.90/5.98 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2879_bijective__Id,axiom,
% 11.90/5.98 ! [A: $tType] : ( bijective @ A @ A @ ( id2 @ A ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2880_euclidean__size__1,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.98 => ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
% 11.90/5.98 = ( one_one @ nat ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2881_fst__swap,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,X: product_prod @ B @ A] :
% 11.90/5.98 ( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
% 11.90/5.98 = ( product_snd @ B @ A @ X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2882_snd__swap,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,X: product_prod @ A @ B] :
% 11.90/5.98 ( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
% 11.90/5.98 = ( product_fst @ A @ B @ X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2883_pair__in__swap__image,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,Y: A,X: B,A4: set @ ( product_prod @ B @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A4 ) )
% 11.90/5.98 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A4 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2884_euclidean__size__integer_Oabs__eq,axiom,
% 11.90/5.98 ! [X: int] :
% 11.90/5.98 ( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.98 = ( euclid6346220572633701492n_size @ int @ X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2885_BNF__Greatest__Fixpoint_OIdD,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
% 11.90/5.98 => ( A3 = B2 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2886_IdE,axiom,
% 11.90/5.98 ! [A: $tType,P5: product_prod @ A @ A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ P5 @ ( id2 @ A ) )
% 11.90/5.98 => ~ ! [X2: A] :
% 11.90/5.98 ( P5
% 11.90/5.98 != ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2887_euclidean__size__natural_Orsp,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( euclid6346220572633701492n_size @ nat )
% 11.90/5.98 @ ( euclid6346220572633701492n_size @ nat ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2888_Linear__order__in__diff__Id,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2889_euclidean__size__unit,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3725896446679973847miring @ A )
% 11.90/5.98 => ! [A3: A] :
% 11.90/5.98 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.98 => ( ( euclid6346220572633701492n_size @ A @ A3 )
% 11.90/5.98 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2890_euclidean__size__integer__def,axiom,
% 11.90/5.98 ( ( euclid6346220572633701492n_size @ code_integer )
% 11.90/5.98 = ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2891_euclidean__size__integer_Orsp,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ int @ nat @ nat
% 11.90/5.98 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( euclid6346220572633701492n_size @ int )
% 11.90/5.98 @ ( euclid6346220572633701492n_size @ int ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2892_Id__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( id2 @ A )
% 11.90/5.98 = ( collect @ ( product_prod @ A @ A )
% 11.90/5.98 @ ^ [P6: product_prod @ A @ A] :
% 11.90/5.98 ? [X3: A] :
% 11.90/5.98 ( P6
% 11.90/5.98 = ( product_Pair @ A @ A @ X3 @ X3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2893_unit__iff__euclidean__size,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3725896446679973847miring @ A )
% 11.90/5.98 => ! [A3: A] :
% 11.90/5.98 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.98 = ( ( ( euclid6346220572633701492n_size @ A @ A3 )
% 11.90/5.98 = ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
% 11.90/5.98 & ( A3
% 11.90/5.98 != ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2894_euclidean__size__times__unit,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3725896446679973847miring @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/5.98 => ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
% 11.90/5.98 = ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2895_product__swap,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,A4: set @ B,B6: set @ A] :
% 11.90/5.98 ( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
% 11.90/5.98 @ ( product_Sigma @ B @ A @ A4
% 11.90/5.98 @ ^ [Uu: B] : B6 ) )
% 11.90/5.98 = ( product_Sigma @ A @ B @ B6
% 11.90/5.98 @ ^ [Uu: A] : A4 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2896_prod_Oswap__def,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( product_swap @ A @ B )
% 11.90/5.98 = ( ^ [P6: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P6 ) @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2897_reflcl__set__eq,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( sup_sup @ ( A > A > $o )
% 11.90/5.98 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R )
% 11.90/5.98 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 11.90/5.98 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2898_divmod__cases,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.98 => ! [B2: A,A3: A] :
% 11.90/5.98 ( ( ( B2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 11.90/5.98 = ( zero_zero @ A ) )
% 11.90/5.98 => ( A3
% 11.90/5.98 != ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
% 11.90/5.98 => ( ( ( B2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ! [Q2: A,R2: A] :
% 11.90/5.98 ( ( ( euclid7384307370059645450egment @ A @ R2 )
% 11.90/5.98 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 11.90/5.98 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 11.90/5.98 => ( ( R2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.98 = Q2 )
% 11.90/5.98 => ( ( ( modulo_modulo @ A @ A3 @ B2 )
% 11.90/5.98 = R2 )
% 11.90/5.98 => ( A3
% 11.90/5.98 != ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R2 ) ) ) ) ) ) ) )
% 11.90/5.98 => ( B2
% 11.90/5.98 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2899_mod__eqI,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.98 => ! [B2: A,R: A,Q3: A,A3: A] :
% 11.90/5.98 ( ( B2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 11.90/5.98 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 11.90/5.98 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 11.90/5.98 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R )
% 11.90/5.98 = A3 )
% 11.90/5.98 => ( ( modulo_modulo @ A @ A3 @ B2 )
% 11.90/5.98 = R ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2900_div__bounded,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.98 => ! [B2: A,R: A,Q3: A] :
% 11.90/5.98 ( ( B2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 11.90/5.98 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 11.90/5.98 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 11.90/5.98 => ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R ) @ B2 )
% 11.90/5.98 = Q3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2901_div__eqI,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.98 => ! [B2: A,R: A,Q3: A,A3: A] :
% 11.90/5.98 ( ( B2
% 11.90/5.98 != ( zero_zero @ A ) )
% 11.90/5.98 => ( ( ( euclid7384307370059645450egment @ A @ R )
% 11.90/5.98 = ( euclid7384307370059645450egment @ A @ B2 ) )
% 11.90/5.98 => ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
% 11.90/5.98 => ( ( ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R )
% 11.90/5.98 = A3 )
% 11.90/5.98 => ( ( divide_divide @ A @ A3 @ B2 )
% 11.90/5.98 = Q3 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2902_division__segment__integer_Orep__eq,axiom,
% 11.90/5.98 ! [X: code_integer] :
% 11.90/5.98 ( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X ) )
% 11.90/5.98 = ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2903_division__segment__1,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.98 => ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
% 11.90/5.98 = ( one_one @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2904_division__segment__numeral,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.98 => ! [K: num] :
% 11.90/5.98 ( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
% 11.90/5.98 = ( one_one @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2905_division__segment__of__nat,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid5411537665997757685th_nat @ A )
% 11.90/5.98 => ! [N: nat] :
% 11.90/5.98 ( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
% 11.90/5.98 = ( one_one @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2906_division__segment__integer_Oabs__eq,axiom,
% 11.90/5.98 ! [X: int] :
% 11.90/5.98 ( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X ) )
% 11.90/5.98 = ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2907_division__segment__nat__def,axiom,
% 11.90/5.98 ( ( euclid7384307370059645450egment @ nat )
% 11.90/5.98 = ( ^ [N5: nat] : ( one_one @ nat ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2908_division__segment__integer_Orsp,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ int @ int @ int
% 11.90/5.98 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.98 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.98 @ ( euclid7384307370059645450egment @ int )
% 11.90/5.98 @ ( euclid7384307370059645450egment @ int ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2909_division__segment__natural_Orsp,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ nat @ nat
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( euclid7384307370059645450egment @ nat )
% 11.90/5.98 @ ( euclid7384307370059645450egment @ nat ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2910_division__segment__integer__def,axiom,
% 11.90/5.98 ( ( euclid7384307370059645450egment @ code_integer )
% 11.90/5.98 = ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2911_is__unit__division__segment,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( euclid3128863361964157862miring @ A )
% 11.90/5.98 => ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2912_Linear__order__wf__diff__Id,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.98 => ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
% 11.90/5.98 = ( ! [A9: set @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( A9
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 => ? [X3: A] :
% 11.90/5.98 ( ( member @ A @ X3 @ A9 )
% 11.90/5.98 & ! [Y3: A] :
% 11.90/5.98 ( ( member @ A @ Y3 @ A9 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2913_bsqr__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( bNF_Wellorder_bsqr @ A )
% 11.90/5.98 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
% 11.90/5.98 @ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
% 11.90/5.98 @ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
% 11.90/5.98 @ ^ [A12: A,A23: A] :
% 11.90/5.98 ( product_case_prod @ A @ A @ $o
% 11.90/5.98 @ ^ [B12: A,B23: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B12 @ ( insert2 @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R3 ) )
% 11.90/5.98 & ( ( ( A12 = B12 )
% 11.90/5.98 & ( A23 = B23 ) )
% 11.90/5.98 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) )
% 11.90/5.98 | ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 )
% 11.90/5.98 = ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B12 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) )
% 11.90/5.98 | ( ( ( bNF_We1388413361240627857o_max2 @ A @ R3 @ A12 @ A23 )
% 11.90/5.98 = ( bNF_We1388413361240627857o_max2 @ A @ R3 @ B12 @ B23 ) )
% 11.90/5.98 & ( A12 = B12 )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2914_wo__rel_Ocases__Total3,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
% 11.90/5.98 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) )
% 11.90/5.98 => ( Phi @ A3 @ B2 ) )
% 11.90/5.98 => ( ( ( A3 = B2 )
% 11.90/5.98 => ( Phi @ A3 @ B2 ) )
% 11.90/5.98 => ( Phi @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2915_ivl__disj__un__two_I5_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( ord_less @ A @ L @ M2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2916_ivl__disj__un__singleton_I4_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2917_finite__greaterThanAtMost,axiom,
% 11.90/5.98 ! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2918_finite__greaterThanAtMost__int,axiom,
% 11.90/5.98 ! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2919_finite__greaterThanAtMost__integer,axiom,
% 11.90/5.98 ! [L: code_integer,U: code_integer] : ( finite_finite2 @ code_integer @ ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2920_card__greaterThanAtMost,axiom,
% 11.90/5.98 ! [L: nat,U: nat] :
% 11.90/5.98 ( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
% 11.90/5.98 = ( minus_minus @ nat @ U @ L ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2921_greaterThanAtMost__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ! [I: A,L: A,U: A] :
% 11.90/5.98 ( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 11.90/5.98 = ( ( ord_less @ A @ L @ I )
% 11.90/5.98 & ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2922_greaterThanAtMost__empty,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,K: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ K )
% 11.90/5.98 => ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2923_greaterThanAtMost__empty__iff2,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( preorder @ A )
% 11.90/5.98 => ! [K: A,L: A] :
% 11.90/5.98 ( ( ( bot_bot @ ( set @ A ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ K @ L ) )
% 11.90/5.98 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2924_greaterThanAtMost__empty__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( preorder @ A )
% 11.90/5.98 => ! [K: A,L: A] :
% 11.90/5.98 ( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 = ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2925_infinite__Ioc__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( dense_linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
% 11.90/5.98 = ( ord_less @ A @ A3 @ B2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2926_image__add__greaterThanAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_semidom @ A )
% 11.90/5.98 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2927_Sup__greaterThanAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( comple6319245703460814977attice @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( ord_less @ A @ X @ Y )
% 11.90/5.98 => ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
% 11.90/5.98 = Y ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2928_Inf__greaterThanAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ( comple6319245703460814977attice @ A )
% 11.90/5.98 & ( dense_linorder @ A ) )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( ord_less @ A @ X @ Y )
% 11.90/5.98 => ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
% 11.90/5.98 = X ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2929_brk__rel__wf,axiom,
% 11.90/5.98 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( wf @ A @ R5 )
% 11.90/5.98 => ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R5 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2930_card__greaterThanAtMost__int,axiom,
% 11.90/5.98 ! [L: int,U: int] :
% 11.90/5.98 ( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
% 11.90/5.98 = ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2931_image__minus__const__greaterThanAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_idom @ A )
% 11.90/5.98 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
% 11.90/5.98 = ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2932_image__diff__atLeastLessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_idom @ A )
% 11.90/5.98 => ! [C2: A,A3: A,B2: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2933_image__uminus__greaterThanAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ordered_ab_group_add @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
% 11.90/5.98 = ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2934_image__uminus__atLeastLessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ordered_ab_group_add @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2935_reduction__pairI,axiom,
% 11.90/5.98 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( wf @ A @ R5 )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S2 ) @ R5 )
% 11.90/5.98 => ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2936_wo__rel_Omax2__equals1,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 11.90/5.98 = A3 )
% 11.90/5.98 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2937_wo__rel_Omax2__equals2,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 11.90/5.98 = B2 )
% 11.90/5.98 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2938_wo__rel_Omax2__greater,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2939_wf__induct__rule,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ( ! [X2: A] :
% 11.90/5.98 ( ! [Y6: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 11.90/5.98 => ( P @ Y6 ) )
% 11.90/5.98 => ( P @ X2 ) )
% 11.90/5.98 => ( P @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2940_wf__eq__minimal,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( wf @ A )
% 11.90/5.98 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ! [Q6: set @ A] :
% 11.90/5.98 ( ? [X3: A] : ( member @ A @ X3 @ Q6 )
% 11.90/5.98 => ? [X3: A] :
% 11.90/5.98 ( ( member @ A @ X3 @ Q6 )
% 11.90/5.98 & ! [Y3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
% 11.90/5.98 => ~ ( member @ A @ Y3 @ Q6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2941_wf__not__refl,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2942_wf__not__sym,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2943_wf__irrefl,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2944_wf__induct,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ( ! [X2: A] :
% 11.90/5.98 ( ! [Y6: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 11.90/5.98 => ( P @ Y6 ) )
% 11.90/5.98 => ( P @ X2 ) )
% 11.90/5.98 => ( P @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2945_wf__asym,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2946_wfUNIVI,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ! [P7: A > $o,X2: A] :
% 11.90/5.98 ( ! [Xa2: A] :
% 11.90/5.98 ( ! [Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
% 11.90/5.98 => ( P7 @ Y4 ) )
% 11.90/5.98 => ( P7 @ Xa2 ) )
% 11.90/5.98 => ( P7 @ X2 ) )
% 11.90/5.98 => ( wf @ A @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2947_wfI__min,axiom,
% 11.90/5.98 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ! [X2: A,Q7: set @ A] :
% 11.90/5.98 ( ( member @ A @ X2 @ Q7 )
% 11.90/5.98 => ? [Xa2: A] :
% 11.90/5.98 ( ( member @ A @ Xa2 @ Q7 )
% 11.90/5.98 & ! [Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R5 )
% 11.90/5.98 => ~ ( member @ A @ Y4 @ Q7 ) ) ) )
% 11.90/5.98 => ( wf @ A @ R5 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2948_wfE__min,axiom,
% 11.90/5.98 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
% 11.90/5.98 ( ( wf @ A @ R5 )
% 11.90/5.98 => ( ( member @ A @ X @ Q )
% 11.90/5.98 => ~ ! [Z3: A] :
% 11.90/5.98 ( ( member @ A @ Z3 @ Q )
% 11.90/5.98 => ~ ! [Y6: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R5 )
% 11.90/5.98 => ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2949_wf__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( wf @ A )
% 11.90/5.98 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ! [P4: A > $o] :
% 11.90/5.98 ( ! [X3: A] :
% 11.90/5.98 ( ! [Y3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
% 11.90/5.98 => ( P4 @ Y3 ) )
% 11.90/5.98 => ( P4 @ X3 ) )
% 11.90/5.98 => ! [X8: A] : ( P4 @ X8 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2950_wo__rel_Omax2__def,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 => ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 11.90/5.98 = B2 ) )
% 11.90/5.98 & ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 => ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
% 11.90/5.98 = A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2951_wo__rel_Owell__order__induct,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ! [X2: A] :
% 11.90/5.98 ( ! [Y6: A] :
% 11.90/5.98 ( ( ( Y6 != X2 )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R ) )
% 11.90/5.98 => ( P @ Y6 ) )
% 11.90/5.98 => ( P @ X2 ) )
% 11.90/5.98 => ( P @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2952_Ioc__inj,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.98 & ( ord_less_eq @ A @ D3 @ C2 ) )
% 11.90/5.98 | ( ( A3 = C2 )
% 11.90/5.98 & ( B2 = D3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2953_wo__rel_OTOTALS,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ! [X7: A] :
% 11.90/5.98 ( ( member @ A @ X7 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ! [Xa2: A] :
% 11.90/5.98 ( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa2 ) @ R )
% 11.90/5.98 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X7 ) @ R ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2954_wo__rel_Omax2__greater__among,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
% 11.90/5.98 & ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2955_wfE__min_H,axiom,
% 11.90/5.98 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),Q: set @ A] :
% 11.90/5.98 ( ( wf @ A @ R5 )
% 11.90/5.98 => ( ( Q
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 => ~ ! [Z3: A] :
% 11.90/5.98 ( ( member @ A @ Z3 @ Q )
% 11.90/5.98 => ~ ! [Y6: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z3 ) @ R5 )
% 11.90/5.98 => ~ ( member @ A @ Y6 @ Q ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2956_infinite__Ioc,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( dense_linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.98 => ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2957_Ioc__subset__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.98 | ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.98 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2958_atLeastSucAtMost__greaterThanAtMost,axiom,
% 11.90/5.98 ! [L: nat,U: nat] :
% 11.90/5.98 ( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2959_ivl__disj__un__two_I6_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2960_well__order__induct__imp,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ! [X2: A] :
% 11.90/5.98 ( ! [Y6: A] :
% 11.90/5.98 ( ( ( Y6 != X2 )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R ) )
% 11.90/5.98 => ( ( member @ A @ Y6 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( P @ Y6 ) ) )
% 11.90/5.98 => ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( P @ X2 ) ) )
% 11.90/5.98 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( P @ A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2961_wf__bounded__measure,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
% 11.90/5.98 ( ! [A5: A,B3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R )
% 11.90/5.98 => ( ( ord_less_eq @ nat @ ( Ub @ B3 ) @ ( Ub @ A5 ) )
% 11.90/5.98 & ( ord_less_eq @ nat @ ( F @ B3 ) @ ( Ub @ A5 ) )
% 11.90/5.98 & ( ord_less @ nat @ ( F @ A5 ) @ ( F @ B3 ) ) ) )
% 11.90/5.98 => ( wf @ A @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2962_wfI,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/5.98 @ ( product_Sigma @ A @ A @ A4
% 11.90/5.98 @ ^ [Uu: A] : B6 ) )
% 11.90/5.98 => ( ! [X2: A,P7: A > $o] :
% 11.90/5.98 ( ! [Xa2: A] :
% 11.90/5.98 ( ! [Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
% 11.90/5.98 => ( P7 @ Y4 ) )
% 11.90/5.98 => ( P7 @ Xa2 ) )
% 11.90/5.98 => ( ( member @ A @ X2 @ A4 )
% 11.90/5.98 => ( ( member @ A @ X2 @ B6 )
% 11.90/5.98 => ( P7 @ X2 ) ) ) )
% 11.90/5.98 => ( wf @ A @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2963_ivl__disj__un__two_I8_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2964_ivl__disj__un__one_I3_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 11.90/5.98 = ( set_ord_atMost @ A @ U ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2965_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 11.90/5.98 ! [L: int,U: int] :
% 11.90/5.98 ( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2966_ivl__disj__un__one_I5_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 11.90/5.98 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2967_wf__eq__minimal2,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( wf @ A )
% 11.90/5.98 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ! [A9: set @ A] :
% 11.90/5.98 ( ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R3 ) )
% 11.90/5.98 & ( A9
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.98 => ? [X3: A] :
% 11.90/5.98 ( ( member @ A @ X3 @ A9 )
% 11.90/5.98 & ! [Y3: A] :
% 11.90/5.98 ( ( member @ A @ Y3 @ A9 )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2968_wf__bounded__set,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
% 11.90/5.98 ( ! [A5: A,B3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R )
% 11.90/5.98 => ( ( finite_finite2 @ B @ ( Ub @ A5 ) )
% 11.90/5.98 & ( ord_less_eq @ ( set @ B ) @ ( Ub @ B3 ) @ ( Ub @ A5 ) )
% 11.90/5.98 & ( ord_less_eq @ ( set @ B ) @ ( F @ B3 ) @ ( Ub @ A5 ) )
% 11.90/5.98 & ( ord_less @ ( set @ B ) @ ( F @ A5 ) @ ( F @ B3 ) ) ) )
% 11.90/5.98 => ( wf @ A @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2969_wo__rel_Ocases__Total,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 => ( Phi @ A3 @ B2 ) )
% 11.90/5.98 => ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
% 11.90/5.98 => ( Phi @ A3 @ B2 ) )
% 11.90/5.98 => ( Phi @ A3 @ B2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2970_sum_Ohead,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.98 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.98 = ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2971_prod_Ohead,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( comm_monoid_mult @ A )
% 11.90/5.98 => ! [M2: nat,N: nat,G: nat > A] :
% 11.90/5.98 ( ( ord_less_eq @ nat @ M2 @ N )
% 11.90/5.98 => ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.90/5.98 = ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2972_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( dense_linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.98 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2973_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( dense_linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.98 & ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2974_ivl__disj__un__two__touch_I3_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( ord_less @ A @ L @ M2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ M2 @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2975_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ( ( set_or3652927894154168847AtMost @ A )
% 11.90/5.98 = ( ^ [A7: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A7 @ B5 ) @ ( insert2 @ A @ A7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2976_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( dense_linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( ( ord_less @ A @ A3 @ B2 )
% 11.90/5.98 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 11.90/5.98 & ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2977_ivl__disj__un__two_I2_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ M2 )
% 11.90/5.98 => ( ( ord_less @ A @ M2 @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 11.90/5.98 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2978_ivl__disj__un__two__touch_I1_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( ord_less @ A @ L @ M2 )
% 11.90/5.98 => ( ( ord_less @ A @ M2 @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 11.90/5.98 = ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2979_wf__bounded__supset,axiom,
% 11.90/5.98 ! [A: $tType,S2: set @ A] :
% 11.90/5.98 ( ( finite_finite2 @ A @ S2 )
% 11.90/5.98 => ( wf @ ( set @ A )
% 11.90/5.98 @ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 11.90/5.98 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 11.90/5.98 @ ^ [Q8: set @ A,Q6: set @ A] :
% 11.90/5.98 ( ( ord_less @ ( set @ A ) @ Q6 @ Q8 )
% 11.90/5.98 & ( ord_less_eq @ ( set @ A ) @ Q8 @ S2 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2980_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
% 11.90/5.98 ! [L: code_integer,U: code_integer] :
% 11.90/5.98 ( ( set_or1337092689740270186AtMost @ code_integer @ ( plus_plus @ code_integer @ L @ ( one_one @ code_integer ) ) @ U )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ code_integer @ L @ U ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2981_ivl__disj__un__singleton_I5_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2982_wf__iff__no__infinite__down__chain,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( wf @ A )
% 11.90/5.98 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ~ ? [F2: nat > A] :
% 11.90/5.98 ! [I3: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) @ R3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2983_wf__no__infinite__down__chainE,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
% 11.90/5.98 ( ( wf @ A @ R )
% 11.90/5.98 => ~ ! [K5: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K5 ) ) @ ( F @ K5 ) ) @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2984_wo__rel_Ominim__least,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ B2 @ B6 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) @ B2 ) @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2985_wo__rel_Oequals__minim,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
% 11.90/5.98 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( member @ A @ A3 @ B6 )
% 11.90/5.98 => ( ! [B3: A] :
% 11.90/5.98 ( ( member @ A @ B3 @ B6 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) )
% 11.90/5.98 => ( A3
% 11.90/5.98 = ( bNF_We6954850376910717587_minim @ A @ R @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2986_dependent__wf__choice,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
% 11.90/5.98 ( ( wf @ A @ R5 )
% 11.90/5.98 => ( ! [F4: A > B,G4: A > B,X2: A,R2: B] :
% 11.90/5.98 ( ! [Z6: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X2 ) @ R5 )
% 11.90/5.98 => ( ( F4 @ Z6 )
% 11.90/5.98 = ( G4 @ Z6 ) ) )
% 11.90/5.98 => ( ( P @ F4 @ X2 @ R2 )
% 11.90/5.98 = ( P @ G4 @ X2 @ R2 ) ) )
% 11.90/5.98 => ( ! [X2: A,F4: A > B] :
% 11.90/5.98 ( ! [Y6: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R5 )
% 11.90/5.98 => ( P @ F4 @ Y6 @ ( F4 @ Y6 ) ) )
% 11.90/5.98 => ? [X_1: B] : ( P @ F4 @ X2 @ X_1 ) )
% 11.90/5.98 => ? [F4: A > B] :
% 11.90/5.98 ! [X7: A] : ( P @ F4 @ X7 @ ( F4 @ X7 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2987_of__rat__neg__one,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( field_char_0 @ A )
% 11.90/5.98 => ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
% 11.90/5.98 = ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2988_Ref__Time_Oupdate__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( heap @ A )
% 11.90/5.98 => ( ( ref_update @ A )
% 11.90/5.98 = ( ^ [R3: ref @ A,V2: A] :
% 11.90/5.98 ( heap_Time_heap @ product_unit
% 11.90/5.98 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R3 @ V2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2989_sum__div__partition,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( euclid4440199948858584721cancel @ A )
% 11.90/5.98 => ! [A4: set @ B,F: B > A,B2: A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A4 ) @ B2 )
% 11.90/5.98 = ( plus_plus @ A
% 11.90/5.98 @ ( groups7311177749621191930dd_sum @ B @ A
% 11.90/5.98 @ ^ [A7: B] : ( divide_divide @ A @ ( F @ A7 ) @ B2 )
% 11.90/5.98 @ ( inf_inf @ ( set @ B ) @ A4
% 11.90/5.98 @ ( collect @ B
% 11.90/5.98 @ ^ [A7: B] : ( dvd_dvd @ A @ B2 @ ( F @ A7 ) ) ) ) )
% 11.90/5.98 @ ( divide_divide @ A
% 11.90/5.98 @ ( groups7311177749621191930dd_sum @ B @ A @ F
% 11.90/5.98 @ ( inf_inf @ ( set @ B ) @ A4
% 11.90/5.98 @ ( collect @ B
% 11.90/5.98 @ ^ [A7: B] :
% 11.90/5.98 ~ ( dvd_dvd @ A @ B2 @ ( F @ A7 ) ) ) ) )
% 11.90/5.98 @ B2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2990_bsqr__max2,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A1: A,A22: A,B1: A,B22: A] :
% 11.90/5.98 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B1 @ B22 ) ) @ ( bNF_Wellorder_bsqr @ A @ R ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B22 ) ) @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2991_UN__atLeast__UNIV,axiom,
% 11.90/5.98 ( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
% 11.90/5.98 = ( top_top @ ( set @ nat ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2992_atLeast__eq__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( ( set_ord_atLeast @ A @ X )
% 11.90/5.98 = ( set_ord_atLeast @ A @ Y ) )
% 11.90/5.98 = ( X = Y ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2993_unit__abs__eta__conv,axiom,
% 11.90/5.98 ! [A: $tType,F: product_unit > A] :
% 11.90/5.98 ( ( ^ [U2: product_unit] : ( F @ product_Unity ) )
% 11.90/5.98 = F ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2994_inf__Some,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( inf @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( inf_inf @ ( option @ A ) @ ( some @ A @ X ) @ ( some @ A @ Y ) )
% 11.90/5.98 = ( some @ A @ ( inf_inf @ A @ X @ Y ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2995_atLeast__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ! [I: A,K: A] :
% 11.90/5.98 ( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
% 11.90/5.98 = ( ord_less_eq @ A @ K @ I ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2996_Inf__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( comple6319245703460814977attice @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X ) )
% 11.90/5.98 = X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2997_atLeast__subset__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( preorder @ A )
% 11.90/5.98 => ! [X: A,Y: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
% 11.90/5.98 = ( ord_less_eq @ A @ Y @ X ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2998_image__add__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_semidom @ A )
% 11.90/5.98 => ! [K: A,I: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_2999_Sup__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( comple6319245703460814977attice @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
% 11.90/5.98 = ( top_top @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3000_inter__compl__diff__conv,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
% 11.90/5.98 = ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3001_Int__atMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_atMost @ A @ B2 ) )
% 11.90/5.98 = ( set_ord_atMost @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3002_Compl__lessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [K: A] :
% 11.90/5.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ K ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3003_Compl__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [K: A] :
% 11.90/5.98 ( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
% 11.90/5.98 = ( set_ord_lessThan @ A @ K ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3004_one__eq__of__rat__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( field_char_0 @ A )
% 11.90/5.98 => ! [A3: rat] :
% 11.90/5.98 ( ( ( one_one @ A )
% 11.90/5.98 = ( field_char_0_of_rat @ A @ A3 ) )
% 11.90/5.98 = ( ( one_one @ rat )
% 11.90/5.98 = A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3005_of__rat__eq__1__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( field_char_0 @ A )
% 11.90/5.98 => ! [A3: rat] :
% 11.90/5.98 ( ( ( field_char_0_of_rat @ A @ A3 )
% 11.90/5.98 = ( one_one @ A ) )
% 11.90/5.98 = ( A3
% 11.90/5.98 = ( one_one @ rat ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3006_of__rat__1,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( field_char_0 @ A )
% 11.90/5.98 => ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
% 11.90/5.98 = ( one_one @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3007_atLeast__0,axiom,
% 11.90/5.98 ( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
% 11.90/5.98 = ( top_top @ ( set @ nat ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3008_Icc__subset__Ici__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( preorder @ A )
% 11.90/5.98 => ! [L: A,H3: A,L4: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H3 ) @ ( set_ord_atLeast @ A @ L4 ) )
% 11.90/5.98 = ( ~ ( ord_less_eq @ A @ L @ H3 )
% 11.90/5.98 | ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3009_Int__atLeastAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3010_Int__atLeastLessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3011_Int__atLeastAtMostR1,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3012_Int__atLeastAtMostL1,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ A3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3013_image__minus__const__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_idom @ A )
% 11.90/5.98 => ! [C2: A,A3: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A3 ) )
% 11.90/5.98 = ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3014_image__minus__const__AtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_idom @ A )
% 11.90/5.98 => ! [C2: A,B2: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B2 ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3015_image__uminus__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ordered_ab_group_add @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
% 11.90/5.98 = ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3016_image__uminus__atMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ordered_ab_group_add @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3017_Int__atLeastAtMostL2,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_ord_atLeast @ A @ C2 ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3018_Int__atLeastAtMostR2,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,C2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ D3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3019_Int__greaterThanAtMost,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3020_Int__greaterThanLessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A3 @ C2 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3021_one__le__of__rat__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_field @ A )
% 11.90/5.98 => ! [R: rat] :
% 11.90/5.98 ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 11.90/5.98 = ( ord_less_eq @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3022_of__rat__le__1__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_field @ A )
% 11.90/5.98 => ! [R: rat] :
% 11.90/5.98 ( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 11.90/5.98 = ( ord_less_eq @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3023_one__less__of__rat__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_field @ A )
% 11.90/5.98 => ! [R: rat] :
% 11.90/5.98 ( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
% 11.90/5.98 = ( ord_less @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3024_of__rat__less__1__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linordered_field @ A )
% 11.90/5.98 => ! [R: rat] :
% 11.90/5.98 ( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
% 11.90/5.98 = ( ord_less @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3025_atLeastLessThan__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_or7035219750837199246ssThan @ A )
% 11.90/5.98 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L3 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3026_well__order__on__domain,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.98 ( ( order_well_order_on @ A @ A4 @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 => ( ( member @ A @ A3 @ A4 )
% 11.90/5.98 & ( member @ A @ B2 @ A4 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3027_Inf__unit__def,axiom,
% 11.90/5.98 ( ( complete_Inf_Inf @ product_unit )
% 11.90/5.98 = ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3028_Sup__unit__def,axiom,
% 11.90/5.98 ( ( complete_Sup_Sup @ product_unit )
% 11.90/5.98 = ( ^ [Uu3: set @ product_unit] : product_Unity ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3029_Times__Int__distrib1,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ A,C5: set @ B] :
% 11.90/5.98 ( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 @ ^ [Uu: A] : C5 )
% 11.90/5.98 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.98 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.98 @ ^ [Uu: A] : C5 )
% 11.90/5.98 @ ( product_Sigma @ A @ B @ B6
% 11.90/5.98 @ ^ [Uu: A] : C5 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3030_Sigma__Int__distrib2,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ B )] :
% 11.90/5.98 ( ( product_Sigma @ A @ B @ I4
% 11.90/5.98 @ ^ [I3: A] : ( inf_inf @ ( set @ B ) @ ( A4 @ I3 ) @ ( B6 @ I3 ) ) )
% 11.90/5.98 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) @ ( product_Sigma @ A @ B @ I4 @ B6 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3031_Times__Int__Times,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
% 11.90/5.98 ( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
% 11.90/5.98 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.98 @ ^ [Uu: A] : B6 )
% 11.90/5.98 @ ( product_Sigma @ A @ B @ C5
% 11.90/5.98 @ ^ [Uu: A] : D4 ) )
% 11.90/5.98 = ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A4 @ C5 )
% 11.90/5.98 @ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B6 @ D4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3032_Sigma__Int__distrib1,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,I4: set @ A,J4: set @ A,C5: A > ( set @ B )] :
% 11.90/5.98 ( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I4 @ J4 ) @ C5 )
% 11.90/5.98 = ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I4 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3033_old_Ounit_Oexhaust,axiom,
% 11.90/5.98 ! [Y: product_unit] : ( Y = product_Unity ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3034_top__unit__def,axiom,
% 11.90/5.98 ( ( top_top @ product_unit )
% 11.90/5.98 = product_Unity ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3035_uminus__unit__def,axiom,
% 11.90/5.98 ( ( uminus_uminus @ product_unit )
% 11.90/5.98 = ( ^ [Uu3: product_unit] : product_Unity ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3036_bot__unit__def,axiom,
% 11.90/5.98 ( ( bot_bot @ product_unit )
% 11.90/5.98 = product_Unity ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3037_ivl__disj__int__one_I8_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3038_sup__unit__def,axiom,
% 11.90/5.98 ( ( sup_sup @ product_unit )
% 11.90/5.98 = ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3039_atLeastAtMost__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_or1337092689740270186AtMost @ A )
% 11.90/5.98 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L3 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3040_ivl__disj__int__one_I6_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3041_disjointI,axiom,
% 11.90/5.98 ! [A: $tType,A3: set @ A,B2: set @ A] :
% 11.90/5.98 ( ! [X2: A] :
% 11.90/5.98 ( ( member @ A @ X2 @ A3 )
% 11.90/5.98 => ~ ( member @ A @ X2 @ B2 ) )
% 11.90/5.98 => ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3042_inter__eq__subsetI,axiom,
% 11.90/5.98 ! [A: $tType,S2: set @ A,S7: set @ A,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ S2 @ S7 )
% 11.90/5.98 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ S7 )
% 11.90/5.98 = ( inf_inf @ ( set @ A ) @ B6 @ S7 ) )
% 11.90/5.98 => ( ( inf_inf @ ( set @ A ) @ A4 @ S2 )
% 11.90/5.98 = ( inf_inf @ ( set @ A ) @ B6 @ S2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3043_not__empty__eq__Ici__eq__empty,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( preorder @ A )
% 11.90/5.98 => ! [L: A] :
% 11.90/5.98 ( ( bot_bot @ ( set @ A ) )
% 11.90/5.98 != ( set_ord_atLeast @ A @ L ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3044_infinite__Ici,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ( linorder @ A )
% 11.90/5.98 & ( no_top @ A ) )
% 11.90/5.98 => ! [A3: A] :
% 11.90/5.98 ~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3045_not__UNIV__eq__Ici,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_bot @ A )
% 11.90/5.98 => ! [L4: A] :
% 11.90/5.98 ( ( top_top @ ( set @ A ) )
% 11.90/5.98 != ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3046_not__Ici__eq__Icc,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_top @ A )
% 11.90/5.98 => ! [L4: A,L: A,H3: A] :
% 11.90/5.98 ( ( set_ord_atLeast @ A @ L4 )
% 11.90/5.98 != ( set_or1337092689740270186AtMost @ A @ L @ H3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3047_not__Iic__eq__Ici,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_top @ A )
% 11.90/5.98 => ! [H3: A,L4: A] :
% 11.90/5.98 ( ( set_ord_atMost @ A @ H3 )
% 11.90/5.98 != ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3048_atMost__Int__atLeast,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [N: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
% 11.90/5.98 = ( insert2 @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3049_atLeast__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_ord_atLeast @ A )
% 11.90/5.98 = ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3050_inf__option__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( inf @ A )
% 11.90/5.98 => ( ( inf_inf @ ( option @ A ) )
% 11.90/5.98 = ( ^ [X3: option @ A,Y3: option @ A] :
% 11.90/5.98 ( case_option @ ( option @ A ) @ A @ ( none @ A )
% 11.90/5.98 @ ^ [Z5: A] :
% 11.90/5.98 ( case_option @ ( option @ A ) @ A @ ( none @ A )
% 11.90/5.98 @ ^ [Aa2: A] : ( some @ A @ ( inf_inf @ A @ Z5 @ Aa2 ) )
% 11.90/5.98 @ Y3 )
% 11.90/5.98 @ X3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3051_translation__Int,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ab_group_add @ A )
% 11.90/5.98 => ! [A3: A,S: set @ A,T2: set @ A] :
% 11.90/5.98 ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
% 11.90/5.98 = ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3052_disjoint__mono,axiom,
% 11.90/5.98 ! [A: $tType,A3: set @ A,A6: set @ A,B2: set @ A,B4: set @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ A3 @ A6 )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ B4 )
% 11.90/5.98 => ( ( ( inf_inf @ ( set @ A ) @ A6 @ B4 )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 => ( ( inf_inf @ ( set @ A ) @ A3 @ B2 )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3053_disjoint__alt__simp2,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 != A4 )
% 11.90/5.98 = ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3054_disjoint__alt__simp1,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( ( minus_minus @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 = A4 )
% 11.90/5.98 = ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3055_ivl__disj__int__two_I3_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3056_ivl__disj__int__two_I6_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3057_greaterThan__Int__greaterThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
% 11.90/5.98 = ( set_ord_lessThan @ A @ ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3058_atLeast__eq__UNIV__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order_bot @ A )
% 11.90/5.98 => ! [X: A] :
% 11.90/5.98 ( ( ( set_ord_atLeast @ A @ X )
% 11.90/5.98 = ( top_top @ ( set @ A ) ) )
% 11.90/5.98 = ( X
% 11.90/5.98 = ( bot_bot @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3059_not__UNIV__le__Ici,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_bot @ A )
% 11.90/5.98 => ! [L: A] :
% 11.90/5.98 ~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3060_UNIV__unit,axiom,
% 11.90/5.98 ( ( top_top @ ( set @ product_unit ) )
% 11.90/5.98 = ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3061_not__Ici__le__Icc,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_top @ A )
% 11.90/5.98 => ! [L: A,L4: A,H4: A] :
% 11.90/5.98 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3062_lessThan__Int__lessThan,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
% 11.90/5.98 = ( set_ord_greaterThan @ A @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3063_not__Iic__le__Ici,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_bot @ A )
% 11.90/5.98 => ! [H3: A,L4: A] :
% 11.90/5.98 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H3 ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3064_not__Ici__le__Iic,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( no_top @ A )
% 11.90/5.98 => ! [L: A,H4: A] :
% 11.90/5.98 ~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3065_translation__subtract__Int,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ab_group_add @ A )
% 11.90/5.98 => ! [A3: A,S: set @ A,T2: set @ A] :
% 11.90/5.98 ( ( image2 @ A @ A
% 11.90/5.98 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.98 @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
% 11.90/5.98 = ( inf_inf @ ( set @ A )
% 11.90/5.98 @ ( image2 @ A @ A
% 11.90/5.98 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.98 @ S )
% 11.90/5.98 @ ( image2 @ A @ A
% 11.90/5.98 @ ^ [X3: A] : ( minus_minus @ A @ X3 @ A3 )
% 11.90/5.98 @ T2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3066_of__rat__add,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( field_char_0 @ A )
% 11.90/5.98 => ! [A3: rat,B2: rat] :
% 11.90/5.98 ( ( field_char_0_of_rat @ A @ ( plus_plus @ rat @ A3 @ B2 ) )
% 11.90/5.98 = ( plus_plus @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3067_Ioi__le__Ico,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( preorder @ A )
% 11.90/5.98 => ! [A3: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_atLeast @ A @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3068_atLeast__Suc__greaterThan,axiom,
% 11.90/5.98 ! [K: nat] :
% 11.90/5.98 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 11.90/5.98 = ( set_ord_greaterThan @ nat @ K ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3069_ivl__disj__int__two_I7_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3070_disjoint__alt__simp3,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ A4 )
% 11.90/5.98 = ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3071_Ioc__disjoint,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A,C2: A,D3: A] :
% 11.90/5.98 ( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D3 ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 = ( ( ord_less_eq @ A @ B2 @ A3 )
% 11.90/5.98 | ( ord_less_eq @ A @ D3 @ C2 )
% 11.90/5.98 | ( ord_less_eq @ A @ B2 @ C2 )
% 11.90/5.98 | ( ord_less_eq @ A @ D3 @ A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3072_ivl__disj__int__one_I4_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3073_ivl__disj__int__one_I2_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3074_wait__def,axiom,
% 11.90/5.98 ( heap_Time_wait
% 11.90/5.98 = ( ^ [N5: nat] :
% 11.90/5.98 ( heap_Time_Heap2 @ product_unit
% 11.90/5.98 @ ^ [H: heap_ext @ product_unit] : ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N5 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3075_ivl__disj__int__two_I8_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3076_ivl__disj__int__two_I5_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3077_ivl__disj__int__two_I4_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3078_ivl__disj__int__two_I1_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3079_ivl__disj__int__one_I3_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3080_ivl__disj__int__one_I7_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3081_ivl__disj__int__one_I1_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3082_ivl__disj__int__two_I2_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,M2: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3083_ivl__disj__int__one_I5_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3084_ivl__disj__un__one_I8_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3085_prod_Ointer__restrict,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_mult @ A )
% 11.90/5.98 => ! [A4: set @ B,G: B > A,B6: set @ B] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 = ( groups7121269368397514597t_prod @ B @ A
% 11.90/5.98 @ ^ [X3: B] : ( if @ A @ ( member @ B @ X3 @ B6 ) @ ( G @ X3 ) @ ( one_one @ A ) )
% 11.90/5.98 @ A4 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3086_greaterThanAtMost__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_or3652927894154168847AtMost @ A )
% 11.90/5.98 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L3 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3087_greaterThanLessThan__eq,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_or5935395276787703475ssThan @ A )
% 11.90/5.98 = ( ^ [A7: A,B5: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A7 ) @ ( set_ord_lessThan @ A @ B5 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3088_greaterThanLessThan__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( ord @ A )
% 11.90/5.98 => ( ( set_or5935395276787703475ssThan @ A )
% 11.90/5.98 = ( ^ [L3: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L3 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3089_Ici__subset__Ioi__iff,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [A3: A,B2: A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
% 11.90/5.98 = ( ord_less @ A @ B2 @ A3 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3090_execute__assert_H,axiom,
% 11.90/5.98 ! [P: $o,H3: heap_ext @ product_unit] :
% 11.90/5.98 ( ( P
% 11.90/5.98 => ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
% 11.90/5.98 = ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) )
% 11.90/5.98 & ( ~ P
% 11.90/5.98 => ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H3 )
% 11.90/5.98 = ( none @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3091_sum_Ounion__inter,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.98 => ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) )
% 11.90/5.98 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3092_sum_OInt__Diff,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [A4: set @ B,G: B > A,B6: set @ B] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
% 11.90/5.98 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3093_prod_Omono__neutral__cong,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_mult @ A )
% 11.90/5.98 => ! [T6: set @ B,S2: set @ B,H3: B > A,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ T6 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ S2 )
% 11.90/5.98 => ( ! [I2: B] :
% 11.90/5.98 ( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
% 11.90/5.98 => ( ( H3 @ I2 )
% 11.90/5.98 = ( one_one @ A ) ) )
% 11.90/5.98 => ( ! [I2: B] :
% 11.90/5.98 ( ( member @ B @ I2 @ ( minus_minus @ ( set @ B ) @ S2 @ T6 ) )
% 11.90/5.98 => ( ( G @ I2 )
% 11.90/5.98 = ( one_one @ A ) ) )
% 11.90/5.98 => ( ! [X2: B] :
% 11.90/5.98 ( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ S2 @ T6 ) )
% 11.90/5.98 => ( ( G @ X2 )
% 11.90/5.98 = ( H3 @ X2 ) ) )
% 11.90/5.98 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
% 11.90/5.98 = ( groups7121269368397514597t_prod @ B @ A @ H3 @ T6 ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3094_Iio__Int__singleton,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( order @ A )
% 11.90/5.98 => ! [X: A,K: A] :
% 11.90/5.98 ( ( ( ord_less @ A @ X @ K )
% 11.90/5.98 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.98 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.98 & ( ~ ( ord_less @ A @ X @ K )
% 11.90/5.98 => ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3095_card__Un__Int,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ A @ B6 )
% 11.90/5.98 => ( ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) )
% 11.90/5.98 = ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3096_sum_OIf__cases,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [A4: set @ B,P: B > $o,H3: B > A,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ B @ A
% 11.90/5.98 @ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( H3 @ X3 ) @ ( G @ X3 ) )
% 11.90/5.98 @ A4 )
% 11.90/5.98 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H3 @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3097_ivl__disj__un__singleton_I1_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A] :
% 11.90/5.98 ( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ L ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3098_ivl__disj__un__one_I7_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less_eq @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
% 11.90/5.98 = ( set_ord_atLeast @ A @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3099_ivl__disj__un__one_I6_J,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( linorder @ A )
% 11.90/5.98 => ! [L: A,U: A] :
% 11.90/5.98 ( ( ord_less @ A @ L @ U )
% 11.90/5.98 => ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
% 11.90/5.98 = ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3100_Linear__order__Well__order__iff,axiom,
% 11.90/5.98 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.98 => ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.98 = ( ! [A9: set @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
% 11.90/5.98 => ( ( A9
% 11.90/5.98 != ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 => ? [X3: A] :
% 11.90/5.98 ( ( member @ A @ X3 @ A9 )
% 11.90/5.98 & ! [Y3: A] :
% 11.90/5.98 ( ( member @ A @ Y3 @ A9 )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3101_sum_Ounion__inter__neutral,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.98 => ( ! [X2: B] :
% 11.90/5.98 ( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 => ( ( G @ X2 )
% 11.90/5.98 = ( zero_zero @ A ) ) )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3102_sum__Un,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( ab_group_add @ A )
% 11.90/5.98 => ! [A4: set @ B,B6: set @ B,F: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3103_sum_Ounion__disjoint,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.98 => ( ( ( inf_inf @ ( set @ B ) @ A4 @ B6 )
% 11.90/5.98 = ( bot_bot @ ( set @ B ) ) )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3104_prod_Ounion__inter__neutral,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_mult @ A )
% 11.90/5.98 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.98 => ( ! [X2: B] :
% 11.90/5.98 ( ( member @ B @ X2 @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 => ( ( G @ X2 )
% 11.90/5.98 = ( one_one @ A ) ) )
% 11.90/5.98 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 = ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B6 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3105_execute__update,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( heap @ A )
% 11.90/5.98 => ! [R: ref @ A,V: A,H3: heap_ext @ product_unit] :
% 11.90/5.98 ( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R @ V ) @ H3 )
% 11.90/5.98 = ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R @ V @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3106_sum_Ounion__diff2,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ A )
% 11.90/5.98 => ! [A4: set @ B,B6: set @ B,G: B > A] :
% 11.90/5.98 ( ( finite_finite2 @ B @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ B @ B6 )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) )
% 11.90/5.98 = ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B6 ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3107_sum__Un2,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType] :
% 11.90/5.98 ( ( comm_monoid_add @ B )
% 11.90/5.98 => ! [A4: set @ A,B6: set @ A,F: A > B] :
% 11.90/5.98 ( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.98 = ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3108_card__Un__disjoint,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/5.98 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ A @ B6 )
% 11.90/5.98 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 => ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.98 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3109_sum__Un__nat,axiom,
% 11.90/5.98 ! [A: $tType,A4: set @ A,B6: set @ A,F: A > nat] :
% 11.90/5.98 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( finite_finite2 @ A @ B6 )
% 11.90/5.98 => ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.98 = ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3110_atLeast__Suc,axiom,
% 11.90/5.98 ! [K: nat] :
% 11.90/5.98 ( ( set_ord_atLeast @ nat @ ( suc @ K ) )
% 11.90/5.98 = ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert2 @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3111_old_Ounit_Orec,axiom,
% 11.90/5.98 ! [T: $tType,F1: T] :
% 11.90/5.98 ( ( product_rec_unit @ T @ F1 @ product_Unity )
% 11.90/5.98 = F1 ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3112_default__unit__def,axiom,
% 11.90/5.98 ( ( default_default @ product_unit )
% 11.90/5.98 = product_Unity ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3113_CODE__ABORT__def,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( cODE_ABORT @ A )
% 11.90/5.98 = ( ^ [F2: product_unit > A] : ( F2 @ product_Unity ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3114_Inf__fin_Oeq__fold_H,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semilattice_inf @ A )
% 11.90/5.98 => ( ( lattic7752659483105999362nf_fin @ A )
% 11.90/5.98 = ( ^ [A9: set @ A] :
% 11.90/5.98 ( the2 @ A
% 11.90/5.98 @ ( finite_fold @ A @ ( option @ A )
% 11.90/5.98 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( inf_inf @ A @ X3 ) @ Y3 ) )
% 11.90/5.98 @ ( none @ A )
% 11.90/5.98 @ A9 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3115_image__split__eq__Sigma,axiom,
% 11.90/5.98 ! [C: $tType,B: $tType,A: $tType,F: C > A,G: C > B,A4: set @ C] :
% 11.90/5.98 ( ( image2 @ C @ ( product_prod @ A @ B )
% 11.90/5.98 @ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ X3 ) )
% 11.90/5.98 @ A4 )
% 11.90/5.98 = ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A4 )
% 11.90/5.98 @ ^ [X3: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3116_atLeast__empty__triv,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 11.90/5.98 = ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3117_inf__None__2,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( inf @ A )
% 11.90/5.98 => ! [X: option @ A] :
% 11.90/5.98 ( ( inf_inf @ ( option @ A ) @ X @ ( none @ A ) )
% 11.90/5.98 = ( none @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3118_inf__None__1,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( inf @ A )
% 11.90/5.98 => ! [Y: option @ A] :
% 11.90/5.98 ( ( inf_inf @ ( option @ A ) @ ( none @ A ) @ Y )
% 11.90/5.98 = ( none @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3119_vimage__id,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( vimage @ A @ A @ ( id @ A ) )
% 11.90/5.98 = ( id @ ( set @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3120_set_Ocomp,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
% 11.90/5.98 ( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
% 11.90/5.98 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3121_set_Ocompositionality,axiom,
% 11.90/5.98 ! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
% 11.90/5.98 ( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
% 11.90/5.98 = ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3122_vimage__comp,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X: set @ C] :
% 11.90/5.98 ( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X ) )
% 11.90/5.98 = ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3123_inf__unit__def,axiom,
% 11.90/5.98 ( ( inf_inf @ product_unit )
% 11.90/5.98 = ( ^ [Uu3: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3124_inf__Int__eq2,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.98 ( ( inf_inf @ ( A > B > $o )
% 11.90/5.98 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R5 )
% 11.90/5.98 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S2 ) )
% 11.90/5.98 = ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3125_surj__image__vimage__eq,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
% 11.90/5.98 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.98 = ( top_top @ ( set @ A ) ) )
% 11.90/5.98 => ( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A4 ) )
% 11.90/5.98 = A4 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3126_set_Oidentity,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( vimage @ A @ A
% 11.90/5.98 @ ^ [X3: A] : X3 )
% 11.90/5.98 = ( id @ ( set @ A ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3127_vimage__Times,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A4: set @ B,B6: set @ C] :
% 11.90/5.98 ( ( vimage @ A @ ( product_prod @ B @ C ) @ F
% 11.90/5.98 @ ( product_Sigma @ B @ C @ A4
% 11.90/5.98 @ ^ [Uu: B] : B6 ) )
% 11.90/5.98 = ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A4 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B6 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3128_Pair__vimage__Sigma,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,X: B,A4: set @ B,F: B > ( set @ A )] :
% 11.90/5.98 ( ( ( member @ B @ X @ A4 )
% 11.90/5.98 => ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F ) )
% 11.90/5.98 = ( F @ X ) ) )
% 11.90/5.98 & ( ~ ( member @ B @ X @ A4 )
% 11.90/5.98 => ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A4 @ F ) )
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3129_surj__vimage__empty,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,F: B > A,A4: set @ A] :
% 11.90/5.98 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.98 = ( top_top @ ( set @ A ) ) )
% 11.90/5.98 => ( ( ( vimage @ B @ A @ F @ A4 )
% 11.90/5.98 = ( bot_bot @ ( set @ B ) ) )
% 11.90/5.98 = ( A4
% 11.90/5.98 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3130_vimage__subsetD,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,F: B > A,B6: set @ A,A4: set @ B] :
% 11.90/5.98 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.98 = ( top_top @ ( set @ A ) ) )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ B6 ) @ A4 )
% 11.90/5.98 => ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A4 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3131_vimage__fst,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,A4: set @ A] :
% 11.90/5.98 ( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A4 )
% 11.90/5.98 = ( product_Sigma @ A @ B @ A4
% 11.90/5.98 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3132_vimage__snd,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,A4: set @ B] :
% 11.90/5.98 ( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A4 )
% 11.90/5.98 = ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 11.90/5.98 @ ^ [Uu: A] : A4 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3133_Inf__fin_Oinfinite,axiom,
% 11.90/5.98 ! [A: $tType] :
% 11.90/5.98 ( ( semilattice_inf @ A )
% 11.90/5.98 => ! [A4: set @ A] :
% 11.90/5.98 ( ~ ( finite_finite2 @ A @ A4 )
% 11.90/5.98 => ( ( lattic7752659483105999362nf_fin @ A @ A4 )
% 11.90/5.98 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3134_rtrancl__last__visit__node,axiom,
% 11.90/5.98 ! [A: $tType,S: A,S3: A,R5: set @ ( product_prod @ A @ A ),Sh: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ S3 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.98 => ( ( ( S != Sh )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ S3 )
% 11.90/5.98 @ ( transitive_rtrancl @ A
% 11.90/5.98 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.98 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.98 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
% 11.90/5.98 | ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S @ Sh ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S3 )
% 11.90/5.98 @ ( transitive_rtrancl @ A
% 11.90/5.98 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.98 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.98 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3135_floor__rat__def,axiom,
% 11.90/5.98 ( ( archim6421214686448440834_floor @ rat )
% 11.90/5.98 = ( ^ [X3: rat] :
% 11.90/5.98 ( the @ int
% 11.90/5.98 @ ^ [Z5: int] :
% 11.90/5.98 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z5 ) @ X3 )
% 11.90/5.98 & ( ord_less @ rat @ X3 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3136_rat__floor__lemma,axiom,
% 11.90/5.98 ! [A3: int,B2: int] :
% 11.90/5.98 ( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
% 11.90/5.98 & ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3137_num__of__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.98 @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
% 11.90/5.98 @ code_num_of_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3138_prod__decode__triangle__add,axiom,
% 11.90/5.98 ! [K: nat,M2: nat] :
% 11.90/5.98 ( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) )
% 11.90/5.98 = ( nat_prod_decode_aux @ K @ M2 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3139_wf__insert,axiom,
% 11.90/5.98 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
% 11.90/5.98 = ( ( wf @ A @ R )
% 11.90/5.98 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3140_add__rat,axiom,
% 11.90/5.98 ! [B2: int,D3: int,A3: int,C2: int] :
% 11.90/5.98 ( ( B2
% 11.90/5.98 != ( zero_zero @ int ) )
% 11.90/5.98 => ( ( D3
% 11.90/5.98 != ( zero_zero @ int ) )
% 11.90/5.98 => ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D3 ) )
% 11.90/5.98 = ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D3 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3141_r__le__rtrancl,axiom,
% 11.90/5.98 ! [A: $tType,S2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S2 ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3142_rtrancl__mono__mp,axiom,
% 11.90/5.98 ! [A: $tType,U4: set @ ( product_prod @ A @ A ),V6: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U4 @ V6 )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ U4 ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ V6 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3143_rtrancl__mono__rightI,axiom,
% 11.90/5.98 ! [A: $tType,S2: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ S7 )
% 11.90/5.98 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S7 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3144_converse__rtranclE_H,axiom,
% 11.90/5.98 ! [A: $tType,U: A,V: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.98 => ( ( U != V )
% 11.90/5.98 => ~ ! [Vh: A] :
% 11.90/5.98 ( ( U != Vh )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R5 )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3145_rtrancl_Ocases,axiom,
% 11.90/5.98 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( A22 != A1 )
% 11.90/5.98 => ~ ! [B3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3146_rtrancl_Osimps,axiom,
% 11.90/5.98 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 = ( ? [A7: A] :
% 11.90/5.98 ( ( A1 = A7 )
% 11.90/5.98 & ( A22 = A7 ) )
% 11.90/5.98 | ? [A7: A,B5: A,C3: A] :
% 11.90/5.98 ( ( A1 = A7 )
% 11.90/5.98 & ( A22 = C3 )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3147_rtrancl_Ortrancl__refl,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3148_rtrancl_Ortrancl__into__rtrancl,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3149_rtranclE,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( A3 != B2 )
% 11.90/5.98 => ~ ! [Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3150_rtrancl__trans,axiom,
% 11.90/5.98 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3151_rtrancl__induct,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( P @ A3 )
% 11.90/5.98 => ( ! [Y4: A,Z3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.90/5.98 => ( ( P @ Y4 )
% 11.90/5.98 => ( P @ Z3 ) ) ) )
% 11.90/5.98 => ( P @ B2 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3152_converse__rtranclE,axiom,
% 11.90/5.98 ! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( X != Z2 )
% 11.90/5.98 => ~ ! [Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
% 11.90/5.98 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3153_converse__rtrancl__induct,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( P @ B2 )
% 11.90/5.98 => ( ! [Y4: A,Z3: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( P @ Z3 )
% 11.90/5.98 => ( P @ Y4 ) ) ) )
% 11.90/5.98 => ( P @ A3 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3154_converse__rtrancl__into__rtrancl,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3155_in__rtrancl__insert,axiom,
% 11.90/5.98 ! [A: $tType,X: product_prod @ A @ A,R5: set @ ( product_prod @ A @ A ),R: product_prod @ A @ A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ R @ R5 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3156_integer_Orep__transfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.98 @ ^ [X3: int] : X3
% 11.90/5.98 @ code_int_of_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3157_integer_Oid__abs__transfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ int @ int @ code_integer
% 11.90/5.98 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.98 @ code_pcr_integer
% 11.90/5.98 @ ^ [X3: int] : X3
% 11.90/5.98 @ code_integer_of_int ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3158_rtrancl__Un__separator__converseE,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
% 11.90/5.98 => ( ! [X2: A,Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ Q )
% 11.90/5.98 => ( Y4 = X2 ) ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3159_rtrancl__Un__separatorE,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
% 11.90/5.98 => ( ! [X2: A,Y4: A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ P ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ Q )
% 11.90/5.98 => ( X2 = Y4 ) ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3160_zero__integer_Otransfer,axiom,
% 11.90/5.98 code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3161_converse__rtrancl__induct2,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 11.90/5.98 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.98 => ( ( P @ Bx @ By )
% 11.90/5.98 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 11.90/5.98 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
% 11.90/5.98 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.98 => ( ( P @ Aa3 @ Ba )
% 11.90/5.98 => ( P @ A5 @ B3 ) ) ) )
% 11.90/5.98 => ( P @ Ax @ Ay ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3162_converse__rtranclE2,axiom,
% 11.90/5.98 ! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
% 11.90/5.98 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.98 => ( ( ( product_Pair @ A @ B @ Xa @ Xb )
% 11.90/5.98 != ( product_Pair @ A @ B @ Za2 @ Zb ) )
% 11.90/5.98 => ~ ! [A5: A,B3: B] :
% 11.90/5.98 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ R )
% 11.90/5.98 => ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3163_rtrancl__induct2,axiom,
% 11.90/5.98 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 11.90/5.98 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.98 => ( ( P @ Ax @ Ay )
% 11.90/5.98 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 11.90/5.98 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.98 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
% 11.90/5.98 => ( ( P @ A5 @ B3 )
% 11.90/5.98 => ( P @ Aa3 @ Ba ) ) ) )
% 11.90/5.98 => ( P @ Bx @ By ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3164_rtrancl__sub__insert__rtrancl,axiom,
% 11.90/5.98 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R5 ) @ ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R5 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3165_plus__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( plus_plus @ int ) @ ( plus_plus @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3166_quotient__of__eq,axiom,
% 11.90/5.98 ! [A3: int,B2: int,P5: int,Q3: int] :
% 11.90/5.98 ( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
% 11.90/5.98 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/5.98 => ( ( fract @ P5 @ Q3 )
% 11.90/5.98 = ( fract @ A3 @ B2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3167_less__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.98 @ ( ord_less @ int )
% 11.90/5.98 @ ( ord_less @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3168_one__integer_Otransfer,axiom,
% 11.90/5.98 code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3169_less__eq__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.90/5.98 @ ( ord_less_eq @ int )
% 11.90/5.98 @ ( ord_less_eq @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3170_uminus__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3171_times__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( times_times @ int ) @ ( times_times @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3172_minus__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( minus_minus @ int ) @ ( minus_minus @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3173_abs__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( abs_abs @ int ) @ ( abs_abs @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3174_nat__of__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ nat2
% 11.90/5.98 @ code_nat_of_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3175_normalize__eq,axiom,
% 11.90/5.98 ! [A3: int,B2: int,P5: int,Q3: int] :
% 11.90/5.98 ( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
% 11.90/5.98 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/5.98 => ( ( fract @ P5 @ Q3 )
% 11.90/5.98 = ( fract @ A3 @ B2 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3176_integer__of__nat_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ code_pcr_integer
% 11.90/5.98 @ ( semiring_1_of_nat @ int )
% 11.90/5.98 @ code_integer_of_nat ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3177_divide__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( divide_divide @ int ) @ ( divide_divide @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3178_modulo__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( modulo_modulo @ int ) @ ( modulo_modulo @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3179_sgn__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( sgn_sgn @ int ) @ ( sgn_sgn @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3180_dup_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
% 11.90/5.98 @ ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 )
% 11.90/5.98 @ code_dup ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3181_take__bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 11.90/5.98 @ ( bit_se2584673776208193580ke_bit @ int )
% 11.90/5.98 @ ( bit_se2584673776208193580ke_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3182_not__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( bit_ri4277139882892585799ns_not @ int ) @ ( bit_ri4277139882892585799ns_not @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3183_and__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344872417868541ns_and @ int ) @ ( bit_se5824344872417868541ns_and @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3184_bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bit_se5641148757651400278ts_bit @ int )
% 11.90/5.98 @ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3185_or__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se1065995026697491101ons_or @ int ) @ ( bit_se1065995026697491101ons_or @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3186_xor__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344971392196577ns_xor @ int ) @ ( bit_se5824344971392196577ns_xor @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3187_drop__bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 11.90/5.98 @ ( bit_se4197421643247451524op_bit @ int )
% 11.90/5.98 @ ( bit_se4197421643247451524op_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3188_push__bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 11.90/5.98 @ ( bit_se4730199178511100633sh_bit @ int )
% 11.90/5.98 @ ( bit_se4730199178511100633sh_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3189_mask__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ int @ code_integer
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ code_pcr_integer
% 11.90/5.98 @ ( bit_se2239418461657761734s_mask @ int )
% 11.90/5.98 @ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3190_unset__bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 11.90/5.98 @ ( bit_se2638667681897837118et_bit @ int )
% 11.90/5.98 @ ( bit_se2638667681897837118et_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3191_set__bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 11.90/5.98 @ ( bit_se5668285175392031749et_bit @ int )
% 11.90/5.98 @ ( bit_se5668285175392031749et_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3192_division__segment__integer_Otransfer,axiom,
% 11.90/5.98 bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3193_flip__bit__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
% 11.90/5.98 @ ( bit_se8732182000553998342ip_bit @ int )
% 11.90/5.98 @ ( bit_se8732182000553998342ip_bit @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3194_euclidean__size__integer_Otransfer,axiom,
% 11.90/5.98 ( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
% 11.90/5.98 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.90/5.98 @ ( euclid6346220572633701492n_size @ int )
% 11.90/5.98 @ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3195_trancl__subset__Sigma__aux,axiom,
% 11.90/5.98 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.98 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/5.98 @ ( product_Sigma @ A @ A @ A4
% 11.90/5.98 @ ^ [Uu: A] : A4 ) )
% 11.90/5.98 => ( ( A3 = B2 )
% 11.90/5.98 | ( member @ A @ A3 @ A4 ) ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3196_Restr__rtrancl__mono,axiom,
% 11.90/5.98 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ A] :
% 11.90/5.98 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
% 11.90/5.98 @ ( transitive_rtrancl @ A
% 11.90/5.98 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 11.90/5.98 @ ( product_Sigma @ A @ A @ U4
% 11.90/5.98 @ ^ [Uu: A] : U4 ) ) ) )
% 11.90/5.98 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ).
% 11.90/5.98
% 11.90/5.98 thf(fact_3197_quotient__of__Fract,axiom,
% 11.90/5.98 ! [A3: int,B2: int] :
% 11.90/5.98 ( ( quotient_of @ ( fract @ A3 @ B2 ) )
% 11.90/5.99 = ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3198_sub_Otransfer,axiom,
% 11.90/5.99 ( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
% 11.90/5.99 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.99 @ ( bNF_rel_fun @ num @ num @ int @ code_integer
% 11.90/5.99 @ ^ [Y5: num,Z4: num] : ( Y5 = Z4 )
% 11.90/5.99 @ code_pcr_integer )
% 11.90/5.99 @ ^ [M: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N5 ) )
% 11.90/5.99 @ code_sub ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3199_rtrancl__mapI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A3: A,B2: A,E4: set @ ( product_prod @ A @ A ),F: A > B] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ E4 ) )
% 11.90/5.99 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A3 ) @ ( F @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F ) @ E4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3200_Fract_Oabs__eq,axiom,
% 11.90/5.99 ( fract
% 11.90/5.99 = ( ^ [Xa4: int,X3: int] :
% 11.90/5.99 ( abs_Rat
% 11.90/5.99 @ ( if @ ( product_prod @ int @ int )
% 11.90/5.99 @ ( X3
% 11.90/5.99 = ( zero_zero @ int ) )
% 11.90/5.99 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.99 @ ( product_Pair @ int @ int @ Xa4 @ X3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3201_rtrancl__last__touch,axiom,
% 11.90/5.99 ! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ( ( member @ A @ Q3 @ S2 )
% 11.90/5.99 => ~ ! [Qt: A] :
% 11.90/5.99 ( ( member @ A @ Qt @ S2 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
% 11.90/5.99 @ ( transitive_rtrancl @ A
% 11.90/5.99 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3202_rtrancl__last__visit_H,axiom,
% 11.90/5.99 ! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 )
% 11.90/5.99 @ ( transitive_rtrancl @ A
% 11.90/5.99 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : S2 ) ) ) )
% 11.90/5.99 => ~ ! [Qt: A] :
% 11.90/5.99 ( ( member @ A @ Qt @ S2 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
% 11.90/5.99 @ ( transitive_rtrancl @ A
% 11.90/5.99 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3203_rtrancl__insert,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
% 11.90/5.99 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R )
% 11.90/5.99 @ ( collect @ ( product_prod @ A @ A )
% 11.90/5.99 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.99 @ ^ [X3: A,Y3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3204_Fract_Otransfer,axiom,
% 11.90/5.99 ( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
% 11.90/5.99 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.99 @ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
% 11.90/5.99 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.90/5.99 @ pcr_rat )
% 11.90/5.99 @ ^ [A7: int,B5: int] :
% 11.90/5.99 ( if @ ( product_prod @ int @ int )
% 11.90/5.99 @ ( B5
% 11.90/5.99 = ( zero_zero @ int ) )
% 11.90/5.99 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.99 @ ( product_Pair @ int @ int @ A7 @ B5 ) )
% 11.90/5.99 @ fract ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3205_Fract__add__one,axiom,
% 11.90/5.99 ! [N: int,M2: int] :
% 11.90/5.99 ( ( N
% 11.90/5.99 != ( zero_zero @ int ) )
% 11.90/5.99 => ( ( fract @ ( plus_plus @ int @ M2 @ N ) @ N )
% 11.90/5.99 = ( plus_plus @ rat @ ( fract @ M2 @ N ) @ ( one_one @ rat ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3206_Fract__def,axiom,
% 11.90/5.99 ( fract
% 11.90/5.99 = ( map_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat ) @ ( id @ int ) @ ( map_fun @ int @ int @ ( product_prod @ int @ int ) @ rat @ ( id @ int ) @ abs_Rat )
% 11.90/5.99 @ ^ [A7: int,B5: int] :
% 11.90/5.99 ( if @ ( product_prod @ int @ int )
% 11.90/5.99 @ ( B5
% 11.90/5.99 = ( zero_zero @ int ) )
% 11.90/5.99 @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
% 11.90/5.99 @ ( product_Pair @ int @ int @ A7 @ B5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3207_old_Orec__unit__def,axiom,
% 11.90/5.99 ! [T: $tType] :
% 11.90/5.99 ( ( product_rec_unit @ T )
% 11.90/5.99 = ( ^ [F12: T,X3: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3208_old_Orec__prod__def,axiom,
% 11.90/5.99 ! [T: $tType,B: $tType,A: $tType] :
% 11.90/5.99 ( ( product_rec_prod @ A @ B @ T )
% 11.90/5.99 = ( ^ [F12: A > B > T,X3: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3209_list__decode_Opinduct,axiom,
% 11.90/5.99 ! [A0: nat,P: nat > $o] :
% 11.90/5.99 ( ( accp @ nat @ nat_list_decode_rel @ A0 )
% 11.90/5.99 => ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 11.90/5.99 => ( P @ ( zero_zero @ nat ) ) )
% 11.90/5.99 => ( ! [N3: nat] :
% 11.90/5.99 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) )
% 11.90/5.99 => ( ! [X7: nat,Y6: nat] :
% 11.90/5.99 ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
% 11.90/5.99 = ( nat_prod_decode @ N3 ) )
% 11.90/5.99 => ( P @ Y6 ) )
% 11.90/5.99 => ( P @ ( suc @ N3 ) ) ) )
% 11.90/5.99 => ( P @ A0 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3210_rtrancl__restrictI,axiom,
% 11.90/5.99 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
% 11.90/5.99 @ ( transitive_rtrancl @ A
% 11.90/5.99 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : R5 ) ) ) )
% 11.90/5.99 => ( ~ ( member @ A @ U @ R5 )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3211_inj__on__vimage__singleton,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,A3: B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A4 )
% 11.90/5.99 @ ( insert2 @ A
% 11.90/5.99 @ ( the @ A
% 11.90/5.99 @ ^ [X3: A] :
% 11.90/5.99 ( ( member @ A @ X3 @ A4 )
% 11.90/5.99 & ( ( F @ X3 )
% 11.90/5.99 = A3 ) ) )
% 11.90/5.99 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3212_inj__on__empty,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3213_inj__uminus,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( ordered_ab_group_add @ A )
% 11.90/5.99 => ! [A4: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3214_rel__restrict__empty,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( rel_restrict @ A @ R5 @ ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 = R5 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3215_The__split__eq,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,X: A,Y: B] :
% 11.90/5.99 ( ( the @ ( product_prod @ A @ B )
% 11.90/5.99 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.99 @ ^ [X9: A,Y8: B] :
% 11.90/5.99 ( ( X = X9 )
% 11.90/5.99 & ( Y = Y8 ) ) ) )
% 11.90/5.99 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3216_inj__apfst,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: A > C] :
% 11.90/5.99 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.99 = ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3217_inj__apsnd,axiom,
% 11.90/5.99 ! [A: $tType,C: $tType,B: $tType,F: B > C] :
% 11.90/5.99 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 11.90/5.99 = ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3218_inj__on__apfst,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: A > C,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F )
% 11.90/5.99 @ ( product_Sigma @ A @ B @ A4
% 11.90/5.99 @ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
% 11.90/5.99 = ( inj_on @ A @ C @ F @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3219_inj__on__apsnd,axiom,
% 11.90/5.99 ! [A: $tType,C: $tType,B: $tType,F: B > C,A4: set @ B] :
% 11.90/5.99 ( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F )
% 11.90/5.99 @ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : A4 ) )
% 11.90/5.99 = ( inj_on @ B @ C @ F @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3220_inj__on__insert,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A3: A,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( insert2 @ A @ A3 @ A4 ) )
% 11.90/5.99 = ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 & ~ ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3221_inj__on__Int,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 | ( inj_on @ A @ B @ F @ B6 ) )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3222_fun_Oinj__map,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,D: $tType,F: A > B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3223_inj__on__imageI2,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F3: C > B,F: A > C,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F3 @ F ) @ A4 )
% 11.90/5.99 => ( inj_on @ A @ C @ F @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3224_inj__on__of__nat,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( semiring_char_0 @ A )
% 11.90/5.99 => ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3225_inj__on__add,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( cancel_semigroup_add @ A )
% 11.90/5.99 => ! [A3: A,A4: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3226_inj__on__image__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: set @ A,G: A > B,F: A > A] :
% 11.90/5.99 ( ! [X2: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ A4 )
% 11.90/5.99 => ! [Xa3: A] :
% 11.90/5.99 ( ( member @ A @ Xa3 @ A4 )
% 11.90/5.99 => ( ( ( G @ ( F @ X2 ) )
% 11.90/5.99 = ( G @ ( F @ Xa3 ) ) )
% 11.90/5.99 = ( ( G @ X2 )
% 11.90/5.99 = ( G @ Xa3 ) ) ) ) )
% 11.90/5.99 => ( ( inj_on @ A @ A @ F @ A4 )
% 11.90/5.99 => ( ( inj_on @ A @ B @ G @ ( image2 @ A @ A @ F @ A4 ) )
% 11.90/5.99 = ( inj_on @ A @ B @ G @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3227_inj__on__id2,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A] :
% 11.90/5.99 ( inj_on @ A @ A
% 11.90/5.99 @ ^ [X3: A] : X3
% 11.90/5.99 @ A4 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3228_inj__on__inverseI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: set @ A,G: B > A,F: A > B] :
% 11.90/5.99 ( ! [X2: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ A4 )
% 11.90/5.99 => ( ( G @ ( F @ X2 ) )
% 11.90/5.99 = X2 ) )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3229_inj__on__contraD,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( X != Y )
% 11.90/5.99 => ( ( member @ A @ X @ A4 )
% 11.90/5.99 => ( ( member @ A @ Y @ A4 )
% 11.90/5.99 => ( ( F @ X )
% 11.90/5.99 != ( F @ Y ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3230_inj__on__eq__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( member @ A @ X @ A4 )
% 11.90/5.99 => ( ( member @ A @ Y @ A4 )
% 11.90/5.99 => ( ( ( F @ X )
% 11.90/5.99 = ( F @ Y ) )
% 11.90/5.99 = ( X = Y ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3231_inj__on__cong,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: set @ A,F: A > B,G: A > B] :
% 11.90/5.99 ( ! [A5: A] :
% 11.90/5.99 ( ( member @ A @ A5 @ A4 )
% 11.90/5.99 => ( ( F @ A5 )
% 11.90/5.99 = ( G @ A5 ) ) )
% 11.90/5.99 => ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 = ( inj_on @ A @ B @ G @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3232_inj__on__def,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( inj_on @ A @ B )
% 11.90/5.99 = ( ^ [F2: A > B,A9: set @ A] :
% 11.90/5.99 ! [X3: A] :
% 11.90/5.99 ( ( member @ A @ X3 @ A9 )
% 11.90/5.99 => ! [Y3: A] :
% 11.90/5.99 ( ( member @ A @ Y3 @ A9 )
% 11.90/5.99 => ( ( ( F2 @ X3 )
% 11.90/5.99 = ( F2 @ Y3 ) )
% 11.90/5.99 => ( X3 = Y3 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3233_inj__onI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 11.90/5.99 ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ A4 )
% 11.90/5.99 => ( ( member @ A @ Y4 @ A4 )
% 11.90/5.99 => ( ( ( F @ X2 )
% 11.90/5.99 = ( F @ Y4 ) )
% 11.90/5.99 => ( X2 = Y4 ) ) ) )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3234_inj__onD,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( ( F @ X )
% 11.90/5.99 = ( F @ Y ) )
% 11.90/5.99 => ( ( member @ A @ X @ A4 )
% 11.90/5.99 => ( ( member @ A @ Y @ A4 )
% 11.90/5.99 => ( X = Y ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3235_inj__on__diff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3236_inj__on__id,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A4 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3237_inj__Some,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3238_inj__on__subset,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ B6 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3239_subset__inj__on,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ B6 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3240_inj__fun,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: A > B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( inj_on @ A @ ( C > B )
% 11.90/5.99 @ ^ [X3: A,Y3: C] : ( F @ X3 )
% 11.90/5.99 @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3241_inj__def,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 = ( ! [X3: A,Y3: A] :
% 11.90/5.99 ( ( ( F @ X3 )
% 11.90/5.99 = ( F @ Y3 ) )
% 11.90/5.99 => ( X3 = Y3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3242_inj__eq,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ( F @ X )
% 11.90/5.99 = ( F @ Y ) )
% 11.90/5.99 = ( X = Y ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3243_injI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/5.99 ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( ( F @ X2 )
% 11.90/5.99 = ( F @ Y4 ) )
% 11.90/5.99 => ( X2 = Y4 ) )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3244_injD,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,X: A,Y: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ( F @ X )
% 11.90/5.99 = ( F @ Y ) )
% 11.90/5.99 => ( X = Y ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3245_inj__swap,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A4 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3246_inj__on__convol__ident,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,X10: set @ A] :
% 11.90/5.99 ( inj_on @ A @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( F @ X3 ) )
% 11.90/5.99 @ X10 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3247_inj__Pair_I1_J,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
% 11.90/5.99 ( inj_on @ A @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( C2 @ X3 ) )
% 11.90/5.99 @ S2 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3248_inj__Pair_I2_J,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
% 11.90/5.99 ( inj_on @ A @ ( product_prod @ B @ A )
% 11.90/5.99 @ ^ [X3: A] : ( product_Pair @ B @ A @ ( C2 @ X3 ) @ X3 )
% 11.90/5.99 @ S2 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3249_inj__on__add_H,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( cancel_semigroup_add @ A )
% 11.90/5.99 => ! [A3: A,A4: set @ A] :
% 11.90/5.99 ( inj_on @ A @ A
% 11.90/5.99 @ ^ [B5: A] : ( plus_plus @ A @ B5 @ A3 )
% 11.90/5.99 @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3250_rel__restrict__lift,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R5 ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3251_rel__restrictI,axiom,
% 11.90/5.99 ! [A: $tType,X: A,R5: set @ A,Y: A,E4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ~ ( member @ A @ X @ R5 )
% 11.90/5.99 => ( ~ ( member @ A @ Y @ R5 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ E4 )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3252_rel__restrict__notR_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R5 ) )
% 11.90/5.99 => ~ ( member @ A @ X @ R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3253_rel__restrict__notR_I2_J,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( rel_restrict @ A @ A4 @ R5 ) )
% 11.90/5.99 => ~ ( member @ A @ Y @ R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3254_rel__restrict__union,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( rel_restrict @ A @ R5 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 = ( rel_restrict @ A @ ( rel_restrict @ A @ R5 @ A4 ) @ B6 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3255_rel__restrict__sub,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 ) @ R5 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3256_rel__restrict__mono,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A4 @ R5 ) @ ( rel_restrict @ A @ B6 @ R5 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3257_finite__rel__restrict,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 11.90/5.99 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
% 11.90/5.99 => ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R5 @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3258_option_Oinj__map,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3259_linorder__inj__onI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( order @ A )
% 11.90/5.99 => ! [A4: set @ A,F: A > B] :
% 11.90/5.99 ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( ord_less @ A @ X2 @ Y4 )
% 11.90/5.99 => ( ( member @ A @ X2 @ A4 )
% 11.90/5.99 => ( ( member @ A @ Y4 @ A4 )
% 11.90/5.99 => ( ( F @ X2 )
% 11.90/5.99 != ( F @ Y4 ) ) ) ) )
% 11.90/5.99 => ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ A4 )
% 11.90/5.99 => ( ( member @ A @ Y4 @ A4 )
% 11.90/5.99 => ( ( ord_less_eq @ A @ X2 @ Y4 )
% 11.90/5.99 | ( ord_less_eq @ A @ Y4 @ X2 ) ) ) )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3260_linorder__injI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [F: A > B] :
% 11.90/5.99 ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( ord_less @ A @ X2 @ Y4 )
% 11.90/5.99 => ( ( F @ X2 )
% 11.90/5.99 != ( F @ Y4 ) ) )
% 11.90/5.99 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3261_inj__add__left,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( cancel_semigroup_add @ A )
% 11.90/5.99 => ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3262_inj__on__image__eq__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ C5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
% 11.90/5.99 => ( ( ( image2 @ A @ B @ F @ A4 )
% 11.90/5.99 = ( image2 @ A @ B @ F @ B6 ) )
% 11.90/5.99 = ( A4 = B6 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3263_inj__on__image__mem__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A3: A,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ B6 )
% 11.90/5.99 => ( ( member @ A @ A3 @ B6 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.99 => ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 = ( member @ A @ A3 @ A4 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3264_inj__img__insertE,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: B,B6: set @ B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ~ ( member @ B @ X @ B6 )
% 11.90/5.99 => ( ( ( insert2 @ B @ X @ B6 )
% 11.90/5.99 = ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 => ~ ! [X5: A,A11: set @ A] :
% 11.90/5.99 ( ~ ( member @ A @ X5 @ A11 )
% 11.90/5.99 => ( ( A4
% 11.90/5.99 = ( insert2 @ A @ X5 @ A11 ) )
% 11.90/5.99 => ( ( X
% 11.90/5.99 = ( F @ X5 ) )
% 11.90/5.99 => ( B6
% 11.90/5.99 != ( image2 @ A @ B @ F @ A11 ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3265_range__ex1__eq,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,B2: B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( member @ B @ B2 @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 11.90/5.99 = ( ? [X3: A] :
% 11.90/5.99 ( ( B2
% 11.90/5.99 = ( F @ X3 ) )
% 11.90/5.99 & ! [Y3: A] :
% 11.90/5.99 ( ( B2
% 11.90/5.99 = ( F @ Y3 ) )
% 11.90/5.99 => ( Y3 = X3 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3266_inj__image__eq__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ( image2 @ A @ B @ F @ A4 )
% 11.90/5.99 = ( image2 @ A @ B @ F @ B6 ) )
% 11.90/5.99 = ( A4 = B6 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3267_inj__image__mem__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A3: A,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 = ( member @ A @ A3 @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3268_comp__inj__on,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: B > C] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( inj_on @ B @ C @ G @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3269_inj__on__imageI,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A4 )
% 11.90/5.99 => ( inj_on @ C @ B @ G @ ( image2 @ A @ C @ F @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3270_comp__inj__on__iff,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,F3: B > C] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( inj_on @ B @ C @ F3 @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3271_inj__on__Un__image__eq__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 => ( ( ( image2 @ A @ B @ F @ A4 )
% 11.90/5.99 = ( image2 @ A @ B @ F @ B6 ) )
% 11.90/5.99 = ( A4 = B6 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3272_inj__compose,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( inj_on @ C @ A @ G @ ( top_top @ ( set @ C ) ) )
% 11.90/5.99 => ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3273_inj__on__strict__subset,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,B6: set @ A,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ B6 )
% 11.90/5.99 => ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.99 => ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3274_inj__on__fun__updI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,Y: B,X: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ~ ( member @ B @ Y @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 => ( inj_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3275_inj__on__fst__map__to__set,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3276_inj__diff__right,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( ab_group_add @ A )
% 11.90/5.99 => ! [A3: A] :
% 11.90/5.99 ( inj_on @ A @ A
% 11.90/5.99 @ ^ [B5: A] : ( minus_minus @ A @ B5 @ A3 )
% 11.90/5.99 @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3277_inj__singleton,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A] :
% 11.90/5.99 ( inj_on @ A @ ( set @ A )
% 11.90/5.99 @ ^ [X3: A] : ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 @ A4 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3278_finite__Collect,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,S2: set @ A,F: B > A] :
% 11.90/5.99 ( ( finite_finite2 @ A @ S2 )
% 11.90/5.99 => ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( finite_finite2 @ B
% 11.90/5.99 @ ( collect @ B
% 11.90/5.99 @ ^ [A7: B] : ( member @ A @ ( F @ A7 ) @ S2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3279_swap__inj__on,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
% 11.90/5.99 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 11.90/5.99 @ ^ [I3: A,J2: B] : ( product_Pair @ B @ A @ J2 @ I3 ) )
% 11.90/5.99 @ A4 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3280_rel__restrict__mono2,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ A,S2: set @ A,A4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ R5 @ S2 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A4 @ S2 ) @ ( rel_restrict @ A @ A4 @ R5 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3281_inj__image__subset__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) )
% 11.90/5.99 = ( ord_less_eq @ ( set @ A ) @ A4 @ B6 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3282_inj__on__image__Int,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ C5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
% 11.90/5.99 => ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 = ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3283_image__Int,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 = ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3284_inj__on__image__set__diff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,C5: set @ A,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ C5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
% 11.90/5.99 => ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 = ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3285_sum_Oreindex,axiom,
% 11.90/5.99 ! [C: $tType,A: $tType,B: $tType] :
% 11.90/5.99 ( ( comm_monoid_add @ A )
% 11.90/5.99 => ! [H3: B > C,A4: set @ B,G: C > A] :
% 11.90/5.99 ( ( inj_on @ B @ C @ H3 @ A4 )
% 11.90/5.99 => ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 11.90/5.99 = ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3286_image__set__diff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 = ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ F @ B6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3287_prod_Oreindex,axiom,
% 11.90/5.99 ! [C: $tType,A: $tType,B: $tType] :
% 11.90/5.99 ( ( comm_monoid_mult @ A )
% 11.90/5.99 => ! [H3: B > C,A4: set @ B,G: C > A] :
% 11.90/5.99 ( ( inj_on @ B @ C @ H3 @ A4 )
% 11.90/5.99 => ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 11.90/5.99 = ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3288_inj__vimage__image__eq,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 = A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3289_rel__restrict__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( rel_restrict @ A )
% 11.90/5.99 = ( ^ [R6: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 11.90/5.99 ( collect @ ( product_prod @ A @ A )
% 11.90/5.99 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.99 @ ^ [V2: A,W2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ R6 )
% 11.90/5.99 & ~ ( member @ A @ V2 @ A9 )
% 11.90/5.99 & ~ ( member @ A @ W2 @ A9 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3290_finite__imp__inj__to__nat__seg_H,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A] :
% 11.90/5.99 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.99 => ~ ! [F4: A > nat] :
% 11.90/5.99 ( ? [N3: nat] :
% 11.90/5.99 ( ( image2 @ A @ nat @ F4 @ A4 )
% 11.90/5.99 = ( collect @ nat
% 11.90/5.99 @ ^ [I3: nat] : ( ord_less @ nat @ I3 @ N3 ) ) )
% 11.90/5.99 => ~ ( inj_on @ A @ nat @ F4 @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3291_set__to__map__inverse,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
% 11.90/5.99 => ( ( map_to_set @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
% 11.90/5.99 = S2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3292_rel__restrict__Int__empty,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( ( inf_inf @ ( set @ A ) @ A4 @ ( field2 @ A @ R5 ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ( rel_restrict @ A @ R5 @ A4 )
% 11.90/5.99 = R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3293_Field__rel__restrict,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( field2 @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( field2 @ A @ R5 ) @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3294_vimage__subsetI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,B6: set @ B,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3295_inj__image__Compl__subset,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3296_The__case__prod,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,P: A > B > $o] :
% 11.90/5.99 ( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
% 11.90/5.99 = ( the @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3297_inj__on__disjoint__Un,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,G: A > B,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( inj_on @ A @ B @ G @ B6 )
% 11.90/5.99 => ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ ( image2 @ A @ B @ G @ B6 ) )
% 11.90/5.99 = ( bot_bot @ ( set @ B ) ) )
% 11.90/5.99 => ( inj_on @ A @ B
% 11.90/5.99 @ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F @ X3 ) @ ( G @ X3 ) )
% 11.90/5.99 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3298_set__to__map__simp,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),K: A,V: B] :
% 11.90/5.99 ( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
% 11.90/5.99 => ( ( ( set_to_map @ A @ B @ S2 @ K )
% 11.90/5.99 = ( some @ B @ V ) )
% 11.90/5.99 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ S2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3299_rel__restrict__compl,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 11.90/5.99 ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 ) @ ( rel_restrict @ A @ R5 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) )
% 11.90/5.99 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3300_homo__rel__restrict__mono,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),B6: set @ A,A4: set @ A] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.99 @ ( product_Sigma @ A @ A @ B6
% 11.90/5.99 @ ^ [Uu: A] : B6 ) )
% 11.90/5.99 => ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R5 @ A4 )
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A4 )
% 11.90/5.99 @ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3301_rel__restrict__alt__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( rel_restrict @ A )
% 11.90/5.99 = ( ^ [R6: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 11.90/5.99 ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R6
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A9 )
% 11.90/5.99 @ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ A9 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3302_inj__on__Un,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/5.99 = ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 & ( inj_on @ A @ B @ F @ B6 )
% 11.90/5.99 & ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A4 @ B6 ) ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B6 @ A4 ) ) )
% 11.90/5.99 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3303_inj__vimage__singleton,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A3: B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
% 11.90/5.99 @ ( insert2 @ A
% 11.90/5.99 @ ( the @ A
% 11.90/5.99 @ ^ [X3: A] :
% 11.90/5.99 ( ( F @ X3 )
% 11.90/5.99 = A3 ) )
% 11.90/5.99 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3304_If__the__inv__into__f__f,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,I: A,C5: set @ A,G: A > B,X: A] :
% 11.90/5.99 ( ( member @ A @ I @ C5 )
% 11.90/5.99 => ( ( inj_on @ A @ B @ G @ C5 )
% 11.90/5.99 => ( ( comp @ B @ A @ A
% 11.90/5.99 @ ^ [I3: B] : ( if @ A @ ( member @ B @ I3 @ ( image2 @ A @ B @ G @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G @ I3 ) @ X )
% 11.90/5.99 @ G
% 11.90/5.99 @ I )
% 11.90/5.99 = ( id @ A @ I ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3305_ran__map__upd__Some,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z2: A] :
% 11.90/5.99 ( ( ( M2 @ X )
% 11.90/5.99 = ( some @ A @ Y ) )
% 11.90/5.99 => ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
% 11.90/5.99 => ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
% 11.90/5.99 => ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z2 ) ) )
% 11.90/5.99 = ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3306_trancl__multi__insert,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),X10: set @ A,M2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
% 11.90/5.99 @ ( transitive_trancl @ A
% 11.90/5.99 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/5.99 @ ( product_Sigma @ A @ A @ X10
% 11.90/5.99 @ ^ [Uu: A] : ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ~ ! [X2: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ X10 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3307_trancl__multi__insert2,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),M2: A,X10: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 )
% 11.90/5.99 @ ( transitive_trancl @ A
% 11.90/5.99 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( insert2 @ A @ M2 @ ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 @ ^ [Uu: A] : X10 ) ) ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ~ ! [X2: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ X10 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ M2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3308_the__inv__f__o__f__id,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,Z2: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F @ Z2 )
% 11.90/5.99 = ( id @ A @ Z2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3309_restrict__map__self,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( restrict_map @ A @ B @ M2 @ ( dom @ A @ B @ M2 ) )
% 11.90/5.99 = M2 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3310_same__fst__trancl,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,P: A > $o,R5: A > ( set @ ( product_prod @ B @ B ) )] :
% 11.90/5.99 ( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R5 ) )
% 11.90/5.99 = ( same_fst @ A @ B @ P
% 11.90/5.99 @ ^ [X3: A] : ( transitive_trancl @ B @ ( R5 @ X3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3311_dom__eq__empty__conv,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > ( option @ B )] :
% 11.90/5.99 ( ( ( dom @ A @ B @ F )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 = ( F
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3312_fun__upd__None__if__notin__dom,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: A,M2: A > ( option @ B )] :
% 11.90/5.99 ( ~ ( member @ A @ K @ ( dom @ A @ B @ M2 ) )
% 11.90/5.99 => ( ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) )
% 11.90/5.99 = M2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3313_the__inv__into__onto,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( image2 @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 = A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3314_dom__empty,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( dom @ A @ B
% 11.90/5.99 @ ^ [X3: A] : ( none @ B ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3315_map__update__eta__repair_I1_J,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: A,V: B,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( dom @ A @ B
% 11.90/5.99 @ ^ [X3: A] : ( if @ ( option @ B ) @ ( X3 = K ) @ ( some @ B @ V ) @ ( M2 @ X3 ) ) )
% 11.90/5.99 = ( insert2 @ A @ K @ ( dom @ A @ B @ M2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3316_dom__const_H,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/5.99 ( ( dom @ A @ B
% 11.90/5.99 @ ^ [X3: A] : ( some @ B @ ( F @ X3 ) ) )
% 11.90/5.99 = ( top_top @ ( set @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3317_dom__mmupd,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B ),K2: set @ A,V: B] :
% 11.90/5.99 ( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M2 @ K2 @ V ) )
% 11.90/5.99 = ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ K2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3318_trancl__single,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A] :
% 11.90/5.99 ( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
% 11.90/5.99 = ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3319_restrict__map__inv,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > ( option @ B )] :
% 11.90/5.99 ( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F ) ) )
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3320_dom__fun__upd,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Y: option @ B,F: A > ( option @ B ),X: A] :
% 11.90/5.99 ( ( ( Y
% 11.90/5.99 = ( none @ B ) )
% 11.90/5.99 => ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
% 11.90/5.99 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 11.90/5.99 & ( ( Y
% 11.90/5.99 != ( none @ B ) )
% 11.90/5.99 => ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y ) )
% 11.90/5.99 = ( insert2 @ A @ X @ ( dom @ A @ B @ F ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3321_domI,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
% 11.90/5.99 ( ( ( M2 @ A3 )
% 11.90/5.99 = ( some @ A @ B2 ) )
% 11.90/5.99 => ( member @ B @ A3 @ ( dom @ B @ A @ M2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3322_domD,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
% 11.90/5.99 => ? [B3: B] :
% 11.90/5.99 ( ( M2 @ A3 )
% 11.90/5.99 = ( some @ B @ B3 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3323_domIff,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
% 11.90/5.99 = ( ( M2 @ A3 )
% 11.90/5.99 != ( none @ B ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3324_nempty__dom,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,E2: A > ( option @ B )] :
% 11.90/5.99 ( ( E2
% 11.90/5.99 != ( ^ [X3: A] : ( none @ B ) ) )
% 11.90/5.99 => ~ ! [M5: A] :
% 11.90/5.99 ~ ( member @ A @ M5 @ ( dom @ A @ B @ E2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3325_trancl__sub,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( transitive_trancl @ A @ R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3326_trancl__mono__mp,axiom,
% 11.90/5.99 ! [A: $tType,U4: set @ ( product_prod @ A @ A ),V6: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U4 @ V6 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ U4 ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ X @ ( transitive_trancl @ A @ V6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3327_trancl_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R )
% 11.90/5.99 => ~ ! [B3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3328_trancl_Osimps,axiom,
% 11.90/5.99 ! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 = ( ? [A7: A,B5: A] :
% 11.90/5.99 ( ( A1 = A7 )
% 11.90/5.99 & ( A22 = B5 )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R ) )
% 11.90/5.99 | ? [A7: A,B5: A,C3: A] :
% 11.90/5.99 ( ( A1 = A7 )
% 11.90/5.99 & ( A22 = C3 )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3329_trancl_Or__into__trancl,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3330_tranclE,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.99 => ~ ! [C4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C4 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C4 @ B2 ) @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3331_trancl__trans,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3332_trancl__induct,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ! [Y4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ R )
% 11.90/5.99 => ( P @ Y4 ) )
% 11.90/5.99 => ( ! [Y4: A,Z3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.90/5.99 => ( ( P @ Y4 )
% 11.90/5.99 => ( P @ Z3 ) ) ) )
% 11.90/5.99 => ( P @ B2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3333_r__r__into__trancl,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R5 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R5 )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3334_converse__tranclE,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Z2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R )
% 11.90/5.99 => ~ ! [Y4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3335_irrefl__trancl__rD,axiom,
% 11.90/5.99 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.90/5.99 ( ! [X2: A] :
% 11.90/5.99 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/5.99 => ( X != Y ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3336_Transitive__Closure_Otrancl__into__trancl,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3337_trancl__into__trancl2,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3338_trancl__trans__induct,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 11.90/5.99 => ( P @ X2 @ Y4 ) )
% 11.90/5.99 => ( ! [X2: A,Y4: A,Z3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( P @ X2 @ Y4 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( P @ Y4 @ Z3 )
% 11.90/5.99 => ( P @ X2 @ Z3 ) ) ) ) )
% 11.90/5.99 => ( P @ X @ Y ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3339_converse__trancl__induct,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ! [Y4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R )
% 11.90/5.99 => ( P @ Y4 ) )
% 11.90/5.99 => ( ! [Y4: A,Z3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( P @ Z3 )
% 11.90/5.99 => ( P @ Y4 ) ) ) )
% 11.90/5.99 => ( P @ A3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3340_dom__def,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( dom @ A @ B )
% 11.90/5.99 = ( ^ [M: A > ( option @ B )] :
% 11.90/5.99 ( collect @ A
% 11.90/5.99 @ ^ [A7: A] :
% 11.90/5.99 ( ( M @ A7 )
% 11.90/5.99 != ( none @ B ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3341_the__inv__into__f__f,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( member @ A @ X @ A4 )
% 11.90/5.99 => ( ( the_inv_into @ A @ B @ A4 @ F @ ( F @ X ) )
% 11.90/5.99 = X ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3342_the__inv__into__f__eq,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A,Y: B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( ( F @ X )
% 11.90/5.99 = Y )
% 11.90/5.99 => ( ( member @ A @ X @ A4 )
% 11.90/5.99 => ( ( the_inv_into @ A @ B @ A4 @ F @ Y )
% 11.90/5.99 = X ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3343_insert__dom,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Y: A] :
% 11.90/5.99 ( ( ( F @ X )
% 11.90/5.99 = ( some @ A @ Y ) )
% 11.90/5.99 => ( ( insert2 @ B @ X @ ( dom @ B @ A @ F ) )
% 11.90/5.99 = ( dom @ B @ A @ F ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3344_trancl__rtrancl__trancl,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3345_rtrancl__trancl__trancl,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3346_rtrancl__into__trancl2,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3347_rtrancl__into__trancl1,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3348_rtrancl__eq__or__trancl,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 = ( ( X = Y )
% 11.90/5.99 | ( ( X != Y )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3349_trancl__into__rtrancl,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3350_tranclD2,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) )
% 11.90/5.99 => ? [Z3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R5 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3351_rtranclD,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ( ( A3 = B2 )
% 11.90/5.99 | ( ( A3 != B2 )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3352_tranclD,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R5 ) )
% 11.90/5.99 => ? [Z3: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R5 )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3353_map__dom__ran__finite,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M3: A > ( option @ B )] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
% 11.90/5.99 => ( finite_finite2 @ B @ ( ran @ A @ B @ M3 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3354_rel__restrict__trancl__notR_I2_J,axiom,
% 11.90/5.99 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) )
% 11.90/5.99 => ~ ( member @ A @ W @ R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3355_rel__restrict__trancl__notR_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) )
% 11.90/5.99 => ~ ( member @ A @ V @ R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3356_rel__restrict__trancl__mem,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,A4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R5 ) ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R5 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3357_trancl__induct2,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 11.90/5.99 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.99 => ( ! [A5: A,B3: B] :
% 11.90/5.99 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ R )
% 11.90/5.99 => ( P @ A5 @ B3 ) )
% 11.90/5.99 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 11.90/5.99 ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) ) @ R )
% 11.90/5.99 => ( ( P @ A5 @ B3 )
% 11.90/5.99 => ( P @ Aa3 @ Ba ) ) ) )
% 11.90/5.99 => ( P @ Bx @ By ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3358_card__map__to__set,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( finite_card @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
% 11.90/5.99 = ( finite_card @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3359_trancl__sub__insert__trancl,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R5 ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ X @ R5 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3360_rel__restrict__trancl__sub,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),R5: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A4 @ R5 ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A4 ) @ R5 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3361_le__map__dom__mono,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( preorder @ B )
% 11.90/5.99 => ! [M2: A > ( option @ B ),M7: A > ( option @ B )] :
% 11.90/5.99 ( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M2 ) @ ( dom @ A @ B @ M7 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3362_the__inv__into__def,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( the_inv_into @ A @ B )
% 11.90/5.99 = ( ^ [A9: set @ A,F2: A > B,X3: B] :
% 11.90/5.99 ( the @ A
% 11.90/5.99 @ ^ [Y3: A] :
% 11.90/5.99 ( ( member @ A @ Y3 @ A9 )
% 11.90/5.99 & ( ( F2 @ Y3 )
% 11.90/5.99 = X3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3363_finite__map__freshness,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: A > ( option @ B )] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( dom @ A @ B @ F ) )
% 11.90/5.99 => ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ? [X2: A] :
% 11.90/5.99 ( ( F @ X2 )
% 11.90/5.99 = ( none @ B ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3364_dom__minus,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,A4: set @ B] :
% 11.90/5.99 ( ( ( F @ X )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 => ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ ( insert2 @ B @ X @ A4 ) )
% 11.90/5.99 = ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3365_restrict__map__eq_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,M2: B > ( option @ A ),A4: set @ B,K: B] :
% 11.90/5.99 ( ( ( restrict_map @ B @ A @ M2 @ A4 @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 = ( ~ ( member @ B @ K @ ( inf_inf @ ( set @ B ) @ ( dom @ B @ A @ M2 ) @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3366_map__card__eq__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M3: A > ( option @ B ),X: A,Y: A] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
% 11.90/5.99 => ( ( ( finite_card @ A @ ( dom @ A @ B @ M3 ) )
% 11.90/5.99 = ( finite_card @ B @ ( ran @ A @ B @ M3 ) ) )
% 11.90/5.99 => ( ( member @ A @ X @ ( dom @ A @ B @ M3 ) )
% 11.90/5.99 => ( ( ( M3 @ X )
% 11.90/5.99 = ( M3 @ Y ) )
% 11.90/5.99 = ( X = Y ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3367_wf__linord__ex__has__least,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M2: B > A] :
% 11.90/5.99 ( ( wf @ A @ R )
% 11.90/5.99 => ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
% 11.90/5.99 => ( ( P @ K )
% 11.90/5.99 => ? [X2: B] :
% 11.90/5.99 ( ( P @ X2 )
% 11.90/5.99 & ! [Y6: B] :
% 11.90/5.99 ( ( P @ Y6 )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X2 ) @ ( M2 @ Y6 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3368_trancl__union__outside,axiom,
% 11.90/5.99 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E4 ) )
% 11.90/5.99 => ? [X2: A,Y4: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ X2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ U4 )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ W ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E4 @ U4 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3369_trancl__over__edgeE,axiom,
% 11.90/5.99 ! [A: $tType,U: A,W: A,V1: A,V22: A,E4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V1 @ V22 ) @ E4 ) ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ W ) @ ( transitive_trancl @ A @ E4 ) )
% 11.90/5.99 => ~ ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V1 ) @ ( transitive_rtrancl @ A @ E4 ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V22 @ W ) @ ( transitive_rtrancl @ A @ E4 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3370_finite__map__to__set,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) )
% 11.90/5.99 = ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3371_map__to__set__dom,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( dom @ A @ B )
% 11.90/5.99 = ( ^ [M: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3372_inj__on__the__inv__into,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3373_f__the__inv__into__f,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,Y: B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( member @ B @ Y @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 => ( ( F @ ( the_inv_into @ A @ B @ A4 @ F @ Y ) )
% 11.90/5.99 = Y ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3374_the__inv__f__f,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,X: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
% 11.90/5.99 = X ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3375_set__to__map__dom,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( dom @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
% 11.90/5.99 = ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3376_Restr__trancl__mono,axiom,
% 11.90/5.99 ! [A: $tType,V: A,W: A,E4: set @ ( product_prod @ A @ A ),U4: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W )
% 11.90/5.99 @ ( transitive_trancl @ A
% 11.90/5.99 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 11.90/5.99 @ ( product_Sigma @ A @ A @ U4
% 11.90/5.99 @ ^ [Uu: A] : U4 ) ) ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V @ W ) @ ( transitive_trancl @ A @ E4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3377_inj__on__map__the,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,D4: set @ A,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ D4 @ ( dom @ A @ B @ M2 ) )
% 11.90/5.99 => ( ( inj_on @ A @ ( option @ B ) @ M2 @ D4 )
% 11.90/5.99 => ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M2 ) @ D4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3378_finite__Map__induct,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
% 11.90/5.99 => ( ( P
% 11.90/5.99 @ ^ [X3: A] : ( none @ B ) )
% 11.90/5.99 => ( ! [K5: A,V4: B,M5: A > ( option @ B )] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( dom @ A @ B @ M5 ) )
% 11.90/5.99 => ( ~ ( member @ A @ K5 @ ( dom @ A @ B @ M5 ) )
% 11.90/5.99 => ( ( P @ M5 )
% 11.90/5.99 => ( P @ ( fun_upd @ A @ ( option @ B ) @ M5 @ K5 @ ( some @ B @ V4 ) ) ) ) ) )
% 11.90/5.99 => ( P @ M2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3379_ran__is__image,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType] :
% 11.90/5.99 ( ( ran @ B @ A )
% 11.90/5.99 = ( ^ [M8: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M8 ) @ ( dom @ B @ A @ M8 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3380_graph__eq__to__snd__dom,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( graph @ A @ B )
% 11.90/5.99 = ( ^ [M: A > ( option @ B )] :
% 11.90/5.99 ( image2 @ A @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ ( the2 @ B @ ( M @ X3 ) ) )
% 11.90/5.99 @ ( dom @ A @ B @ M ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3381_trancl__insert2,axiom,
% 11.90/5.99 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
% 11.90/5.99 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
% 11.90/5.99 @ ( collect @ ( product_prod @ A @ A )
% 11.90/5.99 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.99 @ ^ [X3: A,Y3: A] :
% 11.90/5.99 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ A3 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 | ( X3 = A3 ) )
% 11.90/5.99 & ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y3 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/5.99 | ( Y3 = B2 ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3382_rel__restrict__Sigma__sub,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A,R5: set @ A] :
% 11.90/5.99 ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
% 11.90/5.99 @ ( rel_restrict @ A
% 11.90/5.99 @ ( transitive_trancl @ A
% 11.90/5.99 @ ( product_Sigma @ A @ A @ A4
% 11.90/5.99 @ ^ [Uu: A] : A4 ) )
% 11.90/5.99 @ R5 )
% 11.90/5.99 @ ( transitive_trancl @ A
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A4 @ R5 )
% 11.90/5.99 @ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ A4 @ R5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3383_the__inv__into__into,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: B,B6: set @ A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( member @ B @ X @ ( image2 @ A @ B @ F @ A4 ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/5.99 => ( member @ A @ ( the_inv_into @ A @ B @ A4 @ F @ X ) @ B6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3384_the__inv__into__comp,axiom,
% 11.90/5.99 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A4: set @ C,X: B] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) )
% 11.90/5.99 => ( ( inj_on @ C @ A @ G @ A4 )
% 11.90/5.99 => ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) ) )
% 11.90/5.99 => ( ( the_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F @ G ) @ X )
% 11.90/5.99 = ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A4 @ G ) @ ( the_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A4 ) @ F ) @ X ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3385_dom__eq__singleton__conv,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
% 11.90/5.99 ( ( ( dom @ A @ B @ F )
% 11.90/5.99 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.99 = ( ? [V2: B] :
% 11.90/5.99 ( F
% 11.90/5.99 = ( fun_upd @ A @ ( option @ B )
% 11.90/5.99 @ ^ [X3: A] : ( none @ B )
% 11.90/5.99 @ X
% 11.90/5.99 @ ( some @ B @ V2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3386_rtrancl__last__visit,axiom,
% 11.90/5.99 ! [A: $tType,Q3: A,Q4: A,R5: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Q4 )
% 11.90/5.99 @ ( transitive_rtrancl @ A
% 11.90/5.99 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : S2 ) ) ) )
% 11.90/5.99 => ~ ! [Qt: A] :
% 11.90/5.99 ( ( member @ A @ Qt @ S2 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ Qt ) @ ( transitive_trancl @ A @ R5 ) )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q4 )
% 11.90/5.99 @ ( transitive_rtrancl @ A
% 11.90/5.99 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5
% 11.90/5.99 @ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
% 11.90/5.99 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3387_trancl__insert,axiom,
% 11.90/5.99 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
% 11.90/5.99 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
% 11.90/5.99 @ ( collect @ ( product_prod @ A @ A )
% 11.90/5.99 @ ( product_case_prod @ A @ A @ $o
% 11.90/5.99 @ ^ [A7: A,B5: A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3388_finite__trancl__ntranl,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
% 11.90/5.99 => ( ( transitive_trancl @ A @ R5 )
% 11.90/5.99 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) @ ( one_one @ nat ) ) @ R5 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3389_ran__add,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
% 11.90/5.99 ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( dom @ A @ B @ G ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ran @ A @ B @ ( map_add @ A @ B @ F @ G ) )
% 11.90/5.99 = ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F ) @ ( ran @ A @ B @ G ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3390_eq__f__restr__ss__eq,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,S: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ S @ ( dom @ A @ B @ ( F @ A4 ) ) )
% 11.90/5.99 => ( ( A4
% 11.90/5.99 = ( restrict_map @ A @ B @ ( F @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S ) ) )
% 11.90/5.99 = ( ( map_le @ A @ B @ A4 @ ( F @ A4 ) )
% 11.90/5.99 & ( S
% 11.90/5.99 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3391_eq__f__restr__conv,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,S: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A4: A > ( option @ B )] :
% 11.90/5.99 ( ( ( ord_less_eq @ ( set @ A ) @ S @ ( dom @ A @ B @ ( F @ A4 ) ) )
% 11.90/5.99 & ( A4
% 11.90/5.99 = ( restrict_map @ A @ B @ ( F @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ S ) ) ) )
% 11.90/5.99 = ( ( map_le @ A @ B @ A4 @ ( F @ A4 ) )
% 11.90/5.99 & ( S
% 11.90/5.99 = ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A4 ) ) @ ( dom @ A @ B @ A4 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3392_type__copy__set__map0,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F5: $tType,Rep: A > B,Abs: B > A,S2: B > ( set @ D ),M3: C > B,F: E > D,S7: C > ( set @ E ),G: F5 > C] :
% 11.90/5.99 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( ( ( comp @ B @ ( set @ D ) @ C @ S2 @ M3 )
% 11.90/5.99 = ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F ) @ S7 ) )
% 11.90/5.99 => ( ( comp @ A @ ( set @ D ) @ F5 @ ( comp @ B @ ( set @ D ) @ A @ S2 @ Rep ) @ ( comp @ C @ A @ F5 @ ( comp @ B @ A @ C @ Abs @ M3 ) @ G ) )
% 11.90/5.99 = ( comp @ ( set @ E ) @ ( set @ D ) @ F5 @ ( image2 @ E @ D @ F ) @ ( comp @ C @ ( set @ E ) @ F5 @ S7 @ G ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3393_map__add__find__right,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,N: B > ( option @ A ),K: B,Xx: A,M2: B > ( option @ A )] :
% 11.90/5.99 ( ( ( N @ K )
% 11.90/5.99 = ( some @ A @ Xx ) )
% 11.90/5.99 => ( ( map_add @ B @ A @ M2 @ N @ K )
% 11.90/5.99 = ( some @ A @ Xx ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3394_map__add__eq__empty__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
% 11.90/5.99 ( ( ( map_add @ A @ B @ F @ G )
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) )
% 11.90/5.99 = ( ( F
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) )
% 11.90/5.99 & ( G
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3395_map__add__None,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B] :
% 11.90/5.99 ( ( ( map_add @ B @ A @ M2 @ N @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 = ( ( ( N @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 & ( ( M2 @ K )
% 11.90/5.99 = ( none @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3396_empty__map__add,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( map_add @ A @ B
% 11.90/5.99 @ ^ [X3: A] : ( none @ B )
% 11.90/5.99 @ M2 )
% 11.90/5.99 = M2 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3397_map__add__empty,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( map_add @ A @ B @ M2
% 11.90/5.99 @ ^ [X3: A] : ( none @ B ) )
% 11.90/5.99 = M2 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3398_empty__eq__map__add__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
% 11.90/5.99 ( ( ( ^ [X3: A] : ( none @ B ) )
% 11.90/5.99 = ( map_add @ A @ B @ F @ G ) )
% 11.90/5.99 = ( ( F
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) )
% 11.90/5.99 & ( G
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3399_map__add__upd,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: A > ( option @ B ),G: A > ( option @ B ),X: A,Y: B] :
% 11.90/5.99 ( ( map_add @ A @ B @ F @ ( fun_upd @ A @ ( option @ B ) @ G @ X @ ( some @ B @ Y ) ) )
% 11.90/5.99 = ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F @ G ) @ X @ ( some @ B @ Y ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3400_map__add__upd__left,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,M2: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
% 11.90/5.99 ( ~ ( member @ A @ M2 @ ( dom @ A @ B @ E22 ) )
% 11.90/5.99 => ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M2 @ ( some @ B @ U1 ) ) @ E22 )
% 11.90/5.99 = ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M2 @ ( some @ B @ U1 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3401_le__map__mmupd__not__dom,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,M2: A > ( option @ B ),K2: set @ A,V: B] : ( map_le @ A @ B @ M2 @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3402_map__add__find__left,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,G: B > ( option @ A ),K: B,F: B > ( option @ A )] :
% 11.90/5.99 ( ( ( G @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 => ( ( map_add @ B @ A @ F @ G @ K )
% 11.90/5.99 = ( F @ K ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3403_map__add__left__None,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > ( option @ A ),K: B,G: B > ( option @ A )] :
% 11.90/5.99 ( ( ( F @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 => ( ( map_add @ B @ A @ F @ G @ K )
% 11.90/5.99 = ( G @ K ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3404_map__leI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
% 11.90/5.99 ( ! [X2: A,V4: B] :
% 11.90/5.99 ( ( ( M1 @ X2 )
% 11.90/5.99 = ( some @ B @ V4 ) )
% 11.90/5.99 => ( ( M22 @ X2 )
% 11.90/5.99 = ( some @ B @ V4 ) ) )
% 11.90/5.99 => ( map_le @ A @ B @ M1 @ M22 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3405_map__leD,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),K: A,V: B] :
% 11.90/5.99 ( ( map_le @ A @ B @ M1 @ M22 )
% 11.90/5.99 => ( ( ( M1 @ K )
% 11.90/5.99 = ( some @ B @ V ) )
% 11.90/5.99 => ( ( M22 @ K )
% 11.90/5.99 = ( some @ B @ V ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3406_map__add__first__le,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( order @ B )
% 11.90/5.99 => ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B )] :
% 11.90/5.99 ( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
% 11.90/5.99 => ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3407_map__le__empty,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,G: A > ( option @ B )] :
% 11.90/5.99 ( map_le @ A @ B
% 11.90/5.99 @ ^ [X3: A] : ( none @ B )
% 11.90/5.99 @ G ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3408_map__add__Some__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
% 11.90/5.99 ( ( ( map_add @ B @ A @ M2 @ N @ K )
% 11.90/5.99 = ( some @ A @ X ) )
% 11.90/5.99 = ( ( ( N @ K )
% 11.90/5.99 = ( some @ A @ X ) )
% 11.90/5.99 | ( ( ( N @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 & ( ( M2 @ K )
% 11.90/5.99 = ( some @ A @ X ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3409_map__add__SomeD,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
% 11.90/5.99 ( ( ( map_add @ B @ A @ M2 @ N @ K )
% 11.90/5.99 = ( some @ A @ X ) )
% 11.90/5.99 => ( ( ( N @ K )
% 11.90/5.99 = ( some @ A @ X ) )
% 11.90/5.99 | ( ( ( N @ K )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 & ( ( M2 @ K )
% 11.90/5.99 = ( some @ A @ X ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3410_type__copy__map__comp0__undo,axiom,
% 11.90/5.99 ! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F5: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F5,Abs3: F5 > E,M3: F5 > D,M13: B > D,M24: F5 > B] :
% 11.90/5.99 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
% 11.90/5.99 => ( ( type_definition @ E @ F5 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F5 ) ) )
% 11.90/5.99 => ( ( ( comp @ F5 @ C @ E @ ( comp @ D @ C @ F5 @ Abs2 @ M3 ) @ Rep3 )
% 11.90/5.99 = ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M13 ) @ Rep ) @ ( comp @ F5 @ A @ E @ ( comp @ B @ A @ F5 @ Abs @ M24 ) @ Rep3 ) ) )
% 11.90/5.99 => ( ( comp @ B @ D @ F5 @ M13 @ M24 )
% 11.90/5.99 = M3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3411_type__copy__map__comp0,axiom,
% 11.90/5.99 ! [F5: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M3: C > D,M13: B > D,M24: C > B,F: D > F5,G: E > C] :
% 11.90/5.99 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( ( M3
% 11.90/5.99 = ( comp @ B @ D @ C @ M13 @ M24 ) )
% 11.90/5.99 => ( ( comp @ C @ F5 @ E @ ( comp @ D @ F5 @ C @ F @ M3 ) @ G )
% 11.90/5.99 = ( comp @ A @ F5 @ E @ ( comp @ B @ F5 @ A @ ( comp @ D @ F5 @ B @ F @ M13 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M24 ) @ G ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3412_upd__None__map__le,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > ( option @ B ),X: A] : ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) @ F ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3413_map__add__def,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( map_add @ A @ B )
% 11.90/5.99 = ( ^ [M14: A > ( option @ B ),M25: A > ( option @ B ),X3: A] : ( case_option @ ( option @ B ) @ B @ ( M14 @ X3 ) @ ( some @ B ) @ ( M25 @ X3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3414_map__add__left__comm,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A4: A > ( option @ B ),B6: A > ( option @ B ),C5: A > ( option @ B )] :
% 11.90/5.99 ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A4 ) @ ( dom @ A @ B @ B6 ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ( map_add @ A @ B @ A4 @ ( map_add @ A @ B @ B6 @ C5 ) )
% 11.90/5.99 = ( map_add @ A @ B @ B6 @ ( map_add @ A @ B @ A4 @ C5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3415_type__copy__map__id0,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M3: B > B] :
% 11.90/5.99 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( ( M3
% 11.90/5.99 = ( id @ B ) )
% 11.90/5.99 => ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M3 ) @ Rep )
% 11.90/5.99 = ( id @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3416_type__copy__Abs__o__Rep,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
% 11.90/5.99 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( ( comp @ B @ A @ A @ Abs @ Rep )
% 11.90/5.99 = ( id @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3417_type__copy__Rep__o__Abs,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
% 11.90/5.99 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.90/5.99 => ( ( comp @ A @ B @ B @ Rep @ Abs )
% 11.90/5.99 = ( id @ B ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3418_type__definition__integer,axiom,
% 11.90/5.99 type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3419_map__le__imp__upd__le,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),X: A,Y: B] :
% 11.90/5.99 ( ( map_le @ A @ B @ M1 @ M22 )
% 11.90/5.99 => ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M1 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M22 @ X @ ( some @ B @ Y ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3420_map__add__distinct__le,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( preorder @ B )
% 11.90/5.99 => ! [M2: A > ( option @ B ),M7: A > ( option @ B ),N: A > ( option @ B ),N2: A > ( option @ B )] :
% 11.90/5.99 ( ( ord_less_eq @ ( A > ( option @ B ) ) @ M2 @ M7 )
% 11.90/5.99 => ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N2 )
% 11.90/5.99 => ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M7 ) @ ( dom @ A @ B @ N2 ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M2 @ N ) @ ( map_add @ A @ B @ M7 @ N2 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3421_map__mmupd__update__less,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,K2: set @ A,K9: set @ A,M2: A > ( option @ B ),V: B] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ K2 @ K9 )
% 11.90/5.99 => ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K2 @ ( dom @ A @ B @ M2 ) ) @ V ) @ ( map_mmupd @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ K9 @ ( dom @ A @ B @ M2 ) ) @ V ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3422_trancl__set__ntrancl,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
% 11.90/5.99 = ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3423_pred__nat__trancl__eq__le,axiom,
% 11.90/5.99 ! [M2: nat,N: nat] :
% 11.90/5.99 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
% 11.90/5.99 = ( ord_less_eq @ nat @ M2 @ N ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3424_less__eq,axiom,
% 11.90/5.99 ! [M2: nat,N: nat] :
% 11.90/5.99 ( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
% 11.90/5.99 = ( ord_less @ nat @ M2 @ N ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3425_Gcd__fin__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( semiring_gcd @ A )
% 11.90/5.99 => ( ( semiring_gcd_Gcd_fin @ A )
% 11.90/5.99 = ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3426_relImage__def,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType] :
% 11.90/5.99 ( ( bNF_Gr4221423524335903396lImage @ B @ A )
% 11.90/5.99 = ( ^ [R6: set @ ( product_prod @ B @ B ),F2: B > A] :
% 11.90/5.99 ( collect @ ( product_prod @ A @ A )
% 11.90/5.99 @ ^ [Uu: product_prod @ A @ A] :
% 11.90/5.99 ? [A12: B,A23: B] :
% 11.90/5.99 ( ( Uu
% 11.90/5.99 = ( product_Pair @ A @ A @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) )
% 11.90/5.99 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R6 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3427_set__map__filter,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,G: B > ( option @ A ),Xs: list @ B] :
% 11.90/5.99 ( ( set2 @ A @ ( map_filter @ B @ A @ G @ Xs ) )
% 11.90/5.99 = ( collect @ A
% 11.90/5.99 @ ^ [Y3: A] :
% 11.90/5.99 ? [X3: B] :
% 11.90/5.99 ( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
% 11.90/5.99 & ( ( G @ X3 )
% 11.90/5.99 = ( some @ A @ Y3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3428_ran__map__of,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( map_of @ B @ A @ Xs ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3429_trancl__restrict__reachable,axiom,
% 11.90/5.99 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_trancl @ A @ E4 ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ S2 ) @ S2 )
% 11.90/5.99 => ( ( member @ A @ U @ S2 )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V )
% 11.90/5.99 @ ( transitive_trancl @ A
% 11.90/5.99 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ E4
% 11.90/5.99 @ ( product_Sigma @ A @ A @ S2
% 11.90/5.99 @ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3430_pair__vimage__is__Image,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,U: B,E4: set @ ( product_prod @ B @ A )] :
% 11.90/5.99 ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ U ) @ E4 )
% 11.90/5.99 = ( image @ B @ A @ E4 @ ( insert2 @ B @ U @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3431_set__bind__option,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,X: option @ B,F: B > ( option @ A )] :
% 11.90/5.99 ( ( set_option @ A @ ( bind @ B @ A @ X @ F ) )
% 11.90/5.99 = ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F ) @ ( set_option @ B @ X ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3432_ImageI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),A4: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 11.90/5.99 => ( ( member @ A @ A3 @ A4 )
% 11.90/5.99 => ( member @ B @ B2 @ ( image @ A @ B @ R @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3433_elem__set,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Xo: option @ A] :
% 11.90/5.99 ( ( member @ A @ X @ ( set_option @ A @ Xo ) )
% 11.90/5.99 = ( Xo
% 11.90/5.99 = ( some @ A @ X ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3434_set__empty__eq,axiom,
% 11.90/5.99 ! [A: $tType,Xo: option @ A] :
% 11.90/5.99 ( ( ( set_option @ A @ Xo )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 = ( Xo
% 11.90/5.99 = ( none @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3435_Image__singleton__iff,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A3: B] :
% 11.90/5.99 ( ( member @ A @ B2 @ ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
% 11.90/5.99 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3436_rev__ImageI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,A3: A,A4: set @ A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( member @ A @ A3 @ A4 )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 11.90/5.99 => ( member @ B @ B2 @ ( image @ A @ B @ R @ A4 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3437_Image__iff,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A4: set @ B] :
% 11.90/5.99 ( ( member @ A @ B2 @ ( image @ B @ A @ R @ A4 ) )
% 11.90/5.99 = ( ? [X3: B] :
% 11.90/5.99 ( ( member @ B @ X3 @ A4 )
% 11.90/5.99 & ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ B2 ) @ R ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3438_ImageE,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A4: set @ B] :
% 11.90/5.99 ( ( member @ A @ B2 @ ( image @ B @ A @ R @ A4 ) )
% 11.90/5.99 => ~ ! [X2: B] :
% 11.90/5.99 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R )
% 11.90/5.99 => ~ ( member @ B @ X2 @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3439_ospec,axiom,
% 11.90/5.99 ! [A: $tType,A4: option @ A,P: A > $o,X: A] :
% 11.90/5.99 ( ! [X2: A] :
% 11.90/5.99 ( ( member @ A @ X2 @ ( set_option @ A @ A4 ) )
% 11.90/5.99 => ( P @ X2 ) )
% 11.90/5.99 => ( ( A4
% 11.90/5.99 = ( some @ A @ X ) )
% 11.90/5.99 => ( P @ X ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3440_option_Oset__intros,axiom,
% 11.90/5.99 ! [A: $tType,X22: A] : ( member @ A @ X22 @ ( set_option @ A @ ( some @ A @ X22 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3441_option_Oset__cases,axiom,
% 11.90/5.99 ! [A: $tType,E2: A,A3: option @ A] :
% 11.90/5.99 ( ( member @ A @ E2 @ ( set_option @ A @ A3 ) )
% 11.90/5.99 => ( A3
% 11.90/5.99 = ( some @ A @ E2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3442_map__option__idI,axiom,
% 11.90/5.99 ! [A: $tType,X: option @ A,F: A > A] :
% 11.90/5.99 ( ! [Y4: A] :
% 11.90/5.99 ( ( member @ A @ Y4 @ ( set_option @ A @ X ) )
% 11.90/5.99 => ( ( F @ Y4 )
% 11.90/5.99 = Y4 ) )
% 11.90/5.99 => ( ( map_option @ A @ A @ F @ X )
% 11.90/5.99 = X ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3443_option_Oinj__map__strong,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,X: option @ A,Xa: option @ A,F: A > B,Fa: A > B] :
% 11.90/5.99 ( ! [Z3: A,Za: A] :
% 11.90/5.99 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 11.90/5.99 => ( ( member @ A @ Za @ ( set_option @ A @ Xa ) )
% 11.90/5.99 => ( ( ( F @ Z3 )
% 11.90/5.99 = ( Fa @ Za ) )
% 11.90/5.99 => ( Z3 = Za ) ) ) )
% 11.90/5.99 => ( ( ( map_option @ A @ B @ F @ X )
% 11.90/5.99 = ( map_option @ A @ B @ Fa @ Xa ) )
% 11.90/5.99 => ( X = Xa ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3444_option_Omap__cong0,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,X: option @ A,F: A > B,G: A > B] :
% 11.90/5.99 ( ! [Z3: A] :
% 11.90/5.99 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 11.90/5.99 => ( ( F @ Z3 )
% 11.90/5.99 = ( G @ Z3 ) ) )
% 11.90/5.99 => ( ( map_option @ A @ B @ F @ X )
% 11.90/5.99 = ( map_option @ A @ B @ G @ X ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3445_option_Omap__cong,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
% 11.90/5.99 ( ( X = Ya )
% 11.90/5.99 => ( ! [Z3: A] :
% 11.90/5.99 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 11.90/5.99 => ( ( F @ Z3 )
% 11.90/5.99 = ( G @ Z3 ) ) )
% 11.90/5.99 => ( ( map_option @ A @ B @ F @ X )
% 11.90/5.99 = ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3446_bind__option__cong,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,X: option @ A,Y: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
% 11.90/5.99 ( ( X = Y )
% 11.90/5.99 => ( ! [Z3: A] :
% 11.90/5.99 ( ( member @ A @ Z3 @ ( set_option @ A @ Y ) )
% 11.90/5.99 => ( ( F @ Z3 )
% 11.90/5.99 = ( G @ Z3 ) ) )
% 11.90/5.99 => ( ( bind @ A @ B @ X @ F )
% 11.90/5.99 = ( bind @ A @ B @ Y @ G ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3447_rtrancl__image__advance__rtrancl,axiom,
% 11.90/5.99 ! [A: $tType,Q3: A,R5: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
% 11.90/5.99 ( ( member @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/5.99 => ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3448_rtrancl__image__advance,axiom,
% 11.90/5.99 ! [A: $tType,Q3: A,R5: set @ ( product_prod @ A @ A ),Q0: set @ A,X: A] :
% 11.90/5.99 ( ( member @ A @ Q3 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q3 @ X ) @ R5 )
% 11.90/5.99 => ( member @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ Q0 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3449_finite__Image__subset,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ B,C5: set @ ( product_prod @ B @ A )] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( image @ B @ A @ A4 @ B6 ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C5 @ A4 )
% 11.90/5.99 => ( finite_finite2 @ A @ ( image @ B @ A @ C5 @ B6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3450_rtrancl__image__unfold__right,axiom,
% 11.90/5.99 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V6 ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ V6 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3451_rtrancl__reachable__induct,axiom,
% 11.90/5.99 ! [A: $tType,I4: set @ A,INV: set @ A,E4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ I4 @ INV )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ INV ) @ INV )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I4 ) @ INV ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3452_trancl__Image__unfold__right,axiom,
% 11.90/5.99 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 11.90/5.99 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S2 )
% 11.90/5.99 = ( image @ A @ A @ E4 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ S2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3453_trancl__Image__unfold__left,axiom,
% 11.90/5.99 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),S2: set @ A] :
% 11.90/5.99 ( ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ S2 )
% 11.90/5.99 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( image @ A @ A @ E4 @ S2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3454_option_Osimps_I14_J,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( set_option @ A @ ( none @ A ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3455_option_Oset__map,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,V: option @ A] :
% 11.90/5.99 ( ( set_option @ B @ ( map_option @ A @ B @ F @ V ) )
% 11.90/5.99 = ( image2 @ A @ B @ F @ ( set_option @ A @ V ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3456_option_Oset__sel,axiom,
% 11.90/5.99 ! [A: $tType,A3: option @ A] :
% 11.90/5.99 ( ( A3
% 11.90/5.99 != ( none @ A ) )
% 11.90/5.99 => ( member @ A @ ( the2 @ A @ A3 ) @ ( set_option @ A @ A3 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3457_Image__empty__rtrancl__Image__id,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V: A] :
% 11.90/5.99 ( ( ( image @ A @ A @ R5 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.99 = ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3458_Image__empty__trancl__Image__empty,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V: A] :
% 11.90/5.99 ( ( ( image @ A @ A @ R5 @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ( image @ A @ A @ ( transitive_trancl @ A @ R5 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3459_reachable__mono,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),R7: set @ ( product_prod @ A @ A ),X10: set @ A,X11: set @ A] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ R7 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ X10 @ X11 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ X10 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R7 ) @ X11 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3460_Image__subset__snd__image,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B6: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A4 @ B6 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3461_trancl__image__by__rtrancl,axiom,
% 11.90/5.99 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),Vi: set @ A] :
% 11.90/5.99 ( ( sup_sup @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ Vi ) @ Vi )
% 11.90/5.99 = ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ Vi ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3462_weak__map__of__SomeI,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
% 11.90/5.99 => ? [X2: B] :
% 11.90/5.99 ( ( map_of @ A @ B @ L @ K )
% 11.90/5.99 = ( some @ B @ X2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3463_map__of__SomeD,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
% 11.90/5.99 ( ( ( map_of @ B @ A @ Xs @ K )
% 11.90/5.99 = ( some @ A @ Y ) )
% 11.90/5.99 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3464_Image__singleton,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
% 11.90/5.99 ( ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
% 11.90/5.99 = ( collect @ A
% 11.90/5.99 @ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B5 ) @ R ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3465_option_Osimps_I15_J,axiom,
% 11.90/5.99 ! [A: $tType,X22: A] :
% 11.90/5.99 ( ( set_option @ A @ ( some @ A @ X22 ) )
% 11.90/5.99 = ( insert2 @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3466_rtrancl__apply__insert,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,S2: set @ A] :
% 11.90/5.99 ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( insert2 @ A @ X @ S2 ) )
% 11.90/5.99 = ( insert2 @ A @ X @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ ( sup_sup @ ( set @ A ) @ S2 @ ( image @ A @ A @ R5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3467_rtrancl__Image__in__Field,axiom,
% 11.90/5.99 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R5 ) @ V6 ) @ ( sup_sup @ ( set @ A ) @ ( field2 @ A @ R5 ) @ V6 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3468_E__closed__restr__reach__cases,axiom,
% 11.90/5.99 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R5 ) @ R5 )
% 11.90/5.99 => ( ~ ( member @ A @ V @ R5 )
% 11.90/5.99 => ~ ( ~ ( member @ A @ U @ R5 )
% 11.90/5.99 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3469_rel__restrict__tranclI,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,E4: set @ ( product_prod @ A @ A ),R5: set @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ E4 ) )
% 11.90/5.99 => ( ~ ( member @ A @ X @ R5 )
% 11.90/5.99 => ( ~ ( member @ A @ Y @ R5 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E4 @ R5 ) @ R5 )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E4 @ R5 ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3470_refl__on__reflcl__Image,axiom,
% 11.90/5.99 ! [A: $tType,B6: set @ A,A4: set @ ( product_prod @ A @ A ),C5: set @ A] :
% 11.90/5.99 ( ( refl_on @ A @ B6 @ A4 )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ B6 )
% 11.90/5.99 => ( ( image @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) @ C5 )
% 11.90/5.99 = ( image @ A @ A @ A4 @ C5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3471_map__of__eq__None__iff,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X: B] :
% 11.90/5.99 ( ( ( map_of @ B @ A @ Xys @ X )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 = ( ~ ( member @ B @ X @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3472_finite__reachable__advance,axiom,
% 11.90/5.99 ! [A: $tType,E4: set @ ( product_prod @ A @ A ),V0: A,V: A] :
% 11.90/5.99 ( ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V0 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V0 @ V ) @ ( transitive_rtrancl @ A @ E4 ) )
% 11.90/5.99 => ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3473_rtrancl__Image__advance__ss,axiom,
% 11.90/5.99 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3474_trancl__Image__advance__ss,axiom,
% 11.90/5.99 ! [A: $tType,U: A,V: A,E4: set @ ( product_prod @ A @ A )] :
% 11.90/5.99 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V ) @ E4 )
% 11.90/5.99 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ V @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ ( transitive_trancl @ A @ E4 ) @ ( insert2 @ A @ U @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3475_subset__Image1__Image1__iff,axiom,
% 11.90/5.99 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/5.99 ( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.99 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/5.99 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.90/5.99 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3476_map__of__distinct__upd,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
% 11.90/5.99 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 11.90/5.99 => ( ( map_add @ A @ B
% 11.90/5.99 @ ( fun_upd @ A @ ( option @ B )
% 11.90/5.99 @ ^ [X3: A] : ( none @ B )
% 11.90/5.99 @ X
% 11.90/5.99 @ ( some @ B @ Y ) )
% 11.90/5.99 @ ( map_of @ A @ B @ Xs ) )
% 11.90/5.99 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X @ ( some @ B @ Y ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3477_subset__Image__Image__iff,axiom,
% 11.90/5.99 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 11.90/5.99 ( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ A4 ) @ ( image @ A @ A @ R @ B6 ) )
% 11.90/5.99 = ( ! [X3: A] :
% 11.90/5.99 ( ( member @ A @ X3 @ A4 )
% 11.90/5.99 => ? [Y3: A] :
% 11.90/5.99 ( ( member @ A @ Y3 @ B6 )
% 11.90/5.99 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3478_map__of__map__restrict,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,Ks: list @ A] :
% 11.90/5.99 ( ( map_of @ A @ B
% 11.90/5.99 @ ( map @ A @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( F @ K4 ) )
% 11.90/5.99 @ Ks ) )
% 11.90/5.99 = ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F ) @ ( set2 @ A @ Ks ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3479_map__of__mapk__SomeI,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,C: $tType,F: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X: C] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/5.99 => ( ( ( map_of @ A @ C @ T2 @ K )
% 11.90/5.99 = ( some @ C @ X ) )
% 11.90/5.99 => ( ( map_of @ B @ C
% 11.90/5.99 @ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
% 11.90/5.99 @ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
% 11.90/5.99 @ ^ [K4: A] : ( product_Pair @ B @ C @ ( F @ K4 ) ) )
% 11.90/5.99 @ T2 )
% 11.90/5.99 @ ( F @ K ) )
% 11.90/5.99 = ( some @ C @ X ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3480_map__comp__map,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
% 11.90/5.99 ( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
% 11.90/5.99 = ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3481_map__map,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs: list @ C] :
% 11.90/5.99 ( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs ) )
% 11.90/5.99 = ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3482_List_Omap_Ocompositionality,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
% 11.90/5.99 ( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
% 11.90/5.99 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3483_list_Omap__comp,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
% 11.90/5.99 ( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
% 11.90/5.99 = ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3484_List_Omap_Ocomp,axiom,
% 11.90/5.99 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 11.90/5.99 ( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
% 11.90/5.99 = ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3485_map__fst__mk__snd,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: B,L: list @ A] :
% 11.90/5.99 ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
% 11.90/5.99 @ ( map @ A @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
% 11.90/5.99 @ L ) )
% 11.90/5.99 = L ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3486_map__snd__mk__fst,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: B,L: list @ A] :
% 11.90/5.99 ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K ) @ L ) )
% 11.90/5.99 = L ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3487_set__oo__map__alt,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/5.99 ( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F ) )
% 11.90/5.99 = ( ^ [L3: list @ A] : ( image2 @ A @ B @ F @ ( set2 @ A @ L3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3488_Id__on__set,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A] :
% 11.90/5.99 ( ( id_on @ A @ ( set2 @ A @ Xs ) )
% 11.90/5.99 = ( set2 @ ( product_prod @ A @ A )
% 11.90/5.99 @ ( map @ A @ ( product_prod @ A @ A )
% 11.90/5.99 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 11.90/5.99 @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3489_map__of__map__keys,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
% 11.90/5.99 ( ( ( set2 @ A @ Xs )
% 11.90/5.99 = ( dom @ A @ B @ M2 ) )
% 11.90/5.99 => ( ( map_of @ A @ B
% 11.90/5.99 @ ( map @ A @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [K4: A] : ( product_Pair @ A @ B @ K4 @ ( the2 @ B @ ( M2 @ K4 ) ) )
% 11.90/5.99 @ Xs ) )
% 11.90/5.99 = M2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3490_map__of__map,axiom,
% 11.90/5.99 ! [B: $tType,C: $tType,A: $tType,F: C > B,Xs: list @ ( product_prod @ A @ C )] :
% 11.90/5.99 ( ( map_of @ A @ B
% 11.90/5.99 @ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
% 11.90/5.99 @ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [K4: A,V2: C] : ( product_Pair @ A @ B @ K4 @ ( F @ V2 ) ) )
% 11.90/5.99 @ Xs ) )
% 11.90/5.99 = ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3491_inj__on__map__inv__f,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,L: list @ A,A4: set @ A,F: A > B] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A4 )
% 11.90/5.99 => ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( map @ B @ A @ ( inv_on @ A @ B @ F @ A4 ) @ ( map @ A @ B @ F @ L ) )
% 11.90/5.99 = L ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3492_Misc_Oran__distinct,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
% 11.90/5.99 => ( ( ran @ A @ B @ ( map_of @ A @ B @ Al ) )
% 11.90/5.99 = ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3493_map__of__eq__Some__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
% 11.90/5.99 => ( ( ( map_of @ A @ B @ Xys @ X )
% 11.90/5.99 = ( some @ B @ Y ) )
% 11.90/5.99 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3494_Some__eq__map__of__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
% 11.90/5.99 => ( ( ( some @ B @ Y )
% 11.90/5.99 = ( map_of @ A @ B @ Xys @ X ) )
% 11.90/5.99 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3495_map__of__is__SomeI,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
% 11.90/5.99 => ( ( map_of @ A @ B @ Xys @ X )
% 11.90/5.99 = ( some @ B @ Y ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3496_inv__on__f__f,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,X: A] :
% 11.90/5.99 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/5.99 => ( ( member @ A @ X @ A4 )
% 11.90/5.99 => ( ( inv_on @ A @ B @ F @ A4 @ ( F @ X ) )
% 11.90/5.99 = X ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3497_distinct__mapI,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > A,L: list @ B] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
% 11.90/5.99 => ( distinct @ B @ L ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3498_distinct__finite__set,axiom,
% 11.90/5.99 ! [A: $tType,X: set @ A] :
% 11.90/5.99 ( finite_finite2 @ ( list @ A )
% 11.90/5.99 @ ( collect @ ( list @ A )
% 11.90/5.99 @ ^ [Ys: list @ A] :
% 11.90/5.99 ( ( ( set2 @ A @ Ys )
% 11.90/5.99 = X )
% 11.90/5.99 & ( distinct @ A @ Ys ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3499_distinct__map__eq,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > A,L: list @ B,X: B,Y: B] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
% 11.90/5.99 => ( ( ( F @ X )
% 11.90/5.99 = ( F @ Y ) )
% 11.90/5.99 => ( ( member @ B @ X @ ( set2 @ B @ L ) )
% 11.90/5.99 => ( ( member @ B @ Y @ ( set2 @ B @ L ) )
% 11.90/5.99 => ( X = Y ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3500_finite__set__image,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ ( list @ A )] :
% 11.90/5.99 ( ( finite_finite2 @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A4 ) )
% 11.90/5.99 => ( ! [Xs2: list @ A] :
% 11.90/5.99 ( ( member @ ( list @ A ) @ Xs2 @ A4 )
% 11.90/5.99 => ( distinct @ A @ Xs2 ) )
% 11.90/5.99 => ( finite_finite2 @ ( list @ A ) @ A4 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3501_inv__on__f__range,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Y: A,F: B > A,A4: set @ B] :
% 11.90/5.99 ( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A4 ) )
% 11.90/5.99 => ( member @ B @ ( inv_on @ B @ A @ F @ A4 @ Y ) @ A4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3502_f__inv__on__f,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Y: A,F: B > A,A4: set @ B] :
% 11.90/5.99 ( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A4 ) )
% 11.90/5.99 => ( ( F @ ( inv_on @ B @ A @ F @ A4 @ Y ) )
% 11.90/5.99 = Y ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3503_distinct__finite__subset,axiom,
% 11.90/5.99 ! [A: $tType,X: set @ A] :
% 11.90/5.99 ( ( finite_finite2 @ A @ X )
% 11.90/5.99 => ( finite_finite2 @ ( list @ A )
% 11.90/5.99 @ ( collect @ ( list @ A )
% 11.90/5.99 @ ^ [Ys: list @ A] :
% 11.90/5.99 ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys ) @ X )
% 11.90/5.99 & ( distinct @ A @ Ys ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3504_distinct__map__fstD,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( Y = Z2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3505_eq__key__imp__eq__value,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( V1 = V22 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3506_map__to__set__map__of,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
% 11.90/5.99 => ( ( map_to_set @ A @ B @ ( map_of @ A @ B @ L ) )
% 11.90/5.99 = ( set2 @ ( product_prod @ A @ B ) @ L ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3507_map__of__map__to__set,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B ),M2: A > ( option @ B )] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) )
% 11.90/5.99 => ( ( ( map_of @ A @ B @ L )
% 11.90/5.99 = M2 )
% 11.90/5.99 = ( ( set2 @ ( product_prod @ A @ B ) @ L )
% 11.90/5.99 = ( map_to_set @ A @ B @ M2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3508_set__map__of__compr,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
% 11.90/5.99 = ( collect @ ( product_prod @ A @ B )
% 11.90/5.99 @ ( product_case_prod @ A @ B @ $o
% 11.90/5.99 @ ^ [K4: A,V2: B] :
% 11.90/5.99 ( ( map_of @ A @ B @ Xs @ K4 )
% 11.90/5.99 = ( some @ B @ V2 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3509_card__lists__distinct__length__eq,axiom,
% 11.90/5.99 ! [A: $tType,A4: set @ A,K: nat] :
% 11.90/5.99 ( ( finite_finite2 @ A @ A4 )
% 11.90/5.99 => ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
% 11.90/5.99 => ( ( finite_card @ ( list @ A )
% 11.90/5.99 @ ( collect @ ( list @ A )
% 11.90/5.99 @ ^ [Xs3: list @ A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 11.90/5.99 = K )
% 11.90/5.99 & ( distinct @ A @ Xs3 )
% 11.90/5.99 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
% 11.90/5.99 = ( groups7121269368397514597t_prod @ nat @ nat
% 11.90/5.99 @ ^ [X3: nat] : X3
% 11.90/5.99 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3510_card__lists__distinct__length__eq_H,axiom,
% 11.90/5.99 ! [A: $tType,K: nat,A4: set @ A] :
% 11.90/5.99 ( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
% 11.90/5.99 => ( ( finite_card @ ( list @ A )
% 11.90/5.99 @ ( collect @ ( list @ A )
% 11.90/5.99 @ ^ [Xs3: list @ A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 11.90/5.99 = K )
% 11.90/5.99 & ( distinct @ A @ Xs3 )
% 11.90/5.99 & ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
% 11.90/5.99 = ( groups7121269368397514597t_prod @ nat @ nat
% 11.90/5.99 @ ^ [X3: nat] : X3
% 11.90/5.99 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3511_distinct__foldl__invar,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,S2: list @ A,I4: ( set @ A ) > B > $o,Sigma_0: B,F: B > A > B] :
% 11.90/5.99 ( ( distinct @ A @ S2 )
% 11.90/5.99 => ( ( I4 @ ( set2 @ A @ S2 ) @ Sigma_0 )
% 11.90/5.99 => ( ! [X2: A,It: set @ A,Sigma: B] :
% 11.90/5.99 ( ( member @ A @ X2 @ It )
% 11.90/5.99 => ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S2 ) )
% 11.90/5.99 => ( ( I4 @ It @ Sigma )
% 11.90/5.99 => ( I4 @ ( minus_minus @ ( set @ A ) @ It @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F @ Sigma @ X2 ) ) ) ) )
% 11.90/5.99 => ( I4 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F @ Sigma_0 @ S2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3512_map__of__Some__filter__not__in,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A,P: ( product_prod @ B @ A ) > $o] :
% 11.90/5.99 ( ( ( map_of @ B @ A @ Xs @ K )
% 11.90/5.99 = ( some @ A @ V ) )
% 11.90/5.99 => ( ~ ( P @ ( product_Pair @ B @ A @ K @ V ) )
% 11.90/5.99 => ( ( distinct @ B @ ( map @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ Xs ) )
% 11.90/5.99 => ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ P @ Xs ) @ K )
% 11.90/5.99 = ( none @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3513_map__add__map__of__foldr,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,M2: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( map_add @ A @ B @ M2 @ ( map_of @ A @ B @ Ps ) )
% 11.90/5.99 = ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
% 11.90/5.99 @ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
% 11.90/5.99 @ ^ [K4: A,V2: B,M: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M @ K4 @ ( some @ B @ V2 ) ) )
% 11.90/5.99 @ Ps
% 11.90/5.99 @ M2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3514_foldl__length,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( foldl @ nat @ A
% 11.90/5.99 @ ^ [I3: nat,X3: A] : ( suc @ I3 )
% 11.90/5.99 @ ( zero_zero @ nat )
% 11.90/5.99 @ L )
% 11.90/5.99 = ( size_size @ ( list @ A ) @ L ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3515_foldl__length__aux,axiom,
% 11.90/5.99 ! [A: $tType,A3: nat,L: list @ A] :
% 11.90/5.99 ( ( foldl @ nat @ A
% 11.90/5.99 @ ^ [I3: nat,X3: A] : ( suc @ I3 )
% 11.90/5.99 @ A3
% 11.90/5.99 @ L )
% 11.90/5.99 = ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3516_foldl__A1__eq,axiom,
% 11.90/5.99 ! [A: $tType,F: A > A > A,N: A,I: A,Ww: list @ A] :
% 11.90/5.99 ( ! [E3: A] :
% 11.90/5.99 ( ( F @ N @ E3 )
% 11.90/5.99 = E3 )
% 11.90/5.99 => ( ! [E3: A] :
% 11.90/5.99 ( ( F @ E3 @ N )
% 11.90/5.99 = E3 )
% 11.90/5.99 => ( ! [A5: A,B3: A,C4: A] :
% 11.90/5.99 ( ( F @ A5 @ ( F @ B3 @ C4 ) )
% 11.90/5.99 = ( F @ ( F @ A5 @ B3 ) @ C4 ) )
% 11.90/5.99 => ( ( foldl @ A @ A @ F @ I @ Ww )
% 11.90/5.99 = ( F @ I @ ( foldl @ A @ A @ F @ N @ Ww ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3517_sum__length__filter__compl,axiom,
% 11.90/5.99 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 11.90/5.99 ( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) )
% 11.90/5.99 @ ( size_size @ ( list @ A )
% 11.90/5.99 @ ( filter @ A
% 11.90/5.99 @ ^ [X3: A] :
% 11.90/5.99 ~ ( P @ X3 )
% 11.90/5.99 @ Xs ) ) )
% 11.90/5.99 = ( size_size @ ( list @ A ) @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3518_comp__fun__commute_Ofoldr__conv__foldl,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B > B,Xs: list @ A,A3: B] :
% 11.90/5.99 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 11.90/5.99 => ( ( foldr @ A @ B @ F @ Xs @ A3 )
% 11.90/5.99 = ( foldl @ B @ A
% 11.90/5.99 @ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
% 11.90/5.99 @ A3
% 11.90/5.99 @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3519_comp__fun__commute_Ofoldl__f__commute,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B > B,A3: A,B2: B,Xs: list @ A] :
% 11.90/5.99 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 11.90/5.99 => ( ( F @ A3
% 11.90/5.99 @ ( foldl @ B @ A
% 11.90/5.99 @ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
% 11.90/5.99 @ B2
% 11.90/5.99 @ Xs ) )
% 11.90/5.99 = ( foldl @ B @ A
% 11.90/5.99 @ ^ [A7: B,B5: A] : ( F @ B5 @ A7 )
% 11.90/5.99 @ ( F @ A3 @ B2 )
% 11.90/5.99 @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3520_map__of__None__filterD,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X: B,P: ( product_prod @ B @ A ) > $o] :
% 11.90/5.99 ( ( ( map_of @ B @ A @ Xs @ X )
% 11.90/5.99 = ( none @ A ) )
% 11.90/5.99 => ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ P @ Xs ) @ X )
% 11.90/5.99 = ( none @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3521_fst__foldl,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,C: $tType,F: A > C > A,G: A > B > C > B,A3: A,B2: B,Xs: list @ C] :
% 11.90/5.99 ( ( product_fst @ A @ B
% 11.90/5.99 @ ( foldl @ ( product_prod @ A @ B ) @ C
% 11.90/5.99 @ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ B ) )
% 11.90/5.99 @ ^ [A7: A,B5: B,X3: C] : ( product_Pair @ A @ B @ ( F @ A7 @ X3 ) @ ( G @ A7 @ B5 @ X3 ) ) )
% 11.90/5.99 @ ( product_Pair @ A @ B @ A3 @ B2 )
% 11.90/5.99 @ Xs ) )
% 11.90/5.99 = ( foldl @ A @ C @ F @ A3 @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3522_foldl__absorb1,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( monoid_mult @ A )
% 11.90/5.99 => ! [X: A,Zs: list @ A] :
% 11.90/5.99 ( ( times_times @ A @ X @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
% 11.90/5.99 = ( foldl @ A @ A @ ( times_times @ A ) @ X @ Zs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3523_foldl__un__empty__eq,axiom,
% 11.90/5.99 ! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
% 11.90/5.99 ( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
% 11.90/5.99 = ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3524_distinct__map__fst__filterI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),P: ( product_prod @ A @ B ) > $o] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 11.90/5.99 => ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( filter @ ( product_prod @ A @ B ) @ P @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3525_map__of__filter__in,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Z2: A,P: B > A > $o] :
% 11.90/5.99 ( ( ( map_of @ B @ A @ Xs @ K )
% 11.90/5.99 = ( some @ A @ Z2 ) )
% 11.90/5.99 => ( ( P @ K @ Z2 )
% 11.90/5.99 => ( ( map_of @ B @ A @ ( filter @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs ) @ K )
% 11.90/5.99 = ( some @ A @ Z2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3526_distinct__length__le,axiom,
% 11.90/5.99 ! [A: $tType,Ys2: list @ A,Xs: list @ A] :
% 11.90/5.99 ( ( distinct @ A @ Ys2 )
% 11.90/5.99 => ( ( ( set2 @ A @ Ys2 )
% 11.90/5.99 = ( set2 @ A @ Xs ) )
% 11.90/5.99 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3527_foldl__set,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ ( set @ A )] :
% 11.90/5.99 ( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ L )
% 11.90/5.99 = ( complete_Sup_Sup @ ( set @ A )
% 11.90/5.99 @ ( collect @ ( set @ A )
% 11.90/5.99 @ ^ [X3: set @ A] : ( member @ ( set @ A ) @ X3 @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3528_map__filter__map__filter,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs: list @ B] :
% 11.90/5.99 ( ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) )
% 11.90/5.99 = ( map_filter @ B @ A
% 11.90/5.99 @ ^ [X3: B] : ( if @ ( option @ A ) @ ( P @ X3 ) @ ( some @ A @ ( F @ X3 ) ) @ ( none @ A ) )
% 11.90/5.99 @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3529_map__filter__def,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( map_filter @ A @ B )
% 11.90/5.99 = ( ^ [F2: A > ( option @ B ),Xs3: list @ A] :
% 11.90/5.99 ( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F2 )
% 11.90/5.99 @ ( filter @ A
% 11.90/5.99 @ ^ [X3: A] :
% 11.90/5.99 ( ( F2 @ X3 )
% 11.90/5.99 != ( none @ B ) )
% 11.90/5.99 @ Xs3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3530_execute__of__list,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ! [Xs: list @ A,H3: heap_ext @ product_unit] :
% 11.90/5.99 ( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 )
% 11.90/5.99 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
% 11.90/5.99 @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3531_mergesort__by__rel__split__length,axiom,
% 11.90/5.99 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
% 11.90/5.99 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs1 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( modulo_modulo @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
% 11.90/5.99 & ( ( size_size @ ( list @ A ) @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) )
% 11.90/5.99 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs22 ) @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3532_card__disjoint__shuffles,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.90/5.99 ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
% 11.90/5.99 = ( bot_bot @ ( set @ A ) ) )
% 11.90/5.99 => ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
% 11.90/5.99 = ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3533_of__list__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ( ( array_of_list @ A )
% 11.90/5.99 = ( ^ [Xs3: list @ A] :
% 11.90/5.99 ( heap_Time_heap @ ( array @ A )
% 11.90/5.99 @ ^ [H: heap_ext @ product_unit] :
% 11.90/5.99 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
% 11.90/5.99 @ ( array_alloc @ A @ Xs3 @ H ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3534_slice__len,axiom,
% 11.90/5.99 ! [A: $tType,From: nat,To: nat,Xs: list @ A] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ From @ To )
% 11.90/5.99 => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/5.99 => ( ( size_size @ ( list @ A ) @ ( slice @ A @ From @ To @ Xs ) )
% 11.90/5.99 = ( minus_minus @ nat @ To @ From ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3535_slice__complete,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A] :
% 11.90/5.99 ( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
% 11.90/5.99 = Xs ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3536_foldr__length,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( foldr @ A @ nat
% 11.90/5.99 @ ^ [X3: A] : suc
% 11.90/5.99 @ L
% 11.90/5.99 @ ( zero_zero @ nat ) )
% 11.90/5.99 = ( size_size @ ( list @ A ) @ L ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3537_length__shuffles,axiom,
% 11.90/5.99 ! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
% 11.90/5.99 ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
% 11.90/5.99 => ( ( size_size @ ( list @ A ) @ Zs )
% 11.90/5.99 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3538_foldr__length__aux,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A,A3: nat] :
% 11.90/5.99 ( ( foldr @ A @ nat
% 11.90/5.99 @ ^ [X3: A] : suc
% 11.90/5.99 @ L
% 11.90/5.99 @ A3 )
% 11.90/5.99 = ( plus_plus @ nat @ A3 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3539_effect__of__listI,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ! [A3: array @ A,H4: heap_ext @ product_unit,Xs: list @ A,H3: heap_ext @ product_unit] :
% 11.90/5.99 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
% 11.90/5.99 = ( array_alloc @ A @ Xs @ H3 ) )
% 11.90/5.99 => ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3540_execute__make,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ! [N: nat,F: nat > A,H3: heap_ext @ product_unit] :
% 11.90/5.99 ( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 )
% 11.90/5.99 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 11.90/5.99 @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3541_execute__new,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ! [N: nat,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.99 ( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 )
% 11.90/5.99 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
% 11.90/5.99 @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3542_Array__Time_Omake__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ( ( array_make @ A )
% 11.90/5.99 = ( ^ [N5: nat,F2: nat > A] :
% 11.90/5.99 ( heap_Time_heap @ ( array @ A )
% 11.90/5.99 @ ^ [H: heap_ext @ product_unit] :
% 11.90/5.99 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) )
% 11.90/5.99 @ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) @ H ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3543_new__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ( ( array_new @ A )
% 11.90/5.99 = ( ^ [N5: nat,X3: A] :
% 11.90/5.99 ( heap_Time_heap @ ( array @ A )
% 11.90/5.99 @ ^ [H: heap_ext @ product_unit] :
% 11.90/5.99 ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.90/5.99 @ ^ [R3: array @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) )
% 11.90/5.99 @ ( array_alloc @ A @ ( replicate @ A @ N5 @ X3 ) @ H ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3544_upt__0__eq__Nil__conv,axiom,
% 11.90/5.99 ! [J: nat] :
% 11.90/5.99 ( ( ( upt @ ( zero_zero @ nat ) @ J )
% 11.90/5.99 = ( nil @ nat ) )
% 11.90/5.99 = ( J
% 11.90/5.99 = ( zero_zero @ nat ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3545_map__of__eq__empty__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( ( map_of @ A @ B @ Xys )
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) )
% 11.90/5.99 = ( Xys
% 11.90/5.99 = ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3546_slice__Nil,axiom,
% 11.90/5.99 ! [A: $tType,Begin: nat,End: nat] :
% 11.90/5.99 ( ( slice @ A @ Begin @ End @ ( nil @ A ) )
% 11.90/5.99 = ( nil @ A ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3547_slice__eq__bounds__empty,axiom,
% 11.90/5.99 ! [A: $tType,I: nat,Xs: list @ A] :
% 11.90/5.99 ( ( slice @ A @ I @ I @ Xs )
% 11.90/5.99 = ( nil @ A ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3548_empty__eq__map__of__iff,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( ( ^ [X3: A] : ( none @ B ) )
% 11.90/5.99 = ( map_of @ A @ B @ Xys ) )
% 11.90/5.99 = ( Xys
% 11.90/5.99 = ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3549_length__ge__1__conv,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/5.99 = ( L
% 11.90/5.99 != ( nil @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3550_map__fst__mk__fst,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: A,L: list @ B] :
% 11.90/5.99 ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L ) )
% 11.90/5.99 = ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3551_map__snd__mk__snd,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: A,L: list @ B] :
% 11.90/5.99 ( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
% 11.90/5.99 @ ( map @ B @ ( product_prod @ B @ A )
% 11.90/5.99 @ ^ [X3: B] : ( product_Pair @ B @ A @ X3 @ K )
% 11.90/5.99 @ L ) )
% 11.90/5.99 = ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3552_map__add__upt_H,axiom,
% 11.90/5.99 ! [Ofs: nat,A3: nat,B2: nat] :
% 11.90/5.99 ( ( map @ nat @ nat
% 11.90/5.99 @ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ Ofs )
% 11.90/5.99 @ ( upt @ A3 @ B2 ) )
% 11.90/5.99 = ( upt @ ( plus_plus @ nat @ A3 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3553_len__greater__imp__nonempty,axiom,
% 11.90/5.99 ! [A: $tType,X: nat,L: list @ A] :
% 11.90/5.99 ( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/5.99 => ( L
% 11.90/5.99 != ( nil @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3554_map__add__upt,axiom,
% 11.90/5.99 ! [N: nat,M2: nat] :
% 11.90/5.99 ( ( map @ nat @ nat
% 11.90/5.99 @ ^ [I3: nat] : ( plus_plus @ nat @ I3 @ N )
% 11.90/5.99 @ ( upt @ ( zero_zero @ nat ) @ M2 ) )
% 11.90/5.99 = ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3555_map__of__Cons__code_I1_J,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,K: B] :
% 11.90/5.99 ( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K )
% 11.90/5.99 = ( none @ A ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3556_map__of_Osimps_I1_J,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
% 11.90/5.99 = ( ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3557_mergesort__by__rel__split_Osimps_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,Xs1: list @ A,Xs22: list @ A] :
% 11.90/5.99 ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( nil @ A ) )
% 11.90/5.99 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3558_upt__filter__extend,axiom,
% 11.90/5.99 ! [U: nat,U5: nat,P: nat > $o] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ U @ U5 )
% 11.90/5.99 => ( ! [I2: nat] :
% 11.90/5.99 ( ( ( ord_less_eq @ nat @ U @ I2 )
% 11.90/5.99 & ( ord_less @ nat @ I2 @ U5 ) )
% 11.90/5.99 => ~ ( P @ I2 ) )
% 11.90/5.99 => ( ( filter @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
% 11.90/5.99 = ( filter @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U5 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3559_effect__newI,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ! [A3: array @ A,H4: heap_ext @ product_unit,N: nat,X: A,H3: heap_ext @ product_unit] :
% 11.90/5.99 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
% 11.90/5.99 = ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) )
% 11.90/5.99 => ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3560_effect__makeI,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ! [A3: array @ A,H4: heap_ext @ product_unit,F: nat > A,N: nat,H3: heap_ext @ product_unit] :
% 11.90/5.99 ( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H4 )
% 11.90/5.99 = ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) )
% 11.90/5.99 => ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 @ H4 @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3561_mergesort__by__rel_Opinduct,axiom,
% 11.90/5.99 ! [A: $tType,A0: A > A > $o,A1: list @ A,P: ( A > A > $o ) > ( list @ A ) > $o] :
% 11.90/5.99 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ A0 @ A1 ) )
% 11.90/5.99 => ( ! [R8: A > A > $o,Xs2: list @ A] :
% 11.90/5.99 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs2 ) )
% 11.90/5.99 => ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( P @ R8 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
% 11.90/5.99 => ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( P @ R8 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
% 11.90/5.99 => ( P @ R8 @ Xs2 ) ) ) )
% 11.90/5.99 => ( P @ A0 @ A1 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3562_mergesort__by__rel_Oelims,axiom,
% 11.90/5.99 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
% 11.90/5.99 ( ( ( mergesort_by_rel @ A @ X @ Xa )
% 11.90/5.99 = Y )
% 11.90/5.99 => ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( Y = Xa ) )
% 11.90/5.99 & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( Y
% 11.90/5.99 = ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3563_mergesort__by__rel_Osimps,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( mergesort_by_rel @ A )
% 11.90/5.99 = ( ^ [R6: A > A > $o,Xs3: list @ A] : ( if @ ( list @ A ) @ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Xs3 @ ( merges9089515139780605204_merge @ A @ R6 @ ( mergesort_by_rel @ A @ R6 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R6 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3564_enumerate__replicate__eq,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,M2: nat,A3: A] :
% 11.90/5.99 ( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A3 ) )
% 11.90/5.99 = ( map @ nat @ ( product_prod @ nat @ A )
% 11.90/5.99 @ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A3 )
% 11.90/5.99 @ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3565_nth__step__trancl,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A,R5: set @ ( product_prod @ A @ A ),N: nat,M2: nat] :
% 11.90/5.99 ( ! [N3: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ N3 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N3 ) ) @ ( nth @ A @ Xs @ N3 ) ) @ R5 ) )
% 11.90/5.99 => ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/5.99 => ( ( ord_less @ nat @ M2 @ N )
% 11.90/5.99 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M2 ) ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3566_mergesort__by__rel__simps_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o] :
% 11.90/5.99 ( ( mergesort_by_rel @ A @ R5 @ ( nil @ A ) )
% 11.90/5.99 = ( nil @ A ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3567_set__mergesort__by__rel,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 11.90/5.99 ( ( set2 @ A @ ( mergesort_by_rel @ A @ R5 @ Xs ) )
% 11.90/5.99 = ( set2 @ A @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3568_mergesort__by__rel__merge__simps_I3_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,Ys2: list @ A] :
% 11.90/5.99 ( ( merges9089515139780605204_merge @ A @ R5 @ ( nil @ A ) @ Ys2 )
% 11.90/5.99 = Ys2 ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3569_nth__upt,axiom,
% 11.90/5.99 ! [I: nat,K: nat,J: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
% 11.90/5.99 => ( ( nth @ nat @ ( upt @ I @ J ) @ K )
% 11.90/5.99 = ( plus_plus @ nat @ I @ K ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3570_set__mergesort__by__rel__merge,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 11.90/5.99 ( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
% 11.90/5.99 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3571_map__fst__enumerate,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,Xs: list @ A] :
% 11.90/5.99 ( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
% 11.90/5.99 = ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3572_nth__image__indices,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( image2 @ nat @ A @ ( nth @ A @ L ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
% 11.90/5.99 = ( set2 @ A @ L ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3573_ran__nth__set__encoding__conv,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( ran @ nat @ A
% 11.90/5.99 @ ^ [I3: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I3 ) ) @ ( none @ A ) ) )
% 11.90/5.99 = ( set2 @ A @ L ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3574_mergesort__by__rel_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
% 11.90/5.99 ~ ! [R8: A > A > $o,Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R8 @ Xs2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3575_map__eq__nth__eq,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > A,L: list @ B,L4: list @ B,I: nat] :
% 11.90/5.99 ( ( ( map @ B @ A @ F @ L )
% 11.90/5.99 = ( map @ B @ A @ F @ L4 ) )
% 11.90/5.99 => ( ( F @ ( nth @ B @ L @ I ) )
% 11.90/5.99 = ( F @ ( nth @ B @ L4 @ I ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3576_nth__enumerate__eq,axiom,
% 11.90/5.99 ! [A: $tType,M2: nat,Xs: list @ A,N: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/5.99 => ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M2 )
% 11.90/5.99 = ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3577_mergesort__by__rel__merge__simps_I2_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 11.90/5.99 ( ( merges9089515139780605204_merge @ A @ R5 @ Xs @ ( nil @ A ) )
% 11.90/5.99 = Xs ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3578_obtain__list__from__elements,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,P: A > nat > $o] :
% 11.90/5.99 ( ! [I2: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ I2 @ N )
% 11.90/5.99 => ? [Li: A] : ( P @ Li @ I2 ) )
% 11.90/5.99 => ~ ! [L5: list @ A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ L5 )
% 11.90/5.99 = N )
% 11.90/5.99 => ~ ! [I6: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ I6 @ N )
% 11.90/5.99 => ( P @ ( nth @ A @ L5 @ I6 ) @ I6 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3579_in__set__enumerate__eq,axiom,
% 11.90/5.99 ! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
% 11.90/5.99 ( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
% 11.90/5.99 = ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
% 11.90/5.99 & ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
% 11.90/5.99 & ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
% 11.90/5.99 = ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3580_all__set__conv__nth,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A,P: A > $o] :
% 11.90/5.99 ( ( ! [X3: A] :
% 11.90/5.99 ( ( member @ A @ X3 @ ( set2 @ A @ L ) )
% 11.90/5.99 => ( P @ X3 ) ) )
% 11.90/5.99 = ( ! [I3: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/5.99 => ( P @ ( nth @ A @ L @ I3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3581_set__image__eq__pointwiseI,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F: A > B] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ L )
% 11.90/5.99 = ( size_size @ ( list @ A ) @ L4 ) )
% 11.90/5.99 => ( ! [I2: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/5.99 => ( ( F @ ( nth @ A @ L @ I2 ) )
% 11.90/5.99 = ( F @ ( nth @ A @ L4 @ I2 ) ) ) )
% 11.90/5.99 => ( ( image2 @ A @ B @ F @ ( set2 @ A @ L ) )
% 11.90/5.99 = ( image2 @ A @ B @ F @ ( set2 @ A @ L4 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3582_in__set__image__conv__nth,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: B > A,X: B,L: list @ B] :
% 11.90/5.99 ( ( member @ A @ ( F @ X ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ L ) ) )
% 11.90/5.99 = ( ? [I3: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L ) )
% 11.90/5.99 & ( ( F @ ( nth @ B @ L @ I3 ) )
% 11.90/5.99 = ( F @ X ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3583_distinct__idx,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: B > A,L: list @ B,I: nat,J: nat] :
% 11.90/5.99 ( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
% 11.90/5.99 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
% 11.90/5.99 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
% 11.90/5.99 => ( ( ( F @ ( nth @ B @ L @ I ) )
% 11.90/5.99 = ( F @ ( nth @ B @ L @ J ) ) )
% 11.90/5.99 => ( I = J ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3584_array__of__list__make,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( heap @ A )
% 11.90/5.99 => ( ( array_of_list @ A )
% 11.90/5.99 = ( ^ [Xs3: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( nth @ A @ Xs3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3585_nth__map__upt,axiom,
% 11.90/5.99 ! [A: $tType,I: nat,N: nat,M2: nat,F: nat > A] :
% 11.90/5.99 ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
% 11.90/5.99 => ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M2 @ N ) ) @ I )
% 11.90/5.99 = ( F @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3586_mergesort__by__rel_Opsimps,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 11.90/5.99 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs ) )
% 11.90/5.99 => ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( ( mergesort_by_rel @ A @ R5 @ Xs )
% 11.90/5.99 = Xs ) )
% 11.90/5.99 & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( ( mergesort_by_rel @ A @ R5 @ Xs )
% 11.90/5.99 = ( merges9089515139780605204_merge @ A @ R5 @ ( mergesort_by_rel @ A @ R5 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R5 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3587_mergesort__by__rel_Opelims,axiom,
% 11.90/5.99 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: list @ A] :
% 11.90/5.99 ( ( ( mergesort_by_rel @ A @ X @ Xa )
% 11.90/5.99 = Y )
% 11.90/5.99 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/5.99 => ~ ( ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( Y = Xa ) )
% 11.90/5.99 & ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xa ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ( Y
% 11.90/5.99 = ( merges9089515139780605204_merge @ A @ X @ ( mergesort_by_rel @ A @ X @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) ) ) ) )
% 11.90/5.99 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3588_enumerate__map__upt,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,F: nat > A,M2: nat] :
% 11.90/5.99 ( ( enumerate @ A @ N @ ( map @ nat @ A @ F @ ( upt @ N @ M2 ) ) )
% 11.90/5.99 = ( map @ nat @ ( product_prod @ nat @ A )
% 11.90/5.99 @ ^ [K4: nat] : ( product_Pair @ nat @ A @ K4 @ ( F @ K4 ) )
% 11.90/5.99 @ ( upt @ N @ M2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3589_map__upt__eqI,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A,N: nat,M2: nat,F: nat > A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/5.99 = ( minus_minus @ nat @ N @ M2 ) )
% 11.90/5.99 => ( ! [I2: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/5.99 => ( ( nth @ A @ Xs @ I2 )
% 11.90/5.99 = ( F @ ( plus_plus @ nat @ M2 @ I2 ) ) ) )
% 11.90/5.99 => ( ( map @ nat @ A @ F @ ( upt @ M2 @ N ) )
% 11.90/5.99 = Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3590_slice__nth,axiom,
% 11.90/5.99 ! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
% 11.90/5.99 ( ( ord_less @ nat @ From @ To )
% 11.90/5.99 => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/5.99 => ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
% 11.90/5.99 => ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
% 11.90/5.99 = ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3591_product__nth,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
% 11.90/5.99 ( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
% 11.90/5.99 => ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
% 11.90/5.99 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) @ ( nth @ B @ Ys2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3592_mergesort__def,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( ord @ A )
% 11.90/5.99 => ( ( mergesort @ A )
% 11.90/5.99 = ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3593_sorted__in__between,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [I: nat,J: nat,L: list @ A,X: A] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
% 11.90/5.99 => ( ( ord_less @ nat @ I @ J )
% 11.90/5.99 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/5.99 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 11.90/5.99 => ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X )
% 11.90/5.99 => ( ( ord_less @ A @ X @ ( nth @ A @ L @ J ) )
% 11.90/5.99 => ~ ! [K5: nat] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ I @ K5 )
% 11.90/5.99 => ( ( ord_less @ nat @ K5 @ J )
% 11.90/5.99 => ( ( ord_less_eq @ A @ ( nth @ A @ L @ K5 ) @ X )
% 11.90/5.99 => ~ ( ord_less @ A @ X @ ( nth @ A @ L @ ( plus_plus @ nat @ K5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3594_Un__set__drop__extend,axiom,
% 11.90/5.99 ! [A: $tType,J: nat,L: list @ ( set @ A )] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ J )
% 11.90/5.99 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ ( set @ A ) ) @ L ) )
% 11.90/5.99 => ( ( sup_sup @ ( set @ A ) @ ( nth @ ( set @ A ) @ L @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ J @ L ) ) ) )
% 11.90/5.99 = ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ ( minus_minus @ nat @ J @ ( suc @ ( zero_zero @ nat ) ) ) @ L ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3595_mergesort__by__rel__simps_I3_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,X1: A,X22: A,Xs: list @ A] :
% 11.90/5.99 ( ( mergesort_by_rel @ A @ R5 @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
% 11.90/5.99 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 11.90/5.99 @ ^ [Xs12: list @ A,Xs23: list @ A] : ( merges9089515139780605204_merge @ A @ R5 @ ( mergesort_by_rel @ A @ R5 @ Xs12 ) @ ( mergesort_by_rel @ A @ R5 @ Xs23 ) )
% 11.90/5.99 @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ ( nil @ A ) ) @ ( cons @ A @ X22 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3596_drop__drop,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
% 11.90/5.99 ( ( drop @ A @ N @ ( drop @ A @ M2 @ Xs ) )
% 11.90/5.99 = ( drop @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3597_sorted__wrt__mergesort__by__rel__merge,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 11.90/5.99 ( ! [X2: A,Y4: A] :
% 11.90/5.99 ( ( R5 @ X2 @ Y4 )
% 11.90/5.99 | ( R5 @ Y4 @ X2 ) )
% 11.90/5.99 => ( ! [X2: A,Y4: A,Z3: A] :
% 11.90/5.99 ( ( R5 @ X2 @ Y4 )
% 11.90/5.99 => ( ( R5 @ Y4 @ Z3 )
% 11.90/5.99 => ( R5 @ X2 @ Z3 ) ) )
% 11.90/5.99 => ( ( sorted_wrt @ A @ R5 @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
% 11.90/5.99 = ( ( sorted_wrt @ A @ R5 @ Xs )
% 11.90/5.99 & ( sorted_wrt @ A @ R5 @ Ys2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3598_mergesort__by__rel__simps_I2_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,X: A] :
% 11.90/5.99 ( ( mergesort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( nil @ A ) ) )
% 11.90/5.99 = ( cons @ A @ X @ ( nil @ A ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3599_enumerate__simps_I2_J,axiom,
% 11.90/5.99 ! [B: $tType,N: nat,X: B,Xs: list @ B] :
% 11.90/5.99 ( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
% 11.90/5.99 = ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3600_sorted__wrt__map__linord,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [L: list @ ( product_prod @ A @ B )] :
% 11.90/5.99 ( ( sorted_wrt @ ( product_prod @ A @ B )
% 11.90/5.99 @ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X3 ) @ ( product_fst @ A @ B @ Y3 ) )
% 11.90/5.99 @ L )
% 11.90/5.99 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3601_nth__Cons__numeral,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Xs: list @ A,V: num] :
% 11.90/5.99 ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
% 11.90/5.99 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3602_drop__Cons__numeral,axiom,
% 11.90/5.99 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 11.90/5.99 ( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 11.90/5.99 = ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3603_nth__drop,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,Xs: list @ A,I: nat] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/5.99 => ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
% 11.90/5.99 = ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3604_nth__Cons__pos,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 11.90/5.99 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/5.99 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 11.90/5.99 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3605_Misc_Ofoldr__Cons,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A] :
% 11.90/5.99 ( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
% 11.90/5.99 = Xs ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3606_map__consI_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,W: list @ A,F: B > A,Ww: list @ B,A3: B] :
% 11.90/5.99 ( ( W
% 11.90/5.99 = ( map @ B @ A @ F @ Ww ) )
% 11.90/5.99 => ( ( cons @ A @ ( F @ A3 ) @ W )
% 11.90/5.99 = ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3607_map__eq__consE,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
% 11.90/5.99 ( ( ( map @ B @ A @ F @ Ls )
% 11.90/5.99 = ( cons @ A @ Fa @ Fl ) )
% 11.90/5.99 => ~ ! [A5: B,L5: list @ B] :
% 11.90/5.99 ( ( Ls
% 11.90/5.99 = ( cons @ B @ A5 @ L5 ) )
% 11.90/5.99 => ( ( ( F @ A5 )
% 11.90/5.99 = Fa )
% 11.90/5.99 => ( ( map @ B @ A @ F @ L5 )
% 11.90/5.99 != Fl ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3608_neq__NilE,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( L
% 11.90/5.99 != ( nil @ A ) )
% 11.90/5.99 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/5.99 ( L
% 11.90/5.99 != ( cons @ A @ X2 @ Xs2 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3609_list__2pre__induct,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W22: list @ B] :
% 11.90/5.99 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 11.90/5.99 => ( ! [E3: A,W12: list @ A,W23: list @ B] :
% 11.90/5.99 ( ( P @ W12 @ W23 )
% 11.90/5.99 => ( P @ ( cons @ A @ E3 @ W12 ) @ W23 ) )
% 11.90/5.99 => ( ! [E3: B,W13: list @ A,W24: list @ B] :
% 11.90/5.99 ( ( P @ W13 @ W24 )
% 11.90/5.99 => ( P @ W13 @ ( cons @ B @ E3 @ W24 ) ) )
% 11.90/5.99 => ( P @ W1 @ W22 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3610_list__induct__first2,axiom,
% 11.90/5.99 ! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
% 11.90/5.99 ( ( P @ ( nil @ A ) )
% 11.90/5.99 => ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.90/5.99 => ( ! [X13: A,X23: A,Xs2: list @ A] :
% 11.90/5.99 ( ( P @ Xs2 )
% 11.90/5.99 => ( P @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
% 11.90/5.99 => ( P @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3611_mergesort__by__rel__merge__induct,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R5: A > B > $o,Xs: list @ A,Ys2: list @ B] :
% 11.90/5.99 ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
% 11.90/5.99 => ( ! [X_12: list @ B] : ( P @ ( nil @ A ) @ X_12 )
% 11.90/5.99 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 11.90/5.99 ( ( R5 @ X2 @ Y4 )
% 11.90/5.99 => ( ( P @ Xs2 @ ( cons @ B @ Y4 @ Ys3 ) )
% 11.90/5.99 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
% 11.90/5.99 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 11.90/5.99 ( ~ ( R5 @ X2 @ Y4 )
% 11.90/5.99 => ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
% 11.90/5.99 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y4 @ Ys3 ) ) ) )
% 11.90/5.99 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3612_sorted__wrt__mergesort__by__rel,axiom,
% 11.90/5.99 ! [X14: $tType,R5: X14 > X14 > $o,Xs: list @ X14] :
% 11.90/5.99 ( ! [X2: X14,Y4: X14] :
% 11.90/5.99 ( ( R5 @ X2 @ Y4 )
% 11.90/5.99 | ( R5 @ Y4 @ X2 ) )
% 11.90/5.99 => ( ! [X2: X14,Y4: X14,Z3: X14] :
% 11.90/5.99 ( ( R5 @ X2 @ Y4 )
% 11.90/5.99 => ( ( R5 @ Y4 @ Z3 )
% 11.90/5.99 => ( R5 @ X2 @ Z3 ) ) )
% 11.90/5.99 => ( sorted_wrt @ X14 @ R5 @ ( mergesort_by_rel @ X14 @ R5 @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3613_list__tail__coinc,axiom,
% 11.90/5.99 ! [A: $tType,N12: A,R1: list @ A,N23: A,R22: list @ A] :
% 11.90/5.99 ( ( ( cons @ A @ N12 @ R1 )
% 11.90/5.99 = ( cons @ A @ N23 @ R22 ) )
% 11.90/5.99 => ( ( N12 = N23 )
% 11.90/5.99 & ( R1 = R22 ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3614_drop__eq__ConsD,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,Xs: list @ A,X: A,Xs4: list @ A] :
% 11.90/5.99 ( ( ( drop @ A @ N @ Xs )
% 11.90/5.99 = ( cons @ A @ X @ Xs4 ) )
% 11.90/5.99 => ( ( drop @ A @ ( suc @ N ) @ Xs )
% 11.90/5.99 = Xs4 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3615_drop__Cons_H,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 11.90/5.99 ( ( ( N
% 11.90/5.99 = ( zero_zero @ nat ) )
% 11.90/5.99 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 11.90/5.99 = ( cons @ A @ X @ Xs ) ) )
% 11.90/5.99 & ( ( N
% 11.90/5.99 != ( zero_zero @ nat ) )
% 11.90/5.99 => ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
% 11.90/5.99 = ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3616_mergesort__by__rel__merge__simps_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,X: A,Y: A,Xs: list @ A,Ys2: list @ A] :
% 11.90/5.99 ( ( ( R5 @ X @ Y )
% 11.90/5.99 => ( ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 11.90/5.99 = ( cons @ A @ X @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) ) )
% 11.90/5.99 & ( ~ ( R5 @ X @ Y )
% 11.90/5.99 => ( ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 11.90/5.99 = ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R5 @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3617_sorted__filter_H,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [L: list @ A,P: A > $o] :
% 11.90/5.99 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 11.90/5.99 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( filter @ A @ P @ L ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3618_sorted__mergesort__by__rel,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3619_map__filter__simps_I1_J,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Xs: list @ B] :
% 11.90/5.99 ( ( map_filter @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
% 11.90/5.99 = ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs )
% 11.90/5.99 @ ^ [Y3: A] : ( cons @ A @ Y3 @ ( map_filter @ B @ A @ F @ Xs ) )
% 11.90/5.99 @ ( F @ X ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3620_zipf_Ocases,axiom,
% 11.90/5.99 ! [C: $tType,A: $tType,B: $tType,X: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
% 11.90/5.99 ( ! [F4: A > B > C] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
% 11.90/5.99 => ( ! [F4: A > B > C,A5: A,As: list @ A,B3: B,Bs: list @ B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 11.90/5.99 => ( ! [A5: A > B > C,V4: A,Va: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
% 11.90/5.99 => ~ ! [A5: A > B > C,V4: B,Va: list @ B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3621_merge_Ocases,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/5.99 ( ! [L22: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L22 ) )
% 11.90/5.99 => ( ! [V4: A,Va: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) )
% 11.90/5.99 => ~ ! [X13: A,L12: list @ A,X23: A,L22: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3622_list__all__zip_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,X: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
% 11.90/5.99 ( ! [P7: A > B > $o] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
% 11.90/5.99 => ( ! [P7: A > B > $o,A5: A,As: list @ A,B3: B,Bs: list @ B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 11.90/5.99 => ( ! [P7: A > B > $o,V4: A,Va: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) )
% 11.90/5.99 => ~ ! [P7: A > B > $o,V4: B,Va: list @ B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P7 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3623_partition__rev_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
% 11.90/5.99 ( ! [P7: A > $o,Yes: list @ A,No: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
% 11.90/5.99 => ~ ! [P7: A > $o,Yes: list @ A,No: list @ A,X2: A,Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P7 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3624_quicksort__by__rel_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
% 11.90/5.99 ( ! [R8: A > A > $o,Sl: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
% 11.90/5.99 => ~ ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3625_mergesort__by__rel__merge_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
% 11.90/5.99 ( ! [R8: A > A > $o,X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) )
% 11.90/5.99 => ( ! [R8: A > A > $o,Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
% 11.90/5.99 => ~ ! [R8: A > A > $o,V4: A,Va: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3626_mergesort__by__rel__split_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
% 11.90/5.99 ( ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) )
% 11.90/5.99 => ( ! [Xs13: list @ A,Xs24: list @ A,X2: A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
% 11.90/5.99 => ~ ! [Xs13: list @ A,Xs24: list @ A,X13: A,X23: A,Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3627_shuffles_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/5.99 ( ! [Ys3: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 11.90/5.99 => ( ! [Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
% 11.90/5.99 => ~ ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3628_map__tailrec__rev_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
% 11.90/5.99 ( ! [F4: A > B,Bs: list @ B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs ) ) )
% 11.90/5.99 => ~ ! [F4: A > B,A5: A,As: list @ A,Bs: list @ B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Bs ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3629_foldr_Ofoldr__Cons,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
% 11.90/5.99 ( ( foldr @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
% 11.90/5.99 = ( comp @ B @ B @ B @ ( F @ X ) @ ( foldr @ A @ B @ F @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3630_arg__min__list_Ocases,axiom,
% 11.90/5.99 ! [B: $tType,A: $tType] :
% 11.90/5.99 ( ( linorder @ B )
% 11.90/5.99 => ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
% 11.90/5.99 ( ! [F4: A > B,X2: A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
% 11.90/5.99 => ( ! [F4: A > B,X2: A,Y4: A,Zs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
% 11.90/5.99 => ~ ! [A5: A > B] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > B ) @ ( list @ A ) @ A5 @ ( nil @ A ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3631_successively_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
% 11.90/5.99 ( ! [P7: A > A > $o] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
% 11.90/5.99 => ( ! [P7: A > A > $o,X2: A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
% 11.90/5.99 => ~ ! [P7: A > A > $o,X2: A,Y4: A,Xs2: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3632_sorted__wrt_Ocases,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
% 11.90/5.99 ( ! [P7: A > A > $o] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
% 11.90/5.99 => ~ ! [P7: A > A > $o,X2: A,Ys3: list @ A] :
% 11.90/5.99 ( X
% 11.90/5.99 != ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3633_mergesort__by__rel__merge_Oelims,axiom,
% 11.90/5.99 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 11.90/5.99 ( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
% 11.90/5.99 = Y )
% 11.90/5.99 => ( ! [X2: A,Xs2: list @ A] :
% 11.90/5.99 ( ( Xa
% 11.90/5.99 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/5.99 => ! [Y4: A,Ys3: list @ A] :
% 11.90/5.99 ( ( Xb
% 11.90/5.99 = ( cons @ A @ Y4 @ Ys3 ) )
% 11.90/5.99 => ~ ( ( ( X @ X2 @ Y4 )
% 11.90/5.99 => ( Y
% 11.90/5.99 = ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
% 11.90/5.99 & ( ~ ( X @ X2 @ Y4 )
% 11.90/5.99 => ( Y
% 11.90/5.99 = ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ) ) ) )
% 11.90/5.99 => ( ( ( Xb
% 11.90/5.99 = ( nil @ A ) )
% 11.90/5.99 => ( Y != Xa ) )
% 11.90/5.99 => ~ ( ( Xa
% 11.90/5.99 = ( nil @ A ) )
% 11.90/5.99 => ! [V4: A,Va: list @ A] :
% 11.90/5.99 ( ( Xb
% 11.90/5.99 = ( cons @ A @ V4 @ Va ) )
% 11.90/5.99 => ( Y
% 11.90/5.99 != ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3634_mergesort__by__rel__merge_Osimps_I3_J,axiom,
% 11.90/5.99 ! [A: $tType,R5: A > A > $o,V: A,Va2: list @ A] :
% 11.90/5.99 ( ( merges9089515139780605204_merge @ A @ R5 @ ( nil @ A ) @ ( cons @ A @ V @ Va2 ) )
% 11.90/5.99 = ( cons @ A @ V @ Va2 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3635_mergesort__by__rel__split_Osimps_I3_J,axiom,
% 11.90/5.99 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,X1: A,X22: A,Xs: list @ A] :
% 11.90/5.99 ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X22 @ Xs ) ) )
% 11.90/5.99 = ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs1 ) @ ( cons @ A @ X22 @ Xs22 ) ) @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3636_filter__conv__foldr,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( filter @ A )
% 11.90/5.99 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 11.90/5.99 ( foldr @ A @ ( list @ A )
% 11.90/5.99 @ ^ [X3: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P4 @ X3 ) @ ( cons @ A @ X3 @ Xt ) @ Xt )
% 11.90/5.99 @ Xs3
% 11.90/5.99 @ ( nil @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3637_sorted__wrt01,axiom,
% 11.90/5.99 ! [A: $tType,Xs: list @ A,P: A > A > $o] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 11.90/5.99 => ( sorted_wrt @ A @ P @ Xs ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3638_length__compl__induct,axiom,
% 11.90/5.99 ! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
% 11.90/5.99 ( ( P @ ( nil @ A ) )
% 11.90/5.99 => ( ! [E3: A,L5: list @ A] :
% 11.90/5.99 ( ! [Ll: list @ A] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L5 ) )
% 11.90/5.99 => ( P @ Ll ) )
% 11.90/5.99 => ( P @ ( cons @ A @ E3 @ L5 ) ) )
% 11.90/5.99 => ( P @ L ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3639_list__decomp__1,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ L )
% 11.90/5.99 = ( one_one @ nat ) )
% 11.90/5.99 => ? [A5: A] :
% 11.90/5.99 ( L
% 11.90/5.99 = ( cons @ A @ A5 @ ( nil @ A ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3640_map__of__Cons__code_I2_J,axiom,
% 11.90/5.99 ! [C: $tType,B: $tType,L: B,K: B,V: C,Ps: list @ ( product_prod @ B @ C )] :
% 11.90/5.99 ( ( ( L = K )
% 11.90/5.99 => ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
% 11.90/5.99 = ( some @ C @ V ) ) )
% 11.90/5.99 & ( ( L != K )
% 11.90/5.99 => ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
% 11.90/5.99 = ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3641_mergesort__by__rel__split_Osimps_I2_J,axiom,
% 11.90/5.99 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,X: A] :
% 11.90/5.99 ( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X @ ( nil @ A ) ) )
% 11.90/5.99 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs1 ) @ Xs22 ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3642_mergesort__by__rel__split_Oelims,axiom,
% 11.90/5.99 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/5.99 ( ( ( merges295452479951948502_split @ A @ X @ Xa )
% 11.90/5.99 = Y )
% 11.90/5.99 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/5.99 ( ( X
% 11.90/5.99 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/5.99 => ( ( Xa
% 11.90/5.99 = ( nil @ A ) )
% 11.90/5.99 => ( Y
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) ) ) )
% 11.90/5.99 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/5.99 ( ( X
% 11.90/5.99 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/5.99 => ! [X2: A] :
% 11.90/5.99 ( ( Xa
% 11.90/5.99 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.90/5.99 => ( Y
% 11.90/5.99 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) ) ) )
% 11.90/5.99 => ~ ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/5.99 ( ( X
% 11.90/5.99 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/5.99 => ! [X13: A,X23: A,Xs2: list @ A] :
% 11.90/5.99 ( ( Xa
% 11.90/5.99 = ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) )
% 11.90/5.99 => ( Y
% 11.90/5.99 != ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X23 @ Xs24 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3643_sorted01,axiom,
% 11.90/5.99 ! [A: $tType] :
% 11.90/5.99 ( ( linorder @ A )
% 11.90/5.99 => ! [Xs: list @ A] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 11.90/5.99 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3644_list_Osize_I4_J,axiom,
% 11.90/5.99 ! [A: $tType,X21: A,X222: list @ A] :
% 11.90/5.99 ( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
% 11.90/5.99 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3645_nth__Cons_H,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 11.90/5.99 ( ( ( N
% 11.90/5.99 = ( zero_zero @ nat ) )
% 11.90/5.99 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 11.90/5.99 = X ) )
% 11.90/5.99 & ( ( N
% 11.90/5.99 != ( zero_zero @ nat ) )
% 11.90/5.99 => ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 11.90/5.99 = ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3646_upt__eq__Cons__conv,axiom,
% 11.90/5.99 ! [I: nat,J: nat,X: nat,Xs: list @ nat] :
% 11.90/5.99 ( ( ( upt @ I @ J )
% 11.90/5.99 = ( cons @ nat @ X @ Xs ) )
% 11.90/5.99 = ( ( ord_less @ nat @ I @ J )
% 11.90/5.99 & ( I = X )
% 11.90/5.99 & ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
% 11.90/5.99 = Xs ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3647_in__set__drop__conv__nth,axiom,
% 11.90/5.99 ! [A: $tType,X: A,N: nat,L: list @ A] :
% 11.90/5.99 ( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
% 11.90/5.99 = ( ? [I3: nat] :
% 11.90/5.99 ( ( ord_less_eq @ nat @ N @ I3 )
% 11.90/5.99 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/5.99 & ( X
% 11.90/5.99 = ( nth @ A @ L @ I3 ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3648_set__drop__conv,axiom,
% 11.90/5.99 ! [A: $tType,N: nat,L: list @ A] :
% 11.90/5.99 ( ( set2 @ A @ ( drop @ A @ N @ L ) )
% 11.90/5.99 = ( collect @ A
% 11.90/5.99 @ ^ [Uu: A] :
% 11.90/5.99 ? [I3: nat] :
% 11.90/5.99 ( ( Uu
% 11.90/5.99 = ( nth @ A @ L @ I3 ) )
% 11.90/5.99 & ( ord_less_eq @ nat @ N @ I3 )
% 11.90/5.99 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3649_list__decomp__2,axiom,
% 11.90/5.99 ! [A: $tType,L: list @ A] :
% 11.90/5.99 ( ( ( size_size @ ( list @ A ) @ L )
% 11.90/5.99 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/5.99 => ? [A5: A,B3: A] :
% 11.90/5.99 ( L
% 11.90/5.99 = ( cons @ A @ A5 @ ( cons @ A @ B3 @ ( nil @ A ) ) ) ) ) ).
% 11.90/5.99
% 11.90/5.99 thf(fact_3650_nth__non__equal__first__eq,axiom,
% 11.90/5.99 ! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
% 11.90/5.99 ( ( X != Y )
% 11.90/5.99 => ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
% 11.90/5.99 = Y )
% 11.90/5.99 = ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 11.90/6.00 = Y )
% 11.90/6.00 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3651_Cons__replicate__eq,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
% 11.90/6.00 ( ( ( cons @ A @ X @ Xs )
% 11.90/6.00 = ( replicate @ A @ N @ Y ) )
% 11.90/6.00 = ( ( X = Y )
% 11.90/6.00 & ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/6.00 & ( Xs
% 11.90/6.00 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3652_distinct__sorted__mono,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A,I: nat,J: nat] :
% 11.90/6.00 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 11.90/6.00 => ( ( distinct @ A @ L )
% 11.90/6.00 => ( ( ord_less @ nat @ I @ J )
% 11.90/6.00 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3653_distinct__sorted__strict__mono__iff,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A,I: nat,J: nat] :
% 11.90/6.00 ( ( distinct @ A @ L )
% 11.90/6.00 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 11.90/6.00 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
% 11.90/6.00 = ( ord_less @ nat @ I @ J ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3654_distinct__sorted__mono__iff,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A,I: nat,J: nat] :
% 11.90/6.00 ( ( distinct @ A @ L )
% 11.90/6.00 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 11.90/6.00 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
% 11.90/6.00 = ( ord_less_eq @ nat @ I @ J ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3655_map__of_Osimps_I2_J,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,P5: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
% 11.90/6.00 ( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P5 @ Ps ) )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P5 ) @ ( some @ B @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3656_slice__Cons,axiom,
% 11.90/6.00 ! [A: $tType,Begin: nat,End: nat,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( ( Begin
% 11.90/6.00 = ( zero_zero @ nat ) )
% 11.90/6.00 & ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
% 11.90/6.00 => ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( cons @ A @ X @ ( slice @ A @ Begin @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) )
% 11.90/6.00 & ( ~ ( ( Begin
% 11.90/6.00 = ( zero_zero @ nat ) )
% 11.90/6.00 & ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
% 11.90/6.00 => ( ( slice @ A @ Begin @ End @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3657_mergesort__by__rel__split_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/6.00 ( ( ( merges295452479951948502_split @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( nil @ A ) ) ) ) ) )
% 11.90/6.00 => ( ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/6.00 => ! [X2: A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs13 ) @ Xs24 ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ) )
% 11.90/6.00 => ~ ! [Xs13: list @ A,Xs24: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) )
% 11.90/6.00 => ! [X13: A,X23: A,Xs2: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs13 ) @ ( cons @ A @ X23 @ Xs24 ) ) @ Xs2 ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ ( merges7066485432131860899it_rel @ A ) @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs13 @ Xs24 ) @ ( cons @ A @ X13 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3658_list__decode_Opelims,axiom,
% 11.90/6.00 ! [X: nat,Y: list @ nat] :
% 11.90/6.00 ( ( ( nat_list_decode @ X )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ nat @ nat_list_decode_rel @ X )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( zero_zero @ nat ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( nil @ nat ) )
% 11.90/6.00 => ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
% 11.90/6.00 => ~ ! [N3: nat] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( suc @ N3 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 11.90/6.00 @ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
% 11.90/6.00 @ ( nat_prod_decode @ N3 ) ) )
% 11.90/6.00 => ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3659_mergesort__by__rel__merge_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( merges9089515139780605204_merge @ A @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ! [Y4: A,Ys3: list @ A] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ A @ Y4 @ Ys3 ) )
% 11.90/6.00 => ( ( ( ( X @ X2 @ Y4 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ X @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
% 11.90/6.00 & ( ~ ( X @ X2 @ Y4 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ Y4 @ ( merges9089515139780605204_merge @ A @ X @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ) )
% 11.90/6.00 => ( ( ( Xb
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y = Xa )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
% 11.90/6.00 => ~ ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ! [V4: A,Va: list @ A] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( merges2244889521215249637ge_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3660_subset__eq__mset__impl_Ocases,axiom,
% 11.90/6.00 ! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/6.00 ( ! [Ys3: list @ A] :
% 11.90/6.00 ( X
% 11.90/6.00 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
% 11.90/6.00 ( X
% 11.90/6.00 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3661_drop__upt,axiom,
% 11.90/6.00 ! [M2: nat,I: nat,J: nat] :
% 11.90/6.00 ( ( drop @ nat @ M2 @ ( upt @ I @ J ) )
% 11.90/6.00 = ( upt @ ( plus_plus @ nat @ I @ M2 ) @ J ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3662_natural__zero__minus__one,axiom,
% 11.90/6.00 ( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
% 11.90/6.00 = ( zero_zero @ code_natural ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3663_list__decode_Opsimps_I1_J,axiom,
% 11.90/6.00 ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
% 11.90/6.00 => ( ( nat_list_decode @ ( zero_zero @ nat ) )
% 11.90/6.00 = ( nil @ nat ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3664_list__decode_Opsimps_I2_J,axiom,
% 11.90/6.00 ! [N: nat] :
% 11.90/6.00 ( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
% 11.90/6.00 => ( ( nat_list_decode @ ( suc @ N ) )
% 11.90/6.00 = ( product_case_prod @ nat @ nat @ ( list @ nat )
% 11.90/6.00 @ ^ [X3: nat,Y3: nat] : ( cons @ nat @ X3 @ ( nat_list_decode @ Y3 ) )
% 11.90/6.00 @ ( nat_prod_decode @ N ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3665_list__encode_Oelims,axiom,
% 11.90/6.00 ! [X: list @ nat,Y: nat] :
% 11.90/6.00 ( ( ( nat_list_encode @ X )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ nat ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( zero_zero @ nat ) ) )
% 11.90/6.00 => ~ ! [X2: nat,Xs2: list @ nat] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ nat @ X2 @ Xs2 ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3666_mergesort__remdups__correct,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A] :
% 11.90/6.00 ( ( distinct @ A @ ( mergesort_remdups @ A @ L ) )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( mergesort_remdups @ A @ L ) )
% 11.90/6.00 & ( ( set2 @ A @ ( mergesort_remdups @ A @ L ) )
% 11.90/6.00 = ( set2 @ A @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3667_filter__upt__last,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,L: list @ A,Js: list @ nat,J: nat,I: nat] :
% 11.90/6.00 ( ( ( filter @ nat
% 11.90/6.00 @ ^ [K4: nat] : ( P @ ( nth @ A @ L @ K4 ) )
% 11.90/6.00 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
% 11.90/6.00 = ( append @ nat @ Js @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
% 11.90/6.00 => ( ( ord_less @ nat @ J @ I )
% 11.90/6.00 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3668_empty__append__eq__id,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( append @ A @ ( nil @ A ) )
% 11.90/6.00 = ( ^ [X3: list @ A] : X3 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3669_op__conc__empty__img__id,axiom,
% 11.90/6.00 ! [A: $tType,L6: set @ ( list @ A )] :
% 11.90/6.00 ( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L6 )
% 11.90/6.00 = L6 ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3670_list__e__eq__lel_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,L1: list @ A,E6: A,L2: list @ A,E2: A] :
% 11.90/6.00 ( ( ( append @ A @ L1 @ ( cons @ A @ E6 @ L2 ) )
% 11.90/6.00 = ( cons @ A @ E2 @ ( nil @ A ) ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 & ( E6 = E2 )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3671_list__e__eq__lel_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,E2: A,L1: list @ A,E6: A,L2: list @ A] :
% 11.90/6.00 ( ( ( cons @ A @ E2 @ ( nil @ A ) )
% 11.90/6.00 = ( append @ A @ L1 @ ( cons @ A @ E6 @ L2 ) ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 & ( E6 = E2 )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3672_list__se__match_I4_J,axiom,
% 11.90/6.00 ! [A: $tType,L2: list @ A,A3: A,L1: list @ A] :
% 11.90/6.00 ( ( L2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ( cons @ A @ A3 @ ( nil @ A ) )
% 11.90/6.00 = ( append @ A @ L1 @ L2 ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3673_list__se__match_I3_J,axiom,
% 11.90/6.00 ! [A: $tType,L1: list @ A,A3: A,L2: list @ A] :
% 11.90/6.00 ( ( L1
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ( cons @ A @ A3 @ ( nil @ A ) )
% 11.90/6.00 = ( append @ A @ L1 @ L2 ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( nil @ A ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3674_list__se__match_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,L2: list @ A,L1: list @ A,A3: A] :
% 11.90/6.00 ( ( L2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ( append @ A @ L1 @ L2 )
% 11.90/6.00 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3675_list__se__match_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,L1: list @ A,L2: list @ A,A3: A] :
% 11.90/6.00 ( ( L1
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ( append @ A @ L1 @ L2 )
% 11.90/6.00 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( cons @ A @ A3 @ ( nil @ A ) ) )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( nil @ A ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3676_list__ee__eq__leel_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,L1: list @ A,E12: A,E23: A,L2: list @ A,E1: A,E22: A] :
% 11.90/6.00 ( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) )
% 11.90/6.00 = ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 & ( E1 = E12 )
% 11.90/6.00 & ( E22 = E23 )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3677_list__ee__eq__leel_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,E1: A,E22: A,L1: list @ A,E12: A,E23: A,L2: list @ A] :
% 11.90/6.00 ( ( ( cons @ A @ E1 @ ( cons @ A @ E22 @ ( nil @ A ) ) )
% 11.90/6.00 = ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E23 @ L2 ) ) ) )
% 11.90/6.00 = ( ( L1
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 & ( E1 = E12 )
% 11.90/6.00 & ( E22 = E23 )
% 11.90/6.00 & ( L2
% 11.90/6.00 = ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3678_length__append,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) )
% 11.90/6.00 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3679_upt__merge,axiom,
% 11.90/6.00 ! [I: nat,J: nat,K: nat] :
% 11.90/6.00 ( ( ( ord_less_eq @ nat @ I @ J )
% 11.90/6.00 & ( ord_less_eq @ nat @ J @ K ) )
% 11.90/6.00 => ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
% 11.90/6.00 = ( upt @ I @ K ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3680_nth__append__first,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,L: list @ A,L4: list @ A] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( nth @ A @ ( append @ A @ L @ L4 ) @ I )
% 11.90/6.00 = ( nth @ A @ L @ I ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3681_nth__append__length__plus,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,N: nat] :
% 11.90/6.00 ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
% 11.90/6.00 = ( nth @ A @ Ys2 @ N ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3682_list__match__lel__lel,axiom,
% 11.90/6.00 ! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
% 11.90/6.00 ( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
% 11.90/6.00 = ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
% 11.90/6.00 => ( ! [C21: list @ A] :
% 11.90/6.00 ( ( C1
% 11.90/6.00 = ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
% 11.90/6.00 => ( C23
% 11.90/6.00 != ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
% 11.90/6.00 => ( ( ( C12 = C1 )
% 11.90/6.00 => ( ( Qs2 = Qs )
% 11.90/6.00 => ( C23 != C22 ) ) )
% 11.90/6.00 => ~ ! [C212: list @ A] :
% 11.90/6.00 ( ( C12
% 11.90/6.00 = ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
% 11.90/6.00 => ( C22
% 11.90/6.00 != ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3683_foldl__conc__empty__eq,axiom,
% 11.90/6.00 ! [A: $tType,I: list @ A,Ww: list @ ( list @ A )] :
% 11.90/6.00 ( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I @ Ww )
% 11.90/6.00 = ( append @ A @ I @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3684_Misc_Omap__eq__append__conv,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
% 11.90/6.00 ( ( ( map @ B @ A @ F @ Ls )
% 11.90/6.00 = ( append @ A @ Fl @ Fl2 ) )
% 11.90/6.00 = ( ? [L3: list @ B,L7: list @ B] :
% 11.90/6.00 ( ( Ls
% 11.90/6.00 = ( append @ B @ L3 @ L7 ) )
% 11.90/6.00 & ( ( map @ B @ A @ F @ L3 )
% 11.90/6.00 = Fl )
% 11.90/6.00 & ( ( map @ B @ A @ F @ L7 )
% 11.90/6.00 = Fl2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3685_Misc_Oappend__eq__map__conv,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
% 11.90/6.00 ( ( ( append @ A @ Fl @ Fl2 )
% 11.90/6.00 = ( map @ B @ A @ F @ Ls ) )
% 11.90/6.00 = ( ? [L3: list @ B,L7: list @ B] :
% 11.90/6.00 ( ( Ls
% 11.90/6.00 = ( append @ B @ L3 @ L7 ) )
% 11.90/6.00 & ( ( map @ B @ A @ F @ L3 )
% 11.90/6.00 = Fl )
% 11.90/6.00 & ( ( map @ B @ A @ F @ L7 )
% 11.90/6.00 = Fl2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3686_map__eq__appendE,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
% 11.90/6.00 ( ( ( map @ B @ A @ F @ Ls )
% 11.90/6.00 = ( append @ A @ Fl @ Fl2 ) )
% 11.90/6.00 => ~ ! [L5: list @ B,L8: list @ B] :
% 11.90/6.00 ( ( Ls
% 11.90/6.00 = ( append @ B @ L5 @ L8 ) )
% 11.90/6.00 => ( ( ( map @ B @ A @ F @ L5 )
% 11.90/6.00 = Fl )
% 11.90/6.00 => ( ( map @ B @ A @ F @ L8 )
% 11.90/6.00 != Fl2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3687_append__eq__mapE,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
% 11.90/6.00 ( ( ( append @ A @ Fl @ Fl2 )
% 11.90/6.00 = ( map @ B @ A @ F @ Ls ) )
% 11.90/6.00 => ~ ! [L5: list @ B,L8: list @ B] :
% 11.90/6.00 ( ( Ls
% 11.90/6.00 = ( append @ B @ L5 @ L8 ) )
% 11.90/6.00 => ( ( ( map @ B @ A @ F @ L5 )
% 11.90/6.00 = Fl )
% 11.90/6.00 => ( ( map @ B @ A @ F @ L8 )
% 11.90/6.00 != Fl2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3688_enumerate__append__eq,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys2 ) )
% 11.90/6.00 = ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3689_neq__Nil__revE,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A] :
% 11.90/6.00 ( ( L
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ~ ! [Ll2: list @ A,E3: A] :
% 11.90/6.00 ( L
% 11.90/6.00 != ( append @ A @ Ll2 @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3690_rev__induct2_H,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( P @ ( nil @ A ) @ ( nil @ B ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( nil @ B ) )
% 11.90/6.00 => ( ! [Y4: B,Ys3: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 11.90/6.00 ( ( P @ Xs2 @ Ys3 )
% 11.90/6.00 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
% 11.90/6.00 => ( P @ Xs @ Ys2 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3691_neq__Nil__rev__conv,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A] :
% 11.90/6.00 ( ( L
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 = ( ? [Xs3: list @ A,X3: A] :
% 11.90/6.00 ( L
% 11.90/6.00 = ( append @ A @ Xs3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3692_rev__nonempty__induct2_H,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( Ys2
% 11.90/6.00 != ( nil @ B ) )
% 11.90/6.00 => ( ! [X2: A,Y4: B] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A,Y4: B] :
% 11.90/6.00 ( ( Xs2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) )
% 11.90/6.00 => ( ! [X2: A,Y4: B,Ys3: list @ B] :
% 11.90/6.00 ( ( Ys3
% 11.90/6.00 != ( nil @ B ) )
% 11.90/6.00 => ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A,Y4: B,Ys3: list @ B] :
% 11.90/6.00 ( ( P @ Xs2 @ Ys3 )
% 11.90/6.00 => ( ( Xs2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( Ys3
% 11.90/6.00 != ( nil @ B ) )
% 11.90/6.00 => ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ ( append @ B @ Ys3 @ ( cons @ B @ Y4 @ ( nil @ B ) ) ) ) ) ) )
% 11.90/6.00 => ( P @ Xs @ Ys2 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3693_list__Cons__eq__append__cases,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 11.90/6.00 ( ( ( cons @ A @ X @ Xs )
% 11.90/6.00 = ( append @ A @ Ys2 @ Zs ) )
% 11.90/6.00 => ( ( ( Ys2
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Zs
% 11.90/6.00 != ( cons @ A @ X @ Xs ) ) )
% 11.90/6.00 => ~ ! [Ys4: list @ A] :
% 11.90/6.00 ( ( Ys2
% 11.90/6.00 = ( cons @ A @ X @ Ys4 ) )
% 11.90/6.00 => ( ( append @ A @ Ys4 @ Zs )
% 11.90/6.00 != Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3694_list__append__eq__Cons__cases,axiom,
% 11.90/6.00 ! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( append @ A @ Ys2 @ Zs )
% 11.90/6.00 = ( cons @ A @ X @ Xs ) )
% 11.90/6.00 => ( ( ( Ys2
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Zs
% 11.90/6.00 != ( cons @ A @ X @ Xs ) ) )
% 11.90/6.00 => ~ ! [Ys4: list @ A] :
% 11.90/6.00 ( ( Ys2
% 11.90/6.00 = ( cons @ A @ X @ Ys4 ) )
% 11.90/6.00 => ( ( append @ A @ Ys4 @ Zs )
% 11.90/6.00 != Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3695_in__set__list__format,axiom,
% 11.90/6.00 ! [A: $tType,E2: A,L: list @ A] :
% 11.90/6.00 ( ( member @ A @ E2 @ ( set2 @ A @ L ) )
% 11.90/6.00 => ~ ! [L12: list @ A,L22: list @ A] :
% 11.90/6.00 ( L
% 11.90/6.00 != ( append @ A @ L12 @ ( cons @ A @ E2 @ L22 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3696_xy__in__set__cases,axiom,
% 11.90/6.00 ! [A: $tType,X: A,L: list @ A,Y: A] :
% 11.90/6.00 ( ( member @ A @ X @ ( set2 @ A @ L ) )
% 11.90/6.00 => ( ( member @ A @ Y @ ( set2 @ A @ L ) )
% 11.90/6.00 => ( ( ( X = Y )
% 11.90/6.00 => ! [L12: list @ A,L22: list @ A] :
% 11.90/6.00 ( L
% 11.90/6.00 != ( append @ A @ L12 @ ( cons @ A @ Y @ L22 ) ) ) )
% 11.90/6.00 => ( ( ( X != Y )
% 11.90/6.00 => ! [L12: list @ A,L22: list @ A,L32: list @ A] :
% 11.90/6.00 ( L
% 11.90/6.00 != ( append @ A @ L12 @ ( cons @ A @ X @ ( append @ A @ L22 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
% 11.90/6.00 => ~ ( ( X != Y )
% 11.90/6.00 => ! [L12: list @ A,L22: list @ A,L32: list @ A] :
% 11.90/6.00 ( L
% 11.90/6.00 != ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L22 @ ( cons @ A @ X @ L32 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3697_map__consI_I2_J,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,W: list @ A,L: list @ A,F: B > A,Ww: list @ B,A3: B] :
% 11.90/6.00 ( ( ( append @ A @ W @ L )
% 11.90/6.00 = ( append @ A @ ( map @ B @ A @ F @ Ww ) @ L ) )
% 11.90/6.00 => ( ( cons @ A @ ( F @ A3 ) @ ( append @ A @ W @ L ) )
% 11.90/6.00 = ( append @ A @ ( map @ B @ A @ F @ ( cons @ B @ A3 @ Ww ) ) @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3698_list__rest__coinc,axiom,
% 11.90/6.00 ! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
% 11.90/6.00 => ( ( ( append @ A @ S1 @ R1 )
% 11.90/6.00 = ( append @ A @ S22 @ R22 ) )
% 11.90/6.00 => ? [R1p: list @ A] :
% 11.90/6.00 ( R22
% 11.90/6.00 = ( append @ A @ R1p @ R1 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3699_set__union__code,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
% 11.90/6.00 = ( set2 @ A @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3700_distinct__match,axiom,
% 11.90/6.00 ! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
% 11.90/6.00 ( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) ) )
% 11.90/6.00 => ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
% 11.90/6.00 = ( append @ A @ Al2 @ ( cons @ A @ E2 @ Bl2 ) ) )
% 11.90/6.00 = ( ( Al = Al2 )
% 11.90/6.00 & ( Bl = Bl2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3701_upt__eq__append__conv,axiom,
% 11.90/6.00 ! [I: nat,J: nat,Xs: list @ nat,Ys2: list @ nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/6.00 => ( ( ( upt @ I @ J )
% 11.90/6.00 = ( append @ nat @ Xs @ Ys2 ) )
% 11.90/6.00 = ( ? [K4: nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ I @ K4 )
% 11.90/6.00 & ( ord_less_eq @ nat @ K4 @ J )
% 11.90/6.00 & ( ( upt @ I @ K4 )
% 11.90/6.00 = Xs )
% 11.90/6.00 & ( ( upt @ K4 @ J )
% 11.90/6.00 = Ys2 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3702_replicate__add,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,M2: nat,X: A] :
% 11.90/6.00 ( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X )
% 11.90/6.00 = ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M2 @ X ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3703_upt__append,axiom,
% 11.90/6.00 ! [I: nat,J: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ J )
% 11.90/6.00 => ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
% 11.90/6.00 = ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3704_upt__add__eq__append,axiom,
% 11.90/6.00 ! [I: nat,J: nat,K: nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/6.00 => ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
% 11.90/6.00 = ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3705_foldl__rule,axiom,
% 11.90/6.00 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
% 11.90/6.00 ( ( I4 @ Sigma_0 @ ( nil @ A ) @ L0 )
% 11.90/6.00 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 11.90/6.00 ( ( L0
% 11.90/6.00 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 11.90/6.00 => ( ( I4 @ Sigma @ L12 @ ( cons @ A @ X2 @ L22 ) )
% 11.90/6.00 => ( I4 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L12 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
% 11.90/6.00 => ( I4 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3706_foldl__rule__P,axiom,
% 11.90/6.00 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
% 11.90/6.00 ( ( I4 @ Sigma_0 @ ( nil @ A ) @ L0 )
% 11.90/6.00 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 11.90/6.00 ( ( L0
% 11.90/6.00 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 11.90/6.00 => ( ( I4 @ Sigma @ L12 @ ( cons @ A @ X2 @ L22 ) )
% 11.90/6.00 => ( I4 @ ( F @ Sigma @ X2 ) @ ( append @ A @ L12 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ L22 ) ) )
% 11.90/6.00 => ( ! [Sigma: Sigma2] :
% 11.90/6.00 ( ( I4 @ Sigma @ L0 @ ( nil @ A ) )
% 11.90/6.00 => ( P @ Sigma ) )
% 11.90/6.00 => ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3707_foldl__rule__aux,axiom,
% 11.90/6.00 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
% 11.90/6.00 ( ( I4 @ Sigma_0 @ L0 )
% 11.90/6.00 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 11.90/6.00 ( ( L0
% 11.90/6.00 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 11.90/6.00 => ( ( I4 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
% 11.90/6.00 => ( I4 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
% 11.90/6.00 => ( I4 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3708_foldl__rule__aux__P,axiom,
% 11.90/6.00 ! [Sigma2: $tType,A: $tType,I4: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
% 11.90/6.00 ( ( I4 @ Sigma_0 @ L0 )
% 11.90/6.00 => ( ! [L12: list @ A,L22: list @ A,X2: A,Sigma: Sigma2] :
% 11.90/6.00 ( ( L0
% 11.90/6.00 = ( append @ A @ L12 @ ( cons @ A @ X2 @ L22 ) ) )
% 11.90/6.00 => ( ( I4 @ Sigma @ ( cons @ A @ X2 @ L22 ) )
% 11.90/6.00 => ( I4 @ ( F @ Sigma @ X2 ) @ L22 ) ) )
% 11.90/6.00 => ( ! [Sigma: Sigma2] :
% 11.90/6.00 ( ( I4 @ Sigma @ ( nil @ A ) )
% 11.90/6.00 => ( P @ Sigma ) )
% 11.90/6.00 => ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3709_map__by__foldl,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,F: A > B,L: list @ A] :
% 11.90/6.00 ( ( foldl @ ( list @ B ) @ A
% 11.90/6.00 @ ^ [L3: list @ B,X3: A] : ( append @ B @ L3 @ ( cons @ B @ ( F @ X3 ) @ ( nil @ B ) ) )
% 11.90/6.00 @ ( nil @ B )
% 11.90/6.00 @ L )
% 11.90/6.00 = ( map @ A @ B @ F @ L ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3710_length__Suc__rev__conv,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,N: nat] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( suc @ N ) )
% 11.90/6.00 = ( ? [Ys: list @ A,Y3: A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 = ( append @ A @ Ys @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
% 11.90/6.00 & ( ( size_size @ ( list @ A ) @ Ys )
% 11.90/6.00 = N ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3711_length__compl__rev__induct,axiom,
% 11.90/6.00 ! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
% 11.90/6.00 ( ( P @ ( nil @ A ) )
% 11.90/6.00 => ( ! [L5: list @ A,E3: A] :
% 11.90/6.00 ( ! [Ll: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L5 ) )
% 11.90/6.00 => ( P @ Ll ) )
% 11.90/6.00 => ( P @ ( append @ A @ L5 @ ( cons @ A @ E3 @ ( nil @ A ) ) ) ) )
% 11.90/6.00 => ( P @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3712_not__distinct__split__distinct,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A] :
% 11.90/6.00 ( ~ ( distinct @ A @ Xs )
% 11.90/6.00 => ~ ! [Y4: A,Ys3: list @ A] :
% 11.90/6.00 ( ( distinct @ A @ Ys3 )
% 11.90/6.00 => ( ( member @ A @ Y4 @ ( set2 @ A @ Ys3 ) )
% 11.90/6.00 => ! [Zs2: list @ A] :
% 11.90/6.00 ( Xs
% 11.90/6.00 != ( append @ A @ Ys3 @ ( append @ A @ ( cons @ A @ Y4 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3713_filter__eq__snocD,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,L: list @ A,L4: list @ A,X: A] :
% 11.90/6.00 ( ( ( filter @ A @ P @ L )
% 11.90/6.00 = ( append @ A @ L4 @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 11.90/6.00 => ( ( member @ A @ X @ ( set2 @ A @ L ) )
% 11.90/6.00 & ( P @ X ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3714_replicate__Suc__conv__snoc,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,X: A] :
% 11.90/6.00 ( ( replicate @ A @ ( suc @ N ) @ X )
% 11.90/6.00 = ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3715_slice__prepend,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ I @ K )
% 11.90/6.00 => ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( slice @ A @ I @ K @ Xs )
% 11.90/6.00 = ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( append @ A @ Ys2 @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3716_sorted__append__bigger,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [Xs: list @ A,Y: A] :
% 11.90/6.00 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 11.90/6.00 => ( ! [X2: A] :
% 11.90/6.00 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 => ( ord_less_eq @ A @ X2 @ Y ) )
% 11.90/6.00 => ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3717_upt__eq__lel__conv,axiom,
% 11.90/6.00 ! [L: nat,H3: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
% 11.90/6.00 ( ( ( upt @ L @ H3 )
% 11.90/6.00 = ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
% 11.90/6.00 = ( ( Is1
% 11.90/6.00 = ( upt @ L @ I ) )
% 11.90/6.00 & ( Is2
% 11.90/6.00 = ( upt @ ( suc @ I ) @ H3 ) )
% 11.90/6.00 & ( ord_less_eq @ nat @ L @ I )
% 11.90/6.00 & ( ord_less @ nat @ I @ H3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3718_list__encode_Osimps_I2_J,axiom,
% 11.90/6.00 ! [X: nat,Xs: list @ nat] :
% 11.90/6.00 ( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
% 11.90/6.00 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3719_list__encode_Opelims,axiom,
% 11.90/6.00 ! [X: list @ nat,Y: nat] :
% 11.90/6.00 ( ( ( nat_list_encode @ X )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ nat ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( zero_zero @ nat ) )
% 11.90/6.00 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
% 11.90/6.00 => ~ ! [X2: nat,Xs2: list @ nat] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ nat @ X2 @ Xs2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X2 @ ( nat_list_encode @ Xs2 ) ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X2 @ Xs2 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3720_map__upds__append1,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M2: A > ( option @ B ),X: A] :
% 11.90/6.00 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3721_lexn__conv,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( lexn @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A ),N5: nat] :
% 11.90/6.00 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 11.90/6.00 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 11.90/6.00 = N5 )
% 11.90/6.00 & ( ( size_size @ ( list @ A ) @ Ys )
% 11.90/6.00 = N5 )
% 11.90/6.00 & ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
% 11.90/6.00 ( ( Xs3
% 11.90/6.00 = ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
% 11.90/6.00 & ( Ys
% 11.90/6.00 = ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3722_drop__last__conv,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A] :
% 11.90/6.00 ( ( L
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
% 11.90/6.00 = ( cons @ A @ ( last @ A @ L ) @ ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3723_Misc_Olast__in__set,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A] :
% 11.90/6.00 ( ( L
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( member @ A @ ( last @ A @ L ) @ ( set2 @ A @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3724_last__upt,axiom,
% 11.90/6.00 ! [I: nat,J: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ J )
% 11.90/6.00 => ( ( last @ nat @ ( upt @ I @ J ) )
% 11.90/6.00 = ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3725_map__upds__Cons,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,M2: A > ( option @ B ),A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
% 11.90/6.00 ( ( map_upds @ A @ B @ M2 @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
% 11.90/6.00 = ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs2 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3726_map__upds__twist,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,A3: A,As2: list @ A,M2: A > ( option @ B ),B2: B,Bs2: list @ B] :
% 11.90/6.00 ( ~ ( member @ A @ A3 @ ( set2 @ A @ As2 ) )
% 11.90/6.00 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A3 @ ( some @ B @ B2 ) ) @ As2 @ Bs2 )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As2 @ Bs2 ) @ A3 @ ( some @ B @ B2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3727_last__filter,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,P: A > $o] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( P @ ( last @ A @ Xs ) )
% 11.90/6.00 => ( ( last @ A @ ( filter @ A @ P @ Xs ) )
% 11.90/6.00 = ( last @ A @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3728_lexn__length,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),N: nat] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexn @ A @ R @ N ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = N )
% 11.90/6.00 & ( ( size_size @ ( list @ A ) @ Ys2 )
% 11.90/6.00 = N ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3729_product_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 11.90/6.00 = ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys2 ) @ ( product @ A @ B @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3730_map__of__Some__split,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,V: A] :
% 11.90/6.00 ( ( ( map_of @ B @ A @ Xs @ K )
% 11.90/6.00 = ( some @ A @ V ) )
% 11.90/6.00 => ? [Ys3: list @ ( product_prod @ B @ A ),Zs2: list @ ( product_prod @ B @ A )] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 = ( append @ ( product_prod @ B @ A ) @ Ys3 @ ( cons @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ Zs2 ) ) )
% 11.90/6.00 & ( ( map_of @ B @ A @ Ys3 @ K )
% 11.90/6.00 = ( none @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3731_last__conv__nth,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( last @ A @ Xs )
% 11.90/6.00 = ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3732_map__of__distinct__lookup,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
% 11.90/6.00 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 11.90/6.00 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) @ X )
% 11.90/6.00 = ( some @ B @ Y ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3733_map__of__distinct__upd3,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B,Y7: B] :
% 11.90/6.00 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 11.90/6.00 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y7 ) @ Ys2 ) ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3734_map__of__distinct__upd2,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
% 11.90/6.00 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 11.90/6.00 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3735_map__of__distinct__upd4,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ ( product_prod @ A @ B ),Ys2: list @ ( product_prod @ A @ B ),Y: B] :
% 11.90/6.00 ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 11.90/6.00 => ( ~ ( member @ A @ X @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys2 ) ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys2 ) )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ Ys2 ) ) ) @ X @ ( none @ B ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3736_lex__conv,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( lex @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 11.90/6.00 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ys ) )
% 11.90/6.00 & ? [Xys2: list @ A,X3: A,Y3: A,Xs5: list @ A,Ys5: list @ A] :
% 11.90/6.00 ( ( Xs3
% 11.90/6.00 = ( append @ A @ Xys2 @ ( cons @ A @ X3 @ Xs5 ) ) )
% 11.90/6.00 & ( Ys
% 11.90/6.00 = ( append @ A @ Xys2 @ ( cons @ A @ Y3 @ Ys5 ) ) )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3737_shuffles_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Xa: list @ A,Y: set @ ( list @ A )] :
% 11.90/6.00 ( ( ( shuffles @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 11.90/6.00 => ( ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ! [Y4: A,Ys3: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ Y4 @ Ys3 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y4 ) @ ( shuffles @ A @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3738_set__relcomp,axiom,
% 11.90/6.00 ! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
% 11.90/6.00 ( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
% 11.90/6.00 = ( set2 @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( concat @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ^ [Xy: product_prod @ A @ C] :
% 11.90/6.00 ( concat @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ^ [Yz: product_prod @ C @ B] :
% 11.90/6.00 ( if @ ( list @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ( ( product_snd @ A @ C @ Xy )
% 11.90/6.00 = ( product_fst @ C @ B @ Yz ) )
% 11.90/6.00 @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
% 11.90/6.00 @ ( nil @ ( product_prod @ A @ B ) ) )
% 11.90/6.00 @ Yzs ) )
% 11.90/6.00 @ Xys ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3739_natural_Osimps_I4_J,axiom,
% 11.90/6.00 ! [T: $tType,F1: T,F22: code_natural > T] :
% 11.90/6.00 ( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 11.90/6.00 = F1 ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3740_Cons__in__lex,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( lex @ A @ R ) )
% 11.90/6.00 = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/6.00 & ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ys2 ) ) )
% 11.90/6.00 | ( ( X = Y )
% 11.90/6.00 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3741_concat__filter__neq__Nil,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ ( list @ A )] :
% 11.90/6.00 ( ( concat @ A
% 11.90/6.00 @ ( filter @ ( list @ A )
% 11.90/6.00 @ ^ [Ys: list @ A] :
% 11.90/6.00 ( Ys
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 @ Xs ) )
% 11.90/6.00 = ( concat @ A @ Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3742_Nil__notin__lex,axiom,
% 11.90/6.00 ! [A: $tType,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) @ ( lex @ A @ R ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3743_Nil2__notin__lex,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3744_lex__append__leftI,axiom,
% 11.90/6.00 ! [A: $tType,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3745_foldl__foldl__conv__concat,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,F: A > B > A,A3: A,Xs: list @ ( list @ B )] :
% 11.90/6.00 ( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F ) @ A3 @ Xs )
% 11.90/6.00 = ( foldl @ A @ B @ F @ A3 @ ( concat @ B @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3746_product__concat__map,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType] :
% 11.90/6.00 ( ( product @ A @ B )
% 11.90/6.00 = ( ^ [Xs3: list @ A,Ys: list @ B] :
% 11.90/6.00 ( concat @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys )
% 11.90/6.00 @ Xs3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3747_lex__append__left__iff,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 11.90/6.00 ( ! [X2: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
% 11.90/6.00 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3748_lex__append__leftD,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 11.90/6.00 ( ! [X2: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3749_lex__append__rightI,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ Vs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Us ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3750_map__of__concat,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xss: list @ ( list @ ( product_prod @ A @ B ) )] :
% 11.90/6.00 ( ( map_of @ A @ B @ ( concat @ ( product_prod @ A @ B ) @ Xss ) )
% 11.90/6.00 = ( foldr @ ( list @ ( product_prod @ A @ B ) ) @ ( A > ( option @ B ) )
% 11.90/6.00 @ ^ [Xs3: list @ ( product_prod @ A @ B ),F2: A > ( option @ B )] : ( map_add @ A @ B @ F2 @ ( map_of @ A @ B @ Xs3 ) )
% 11.90/6.00 @ Xss
% 11.90/6.00 @ ^ [X3: A] : ( none @ B ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3751_comm__append__is__replicate,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( Ys2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ( append @ A @ Xs @ Ys2 )
% 11.90/6.00 = ( append @ A @ Ys2 @ Xs ) )
% 11.90/6.00 => ? [N3: nat,Zs2: list @ A] :
% 11.90/6.00 ( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
% 11.90/6.00 & ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
% 11.90/6.00 = ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3752_shuffles_Opinduct,axiom,
% 11.90/6.00 ! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
% 11.90/6.00 => ( ! [Ys3: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 11.90/6.00 => ( P @ ( nil @ A ) @ Ys3 ) )
% 11.90/6.00 => ( ! [Xs2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
% 11.90/6.00 => ( P @ Xs2 @ ( nil @ A ) ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A,Y4: A,Ys3: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) )
% 11.90/6.00 => ( ( P @ Xs2 @ ( cons @ A @ Y4 @ Ys3 ) )
% 11.90/6.00 => ( ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 )
% 11.90/6.00 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y4 @ Ys3 ) ) ) ) )
% 11.90/6.00 => ( P @ A0 @ A1 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3753_shuffles_Opsimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,Ys2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
% 11.90/6.00 => ( ( shuffles @ A @ ( nil @ A ) @ Ys2 )
% 11.90/6.00 = ( insert2 @ ( list @ A ) @ Ys2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3754_shuffles_Opsimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
% 11.90/6.00 => ( ( shuffles @ A @ Xs @ ( nil @ A ) )
% 11.90/6.00 = ( insert2 @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3755_distinct__concat_H,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ ( list @ A )] :
% 11.90/6.00 ( ( distinct @ ( list @ A )
% 11.90/6.00 @ ( filter @ ( list @ A )
% 11.90/6.00 @ ^ [Ys: list @ A] :
% 11.90/6.00 ( Ys
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 @ Xs ) )
% 11.90/6.00 => ( ! [Ys3: list @ A] :
% 11.90/6.00 ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( distinct @ A @ Ys3 ) )
% 11.90/6.00 => ( ! [Ys3: list @ A,Zs2: list @ A] :
% 11.90/6.00 ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( Ys3 != Zs2 )
% 11.90/6.00 => ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs2 ) )
% 11.90/6.00 = ( bot_bot @ ( set @ A ) ) ) ) ) )
% 11.90/6.00 => ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3756_shuffles_Opsimps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) )
% 11.90/6.00 => ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
% 11.90/6.00 = ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys2 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3757_product__code,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys2 ) )
% 11.90/6.00 = ( set2 @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( concat @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ^ [X3: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 ) @ Ys2 )
% 11.90/6.00 @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3758_lenlex__conv,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( lenlex @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 11.90/6.00 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 11.90/6.00 ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys ) )
% 11.90/6.00 | ( ( ( size_size @ ( list @ A ) @ Xs3 )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ys ) )
% 11.90/6.00 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lex @ A @ R3 ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3759_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 11.90/6.00 ! [N: nat,J: nat,I: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
% 11.90/6.00 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
% 11.90/6.00 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3760_splice_Opinduct,axiom,
% 11.90/6.00 ! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
% 11.90/6.00 => ( ! [Ys3: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys3 ) )
% 11.90/6.00 => ( P @ ( nil @ A ) @ Ys3 ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A,Ys3: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) )
% 11.90/6.00 => ( ( P @ Ys3 @ Xs2 )
% 11.90/6.00 => ( P @ ( cons @ A @ X2 @ Xs2 ) @ Ys3 ) ) )
% 11.90/6.00 => ( P @ A0 @ A1 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3761_Nil__lenlex__iff1,axiom,
% 11.90/6.00 ! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
% 11.90/6.00 = ( Ns
% 11.90/6.00 != ( nil @ A ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3762_lenlex__irreflexive,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 11.90/6.00 ( ! [X2: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 11.90/6.00 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3763_Nil__lenlex__iff2,axiom,
% 11.90/6.00 ! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3764_lenlex__length,axiom,
% 11.90/6.00 ! [A: $tType,Ms: list @ A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) )
% 11.90/6.00 => ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3765_lenlex__append1,axiom,
% 11.90/6.00 ! [A: $tType,Us: list @ A,Xs: list @ A,R5: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R5 ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ Vs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys2 ) ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3766_member__product,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,X: product_prod @ A @ B,A4: set @ A,B6: set @ B] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A4 @ B6 ) )
% 11.90/6.00 = ( member @ ( product_prod @ A @ B ) @ X
% 11.90/6.00 @ ( product_Sigma @ A @ B @ A4
% 11.90/6.00 @ ^ [Uu: A] : B6 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3767_Product__Type_Oproduct__def,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType] :
% 11.90/6.00 ( ( product_product @ A @ B )
% 11.90/6.00 = ( ^ [A9: set @ A,B9: set @ B] :
% 11.90/6.00 ( product_Sigma @ A @ B @ A9
% 11.90/6.00 @ ^ [Uu: A] : B9 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3768_lenlex__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( lenlex @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R3 ) )
% 11.90/6.00 @ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3769_Cons__lenlex__iff,axiom,
% 11.90/6.00 ! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
% 11.90/6.00 = ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
% 11.90/6.00 | ( ( ( size_size @ ( list @ A ) @ Ms )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ns ) )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M2 @ N ) @ R ) )
% 11.90/6.00 | ( ( M2 = N )
% 11.90/6.00 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3770_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 11.90/6.00 ! [N: nat,J: nat,I: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
% 11.90/6.00 => ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
% 11.90/6.00 = ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3771_splice_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( splice @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y = Xa )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( cons @ A @ X2 @ ( splice @ A @ Xa @ Xs2 ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3772_sorted__find__Min,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [Xs: list @ A,P: A > $o] :
% 11.90/6.00 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 11.90/6.00 => ( ? [X7: A] :
% 11.90/6.00 ( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 & ( P @ X7 ) )
% 11.90/6.00 => ( ( find @ A @ P @ Xs )
% 11.90/6.00 = ( some @ A
% 11.90/6.00 @ ( lattic643756798350308766er_Min @ A
% 11.90/6.00 @ ( collect @ A
% 11.90/6.00 @ ^ [X3: A] :
% 11.90/6.00 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 & ( P @ X3 ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3773_part__def,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ( ( linorder_part @ B @ A )
% 11.90/6.00 = ( ^ [F2: B > A,Pivot: A,Xs3: list @ B] :
% 11.90/6.00 ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
% 11.90/6.00 @ ( filter @ B
% 11.90/6.00 @ ^ [X3: B] : ( ord_less @ A @ ( F2 @ X3 ) @ Pivot )
% 11.90/6.00 @ Xs3 )
% 11.90/6.00 @ ( product_Pair @ ( list @ B ) @ ( list @ B )
% 11.90/6.00 @ ( filter @ B
% 11.90/6.00 @ ^ [X3: B] :
% 11.90/6.00 ( ( F2 @ X3 )
% 11.90/6.00 = Pivot )
% 11.90/6.00 @ Xs3 )
% 11.90/6.00 @ ( filter @ B
% 11.90/6.00 @ ^ [X3: B] : ( ord_less @ A @ Pivot @ ( F2 @ X3 ) )
% 11.90/6.00 @ Xs3 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3774_merge__list__correct,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [Ls: list @ ( list @ A ),As2: list @ ( list @ A )] :
% 11.90/6.00 ( ! [L5: list @ A] :
% 11.90/6.00 ( ( member @ ( list @ A ) @ L5 @ ( set2 @ ( list @ A ) @ Ls ) )
% 11.90/6.00 => ( ( distinct @ A @ L5 )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L5 ) ) )
% 11.90/6.00 => ( ! [L5: list @ A] :
% 11.90/6.00 ( ( member @ ( list @ A ) @ L5 @ ( set2 @ ( list @ A ) @ As2 ) )
% 11.90/6.00 => ( ( distinct @ A @ L5 )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L5 ) ) )
% 11.90/6.00 => ( ( distinct @ A @ ( merge_list @ A @ As2 @ Ls ) )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As2 @ Ls ) )
% 11.90/6.00 & ( ( set2 @ A @ ( merge_list @ A @ As2 @ Ls ) )
% 11.90/6.00 = ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As2 @ Ls ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3775_length__splice,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) )
% 11.90/6.00 = ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3776_splice__replicate,axiom,
% 11.90/6.00 ! [A: $tType,M2: nat,X: A,N: nat] :
% 11.90/6.00 ( ( splice @ A @ ( replicate @ A @ M2 @ X ) @ ( replicate @ A @ N @ X ) )
% 11.90/6.00 = ( replicate @ A @ ( plus_plus @ nat @ M2 @ N ) @ X ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3777_merge__list_Osimps_I4_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [La: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
% 11.90/6.00 ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3778_merge__list_Osimps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [La: list @ A,Acc22: list @ ( list @ A )] :
% 11.90/6.00 ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3779_merge__list_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A] :
% 11.90/6.00 ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 = L ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3780_merge__list_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 = ( nil @ A ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3781_find__SomeD_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 11.90/6.00 ( ( ( find @ A @ P @ Xs )
% 11.90/6.00 = ( some @ A @ X ) )
% 11.90/6.00 => ( P @ X ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3782_find_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( P @ X )
% 11.90/6.00 => ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( some @ A @ X ) ) )
% 11.90/6.00 & ( ~ ( P @ X )
% 11.90/6.00 => ( ( find @ A @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( find @ A @ P @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3783_find_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,Uu2: A > $o] :
% 11.90/6.00 ( ( find @ A @ Uu2 @ ( nil @ A ) )
% 11.90/6.00 = ( none @ A ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3784_find__SomeD_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 11.90/6.00 ( ( ( find @ A @ P @ Xs )
% 11.90/6.00 = ( some @ A @ X ) )
% 11.90/6.00 => ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3785_find__None__iff2,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 11.90/6.00 ( ( ( none @ A )
% 11.90/6.00 = ( find @ A @ P @ Xs ) )
% 11.90/6.00 = ( ~ ? [X3: A] :
% 11.90/6.00 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 & ( P @ X3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3786_find__None__iff,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A] :
% 11.90/6.00 ( ( ( find @ A @ P @ Xs )
% 11.90/6.00 = ( none @ A ) )
% 11.90/6.00 = ( ~ ? [X3: A] :
% 11.90/6.00 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 & ( P @ X3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3787_part__code_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [F: B > A,Pivot2: A] :
% 11.90/6.00 ( ( linorder_part @ B @ A @ F @ Pivot2 @ ( nil @ B ) )
% 11.90/6.00 = ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3788_find__Some__iff,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
% 11.90/6.00 ( ( ( find @ A @ P @ Xs )
% 11.90/6.00 = ( some @ A @ X ) )
% 11.90/6.00 = ( ? [I3: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 & ( P @ ( nth @ A @ Xs @ I3 ) )
% 11.90/6.00 & ( X
% 11.90/6.00 = ( nth @ A @ Xs @ I3 ) )
% 11.90/6.00 & ! [J2: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ J2 @ I3 )
% 11.90/6.00 => ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3789_find__Some__iff2,axiom,
% 11.90/6.00 ! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
% 11.90/6.00 ( ( ( some @ A @ X )
% 11.90/6.00 = ( find @ A @ P @ Xs ) )
% 11.90/6.00 = ( ? [I3: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 & ( P @ ( nth @ A @ Xs @ I3 ) )
% 11.90/6.00 & ( X
% 11.90/6.00 = ( nth @ A @ Xs @ I3 ) )
% 11.90/6.00 & ! [J2: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ J2 @ I3 )
% 11.90/6.00 => ~ ( P @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3790_mergesort__remdups__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ( ( mergesort_remdups @ A )
% 11.90/6.00 = ( ^ [Xs3: list @ A] :
% 11.90/6.00 ( merge_list @ A @ ( nil @ ( list @ A ) )
% 11.90/6.00 @ ( map @ A @ ( list @ A )
% 11.90/6.00 @ ^ [X3: A] : ( cons @ A @ X3 @ ( nil @ A ) )
% 11.90/6.00 @ Xs3 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3791_splice_Opsimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys2 ) )
% 11.90/6.00 => ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 11.90/6.00 = ( cons @ A @ X @ ( splice @ A @ Ys2 @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3792_splice_Opsimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,Ys2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
% 11.90/6.00 => ( ( splice @ A @ ( nil @ A ) @ Ys2 )
% 11.90/6.00 = Ys2 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3793_part__code_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [F: B > A,Pivot2: A,X: B,Xs: list @ B] :
% 11.90/6.00 ( ( linorder_part @ B @ A @ F @ Pivot2 @ ( cons @ B @ X @ Xs ) )
% 11.90/6.00 = ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
% 11.90/6.00 @ ^ [Lts: list @ B] :
% 11.90/6.00 ( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
% 11.90/6.00 @ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F @ X ) @ Pivot2 ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot2 @ ( F @ X ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X @ Eqs ) @ Gts ) ) ) ) )
% 11.90/6.00 @ ( linorder_part @ B @ A @ F @ Pivot2 @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3794_extract__SomeE,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,Ys2: list @ A,Y: A,Zs: list @ A] :
% 11.90/6.00 ( ( ( extract @ A @ P @ Xs )
% 11.90/6.00 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
% 11.90/6.00 => ( ( Xs
% 11.90/6.00 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ Zs ) ) )
% 11.90/6.00 & ( P @ Y )
% 11.90/6.00 & ~ ? [X7: A] :
% 11.90/6.00 ( ( member @ A @ X7 @ ( set2 @ A @ Ys2 ) )
% 11.90/6.00 & ( P @ X7 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3795_extract__Some__iff,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,Ys2: list @ A,Y: A,Zs: list @ A] :
% 11.90/6.00 ( ( ( extract @ A @ P @ Xs )
% 11.90/6.00 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
% 11.90/6.00 = ( ( Xs
% 11.90/6.00 = ( append @ A @ Ys2 @ ( cons @ A @ Y @ Zs ) ) )
% 11.90/6.00 & ( P @ Y )
% 11.90/6.00 & ~ ? [X3: A] :
% 11.90/6.00 ( ( member @ A @ X3 @ ( set2 @ A @ Ys2 ) )
% 11.90/6.00 & ( P @ X3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3796_extract__Cons__code,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( P @ X )
% 11.90/6.00 => ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
% 11.90/6.00 & ( ~ ( P @ X )
% 11.90/6.00 => ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 11.90/6.00 @ ^ [Ys: list @ A] :
% 11.90/6.00 ( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 11.90/6.00 @ ^ [Y3: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs3 ) ) ) ) )
% 11.90/6.00 @ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3797_merge__list_Ocases,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [X: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ! [L5: list @ A] :
% 11.90/6.00 ( X
% 11.90/6.00 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 11.90/6.00 ( X
% 11.90/6.00 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A ),L5: list @ A] :
% 11.90/6.00 ( X
% 11.90/6.00 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ~ ! [Acc23: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 11.90/6.00 ( X
% 11.90/6.00 != ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3798_subset__eq__mset__impl_Oelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
% 11.90/6.00 ( ( ( subset_eq_mset_impl @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( some @ $o
% 11.90/6.00 @ ( Xa
% 11.90/6.00 != ( nil @ A ) ) ) ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
% 11.90/6.00 @ ^ [Ys1: list @ A] :
% 11.90/6.00 ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
% 11.90/6.00 @ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
% 11.90/6.00 @ ( extract @ A
% 11.90/6.00 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.90/6.00 @ X2 )
% 11.90/6.00 @ Xa ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3799_lexord__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( lexord @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 11.90/6.00 @ ^ [X3: list @ A,Y3: list @ A] :
% 11.90/6.00 ? [A7: A,V2: list @ A] :
% 11.90/6.00 ( ( Y3
% 11.90/6.00 = ( append @ A @ X3 @ ( cons @ A @ A7 @ V2 ) ) )
% 11.90/6.00 | ? [U2: list @ A,B5: A,C3: A,W2: list @ A,Z5: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R3 )
% 11.90/6.00 & ( X3
% 11.90/6.00 = ( append @ A @ U2 @ ( cons @ A @ B5 @ W2 ) ) )
% 11.90/6.00 & ( Y3
% 11.90/6.00 = ( append @ A @ U2 @ ( cons @ A @ C3 @ Z5 ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3800_list__collect__set__alt,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType] :
% 11.90/6.00 ( ( list_collect_set @ B @ A )
% 11.90/6.00 = ( ^ [F2: B > ( set @ A ),L3: list @ B] :
% 11.90/6.00 ( complete_Sup_Sup @ ( set @ A )
% 11.90/6.00 @ ( collect @ ( set @ A )
% 11.90/6.00 @ ^ [Uu: set @ A] :
% 11.90/6.00 ? [I3: nat] :
% 11.90/6.00 ( ( Uu
% 11.90/6.00 = ( F2 @ ( nth @ B @ L3 @ I3 ) ) )
% 11.90/6.00 & ( ord_less @ nat @ I3 @ ( size_size @ ( list @ B ) @ L3 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3801_subset__eq__mset__impl_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Xa: list @ A,Y: option @ $o] :
% 11.90/6.00 ( ( ( subset_eq_mset_impl @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( some @ $o
% 11.90/6.00 @ ( Xa
% 11.90/6.00 != ( nil @ A ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
% 11.90/6.00 @ ^ [Ys1: list @ A] :
% 11.90/6.00 ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
% 11.90/6.00 @ ^ [Y3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs2 @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
% 11.90/6.00 @ ( extract @ A
% 11.90/6.00 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.90/6.00 @ X2 )
% 11.90/6.00 @ Xa ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xa ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3802_list__collect__set__simps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ ( nil @ B ) ) )
% 11.90/6.00 = ( F @ A3 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3803_list__collect__set__simps_I1_J,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,F: B > ( set @ A )] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( nil @ B ) )
% 11.90/6.00 = ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3804_list__collect__set__simps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,F: B > ( set @ A ),A3: B,L: list @ B] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A3 @ L ) )
% 11.90/6.00 = ( sup_sup @ ( set @ A ) @ ( F @ A3 ) @ ( list_collect_set @ B @ A @ F @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3805_list__collect__set__simps_I4_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,F: B > ( set @ A ),L: list @ B,L4: list @ B] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( append @ B @ L @ L4 ) )
% 11.90/6.00 = ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ L ) @ ( list_collect_set @ B @ A @ F @ L4 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3806_lexord__cons__cons,axiom,
% 11.90/6.00 ! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R ) )
% 11.90/6.00 = ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/6.00 | ( ( A3 = B2 )
% 11.90/6.00 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3807_lexord__Nil__left,axiom,
% 11.90/6.00 ! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
% 11.90/6.00 = ( ? [A7: A,X3: list @ A] :
% 11.90/6.00 ( Y
% 11.90/6.00 = ( cons @ A @ A7 @ X3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3808_list__collect__set__map__simps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ ( nil @ C ) ) ) )
% 11.90/6.00 = ( F @ ( X @ A3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3809_list__collect__set__map__simps_I1_J,axiom,
% 11.90/6.00 ! [C: $tType,B: $tType,A: $tType,F: B > ( set @ A ),X: C > B] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( nil @ C ) ) )
% 11.90/6.00 = ( bot_bot @ ( set @ A ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3810_list__collect__set__map__simps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,A3: C,L: list @ C] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( cons @ C @ A3 @ L ) ) )
% 11.90/6.00 = ( sup_sup @ ( set @ A ) @ ( F @ ( X @ A3 ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3811_list__collect__set__map__simps_I4_J,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X: C > B,L: list @ C,L4: list @ C] :
% 11.90/6.00 ( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ ( append @ C @ L @ L4 ) ) )
% 11.90/6.00 = ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X @ L4 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3812_lexord__irreflexive,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 11.90/6.00 ( ! [X2: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R )
% 11.90/6.00 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3813_lexord__linear,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
% 11.90/6.00 ( ! [A5: A,B3: A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R )
% 11.90/6.00 | ( A5 = B3 )
% 11.90/6.00 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R ) )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 11.90/6.00 | ( X = Y )
% 11.90/6.00 | ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3814_lexord__Nil__right,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3815_lexord__append__leftI,axiom,
% 11.90/6.00 ! [A: $tType,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3816_lexord__partial__trans,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Ys2: list @ A,Zs: list @ A] :
% 11.90/6.00 ( ! [X2: A,Y4: A,Z3: A] :
% 11.90/6.00 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 11.90/6.00 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.90/6.00 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) ) )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R ) )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3817_lexord__append__leftD,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) )
% 11.90/6.00 => ( ! [A5: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3818_lexord__append__rightI,axiom,
% 11.90/6.00 ! [A: $tType,Y: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ? [B8: A,Z6: list @ A] :
% 11.90/6.00 ( Y
% 11.90/6.00 = ( cons @ A @ B8 @ Z6 ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3819_lexord__sufE,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Zs: list @ A,Ys2: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R ) )
% 11.90/6.00 => ( ( Xs != Ys2 )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ys2 ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ Zs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Qs ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3820_list__collect__set__as__map,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType] :
% 11.90/6.00 ( ( list_collect_set @ B @ A )
% 11.90/6.00 = ( ^ [F2: B > ( set @ A ),L3: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F2 @ L3 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3821_lexord__lex,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R ) )
% 11.90/6.00 = ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 11.90/6.00 & ( ( size_size @ ( list @ A ) @ X )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3822_lexord__append__left__rightI,axiom,
% 11.90/6.00 ! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3823_lexord__same__pref__iff,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
% 11.90/6.00 = ( ? [X3: A] :
% 11.90/6.00 ( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R ) )
% 11.90/6.00 | ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3824_lexord__sufI,axiom,
% 11.90/6.00 ! [A: $tType,U: list @ A,W: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z2: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R ) )
% 11.90/6.00 => ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W @ Z2 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3825_list__collect__set__def,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType] :
% 11.90/6.00 ( ( list_collect_set @ B @ A )
% 11.90/6.00 = ( ^ [F2: B > ( set @ A ),L3: list @ B] :
% 11.90/6.00 ( complete_Sup_Sup @ ( set @ A )
% 11.90/6.00 @ ( collect @ ( set @ A )
% 11.90/6.00 @ ^ [Uu: set @ A] :
% 11.90/6.00 ? [A7: B] :
% 11.90/6.00 ( ( Uu
% 11.90/6.00 = ( F2 @ A7 ) )
% 11.90/6.00 & ( member @ B @ A7 @ ( set2 @ B @ L3 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3826_subset__eq__mset__impl_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( subset_eq_mset_impl @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 11.90/6.00 = ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
% 11.90/6.00 @ ^ [Ys1: list @ A] :
% 11.90/6.00 ( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
% 11.90/6.00 @ ^ [X3: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
% 11.90/6.00 @ ( extract @ A
% 11.90/6.00 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.90/6.00 @ X )
% 11.90/6.00 @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3827_List_Olexordp__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( lexordp @ A )
% 11.90/6.00 = ( ^ [R3: A > A > $o,Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3828_upto_Opelims,axiom,
% 11.90/6.00 ! [X: int,Xa: int,Y: list @ int] :
% 11.90/6.00 ( ( ( upto @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
% 11.90/6.00 => ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ X @ Xa )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( nil @ int ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3829_upto_Opsimps,axiom,
% 11.90/6.00 ! [I: int,J: int] :
% 11.90/6.00 ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
% 11.90/6.00 => ( ( ( ord_less_eq @ int @ I @ J )
% 11.90/6.00 => ( ( upto @ I @ J )
% 11.90/6.00 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ I @ J )
% 11.90/6.00 => ( ( upto @ I @ J )
% 11.90/6.00 = ( nil @ int ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3830_listrel1__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( listrel1 @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 11.90/6.00 @ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
% 11.90/6.00 @ ^ [Xs3: list @ A,Ys: list @ A] :
% 11.90/6.00 ? [Us2: list @ A,Z5: A,Z8: A,Vs2: list @ A] :
% 11.90/6.00 ( ( Xs3
% 11.90/6.00 = ( append @ A @ Us2 @ ( cons @ A @ Z5 @ Vs2 ) ) )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ Z8 ) @ R3 )
% 11.90/6.00 & ( Ys
% 11.90/6.00 = ( append @ A @ Us2 @ ( cons @ A @ Z8 @ Vs2 ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3831_Cons__listrel1__Cons,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 = ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/6.00 & ( Xs = Ys2 ) )
% 11.90/6.00 | ( ( X = Y )
% 11.90/6.00 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3832_nth__upto,axiom,
% 11.90/6.00 ! [I: int,K: nat,J: int] :
% 11.90/6.00 ( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
% 11.90/6.00 => ( ( nth @ int @ ( upto @ I @ J ) @ K )
% 11.90/6.00 = ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3833_length__upto,axiom,
% 11.90/6.00 ! [I: int,J: int] :
% 11.90/6.00 ( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
% 11.90/6.00 = ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3834_upto__rec__numeral_I1_J,axiom,
% 11.90/6.00 ! [M2: num,N: num] :
% 11.90/6.00 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 = ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 = ( nil @ int ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3835_upto__rec__numeral_I2_J,axiom,
% 11.90/6.00 ! [M2: num,N: num] :
% 11.90/6.00 ( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 = ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 => ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 = ( nil @ int ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3836_upto__rec__numeral_I3_J,axiom,
% 11.90/6.00 ! [M2: num,N: num] :
% 11.90/6.00 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
% 11.90/6.00 = ( nil @ int ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3837_upto__rec__numeral_I4_J,axiom,
% 11.90/6.00 ! [M2: num,N: num] :
% 11.90/6.00 ( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 = ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 => ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.00 = ( nil @ int ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3838_listrel1I2,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3839_rtrancl__listrel1__ConsI1,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3840_not__listrel1__Nil,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3841_not__Nil__listrel1,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3842_rtrancl__listrel1__eq__len,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
% 11.90/6.00 => ( ( size_size @ ( list @ A ) @ X )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Y ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3843_listrel1__eq__len,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 => ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3844_append__listrel1I,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
% 11.90/6.00 ( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 & ( Us = Vs ) )
% 11.90/6.00 | ( ( Xs = Ys2 )
% 11.90/6.00 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3845_upto__split2,axiom,
% 11.90/6.00 ! [I: int,J: int,K: int] :
% 11.90/6.00 ( ( ord_less_eq @ int @ I @ J )
% 11.90/6.00 => ( ( ord_less_eq @ int @ J @ K )
% 11.90/6.00 => ( ( upto @ I @ K )
% 11.90/6.00 = ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3846_upto__rec1,axiom,
% 11.90/6.00 ! [I: int,J: int] :
% 11.90/6.00 ( ( ord_less_eq @ int @ I @ J )
% 11.90/6.00 => ( ( upto @ I @ J )
% 11.90/6.00 = ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3847_upto_Osimps,axiom,
% 11.90/6.00 ( upto
% 11.90/6.00 = ( ^ [I3: int,J2: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I3 @ J2 ) @ ( cons @ int @ I3 @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) @ ( nil @ int ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3848_upto_Oelims,axiom,
% 11.90/6.00 ! [X: int,Xa: int,Y: list @ int] :
% 11.90/6.00 ( ( ( upto @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( ord_less_eq @ int @ X @ Xa )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less_eq @ int @ X @ Xa )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( nil @ int ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3849_Cons__listrel1E2,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Y: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 => ( ! [X2: A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 = ( cons @ A @ X2 @ Ys2 ) )
% 11.90/6.00 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
% 11.90/6.00 => ~ ! [Zs2: list @ A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 = ( cons @ A @ Y @ Zs2 ) )
% 11.90/6.00 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3850_Cons__listrel1E1,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 => ( ! [Y4: A] :
% 11.90/6.00 ( ( Ys2
% 11.90/6.00 = ( cons @ A @ Y4 @ Xs ) )
% 11.90/6.00 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R ) )
% 11.90/6.00 => ~ ! [Zs2: list @ A] :
% 11.90/6.00 ( ( Ys2
% 11.90/6.00 = ( cons @ A @ X @ Zs2 ) )
% 11.90/6.00 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3851_listrel1I1,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3852_greaterThanAtMost__upto,axiom,
% 11.90/6.00 ( ( set_or3652927894154168847AtMost @ int )
% 11.90/6.00 = ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3853_listrel1I,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/6.00 => ( ( Xs
% 11.90/6.00 = ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
% 11.90/6.00 => ( ( Ys2
% 11.90/6.00 = ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3854_listrel1E,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 => ~ ! [X2: A,Y4: A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 11.90/6.00 => ! [Us3: list @ A,Vs3: list @ A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 = ( append @ A @ Us3 @ ( cons @ A @ X2 @ Vs3 ) ) )
% 11.90/6.00 => ( Ys2
% 11.90/6.00 != ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3855_upto__split3,axiom,
% 11.90/6.00 ! [I: int,J: int,K: int] :
% 11.90/6.00 ( ( ord_less_eq @ int @ I @ J )
% 11.90/6.00 => ( ( ord_less_eq @ int @ J @ K )
% 11.90/6.00 => ( ( upto @ I @ K )
% 11.90/6.00 = ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3856_rtrancl__listrel1__ConsI2,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
% 11.90/6.00 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3857_greaterThanLessThan__upto,axiom,
% 11.90/6.00 ( ( set_or5935395276787703475ssThan @ int )
% 11.90/6.00 = ( ^ [I3: int,J2: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3858_snoc__listrel1__snoc__iff,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys2 @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 = ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 & ( X = Y ) )
% 11.90/6.00 | ( ( Xs = Ys2 )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3859_listrel1p__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( listrel1p @ A )
% 11.90/6.00 = ( ^ [R3: A > A > $o,Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3860_sorted__wrt__map__rev__linord,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ ( product_prod @ A @ B )] :
% 11.90/6.00 ( ( sorted_wrt @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: product_prod @ A @ B,Y3: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y3 ) @ ( product_fst @ A @ B @ X3 ) )
% 11.90/6.00 @ L )
% 11.90/6.00 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ L ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3861_set__zip,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 11.90/6.00 = ( collect @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [Uu: product_prod @ A @ B] :
% 11.90/6.00 ? [I3: nat] :
% 11.90/6.00 ( ( Uu
% 11.90/6.00 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I3 ) @ ( nth @ B @ Ys2 @ I3 ) ) )
% 11.90/6.00 & ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3862_arg__min__list_Opelims,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType] :
% 11.90/6.00 ( ( linorder @ B )
% 11.90/6.00 => ! [X: A > B,Xa: list @ A,Y: A] :
% 11.90/6.00 ( ( ( arg_min_list @ A @ B @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ! [X2: A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.90/6.00 => ( ( Y = X2 )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 11.90/6.00 => ( ! [X2: A,Y4: A,Zs2: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( if @ A @ ( ord_less_eq @ B @ ( X @ X2 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) ) )
% 11.90/6.00 => ~ ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( undefined @ A ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3863_zip__eq__zip__same__len,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A6: list @ A,B4: list @ B] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ A3 )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ B2 ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ A6 )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ B4 ) )
% 11.90/6.00 => ( ( ( zip @ A @ B @ A3 @ B2 )
% 11.90/6.00 = ( zip @ A @ B @ A6 @ B4 ) )
% 11.90/6.00 = ( ( A3 = A6 )
% 11.90/6.00 & ( B2 = B4 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3864_sorted__wrt__rev__linord,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A] :
% 11.90/6.00 ( ( sorted_wrt @ A
% 11.90/6.00 @ ^ [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ X3 )
% 11.90/6.00 @ L )
% 11.90/6.00 = ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3865_zip__Cons__Cons,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys2: list @ B] :
% 11.90/6.00 ( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) )
% 11.90/6.00 = ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3866_map__of__rev__distinct,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,M2: list @ ( product_prod @ A @ B )] :
% 11.90/6.00 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ M2 ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( rev @ ( product_prod @ A @ B ) @ M2 ) )
% 11.90/6.00 = ( map_of @ A @ B @ M2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3867_zip__replicate,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y: B] :
% 11.90/6.00 ( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y ) )
% 11.90/6.00 = ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3868_map__of__zip__is__None,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
% 11.90/6.00 = ( none @ B ) )
% 11.90/6.00 = ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3869_nth__zip,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
% 11.90/6.00 = ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3870_zip__assoc,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C] :
% 11.90/6.00 ( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys2 @ Zs ) )
% 11.90/6.00 = ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 11.90/6.00 @ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 11.90/6.00 @ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
% 11.90/6.00 @ ^ [X3: A,Y3: B,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
% 11.90/6.00 @ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys2 ) @ Zs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3871_pair__list__split,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,L: list @ ( product_prod @ A @ B )] :
% 11.90/6.00 ~ ! [L12: list @ A,L22: list @ B] :
% 11.90/6.00 ( ( L
% 11.90/6.00 = ( zip @ A @ B @ L12 @ L22 ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ L12 )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ L22 ) )
% 11.90/6.00 => ( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ L )
% 11.90/6.00 != ( size_size @ ( list @ B ) @ L22 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3872_zip__inj,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,A6: list @ A,B4: list @ B] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ A3 )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ B2 ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ A6 )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ B4 ) )
% 11.90/6.00 => ( ( ( zip @ A @ B @ A3 @ B2 )
% 11.90/6.00 = ( zip @ A @ B @ A6 @ B4 ) )
% 11.90/6.00 => ( ( A3 = A6 )
% 11.90/6.00 & ( B2 = B4 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3873_zip__left__commute,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C] :
% 11.90/6.00 ( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys2 @ Zs ) )
% 11.90/6.00 = ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 11.90/6.00 @ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 11.90/6.00 @ ^ [Y3: B] :
% 11.90/6.00 ( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
% 11.90/6.00 @ ^ [X3: A,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X3 @ ( product_Pair @ B @ C @ Y3 @ Z5 ) ) ) )
% 11.90/6.00 @ ( zip @ B @ ( product_prod @ A @ C ) @ Ys2 @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3874_zip__same__conv__map,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A] :
% 11.90/6.00 ( ( zip @ A @ A @ Xs @ Xs )
% 11.90/6.00 = ( map @ A @ ( product_prod @ A @ A )
% 11.90/6.00 @ ^ [X3: A] : ( product_Pair @ A @ A @ X3 @ X3 )
% 11.90/6.00 @ Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3875_set__zip__rightD,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 11.90/6.00 => ( member @ B @ Y @ ( set2 @ B @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3876_set__zip__leftD,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 11.90/6.00 => ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3877_in__set__zipE,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 11.90/6.00 => ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.00 => ~ ( member @ B @ Y @ ( set2 @ B @ Ys2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3878_zip__same,axiom,
% 11.90/6.00 ! [A: $tType,A3: A,B2: A,Xs: list @ A] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
% 11.90/6.00 = ( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 & ( A3 = B2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3879_zip__eq__ConsE,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
% 11.90/6.00 ( ( ( zip @ A @ B @ Xs @ Ys2 )
% 11.90/6.00 = ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
% 11.90/6.00 => ~ ! [X2: A,Xs6: list @ A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 = ( cons @ A @ X2 @ Xs6 ) )
% 11.90/6.00 => ! [Y4: B,Ys4: list @ B] :
% 11.90/6.00 ( ( Ys2
% 11.90/6.00 = ( cons @ B @ Y4 @ Ys4 ) )
% 11.90/6.00 => ( ( Xy2
% 11.90/6.00 = ( product_Pair @ A @ B @ X2 @ Y4 ) )
% 11.90/6.00 => ( Xys
% 11.90/6.00 != ( zip @ A @ B @ Xs6 @ Ys4 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3880_set__zip__cart,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,X: product_prod @ A @ B,L: list @ A,L4: list @ B] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L4 ) ) )
% 11.90/6.00 => ( member @ ( product_prod @ A @ B ) @ X
% 11.90/6.00 @ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
% 11.90/6.00 @ ^ [Uu: A] : ( set2 @ B @ L4 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3881_zip__commute,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType] :
% 11.90/6.00 ( ( zip @ A @ B )
% 11.90/6.00 = ( ^ [Xs3: list @ A,Ys: list @ B] :
% 11.90/6.00 ( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: B,Y3: A] : ( product_Pair @ A @ B @ Y3 @ X3 ) )
% 11.90/6.00 @ ( zip @ B @ A @ Ys @ Xs3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3882_last__zip,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( Ys2
% 11.90/6.00 != ( nil @ B ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 11.90/6.00 = ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3883_in__set__impl__in__set__zip1,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.00 => ~ ! [Y4: B] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3884_in__set__impl__in__set__zip2,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Y: B] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( member @ B @ Y @ ( set2 @ B @ Ys2 ) )
% 11.90/6.00 => ~ ! [X2: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3885_map__of__zip__is__Some,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.00 = ( ? [Y3: B] :
% 11.90/6.00 ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
% 11.90/6.00 = ( some @ B @ Y3 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3886_map__upds__fold__map__upd,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType] :
% 11.90/6.00 ( ( map_upds @ A @ B )
% 11.90/6.00 = ( ^ [M: A > ( option @ B ),Ks2: list @ A,Vs2: list @ B] :
% 11.90/6.00 ( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [N5: A > ( option @ B )] :
% 11.90/6.00 ( product_case_prod @ A @ B @ ( A > ( option @ B ) )
% 11.90/6.00 @ ^ [K4: A,V2: B] : ( fun_upd @ A @ ( option @ B ) @ N5 @ K4 @ ( some @ B @ V2 ) ) )
% 11.90/6.00 @ M
% 11.90/6.00 @ ( zip @ A @ B @ Ks2 @ Vs2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3887_enumerate__eq__zip,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( enumerate @ A )
% 11.90/6.00 = ( ^ [N5: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N5 @ ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3888_map__zip__map,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys2: list @ C] :
% 11.90/6.00 ( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys2 ) )
% 11.90/6.00 = ( map @ ( product_prod @ D @ C ) @ A
% 11.90/6.00 @ ( product_case_prod @ D @ C @ A
% 11.90/6.00 @ ^ [X3: D,Y3: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X3 ) @ Y3 ) ) )
% 11.90/6.00 @ ( zip @ D @ C @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3889_map__zip__map2,axiom,
% 11.90/6.00 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys2: list @ D] :
% 11.90/6.00 ( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys2 ) ) )
% 11.90/6.00 = ( map @ ( product_prod @ B @ D ) @ A
% 11.90/6.00 @ ( product_case_prod @ B @ D @ A
% 11.90/6.00 @ ^ [X3: B,Y3: D] : ( F @ ( product_Pair @ B @ C @ X3 @ ( G @ Y3 ) ) ) )
% 11.90/6.00 @ ( zip @ B @ D @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3890_foldl__snd__zip,axiom,
% 11.90/6.00 ! [B: $tType,C: $tType,A: $tType,Ys2: list @ A,Xs: list @ B,F: C > A > C,B2: C] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ B ) @ Xs ) )
% 11.90/6.00 => ( ( foldl @ C @ ( product_prod @ B @ A )
% 11.90/6.00 @ ^ [B5: C] :
% 11.90/6.00 ( product_case_prod @ B @ A @ C
% 11.90/6.00 @ ^ [X3: B] : ( F @ B5 ) )
% 11.90/6.00 @ B2
% 11.90/6.00 @ ( zip @ B @ A @ Xs @ Ys2 ) )
% 11.90/6.00 = ( foldl @ C @ A @ F @ B2 @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3891_map__zip2,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,K: A,L: list @ B] :
% 11.90/6.00 ( ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K ) @ L )
% 11.90/6.00 = ( zip @ A @ B @ ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) @ L ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3892_map__zip1,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,K: B,L: list @ A] :
% 11.90/6.00 ( ( map @ A @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ K )
% 11.90/6.00 @ L )
% 11.90/6.00 = ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3893_map__prod__fun__zip,axiom,
% 11.90/6.00 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,Xs: list @ C,Ys2: list @ D] :
% 11.90/6.00 ( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: C,Y3: D] : ( product_Pair @ A @ B @ ( F @ X3 ) @ ( G @ Y3 ) ) )
% 11.90/6.00 @ ( zip @ C @ D @ Xs @ Ys2 ) )
% 11.90/6.00 = ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ ( map @ D @ B @ G @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3894_zip__map2,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F: C > B,Ys2: list @ C] :
% 11.90/6.00 ( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F @ Ys2 ) )
% 11.90/6.00 = ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: A,Y3: C] : ( product_Pair @ A @ B @ X3 @ ( F @ Y3 ) ) )
% 11.90/6.00 @ ( zip @ A @ C @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3895_zip__map1,axiom,
% 11.90/6.00 ! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: list @ C,Ys2: list @ B] :
% 11.90/6.00 ( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ Ys2 )
% 11.90/6.00 = ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
% 11.90/6.00 @ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: C] : ( product_Pair @ A @ B @ ( F @ X3 ) ) )
% 11.90/6.00 @ ( zip @ C @ B @ Xs @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3896_map__of__zip__map,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B] :
% 11.90/6.00 ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F @ Xs ) ) )
% 11.90/6.00 = ( ^ [X3: A] : ( if @ ( option @ B ) @ ( member @ A @ X3 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F @ X3 ) ) @ ( none @ B ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3897_map__of__zip__upd,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Ys2: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y: B,Z2: B] :
% 11.90/6.00 ( ( ( size_size @ ( list @ B ) @ Ys2 )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( ( size_size @ ( list @ B ) @ Zs )
% 11.90/6.00 = ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.00 => ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y ) )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z2 ) ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 11.90/6.00 = ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3898_map__of__zip__nth,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( distinct @ A @ Xs )
% 11.90/6.00 => ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.00 => ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
% 11.90/6.00 = ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3899_foldr__snd__zip,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,C: $tType,Ys2: list @ A,Xs: list @ B,F: A > C > C,B2: C] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ B ) @ Xs ) )
% 11.90/6.00 => ( ( foldr @ ( product_prod @ B @ A ) @ C
% 11.90/6.00 @ ( product_case_prod @ B @ A @ ( C > C )
% 11.90/6.00 @ ^ [X3: B] : F )
% 11.90/6.00 @ ( zip @ B @ A @ Xs @ Ys2 )
% 11.90/6.00 @ B2 )
% 11.90/6.00 = ( foldr @ A @ C @ F @ Ys2 @ B2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3900_nths__shift__lemma,axiom,
% 11.90/6.00 ! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
% 11.90/6.00 ( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 11.90/6.00 @ ( filter @ ( product_prod @ A @ nat )
% 11.90/6.00 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A4 )
% 11.90/6.00 @ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
% 11.90/6.00 = ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
% 11.90/6.00 @ ( filter @ ( product_prod @ A @ nat )
% 11.90/6.00 @ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A4 )
% 11.90/6.00 @ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3901_remove__rev__alt__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( remove_rev @ A )
% 11.90/6.00 = ( ^ [X3: A,Xs3: list @ A] :
% 11.90/6.00 ( filter @ A
% 11.90/6.00 @ ^ [Y3: A] : ( Y3 != X3 )
% 11.90/6.00 @ ( rev @ A @ Xs3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3902_zip__Cons1,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.00 ( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys2 )
% 11.90/6.00 = ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ^ [Y3: B,Ys: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( zip @ A @ B @ Xs @ Ys ) )
% 11.90/6.00 @ Ys2 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3903_zip__Cons,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys2: list @ B] :
% 11.90/6.00 ( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys2 ) )
% 11.90/6.00 = ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
% 11.90/6.00 @ ^ [Z5: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z5 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys2 ) )
% 11.90/6.00 @ Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3904_filter__rev__alt,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( filter_rev @ A )
% 11.90/6.00 = ( ^ [P4: A > $o,L3: list @ A] : ( filter @ A @ P4 @ ( rev @ A @ L3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3905_remove__rev__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( remove_rev @ A )
% 11.90/6.00 = ( ^ [X3: A] :
% 11.90/6.00 ( filter_rev @ A
% 11.90/6.00 @ ( comp @ $o @ $o @ A @ (~)
% 11.90/6.00 @ ( ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.90/6.00 @ X3 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3906_min__list_Opelims,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( ord @ A )
% 11.90/6.00 => ! [X: list @ A,Y: A] :
% 11.90/6.00 ( ( ( min_list @ A @ X )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( case_list @ A @ A @ X2
% 11.90/6.00 @ ^ [A7: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs2 ) )
% 11.90/6.00 @ Xs2 ) )
% 11.90/6.00 => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 11.90/6.00 => ~ ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( undefined @ A ) )
% 11.90/6.00 => ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3907_merge__list_Opsimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 = L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3908_merge__list_Opsimps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [La: list @ A,Acc22: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3909_merge__list_Opsimps_I4_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [La: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc22 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3910_merge__list_Opsimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 = ( nil @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3911_merge__list_Opelims,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
% 11.90/6.00 ( ( ( merge_list @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ Xa ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ! [L5: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ( Y = L5 )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 11.90/6.00 => ! [L5: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
% 11.90/6.00 => ~ ! [L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls2 ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3912_merge__list_Opsimps_I5_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) ) )
% 11.90/6.00 => ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) )
% 11.90/6.00 = ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3913_partition__filter__conv,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( partition @ A )
% 11.90/6.00 = ( ^ [F2: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter @ A @ F2 @ Xs3 ) @ ( filter @ A @ ( comp @ $o @ $o @ A @ (~) @ F2 ) @ Xs3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3914_partition__rev__filter__conv,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,Xs: list @ A] :
% 11.90/6.00 ( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ Xs )
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( rev @ A @ ( filter @ A @ P @ Xs ) ) @ Yes2 ) @ ( append @ A @ ( rev @ A @ ( filter @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs ) ) @ No2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3915_merge_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L2: list @ A] :
% 11.90/6.00 ( ( merge @ A @ ( nil @ A ) @ L2 )
% 11.90/6.00 = L2 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3916_merge_Osimps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [X1: A,X22: A,L1: list @ A,L2: list @ A] :
% 11.90/6.00 ( ( ( ord_less @ A @ X1 @ X22 )
% 11.90/6.00 => ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 11.90/6.00 = ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X22 @ L2 ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less @ A @ X1 @ X22 )
% 11.90/6.00 => ( ( ( X1 = X22 )
% 11.90/6.00 => ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 11.90/6.00 = ( cons @ A @ X1 @ ( merge @ A @ L1 @ L2 ) ) ) )
% 11.90/6.00 & ( ( X1 != X22 )
% 11.90/6.00 => ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X22 @ L2 ) )
% 11.90/6.00 = ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3917_merge_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [V: A,Va2: list @ A] :
% 11.90/6.00 ( ( merge @ A @ ( cons @ A @ V @ Va2 ) @ ( nil @ A ) )
% 11.90/6.00 = ( cons @ A @ V @ Va2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3918_partition__rev_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) ) @ Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3919_partition__rev_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A] :
% 11.90/6.00 ( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( nil @ A ) )
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3920_merge__list_Osimps_I5_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls: list @ ( list @ A )] :
% 11.90/6.00 ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls ) ) )
% 11.90/6.00 = ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3921_partition_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o] :
% 11.90/6.00 ( ( partition @ A @ P @ ( nil @ A ) )
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3922_partition__P,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
% 11.90/6.00 ( ( ( partition @ A @ P @ Xs )
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
% 11.90/6.00 => ( ! [X7: A] :
% 11.90/6.00 ( ( member @ A @ X7 @ ( set2 @ A @ Yes2 ) )
% 11.90/6.00 => ( P @ X7 ) )
% 11.90/6.00 & ! [X7: A] :
% 11.90/6.00 ( ( member @ A @ X7 @ ( set2 @ A @ No2 ) )
% 11.90/6.00 => ~ ( P @ X7 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3923_merge_Oelims,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [X: list @ A,Xa: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( merge @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Y != Xa ) )
% 11.90/6.00 => ( ! [V4: A,Va: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( cons @ A @ V4 @ Va ) ) ) )
% 11.90/6.00 => ~ ! [X13: A,L12: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X13 @ L12 ) )
% 11.90/6.00 => ! [X23: A,L22: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X23 @ L22 ) )
% 11.90/6.00 => ~ ( ( ( ord_less @ A @ X13 @ X23 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less @ A @ X13 @ X23 )
% 11.90/6.00 => ( ( ( X13 = X23 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ L22 ) ) ) )
% 11.90/6.00 & ( ( X13 != X23 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X13 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3924_partition_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( partition @ A @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
% 11.90/6.00 @ ^ [Yes3: list @ A,No3: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes3 ) @ No3 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes3 @ ( cons @ A @ X @ No3 ) ) )
% 11.90/6.00 @ ( partition @ A @ P @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3925_partition__rev_Oelims,axiom,
% 11.90/6.00 ! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/6.00 ( ( ( partition_rev @ A @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ! [Yes: list @ A,No: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 11.90/6.00 => ( ( Xb
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) ) ) )
% 11.90/6.00 => ~ ! [Yes: list @ A,No: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 11.90/6.00 => ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3926_partition__set,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,Xs: list @ A,Yes2: list @ A,No2: list @ A] :
% 11.90/6.00 ( ( ( partition @ A @ P @ Xs )
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) )
% 11.90/6.00 => ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes2 ) @ ( set2 @ A @ No2 ) )
% 11.90/6.00 = ( set2 @ A @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3927_inv__image__partition,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,P: A > $o,Ys2: list @ A] :
% 11.90/6.00 ( ! [X2: A] :
% 11.90/6.00 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.90/6.00 => ( P @ X2 ) )
% 11.90/6.00 => ( ! [Y4: A] :
% 11.90/6.00 ( ( member @ A @ Y4 @ ( set2 @ A @ Ys2 ) )
% 11.90/6.00 => ~ ( P @ Y4 ) )
% 11.90/6.00 => ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
% 11.90/6.00 = ( shuffles @ A @ Xs @ Ys2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3928_merge__list_Oelims,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
% 11.90/6.00 ( ( ( merge_list @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( nil @ A ) ) ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ! [L5: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( Y != L5 ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 11.90/6.00 => ! [L5: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) ) ) ) )
% 11.90/6.00 => ~ ! [L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls2 ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3929_merge__correct,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [L1: list @ A,L2: list @ A] :
% 11.90/6.00 ( ( ( distinct @ A @ L1 )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L1 ) )
% 11.90/6.00 => ( ( ( distinct @ A @ L2 )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L2 ) )
% 11.90/6.00 => ( ( distinct @ A @ ( merge @ A @ L1 @ L2 ) )
% 11.90/6.00 & ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge @ A @ L1 @ L2 ) )
% 11.90/6.00 & ( ( set2 @ A @ ( merge @ A @ L1 @ L2 ) )
% 11.90/6.00 = ( sup_sup @ ( set @ A ) @ ( set2 @ A @ L1 ) @ ( set2 @ A @ L2 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3930_merge__list_Opinduct,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [A0: list @ ( list @ A ),A1: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ A0 @ A1 ) )
% 11.90/6.00 => ( ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ! [L5: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) )
% 11.90/6.00 => ( P @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ! [La2: list @ A,Acc23: list @ ( list @ A ),L5: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L5 @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) ) )
% 11.90/6.00 => ( P @ ( cons @ ( list @ A ) @ La2 @ Acc23 ) @ ( cons @ ( list @ A ) @ L5 @ ( nil @ ( list @ A ) ) ) ) ) )
% 11.90/6.00 => ( ! [Acc23: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls2: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) )
% 11.90/6.00 => ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ Acc23 ) @ Ls2 )
% 11.90/6.00 => ( P @ Acc23 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls2 ) ) ) ) )
% 11.90/6.00 => ( P @ A0 @ A1 ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3931_partition__rev_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
% 11.90/6.00 ( ( ( partition_rev @ A @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ Xa @ Xb ) ) )
% 11.90/6.00 => ( ! [Yes: list @ A,No: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 11.90/6.00 => ( ( Xb
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) ) ) ) )
% 11.90/6.00 => ~ ! [Yes: list @ A,No: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
% 11.90/6.00 => ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( partition_rev @ A @ X @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ ( cons @ A @ X2 @ No ) ) ) @ Xs2 ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) ) @ ( partition_rev_rel @ A ) @ ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3932_merge_Opelims,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [X: list @ A,Xa: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( merge @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y = Xa )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 11.90/6.00 => ( ! [V4: A,Va: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ A ) ) ) ) ) )
% 11.90/6.00 => ~ ! [X13: A,L12: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ X13 @ L12 ) )
% 11.90/6.00 => ! [X23: A,L22: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ X23 @ L22 ) )
% 11.90/6.00 => ( ( ( ( ord_less @ A @ X13 @ X23 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
% 11.90/6.00 & ( ~ ( ord_less @ A @ X13 @ X23 )
% 11.90/6.00 => ( ( ( X13 = X23 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X13 @ ( merge @ A @ L12 @ L22 ) ) ) )
% 11.90/6.00 & ( ( X13 != X23 )
% 11.90/6.00 => ( Y
% 11.90/6.00 = ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X13 @ L12 ) @ L22 ) ) ) ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3933_transpose_Opinduct,axiom,
% 11.90/6.00 ! [A: $tType,A0: list @ ( list @ A ),P: ( list @ ( list @ A ) ) > $o] :
% 11.90/6.00 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
% 11.90/6.00 => ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( P @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 => ( ! [Xss2: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
% 11.90/6.00 => ( ( P @ Xss2 )
% 11.90/6.00 => ( P @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
% 11.90/6.00 => ( ! [X2: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) )
% 11.90/6.00 => ( ( P
% 11.90/6.00 @ ( cons @ ( list @ A ) @ Xs2
% 11.90/6.00 @ ( concat @ ( list @ A )
% 11.90/6.00 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 11.90/6.00 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
% 11.90/6.00 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 @ Xss2 ) ) ) )
% 11.90/6.00 => ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) ) ) )
% 11.90/6.00 => ( P @ A0 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3934_quicksort__by__rel_Oelims,axiom,
% 11.90/6.00 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( Xb
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Y != Xa ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 11.90/6.00 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
% 11.90/6.00 @ ( partition_rev @ A
% 11.90/6.00 @ ^ [Y3: A] : ( X @ Y3 @ X2 )
% 11.90/6.00 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 11.90/6.00 @ Xs2 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3935_set__quicksort__by__rel,axiom,
% 11.90/6.00 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,Xs: list @ A] :
% 11.90/6.00 ( ( set2 @ A @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs ) )
% 11.90/6.00 = ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3936_quicksort__by__rel_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,R5: A > A > $o,Sl2: list @ A] :
% 11.90/6.00 ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( nil @ A ) )
% 11.90/6.00 = Sl2 ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3937_quicksort__by__rel__remove__acc,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( quicksort_by_rel @ A )
% 11.90/6.00 = ( ^ [R6: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R6 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3938_quicksort__by__rel__remove__acc__guared,axiom,
% 11.90/6.00 ! [A: $tType,Sl2: list @ A,R5: A > A > $o,Xs: list @ A] :
% 11.90/6.00 ( ( Sl2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs )
% 11.90/6.00 = ( append @ A @ ( quicksort_by_rel @ A @ R5 @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3939_sorted__wrt__quicksort__by__rel,axiom,
% 11.90/6.00 ! [X14: $tType,R5: X14 > X14 > $o,Xs: list @ X14] :
% 11.90/6.00 ( ! [X2: X14,Y4: X14] :
% 11.90/6.00 ( ( R5 @ X2 @ Y4 )
% 11.90/6.00 | ( R5 @ Y4 @ X2 ) )
% 11.90/6.00 => ( ! [X2: X14,Y4: X14,Z3: X14] :
% 11.90/6.00 ( ( R5 @ X2 @ Y4 )
% 11.90/6.00 => ( ( R5 @ Y4 @ Z3 )
% 11.90/6.00 => ( R5 @ X2 @ Z3 ) ) )
% 11.90/6.00 => ( sorted_wrt @ X14 @ R5 @ ( quicksort_by_rel @ X14 @ R5 @ ( nil @ X14 ) @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3940_sorted__quicksort__by__rel,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( linorder @ A )
% 11.90/6.00 => ! [Xs: list @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3941_quicksort__by__rel_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 11.90/6.00 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs_b ) ) @ Xs_s )
% 11.90/6.00 @ ( partition_rev @ A
% 11.90/6.00 @ ^ [Y3: A] : ( R5 @ Y3 @ X )
% 11.90/6.00 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 11.90/6.00 @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3942_transpose_Opsimps_I3_J,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
% 11.90/6.00 => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
% 11.90/6.00 = ( cons @ ( list @ A )
% 11.90/6.00 @ ( cons @ A @ X
% 11.90/6.00 @ ( concat @ A
% 11.90/6.00 @ ( map @ ( list @ A ) @ ( list @ A )
% 11.90/6.00 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
% 11.90/6.00 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
% 11.90/6.00 @ Xss ) ) )
% 11.90/6.00 @ ( transpose @ A
% 11.90/6.00 @ ( cons @ ( list @ A ) @ Xs
% 11.90/6.00 @ ( concat @ ( list @ A )
% 11.90/6.00 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 11.90/6.00 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
% 11.90/6.00 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 @ Xss ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3943_transpose_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
% 11.90/6.00 ( ( ( transpose @ A @ X )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
% 11.90/6.00 => ( ! [Xss2: list @ ( list @ A )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( transpose @ A @ Xss2 ) )
% 11.90/6.00 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( cons @ ( list @ A )
% 11.90/6.00 @ ( cons @ A @ X2
% 11.90/6.00 @ ( concat @ A
% 11.90/6.00 @ ( map @ ( list @ A ) @ ( list @ A )
% 11.90/6.00 @ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
% 11.90/6.00 @ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
% 11.90/6.00 @ Xss2 ) ) )
% 11.90/6.00 @ ( transpose @ A
% 11.90/6.00 @ ( cons @ ( list @ A ) @ Xs2
% 11.90/6.00 @ ( concat @ ( list @ A )
% 11.90/6.00 @ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
% 11.90/6.00 @ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
% 11.90/6.00 @ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
% 11.90/6.00 @ Xss2 ) ) ) ) ) )
% 11.90/6.00 => ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3944_quicksort__by__rel_Opsimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X @ Xs ) ) ) )
% 11.90/6.00 => ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 11.90/6.00 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R5 @ ( cons @ A @ X @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs_b ) ) @ Xs_s )
% 11.90/6.00 @ ( partition_rev @ A
% 11.90/6.00 @ ^ [Y3: A] : ( R5 @ Y3 @ X )
% 11.90/6.00 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 11.90/6.00 @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3945_quicksort__by__rel_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( quicksort_by_rel @ A @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
% 11.90/6.00 => ( ( ( Xb
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y = Xa )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
% 11.90/6.00 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
% 11.90/6.00 @ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ X @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ X @ Xa @ Xs_b ) ) @ Xs_s )
% 11.90/6.00 @ ( partition_rev @ A
% 11.90/6.00 @ ^ [Y3: A] : ( X @ Y3 @ X2 )
% 11.90/6.00 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 11.90/6.00 @ Xs2 ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3946_transpose_Opsimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 => ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
% 11.90/6.00 = ( nil @ ( list @ A ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3947_transpose_Opsimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,Xss: list @ ( list @ A )] :
% 11.90/6.00 ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 11.90/6.00 => ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
% 11.90/6.00 = ( transpose @ A @ Xss ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3948_quicksort__by__rel_Opsimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,R5: A > A > $o,Sl2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
% 11.90/6.00 => ( ( quicksort_by_rel @ A @ R5 @ Sl2 @ ( nil @ A ) )
% 11.90/6.00 = Sl2 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3949_quicksort__by__rel_Opinduct,axiom,
% 11.90/6.00 ! [A: $tType,A0: A > A > $o,A1: list @ A,A22: list @ A,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ A0 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A1 @ A22 ) ) )
% 11.90/6.00 => ( ! [R8: A > A > $o,Sl: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
% 11.90/6.00 => ( P @ R8 @ Sl @ ( nil @ A ) ) )
% 11.90/6.00 => ( ! [R8: A > A > $o,Sl: list @ A,X2: A,Xs2: list @ A] :
% 11.90/6.00 ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R8 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 11.90/6.00 => ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
% 11.90/6.00 ( ( Xa2
% 11.90/6.00 = ( partition_rev @ A
% 11.90/6.00 @ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
% 11.90/6.00 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 11.90/6.00 @ Xs2 ) )
% 11.90/6.00 => ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
% 11.90/6.00 = Xa2 )
% 11.90/6.00 => ( P @ R8 @ Sl @ Y6 ) ) )
% 11.90/6.00 => ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y6: list @ A] :
% 11.90/6.00 ( ( Xa2
% 11.90/6.00 = ( partition_rev @ A
% 11.90/6.00 @ ^ [Z5: A] : ( R8 @ Z5 @ X2 )
% 11.90/6.00 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
% 11.90/6.00 @ Xs2 ) )
% 11.90/6.00 => ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y6 )
% 11.90/6.00 = Xa2 )
% 11.90/6.00 => ( P @ R8 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R8 @ Sl @ Y6 ) ) @ Xb2 ) ) )
% 11.90/6.00 => ( P @ R8 @ Sl @ ( cons @ A @ X2 @ Xs2 ) ) ) ) )
% 11.90/6.00 => ( P @ A0 @ A1 @ A22 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3950_filter__rev__aux__alt,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( filter_rev_aux @ A )
% 11.90/6.00 = ( ^ [A7: list @ A,P4: A > $o,L3: list @ A] : ( append @ A @ ( filter @ A @ P4 @ ( rev @ A @ L3 ) ) @ A7 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3951_lexord__take__index__conv,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 11.90/6.00 = ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
% 11.90/6.00 & ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
% 11.90/6.00 = X ) )
% 11.90/6.00 | ? [I3: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ I3 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
% 11.90/6.00 & ( ( take @ A @ I3 @ X )
% 11.90/6.00 = ( take @ A @ I3 @ Y ) )
% 11.90/6.00 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I3 ) @ ( nth @ A @ Y @ I3 ) ) @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3952_last__take__nth__conv,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,L: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( N
% 11.90/6.00 != ( zero_zero @ nat ) )
% 11.90/6.00 => ( ( last @ A @ ( take @ A @ N @ L ) )
% 11.90/6.00 = ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3953_revg__fun,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( revg @ A )
% 11.90/6.00 = ( ^ [A7: list @ A] : ( append @ A @ ( rev @ A @ A7 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3954_take__upt,axiom,
% 11.90/6.00 ! [I: nat,M2: nat,N: nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M2 ) @ N )
% 11.90/6.00 => ( ( take @ nat @ M2 @ ( upt @ I @ N ) )
% 11.90/6.00 = ( upt @ I @ ( plus_plus @ nat @ I @ M2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3955_take__Cons__numeral,axiom,
% 11.90/6.00 ! [A: $tType,V: num,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3956_take__drop,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
% 11.90/6.00 ( ( take @ A @ N @ ( drop @ A @ M2 @ Xs ) )
% 11.90/6.00 = ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3957_revg_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,A3: A,As2: list @ A,B2: list @ A] :
% 11.90/6.00 ( ( revg @ A @ ( cons @ A @ A3 @ As2 ) @ B2 )
% 11.90/6.00 = ( revg @ A @ As2 @ ( cons @ A @ A3 @ B2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3958_revg_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,B2: list @ A] :
% 11.90/6.00 ( ( revg @ A @ ( nil @ A ) @ B2 )
% 11.90/6.00 = B2 ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3959_filter__rev__aux_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,X: A,A3: list @ A,Xs: list @ A] :
% 11.90/6.00 ( ( ( P @ X )
% 11.90/6.00 => ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( filter_rev_aux @ A @ ( cons @ A @ X @ A3 ) @ P @ Xs ) ) )
% 11.90/6.00 & ( ~ ( P @ X )
% 11.90/6.00 => ( ( filter_rev_aux @ A @ A3 @ P @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( filter_rev_aux @ A @ A3 @ P @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3960_filter__rev__aux_Osimps_I1_J,axiom,
% 11.90/6.00 ! [A: $tType,A3: list @ A,P: A > $o] :
% 11.90/6.00 ( ( filter_rev_aux @ A @ A3 @ P @ ( nil @ A ) )
% 11.90/6.00 = A3 ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3961_drop__take__drop__unsplit,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,J: nat,L: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/6.00 => ( ( append @ A @ ( drop @ A @ I @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
% 11.90/6.00 = ( drop @ A @ I @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3962_take__add,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,J: nat,Xs: list @ A] :
% 11.90/6.00 ( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
% 11.90/6.00 = ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3963_slice__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( slice @ A )
% 11.90/6.00 = ( ^ [From2: nat,To2: nat,List2: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List2 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3964_filter__upt__take__conv,axiom,
% 11.90/6.00 ! [A: $tType,P: A > $o,M2: nat,L: list @ A,N: nat] :
% 11.90/6.00 ( ( filter @ nat
% 11.90/6.00 @ ^ [I3: nat] : ( P @ ( nth @ A @ ( take @ A @ M2 @ L ) @ I3 ) )
% 11.90/6.00 @ ( upt @ N @ M2 ) )
% 11.90/6.00 = ( filter @ nat
% 11.90/6.00 @ ^ [I3: nat] : ( P @ ( nth @ A @ L @ I3 ) )
% 11.90/6.00 @ ( upt @ N @ M2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3965_zip__replicate1,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,N: nat,X: A,Ys2: list @ B] :
% 11.90/6.00 ( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys2 )
% 11.90/6.00 = ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3966_revg_Oelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( revg @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( Y != Xa ) )
% 11.90/6.00 => ~ ! [A5: A,As: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ A5 @ As ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( revg @ A @ As @ ( cons @ A @ A5 @ Xa ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3967_zip__replicate2,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
% 11.90/6.00 ( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
% 11.90/6.00 = ( map @ A @ ( product_prod @ A @ B )
% 11.90/6.00 @ ^ [X3: A] : ( product_Pair @ A @ B @ X3 @ Y )
% 11.90/6.00 @ ( take @ A @ N @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3968_take__Cons_H,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( N
% 11.90/6.00 = ( zero_zero @ nat ) )
% 11.90/6.00 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( nil @ A ) ) )
% 11.90/6.00 & ( ( N
% 11.90/6.00 != ( zero_zero @ nat ) )
% 11.90/6.00 => ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3969_filter__nth__ex__nth,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,P: A > $o,Xs: list @ A] :
% 11.90/6.00 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) )
% 11.90/6.00 => ? [M5: nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ N @ M5 )
% 11.90/6.00 & ( ord_less @ nat @ M5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 & ( ( nth @ A @ ( filter @ A @ P @ Xs ) @ N )
% 11.90/6.00 = ( nth @ A @ Xs @ M5 ) )
% 11.90/6.00 & ( ( filter @ A @ P @ ( take @ A @ M5 @ Xs ) )
% 11.90/6.00 = ( take @ A @ N @ ( filter @ A @ P @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3970_map__upd__upds__conv__if,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A,Ys2: list @ B,Xs: list @ A,F: A > ( option @ B ),Y: B] :
% 11.90/6.00 ( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
% 11.90/6.00 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y ) ) @ Xs @ Ys2 )
% 11.90/6.00 = ( map_upds @ A @ B @ F @ Xs @ Ys2 ) ) )
% 11.90/6.00 & ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
% 11.90/6.00 => ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y ) ) @ Xs @ Ys2 )
% 11.90/6.00 = ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F @ Xs @ Ys2 ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3971_Union__take__drop__id,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,L: list @ ( set @ A )] :
% 11.90/6.00 ( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
% 11.90/6.00 = ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3972_lex__take__index,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
% 11.90/6.00 => ~ ! [I2: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys2 ) )
% 11.90/6.00 => ( ( ( take @ A @ I2 @ Xs )
% 11.90/6.00 = ( take @ A @ I2 @ Ys2 ) )
% 11.90/6.00 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Ys2 @ I2 ) ) @ R ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3973_map__nth__upt__drop__take__conv,axiom,
% 11.90/6.00 ! [A: $tType,N6: nat,L: list @ A,M3: nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ N6 @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M3 @ N6 ) )
% 11.90/6.00 = ( drop @ A @ M3 @ ( take @ A @ N6 @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3974_filter__rev__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( filter_rev @ A )
% 11.90/6.00 = ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3975_revg_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
% 11.90/6.00 ( ( ( revg @ A @ X @ Xa )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.00 => ( ( ( X
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y = Xa )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
% 11.90/6.00 => ~ ! [A5: A,As: list @ A] :
% 11.90/6.00 ( ( X
% 11.90/6.00 = ( cons @ A @ A5 @ As ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( revg @ A @ As @ ( cons @ A @ A5 @ Xa ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A5 @ As ) @ Xa ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3976_take__butlast__conv,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A] :
% 11.90/6.00 ( ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
% 11.90/6.00 = ( butlast @ A @ L ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3977_take__minus__one__conv__butlast,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,L: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( take @ A @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
% 11.90/6.00 = ( butlast @ A @ ( take @ A @ N @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3978_take__update__last,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,List: list @ A,X: A] :
% 11.90/6.00 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
% 11.90/6.00 => ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X )
% 11.90/6.00 = ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3979_take__update,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,L: list @ A,I: nat,X: A] :
% 11.90/6.00 ( ( take @ A @ N @ ( list_update @ A @ L @ I @ X ) )
% 11.90/6.00 = ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3980_drop__upd__irrelevant,axiom,
% 11.90/6.00 ! [A: $tType,M2: nat,N: nat,L: list @ A,X: A] :
% 11.90/6.00 ( ( ord_less @ nat @ M2 @ N )
% 11.90/6.00 => ( ( drop @ A @ N @ ( list_update @ A @ L @ M2 @ X ) )
% 11.90/6.00 = ( drop @ A @ N @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3981_nth__update__invalid,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,L: list @ A,J: nat,X: A] :
% 11.90/6.00 ( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( nth @ A @ ( list_update @ A @ L @ J @ X ) @ I )
% 11.90/6.00 = ( nth @ A @ L @ I ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3982_length__butlast,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A] :
% 11.90/6.00 ( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
% 11.90/6.00 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3983_butlast__upd__last__eq,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A,X: A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( list_update @ A @ ( butlast @ A @ L ) @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ X )
% 11.90/6.00 = ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3984_butlast__update_H,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A,I: nat,X: A] :
% 11.90/6.00 ( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X )
% 11.90/6.00 = ( butlast @ A @ ( list_update @ A @ L @ I @ X ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3985_distinct__butlast__swap,axiom,
% 11.90/6.00 ! [A: $tType,Pq: list @ A,I: nat] :
% 11.90/6.00 ( ( distinct @ A @ Pq )
% 11.90/6.00 => ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I @ ( last @ A @ Pq ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3986_zip__update,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,Xs: list @ A,I: nat,X: A,Ys2: list @ B,Y: B] :
% 11.90/6.00 ( ( zip @ A @ B @ ( list_update @ A @ Xs @ I @ X ) @ ( list_update @ B @ Ys2 @ I @ Y ) )
% 11.90/6.00 = ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3987_butlast__list__update,axiom,
% 11.90/6.00 ! [A: $tType,K: nat,Xs: list @ A,X: A] :
% 11.90/6.00 ( ( ( K
% 11.90/6.00 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 11.90/6.00 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 11.90/6.00 = ( butlast @ A @ Xs ) ) )
% 11.90/6.00 & ( ( K
% 11.90/6.00 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 11.90/6.00 => ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 11.90/6.00 = ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3988_in__set__upd__eq__aux,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
% 11.90/6.00 = ( ( X = Y )
% 11.90/6.00 | ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3989_in__set__upd__cases,axiom,
% 11.90/6.00 ! [A: $tType,X: A,L: list @ A,I: nat,Y: A] :
% 11.90/6.00 ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
% 11.90/6.00 => ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( X != Y ) )
% 11.90/6.00 => ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3990_in__set__upd__eq,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,L: list @ A,X: A,Y: A] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
% 11.90/6.00 = ( ( X = Y )
% 11.90/6.00 | ( ( member @ A @ X @ ( set2 @ A @ L ) )
% 11.90/6.00 & ! [Y3: A] : ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y3 ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3991_nth__list__update_H,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,J: nat,L: list @ A,X: A] :
% 11.90/6.00 ( ( ( ( I = J )
% 11.90/6.00 & ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
% 11.90/6.00 => ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
% 11.90/6.00 = X ) )
% 11.90/6.00 & ( ~ ( ( I = J )
% 11.90/6.00 & ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
% 11.90/6.00 => ( ( nth @ A @ ( list_update @ A @ L @ I @ X ) @ J )
% 11.90/6.00 = ( nth @ A @ L @ J ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3992_butlast__subset,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,A4: set @ A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
% 11.90/6.00 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( butlast @ A @ Xs ) ) @ A4 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3993_butlast__eq__cons__conv,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( butlast @ A @ L )
% 11.90/6.00 = ( cons @ A @ X @ Xs ) )
% 11.90/6.00 = ( ? [Xl: A] :
% 11.90/6.00 ( L
% 11.90/6.00 = ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3994_butlast__eq__consE,axiom,
% 11.90/6.00 ! [A: $tType,L: list @ A,X: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( butlast @ A @ L )
% 11.90/6.00 = ( cons @ A @ X @ Xs ) )
% 11.90/6.00 => ~ ! [Xl2: A] :
% 11.90/6.00 ( L
% 11.90/6.00 != ( cons @ A @ X @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3995_butlast__upt,axiom,
% 11.90/6.00 ! [M2: nat,N: nat] :
% 11.90/6.00 ( ( butlast @ nat @ ( upt @ M2 @ N ) )
% 11.90/6.00 = ( upt @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3996_map__upd__eq,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: A] :
% 11.90/6.00 ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( F @ ( nth @ A @ L @ I ) )
% 11.90/6.00 = ( F @ X ) ) )
% 11.90/6.00 => ( ( map @ A @ B @ F @ ( list_update @ A @ L @ I @ X ) )
% 11.90/6.00 = ( map @ A @ B @ F @ L ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3997_snoc__eq__iff__butlast_H,axiom,
% 11.90/6.00 ! [A: $tType,Ys2: list @ A,Xs: list @ A,X: A] :
% 11.90/6.00 ( ( Ys2
% 11.90/6.00 = ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
% 11.90/6.00 = ( ( Ys2
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 & ( ( butlast @ A @ Ys2 )
% 11.90/6.00 = Xs )
% 11.90/6.00 & ( ( last @ A @ Ys2 )
% 11.90/6.00 = X ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3998_butlast__conv__take,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( butlast @ A )
% 11.90/6.00 = ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_3999_insert__swap__set__eq,axiom,
% 11.90/6.00 ! [A: $tType,I: nat,L: list @ A,X: A] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( insert2 @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X ) ) )
% 11.90/6.00 = ( insert2 @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4000_rev__update,axiom,
% 11.90/6.00 ! [A: $tType,K: nat,Xs: list @ A,Y: A] :
% 11.90/6.00 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y ) )
% 11.90/6.00 = ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4001_last__list__update,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,K: nat,X: A] :
% 11.90/6.00 ( ( Xs
% 11.90/6.00 != ( nil @ A ) )
% 11.90/6.00 => ( ( ( K
% 11.90/6.00 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 11.90/6.00 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 11.90/6.00 = X ) )
% 11.90/6.00 & ( ( K
% 11.90/6.00 != ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
% 11.90/6.00 => ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 11.90/6.00 = ( last @ A @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4002_butlast__take,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,Xs: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
% 11.90/6.00 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4003_listrel1__iff__update,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.00 = ( ? [Y3: A,N5: nat] :
% 11.90/6.00 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N5 ) @ Y3 ) @ R )
% 11.90/6.00 & ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 & ( Ys2
% 11.90/6.00 = ( list_update @ A @ Xs @ N5 @ Y3 ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4004_map__distinct__upd__conv,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X: B] :
% 11.90/6.00 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.00 => ( ( distinct @ A @ L )
% 11.90/6.00 => ( ( list_update @ B @ ( map @ A @ B @ F @ L ) @ I @ X )
% 11.90/6.00 = ( map @ A @ B @ ( fun_upd @ A @ B @ F @ ( nth @ A @ L @ I ) @ X ) @ L ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4005_foldl__list__update,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: B > A > B,A3: B,X: A] :
% 11.90/6.00 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( foldl @ B @ A @ F @ A3 @ ( list_update @ A @ Xs @ N @ X ) )
% 11.90/6.00 = ( foldl @ B @ A @ F @ ( F @ ( foldl @ B @ A @ F @ A3 @ ( take @ A @ N @ Xs ) ) @ X ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4006_case__natural__def,axiom,
% 11.90/6.00 ! [T: $tType] :
% 11.90/6.00 ( ( code_case_natural @ T )
% 11.90/6.00 = ( ^ [F12: T,F23: code_natural > T] :
% 11.90/6.00 ( code_rec_natural @ T @ F12
% 11.90/6.00 @ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4007_nth__rotate,axiom,
% 11.90/6.00 ! [A: $tType,N: nat,Xs: list @ A,M2: nat] :
% 11.90/6.00 ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.00 => ( ( nth @ A @ ( rotate @ A @ M2 @ Xs ) @ N )
% 11.90/6.00 = ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4008_list_Osize__gen_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
% 11.90/6.00 ( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
% 11.90/6.00 = ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4009_rotate1__length01,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 11.90/6.00 => ( ( rotate1 @ A @ Xs )
% 11.90/6.00 = Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4010_natural_Osimps_I6_J,axiom,
% 11.90/6.00 ! [T: $tType,F1: T,F22: code_natural > T > T] :
% 11.90/6.00 ( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
% 11.90/6.00 = F1 ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4011_size__list__append,axiom,
% 11.90/6.00 ! [A: $tType,F: A > nat,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( size_list @ A @ F @ ( append @ A @ Xs @ Ys2 ) )
% 11.90/6.00 = ( plus_plus @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ F @ Ys2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4012_rotate__length01,axiom,
% 11.90/6.00 ! [A: $tType,Xs: list @ A,N: nat] :
% 11.90/6.00 ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
% 11.90/6.00 => ( ( rotate @ A @ N @ Xs )
% 11.90/6.00 = Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4013_rotate__rotate,axiom,
% 11.90/6.00 ! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
% 11.90/6.00 ( ( rotate @ A @ M2 @ ( rotate @ A @ N @ Xs ) )
% 11.90/6.00 = ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) @ Xs ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4014_rotate__add,axiom,
% 11.90/6.00 ! [A: $tType,M2: nat,N: nat] :
% 11.90/6.00 ( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) )
% 11.90/6.00 = ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4015_rec__natural__def,axiom,
% 11.90/6.00 ! [T: $tType] :
% 11.90/6.00 ( ( code_rec_natural @ T )
% 11.90/6.00 = ( ^ [F12: T,F23: code_natural > T > T,X3: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4016_lenlex__append2,axiom,
% 11.90/6.00 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.00 ( ( irrefl @ A @ R5 )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys2 ) ) @ ( lenlex @ A @ R5 ) )
% 11.90/6.00 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4017_zipf__zip,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,L1: list @ A,L2: list @ B] :
% 11.90/6.00 ( ( ( size_size @ ( list @ A ) @ L1 )
% 11.90/6.00 = ( size_size @ ( list @ B ) @ L2 ) )
% 11.90/6.00 => ( ( zipf @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ L1 @ L2 )
% 11.90/6.00 = ( zip @ A @ B @ L1 @ L2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4018_map__tailrec__rev_Opelims,axiom,
% 11.90/6.00 ! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
% 11.90/6.00 ( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 11.90/6.00 => ( ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Y = Xb )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
% 11.90/6.00 => ~ ! [A5: A,As: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ A5 @ As ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A5 ) @ Xb ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4019_lexord__same__pref__if__irrefl,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
% 11.90/6.00 ( ( irrefl @ A @ R )
% 11.90/6.00 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
% 11.90/6.00 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4020_irreflI,axiom,
% 11.90/6.00 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ( ! [A5: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A5 ) @ R5 )
% 11.90/6.00 => ( irrefl @ A @ R5 ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4021_irrefl__def,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( irrefl @ A )
% 11.90/6.00 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.00 ! [A7: A] :
% 11.90/6.00 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ A7 ) @ R3 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4022_zipf_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,C: $tType,B: $tType,F: A > B > C,A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
% 11.90/6.00 ( ( zipf @ A @ B @ C @ F @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
% 11.90/6.00 = ( cons @ C @ ( F @ A3 @ B2 ) @ ( zipf @ A @ B @ C @ F @ As2 @ Bs2 ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4023_zipf_Osimps_I1_J,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,C: $tType,F: A > B > C] :
% 11.90/6.00 ( ( zipf @ A @ B @ C @ F @ ( nil @ A ) @ ( nil @ B ) )
% 11.90/6.00 = ( nil @ C ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4024_lexl__not__refl,axiom,
% 11.90/6.00 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A] :
% 11.90/6.00 ( ( irrefl @ A @ R )
% 11.90/6.00 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4025_zipf_Oelims,axiom,
% 11.90/6.00 ! [B: $tType,A: $tType,C: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
% 11.90/6.00 ( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Xb
% 11.90/6.00 = ( nil @ B ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( nil @ C ) ) ) )
% 11.90/6.00 => ( ! [A5: A,As: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ A5 @ As ) )
% 11.90/6.00 => ! [B3: B,Bs: list @ B] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ B @ B3 @ Bs ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( cons @ C @ ( X @ A5 @ B3 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs ) ) ) ) )
% 11.90/6.00 => ( ( ? [V4: A,Va: list @ A] :
% 11.90/6.00 ( Xa
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ( ( Xb
% 11.90/6.00 = ( nil @ B ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( undefined @ ( list @ C ) ) ) ) )
% 11.90/6.00 => ~ ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ? [V4: B,Va: list @ B] :
% 11.90/6.00 ( Xb
% 11.90/6.00 = ( cons @ B @ V4 @ Va ) )
% 11.90/6.00 => ( Y
% 11.90/6.00 != ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4026_zipf_Opelims,axiom,
% 11.90/6.00 ! [C: $tType,A: $tType,B: $tType,X: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
% 11.90/6.00 ( ( ( zipf @ A @ B @ C @ X @ Xa @ Xb )
% 11.90/6.00 = Y )
% 11.90/6.00 => ( ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 11.90/6.00 => ( ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ( ( Xb
% 11.90/6.00 = ( nil @ B ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( nil @ C ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
% 11.90/6.00 => ( ! [A5: A,As: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ A5 @ As ) )
% 11.90/6.00 => ! [B3: B,Bs: list @ B] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ B @ B3 @ Bs ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( cons @ C @ ( X @ A5 @ B3 ) @ ( zipf @ A @ B @ C @ X @ As @ Bs ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) ) ) ) )
% 11.90/6.00 => ( ! [V4: A,Va: list @ A] :
% 11.90/6.00 ( ( Xa
% 11.90/6.00 = ( cons @ A @ V4 @ Va ) )
% 11.90/6.00 => ( ( Xb
% 11.90/6.00 = ( nil @ B ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( undefined @ ( list @ C ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
% 11.90/6.00 => ~ ( ( Xa
% 11.90/6.00 = ( nil @ A ) )
% 11.90/6.00 => ! [V4: B,Va: list @ B] :
% 11.90/6.00 ( ( Xb
% 11.90/6.00 = ( cons @ B @ V4 @ Va ) )
% 11.90/6.00 => ( ( Y
% 11.90/6.00 = ( undefined @ ( list @ C ) ) )
% 11.90/6.00 => ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4027_count__list_Osimps_I2_J,axiom,
% 11.90/6.00 ! [A: $tType,X: A,Y: A,Xs: list @ A] :
% 11.90/6.00 ( ( ( X = Y )
% 11.90/6.00 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
% 11.90/6.00 = ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
% 11.90/6.00 & ( ( X != Y )
% 11.90/6.00 => ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
% 11.90/6.00 = ( count_list @ A @ Xs @ Y ) ) ) ) ).
% 11.90/6.00
% 11.90/6.00 thf(fact_4028_effect__makeE,axiom,
% 11.90/6.00 ! [A: $tType] :
% 11.90/6.00 ( ( heap @ A )
% 11.90/6.00 => ! [N: nat,F: nat > A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 11.90/6.00 ( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H3 @ H4 @ R @ N2 )
% 11.90/6.00 => ~ ( ( R
% 11.90/6.00 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H3 ) ) )
% 11.90/6.01 => ( ( ( array_get @ A @ H4 @ R )
% 11.90/6.01 = ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
% 11.90/6.01 => ( ( array_present @ A @ H4 @ R )
% 11.90/6.01 => ( ~ ( array_present @ A @ H3 @ R )
% 11.90/6.01 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 11.90/6.01 != N2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4029_listrel__iff__nth,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 = ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.01 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.01 & ! [N5: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N5 ) @ ( nth @ B @ Ys2 @ N5 ) ) @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4030_listrel__rtrancl__refl,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4031_Array__Time_Oget__alloc,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [Xs: list @ A,H3: heap_ext @ product_unit,Ys2: list @ A] :
% 11.90/6.01 ( ( array_get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H3 ) ) )
% 11.90/6.01 = Xs ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4032_Array__Time_Opresent__alloc,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [Xs: list @ A,H3: heap_ext @ product_unit] : ( array_present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4033_Array__Time_Onot__present__alloc,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H3: heap_ext @ product_unit,Xs: list @ A] :
% 11.90/6.01 ~ ( array_present @ A @ H3 @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4034_listrel_ONil,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4035_listrel__Nil1,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ( Xs
% 11.90/6.01 = ( nil @ B ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4036_listrel__Nil2,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ( Xs
% 11.90/6.01 = ( nil @ A ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4037_listrel__eq__len,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.01 = ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4038_listrel__rtrancl__trans,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Zs: list @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4039_listrel_OCons,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,X: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys2: list @ B] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4040_listrel__Cons1,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,Y: A,Ys2: list @ A,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys2 ) @ Xs ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ~ ! [Y4: B,Ys3: list @ B] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 = ( cons @ B @ Y4 @ Ys3 ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y4 ) @ R )
% 11.90/6.01 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys2 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4041_listrel__Cons2,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys2 ) ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
% 11.90/6.01 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4042_listrel__reflcl__if__listrel1,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4043_rtrancl__listrel1__if__listrel,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel @ A @ A @ R ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4044_effect__of__listE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [Xs: list @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H3 @ H4 @ R @ N2 )
% 11.90/6.01 => ~ ( ( R
% 11.90/6.01 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) )
% 11.90/6.01 => ( ( ( array_get @ A @ H4 @ R )
% 11.90/6.01 = Xs )
% 11.90/6.01 => ( ( array_present @ A @ H4 @ R )
% 11.90/6.01 => ( ~ ( array_present @ A @ H3 @ R )
% 11.90/6.01 => ( N2
% 11.90/6.01 != ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4045_effect__newE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [N: nat,X: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N2: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H3 @ H4 @ R @ N2 )
% 11.90/6.01 => ~ ( ( R
% 11.90/6.01 = ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H3 ) ) )
% 11.90/6.01 => ( ( ( array_get @ A @ H4 @ R )
% 11.90/6.01 = ( replicate @ A @ N @ X ) )
% 11.90/6.01 => ( ( array_present @ A @ H4 @ R )
% 11.90/6.01 => ( ~ ( array_present @ A @ H3 @ R )
% 11.90/6.01 => ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
% 11.90/6.01 != N2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4046_listrel_Ocases,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 => ( ( ( A1
% 11.90/6.01 = ( nil @ A ) )
% 11.90/6.01 => ( A22
% 11.90/6.01 != ( nil @ B ) ) )
% 11.90/6.01 => ~ ! [X2: A,Y4: B,Xs2: list @ A] :
% 11.90/6.01 ( ( A1
% 11.90/6.01 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.01 => ! [Ys3: list @ B] :
% 11.90/6.01 ( ( A22
% 11.90/6.01 = ( cons @ B @ Y4 @ Ys3 ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 11.90/6.01 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4047_listrel_Osimps,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 = ( ( ( A1
% 11.90/6.01 = ( nil @ A ) )
% 11.90/6.01 & ( A22
% 11.90/6.01 = ( nil @ B ) ) )
% 11.90/6.01 | ? [X3: A,Y3: B,Xs3: list @ A,Ys: list @ B] :
% 11.90/6.01 ( ( A1
% 11.90/6.01 = ( cons @ A @ X3 @ Xs3 ) )
% 11.90/6.01 & ( A22
% 11.90/6.01 = ( cons @ B @ Y3 @ Ys ) )
% 11.90/6.01 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 11.90/6.01 & ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4048_listrel__iff__zip,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
% 11.90/6.01 = ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.01 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.01 & ! [X3: product_prod @ A @ B] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
% 11.90/6.01 => ( product_case_prod @ A @ B @ $o
% 11.90/6.01 @ ^ [Y3: A,Z5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Z5 ) @ R )
% 11.90/6.01 @ X3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4049_present__alloc__noteq,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType] :
% 11.90/6.01 ( ( ( heap @ B )
% 11.90/6.01 & ( heap @ A ) )
% 11.90/6.01 => ! [H3: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
% 11.90/6.01 ( ( array_present @ A @ H3 @ A3 )
% 11.90/6.01 => ( array_noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ B @ Xs @ H3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4050_listrel__def,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( listrel @ A @ B )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
% 11.90/6.01 @ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
% 11.90/6.01 @ ( listrelp @ A @ B
% 11.90/6.01 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4051_set,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [Ls: list @ A,H3: heap_ext @ product_unit,New_ls: list @ A] :
% 11.90/6.01 ( ( array_set @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H3 ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H3 ) ) )
% 11.90/6.01 = ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ New_ls @ H3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4052_Array__Time_Oset__same,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [R: array @ A,X: list @ A,Y: list @ A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_set @ A @ R @ X @ ( array_set @ A @ R @ Y @ H3 ) )
% 11.90/6.01 = ( array_set @ A @ R @ X @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4053_Array__Time_Ounequal,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,A6: array @ A] :
% 11.90/6.01 ( ( A3 != A6 )
% 11.90/6.01 = ( array_noteq @ A @ A @ A3 @ A6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4054_Array__Time_Oget__set__eq,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [R: array @ A,X: list @ A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_get @ A @ ( array_set @ A @ R @ X @ H3 ) @ R )
% 11.90/6.01 = X ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4055_Array__Time_Oget__set__neq,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [R: array @ A,S: array @ B,X: list @ B,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_noteq @ A @ B @ R @ S )
% 11.90/6.01 => ( ( array_get @ A @ ( array_set @ B @ S @ X @ H3 ) @ R )
% 11.90/6.01 = ( array_get @ A @ H3 @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4056_Array__Time_Oset__set__swap,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [R: array @ A,R4: array @ B,X: list @ A,X6: list @ B,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_noteq @ A @ B @ R @ R4 )
% 11.90/6.01 => ( ( array_set @ A @ R @ X @ ( array_set @ B @ R4 @ X6 @ H3 ) )
% 11.90/6.01 = ( array_set @ B @ R4 @ X6 @ ( array_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4057_Array__Time_Onoteq__irrefl,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [R: array @ A] :
% 11.90/6.01 ~ ( array_noteq @ A @ A @ R @ R ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4058_Array__Time_Onoteq__sym,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [A3: array @ A,B2: array @ B] :
% 11.90/6.01 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 11.90/6.01 => ( array_noteq @ B @ A @ B2 @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4059_Field__not__elem,axiom,
% 11.90/6.01 ! [A: $tType,V: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ~ ( member @ A @ V @ ( field2 @ A @ R5 ) )
% 11.90/6.01 => ! [X7: product_prod @ A @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ X7 @ R5 )
% 11.90/6.01 => ( product_case_prod @ A @ A @ $o
% 11.90/6.01 @ ^ [Y3: A,Z5: A] :
% 11.90/6.01 ( ( Y3 != V )
% 11.90/6.01 & ( Z5 != V ) )
% 11.90/6.01 @ X7 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4060_refl__on__def_H,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( refl_on @ A )
% 11.90/6.01 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ! [X3: product_prod @ A @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ X3 @ R3 )
% 11.90/6.01 => ( product_case_prod @ A @ A @ $o
% 11.90/6.01 @ ^ [Y3: A,Z5: A] :
% 11.90/6.01 ( ( member @ A @ Y3 @ A9 )
% 11.90/6.01 & ( member @ A @ Z5 @ A9 ) )
% 11.90/6.01 @ X3 ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ A9 )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4061_listrelp__listrel__eq,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( listrelp @ A @ B
% 11.90/6.01 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 11.90/6.01 = ( ^ [X3: list @ A,Y3: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X3 @ Y3 ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4062_Under__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( order_Under @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 11.90/6.01 ( collect @ A
% 11.90/6.01 @ ^ [B5: A] :
% 11.90/6.01 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ A9 )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X3 ) @ R3 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4063_UnderS__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( order_UnderS @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 11.90/6.01 ( collect @ A
% 11.90/6.01 @ ^ [B5: A] :
% 11.90/6.01 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ A9 )
% 11.90/6.01 => ( ( B5 != X3 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4064_sorted__wrt_Opelims_I2_J,axiom,
% 11.90/6.01 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 11.90/6.01 ( ( sorted_wrt @ A @ X @ Xa )
% 11.90/6.01 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.01 => ( ( ( Xa
% 11.90/6.01 = ( nil @ A ) )
% 11.90/6.01 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
% 11.90/6.01 => ~ ! [X2: A,Ys3: list @ A] :
% 11.90/6.01 ( ( Xa
% 11.90/6.01 = ( cons @ A @ X2 @ Ys3 ) )
% 11.90/6.01 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
% 11.90/6.01 => ~ ( ! [Xa2: A] :
% 11.90/6.01 ( ( member @ A @ Xa2 @ ( set2 @ A @ Ys3 ) )
% 11.90/6.01 => ( X @ X2 @ Xa2 ) )
% 11.90/6.01 & ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4065_sorted__wrt_Opelims_I1_J,axiom,
% 11.90/6.01 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
% 11.90/6.01 ( ( ( sorted_wrt @ A @ X @ Xa )
% 11.90/6.01 = Y )
% 11.90/6.01 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.01 => ( ( ( Xa
% 11.90/6.01 = ( nil @ A ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
% 11.90/6.01 => ~ ! [X2: A,Ys3: list @ A] :
% 11.90/6.01 ( ( Xa
% 11.90/6.01 = ( cons @ A @ X2 @ Ys3 ) )
% 11.90/6.01 => ( ( Y
% 11.90/6.01 = ( ! [Y3: A] :
% 11.90/6.01 ( ( member @ A @ Y3 @ ( set2 @ A @ Ys3 ) )
% 11.90/6.01 => ( X @ X2 @ Y3 ) )
% 11.90/6.01 & ( sorted_wrt @ A @ X @ Ys3 ) ) )
% 11.90/6.01 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4066_sorted__wrt_Opelims_I3_J,axiom,
% 11.90/6.01 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 11.90/6.01 ( ~ ( sorted_wrt @ A @ X @ Xa )
% 11.90/6.01 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.90/6.01 => ~ ! [X2: A,Ys3: list @ A] :
% 11.90/6.01 ( ( Xa
% 11.90/6.01 = ( cons @ A @ X2 @ Ys3 ) )
% 11.90/6.01 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ Ys3 ) ) )
% 11.90/6.01 => ( ! [Xa3: A] :
% 11.90/6.01 ( ( member @ A @ Xa3 @ ( set2 @ A @ Ys3 ) )
% 11.90/6.01 => ( X @ X2 @ Xa3 ) )
% 11.90/6.01 & ( sorted_wrt @ A @ X @ Ys3 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4067_Above__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( order_Above @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 11.90/6.01 ( collect @ A
% 11.90/6.01 @ ^ [B5: A] :
% 11.90/6.01 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ A9 )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B5 ) @ R3 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4068_execute__nth_I1_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( heap_Time_execute @ A @ ( array_nth @ A @ A3 @ I ) @ H3 )
% 11.90/6.01 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4069_Array__Time_Olength__alloc,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [Xs: list @ A,H3: heap_ext @ product_unit,Ys2: list @ A] :
% 11.90/6.01 ( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H3 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H3 ) ) )
% 11.90/6.01 = ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4070_Array__Time_Onth__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_nth @ A )
% 11.90/6.01 = ( ^ [A7: array @ A,I3: nat] :
% 11.90/6.01 ( heap_Time_guard @ A
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4071_success__nthI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( heap_Time_success @ A @ ( array_nth @ A @ A3 @ I ) @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4072_fails__array__nth,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [P5: array @ A,I: nat,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( time_fails @ A @ ( array_nth @ A @ P5 @ I ) @ H3 )
% 11.90/6.01 = ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ P5 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4073_execute__nth_I2_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 11.90/6.01 => ( ( heap_Time_execute @ A @ ( array_nth @ A @ A3 @ I ) @ H3 )
% 11.90/6.01 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4074_length__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_length @ A )
% 11.90/6.01 = ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H @ A7 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4075_effect__nthE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,I: nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ A @ ( array_nth @ A @ A3 @ I ) @ H3 @ H4 @ R @ N )
% 11.90/6.01 => ~ ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( R
% 11.90/6.01 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 11.90/6.01 => ( ( H4 = H3 )
% 11.90/6.01 => ( N
% 11.90/6.01 != ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4076_effect__nthI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( H4 = H3 )
% 11.90/6.01 => ( ( R
% 11.90/6.01 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 11.90/6.01 => ( ( N
% 11.90/6.01 = ( one_one @ nat ) )
% 11.90/6.01 => ( heap_Time_effect @ A @ ( array_nth @ A @ A3 @ I ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4077_nth_H__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_nth2 @ A )
% 11.90/6.01 = ( ^ [A7: array @ A] : ( comp @ nat @ ( heap_Time_Heap @ A ) @ code_integer @ ( array_nth @ A @ A7 ) @ code_nat_of_integer ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4078_freeze__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_freeze @ A )
% 11.90/6.01 = ( ^ [A7: array @ A] :
% 11.90/6.01 ( heap_Time_heap @ ( list @ A )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H @ A7 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H @ A7 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4079_execute__map__entry_I2_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,F: A > A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 11.90/6.01 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 )
% 11.90/6.01 = ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4080_execute__freeze,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( heap_Time_execute @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 )
% 11.90/6.01 = ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H3 @ A3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4081_success__freezeI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4082_success__map__entryI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,F: A > A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( heap_Time_success @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4083_effect__freezeE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: list @ A,N: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 @ H4 @ R @ N )
% 11.90/6.01 => ~ ( ( H4 = H3 )
% 11.90/6.01 => ( ( R
% 11.90/6.01 = ( array_get @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( N
% 11.90/6.01 != ( plus_plus @ nat @ ( array_length @ A @ H3 @ A3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4084_effect__freezeI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: list @ A,A3: array @ A,N: nat] :
% 11.90/6.01 ( ( H4 = H3 )
% 11.90/6.01 => ( ( R
% 11.90/6.01 = ( array_get @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( N
% 11.90/6.01 = ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A3 ) @ H3 @ H4 @ R @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4085_execute__map__entry_I1_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,F: A > A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 )
% 11.90/6.01 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4086_map__entry__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_map_entry @ A )
% 11.90/6.01 = ( ^ [I3: nat,F2: A > A,A7: array @ A] :
% 11.90/6.01 ( heap_Time_guard @ ( array @ A )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) ) @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4087_effect__map__entryI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,F: A > A,R: array @ A,N: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) )
% 11.90/6.01 => ( ( R = A3 )
% 11.90/6.01 => ( ( N
% 11.90/6.01 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/6.01 => ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4088_effect__map__entryE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,F: A > A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A3 ) @ H3 @ H4 @ R @ N )
% 11.90/6.01 => ~ ( ( R = A3 )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( array_update @ A @ A3 @ I @ ( F @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) @ H3 ) )
% 11.90/6.01 => ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( N
% 11.90/6.01 != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4089_length__update,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [B2: array @ B,I: nat,V: B,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_length @ A @ ( array_update @ B @ B2 @ I @ V @ H3 ) )
% 11.90/6.01 = ( array_length @ A @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4090_Array__Time_Opresent__update,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [B2: array @ B,I: nat,V: B,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_present @ A @ ( array_update @ B @ B2 @ I @ V @ H3 ) )
% 11.90/6.01 = ( array_present @ A @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4091_get__update__elem__neqIndex,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,J: nat,A3: array @ A,V: A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( I != J )
% 11.90/6.01 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A3 @ J @ V @ H3 ) @ A3 ) @ I )
% 11.90/6.01 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4092_get__update__eq,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,I: nat,V: A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_get @ A @ ( array_update @ A @ A3 @ I @ V @ H3 ) @ A3 )
% 11.90/6.01 = ( list_update @ A @ ( array_get @ A @ H3 @ A3 ) @ I @ V ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4093_nth__update__neq,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [A3: array @ A,B2: array @ B,J: nat,V: B,H3: heap_ext @ product_unit,I: nat] :
% 11.90/6.01 ( ( array_noteq @ A @ B @ A3 @ B2 )
% 11.90/6.01 => ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B2 @ J @ V @ H3 ) @ A3 ) @ I )
% 11.90/6.01 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4094_update__swap__neqIndex,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,I7: nat,A3: array @ A,V: A,V7: A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( I != I7 )
% 11.90/6.01 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ A @ A3 @ I7 @ V7 @ H3 ) )
% 11.90/6.01 = ( array_update @ A @ A3 @ I7 @ V7 @ ( array_update @ A @ A3 @ I @ V @ H3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4095_update__swap__neq,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( ( heap @ A )
% 11.90/6.01 & ( heap @ B ) )
% 11.90/6.01 => ! [A3: array @ A,A6: array @ B,I: nat,V: A,I7: nat,V7: B,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( array_noteq @ A @ B @ A3 @ A6 )
% 11.90/6.01 => ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ B @ A6 @ I7 @ V7 @ H3 ) )
% 11.90/6.01 = ( array_update @ B @ A6 @ I7 @ V7 @ ( array_update @ A @ A3 @ I @ V @ H3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4096_Array__Time_Oupdate__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_update @ A )
% 11.90/6.01 = ( ^ [A7: array @ A,I3: nat,X3: A,H: heap_ext @ product_unit] : ( array_set @ A @ A7 @ ( list_update @ A @ ( array_get @ A @ H @ A7 ) @ I3 @ X3 ) @ H ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4097_execute__swap_I1_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 )
% 11.90/6.01 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ X @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4098_swap__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_swap @ A )
% 11.90/6.01 = ( ^ [I3: nat,X3: A,A7: array @ A] :
% 11.90/6.01 ( heap_Time_guard @ A
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A7 ) @ I3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ X3 @ H ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4099_effect__swapI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,X: A,R: A,N: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( array_update @ A @ A3 @ I @ X @ H3 ) )
% 11.90/6.01 => ( ( R
% 11.90/6.01 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 11.90/6.01 => ( ( N
% 11.90/6.01 = ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/6.01 => ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4100_effect__swapE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,X: A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 @ H4 @ R @ N )
% 11.90/6.01 => ~ ( ( R
% 11.90/6.01 = ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( array_update @ A @ A3 @ I @ X @ H3 ) )
% 11.90/6.01 => ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( N
% 11.90/6.01 != ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4101_success__swapI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( heap_Time_success @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4102_execute__swap_I2_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,X: A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 11.90/6.01 => ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X @ A3 ) @ H3 )
% 11.90/6.01 = ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4103_execute__upd_I1_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 )
% 11.90/6.01 = ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I @ X @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4104_upd__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_upd @ A )
% 11.90/6.01 = ( ^ [I3: nat,X3: A,A7: array @ A] :
% 11.90/6.01 ( heap_Time_guard @ ( array @ A )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( ord_less @ nat @ I3 @ ( array_length @ A @ H @ A7 ) )
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A7 @ I3 @ X3 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4105_execute__upd_I2_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H3: heap_ext @ product_unit,A3: array @ A,I: nat,X: A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( array_length @ A @ H3 @ A3 ) @ I )
% 11.90/6.01 => ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 )
% 11.90/6.01 = ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4106_effect__updI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,H4: heap_ext @ product_unit,V: A,N: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( array_update @ A @ A3 @ I @ V @ H3 ) )
% 11.90/6.01 => ( ( N
% 11.90/6.01 = ( one_one @ nat ) )
% 11.90/6.01 => ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V @ A3 ) @ H3 @ H4 @ A3 @ N ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4107_upd__ureturn,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,X: A,A3: array @ A] :
% 11.90/6.01 ( ( heap_Time_bind @ ( array @ A ) @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 )
% 11.90/6.01 @ ^ [Uu: array @ A] : ( heap_Time_ureturn @ ( array @ A ) @ A3 ) )
% 11.90/6.01 = ( array_upd @ A @ I @ X @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4108_success__updI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,H3: heap_ext @ product_unit,A3: array @ A,X: A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( heap_Time_success @ ( array @ A ) @ ( array_upd @ A @ I @ X @ A3 ) @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4109_fails__array__upd,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,X: A,P5: array @ A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X @ P5 ) @ H3 )
% 11.90/6.01 = ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ P5 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4110_effect__updE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [I: nat,V: A,A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V @ A3 ) @ H3 @ H4 @ R @ N )
% 11.90/6.01 => ~ ( ( R = A3 )
% 11.90/6.01 => ( ( H4
% 11.90/6.01 = ( array_update @ A @ A3 @ I @ V @ H3 ) )
% 11.90/6.01 => ( ( ord_less @ nat @ I @ ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( N
% 11.90/6.01 != ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4111_upd_H__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_upd2 @ A )
% 11.90/6.01 = ( ^ [A7: array @ A,I3: code_integer,X3: A] :
% 11.90/6.01 ( heap_Time_bind @ ( array @ A ) @ product_unit @ ( array_upd @ A @ ( code_nat_of_integer @ I3 ) @ X3 @ A7 )
% 11.90/6.01 @ ^ [Uu: array @ A] : ( heap_Time_ureturn @ product_unit @ product_Unity ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4112_execute__len,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ( ( heap_Time_execute @ nat @ ( array_len @ A @ A3 ) @ H3 )
% 11.90/6.01 = ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4113_in__measures_I2_J,axiom,
% 11.90/6.01 ! [A: $tType,X: A,Y: A,F: A > nat,Fs: list @ ( A > nat )] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
% 11.90/6.01 = ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 11.90/6.01 | ( ( ( F @ X )
% 11.90/6.01 = ( F @ Y ) )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4114_mset__mergesort__by__rel__split,axiom,
% 11.90/6.01 ! [A: $tType,Xs1: list @ A,Xs22: list @ A,Xs: list @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) @ ( mset @ A @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ Xs ) ) ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Xs1 ) ) @ ( mset @ A @ Xs22 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4115_union__eq__empty,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N6 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ( M3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4116_empty__eq__union,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( ( zero_zero @ ( multiset @ A ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( ( M3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4117_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 11.90/6.01 ( ( ( zero_zero @ ( multiset @ A ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ X @ Y ) )
% 11.90/6.01 = ( ( X
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( Y
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4118_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ( X
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( Y
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4119_diff__diff__add__mset,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,P: multiset @ A] :
% 11.90/6.01 ( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) @ P )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ P ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4120_mergesort__by__rel__permutes,axiom,
% 11.90/6.01 ! [A: $tType,R5: A > A > $o,Xs: list @ A] :
% 11.90/6.01 ( ( mset @ A @ ( mergesort_by_rel @ A @ R5 @ Xs ) )
% 11.90/6.01 = ( mset @ A @ Xs ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4121_in__measures_I1_J,axiom,
% 11.90/6.01 ! [A: $tType,X: A,Y: A] :
% 11.90/6.01 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4122_mset__append,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.01 ( ( mset @ A @ ( append @ A @ Xs @ Ys2 ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4123_quicksort__by__rel__permutes,axiom,
% 11.90/6.01 ! [A: $tType,R5: A > A > $o,Sl2: list @ A,Xs: list @ A] :
% 11.90/6.01 ( ( mset @ A @ ( quicksort_by_rel @ A @ R5 @ Sl2 @ Xs ) )
% 11.90/6.01 = ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4124_mset__mergesort__by__rel__merge,axiom,
% 11.90/6.01 ! [A: $tType,R5: A > A > $o,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.01 ( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R5 @ Xs @ Ys2 ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4125_success__lenI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,H3: heap_ext @ product_unit] : ( heap_Time_success @ nat @ ( array_len @ A @ A3 ) @ H3 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4126_fails__array__len,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [P5: array @ A,H3: heap_ext @ product_unit] :
% 11.90/6.01 ~ ( time_fails @ nat @ ( array_len @ A @ P5 ) @ H3 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4127_mset__distrib,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ A4 @ B6 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 => ~ ! [Am: multiset @ A,An: multiset @ A] :
% 11.90/6.01 ( ( A4
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
% 11.90/6.01 => ! [Bm: multiset @ A,Bn: multiset @ A] :
% 11.90/6.01 ( ( B6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
% 11.90/6.01 => ( ( M3
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
% 11.90/6.01 => ( N6
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4128_mset__shuffles,axiom,
% 11.90/6.01 ! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
% 11.90/6.01 ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
% 11.90/6.01 => ( ( mset @ A @ Zs )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4129_union__less__mono,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( preorder @ A )
% 11.90/6.01 => ! [A4: multiset @ A,C5: multiset @ A,B6: multiset @ A,D4: multiset @ A] :
% 11.90/6.01 ( ( ord_less @ ( multiset @ A ) @ A4 @ C5 )
% 11.90/6.01 => ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
% 11.90/6.01 => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4130_union__le__mono2,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( preorder @ A )
% 11.90/6.01 => ! [B6: multiset @ A,D4: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
% 11.90/6.01 => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4131_union__le__mono1,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( preorder @ A )
% 11.90/6.01 => ! [B6: multiset @ A,D4: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( ord_less @ ( multiset @ A ) @ B6 @ D4 )
% 11.90/6.01 => ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ D4 @ C5 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4132_multi__union__self__other__eq,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,X10: multiset @ A,Y9: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ A4 @ X10 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ A4 @ Y9 ) )
% 11.90/6.01 => ( X10 = Y9 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4133_union__right__cancel,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,K2: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ M3 @ K2 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) )
% 11.90/6.01 = ( M3 = N6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4134_union__left__cancel,axiom,
% 11.90/6.01 ! [A: $tType,K2: multiset @ A,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ K2 @ M3 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ K2 @ N6 ) )
% 11.90/6.01 = ( M3 = N6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4135_union__commute,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) )
% 11.90/6.01 = ( ^ [M8: multiset @ A,N9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N9 @ M8 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4136_union__lcomm,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,K2: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ N6 @ ( plus_plus @ ( multiset @ A ) @ M3 @ K2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4137_union__assoc,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,K2: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ K2 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ K2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4138_empty__neutral_I1_J,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 11.90/6.01 = X ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4139_empty__neutral_I2_J,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = X ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4140_diff__union__cancelR,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ N6 )
% 11.90/6.01 = M3 ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4141_diff__union__cancelL,axiom,
% 11.90/6.01 ! [A: $tType,N6: multiset @ A,M3: multiset @ A] :
% 11.90/6.01 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N6 @ M3 ) @ N6 )
% 11.90/6.01 = M3 ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4142_Multiset_Odiff__add,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,Q: multiset @ A] :
% 11.90/6.01 ( ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N6 @ Q ) )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) @ Q ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4143_union__diff__assoc,axiom,
% 11.90/6.01 ! [A: $tType,C5: multiset @ A,B6: multiset @ A,A4: multiset @ A] :
% 11.90/6.01 ( ( ( minus_minus @ ( multiset @ A ) @ C5 @ B6 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ A4 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4144_comp__fun__commute__plus__mset,axiom,
% 11.90/6.01 ! [A: $tType] : ( finite6289374366891150609ommute @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4145_len__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ( ( array_len @ A )
% 11.90/6.01 = ( ^ [A7: array @ A] :
% 11.90/6.01 ( heap_Time_tap @ nat
% 11.90/6.01 @ ^ [H: heap_ext @ product_unit] : ( array_length @ A @ H @ A7 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4146_measures__less,axiom,
% 11.90/6.01 ! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
% 11.90/6.01 ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4147_measures__lesseq,axiom,
% 11.90/6.01 ! [A: $tType,F: A > nat,X: A,Y: A,Fs: list @ ( A > nat )] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ Fs ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4148_effect__lengthE,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [A3: array @ A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: nat,N: nat] :
% 11.90/6.01 ( ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H3 @ H4 @ R @ N )
% 11.90/6.01 => ~ ( ( R
% 11.90/6.01 = ( array_length @ A @ H4 @ A3 ) )
% 11.90/6.01 => ( ( H4 = H3 )
% 11.90/6.01 => ( N
% 11.90/6.01 != ( one_one @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4149_effect__lengthI,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( heap @ A )
% 11.90/6.01 => ! [H4: heap_ext @ product_unit,H3: heap_ext @ product_unit,R: nat,A3: array @ A,N: nat] :
% 11.90/6.01 ( ( H4 = H3 )
% 11.90/6.01 => ( ( R
% 11.90/6.01 = ( array_length @ A @ H3 @ A3 ) )
% 11.90/6.01 => ( ( N
% 11.90/6.01 = ( one_one @ nat ) )
% 11.90/6.01 => ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H3 @ H4 @ R @ N ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4150_len_H__def,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType] :
% 11.90/6.01 ( ( ( heap @ B )
% 11.90/6.01 & ( semiring_1 @ A ) )
% 11.90/6.01 => ( ( array_len2 @ B @ A )
% 11.90/6.01 = ( ^ [A7: array @ B] :
% 11.90/6.01 ( heap_Time_bind @ nat @ A @ ( array_len @ B @ A7 )
% 11.90/6.01 @ ^ [N5: nat] : ( heap_Time_ureturn @ A @ ( semiring_1_of_nat @ A @ N5 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4151_mset__zip__take__Cons__drop__twice,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,J: nat,X: A,Y: B] :
% 11.90/6.01 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.01 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.01 => ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.01 => ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys2 ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys2 ) ) ) ) )
% 11.90/6.01 = ( add_mset @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4152_image__mset__map__of,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
% 11.90/6.01 ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
% 11.90/6.01 => ( ( image_mset @ A @ B
% 11.90/6.01 @ ^ [I3: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I3 ) )
% 11.90/6.01 @ ( mset @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
% 11.90/6.01 = ( mset @ B @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4153_mset__set__Union,axiom,
% 11.90/6.01 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/6.01 ( ( finite_finite2 @ A @ A4 )
% 11.90/6.01 => ( ( finite_finite2 @ A @ B6 )
% 11.90/6.01 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/6.01 = ( bot_bot @ ( set @ A ) ) )
% 11.90/6.01 => ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A4 ) @ ( mset_set @ A @ B6 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4154_union__mset__add__mset__right,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,A3: A,B6: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ A3 @ B6 ) )
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4155_union__mset__add__mset__left,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ A4 ) @ B6 )
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4156_image__mset__union,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,F: B > A,M3: multiset @ B,N6: multiset @ B] :
% 11.90/6.01 ( ( image_mset @ B @ A @ F @ ( plus_plus @ ( multiset @ B ) @ M3 @ N6 ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M3 ) @ ( image_mset @ B @ A @ F @ N6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4157_mset__map__split__orig,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M13: multiset @ A,M24: multiset @ A] :
% 11.90/6.01 ( ( ( image_mset @ B @ A @ F @ P )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M13 @ M24 ) )
% 11.90/6.01 => ~ ! [P1: multiset @ B,P22: multiset @ B] :
% 11.90/6.01 ( ( P
% 11.90/6.01 = ( plus_plus @ ( multiset @ B ) @ P1 @ P22 ) )
% 11.90/6.01 => ( ( ( image_mset @ B @ A @ F @ P1 )
% 11.90/6.01 = M13 )
% 11.90/6.01 => ( ( image_mset @ B @ A @ F @ P22 )
% 11.90/6.01 != M24 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4158_add__mset__add__single,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( add_mset @ A )
% 11.90/6.01 = ( ^ [A7: A,A9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A9 @ ( add_mset @ A @ A7 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4159_union__is__single,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,A3: A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N6 )
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = ( ( ( M3
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 | ( ( M3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4160_single__is__union,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = M3 )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 | ( ( M3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = N6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4161_add__mset__replicate__mset__safe,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,M3: multiset @ B,A3: B] :
% 11.90/6.01 ( ( nO_MATCH @ ( multiset @ A ) @ ( multiset @ B ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
% 11.90/6.01 => ( ( add_mset @ B @ A3 @ M3 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ B ) @ ( add_mset @ B @ A3 @ ( zero_zero @ ( multiset @ B ) ) ) @ M3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4162_mset__single__cases,axiom,
% 11.90/6.01 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ S @ C2 )
% 11.90/6.01 = ( add_mset @ A @ R4 @ C6 ) )
% 11.90/6.01 => ( ( ( S = R4 )
% 11.90/6.01 => ( C2 != C6 ) )
% 11.90/6.01 => ~ ( ( C6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( ( C2
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 != ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4163_mset__single__cases_H,axiom,
% 11.90/6.01 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ S @ C2 )
% 11.90/6.01 = ( add_mset @ A @ R4 @ C6 ) )
% 11.90/6.01 => ( ( ( S = R4 )
% 11.90/6.01 => ( C2 != C6 ) )
% 11.90/6.01 => ~ ! [Cc: multiset @ A] :
% 11.90/6.01 ( ( C6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
% 11.90/6.01 => ( ( C2
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
% 11.90/6.01 => ( ( ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = Cc )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 != Cc ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4164_mset__single__cases2,axiom,
% 11.90/6.01 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ S @ C2 )
% 11.90/6.01 = ( add_mset @ A @ R4 @ C6 ) )
% 11.90/6.01 => ( ( ( S = R4 )
% 11.90/6.01 => ( C2 != C6 ) )
% 11.90/6.01 => ~ ( ( C6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 => ( ( C2
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 != ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4165_mset__single__cases2_H,axiom,
% 11.90/6.01 ! [A: $tType,S: A,C2: multiset @ A,R4: A,C6: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ S @ C2 )
% 11.90/6.01 = ( add_mset @ A @ R4 @ C6 ) )
% 11.90/6.01 => ( ( ( S = R4 )
% 11.90/6.01 => ( C2 != C6 ) )
% 11.90/6.01 => ~ ! [Cc: multiset @ A] :
% 11.90/6.01 ( ( C6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 => ( ( C2
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 => ( ( ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = Cc )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 != Cc ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4166_mset__unplusm__dist__cases,axiom,
% 11.90/6.01 ! [A: $tType,S: A,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 11.90/6.01 => ( ( ( B6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( A4
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
% 11.90/6.01 => ~ ( ( C5
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( A4
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4167_mset__unplusm__dist__cases2,axiom,
% 11.90/6.01 ! [A: $tType,B6: multiset @ A,C5: multiset @ A,S: A,A4: multiset @ A] :
% 11.90/6.01 ( ( ( plus_plus @ ( multiset @ A ) @ B6 @ C5 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 ) )
% 11.90/6.01 => ( ( ( B6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( A4
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
% 11.90/6.01 => ~ ( ( C5
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( A4
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ S @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4168_image__mset_Ocompositionality,axiom,
% 11.90/6.01 ! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Multiset: multiset @ A] :
% 11.90/6.01 ( ( image_mset @ B @ C @ F @ ( image_mset @ A @ B @ G @ Multiset ) )
% 11.90/6.01 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Multiset ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4169_multiset_Omap__comp,axiom,
% 11.90/6.01 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: multiset @ A] :
% 11.90/6.01 ( ( image_mset @ B @ C @ G @ ( image_mset @ A @ B @ F @ V ) )
% 11.90/6.01 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4170_image__mset_Ocomp,axiom,
% 11.90/6.01 ! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
% 11.90/6.01 ( ( comp @ ( multiset @ B ) @ ( multiset @ C ) @ ( multiset @ A ) @ ( image_mset @ B @ C @ F ) @ ( image_mset @ A @ B @ G ) )
% 11.90/6.01 = ( image_mset @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4171_mset__map__id,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,F: B > A,G: A > B,X10: multiset @ A] :
% 11.90/6.01 ( ! [X2: A] :
% 11.90/6.01 ( ( F @ ( G @ X2 ) )
% 11.90/6.01 = X2 )
% 11.90/6.01 => ( ( image_mset @ B @ A @ F @ ( image_mset @ A @ B @ G @ X10 ) )
% 11.90/6.01 = X10 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4172_mset__union__2__elem,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,B2: A,C2: A,M3: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ A3 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = ( add_mset @ A @ C2 @ M3 ) )
% 11.90/6.01 => ( ( ( ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = M3 )
% 11.90/6.01 & ( B2 = C2 ) )
% 11.90/6.01 | ( ( A3 = C2 )
% 11.90/6.01 & ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = M3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4173_wcount__add__mset,axiom,
% 11.90/6.01 ! [A: $tType,F: A > nat,X: A,M3: multiset @ A,A3: A] :
% 11.90/6.01 ( ( wcount @ A @ F @ ( add_mset @ A @ X @ M3 ) @ A3 )
% 11.90/6.01 = ( plus_plus @ nat @ ( if @ nat @ ( X = A3 ) @ ( suc @ ( F @ A3 ) ) @ ( zero_zero @ nat ) ) @ ( wcount @ A @ F @ M3 @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4174_subset__mset_Osum__list__update,axiom,
% 11.90/6.01 ! [A: $tType,K: nat,Xs: list @ ( multiset @ A ),X: multiset @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Xs ) )
% 11.90/6.01 => ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( list_update @ ( multiset @ A ) @ Xs @ K @ X ) )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X ) @ ( nth @ ( multiset @ A ) @ Xs @ K ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4175_size__multiset__add__mset,axiom,
% 11.90/6.01 ! [A: $tType,F: A > nat,A3: A,M3: multiset @ A] :
% 11.90/6.01 ( ( size_multiset @ A @ F @ ( add_mset @ A @ A3 @ M3 ) )
% 11.90/6.01 = ( plus_plus @ nat @ ( suc @ ( F @ A3 ) ) @ ( size_multiset @ A @ F @ M3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4176_subset__mset_Osum__eq__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,F6: set @ B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ F6 )
% 11.90/6.01 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ F6 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ! [X3: B] :
% 11.90/6.01 ( ( member @ B @ X3 @ F6 )
% 11.90/6.01 => ( ( F @ X3 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4177_size__multiset__union,axiom,
% 11.90/6.01 ! [A: $tType,F: A > nat,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( size_multiset @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( plus_plus @ nat @ ( size_multiset @ A @ F @ M3 ) @ ( size_multiset @ A @ F @ N6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4178_subset__mset_Osum__list__eq__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,Ns: list @ ( multiset @ A )] :
% 11.90/6.01 ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ! [X3: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Ns ) )
% 11.90/6.01 => ( X3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4179_wcount__union,axiom,
% 11.90/6.01 ! [A: $tType,F: A > nat,M3: multiset @ A,N6: multiset @ A,A3: A] :
% 11.90/6.01 ( ( wcount @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ A3 )
% 11.90/6.01 = ( plus_plus @ nat @ ( wcount @ A @ F @ M3 @ A3 ) @ ( wcount @ A @ F @ N6 @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4180_subset__mset_Oelem__le__sum__list,axiom,
% 11.90/6.01 ! [A: $tType,K: nat,Ns: list @ ( multiset @ A )] :
% 11.90/6.01 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ ( multiset @ A ) ) @ Ns ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( nth @ ( multiset @ A ) @ Ns @ K ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4181_subset__mset_Osum__mono2,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,B6: set @ B,A4: set @ B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ B6 )
% 11.90/6.01 => ( ( ord_less_eq @ ( set @ B ) @ A4 @ B6 )
% 11.90/6.01 => ( ! [B3: B] :
% 11.90/6.01 ( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B6 @ A4 ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ B3 ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ B6 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4182_next_Osimps,axiom,
% 11.90/6.01 ! [V: code_natural,W: code_natural] :
% 11.90/6.01 ( ( next @ ( product_Pair @ code_natural @ code_natural @ V @ W ) )
% 11.90/6.01 = ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4183_find__dropWhile,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( find @ A )
% 11.90/6.01 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 11.90/6.01 ( case_list @ ( option @ A ) @ A @ ( none @ A )
% 11.90/6.01 @ ^ [X3: A,Xa4: list @ A] : ( some @ A @ X3 )
% 11.90/6.01 @ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4184_mset__subset__eq__mono__add__right__cancel,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,C5: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ A4 @ B6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4185_mset__subset__eq__mono__add__left__cancel,axiom,
% 11.90/6.01 ! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ A4 @ B6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4186_subset__mset_Oadd__le__cancel__right,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4187_subset__mset_Oadd__le__cancel__left,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4188_subset__mset_Ole__add__same__cancel2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4189_subset__mset_Ole__add__same__cancel1,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4190_subset__mset_Oadd__le__same__cancel2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ B2 )
% 11.90/6.01 = ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4191_subset__mset_Oadd__le__same__cancel1,axiom,
% 11.90/6.01 ! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) @ B2 )
% 11.90/6.01 = ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4192_mset__subset__eq__multiset__union__diff__commute,axiom,
% 11.90/6.01 ! [A: $tType,B6: multiset @ A,A4: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ B6 @ A4 )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4193_subset__mset_Oadd__diff__assoc2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ C2 )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4194_subset__mset_Oadd__diff__assoc,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4195_subset__mset_Odiff__add,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ A3 )
% 11.90/6.01 = B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4196_mset__le__add__mset__decr__left1,axiom,
% 11.90/6.01 ! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4197_mset__le__incr__right2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4198_mset__le__incr__right1,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4199_mset__le__decr__left2,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4200_mset__le__decr__left1,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4201_mset__union__subset,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ A4 @ C5 )
% 11.90/6.01 & ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4202_mset__le__distrib,axiom,
% 11.90/6.01 ! [A: $tType,X10: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ X10 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 11.90/6.01 => ~ ! [Xa5: multiset @ A,Xb4: multiset @ A] :
% 11.90/6.01 ( ( X10
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ Xa5 @ Xb4 ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ Xa5 @ A4 )
% 11.90/6.01 => ~ ( subseteq_mset @ A @ Xb4 @ B6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4203_mset__le__addE,axiom,
% 11.90/6.01 ! [A: $tType,Xs: multiset @ A,Ys2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ Xs @ Ys2 )
% 11.90/6.01 => ~ ! [Zs2: multiset @ A] :
% 11.90/6.01 ( Ys2
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4204_mset__subset__eq__exists__conv,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( subseteq_mset @ A )
% 11.90/6.01 = ( ^ [A9: multiset @ A,B9: multiset @ A] :
% 11.90/6.01 ? [C7: multiset @ A] :
% 11.90/6.01 ( B9
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ A9 @ C7 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4205_mset__subset__eq__add__right,axiom,
% 11.90/6.01 ! [A: $tType,B6: multiset @ A,A4: multiset @ A] : ( subseteq_mset @ A @ B6 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4206_mset__subset__eq__mono__add,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A,D4: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A4 @ B6 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ C5 @ D4 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ D4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4207_mset__subset__eq__add__left,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] : ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4208_subset__mset_Oadd__le__imp__le__right,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4209_subset__mset_Oadd__le__imp__le__left,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 11.90/6.01 => ( subseteq_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4210_subset__mset_Oadd__right__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4211_subset__mset_Oadd__left__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4212_subset__mset_Ole__iff__add,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( subseteq_mset @ A )
% 11.90/6.01 = ( ^ [A7: multiset @ A,B5: multiset @ A] :
% 11.90/6.01 ? [C3: multiset @ A] :
% 11.90/6.01 ( B5
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ A7 @ C3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4213_subset__mset_Oless__eqE,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ~ ! [C4: multiset @ A] :
% 11.90/6.01 ( B2
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4214_subset__mset_Oadd__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ C2 @ D3 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4215_full__exhaustive__natural_H_Ocases,axiom,
% 11.90/6.01 ! [X: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 11.90/6.01 ~ ! [F4: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4216_full__exhaustive__fun_H_Ocases,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType] :
% 11.90/6.01 ( ( ( quickc3360725361186068524ustive @ B )
% 11.90/6.01 & ( cl_HOL_Oequal @ A )
% 11.90/6.01 & ( quickc3360725361186068524ustive @ A ) )
% 11.90/6.01 => ! [X: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 11.90/6.01 ~ ! [F4: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4217_exhaustive__natural_H_Ocases,axiom,
% 11.90/6.01 ! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 11.90/6.01 ~ ! [F4: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I2: code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4218_exhaustive__fun_H_Ocases,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType] :
% 11.90/6.01 ( ( ( quickc658316121487927005ustive @ B )
% 11.90/6.01 & ( cl_HOL_Oequal @ A )
% 11.90/6.01 & ( quickc658316121487927005ustive @ A ) )
% 11.90/6.01 => ! [X: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
% 11.90/6.01 ~ ! [F4: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I2: code_natural,D2: code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I2 @ D2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4219_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
% 11.90/6.01 ! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
% 11.90/6.01 ~ ! [F4: code_natural > A,N3: code_natural,M5: code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4220_iter_Ocases,axiom,
% 11.90/6.01 ! [A: $tType,X: product_prod @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )] :
% 11.90/6.01 ~ ! [Random: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ),Nrandom: code_natural,Seed: product_prod @ code_natural @ code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Random @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Nrandom @ Seed ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4221_log_Ocases,axiom,
% 11.90/6.01 ! [X: product_prod @ code_natural @ code_natural] :
% 11.90/6.01 ~ ! [B3: code_natural,I2: code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ code_natural @ code_natural @ B3 @ I2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4222_mset__le__subtract,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A4 @ B6 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4223_mset__le__add__mset__decr__left2,axiom,
% 11.90/6.01 ! [A: $tType,C2: A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A3 ) @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4224_mset__le__single__cases,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,A3: A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ M3 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 => ( ( M3
% 11.90/6.01 != ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( M3
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4225_mset__le__add__mset,axiom,
% 11.90/6.01 ! [A: $tType,X: A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ C5 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ C5 )
% 11.90/6.01 & ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4226_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ( X
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( Y
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4227_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A,Y: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ Y )
% 11.90/6.01 => ( ( ( plus_plus @ ( multiset @ A ) @ X @ Y )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ( X
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( Y
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4228_subset__mset_Oadd__nonpos__nonpos,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4229_subset__mset_Oadd__nonneg__nonneg,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4230_subset__mset_Oadd__increasing2,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,B2: multiset @ A,A3: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ C2 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B2 @ A3 )
% 11.90/6.01 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4231_subset__mset_Oadd__decreasing2,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4232_subset__mset_Oadd__increasing,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 11.90/6.01 => ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4233_subset__mset_Oadd__decreasing,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ C2 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4234_mset__le__subtract__left,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,X10: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ X10 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B6 @ ( minus_minus @ ( multiset @ A ) @ X10 @ A4 ) )
% 11.90/6.01 & ( subseteq_mset @ A @ A4 @ X10 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4235_mset__le__subtract__right,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,X10: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ X10 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ A4 @ ( minus_minus @ ( multiset @ A ) @ X10 @ B6 ) )
% 11.90/6.01 & ( subseteq_mset @ A @ B6 @ X10 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4236_multiset__diff__union__assoc,axiom,
% 11.90/6.01 ! [A: $tType,C5: multiset @ A,B6: multiset @ A,A4: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ C5 @ B6 )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ A4 @ ( minus_minus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4237_subset__eq__diff__conv,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,C5: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 )
% 11.90/6.01 = ( subseteq_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4238_subset__mset_Ole__imp__diff__is__add,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( ( minus_minus @ ( multiset @ A ) @ B2 @ A3 )
% 11.90/6.01 = C2 )
% 11.90/6.01 = ( B2
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4239_subset__mset_Oadd__diff__inverse,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ A3 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 11.90/6.01 = B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4240_subset__mset_Odiff__diff__right,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4241_subset__mset_Odiff__add__assoc2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) @ C2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4242_subset__mset_Odiff__add__assoc,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A3 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4243_subset__mset_Ole__diff__conv2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A3 ) )
% 11.90/6.01 = ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4244_subset__mset_Ole__add__diff,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4245_mset__map__split__orig__le,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M13: multiset @ A,M24: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P ) @ ( plus_plus @ ( multiset @ A ) @ M13 @ M24 ) )
% 11.90/6.01 => ~ ! [P1: multiset @ B,P22: multiset @ B] :
% 11.90/6.01 ( ( P
% 11.90/6.01 = ( plus_plus @ ( multiset @ B ) @ P1 @ P22 ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P1 ) @ M13 )
% 11.90/6.01 => ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P22 ) @ M24 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4246_mset__le__subtract__add__mset__right,axiom,
% 11.90/6.01 ! [A: $tType,X: A,B6: multiset @ A,X10: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ X10 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ B6 ) )
% 11.90/6.01 & ( subseteq_mset @ A @ B6 @ X10 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4247_mset__le__subtract__add__mset__left,axiom,
% 11.90/6.01 ! [A: $tType,X: A,B6: multiset @ A,X10: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( add_mset @ A @ X @ B6 ) @ X10 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B6 @ ( minus_minus @ ( multiset @ A ) @ X10 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 & ( subseteq_mset @ A @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ X10 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4248_subset__mset_Osum__nonneg,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ! [X2: B] :
% 11.90/6.01 ( ( member @ B @ X2 @ A4 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X2 ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4249_subset__mset_Osum__nonpos,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,A4: set @ B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ! [X2: B] :
% 11.90/6.01 ( ( member @ B @ X2 @ A4 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( F @ X2 ) @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4250_subset__mset_Osum__list__nonneg,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ ( multiset @ A )] :
% 11.90/6.01 ( ! [X2: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4251_subset__mset_Osum__list__nonpos,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ ( multiset @ A )] :
% 11.90/6.01 ( ! [X2: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 11.90/6.01 => ( subseteq_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4252_subset__mset_Omember__le__sum__list,axiom,
% 11.90/6.01 ! [A: $tType,X: multiset @ A,Xs: list @ ( multiset @ A )] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 11.90/6.01 => ( subseteq_mset @ A @ X @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4253_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ ( multiset @ A )] :
% 11.90/6.01 ( ! [X2: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 ) )
% 11.90/6.01 => ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ! [X3: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
% 11.90/6.01 => ( X3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4254_subset__mset_Osum__mono,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,K2: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
% 11.90/6.01 ( ! [I2: B] :
% 11.90/6.01 ( ( member @ B @ I2 @ K2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ K2 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ K2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4255_subset__mset_Osum__le__included,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,C: $tType,S: set @ B,T2: set @ C,G: C > ( multiset @ A ),I: C > B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ S )
% 11.90/6.01 => ( ( finite_finite2 @ C @ T2 )
% 11.90/6.01 => ( ! [X2: C] :
% 11.90/6.01 ( ( member @ C @ X2 @ T2 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( G @ X2 ) ) )
% 11.90/6.01 => ( ! [X2: B] :
% 11.90/6.01 ( ( member @ B @ X2 @ S )
% 11.90/6.01 => ? [Xa2: C] :
% 11.90/6.01 ( ( member @ C @ Xa2 @ T2 )
% 11.90/6.01 & ( ( I @ Xa2 )
% 11.90/6.01 = X2 )
% 11.90/6.01 & ( subseteq_mset @ A @ ( F @ X2 ) @ ( G @ Xa2 ) ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ C @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ T2 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4256_subset__mset_Osum__nonneg__eq__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ A4 )
% 11.90/6.01 => ( ! [X2: B] :
% 11.90/6.01 ( ( member @ B @ X2 @ A4 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X2 ) ) )
% 11.90/6.01 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ! [X3: B] :
% 11.90/6.01 ( ( member @ B @ X3 @ A4 )
% 11.90/6.01 => ( ( F @ X3 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4257_subset__mset_Osum__nonneg__0,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,S: set @ B,F: B > ( multiset @ A ),I: B] :
% 11.90/6.01 ( ( finite_finite2 @ B @ S )
% 11.90/6.01 => ( ! [I2: B] :
% 11.90/6.01 ( ( member @ B @ I2 @ S )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 11.90/6.01 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( member @ B @ I @ S )
% 11.90/6.01 => ( ( F @ I )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4258_subset__mset_Osum__nonneg__leq__bound,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,S: set @ B,F: B > ( multiset @ A ),B6: multiset @ A,I: B] :
% 11.90/6.01 ( ( finite_finite2 @ B @ S )
% 11.90/6.01 => ( ! [I2: B] :
% 11.90/6.01 ( ( member @ B @ I2 @ S )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 11.90/6.01 => ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S )
% 11.90/6.01 = B6 )
% 11.90/6.01 => ( ( member @ B @ I @ S )
% 11.90/6.01 => ( subseteq_mset @ A @ ( F @ I ) @ B6 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4259_minus__shift__def,axiom,
% 11.90/6.01 ( minus_shift
% 11.90/6.01 = ( ^ [R3: code_natural,K4: code_natural,L3: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K4 @ L3 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R3 @ K4 ) @ L3 ) @ ( minus_minus @ code_natural @ K4 @ L3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4260_split__seed__def,axiom,
% 11.90/6.01 ( split_seed
% 11.90/6.01 = ( ^ [S4: product_prod @ code_natural @ code_natural] :
% 11.90/6.01 ( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
% 11.90/6.01 @ ^ [V2: code_natural,W2: code_natural] :
% 11.90/6.01 ( product_case_prod @ code_natural @ code_natural @ ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) )
% 11.90/6.01 @ ^ [V5: code_natural,W3: code_natural] : ( product_Pair @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_Pair @ code_natural @ code_natural @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V2 ) @ W3 ) @ ( product_Pair @ code_natural @ code_natural @ V5 @ ( inc_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W2 ) ) )
% 11.90/6.01 @ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S4 ) ) )
% 11.90/6.01 @ S4 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4261_log_Osimps,axiom,
% 11.90/6.01 ( log
% 11.90/6.01 = ( ^ [B5: code_natural,I3: code_natural] :
% 11.90/6.01 ( if @ code_natural
% 11.90/6.01 @ ( ( ord_less_eq @ code_natural @ B5 @ ( one_one @ code_natural ) )
% 11.90/6.01 | ( ord_less @ code_natural @ I3 @ B5 ) )
% 11.90/6.01 @ ( one_one @ code_natural )
% 11.90/6.01 @ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B5 @ ( divide_divide @ code_natural @ I3 @ B5 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4262_log_Oelims,axiom,
% 11.90/6.01 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 11.90/6.01 ( ( ( log @ X @ Xa )
% 11.90/6.01 = Y )
% 11.90/6.01 => ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 11.90/6.01 | ( ord_less @ code_natural @ Xa @ X ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( one_one @ code_natural ) ) )
% 11.90/6.01 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 11.90/6.01 | ( ord_less @ code_natural @ Xa @ X ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4263_extract__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( extract @ A )
% 11.90/6.01 = ( ^ [P4: A > $o,Xs3: list @ A] :
% 11.90/6.01 ( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
% 11.90/6.01 @ ^ [Y3: A,Ys: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Ys ) ) )
% 11.90/6.01 @ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4264_drop__takeWhile,axiom,
% 11.90/6.01 ! [A: $tType,I: nat,P: A > $o,L: list @ A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
% 11.90/6.01 => ( ( drop @ A @ I @ ( takeWhile @ A @ P @ L ) )
% 11.90/6.01 = ( takeWhile @ A @ P @ ( drop @ A @ I @ L ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4265_eq__len__takeWhile__conv,axiom,
% 11.90/6.01 ! [A: $tType,I: nat,P: A > $o,L: list @ A] :
% 11.90/6.01 ( ( I
% 11.90/6.01 = ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
% 11.90/6.01 = ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.01 & ! [J2: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ J2 @ I )
% 11.90/6.01 => ( P @ ( nth @ A @ L @ J2 ) ) )
% 11.90/6.01 & ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.01 => ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4266_less__length__takeWhile__conv,axiom,
% 11.90/6.01 ! [A: $tType,I: nat,P: A > $o,L: list @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
% 11.90/6.01 = ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
% 11.90/6.01 & ! [J2: nat] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ J2 @ I )
% 11.90/6.01 => ( P @ ( nth @ A @ L @ J2 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4267_length__dropWhile__takeWhile,axiom,
% 11.90/6.01 ! [A: $tType,X: nat,P: A > $o,Xs: list @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ X @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
% 11.90/6.01 => ( ord_less @ nat @ ( plus_plus @ nat @ X @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4268_dropWhile__nth,axiom,
% 11.90/6.01 ! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
% 11.90/6.01 => ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
% 11.90/6.01 = ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4269_inc__shift__def,axiom,
% 11.90/6.01 ( inc_shift
% 11.90/6.01 = ( ^ [V2: code_natural,K4: code_natural] : ( if @ code_natural @ ( V2 = K4 ) @ ( one_one @ code_natural ) @ ( plus_plus @ code_natural @ K4 @ ( one_one @ code_natural ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4270_log_Opelims,axiom,
% 11.90/6.01 ! [X: code_natural,Xa: code_natural,Y: code_natural] :
% 11.90/6.01 ( ( ( log @ X @ Xa )
% 11.90/6.01 = Y )
% 11.90/6.01 => ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
% 11.90/6.01 => ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 11.90/6.01 | ( ord_less @ code_natural @ Xa @ X ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( one_one @ code_natural ) ) )
% 11.90/6.01 & ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
% 11.90/6.01 | ( ord_less @ code_natural @ Xa @ X ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
% 11.90/6.01 => ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4271_Random_Orange__def,axiom,
% 11.90/6.01 ( range
% 11.90/6.01 = ( ^ [K4: code_natural] :
% 11.90/6.01 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )
% 11.90/6.01 @ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K4 )
% 11.90/6.01 @ ^ [L3: code_natural] :
% 11.90/6.01 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
% 11.90/6.01 @ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V2 @ ( times_times @ code_natural @ L3 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 11.90/6.01 @ ( one_one @ code_natural ) )
% 11.90/6.01 @ ^ [V2: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V2 @ K4 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4272_Predicate_Oiterate__upto_Opinduct,axiom,
% 11.90/6.01 ! [A: $tType,A0: code_natural > A,A1: code_natural,A22: code_natural,P: ( code_natural > A ) > code_natural > code_natural > $o] :
% 11.90/6.01 ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ A0 @ ( product_Pair @ code_natural @ code_natural @ A1 @ A22 ) ) )
% 11.90/6.01 => ( ! [F4: code_natural > A,N3: code_natural,M5: code_natural] :
% 11.90/6.01 ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N3 @ M5 ) ) )
% 11.90/6.01 => ( ! [X7: product_unit] :
% 11.90/6.01 ( ~ ( ord_less @ code_natural @ M5 @ N3 )
% 11.90/6.01 => ( P @ F4 @ ( plus_plus @ code_natural @ N3 @ ( one_one @ code_natural ) ) @ M5 ) )
% 11.90/6.01 => ( P @ F4 @ N3 @ M5 ) ) )
% 11.90/6.01 => ( P @ A0 @ A1 @ A22 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4273_mset__size2elem,axiom,
% 11.90/6.01 ! [A: $tType,P: multiset @ A,Q3: A,Q4: A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
% 11.90/6.01 => ( P
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4274_scomp__apply,axiom,
% 11.90/6.01 ! [A: $tType,D: $tType,C: $tType,B: $tType] :
% 11.90/6.01 ( ( product_scomp @ B @ C @ D @ A )
% 11.90/6.01 = ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X3: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4275_size__union,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ N6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4276_scomp__scomp,axiom,
% 11.90/6.01 ! [A: $tType,C: $tType,D: $tType,B: $tType,F5: $tType,E: $tType,F: A > ( product_prod @ E @ F5 ),G: E > F5 > ( product_prod @ C @ D ),H3: C > D > B] :
% 11.90/6.01 ( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F5 @ ( product_prod @ C @ D ) @ F @ G ) @ H3 )
% 11.90/6.01 = ( product_scomp @ A @ E @ F5 @ B @ F
% 11.90/6.01 @ ^ [X3: E] : ( product_scomp @ F5 @ C @ D @ B @ ( G @ X3 ) @ H3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4277_scomp__Pair,axiom,
% 11.90/6.01 ! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
% 11.90/6.01 ( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
% 11.90/6.01 = X ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4278_Pair__scomp,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
% 11.90/6.01 ( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
% 11.90/6.01 = ( F @ X ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4279_scomp__def,axiom,
% 11.90/6.01 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 11.90/6.01 ( ( product_scomp @ A @ B @ C @ D )
% 11.90/6.01 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X3: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4280_size__1__singleton__mset,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A] :
% 11.90/6.01 ( ( ( size_size @ ( multiset @ A ) @ M3 )
% 11.90/6.01 = ( one_one @ nat ) )
% 11.90/6.01 => ? [A5: A] :
% 11.90/6.01 ( M3
% 11.90/6.01 = ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4281_size__single,axiom,
% 11.90/6.01 ! [A: $tType,B2: A] :
% 11.90/6.01 ( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = ( one_one @ nat ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4282_iterate_Oelims,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
% 11.90/6.01 ( ( ( iterate @ B @ A @ X @ Xa @ Xb )
% 11.90/6.01 = Y )
% 11.90/6.01 => ( ( ( X
% 11.90/6.01 = ( zero_zero @ code_natural ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( product_Pair @ B @ A @ Xb ) ) )
% 11.90/6.01 & ( ( X
% 11.90/6.01 != ( zero_zero @ code_natural ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4283_iterate_Osimps,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( iterate @ B @ A )
% 11.90/6.01 = ( ^ [K4: code_natural,F2: B > A > ( product_prod @ B @ A ),X3: B] :
% 11.90/6.01 ( if @ ( A > ( product_prod @ B @ A ) )
% 11.90/6.01 @ ( K4
% 11.90/6.01 = ( zero_zero @ code_natural ) )
% 11.90/6.01 @ ( product_Pair @ B @ A @ X3 )
% 11.90/6.01 @ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F2 @ X3 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K4 @ ( one_one @ code_natural ) ) @ F2 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4284_size__mset__SucE,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,N: nat] :
% 11.90/6.01 ( ( ( size_size @ ( multiset @ A ) @ A4 )
% 11.90/6.01 = ( suc @ N ) )
% 11.90/6.01 => ~ ! [A5: A,B10: multiset @ A] :
% 11.90/6.01 ( ( A4
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A5 @ ( zero_zero @ ( multiset @ A ) ) ) @ B10 ) )
% 11.90/6.01 => ( ( size_size @ ( multiset @ A ) @ B10 )
% 11.90/6.01 != N ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4285_scomp__unfold,axiom,
% 11.90/6.01 ! [D: $tType,C: $tType,B: $tType,A: $tType] :
% 11.90/6.01 ( ( product_scomp @ A @ B @ C @ D )
% 11.90/6.01 = ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X3: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X3 ) ) @ ( product_snd @ B @ C @ ( F2 @ X3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4286_mset__size__le1__cases,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( suc @ ( zero_zero @ nat ) ) )
% 11.90/6.01 => ( ( M3
% 11.90/6.01 != ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ~ ! [M5: A] :
% 11.90/6.01 ( M3
% 11.90/6.01 != ( add_mset @ A @ M5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4287_iterate_Opelims,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
% 11.90/6.01 ( ( ( iterate @ B @ A @ X @ Xa @ Xb )
% 11.90/6.01 = Y )
% 11.90/6.01 => ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
% 11.90/6.01 => ~ ( ( ( ( X
% 11.90/6.01 = ( zero_zero @ code_natural ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( product_Pair @ B @ A @ Xb ) ) )
% 11.90/6.01 & ( ( X
% 11.90/6.01 != ( zero_zero @ code_natural ) )
% 11.90/6.01 => ( Y
% 11.90/6.01 = ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
% 11.90/6.01 => ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4288_size__Diff__singleton__if,axiom,
% 11.90/6.01 ! [A: $tType,X: A,A4: multiset @ A] :
% 11.90/6.01 ( ( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 => ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 = ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( one_one @ nat ) ) ) )
% 11.90/6.01 & ( ~ ( member @ A @ X @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 => ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 = ( size_size @ ( multiset @ A ) @ A4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4289_size__Diff__singleton,axiom,
% 11.90/6.01 ! [A: $tType,X: A,M3: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
% 11.90/6.01 => ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 = ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( one_one @ nat ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4290_size__diff__se,axiom,
% 11.90/6.01 ! [A: $tType,T2: A,S2: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
% 11.90/6.01 => ( ( size_size @ ( multiset @ A ) @ S2 )
% 11.90/6.01 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4291_set__mset__union,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ M3 ) @ ( set_mset @ A @ N6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4292_mset__diff__cancel1elem,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,B6: multiset @ A] :
% 11.90/6.01 ( ~ ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B6 )
% 11.90/6.01 = ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4293_ex__Melem__conv,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A] :
% 11.90/6.01 ( ( ? [X3: A] : ( member @ A @ X3 @ ( set_mset @ A @ A4 ) ) )
% 11.90/6.01 = ( A4
% 11.90/6.01 != ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4294_union__iff,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 11.90/6.01 = ( ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 | ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4295_mset__un__cases,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 11.90/6.01 => ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 => ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4296_mset__left__cancel__union,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 11.90/6.01 => ( ~ ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 => ( member @ A @ A3 @ ( set_mset @ A @ B6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4297_mset__right__cancel__union,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) )
% 11.90/6.01 => ( ~ ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 11.90/6.01 => ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4298_iterate_Ocases,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
% 11.90/6.01 ~ ! [K5: code_natural,F4: B > A > ( product_prod @ B @ A ),X2: B] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K5 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F4 @ X2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4299_image__mset__cong__pair,axiom,
% 11.90/6.01 ! [C: $tType,B: $tType,A: $tType,M3: multiset @ ( product_prod @ A @ B ),F: A > B > C,G: A > B > C] :
% 11.90/6.01 ( ! [X2: A,Y4: B] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M3 ) )
% 11.90/6.01 => ( ( F @ X2 @ Y4 )
% 11.90/6.01 = ( G @ X2 @ Y4 ) ) )
% 11.90/6.01 => ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ M3 )
% 11.90/6.01 = ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G ) @ M3 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4300_multi__member__skip,axiom,
% 11.90/6.01 ! [A: $tType,X: A,XS: multiset @ A,Y: A] :
% 11.90/6.01 ( ( member @ A @ X @ ( set_mset @ A @ XS ) )
% 11.90/6.01 => ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4301_multi__member__this,axiom,
% 11.90/6.01 ! [A: $tType,X: A,XS: multiset @ A] : ( member @ A @ X @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) @ XS ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4302_mset__right__cancel__elem,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B2: A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
% 11.90/6.01 => ( ( A3 != B2 )
% 11.90/6.01 => ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4303_mset__left__cancel__elem,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,B2: A,A4: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A4 ) ) )
% 11.90/6.01 => ( ( A3 != B2 )
% 11.90/6.01 => ( member @ A @ A3 @ ( set_mset @ A @ A4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4304_mset__le__mono__add__single,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,Ys2: multiset @ A,B2: A,Ws: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ Ys2 ) )
% 11.90/6.01 => ( ( member @ A @ B2 @ ( set_mset @ A @ Ws ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys2 @ Ws ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4305_mset__union__subset__s,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ B6 ) @ C5 )
% 11.90/6.01 => ( ( member @ A @ A3 @ ( set_mset @ A @ C5 ) )
% 11.90/6.01 & ( subseteq_mset @ A @ B6 @ C5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4306_mset__2dist2__cases,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,B2: A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 11.90/6.01 => ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A4 )
% 11.90/6.01 => ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B6 )
% 11.90/6.01 => ( ( ( member @ A @ A3 @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 => ~ ( member @ A @ B2 @ ( set_mset @ A @ B6 ) ) )
% 11.90/6.01 => ~ ( ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 11.90/6.01 => ~ ( member @ A @ B2 @ ( set_mset @ A @ A4 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4307_insert__DiffM2,axiom,
% 11.90/6.01 ! [A: $tType,X: A,M3: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = M3 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4308_diff__union__single__conv,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,J4: multiset @ A,I4: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ J4 ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ I4 @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4309_mset__un__single__un__cases,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( ( add_mset @ A @ A3 @ A4 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 11.90/6.01 => ( ( ( member @ A @ A3 @ ( set_mset @ A @ B6 ) )
% 11.90/6.01 => ( A4
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B6 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C5 ) ) )
% 11.90/6.01 => ~ ( ( member @ A @ A3 @ ( set_mset @ A @ C5 ) )
% 11.90/6.01 => ( A4
% 11.90/6.01 != ( plus_plus @ ( multiset @ A ) @ B6 @ ( minus_minus @ ( multiset @ A ) @ C5 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4310_diff__union__single__conv2,axiom,
% 11.90/6.01 ! [A: $tType,A3: A,J4: multiset @ A,I4: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ A3 @ ( set_mset @ A @ J4 ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J4 @ I4 ) @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I4 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4311_mset__union__diff__comm,axiom,
% 11.90/6.01 ! [A: $tType,T2: A,S2: multiset @ A,T6: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
% 11.90/6.01 => ( ( plus_plus @ ( multiset @ A ) @ T6 @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 = ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T6 @ S2 ) @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4312_mset__contains__eq,axiom,
% 11.90/6.01 ! [A: $tType,M2: A,M3: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ M2 @ ( set_mset @ A @ M3 ) )
% 11.90/6.01 = ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ M2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
% 11.90/6.01 = M3 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4313_mset__size1elem,axiom,
% 11.90/6.01 ! [A: $tType,P: multiset @ A,Q3: A] :
% 11.90/6.01 ( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
% 11.90/6.01 => ( ( member @ A @ Q3 @ ( set_mset @ A @ P ) )
% 11.90/6.01 => ( P
% 11.90/6.01 = ( add_mset @ A @ Q3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4314_subset__mset_Osum__mset__mono,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,K2: multiset @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
% 11.90/6.01 ( ! [I2: B] :
% 11.90/6.01 ( ( member @ B @ I2 @ ( set_mset @ B @ K2 ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( F @ I2 ) @ ( G @ I2 ) ) )
% 11.90/6.01 => ( subseteq_mset @ A @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ F @ K2 ) ) @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ G @ K2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4315_subset__mset_Osum__mset__0__iff,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ ( multiset @ A )] :
% 11.90/6.01 ( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ! [X3: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ X3 @ ( set_mset @ ( multiset @ A ) @ M3 ) )
% 11.90/6.01 => ( X3
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4316_wmsI,axiom,
% 11.90/6.01 ! [A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 11.90/6.01 | ( ( A4
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 11.90/6.01 & ( B6
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ B6 ) ) @ ms_weak ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4317_smsI,axiom,
% 11.90/6.01 ! [A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat ),Z9: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ B6 ) ) @ ms_strict ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4318_ms__reduction__pair,axiom,
% 11.90/6.01 fun_reduction_pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ms_strict @ ms_weak ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4319_ms__strictI,axiom,
% 11.90/6.01 ! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( pw_leq @ Z9 @ Z10 )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B6 ) ) @ ms_strict ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4320_ms__weakI1,axiom,
% 11.90/6.01 ! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat ),A4: multiset @ ( product_prod @ nat @ nat ),B6: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( pw_leq @ Z9 @ Z10 )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A4 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B6 ) ) @ fun_max_strict )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ A4 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ B6 ) ) @ ms_weak ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4321_pw__leq__split,axiom,
% 11.90/6.01 ! [X10: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( pw_leq @ X10 @ Y9 )
% 11.90/6.01 => ? [A15: multiset @ ( product_prod @ nat @ nat ),B10: multiset @ ( product_prod @ nat @ nat ),Z11: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( X10
% 11.90/6.01 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ A15 @ Z11 ) )
% 11.90/6.01 & ( Y9
% 11.90/6.01 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ B10 @ Z11 ) )
% 11.90/6.01 & ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ A15 ) @ ( set_mset @ ( product_prod @ nat @ nat ) @ B10 ) ) @ fun_max_strict )
% 11.90/6.01 | ( ( B10
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 11.90/6.01 & ( A15
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4322_ms__weakI2,axiom,
% 11.90/6.01 ! [Z9: multiset @ ( product_prod @ nat @ nat ),Z10: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( pw_leq @ Z9 @ Z10 )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z9 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ Z10 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) @ ms_weak ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4323_pw__leq__lstep,axiom,
% 11.90/6.01 ! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 11.90/6.01 => ( pw_leq @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4324_pw__leq__step,axiom,
% 11.90/6.01 ! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat,X10: multiset @ ( product_prod @ nat @ nat ),Y9: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
% 11.90/6.01 => ( ( pw_leq @ X10 @ Y9 )
% 11.90/6.01 => ( pw_leq @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X10 ) @ ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y9 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4325_pw__leq_Osimps,axiom,
% 11.90/6.01 ( pw_leq
% 11.90/6.01 = ( ^ [A12: multiset @ ( product_prod @ nat @ nat ),A23: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( ( A12
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 11.90/6.01 & ( A23
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
% 11.90/6.01 | ? [X3: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat,X8: multiset @ ( product_prod @ nat @ nat ),Y10: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( A12
% 11.90/6.01 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X8 ) )
% 11.90/6.01 & ( A23
% 11.90/6.01 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y3 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y10 ) )
% 11.90/6.01 & ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X3 @ Y3 ) @ fun_pair_leq )
% 11.90/6.01 & ( pw_leq @ X8 @ Y10 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4326_pw__leq_Ocases,axiom,
% 11.90/6.01 ! [A1: multiset @ ( product_prod @ nat @ nat ),A22: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( pw_leq @ A1 @ A22 )
% 11.90/6.01 => ( ( ( A1
% 11.90/6.01 = ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) )
% 11.90/6.01 => ( A22
% 11.90/6.01 != ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) )
% 11.90/6.01 => ~ ! [X2: product_prod @ nat @ nat,Y4: product_prod @ nat @ nat,X15: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( A1
% 11.90/6.01 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ X2 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ X15 ) )
% 11.90/6.01 => ! [Y11: multiset @ ( product_prod @ nat @ nat )] :
% 11.90/6.01 ( ( A22
% 11.90/6.01 = ( plus_plus @ ( multiset @ ( product_prod @ nat @ nat ) ) @ ( add_mset @ ( product_prod @ nat @ nat ) @ Y4 @ ( zero_zero @ ( multiset @ ( product_prod @ nat @ nat ) ) ) ) @ Y11 ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X2 @ Y4 ) @ fun_pair_leq )
% 11.90/6.01 => ~ ( pw_leq @ X15 @ Y11 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4327_mult1E,axiom,
% 11.90/6.01 ! [A: $tType,N6: multiset @ A,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ R ) )
% 11.90/6.01 => ~ ! [A5: A,M0: multiset @ A] :
% 11.90/6.01 ( ( M3
% 11.90/6.01 = ( add_mset @ A @ A5 @ M0 ) )
% 11.90/6.01 => ! [K10: multiset @ A] :
% 11.90/6.01 ( ( N6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M0 @ K10 ) )
% 11.90/6.01 => ~ ! [B8: A] :
% 11.90/6.01 ( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A5 ) @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4328_mult1I,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,A3: A,M02: multiset @ A,N6: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( M3
% 11.90/6.01 = ( add_mset @ A @ A3 @ M02 ) )
% 11.90/6.01 => ( ( N6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M02 @ K2 ) )
% 11.90/6.01 => ( ! [B3: A] :
% 11.90/6.01 ( ( member @ A @ B3 @ ( set_mset @ A @ K2 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4329_less__add,axiom,
% 11.90/6.01 ! [A: $tType,N6: multiset @ A,A3: A,M02: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ ( add_mset @ A @ A3 @ M02 ) ) @ ( mult1 @ A @ R ) )
% 11.90/6.01 => ( ? [M9: multiset @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M9 @ M02 ) @ ( mult1 @ A @ R ) )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( add_mset @ A @ A3 @ M9 ) ) )
% 11.90/6.01 | ? [K10: multiset @ A] :
% 11.90/6.01 ( ! [B8: A] :
% 11.90/6.01 ( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A3 ) @ R ) )
% 11.90/6.01 & ( N6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M02 @ K10 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4330_one__step__implies__mult,axiom,
% 11.90/6.01 ! [A: $tType,J4: multiset @ A,K2: multiset @ A,R: set @ ( product_prod @ A @ A ),I4: multiset @ A] :
% 11.90/6.01 ( ( J4
% 11.90/6.01 != ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ! [X2: A] :
% 11.90/6.01 ( ( member @ A @ X2 @ ( set_mset @ A @ K2 ) )
% 11.90/6.01 => ? [Xa2: A] :
% 11.90/6.01 ( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R ) ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) ) @ ( mult @ A @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4331_not__less__empty,axiom,
% 11.90/6.01 ! [A: $tType,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ~ ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( mult1 @ A @ R ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4332_mult1__union,axiom,
% 11.90/6.01 ! [A: $tType,B6: multiset @ A,D4: multiset @ A,R: set @ ( product_prod @ A @ A ),C5: multiset @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B6 @ D4 ) @ ( mult1 @ A @ R ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ D4 ) ) @ ( mult1 @ A @ R ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4333_mult1__lessE,axiom,
% 11.90/6.01 ! [A: $tType,N6: multiset @ A,M3: multiset @ A,R: A > A > $o] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) )
% 11.90/6.01 => ( ( asymp @ A @ R )
% 11.90/6.01 => ~ ! [A5: A,M0: multiset @ A] :
% 11.90/6.01 ( ( M3
% 11.90/6.01 = ( add_mset @ A @ A5 @ M0 ) )
% 11.90/6.01 => ! [K10: multiset @ A] :
% 11.90/6.01 ( ( N6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M0 @ K10 ) )
% 11.90/6.01 => ( ~ ( member @ A @ A5 @ ( set_mset @ A @ K10 ) )
% 11.90/6.01 => ~ ! [B8: A] :
% 11.90/6.01 ( ( member @ A @ B8 @ ( set_mset @ A @ K10 ) )
% 11.90/6.01 => ( R @ B8 @ A5 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4334_in__mset__fold__plus__iff,axiom,
% 11.90/6.01 ! [A: $tType,X: A,M3: multiset @ A,NN: multiset @ ( multiset @ A )] :
% 11.90/6.01 ( ( member @ A @ X @ ( set_mset @ A @ ( fold_mset @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ M3 @ NN ) ) )
% 11.90/6.01 = ( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
% 11.90/6.01 | ? [N9: multiset @ A] :
% 11.90/6.01 ( ( member @ ( multiset @ A ) @ N9 @ ( set_mset @ ( multiset @ A ) @ NN ) )
% 11.90/6.01 & ( member @ A @ X @ ( set_mset @ A @ N9 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4335_mult__implies__one__step,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( trans @ A @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ N6 ) @ ( mult @ A @ R ) )
% 11.90/6.01 => ? [I8: multiset @ A,J5: multiset @ A] :
% 11.90/6.01 ( ( N6
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
% 11.90/6.01 & ? [K10: multiset @ A] :
% 11.90/6.01 ( ( M3
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ I8 @ K10 ) )
% 11.90/6.01 & ( J5
% 11.90/6.01 != ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ! [X7: A] :
% 11.90/6.01 ( ( member @ A @ X7 @ ( set_mset @ A @ K10 ) )
% 11.90/6.01 => ? [Xa3: A] :
% 11.90/6.01 ( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4336_mult1__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( mult1 @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
% 11.90/6.01 @ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
% 11.90/6.01 @ ^ [N9: multiset @ A,M8: multiset @ A] :
% 11.90/6.01 ? [A7: A,M03: multiset @ A,K3: multiset @ A] :
% 11.90/6.01 ( ( M8
% 11.90/6.01 = ( add_mset @ A @ A7 @ M03 ) )
% 11.90/6.01 & ( N9
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ M03 @ K3 ) )
% 11.90/6.01 & ! [B5: A] :
% 11.90/6.01 ( ( member @ A @ B5 @ ( set_mset @ A @ K3 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4337_trans__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( trans @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ! [X3: A,Y3: A,Z5: A] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z5 ) @ R3 )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Z5 ) @ R3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4338_transI,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ! [X2: A,Y4: A,Z3: A] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R ) ) )
% 11.90/6.01 => ( trans @ A @ R ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4339_transE,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 11.90/6.01 ( ( trans @ A @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4340_transD,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 11.90/6.01 ( ( trans @ A @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4341_lexord__trans,axiom,
% 11.90/6.01 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A ),Z2: list @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lexord @ A @ R ) )
% 11.90/6.01 => ( ( trans @ A @ R )
% 11.90/6.01 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4342_lenlex__trans,axiom,
% 11.90/6.01 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A ),Z2: list @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lenlex @ A @ R ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ Z2 ) @ ( lenlex @ A @ R ) )
% 11.90/6.01 => ( ( trans @ A @ R )
% 11.90/6.01 => ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z2 ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4343_union__fold__mset__add__mset,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) )
% 11.90/6.01 = ( fold_mset @ A @ ( multiset @ A ) @ ( add_mset @ A ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4344_trans__singleton,axiom,
% 11.90/6.01 ! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4345_trans__rtrancl__eq__reflcl,axiom,
% 11.90/6.01 ! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( trans @ A @ A4 )
% 11.90/6.01 => ( ( transitive_rtrancl @ A @ A4 )
% 11.90/6.01 = ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ ( id2 @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4346_comp__fun__commute_Ofold__mset__union,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,F: A > B > B,S: B,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( finite6289374366891150609ommute @ A @ B @ F )
% 11.90/6.01 => ( ( fold_mset @ A @ B @ F @ S @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( fold_mset @ A @ B @ F @ ( fold_mset @ A @ B @ F @ S @ M3 ) @ N6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4347_trans__join,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( trans @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ! [X3: product_prod @ A @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ X3 @ R3 )
% 11.90/6.01 => ( product_case_prod @ A @ A @ $o
% 11.90/6.01 @ ^ [Y3: A,Y13: A] :
% 11.90/6.01 ! [Z5: product_prod @ A @ A] :
% 11.90/6.01 ( ( member @ ( product_prod @ A @ A ) @ Z5 @ R3 )
% 11.90/6.01 => ( product_case_prod @ A @ A @ $o
% 11.90/6.01 @ ^ [Y23: A,Aa2: A] :
% 11.90/6.01 ( ( Y13 = Y23 )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Aa2 ) @ R3 ) )
% 11.90/6.01 @ Z5 ) )
% 11.90/6.01 @ X3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4348_Image__absorb__rtrancl,axiom,
% 11.90/6.01 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ A,C5: set @ A] :
% 11.90/6.01 ( ( trans @ A @ A4 )
% 11.90/6.01 => ( ( refl_on @ A @ B6 @ A4 )
% 11.90/6.01 => ( ( ord_less_eq @ ( set @ A ) @ C5 @ B6 )
% 11.90/6.01 => ( ( image @ A @ A @ ( transitive_rtrancl @ A @ A4 ) @ C5 )
% 11.90/6.01 = ( image @ A @ A @ A4 @ C5 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4349_wf__finite__segments,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( irrefl @ A @ R )
% 11.90/6.01 => ( ( trans @ A @ R )
% 11.90/6.01 => ( ! [X2: A] :
% 11.90/6.01 ( finite_finite2 @ A
% 11.90/6.01 @ ( collect @ A
% 11.90/6.01 @ ^ [Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X2 ) @ R ) ) )
% 11.90/6.01 => ( wf @ A @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4350_mult__cancel__add__mset,axiom,
% 11.90/6.01 ! [A: $tType,S: set @ ( product_prod @ A @ A ),Uu2: A,X10: multiset @ A,Y9: multiset @ A] :
% 11.90/6.01 ( ( trans @ A @ S )
% 11.90/6.01 => ( ( irrefl @ A @ S )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( add_mset @ A @ Uu2 @ X10 ) @ ( add_mset @ A @ Uu2 @ Y9 ) ) @ ( mult @ A @ S ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4351_mult__cancel,axiom,
% 11.90/6.01 ! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Z9: multiset @ A,Y9: multiset @ A] :
% 11.90/6.01 ( ( trans @ A @ S )
% 11.90/6.01 => ( ( irrefl @ A @ S )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X10 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y9 @ Z9 ) ) @ ( mult @ A @ S ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4352_mult__cancel__max,axiom,
% 11.90/6.01 ! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Y9: multiset @ A] :
% 11.90/6.01 ( ( trans @ A @ S )
% 11.90/6.01 => ( ( irrefl @ A @ S )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ Y9 ) @ ( minus_minus @ ( multiset @ A ) @ Y9 @ X10 ) ) @ ( mult @ A @ S ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4353_multeqp__code__iff__reflcl__mult,axiom,
% 11.90/6.01 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),P: A > A > $o,N6: multiset @ A,M3: multiset @ A] :
% 11.90/6.01 ( ( irrefl @ A @ R5 )
% 11.90/6.01 => ( ( trans @ A @ R5 )
% 11.90/6.01 => ( ! [X2: A,Y4: A] :
% 11.90/6.01 ( ( P @ X2 @ Y4 )
% 11.90/6.01 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R5 ) )
% 11.90/6.01 => ( ( multeqp_code @ A @ P @ N6 @ M3 )
% 11.90/6.01 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( sup_sup @ ( set @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) ) @ ( mult @ A @ R5 ) @ ( id2 @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4354_multp__code__iff__mult,axiom,
% 11.90/6.01 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),P: A > A > $o,N6: multiset @ A,M3: multiset @ A] :
% 11.90/6.01 ( ( irrefl @ A @ R5 )
% 11.90/6.01 => ( ( trans @ A @ R5 )
% 11.90/6.01 => ( ! [X2: A,Y4: A] :
% 11.90/6.01 ( ( P @ X2 @ Y4 )
% 11.90/6.01 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R5 ) )
% 11.90/6.01 => ( ( multp_code @ A @ P @ N6 @ M3 )
% 11.90/6.01 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N6 @ M3 ) @ ( mult @ A @ R5 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4355_max__ext__eq,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( max_ext @ A )
% 11.90/6.01 = ( ^ [R6: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 11.90/6.01 @ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
% 11.90/6.01 @ ^ [X8: set @ A,Y10: set @ A] :
% 11.90/6.01 ( ( finite_finite2 @ A @ X8 )
% 11.90/6.01 & ( finite_finite2 @ A @ Y10 )
% 11.90/6.01 & ( Y10
% 11.90/6.01 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ X8 )
% 11.90/6.01 => ? [Y3: A] :
% 11.90/6.01 ( ( member @ A @ Y3 @ Y10 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4356_init__seg__of__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( init_seg_of @ A )
% 11.90/6.01 = ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/6.01 @ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
% 11.90/6.01 @ ^ [R3: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S4 )
% 11.90/6.01 & ! [A7: A,B5: A,C3: A] :
% 11.90/6.01 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ S4 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C3 ) @ R3 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4357_refl__on__init__seg__of,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( init_seg_of @ A ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4358_antisym__init__seg__of,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ R ) @ ( init_seg_of @ A ) )
% 11.90/6.01 => ( R = S ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4359_trans__init__seg__of,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ T2 ) @ ( init_seg_of @ A ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4360_Image__def,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( image @ A @ B )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ B ),S4: set @ A] :
% 11.90/6.01 ( collect @ B
% 11.90/6.01 @ ^ [Y3: B] :
% 11.90/6.01 ? [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ S4 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4361_initial__segment__of__Diff,axiom,
% 11.90/6.01 ! [A: $tType,P5: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ Q3 ) @ ( init_seg_of @ A ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ S ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q3 @ S ) ) @ ( init_seg_of @ A ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4362_min__ext__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( min_ext @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
% 11.90/6.01 @ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
% 11.90/6.01 ? [X8: set @ A,Y10: set @ A] :
% 11.90/6.01 ( ( Uu
% 11.90/6.01 = ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X8 @ Y10 ) )
% 11.90/6.01 & ( X8
% 11.90/6.01 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ Y10 )
% 11.90/6.01 => ? [Y3: A] :
% 11.90/6.01 ( ( member @ A @ Y3 @ X8 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4363_map__project__def,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType] :
% 11.90/6.01 ( ( map_project @ A @ B )
% 11.90/6.01 = ( ^ [F2: A > ( option @ B ),A9: set @ A] :
% 11.90/6.01 ( collect @ B
% 11.90/6.01 @ ^ [B5: B] :
% 11.90/6.01 ? [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ A9 )
% 11.90/6.01 & ( ( F2 @ X3 )
% 11.90/6.01 = ( some @ B @ B5 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4364_Chains__init__seg__of__Union,axiom,
% 11.90/6.01 ! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R5 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
% 11.90/6.01 => ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R @ R5 )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R5 ) ) @ ( init_seg_of @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4365_mult__cancel__max0,axiom,
% 11.90/6.01 ! [A: $tType,S: set @ ( product_prod @ A @ A ),X10: multiset @ A,Y9: multiset @ A] :
% 11.90/6.01 ( ( trans @ A @ S )
% 11.90/6.01 => ( ( irrefl @ A @ S )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X10 @ Y9 ) @ ( mult @ A @ S ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ X10 @ ( inter_mset @ A @ X10 @ Y9 ) ) @ ( minus_minus @ ( multiset @ A ) @ Y9 @ ( inter_mset @ A @ X10 @ Y9 ) ) ) @ ( mult @ A @ S ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4366_sum__mset_Oremove,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( comm_monoid_add @ A )
% 11.90/6.01 => ! [X: A,A4: multiset @ A] :
% 11.90/6.01 ( ( member @ A @ X @ ( set_mset @ A @ A4 ) )
% 11.90/6.01 => ( ( comm_m7189776963980413722m_mset @ A @ A4 )
% 11.90/6.01 = ( plus_plus @ A @ X @ ( comm_m7189776963980413722m_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A4 @ ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4367_sum__mset_Oadd__mset,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( comm_monoid_add @ A )
% 11.90/6.01 => ! [X: A,N6: multiset @ A] :
% 11.90/6.01 ( ( comm_m7189776963980413722m_mset @ A @ ( add_mset @ A @ X @ N6 ) )
% 11.90/6.01 = ( plus_plus @ A @ X @ ( comm_m7189776963980413722m_mset @ A @ N6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4368_sum__mset_Ounion,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( comm_monoid_add @ A )
% 11.90/6.01 => ! [M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.01 ( ( comm_m7189776963980413722m_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.01 = ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ M3 ) @ ( comm_m7189776963980413722m_mset @ A @ N6 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4369_sum__mset_Oinsert,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType] :
% 11.90/6.01 ( ( comm_monoid_add @ A )
% 11.90/6.01 => ! [G: B > A,X: B,A4: multiset @ B] :
% 11.90/6.01 ( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X @ A4 ) ) )
% 11.90/6.01 = ( plus_plus @ A @ ( G @ X ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4370_inter__union__distrib__right,axiom,
% 11.90/6.01 ! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ C5 @ ( inter_mset @ A @ A4 @ B6 ) )
% 11.90/6.01 = ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4371_inter__union__distrib__left,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( plus_plus @ ( multiset @ A ) @ ( inter_mset @ A @ A4 @ B6 ) @ C5 )
% 11.90/6.01 = ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4372_inter__subset__eq__union,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] : ( subseteq_mset @ A @ ( inter_mset @ A @ A4 @ B6 ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4373_inter__mset__empty__distrib__left,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ( ( inter_mset @ A @ A4 @ C5 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( ( inter_mset @ A @ B6 @ C5 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4374_inter__mset__empty__distrib__right,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.01 ( ( ( inter_mset @ A @ A4 @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 = ( ( ( inter_mset @ A @ A4 @ B6 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 & ( ( inter_mset @ A @ A4 @ C5 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4375_sum__mset_Odistrib,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType] :
% 11.90/6.01 ( ( comm_monoid_add @ A )
% 11.90/6.01 => ! [G: B > A,H3: B > A,A4: multiset @ B] :
% 11.90/6.01 ( ( comm_m7189776963980413722m_mset @ A
% 11.90/6.01 @ ( image_mset @ B @ A
% 11.90/6.01 @ ^ [X3: B] : ( plus_plus @ A @ ( G @ X3 ) @ ( H3 @ X3 ) )
% 11.90/6.01 @ A4 ) )
% 11.90/6.01 = ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ H3 @ A4 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4376_size__eq__sum__mset,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( size_size @ ( multiset @ A ) )
% 11.90/6.01 = ( ^ [M8: multiset @ A] :
% 11.90/6.01 ( comm_m7189776963980413722m_mset @ nat
% 11.90/6.01 @ ( image_mset @ A @ nat
% 11.90/6.01 @ ^ [A7: A] : ( one_one @ nat )
% 11.90/6.01 @ M8 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4377_multiset__union__diff__commute,axiom,
% 11.90/6.01 ! [A: $tType,B6: multiset @ A,C5: multiset @ A,A4: multiset @ A] :
% 11.90/6.01 ( ( ( inter_mset @ A @ B6 @ C5 )
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ C5 )
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A4 @ C5 ) @ B6 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4378_sum__mset_Oeq__fold,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( comm_monoid_add @ A )
% 11.90/6.01 => ( ( comm_m7189776963980413722m_mset @ A )
% 11.90/6.01 = ( fold_mset @ A @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4379_Chains__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( chains @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( collect @ ( set @ A )
% 11.90/6.01 @ ^ [C7: set @ A] :
% 11.90/6.01 ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ C7 )
% 11.90/6.01 => ! [Y3: A] :
% 11.90/6.01 ( ( member @ A @ Y3 @ C7 )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 11.90/6.01 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4380_Chains__subset_H,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 11.90/6.01 => ( ord_less_eq @ ( set @ ( set @ A ) )
% 11.90/6.01 @ ( collect @ ( set @ A )
% 11.90/6.01 @ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
% 11.90/6.01 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) )
% 11.90/6.01 @ ( chains @ A @ R ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4381_Chains__subset,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R )
% 11.90/6.01 @ ( collect @ ( set @ A )
% 11.90/6.01 @ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
% 11.90/6.01 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4382_Chains__alt__def,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
% 11.90/6.01 => ( ( chains @ A @ R )
% 11.90/6.01 = ( collect @ ( set @ A )
% 11.90/6.01 @ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
% 11.90/6.01 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4383_iter_H_Ocases,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( quickcheck_random @ A )
% 11.90/6.01 => ! [X: product_prod @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) )] :
% 11.90/6.01 ~ ! [T4: itself @ A,Nrandom: code_natural,Sz: code_natural,Seed: product_prod @ code_natural @ code_natural] :
% 11.90/6.01 ( X
% 11.90/6.01 != ( product_Pair @ ( itself @ A ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ) @ T4 @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ Nrandom @ ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ Sz @ Seed ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4384_multp__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( multp @ A )
% 11.90/6.01 = ( ^ [R3: A > A > $o,M8: multiset @ A,N9: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M8 @ N9 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4385_AboveS__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( order_AboveS @ A )
% 11.90/6.01 = ( ^ [R3: set @ ( product_prod @ A @ A ),A9: set @ A] :
% 11.90/6.01 ( collect @ A
% 11.90/6.01 @ ^ [B5: A] :
% 11.90/6.01 ( ( member @ A @ B5 @ ( field2 @ A @ R3 ) )
% 11.90/6.01 & ! [X3: A] :
% 11.90/6.01 ( ( member @ A @ X3 @ A9 )
% 11.90/6.01 => ( ( B5 != X3 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B5 ) @ R3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4386_lexordp__conv__lexord,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( linorder @ A )
% 11.90/6.01 => ( ( ord_lexordp @ A )
% 11.90/6.01 = ( ^ [Xs3: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4387_select__weight__cons__zero,axiom,
% 11.90/6.01 ! [A: $tType,X: A,Xs: list @ ( product_prod @ code_natural @ A )] :
% 11.90/6.01 ( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X ) @ Xs ) )
% 11.90/6.01 = ( select_weight @ A @ Xs ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4388_one__step__implies__multp,axiom,
% 11.90/6.01 ! [A: $tType,J4: multiset @ A,K2: multiset @ A,R: A > A > $o,I4: multiset @ A] :
% 11.90/6.01 ( ( J4
% 11.90/6.01 != ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ! [X2: A] :
% 11.90/6.01 ( ( member @ A @ X2 @ ( set_mset @ A @ K2 ) )
% 11.90/6.01 => ? [Xa2: A] :
% 11.90/6.01 ( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
% 11.90/6.01 & ( R @ X2 @ Xa2 ) ) )
% 11.90/6.01 => ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ I4 @ K2 ) @ ( plus_plus @ ( multiset @ A ) @ I4 @ J4 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4389_select__weight__select,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) )
% 11.90/6.01 = ( select @ A @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4390_wo__rel_Osuc__greater,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,B2: A] :
% 11.90/6.01 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/6.01 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 11.90/6.01 => ( ( ( order_AboveS @ A @ R @ B6 )
% 11.90/6.01 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.01 => ( ( member @ A @ B2 @ B6 )
% 11.90/6.01 => ( ( ( bNF_Wellorder_wo_suc @ A @ R @ B6 )
% 11.90/6.01 != B2 )
% 11.90/6.01 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) ) @ R ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4391_wo__rel_Oequals__suc__AboveS,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ A,A3: A] :
% 11.90/6.01 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/6.01 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R ) )
% 11.90/6.01 => ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B6 ) )
% 11.90/6.01 => ( ! [A16: A] :
% 11.90/6.01 ( ( member @ A @ A16 @ ( order_AboveS @ A @ R @ B6 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A16 ) @ R ) )
% 11.90/6.01 => ( A3
% 11.90/6.01 = ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4392_wo__rel_Osuc__least__AboveS,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B6: set @ A] :
% 11.90/6.01 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/6.01 => ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B6 ) )
% 11.90/6.01 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R @ B6 ) @ A3 ) @ R ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4393_wo__rel_Osuc__ofilter__in,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
% 11.90/6.01 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/6.01 => ( ( order_ofilter @ A @ R @ A4 )
% 11.90/6.01 => ( ( ( order_AboveS @ A @ R @ A4 )
% 11.90/6.01 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ A4 ) ) @ R )
% 11.90/6.01 => ( ( B2
% 11.90/6.01 != ( bNF_Wellorder_wo_suc @ A @ R @ A4 ) )
% 11.90/6.01 => ( member @ A @ B2 @ A4 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4394_slice__head,axiom,
% 11.90/6.01 ! [A: $tType,From: nat,To: nat,Xs: list @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ From @ To )
% 11.90/6.01 => ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.01 => ( ( hd @ A @ ( slice @ A @ From @ To @ Xs ) )
% 11.90/6.01 = ( nth @ A @ Xs @ From ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4395_gen__length__def,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( gen_length @ A )
% 11.90/6.01 = ( ^ [N5: nat,Xs3: list @ A] : ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4396_subset__mset_Osum__pos2,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,I4: set @ B,I: B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ I4 )
% 11.90/6.01 => ( ( member @ B @ I @ I4 )
% 11.90/6.01 => ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I ) )
% 11.90/6.01 => ( ! [I2: B] :
% 11.90/6.01 ( ( member @ B @ I2 @ I4 )
% 11.90/6.01 => ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 11.90/6.01 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I4 ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4397_subset__mset_Oadd__less__cancel__right,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 11.90/6.01 = ( subset_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4398_subset__mset_Oadd__less__cancel__left,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 11.90/6.01 = ( subset_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4399_subset__mset_Oadd__less__same__cancel1,axiom,
% 11.90/6.01 ! [A: $tType,B2: multiset @ A,A3: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) @ B2 )
% 11.90/6.01 = ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4400_subset__mset_Oadd__less__same__cancel2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ B2 )
% 11.90/6.01 = ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4401_subset__mset_Oless__add__same__cancel1,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) )
% 11.90/6.01 = ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4402_subset__mset_Oless__add__same__cancel2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A3 ) )
% 11.90/6.01 = ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4403_subset__mset_Oadd__less__imp__less__right,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
% 11.90/6.01 => ( subset_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4404_subset__mset_Oadd__less__imp__less__left,axiom,
% 11.90/6.01 ! [A: $tType,C2: multiset @ A,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
% 11.90/6.01 => ( subset_mset @ A @ A3 @ B2 ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4405_subset__mset_Oadd__strict__right__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4406_subset__mset_Oadd__strict__left__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A3 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4407_subset__mset_Oadd__strict__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( subset_mset @ A @ C2 @ D3 )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4408_hd__zip,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( Ys2
% 11.90/6.01 != ( nil @ B ) )
% 11.90/6.01 => ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
% 11.90/6.01 = ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys2 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4409_subset__mset_Oadd__le__less__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( subset_mset @ A @ C2 @ D3 )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4410_subset__mset_Oadd__less__le__mono,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A,D3: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ C2 @ D3 )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D3 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4411_subset__mset_OlessE,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ B2 )
% 11.90/6.01 => ~ ! [C4: multiset @ A] :
% 11.90/6.01 ( ( B2
% 11.90/6.01 = ( plus_plus @ ( multiset @ A ) @ A3 @ C4 ) )
% 11.90/6.01 => ( C4
% 11.90/6.01 = ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4412_subset__mset_Oadd__neg__neg,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4413_subset__mset_Oadd__pos__pos,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 11.90/6.01 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4414_subset__mset_Opos__add__strict,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subset_mset @ A @ B2 @ C2 )
% 11.90/6.01 => ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4415_subset__implies__mult,axiom,
% 11.90/6.01 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( subset_mset @ A @ A4 @ B6 )
% 11.90/6.01 => ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ A4 @ B6 ) @ ( mult @ A @ R ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4416_subset__mset_Oadd__strict__increasing2,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subset_mset @ A @ B2 @ C2 )
% 11.90/6.01 => ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4417_subset__mset_Oadd__strict__increasing,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B2 @ C2 )
% 11.90/6.01 => ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A3 @ C2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4418_subset__mset_Oadd__pos__nonneg,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 11.90/6.01 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4419_subset__mset_Oadd__nonpos__neg,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4420_subset__mset_Oadd__nonneg__pos,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A3 )
% 11.90/6.01 => ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
% 11.90/6.01 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4421_subset__mset_Oadd__neg__nonpos,axiom,
% 11.90/6.01 ! [A: $tType,A3: multiset @ A,B2: multiset @ A] :
% 11.90/6.01 ( ( subset_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.01 => ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A3 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4422_sorted__hd__min,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( linorder @ A )
% 11.90/6.01 => ! [Xs: list @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
% 11.90/6.01 => ! [X7: A] :
% 11.90/6.01 ( ( member @ A @ X7 @ ( set2 @ A @ Xs ) )
% 11.90/6.01 => ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X7 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4423_sorted__hd__last,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( linorder @ A )
% 11.90/6.01 => ! [L: list @ A] :
% 11.90/6.01 ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
% 11.90/6.01 => ( ( L
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ord_less_eq @ A @ ( hd @ A @ L ) @ ( last @ A @ L ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4424_hd__last__singletonI,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( ( hd @ A @ Xs )
% 11.90/6.01 = ( last @ A @ Xs ) )
% 11.90/6.01 => ( ( distinct @ A @ Xs )
% 11.90/6.01 => ( Xs
% 11.90/6.01 = ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4425_hd__butlast,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( ord_less @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.01 => ( ( hd @ A @ ( butlast @ A @ Xs ) )
% 11.90/6.01 = ( hd @ A @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4426_subset__mset_Osum__pos,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,I4: set @ B,F: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ I4 )
% 11.90/6.01 => ( ( I4
% 11.90/6.01 != ( bot_bot @ ( set @ B ) ) )
% 11.90/6.01 => ( ! [I2: B] :
% 11.90/6.01 ( ( member @ B @ I2 @ I4 )
% 11.90/6.01 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I2 ) ) )
% 11.90/6.01 => ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I4 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4427_subset__mset_Osum__strict__mono,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,A4: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
% 11.90/6.01 ( ( finite_finite2 @ B @ A4 )
% 11.90/6.01 => ( ( A4
% 11.90/6.01 != ( bot_bot @ ( set @ B ) ) )
% 11.90/6.01 => ( ! [X2: B] :
% 11.90/6.01 ( ( member @ B @ X2 @ A4 )
% 11.90/6.01 => ( subset_mset @ A @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 11.90/6.01 => ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A4 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ A4 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4428_ofilter__ordLess,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_ofilter @ A @ R @ A4 )
% 11.90/6.01 => ( ( ord_less @ ( set @ A ) @ A4 @ ( field2 @ A @ R ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/6.01 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.90/6.01 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.01 @ ( product_Sigma @ A @ A @ A4
% 11.90/6.01 @ ^ [Uu: A] : A4 ) )
% 11.90/6.01 @ R )
% 11.90/6.01 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4429_ofilter__subset__ordLess,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_ofilter @ A @ R @ A4 )
% 11.90/6.01 => ( ( order_ofilter @ A @ R @ B6 )
% 11.90/6.01 => ( ( ord_less @ ( set @ A ) @ A4 @ B6 )
% 11.90/6.01 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/6.01 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.90/6.01 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.01 @ ( product_Sigma @ A @ A @ A4
% 11.90/6.01 @ ^ [Uu: A] : A4 ) )
% 11.90/6.01 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.01 @ ( product_Sigma @ A @ A @ B6
% 11.90/6.01 @ ^ [Uu: A] : B6 ) ) )
% 11.90/6.01 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4430_ofilter__subset__ordLeq,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B6: set @ A] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_ofilter @ A @ R @ A4 )
% 11.90/6.01 => ( ( order_ofilter @ A @ R @ B6 )
% 11.90/6.01 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.90/6.01 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/6.01 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.90/6.01 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.01 @ ( product_Sigma @ A @ A @ A4
% 11.90/6.01 @ ^ [Uu: A] : A4 ) )
% 11.90/6.01 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.01 @ ( product_Sigma @ A @ A @ B6
% 11.90/6.01 @ ^ [Uu: A] : B6 ) ) )
% 11.90/6.01 @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4431_Nitpick_Osize__list__simp_I1_J,axiom,
% 11.90/6.01 ! [A: $tType] :
% 11.90/6.01 ( ( size_list @ A )
% 11.90/6.01 = ( ^ [F2: A > nat,Xs3: list @ A] :
% 11.90/6.01 ( if @ nat
% 11.90/6.01 @ ( Xs3
% 11.90/6.01 = ( nil @ A ) )
% 11.90/6.01 @ ( zero_zero @ nat )
% 11.90/6.01 @ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4432_length__tl,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
% 11.90/6.01 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4433_in__hd__or__tl__conv,axiom,
% 11.90/6.01 ! [A: $tType,L: list @ A,X: A] :
% 11.90/6.01 ( ( L
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( ( X
% 11.90/6.01 = ( hd @ A @ L ) )
% 11.90/6.01 | ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
% 11.90/6.01 = ( member @ A @ X @ ( set2 @ A @ L ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4434_tl__replicate,axiom,
% 11.90/6.01 ! [A: $tType,N: nat,X: A] :
% 11.90/6.01 ( ( tl @ A @ ( replicate @ A @ N @ X ) )
% 11.90/6.01 = ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4435_rev__split__conv,axiom,
% 11.90/6.01 ! [A: $tType,L: list @ A] :
% 11.90/6.01 ( ( L
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( append @ A @ ( rev @ A @ ( tl @ A @ L ) ) @ ( cons @ A @ ( hd @ A @ L ) @ ( nil @ A ) ) )
% 11.90/6.01 = ( rev @ A @ L ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4436_ordLess__irreflexive,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4437_ordLess__transitive,axiom,
% 11.90/6.01 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4438_ordLeq__transitive,axiom,
% 11.90/6.01 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4439_not__ordLess__ordLeq,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.01 => ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4440_ordLess__imp__ordLeq,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4441_ordLeq__ordLess__trans,axiom,
% 11.90/6.01 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4442_ordLess__ordLeq__trans,axiom,
% 11.90/6.01 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4443_not__ordLess__iff__ordLeq,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.01 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4444_not__ordLeq__iff__ordLess,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.01 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
% 11.90/6.01 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4445_ordLess__or__ordLeq,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.01 | ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4446_in__set__tlD,axiom,
% 11.90/6.01 ! [A: $tType,X: A,Xs: list @ A] :
% 11.90/6.01 ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
% 11.90/6.01 => ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4447_tl__obtain__elem,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( ( tl @ A @ Xs )
% 11.90/6.01 = ( nil @ A ) )
% 11.90/6.01 => ~ ! [E3: A] :
% 11.90/6.01 ( Xs
% 11.90/6.01 != ( cons @ A @ E3 @ ( nil @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4448_not__hd__in__tl,axiom,
% 11.90/6.01 ! [A: $tType,X: A,Xs: list @ A] :
% 11.90/6.01 ( ( X
% 11.90/6.01 != ( hd @ A @ Xs ) )
% 11.90/6.01 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.01 => ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4449_tl__last,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( ( tl @ A @ Xs )
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( last @ A @ Xs )
% 11.90/6.01 = ( last @ A @ ( tl @ A @ Xs ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4450_rev__butlast__is__tl__rev,axiom,
% 11.90/6.01 ! [A: $tType,L: list @ A] :
% 11.90/6.01 ( ( rev @ A @ ( butlast @ A @ L ) )
% 11.90/6.01 = ( tl @ A @ ( rev @ A @ L ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4451_tl__subset,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,A4: set @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
% 11.90/6.01 => ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( tl @ A @ Xs ) ) @ A4 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4452_Misc_Onth__tl,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,N: nat] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
% 11.90/6.01 = ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4453_list__take__induct__tl2,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,P: B > A > $o] :
% 11.90/6.01 ( ( ( size_size @ ( list @ A ) @ Xs )
% 11.90/6.01 = ( size_size @ ( list @ B ) @ Ys2 ) )
% 11.90/6.01 => ( ! [N3: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.01 => ( P @ ( nth @ B @ Ys2 @ N3 ) @ ( nth @ A @ Xs @ N3 ) ) )
% 11.90/6.01 => ! [N10: nat] :
% 11.90/6.01 ( ( ord_less @ nat @ N10 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
% 11.90/6.01 => ( P @ ( nth @ B @ ( tl @ B @ Ys2 ) @ N10 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N10 ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4454_exists__minim__Well__order,axiom,
% 11.90/6.01 ! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
% 11.90/6.01 ( ( R5
% 11.90/6.01 != ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
% 11.90/6.01 => ( ! [X2: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 11.90/6.01 => ( order_well_order_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
% 11.90/6.01 => ? [X2: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 11.90/6.01 & ! [Xa2: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4455_ordLeq__total,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.01 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.01 | ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4456_ordLeq__reflexive,axiom,
% 11.90/6.01 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4457_ordLeq__Well__order__simp,axiom,
% 11.90/6.01 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.01 => ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 & ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4458_tl__take,axiom,
% 11.90/6.01 ! [A: $tType,N: nat,Xs: list @ A] :
% 11.90/6.01 ( ( tl @ A @ ( take @ A @ N @ Xs ) )
% 11.90/6.01 = ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4459_distinct__hd__tl,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A,X: A] :
% 11.90/6.01 ( ( distinct @ A @ Xs )
% 11.90/6.01 => ( ( X
% 11.90/6.01 = ( hd @ A @ Xs ) )
% 11.90/6.01 => ~ ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4460_butlast__rev__tl,axiom,
% 11.90/6.01 ! [A: $tType,Xs: list @ A] :
% 11.90/6.01 ( ( Xs
% 11.90/6.01 != ( nil @ A ) )
% 11.90/6.01 => ( ( butlast @ A @ ( rev @ A @ Xs ) )
% 11.90/6.01 = ( rev @ A @ ( tl @ A @ Xs ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4461_finite__ordLess__infinite,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.01 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.01 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.01 => ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.90/6.01 => ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R4 ) )
% 11.90/6.01 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% 11.90/6.01
% 11.90/6.01 thf(fact_4462_ordLeq__iff__ordLess__Restr,axiom,
% 11.90/6.01 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.02 = ( ! [X3: A] :
% 11.90/6.02 ( ( member @ A @ X3 @ ( field2 @ A @ R ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
% 11.90/6.02 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
% 11.90/6.02 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.02 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
% 11.90/6.02 @ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) )
% 11.90/6.02 @ R4 )
% 11.90/6.02 @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4463_underS__Restr__ordLess,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.90/6.02 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( ( ( field2 @ A @ R )
% 11.90/6.02 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/6.02 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.90/6.02 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.02 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ A3 )
% 11.90/6.02 @ ^ [Uu: A] : ( order_underS @ A @ R @ A3 ) ) )
% 11.90/6.02 @ R )
% 11.90/6.02 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4464_horner__sum__append,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_semiring_1 @ A )
% 11.90/6.02 => ! [F: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
% 11.90/6.02 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys2 ) )
% 11.90/6.02 = ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Ys2 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4465_Code__Numeral_Osub__code_I4_J,axiom,
% 11.90/6.02 ! [N: num] :
% 11.90/6.02 ( ( code_sub @ one2 @ ( bit0 @ N ) )
% 11.90/6.02 = ( code_Neg @ ( bitM @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4466_horner__sum__simps_I2_J,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_semiring_0 @ A )
% 11.90/6.02 => ! [F: B > A,A3: A,X: B,Xs: list @ B] :
% 11.90/6.02 ( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
% 11.90/6.02 = ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4467_underS__E,axiom,
% 11.90/6.02 ! [A: $tType,I: A,R5: set @ ( product_prod @ A @ A ),J: A] :
% 11.90/6.02 ( ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) )
% 11.90/6.02 => ( ( I != J )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4468_underS__I,axiom,
% 11.90/6.02 ! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( I != J )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
% 11.90/6.02 => ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4469_underS__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( order_underS @ A )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 11.90/6.02 ( collect @ A
% 11.90/6.02 @ ^ [B5: A] :
% 11.90/6.02 ( ( B5 != A7 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4470_Code__Numeral_Odup__code_I3_J,axiom,
% 11.90/6.02 ! [N: num] :
% 11.90/6.02 ( ( code_dup @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4471_less__integer__code_I3_J,axiom,
% 11.90/6.02 ! [L: num] :
% 11.90/6.02 ~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4472_less__integer__code_I7_J,axiom,
% 11.90/6.02 ! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4473_plus__integer__code_I6_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( plus_plus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_Neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4474_less__eq__integer__code_I7_J,axiom,
% 11.90/6.02 ! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4475_less__eq__integer__code_I3_J,axiom,
% 11.90/6.02 ! [L: num] :
% 11.90/6.02 ~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4476_less__integer__code_I9_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 11.90/6.02 = ( ord_less @ num @ L @ K ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4477_less__eq__integer__code_I9_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 11.90/6.02 = ( ord_less_eq @ num @ L @ K ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4478_minus__integer__code_I6_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( minus_minus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_sub @ N @ M2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4479_divmod__abs__code_I2_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
% 11.90/6.02 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4480_horner__sum__foldr,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_semiring_0 @ A )
% 11.90/6.02 => ( ( groups4207007520872428315er_sum @ B @ A )
% 11.90/6.02 = ( ^ [F2: B > A,A7: A,Xs3: list @ B] :
% 11.90/6.02 ( foldr @ B @ A
% 11.90/6.02 @ ^ [X3: B,B5: A] : ( plus_plus @ A @ ( F2 @ X3 ) @ ( times_times @ A @ A7 @ B5 ) )
% 11.90/6.02 @ Xs3
% 11.90/6.02 @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4481_underS__incl__iff,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/6.02 ( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.90/6.02 => ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) )
% 11.90/6.02 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4482_Code__Numeral_Osub__code_I5_J,axiom,
% 11.90/6.02 ! [N: num] :
% 11.90/6.02 ( ( code_sub @ one2 @ ( bit1 @ N ) )
% 11.90/6.02 = ( code_Neg @ ( bit0 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4483_ordLess__iff__ordIso__Restr,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 11.90/6.02 = ( ? [X3: A] :
% 11.90/6.02 ( ( member @ A @ X3 @ ( field2 @ A @ R ) )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.90/6.02 @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4
% 11.90/6.02 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.90/6.02 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
% 11.90/6.02 @ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) ) )
% 11.90/6.02 @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4484_sum__list__update,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( ordere1170586879665033532d_diff @ A )
% 11.90/6.02 => ! [K: nat,Xs: list @ A,X: A] :
% 11.90/6.02 ( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.02 => ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
% 11.90/6.02 = ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4485_length__product__lists,axiom,
% 11.90/6.02 ! [B: $tType,Xss: list @ ( list @ B )] :
% 11.90/6.02 ( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
% 11.90/6.02 = ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4486_horner__sum__transfer,axiom,
% 11.90/6.02 ! [C: $tType,A: $tType,B: $tType,D: $tType] :
% 11.90/6.02 ( ( ( comm_semiring_0 @ B )
% 11.90/6.02 & ( comm_semiring_0 @ A ) )
% 11.90/6.02 => ! [A4: A > B > $o,B6: C > D > $o] :
% 11.90/6.02 ( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 11.90/6.02 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 11.90/6.02 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 11.90/6.02 => ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B6 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B6 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4487_sum__list_OCons,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_add @ A )
% 11.90/6.02 => ! [X: A,Xs: list @ A] :
% 11.90/6.02 ( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
% 11.90/6.02 = ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4488_sum__list__append,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_add @ A )
% 11.90/6.02 => ! [Xs: list @ A,Ys2: list @ A] :
% 11.90/6.02 ( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys2 ) )
% 11.90/6.02 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys2 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4489_ordIso__transitive,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4490_ordIso__symmetric,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4491_sum__list__transfer,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( ( monoid_add @ B )
% 11.90/6.02 & ( monoid_add @ A ) )
% 11.90/6.02 => ! [A4: A > B > $o] :
% 11.90/6.02 ( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
% 11.90/6.02 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
% 11.90/6.02 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4492_Misc_Olist__all2__induct,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L4: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
% 11.90/6.02 ( ( list_all2 @ A @ B @ P @ L @ L4 )
% 11.90/6.02 => ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
% 11.90/6.02 => ( ! [X2: A,X5: B,Ls2: list @ A,Ls3: list @ B] :
% 11.90/6.02 ( ( P @ X2 @ X5 )
% 11.90/6.02 => ( ( list_all2 @ A @ B @ P @ Ls2 @ Ls3 )
% 11.90/6.02 => ( ( Q @ Ls2 @ Ls3 )
% 11.90/6.02 => ( Q @ ( cons @ A @ X2 @ Ls2 ) @ ( cons @ B @ X5 @ Ls3 ) ) ) ) )
% 11.90/6.02 => ( Q @ L @ L4 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4493_ordIso__iff__ordLeq,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 = ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4494_ordIso__imp__ordLeq,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4495_ordIso__ordLeq__trans,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4496_ordLeq__ordIso__trans,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4497_not__ordLess__ordIso,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.02 => ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4498_ordIso__ordLess__trans,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4499_ordLess__ordIso__trans,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),R9: set @ ( product_prod @ C @ C )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4500_sum__list__addf,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_monoid_add @ A )
% 11.90/6.02 => ! [F: B > A,G: B > A,Xs: list @ B] :
% 11.90/6.02 ( ( groups8242544230860333062m_list @ A
% 11.90/6.02 @ ( map @ B @ A
% 11.90/6.02 @ ^ [X3: B] : ( plus_plus @ A @ ( F @ X3 ) @ ( G @ X3 ) )
% 11.90/6.02 @ Xs ) )
% 11.90/6.02 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4501_finite__well__order__on__ordIso,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 => ( ( order_well_order_on @ A @ A4 @ R )
% 11.90/6.02 => ( ( order_well_order_on @ A @ A4 @ R4 )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4502_ordIso__reflexive,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4503_uminus__sum__list__map,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( ab_group_add @ A )
% 11.90/6.02 => ! [F: B > A,Xs: list @ B] :
% 11.90/6.02 ( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
% 11.90/6.02 = ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4504_ordLeq__iff__ordLess__or__ordIso,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.02 = ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.02 | ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4505_sum__list_Oeq__foldr,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_add @ A )
% 11.90/6.02 => ( ( groups8242544230860333062m_list @ A )
% 11.90/6.02 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4506_internalize__ordLeq,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.90/6.02 = ( ? [P6: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4507_internalize__ordLess,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.02 = ( ? [P6: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( field2 @ B @ R ) )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R4 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4508_card__length__sum__list__rec,axiom,
% 11.90/6.02 ! [M2: nat,N6: nat] :
% 11.90/6.02 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
% 11.90/6.02 => ( ( finite_card @ ( list @ nat )
% 11.90/6.02 @ ( collect @ ( list @ nat )
% 11.90/6.02 @ ^ [L3: list @ nat] :
% 11.90/6.02 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 11.90/6.02 = M2 )
% 11.90/6.02 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 11.90/6.02 = N6 ) ) ) )
% 11.90/6.02 = ( plus_plus @ nat
% 11.90/6.02 @ ( finite_card @ ( list @ nat )
% 11.90/6.02 @ ( collect @ ( list @ nat )
% 11.90/6.02 @ ^ [L3: list @ nat] :
% 11.90/6.02 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 11.90/6.02 = ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) )
% 11.90/6.02 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 11.90/6.02 = N6 ) ) ) )
% 11.90/6.02 @ ( finite_card @ ( list @ nat )
% 11.90/6.02 @ ( collect @ ( list @ nat )
% 11.90/6.02 @ ^ [L3: list @ nat] :
% 11.90/6.02 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 11.90/6.02 = M2 )
% 11.90/6.02 & ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L3 ) @ ( one_one @ nat ) )
% 11.90/6.02 = N6 ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4509_size__list__conv__sum__list,axiom,
% 11.90/6.02 ! [B: $tType] :
% 11.90/6.02 ( ( size_list @ B )
% 11.90/6.02 = ( ^ [F2: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F2 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4510_card__length__sum__list,axiom,
% 11.90/6.02 ! [M2: nat,N6: nat] :
% 11.90/6.02 ( ( finite_card @ ( list @ nat )
% 11.90/6.02 @ ( collect @ ( list @ nat )
% 11.90/6.02 @ ^ [L3: list @ nat] :
% 11.90/6.02 ( ( ( size_size @ ( list @ nat ) @ L3 )
% 11.90/6.02 = M2 )
% 11.90/6.02 & ( ( groups8242544230860333062m_list @ nat @ L3 )
% 11.90/6.02 = N6 ) ) ) )
% 11.90/6.02 = ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M2 ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4511_sum__list__Suc,axiom,
% 11.90/6.02 ! [A: $tType,F: A > nat,Xs: list @ A] :
% 11.90/6.02 ( ( groups8242544230860333062m_list @ nat
% 11.90/6.02 @ ( map @ A @ nat
% 11.90/6.02 @ ^ [X3: A] : ( suc @ ( F @ X3 ) )
% 11.90/6.02 @ Xs ) )
% 11.90/6.02 = ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4512_select__weight__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( select_weight @ A )
% 11.90/6.02 = ( ^ [Xs3: list @ ( product_prod @ code_natural @ A )] :
% 11.90/6.02 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs3 ) ) )
% 11.90/6.02 @ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K4 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4513_prod__list__transfer,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( ( monoid_mult @ B )
% 11.90/6.02 & ( monoid_mult @ A ) )
% 11.90/6.02 => ! [A4: A > B > $o] :
% 11.90/6.02 ( ( A4 @ ( one_one @ A ) @ ( one_one @ B ) )
% 11.90/6.02 => ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
% 11.90/6.02 => ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4514_dir__image__ordIso,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
% 11.90/6.02 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( ( inj_on @ A @ B @ F @ ( field2 @ A @ R ) )
% 11.90/6.02 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4515_underS__incr,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/6.02 ( ( trans @ A @ R )
% 11.90/6.02 => ( ( antisym @ A @ R )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/6.02 => ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4516_prod__list_ONil,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_mult @ A )
% 11.90/6.02 => ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
% 11.90/6.02 = ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4517_antisymD,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.90/6.02 ( ( antisym @ A @ R )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
% 11.90/6.02 => ( A3 = B2 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4518_antisymI,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ! [X2: A,Y4: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X2 ) @ R )
% 11.90/6.02 => ( X2 = Y4 ) ) )
% 11.90/6.02 => ( antisym @ A @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4519_antisym__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( antisym @ A )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ! [X3: A,Y3: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R3 )
% 11.90/6.02 => ( X3 = Y3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4520_dir__image__def,axiom,
% 11.90/6.02 ! [A2: $tType,A: $tType] :
% 11.90/6.02 ( ( bNF_We2720479622203943262_image @ A @ A2 )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ A ),F2: A > A2] :
% 11.90/6.02 ( collect @ ( product_prod @ A2 @ A2 )
% 11.90/6.02 @ ^ [Uu: product_prod @ A2 @ A2] :
% 11.90/6.02 ? [A7: A,B5: A] :
% 11.90/6.02 ( ( Uu
% 11.90/6.02 = ( product_Pair @ A2 @ A2 @ ( F2 @ A7 ) @ ( F2 @ B5 ) ) )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4521_prod__list_Oeq__foldr,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_mult @ A )
% 11.90/6.02 => ( ( groups5270119922927024881d_list @ A )
% 11.90/6.02 = ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4522_pick__same,axiom,
% 11.90/6.02 ! [A: $tType,L: nat,Xs: list @ A] :
% 11.90/6.02 ( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.90/6.02 => ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
% 11.90/6.02 = ( nth @ A @ Xs @ L ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4523_flip__pred,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A4: set @ ( product_prod @ A @ B ),R5: B > A > $o] :
% 11.90/6.02 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( conversep @ B @ A @ R5 ) ) ) )
% 11.90/6.02 => ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
% 11.90/6.02 @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
% 11.90/6.02 @ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
% 11.90/6.02 @ ^ [X3: A,Y3: B] : ( product_Pair @ B @ A @ Y3 @ X3 ) )
% 11.90/6.02 @ A4 )
% 11.90/6.02 @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4524_prod__set__simps_I1_J,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,X: A,Y: B] :
% 11.90/6.02 ( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 11.90/6.02 = ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4525_prod__set__simps_I2_J,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,X: A,Y: B] :
% 11.90/6.02 ( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
% 11.90/6.02 = ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4526_natural__of__nat__of__nat,axiom,
% 11.90/6.02 ( code_natural_of_nat
% 11.90/6.02 = ( semiring_1_of_nat @ code_natural ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4527_division__segment__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [X: nat] :
% 11.90/6.02 ( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4528_set__bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se5668285175392031749et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4529_flip__bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4530_unset__bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se2638667681897837118et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4531_euclidean__size__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [X: nat] :
% 11.90/6.02 ( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( euclid6346220572633701492n_size @ nat @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4532_bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [X: nat] :
% 11.90/6.02 ( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( bit_se5641148757651400278ts_bit @ nat @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4533_natural__of__nat__cases,axiom,
% 11.90/6.02 ! [X: code_natural] :
% 11.90/6.02 ~ ! [Y4: nat] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( code_natural_of_nat @ Y4 ) )
% 11.90/6.02 => ~ ( member @ nat @ Y4 @ ( top_top @ ( set @ nat ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4534_natural__of__nat__induct,axiom,
% 11.90/6.02 ! [P: code_natural > $o,X: code_natural] :
% 11.90/6.02 ( ! [Y4: nat] :
% 11.90/6.02 ( ( member @ nat @ Y4 @ ( top_top @ ( set @ nat ) ) )
% 11.90/6.02 => ( P @ ( code_natural_of_nat @ Y4 ) ) )
% 11.90/6.02 => ( P @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4535_natural__of__nat__inject,axiom,
% 11.90/6.02 ! [X: nat,Y: nat] :
% 11.90/6.02 ( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
% 11.90/6.02 => ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 11.90/6.02 => ( ( ( code_natural_of_nat @ X )
% 11.90/6.02 = ( code_natural_of_nat @ Y ) )
% 11.90/6.02 = ( X = Y ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4536_take__bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4537_and__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4538_xor__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4539_or__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4540_mask__natural_Oabs__eq,axiom,
% 11.90/6.02 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 11.90/6.02 = ( ^ [X3: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4541_push__bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4542_drop__bit__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( bit_se4197421643247451524op_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4543_zero__natural__def,axiom,
% 11.90/6.02 ( ( zero_zero @ code_natural )
% 11.90/6.02 = ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4544_one__natural__def,axiom,
% 11.90/6.02 ( ( one_one @ code_natural )
% 11.90/6.02 = ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4545_less__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( ord_less @ nat @ Xa @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4546_plus__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( plus_plus @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4547_minus__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( minus_minus @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4548_less__eq__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( ord_less_eq @ nat @ Xa @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4549_times__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( times_times @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4550_divide__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( divide_divide @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4551_mask__natural__def,axiom,
% 11.90/6.02 ( ( bit_se2239418461657761734s_mask @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4552_modulo__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [Xa: nat,X: nat] :
% 11.90/6.02 ( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4553_select__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( select @ A )
% 11.90/6.02 = ( ^ [Xs3: list @ A] :
% 11.90/6.02 ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( code_natural_of_nat @ ( size_size @ ( list @ A ) @ Xs3 ) ) )
% 11.90/6.02 @ ^ [K4: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K4 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4554_Nat_Oabs__eq,axiom,
% 11.90/6.02 ! [X: int] :
% 11.90/6.02 ( ( code_Nat @ ( code_integer_of_int @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4555_Nat__def,axiom,
% 11.90/6.02 ( code_Nat
% 11.90/6.02 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4556_natural__of__integer_Oabs__eq,axiom,
% 11.90/6.02 ! [X: int] :
% 11.90/6.02 ( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4557_natural__of__nat__of__natural__inverse,axiom,
% 11.90/6.02 ! [N: code_natural] :
% 11.90/6.02 ( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
% 11.90/6.02 = N ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4558_nat__of__natural__of__nat__inverse,axiom,
% 11.90/6.02 ! [N: nat] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
% 11.90/6.02 = N ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4559_nat__of__natural__inverse,axiom,
% 11.90/6.02 ! [X: code_natural] :
% 11.90/6.02 ( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
% 11.90/6.02 = X ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4560_nat__of__natural__min,axiom,
% 11.90/6.02 ! [K: code_natural,L: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
% 11.90/6.02 = ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4561_nat__of__natural__max,axiom,
% 11.90/6.02 ! [K: code_natural,L: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
% 11.90/6.02 = ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4562_of__nat__of__natural,axiom,
% 11.90/6.02 ! [N: code_natural] :
% 11.90/6.02 ( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
% 11.90/6.02 = N ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4563_nat__of__natural__of__nat,axiom,
% 11.90/6.02 ! [N: nat] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 11.90/6.02 = N ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4564_zero__natural_Orep__eq,axiom,
% 11.90/6.02 ( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
% 11.90/6.02 = ( zero_zero @ nat ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4565_one__natural_Orep__eq,axiom,
% 11.90/6.02 ( ( code_nat_of_natural @ ( one_one @ code_natural ) )
% 11.90/6.02 = ( one_one @ nat ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4566_plus__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4567_minus__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4568_nat__of__natural__numeral,axiom,
% 11.90/6.02 ! [K: num] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
% 11.90/6.02 = ( numeral_numeral @ nat @ K ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4569_times__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4570_divide__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4571_modulo__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( modulo_modulo @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4572_division__segment__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X ) )
% 11.90/6.02 = ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4573_euclidean__size__natural_Orep__eq,axiom,
% 11.90/6.02 ( ( euclid6346220572633701492n_size @ code_natural )
% 11.90/6.02 = ( ^ [X3: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4574_unset__bit__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se2638667681897837118et_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4575_flip__bit__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se8732182000553998342ip_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4576_set__bit__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se5668285175392031749et_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4577_Nat_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_integer] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( code_Nat @ X ) )
% 11.90/6.02 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4578_push__bit__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se4730199178511100633sh_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4579_mask__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X ) )
% 11.90/6.02 = ( bit_se2239418461657761734s_mask @ nat @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4580_and__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4581_bit__natural_Orep__eq,axiom,
% 11.90/6.02 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 11.90/6.02 = ( ^ [X3: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4582_nat__of__natural__inject,axiom,
% 11.90/6.02 ! [X: code_natural,Y: code_natural] :
% 11.90/6.02 ( ( ( code_nat_of_natural @ X )
% 11.90/6.02 = ( code_nat_of_natural @ Y ) )
% 11.90/6.02 = ( X = Y ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4583_natural__eqI,axiom,
% 11.90/6.02 ! [M2: code_natural,N: code_natural] :
% 11.90/6.02 ( ( ( code_nat_of_natural @ M2 )
% 11.90/6.02 = ( code_nat_of_natural @ N ) )
% 11.90/6.02 => ( M2 = N ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4584_natural__eq__iff,axiom,
% 11.90/6.02 ( ( ^ [Y5: code_natural,Z4: code_natural] : ( Y5 = Z4 ) )
% 11.90/6.02 = ( ^ [M: code_natural,N5: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ M )
% 11.90/6.02 = ( code_nat_of_natural @ N5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4585_natural__cases,axiom,
% 11.90/6.02 ! [M2: code_natural] :
% 11.90/6.02 ~ ! [N3: nat] :
% 11.90/6.02 ( M2
% 11.90/6.02 != ( semiring_1_of_nat @ code_natural @ N3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4586_nat__of__natural,axiom,
% 11.90/6.02 ! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4587_nat__of__natural__cases,axiom,
% 11.90/6.02 ! [Y: nat] :
% 11.90/6.02 ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 11.90/6.02 => ~ ! [X2: code_natural] :
% 11.90/6.02 ( Y
% 11.90/6.02 != ( code_nat_of_natural @ X2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4588_nat__of__natural__induct,axiom,
% 11.90/6.02 ! [Y: nat,P: nat > $o] :
% 11.90/6.02 ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 11.90/6.02 => ( ! [X2: code_natural] : ( P @ ( code_nat_of_natural @ X2 ) )
% 11.90/6.02 => ( P @ Y ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4589_take__bit__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se2584673776208193580ke_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4590_or__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4591_xor__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4592_drop__bit__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: nat,Xa: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X @ Xa ) )
% 11.90/6.02 = ( bit_se4197421643247451524op_bit @ nat @ X @ ( code_nat_of_natural @ Xa ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4593_natural__of__integer_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_integer] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
% 11.90/6.02 = ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4594_less__natural_Orep__eq,axiom,
% 11.90/6.02 ( ( ord_less @ code_natural )
% 11.90/6.02 = ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4595_natural__of__nat__inverse,axiom,
% 11.90/6.02 ! [Y: nat] :
% 11.90/6.02 ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
% 11.90/6.02 => ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
% 11.90/6.02 = Y ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4596_type__definition__natural,axiom,
% 11.90/6.02 type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4597_or__natural__def,axiom,
% 11.90/6.02 ( ( bit_se1065995026697491101ons_or @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se1065995026697491101ons_or @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4598_xor__natural__def,axiom,
% 11.90/6.02 ( ( bit_se5824344971392196577ns_xor @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344971392196577ns_xor @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4599_and__natural__def,axiom,
% 11.90/6.02 ( ( bit_se5824344872417868541ns_and @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344872417868541ns_and @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4600_less__eq__natural_Orep__eq,axiom,
% 11.90/6.02 ( ( ord_less_eq @ code_natural )
% 11.90/6.02 = ( ^ [X3: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X3 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4601_division__segment__natural__def,axiom,
% 11.90/6.02 ( ( euclid7384307370059645450egment @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4602_euclidean__size__natural__def,axiom,
% 11.90/6.02 ( ( euclid6346220572633701492n_size @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4603_plus__natural__def,axiom,
% 11.90/6.02 ( ( plus_plus @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( plus_plus @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4604_minus__natural__def,axiom,
% 11.90/6.02 ( ( minus_minus @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( minus_minus @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4605_bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se5641148757651400278ts_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4606_less__natural__def,axiom,
% 11.90/6.02 ( ( ord_less @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4607_times__natural__def,axiom,
% 11.90/6.02 ( ( times_times @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( times_times @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4608_divide__natural__def,axiom,
% 11.90/6.02 ( ( divide_divide @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( divide_divide @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4609_take__bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se2584673776208193580ke_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2584673776208193580ke_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4610_push__bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se4730199178511100633sh_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4730199178511100633sh_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4611_less__eq__natural__def,axiom,
% 11.90/6.02 ( ( ord_less_eq @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( code_natural > $o ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ $o @ $o @ code_nat_of_natural @ ( id @ $o ) ) @ ( ord_less_eq @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4612_modulo__natural__def,axiom,
% 11.90/6.02 ( ( modulo_modulo @ code_natural )
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( modulo_modulo @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4613_drop__bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se4197421643247451524op_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4197421643247451524op_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4614_unset__bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se2638667681897837118et_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2638667681897837118et_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4615_flip__bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se8732182000553998342ip_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se8732182000553998342ip_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4616_set__bit__natural__def,axiom,
% 11.90/6.02 ( ( bit_se5668285175392031749et_bit @ code_natural )
% 11.90/6.02 = ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5668285175392031749et_bit @ nat ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4617_natural__decr,axiom,
% 11.90/6.02 ! [N: code_natural] :
% 11.90/6.02 ( ( N
% 11.90/6.02 != ( zero_zero @ code_natural ) )
% 11.90/6.02 => ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4618_natural__of__integer__def,axiom,
% 11.90/6.02 ( code_n4118661773612635043nteger
% 11.90/6.02 = ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4619_integer__of__natural__of__integer,axiom,
% 11.90/6.02 ! [K: code_integer] :
% 11.90/6.02 ( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
% 11.90/6.02 = ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4620_integer__of__natural_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural] :
% 11.90/6.02 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
% 11.90/6.02 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4621_int__of__integer__of__natural,axiom,
% 11.90/6.02 ! [N: code_natural] :
% 11.90/6.02 ( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
% 11.90/6.02 = ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4622_integer__of__natural__of__nat,axiom,
% 11.90/6.02 ! [N: nat] :
% 11.90/6.02 ( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
% 11.90/6.02 = ( semiring_1_of_nat @ code_integer @ N ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4623_natural__of__integer__of__natural,axiom,
% 11.90/6.02 ! [N: code_natural] :
% 11.90/6.02 ( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
% 11.90/6.02 = N ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4624_integer__of__natural__def,axiom,
% 11.90/6.02 ( code_i5400310926305786745atural
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4625_integer__of__natural_Oabs__eq,axiom,
% 11.90/6.02 ! [X: nat] :
% 11.90/6.02 ( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4626_Code__Numeral_Osize__nat__def,axiom,
% 11.90/6.02 code_size_nat = code_nat_of_natural ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4627_Code__Numeral_OSuc__def,axiom,
% 11.90/6.02 ( code_Suc
% 11.90/6.02 = ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4628_sum__mset_Ounion__disjoint,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_monoid_add @ A )
% 11.90/6.02 => ! [A4: multiset @ B,B6: multiset @ B,G: B > A] :
% 11.90/6.02 ( ( ( inter_mset @ B @ A4 @ B6 )
% 11.90/6.02 = ( zero_zero @ ( multiset @ B ) ) )
% 11.90/6.02 => ( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A4 @ B6 ) ) )
% 11.90/6.02 = ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A4 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ B6 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4629_natural_Oinject,axiom,
% 11.90/6.02 ! [Natural: code_natural,Natural2: code_natural] :
% 11.90/6.02 ( ( ( code_Suc @ Natural )
% 11.90/6.02 = ( code_Suc @ Natural2 ) )
% 11.90/6.02 = ( Natural = Natural2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4630_natural_Osimps_I7_J,axiom,
% 11.90/6.02 ! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
% 11.90/6.02 ( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 11.90/6.02 = ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4631_natural_Osimps_I5_J,axiom,
% 11.90/6.02 ! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
% 11.90/6.02 ( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
% 11.90/6.02 = ( F22 @ Natural ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4632_Suc_Orep__eq,axiom,
% 11.90/6.02 ! [X: code_natural] :
% 11.90/6.02 ( ( code_nat_of_natural @ ( code_Suc @ X ) )
% 11.90/6.02 = ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4633_union__sup__distrib__right,axiom,
% 11.90/6.02 ! [A: $tType,C5: multiset @ A,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( plus_plus @ ( multiset @ A ) @ C5 @ ( union_mset @ A @ A4 @ B6 ) )
% 11.90/6.02 = ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C5 @ A4 ) @ ( plus_plus @ ( multiset @ A ) @ C5 @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4634_sup__union__distrib__left,axiom,
% 11.90/6.02 ! [A: $tType,A4: multiset @ A,B6: multiset @ A,C5: multiset @ A] :
% 11.90/6.02 ( ( plus_plus @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) @ C5 )
% 11.90/6.02 = ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ C5 ) @ ( plus_plus @ ( multiset @ A ) @ B6 @ C5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4635_natural_Oexhaust,axiom,
% 11.90/6.02 ! [Y: code_natural] :
% 11.90/6.02 ( ( Y
% 11.90/6.02 != ( zero_zero @ code_natural ) )
% 11.90/6.02 => ~ ! [Natural3: code_natural] :
% 11.90/6.02 ( Y
% 11.90/6.02 != ( code_Suc @ Natural3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4636_natural_Odistinct_I1_J,axiom,
% 11.90/6.02 ! [Natural2: code_natural] :
% 11.90/6.02 ( ( zero_zero @ code_natural )
% 11.90/6.02 != ( code_Suc @ Natural2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4637_natural_Odistinct_I2_J,axiom,
% 11.90/6.02 ! [Natural4: code_natural] :
% 11.90/6.02 ( ( code_Suc @ Natural4 )
% 11.90/6.02 != ( zero_zero @ code_natural ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4638_union__mset__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( union_mset @ A )
% 11.90/6.02 = ( ^ [A9: multiset @ A,B9: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A9 @ ( minus_minus @ ( multiset @ A ) @ B9 @ A9 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4639_union__diff__inter__eq__sup,axiom,
% 11.90/6.02 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( inter_mset @ A @ A4 @ B6 ) )
% 11.90/6.02 = ( union_mset @ A @ A4 @ B6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4640_union__diff__sup__eq__inter,axiom,
% 11.90/6.02 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) @ ( union_mset @ A @ A4 @ B6 ) )
% 11.90/6.02 = ( inter_mset @ A @ A4 @ B6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4641_size__Un__Int,axiom,
% 11.90/6.02 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( size_size @ ( multiset @ A ) @ B6 ) )
% 11.90/6.02 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) ) @ ( size_size @ ( multiset @ A ) @ ( inter_mset @ A @ A4 @ B6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4642_Suc_Oabs__eq,axiom,
% 11.90/6.02 ! [X: nat] :
% 11.90/6.02 ( ( code_Suc @ ( code_natural_of_nat @ X ) )
% 11.90/6.02 = ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4643_Suc__natural__minus__one,axiom,
% 11.90/6.02 ! [N: code_natural] :
% 11.90/6.02 ( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
% 11.90/6.02 = N ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4644_size__Un__disjoint,axiom,
% 11.90/6.02 ! [A: $tType,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( ( inter_mset @ A @ A4 @ B6 )
% 11.90/6.02 = ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.02 => ( ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A4 @ B6 ) )
% 11.90/6.02 = ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A4 ) @ ( size_size @ ( multiset @ A ) @ B6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4645_random__aux__set_Ocases,axiom,
% 11.90/6.02 ! [X: product_prod @ code_natural @ code_natural] :
% 11.90/6.02 ( ! [J3: code_natural] :
% 11.90/6.02 ( X
% 11.90/6.02 != ( product_Pair @ code_natural @ code_natural @ ( zero_zero @ code_natural ) @ J3 ) )
% 11.90/6.02 => ~ ! [I2: code_natural,J3: code_natural] :
% 11.90/6.02 ( X
% 11.90/6.02 != ( product_Pair @ code_natural @ code_natural @ ( code_Suc @ I2 ) @ J3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4646_Eps__Opt__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( eps_Opt @ A )
% 11.90/6.02 = ( ^ [P4: A > $o] :
% 11.90/6.02 ( if @ ( option @ A )
% 11.90/6.02 @ ? [X8: A] : ( P4 @ X8 )
% 11.90/6.02 @ ( some @ A @ ( fChoice @ A @ P4 ) )
% 11.90/6.02 @ ( none @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4647_acyclic__insert,axiom,
% 11.90/6.02 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R ) )
% 11.90/6.02 = ( ( transitive_acyclic @ A @ R )
% 11.90/6.02 & ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4648_acyclic__empty,axiom,
% 11.90/6.02 ! [A: $tType] : ( transitive_acyclic @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4649_Eps__case__prod__eq,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,X: A,Y: B] :
% 11.90/6.02 ( ( fChoice @ ( product_prod @ A @ B )
% 11.90/6.02 @ ( product_case_prod @ A @ B @ $o
% 11.90/6.02 @ ^ [X9: A,Y8: B] :
% 11.90/6.02 ( ( X = X9 )
% 11.90/6.02 & ( Y = Y8 ) ) ) )
% 11.90/6.02 = ( product_Pair @ A @ B @ X @ Y ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4650_some__insert__self,axiom,
% 11.90/6.02 ! [A: $tType,S2: set @ A] :
% 11.90/6.02 ( ( S2
% 11.90/6.02 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( ( insert2 @ A
% 11.90/6.02 @ ( fChoice @ A
% 11.90/6.02 @ ^ [X3: A] : ( member @ A @ X3 @ S2 ) )
% 11.90/6.02 @ S2 )
% 11.90/6.02 = S2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4651_some__elem,axiom,
% 11.90/6.02 ! [A: $tType,S2: set @ A] :
% 11.90/6.02 ( ( S2
% 11.90/6.02 != ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( member @ A
% 11.90/6.02 @ ( fChoice @ A
% 11.90/6.02 @ ^ [X3: A] : ( member @ A @ X3 @ S2 ) )
% 11.90/6.02 @ S2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4652_cyclic__subset,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ~ ( transitive_acyclic @ A @ R5 )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S2 )
% 11.90/6.02 => ~ ( transitive_acyclic @ A @ S2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4653_acyclic__union_I1_J,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) )
% 11.90/6.02 => ( transitive_acyclic @ A @ A4 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4654_acyclic__union_I2_J,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( transitive_acyclic @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B6 ) )
% 11.90/6.02 => ( transitive_acyclic @ A @ B6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4655_inv__on__def,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( inv_on @ A @ B )
% 11.90/6.02 = ( ^ [F2: A > B,A9: set @ A,X3: B] :
% 11.90/6.02 ( fChoice @ A
% 11.90/6.02 @ ^ [Y3: A] :
% 11.90/6.02 ( ( member @ A @ Y3 @ A9 )
% 11.90/6.02 & ( ( F2 @ Y3 )
% 11.90/6.02 = X3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4656_split__paired__Eps,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( fChoice @ ( product_prod @ A @ B ) )
% 11.90/6.02 = ( ^ [P4: ( product_prod @ A @ B ) > $o] :
% 11.90/6.02 ( fChoice @ ( product_prod @ A @ B )
% 11.90/6.02 @ ( product_case_prod @ A @ B @ $o
% 11.90/6.02 @ ^ [A7: A,B5: B] : ( P4 @ ( product_Pair @ A @ B @ A7 @ B5 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4657_acyclicI__order,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( preorder @ A )
% 11.90/6.02 => ! [R: set @ ( product_prod @ B @ B ),F: B > A] :
% 11.90/6.02 ( ! [A5: B,B3: B] :
% 11.90/6.02 ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B3 ) @ R )
% 11.90/6.02 => ( ord_less @ A @ ( F @ B3 ) @ ( F @ A5 ) ) )
% 11.90/6.02 => ( transitive_acyclic @ B @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4658_subset__mset_OacyclicI__order,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B ),F: B > ( multiset @ A )] :
% 11.90/6.02 ( ! [A5: B,B3: B] :
% 11.90/6.02 ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A5 @ B3 ) @ R )
% 11.90/6.02 => ( subset_mset @ A @ ( F @ B3 ) @ ( F @ A5 ) ) )
% 11.90/6.02 => ( transitive_acyclic @ B @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4659_acyclicI,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ! [X2: A] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R ) )
% 11.90/6.02 => ( transitive_acyclic @ A @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4660_acyclic__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( transitive_acyclic @ A )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ! [X3: A] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4661_cyclicE,axiom,
% 11.90/6.02 ! [A: $tType,G: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ~ ( transitive_acyclic @ A @ G )
% 11.90/6.02 => ~ ! [X2: A] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ G ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4662_some__theI,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,P: A > B > $o] :
% 11.90/6.02 ( ? [A17: A,X_1: B] : ( P @ A17 @ X_1 )
% 11.90/6.02 => ( ! [B13: B,B24: B] :
% 11.90/6.02 ( ? [A5: A] : ( P @ A5 @ B13 )
% 11.90/6.02 => ( ? [A5: A] : ( P @ A5 @ B24 )
% 11.90/6.02 => ( B13 = B24 ) ) )
% 11.90/6.02 => ( P
% 11.90/6.02 @ ( fChoice @ A
% 11.90/6.02 @ ^ [A7: A] :
% 11.90/6.02 ? [X8: B] : ( P @ A7 @ X8 ) )
% 11.90/6.02 @ ( the @ B
% 11.90/6.02 @ ^ [B5: B] :
% 11.90/6.02 ? [A7: A] : ( P @ A7 @ B5 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4663_acyclic__insert__cyclic,axiom,
% 11.90/6.02 ! [A: $tType,G: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.90/6.02 ( ( transitive_acyclic @ A @ G )
% 11.90/6.02 => ( ~ ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ G ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ G ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4664_fun__of__rel__def,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( fun_of_rel @ B @ A )
% 11.90/6.02 = ( ^ [R6: set @ ( product_prod @ B @ A ),X3: B] :
% 11.90/6.02 ( fChoice @ A
% 11.90/6.02 @ ^ [Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4665_irrefl__tranclI,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A] :
% 11.90/6.02 ( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/6.02 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4666_quicksort_Opelims,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linorder @ A )
% 11.90/6.02 => ! [X: list @ A,Y: list @ A] :
% 11.90/6.02 ( ( ( linorder_quicksort @ A @ X )
% 11.90/6.02 = Y )
% 11.90/6.02 => ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X )
% 11.90/6.02 => ( ( ( X
% 11.90/6.02 = ( nil @ A ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( nil @ A ) )
% 11.90/6.02 => ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
% 11.90/6.02 => ~ ! [X2: A,Xs2: list @ A] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( cons @ A @ X2 @ Xs2 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( append @ A
% 11.90/6.02 @ ( linorder_quicksort @ A
% 11.90/6.02 @ ( filter @ A
% 11.90/6.02 @ ^ [Y3: A] :
% 11.90/6.02 ~ ( ord_less_eq @ A @ X2 @ Y3 )
% 11.90/6.02 @ Xs2 ) )
% 11.90/6.02 @ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter @ A @ ( ord_less_eq @ A @ X2 ) @ Xs2 ) ) ) ) )
% 11.90/6.02 => ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4667_converse__iff,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
% 11.90/6.02 = ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4668_pair__set__inverse,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,P: B > A > $o] :
% 11.90/6.02 ( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
% 11.90/6.02 = ( collect @ ( product_prod @ A @ B )
% 11.90/6.02 @ ( product_case_prod @ A @ B @ $o
% 11.90/6.02 @ ^ [B5: A,A7: B] : ( P @ A7 @ B5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4669_below__Id__inv,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R5 ) @ ( id2 @ A ) )
% 11.90/6.02 = ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4670_converseI,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 11.90/6.02 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4671_converseE,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R ) )
% 11.90/6.02 => ~ ! [X2: B,Y4: A] :
% 11.90/6.02 ( ( Yx
% 11.90/6.02 = ( product_Pair @ A @ B @ Y4 @ X2 ) )
% 11.90/6.02 => ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y4 ) @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4672_converseD,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
% 11.90/6.02 => ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4673_converse_Osimps,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 11.90/6.02 = ( ? [A7: A,B5: B] :
% 11.90/6.02 ( ( A1 = B5 )
% 11.90/6.02 & ( A22 = A7 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4674_converse_Ocases,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4675_in__listrel1__converse,axiom,
% 11.90/6.02 ! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( listrel1 @ A @ ( converse @ A @ A @ R ) ) )
% 11.90/6.02 = ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4676_rtrancl__converseI,axiom,
% 11.90/6.02 ! [A: $tType,Y: A,X: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4677_rtrancl__converseD,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4678_trancl__converseI,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4679_trancl__converseD,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4680_finite__wf__eq__wf__converse,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
% 11.90/6.02 => ( ( wf @ A @ ( converse @ A @ A @ R5 ) )
% 11.90/6.02 = ( wf @ A @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4681_converse__unfold,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( converse @ B @ A )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( collect @ ( product_prod @ A @ B )
% 11.90/6.02 @ ( product_case_prod @ A @ B @ $o
% 11.90/6.02 @ ^ [Y3: A,X3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4682_conversep__converse__eq,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( conversep @ A @ B
% 11.90/6.02 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 11.90/6.02 = ( ^ [X3: B,Y3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y3 ) @ ( converse @ A @ B @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4683_converse__def,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( converse @ A @ B )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( collect @ ( product_prod @ B @ A )
% 11.90/6.02 @ ( product_case_prod @ B @ A @ $o
% 11.90/6.02 @ ( conversep @ A @ B
% 11.90/6.02 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4684_is__singleton__altdef,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( is_singleton @ A )
% 11.90/6.02 = ( ^ [A9: set @ A] :
% 11.90/6.02 ( ( finite_card @ A @ A9 )
% 11.90/6.02 = ( one_one @ nat ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4685_Code__Numeral_Osub__code_I2_J,axiom,
% 11.90/6.02 ! [M2: num] :
% 11.90/6.02 ( ( code_sub @ ( bit0 @ M2 ) @ one2 )
% 11.90/6.02 = ( code_Pos @ ( bitM @ M2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4686_su__rel__fun__def,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B )
% 11.90/6.02 = ( ^ [F7: set @ ( product_prod @ A @ B ),F2: A > B] :
% 11.90/6.02 ( ! [A9: A,B9: B,B11: B] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B11 ) @ F7 )
% 11.90/6.02 => ( B9 = B11 ) ) )
% 11.90/6.02 & ! [A9: A,P4: $o] :
% 11.90/6.02 ( ! [B9: B] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 )
% 11.90/6.02 => P4 )
% 11.90/6.02 => P4 )
% 11.90/6.02 & ! [A9: A] :
% 11.90/6.02 ( ( F2 @ A9 )
% 11.90/6.02 = ( the @ B
% 11.90/6.02 @ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A9 @ B9 ) @ F7 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4687_Code__Numeral_OPos__def,axiom,
% 11.90/6.02 ( code_Pos
% 11.90/6.02 = ( numeral_numeral @ code_integer ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4688_su__rel__fun_Osurjective,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 11.90/6.02 => ~ ! [B10: B] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B10 ) @ F6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4689_su__rel__fun_Ounique,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B,B14: B] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B14 ) @ F6 )
% 11.90/6.02 => ( B6 = B14 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4690_su__rel__fun_Orepr2,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 )
% 11.90/6.02 => ( B6
% 11.90/6.02 = ( F @ A4 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4691_su__rel__fun_Orepr1,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 11.90/6.02 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ ( F @ A4 ) ) @ F6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4692_su__rel__fun_Orepr,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A,B6: B] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 11.90/6.02 => ( ( ( F @ A4 )
% 11.90/6.02 = B6 )
% 11.90/6.02 = ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B6 ) @ F6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4693_Pos__fold_I2_J,axiom,
% 11.90/6.02 ! [K: num] :
% 11.90/6.02 ( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
% 11.90/6.02 = ( code_Pos @ ( bit0 @ K ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4694_Pos__fold_I1_J,axiom,
% 11.90/6.02 ( ( numeral_numeral @ code_integer @ one2 )
% 11.90/6.02 = ( code_Pos @ one2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4695_Pos__fold_I3_J,axiom,
% 11.90/6.02 ! [K: num] :
% 11.90/6.02 ( ( numeral_numeral @ code_integer @ ( bit1 @ K ) )
% 11.90/6.02 = ( code_Pos @ ( bit1 @ K ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4696_one__integer__code,axiom,
% 11.90/6.02 ( ( one_one @ code_integer )
% 11.90/6.02 = ( code_Pos @ one2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4697_Code__Numeral_Odup__code_I2_J,axiom,
% 11.90/6.02 ! [N: num] :
% 11.90/6.02 ( ( code_dup @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_Pos @ ( bit0 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4698_less__integer__code_I4_J,axiom,
% 11.90/6.02 ! [K: num] :
% 11.90/6.02 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4699_less__integer__code_I2_J,axiom,
% 11.90/6.02 ! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4700_plus__integer__code_I3_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( plus_plus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_Pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4701_less__eq__integer__code_I2_J,axiom,
% 11.90/6.02 ! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4702_less__eq__integer__code_I4_J,axiom,
% 11.90/6.02 ! [K: num] :
% 11.90/6.02 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4703_less__integer__code_I8_J,axiom,
% 11.90/6.02 ! [K: num,L: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4704_less__integer__code_I6_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4705_uminus__integer__code_I3_J,axiom,
% 11.90/6.02 ! [M2: num] :
% 11.90/6.02 ( ( uminus_uminus @ code_integer @ ( code_Neg @ M2 ) )
% 11.90/6.02 = ( code_Pos @ M2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4706_uminus__integer__code_I2_J,axiom,
% 11.90/6.02 ! [M2: num] :
% 11.90/6.02 ( ( uminus_uminus @ code_integer @ ( code_Pos @ M2 ) )
% 11.90/6.02 = ( code_Neg @ M2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4707_Code__Numeral_ONeg__def,axiom,
% 11.90/6.02 ( code_Neg
% 11.90/6.02 = ( ^ [N5: num] : ( uminus_uminus @ code_integer @ ( code_Pos @ N5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4708_less__eq__integer__code_I8_J,axiom,
% 11.90/6.02 ! [K: num,L: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4709_less__eq__integer__code_I6_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4710_times__integer__code_I3_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( times_times @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_Pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4711_less__integer__code_I5_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 11.90/6.02 = ( ord_less @ num @ K @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4712_less__eq__integer__code_I5_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 11.90/6.02 = ( ord_less_eq @ num @ K @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4713_minus__integer__code_I3_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( minus_minus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_sub @ M2 @ N ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4714_divmod__abs__code_I1_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
% 11.90/6.02 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4715_minus__integer__code_I4_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( minus_minus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_Pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4716_minus__integer__code_I5_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( minus_minus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_Neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4717_times__integer__code_I4_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( times_times @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_Neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4718_times__integer__code_I5_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( times_times @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_Neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4719_times__integer__code_I6_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( times_times @ code_integer @ ( code_Neg @ M2 ) @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_Pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4720_plus__integer__code_I4_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( plus_plus @ code_integer @ ( code_Pos @ M2 ) @ ( code_Neg @ N ) )
% 11.90/6.02 = ( code_sub @ M2 @ N ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4721_plus__integer__code_I5_J,axiom,
% 11.90/6.02 ! [M2: num,N: num] :
% 11.90/6.02 ( ( plus_plus @ code_integer @ ( code_Neg @ M2 ) @ ( code_Pos @ N ) )
% 11.90/6.02 = ( code_sub @ N @ M2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4722_divmod__abs__code_I3_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
% 11.90/6.02 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4723_divmod__abs__code_I4_J,axiom,
% 11.90/6.02 ! [K: num,L: num] :
% 11.90/6.02 ( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
% 11.90/6.02 = ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4724_Code__Numeral_Osub__code_I3_J,axiom,
% 11.90/6.02 ! [M2: num] :
% 11.90/6.02 ( ( code_sub @ ( bit1 @ M2 ) @ one2 )
% 11.90/6.02 = ( code_Pos @ ( bit0 @ M2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4725_su__rel__fun_Of__def,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B,A4: A] :
% 11.90/6.02 ( ( su_rel_fun @ A @ B @ F6 @ F )
% 11.90/6.02 => ( ( F @ A4 )
% 11.90/6.02 = ( the @ B
% 11.90/6.02 @ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B9 ) @ F6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4726_su__rel__fun_Ointro,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F6: set @ ( product_prod @ A @ B ),F: A > B] :
% 11.90/6.02 ( ! [A15: A,B10: B,B15: B] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B10 ) @ F6 )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B15 ) @ F6 )
% 11.90/6.02 => ( B10 = B15 ) ) )
% 11.90/6.02 => ( ! [A15: A,P7: $o] :
% 11.90/6.02 ( ! [B16: B] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B16 ) @ F6 )
% 11.90/6.02 => P7 )
% 11.90/6.02 => P7 )
% 11.90/6.02 => ( ! [A15: A] :
% 11.90/6.02 ( ( F @ A15 )
% 11.90/6.02 = ( the @ B
% 11.90/6.02 @ ^ [B9: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A15 @ B9 ) @ F6 ) ) )
% 11.90/6.02 => ( su_rel_fun @ A @ B @ F6 @ F ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4727_aboveS__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( order_aboveS @ A )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 11.90/6.02 ( collect @ A
% 11.90/6.02 @ ^ [B5: A] :
% 11.90/6.02 ( ( B5 != A7 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4728_relInvImage__def,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( bNF_Gr7122648621184425601vImage @ A @ B )
% 11.90/6.02 = ( ^ [A9: set @ A,R6: set @ ( product_prod @ B @ B ),F2: A > B] :
% 11.90/6.02 ( collect @ ( product_prod @ A @ A )
% 11.90/6.02 @ ^ [Uu: product_prod @ A @ A] :
% 11.90/6.02 ? [A12: A,A23: A] :
% 11.90/6.02 ( ( Uu
% 11.90/6.02 = ( product_Pair @ A @ A @ A12 @ A23 ) )
% 11.90/6.02 & ( member @ A @ A12 @ A9 )
% 11.90/6.02 & ( member @ A @ A23 @ A9 )
% 11.90/6.02 & ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R6 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4729_prod__list__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_mult @ A )
% 11.90/6.02 => ( ( groups5270119922927024881d_list @ A )
% 11.90/6.02 = ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4730_sum__list__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_add @ A )
% 11.90/6.02 => ( ( groups8242544230860333062m_list @ A )
% 11.90/6.02 = ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4731_sum__list__map__remove1,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_monoid_add @ A )
% 11.90/6.02 => ! [X: B,Xs: list @ B,F: B > A] :
% 11.90/6.02 ( ( member @ B @ X @ ( set2 @ B @ Xs ) )
% 11.90/6.02 => ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) )
% 11.90/6.02 = ( plus_plus @ A @ ( F @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4732_normalize__stable,axiom,
% 11.90/6.02 ! [Q3: int,P5: int] :
% 11.90/6.02 ( ( ord_less @ int @ ( zero_zero @ int ) @ Q3 )
% 11.90/6.02 => ( ( algebr8660921524188924756oprime @ int @ P5 @ Q3 )
% 11.90/6.02 => ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/6.02 = ( product_Pair @ int @ int @ P5 @ Q3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4733_distinct__count__atmost__1,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( distinct @ A )
% 11.90/6.02 = ( ^ [X3: list @ A] :
% 11.90/6.02 ! [A7: A] :
% 11.90/6.02 ( ( ( member @ A @ A7 @ ( set2 @ A @ X3 ) )
% 11.90/6.02 => ( ( count @ A @ ( mset @ A @ X3 ) @ A7 )
% 11.90/6.02 = ( one_one @ nat ) ) )
% 11.90/6.02 & ( ~ ( member @ A @ A7 @ ( set2 @ A @ X3 ) )
% 11.90/6.02 => ( ( count @ A @ ( mset @ A @ X3 ) @ A7 )
% 11.90/6.02 = ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4734_coprime__self,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
% 11.90/6.02 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4735_mset__empty__count,axiom,
% 11.90/6.02 ! [A: $tType,M3: multiset @ A] :
% 11.90/6.02 ( ( ! [P6: A] :
% 11.90/6.02 ( ( count @ A @ M3 @ P6 )
% 11.90/6.02 = ( zero_zero @ nat ) ) )
% 11.90/6.02 = ( M3
% 11.90/6.02 = ( zero_zero @ ( multiset @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4736_coprime__imp__gcd__eq__1,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_gcd @ A )
% 11.90/6.02 => ! [A3: A,B2: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 11.90/6.02 => ( ( gcd_gcd @ A @ A3 @ B2 )
% 11.90/6.02 = ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4737_count__union,axiom,
% 11.90/6.02 ! [A: $tType,M3: multiset @ A,N6: multiset @ A,A3: A] :
% 11.90/6.02 ( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) @ A3 )
% 11.90/6.02 = ( plus_plus @ nat @ ( count @ A @ M3 @ A3 ) @ ( count @ A @ N6 @ A3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4738_coprime__0__right__iff,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
% 11.90/6.02 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4739_coprime__0__left__iff,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/6.02 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4740_coprime__mult__self__right__iff,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A,C2: A,B2: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
% 11.90/6.02 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 11.90/6.02 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4741_coprime__mult__self__left__iff,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [C2: A,A3: A,B2: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
% 11.90/6.02 = ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 11.90/6.02 & ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4742_is__unit__gcd,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_gcd @ A )
% 11.90/6.02 => ! [A3: A,B2: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
% 11.90/6.02 = ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4743_count__greater__eq__one__iff,axiom,
% 11.90/6.02 ! [A: $tType,M3: multiset @ A,X: A] :
% 11.90/6.02 ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( count @ A @ M3 @ X ) )
% 11.90/6.02 = ( member @ A @ X @ ( set_mset @ A @ M3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4744_count__mset__set_I1_J,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ A,X: A] :
% 11.90/6.02 ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 => ( ( member @ A @ X @ A4 )
% 11.90/6.02 => ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
% 11.90/6.02 = ( one_one @ nat ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4745_coprime__1__left,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4746_coprime__1__right,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4747_coprime__add__one__left,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_gcd @ A )
% 11.90/6.02 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4748_coprime__add__one__right,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_gcd @ A )
% 11.90/6.02 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4749_coprime__doff__one__right,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( ring_gcd @ A )
% 11.90/6.02 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4750_coprime__diff__one__left,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( ring_gcd @ A )
% 11.90/6.02 => ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4751_is__unit__right__imp__coprime,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [B2: A,A3: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4752_is__unit__left__imp__coprime,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A,B2: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4753_coprime__common__divisor,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A,B2: A,C2: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ C2 @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 11.90/6.02 => ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4754_coprime__absorb__right,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [Y: A,X: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ Y @ X )
% 11.90/6.02 => ( ( algebr8660921524188924756oprime @ A @ X @ Y )
% 11.90/6.02 = ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4755_coprime__imp__coprime,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [C2: A,D3: A,A3: A,B2: A] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
% 11.90/6.02 => ( ! [E3: A] :
% 11.90/6.02 ( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
% 11.90/6.02 => ( ( dvd_dvd @ A @ E3 @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ E3 @ B2 )
% 11.90/6.02 => ( dvd_dvd @ A @ E3 @ C2 ) ) ) )
% 11.90/6.02 => ( ! [E3: A] :
% 11.90/6.02 ( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
% 11.90/6.02 => ( ( dvd_dvd @ A @ E3 @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ E3 @ B2 )
% 11.90/6.02 => ( dvd_dvd @ A @ E3 @ D3 ) ) ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4756_coprime__absorb__left,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [X: A,Y: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ X @ Y )
% 11.90/6.02 => ( ( algebr8660921524188924756oprime @ A @ X @ Y )
% 11.90/6.02 = ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4757_not__coprimeI,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [C2: A,A3: A,B2: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ C2 @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ C2 @ B2 )
% 11.90/6.02 => ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
% 11.90/6.02 => ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4758_not__coprimeE,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A,B2: A] :
% 11.90/6.02 ( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
% 11.90/6.02 => ~ ! [C4: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ C4 @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ C4 @ B2 )
% 11.90/6.02 => ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4759_coprime__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ( ( algebr8660921524188924756oprime @ A )
% 11.90/6.02 = ( ^ [A7: A,B5: A] :
% 11.90/6.02 ! [C3: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ C3 @ A7 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ C3 @ B5 )
% 11.90/6.02 => ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4760_coprimeI,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A3: A,B2: A] :
% 11.90/6.02 ( ! [C4: A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ C4 @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ A @ C4 @ B2 )
% 11.90/6.02 => ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4761_plus__multiset_Orep__eq,axiom,
% 11.90/6.02 ! [A: $tType,X: multiset @ A,Xa: multiset @ A] :
% 11.90/6.02 ( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ X @ Xa ) )
% 11.90/6.02 = ( ^ [A7: A] : ( plus_plus @ nat @ ( count @ A @ X @ A7 ) @ ( count @ A @ Xa @ A7 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4762_coprime__iff__gcd__eq__1,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_gcd @ A )
% 11.90/6.02 => ( ( algebr8660921524188924756oprime @ A )
% 11.90/6.02 = ( ^ [A7: A,B5: A] :
% 11.90/6.02 ( ( gcd_gcd @ A @ A7 @ B5 )
% 11.90/6.02 = ( one_one @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4763_gcd__eq__1__imp__coprime,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_gcd @ A )
% 11.90/6.02 => ! [A3: A,B2: A] :
% 11.90/6.02 ( ( ( gcd_gcd @ A @ A3 @ B2 )
% 11.90/6.02 = ( one_one @ A ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4764_quotient__of__coprime,axiom,
% 11.90/6.02 ! [R: rat,P5: int,Q3: int] :
% 11.90/6.02 ( ( ( quotient_of @ R )
% 11.90/6.02 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ int @ P5 @ Q3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4765_normalize__coprime,axiom,
% 11.90/6.02 ! [R: product_prod @ int @ int,P5: int,Q3: int] :
% 11.90/6.02 ( ( ( normalize @ R )
% 11.90/6.02 = ( product_Pair @ int @ int @ P5 @ Q3 ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ int @ P5 @ Q3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4766_invertible__coprime,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( euclid8851590272496341667cancel @ A )
% 11.90/6.02 => ! [A3: A,B2: A,C2: A] :
% 11.90/6.02 ( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
% 11.90/6.02 = ( one_one @ A ) )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4767_count__ne__remove,axiom,
% 11.90/6.02 ! [A: $tType,X: A,T2: A,S2: multiset @ A] :
% 11.90/6.02 ( ( X != T2 )
% 11.90/6.02 => ( ( count @ A @ S2 @ X )
% 11.90/6.02 = ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4768_count__in__diffI,axiom,
% 11.90/6.02 ! [A: $tType,N6: multiset @ A,X: A,M3: multiset @ A] :
% 11.90/6.02 ( ! [N3: nat] :
% 11.90/6.02 ( ( count @ A @ N6 @ X )
% 11.90/6.02 != ( plus_plus @ nat @ N3 @ ( count @ A @ M3 @ X ) ) )
% 11.90/6.02 => ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4769_count__single,axiom,
% 11.90/6.02 ! [A: $tType,B2: A,A3: A] :
% 11.90/6.02 ( ( ( B2 = A3 )
% 11.90/6.02 => ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
% 11.90/6.02 = ( one_one @ nat ) ) )
% 11.90/6.02 & ( ( B2 != A3 )
% 11.90/6.02 => ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A3 )
% 11.90/6.02 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4770_in__diff__countE,axiom,
% 11.90/6.02 ! [A: $tType,X: A,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.02 ( ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N6 ) ) )
% 11.90/6.02 => ~ ! [N3: nat] :
% 11.90/6.02 ( ( count @ A @ M3 @ X )
% 11.90/6.02 != ( plus_plus @ nat @ ( suc @ N3 ) @ ( count @ A @ N6 @ X ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4771_remove1__tl,axiom,
% 11.90/6.02 ! [A: $tType,Xs: list @ A] :
% 11.90/6.02 ( ( Xs
% 11.90/6.02 != ( nil @ A ) )
% 11.90/6.02 => ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
% 11.90/6.02 = ( tl @ A @ Xs ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4772_count__mset__set__finite__iff,axiom,
% 11.90/6.02 ! [A: $tType,S2: set @ A,A3: A] :
% 11.90/6.02 ( ( finite_finite2 @ A @ S2 )
% 11.90/6.02 => ( ( ( member @ A @ A3 @ S2 )
% 11.90/6.02 => ( ( count @ A @ ( mset_set @ A @ S2 ) @ A3 )
% 11.90/6.02 = ( one_one @ nat ) ) )
% 11.90/6.02 & ( ~ ( member @ A @ A3 @ S2 )
% 11.90/6.02 => ( ( count @ A @ ( mset_set @ A @ S2 ) @ A3 )
% 11.90/6.02 = ( zero_zero @ nat ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4773_count__mset__set_H,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ A,X: A] :
% 11.90/6.02 ( ( ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 & ( member @ A @ X @ A4 ) )
% 11.90/6.02 => ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
% 11.90/6.02 = ( one_one @ nat ) ) )
% 11.90/6.02 & ( ~ ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 & ( member @ A @ X @ A4 ) )
% 11.90/6.02 => ( ( count @ A @ ( mset_set @ A @ A4 ) @ X )
% 11.90/6.02 = ( zero_zero @ nat ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4774_length__remove1,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Xs: list @ A] :
% 11.90/6.02 ( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.02 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 11.90/6.02 = ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
% 11.90/6.02 & ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.90/6.02 => ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
% 11.90/6.02 = ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4775_prod__mset__delta_H,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [Y: B,C2: A,A4: multiset @ B] :
% 11.90/6.02 ( ( comm_m9189036328036947845d_mset @ A
% 11.90/6.02 @ ( image_mset @ B @ A
% 11.90/6.02 @ ^ [X3: B] : ( if @ A @ ( Y = X3 ) @ C2 @ ( one_one @ A ) )
% 11.90/6.02 @ A4 ) )
% 11.90/6.02 = ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4776_prod__mset__delta,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [Y: B,C2: A,A4: multiset @ B] :
% 11.90/6.02 ( ( comm_m9189036328036947845d_mset @ A
% 11.90/6.02 @ ( image_mset @ B @ A
% 11.90/6.02 @ ^ [X3: B] : ( if @ A @ ( X3 = Y ) @ C2 @ ( one_one @ A ) )
% 11.90/6.02 @ A4 ) )
% 11.90/6.02 = ( power_power @ A @ C2 @ ( count @ B @ A4 @ Y ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4777_prod_OatLeastLessThan__reindex,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( ( comm_monoid_mult @ A )
% 11.90/6.02 & ( ord @ B ) )
% 11.90/6.02 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 11.90/6.02 ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4778_bij__betw__id,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ A4 ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4779_bij__betw__add,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( cancel_semigroup_add @ A )
% 11.90/6.02 => ! [A3: A,A4: set @ A,B6: set @ A] :
% 11.90/6.02 ( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 @ B6 )
% 11.90/6.02 = ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A4 )
% 11.90/6.02 = B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4780_prod__mset__empty,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ( ( comm_m9189036328036947845d_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.02 = ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4781_prod__mset_Ounion,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.02 ( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.90/6.02 = ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M3 ) @ ( comm_m9189036328036947845d_mset @ A @ N6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4782_prod__mset__Un,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 11.90/6.02 = ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( comm_m9189036328036947845d_mset @ A @ B6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4783_bij__betw__of__nat,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_char_0 @ A )
% 11.90/6.02 => ! [N6: set @ nat,A4: set @ A] :
% 11.90/6.02 ( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A4 )
% 11.90/6.02 = ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
% 11.90/6.02 = A4 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4784_prod__mset_Oneutral__const,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [A4: multiset @ B] :
% 11.90/6.02 ( ( comm_m9189036328036947845d_mset @ A
% 11.90/6.02 @ ( image_mset @ B @ A
% 11.90/6.02 @ ^ [Uu: B] : ( one_one @ A )
% 11.90/6.02 @ A4 ) )
% 11.90/6.02 = ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4785_bij__betw__imp__surj__on,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( ( image2 @ A @ B @ F @ A4 )
% 11.90/6.02 = B6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4786_f__the__inv__into__f__bij__betw,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,X: B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( member @ B @ X @ B6 ) )
% 11.90/6.02 => ( ( F @ ( the_inv_into @ A @ B @ A4 @ F @ X ) )
% 11.90/6.02 = X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4787_bij__betw__the__inv__into,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A4 @ F ) @ B6 @ A4 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4788_bij__betw__id__iff,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/6.02 ( ( bij_betw @ A @ A @ ( id @ A ) @ A4 @ B6 )
% 11.90/6.02 = ( A4 = B6 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4789_bij__betwE,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ! [X7: A] :
% 11.90/6.02 ( ( member @ A @ X7 @ A4 )
% 11.90/6.02 => ( member @ B @ ( F @ X7 ) @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4790_bij__betw__inv,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B6 @ A4 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4791_bij__betw__cong,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A4: set @ A,F: A > B,G: A > B,A18: set @ B] :
% 11.90/6.02 ( ! [A5: A] :
% 11.90/6.02 ( ( member @ A @ A5 @ A4 )
% 11.90/6.02 => ( ( F @ A5 )
% 11.90/6.02 = ( G @ A5 ) ) )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 11.90/6.02 = ( bij_betw @ A @ B @ G @ A4 @ A18 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4792_bij__betw__apply,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,A3: A] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( ( member @ A @ A3 @ A4 )
% 11.90/6.02 => ( member @ B @ ( F @ A3 ) @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4793_bij__betw__iff__bijections,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( bij_betw @ A @ B )
% 11.90/6.02 = ( ^ [F2: A > B,A9: set @ A,B9: set @ B] :
% 11.90/6.02 ? [G2: B > A] :
% 11.90/6.02 ( ! [X3: A] :
% 11.90/6.02 ( ( member @ A @ X3 @ A9 )
% 11.90/6.02 => ( ( member @ B @ ( F2 @ X3 ) @ B9 )
% 11.90/6.02 & ( ( G2 @ ( F2 @ X3 ) )
% 11.90/6.02 = X3 ) ) )
% 11.90/6.02 & ! [X3: B] :
% 11.90/6.02 ( ( member @ B @ X3 @ B9 )
% 11.90/6.02 => ( ( member @ A @ ( G2 @ X3 ) @ A9 )
% 11.90/6.02 & ( ( F2 @ ( G2 @ X3 ) )
% 11.90/6.02 = X3 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4794_involuntory__imp__bij,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A] :
% 11.90/6.02 ( ! [X2: A] :
% 11.90/6.02 ( ( F @ ( F @ X2 ) )
% 11.90/6.02 = X2 )
% 11.90/6.02 => ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4795_bij__pointE,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,Y: B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ~ ! [X2: A] :
% 11.90/6.02 ( ( Y
% 11.90/6.02 = ( F @ X2 ) )
% 11.90/6.02 => ~ ! [X16: A] :
% 11.90/6.02 ( ( Y
% 11.90/6.02 = ( F @ X16 ) )
% 11.90/6.02 => ( X16 = X2 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4796_bij__iff,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 = ( ! [X3: B] :
% 11.90/6.02 ? [Y3: A] :
% 11.90/6.02 ( ( ( F @ Y3 )
% 11.90/6.02 = X3 )
% 11.90/6.02 & ! [Z5: A] :
% 11.90/6.02 ( ( ( F @ Z5 )
% 11.90/6.02 = X3 )
% 11.90/6.02 => ( Z5 = Y3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4797_bij__betw__imp__inj__on,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( inj_on @ A @ B @ F @ A4 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4798_bij__betw__trans,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,C: $tType,F: A > B,A4: set @ A,B6: set @ B,G: B > C,C5: set @ C] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( ( bij_betw @ B @ C @ G @ B6 @ C5 )
% 11.90/6.02 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A4 @ C5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4799_bij__betw__comp__iff,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,C: $tType,F: A > B,A4: set @ A,A18: set @ B,F3: B > C,A19: set @ C] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 11.90/6.02 => ( ( bij_betw @ B @ C @ F3 @ A18 @ A19 )
% 11.90/6.02 = ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F3 @ F ) @ A4 @ A19 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4800_bij__betw__empty2,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ ( bot_bot @ ( set @ B ) ) )
% 11.90/6.02 => ( A4
% 11.90/6.02 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4801_bij__betw__empty1,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) @ A4 )
% 11.90/6.02 => ( A4
% 11.90/6.02 = ( bot_bot @ ( set @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4802_bij__betw__byWitness,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A4: set @ A,F3: B > A,F: A > B,A18: set @ B] :
% 11.90/6.02 ( ! [X2: A] :
% 11.90/6.02 ( ( member @ A @ X2 @ A4 )
% 11.90/6.02 => ( ( F3 @ ( F @ X2 ) )
% 11.90/6.02 = X2 ) )
% 11.90/6.02 => ( ! [X2: B] :
% 11.90/6.02 ( ( member @ B @ X2 @ A18 )
% 11.90/6.02 => ( ( F @ ( F3 @ X2 ) )
% 11.90/6.02 = X2 ) )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) @ A18 )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A18 ) @ A4 )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ A4 @ A18 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4803_bij__betw__subset,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,A18: set @ B,B6: set @ A,B14: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 11.90/6.02 => ( ( ( image2 @ A @ B @ F @ B6 )
% 11.90/6.02 = B14 )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ B6 @ B14 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4804_bij__betw__imp__surj,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( top_top @ ( set @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4805_bij__is__surj,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( top_top @ ( set @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4806_bij__uminus,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( ab_group_add @ A )
% 11.90/6.02 => ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4807_bij__comp,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( bij_betw @ B @ C @ G @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
% 11.90/6.02 => ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4808_inj__on__imp__bij__betw,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ A4 @ ( image2 @ A @ B @ F @ A4 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4809_bij__betw__imageI,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/6.02 => ( ( ( image2 @ A @ B @ F @ A4 )
% 11.90/6.02 = B6 )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ A4 @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4810_bij__betw__def,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( bij_betw @ A @ B )
% 11.90/6.02 = ( ^ [F2: A > B,A9: set @ A,B9: set @ B] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F2 @ A9 )
% 11.90/6.02 & ( ( image2 @ A @ B @ F2 @ A9 )
% 11.90/6.02 = B9 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4811_bij__betw__iff__card,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 => ( ( finite_finite2 @ B @ B6 )
% 11.90/6.02 => ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B6 ) )
% 11.90/6.02 = ( ( finite_card @ A @ A4 )
% 11.90/6.02 = ( finite_card @ B @ B6 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4812_finite__same__card__bij,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.90/6.02 ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 => ( ( finite_finite2 @ B @ B6 )
% 11.90/6.02 => ( ( ( finite_card @ A @ A4 )
% 11.90/6.02 = ( finite_card @ B @ B6 ) )
% 11.90/6.02 => ? [H2: A > B] : ( bij_betw @ A @ B @ H2 @ A4 @ B6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4813_bij__is__inj,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4814_bij__id,axiom,
% 11.90/6.02 ! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4815_coprime__common__divisor__nat,axiom,
% 11.90/6.02 ! [A3: nat,B2: nat,X: nat] :
% 11.90/6.02 ( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
% 11.90/6.02 => ( ( dvd_dvd @ nat @ X @ A3 )
% 11.90/6.02 => ( ( dvd_dvd @ nat @ X @ B2 )
% 11.90/6.02 => ( X
% 11.90/6.02 = ( one_one @ nat ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4816_prod__mset_Oneutral,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [A4: multiset @ A] :
% 11.90/6.02 ( ! [X2: A] :
% 11.90/6.02 ( ( member @ A @ X2 @ ( set_mset @ A @ A4 ) )
% 11.90/6.02 => ( X2
% 11.90/6.02 = ( one_one @ A ) ) )
% 11.90/6.02 => ( ( comm_m9189036328036947845d_mset @ A @ A4 )
% 11.90/6.02 = ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4817_bij__betw__comp__iff2,axiom,
% 11.90/6.02 ! [C: $tType,A: $tType,B: $tType,F3: A > B,A18: set @ A,A19: set @ B,F: C > A,A4: set @ C] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F3 @ A18 @ A19 )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F @ A4 ) @ A18 )
% 11.90/6.02 => ( ( bij_betw @ C @ A @ F @ A4 @ A18 )
% 11.90/6.02 = ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F3 @ F ) @ A4 @ A19 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4818_notIn__Un__bij__betw,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,B2: A,A4: set @ A,F: A > B,A18: set @ B] :
% 11.90/6.02 ( ~ ( member @ A @ B2 @ A4 )
% 11.90/6.02 => ( ~ ( member @ B @ ( F @ B2 ) @ A18 )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A18 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4819_notIn__Un__bij__betw3,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,B2: A,A4: set @ A,F: A > B,A18: set @ B] :
% 11.90/6.02 ( ~ ( member @ A @ B2 @ A4 )
% 11.90/6.02 => ( ~ ( member @ B @ ( F @ B2 ) @ A18 )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 11.90/6.02 = ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A18 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4820_bijI,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4821_bij__def,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 = ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 & ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( top_top @ ( set @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4822_bij__betw__partition,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,C5: set @ A,B6: set @ B,D4: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D4 ) )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ F @ C5 @ D4 )
% 11.90/6.02 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ C5 )
% 11.90/6.02 = ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( ( ( inf_inf @ ( set @ B ) @ B6 @ D4 )
% 11.90/6.02 = ( bot_bot @ ( set @ B ) ) )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ A4 @ B6 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4823_bij__betw__combine,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,B6: set @ B,C5: set @ A,D4: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ F @ C5 @ D4 )
% 11.90/6.02 => ( ( ( inf_inf @ ( set @ B ) @ B6 @ D4 )
% 11.90/6.02 = ( bot_bot @ ( set @ B ) ) )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A4 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D4 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4824_o__bij,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,G: B > A,F: A > B] :
% 11.90/6.02 ( ( ( comp @ B @ A @ A @ G @ F )
% 11.90/6.02 = ( id @ A ) )
% 11.90/6.02 => ( ( ( comp @ A @ B @ B @ F @ G )
% 11.90/6.02 = ( id @ B ) )
% 11.90/6.02 => ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4825_bij__image__Compl__eq,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
% 11.90/6.02 = ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4826_Well__order__iso__copy,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,A18: set @ B] :
% 11.90/6.02 ( ( order_well_order_on @ A @ A4 @ R )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ F @ A4 @ A18 )
% 11.90/6.02 => ? [R10: set @ ( product_prod @ B @ B )] :
% 11.90/6.02 ( ( order_well_order_on @ B @ A18 @ R10 )
% 11.90/6.02 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R10 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4827_bij__betw__disjoint__Un,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,F: A > B,A4: set @ A,C5: set @ B,G: A > B,B6: set @ A,D4: set @ B] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ A4 @ C5 )
% 11.90/6.02 => ( ( bij_betw @ A @ B @ G @ B6 @ D4 )
% 11.90/6.02 => ( ( ( inf_inf @ ( set @ A ) @ A4 @ B6 )
% 11.90/6.02 = ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( ( ( inf_inf @ ( set @ B ) @ C5 @ D4 )
% 11.90/6.02 = ( bot_bot @ ( set @ B ) ) )
% 11.90/6.02 => ( bij_betw @ A @ B
% 11.90/6.02 @ ^ [X3: A] : ( if @ B @ ( member @ A @ X3 @ A4 ) @ ( F @ X3 ) @ ( G @ X3 ) )
% 11.90/6.02 @ ( sup_sup @ ( set @ A ) @ A4 @ B6 )
% 11.90/6.02 @ ( sup_sup @ ( set @ B ) @ C5 @ D4 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4828_is__unit__prod__mset__iff,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( algebraic_semidom @ A )
% 11.90/6.02 => ! [A4: multiset @ A] :
% 11.90/6.02 ( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A4 ) @ ( one_one @ A ) )
% 11.90/6.02 = ( ! [X3: A] :
% 11.90/6.02 ( ( member @ A @ X3 @ ( set_mset @ A @ A4 ) )
% 11.90/6.02 => ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4829_prod__mset_Oeq__fold,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ( ( comm_m9189036328036947845d_mset @ A )
% 11.90/6.02 = ( fold_mset @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4830_vimage__subset__eq,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,B6: set @ B,A4: set @ A] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B6 ) @ A4 )
% 11.90/6.02 = ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F @ A4 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4831_ex__bij__betw__nat__finite,axiom,
% 11.90/6.02 ! [A: $tType,M3: set @ A] :
% 11.90/6.02 ( ( finite_finite2 @ A @ M3 )
% 11.90/6.02 => ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4832_ex__bij__betw__nat__finite__1,axiom,
% 11.90/6.02 ! [A: $tType,M3: set @ A] :
% 11.90/6.02 ( ( finite_finite2 @ A @ M3 )
% 11.90/6.02 => ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4833_coprime__diff__one__right__nat,axiom,
% 11.90/6.02 ! [N: nat] :
% 11.90/6.02 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4834_coprime__diff__one__left__nat,axiom,
% 11.90/6.02 ! [N: nat] :
% 11.90/6.02 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
% 11.90/6.02 => ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4835_prod_Oreindex__bij__betw__not__neutral,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,C: $tType] :
% 11.90/6.02 ( ( comm_monoid_mult @ A )
% 11.90/6.02 => ! [S7: set @ B,T5: set @ C,H3: B > C,S2: set @ B,T6: set @ C,G: C > A] :
% 11.90/6.02 ( ( finite_finite2 @ B @ S7 )
% 11.90/6.02 => ( ( finite_finite2 @ C @ T5 )
% 11.90/6.02 => ( ( bij_betw @ B @ C @ H3 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
% 11.90/6.02 => ( ! [A5: B] :
% 11.90/6.02 ( ( member @ B @ A5 @ S7 )
% 11.90/6.02 => ( ( G @ ( H3 @ A5 ) )
% 11.90/6.02 = ( one_one @ A ) ) )
% 11.90/6.02 => ( ! [B3: C] :
% 11.90/6.02 ( ( member @ C @ B3 @ T5 )
% 11.90/6.02 => ( ( G @ B3 )
% 11.90/6.02 = ( one_one @ A ) ) )
% 11.90/6.02 => ( ( groups7121269368397514597t_prod @ B @ A
% 11.90/6.02 @ ^ [X3: B] : ( G @ ( H3 @ X3 ) )
% 11.90/6.02 @ S2 )
% 11.90/6.02 = ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4836_sum_OatLeastAtMost__reindex,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( ( comm_monoid_add @ A )
% 11.90/6.02 & ( ord @ B ) )
% 11.90/6.02 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 11.90/6.02 ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4837_sum_OatLeastLessThan__reindex,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( ( comm_monoid_add @ A )
% 11.90/6.02 & ( ord @ B ) )
% 11.90/6.02 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 11.90/6.02 ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 => ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 = ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4838_prod_OatLeastAtMost__reindex,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( ( comm_monoid_mult @ A )
% 11.90/6.02 & ( ord @ B ) )
% 11.90/6.02 => ! [H3: nat > B,M2: nat,N: nat,G: B > A] :
% 11.90/6.02 ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 => ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.90/6.02 = ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4839_antisymp__antisym__eq,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( antisymp @ A
% 11.90/6.02 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.90/6.02 = ( antisym @ A @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4840_convol__expand__snd_H,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H3: A > C] :
% 11.90/6.02 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 11.90/6.02 = G )
% 11.90/6.02 => ( ( H3
% 11.90/6.02 = ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 11.90/6.02 = ( ( bNF_convol @ A @ B @ C @ G @ H3 )
% 11.90/6.02 = F ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4841_convol__expand__snd,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
% 11.90/6.02 ( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
% 11.90/6.02 = G )
% 11.90/6.02 => ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
% 11.90/6.02 = F ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4842_convol__def,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/6.02 ( ( bNF_convol @ A @ B @ C )
% 11.90/6.02 = ( ^ [F2: A > B,G2: A > C,A7: A] : ( product_Pair @ B @ C @ ( F2 @ A7 ) @ ( G2 @ A7 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4843_bij__swap,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4844_fst__convol,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
% 11.90/6.02 ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
% 11.90/6.02 = F ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4845_snd__convol,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
% 11.90/6.02 ( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
% 11.90/6.02 = G ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4846_convol__o,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H3: A > D] :
% 11.90/6.02 ( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H3 )
% 11.90/6.02 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H3 ) @ ( comp @ D @ C @ A @ G @ H3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4847_ex__bij__betw__finite__nat,axiom,
% 11.90/6.02 ! [A: $tType,M3: set @ A] :
% 11.90/6.02 ( ( finite_finite2 @ A @ M3 )
% 11.90/6.02 => ? [H2: A > nat] : ( bij_betw @ A @ nat @ H2 @ M3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4848_map__prod__o__convol__id,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
% 11.90/6.02 ( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X )
% 11.90/6.02 = ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4849_Heap__lub__empty,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
% 11.90/6.02 = ( heap_Time_Heap2 @ A
% 11.90/6.02 @ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4850_and_Osemilattice__neutr__axioms,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/6.02 => ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4851_map__prod__ident,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( product_map_prod @ A @ A @ B @ B
% 11.90/6.02 @ ^ [X3: A] : X3
% 11.90/6.02 @ ^ [Y3: B] : Y3 )
% 11.90/6.02 = ( ^ [Z5: product_prod @ A @ B] : Z5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4852_map__prod__simp,axiom,
% 11.90/6.02 ! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A3: C,B2: D] :
% 11.90/6.02 ( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A3 @ B2 ) )
% 11.90/6.02 = ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4853_fst__map__prod,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
% 11.90/6.02 ( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
% 11.90/6.02 = ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4854_snd__map__prod,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
% 11.90/6.02 ( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
% 11.90/6.02 = ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4855_map__prod__imageI,axiom,
% 11.90/6.02 ! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R5: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R5 )
% 11.90/6.02 => ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4856_fst__comp__map__prod,axiom,
% 11.90/6.02 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
% 11.90/6.02 ( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
% 11.90/6.02 = ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4857_snd__comp__map__prod,axiom,
% 11.90/6.02 ! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
% 11.90/6.02 ( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
% 11.90/6.02 = ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4858_prod_Omap__comp,axiom,
% 11.90/6.02 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F5,F1: A > C,F22: B > D,V: product_prod @ A @ B] :
% 11.90/6.02 ( ( product_map_prod @ C @ E @ D @ F5 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 11.90/6.02 = ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F5 @ B @ G22 @ F22 ) @ V ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4859_map__prod_Ocompositionality,axiom,
% 11.90/6.02 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D,Prod: product_prod @ A @ B] :
% 11.90/6.02 ( ( product_map_prod @ C @ E @ D @ F5 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H3 @ I @ Prod ) )
% 11.90/6.02 = ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Prod ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4860_map__prod_Oidentity,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( product_map_prod @ A @ A @ B @ B
% 11.90/6.02 @ ^ [X3: A] : X3
% 11.90/6.02 @ ^ [X3: B] : X3 )
% 11.90/6.02 = ( id @ ( product_prod @ A @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4861_prod__fun__imageE,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G: D > B,R5: set @ ( product_prod @ C @ D )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R5 ) )
% 11.90/6.02 => ~ ! [X2: C,Y4: D] :
% 11.90/6.02 ( ( C2
% 11.90/6.02 = ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ Y4 ) ) )
% 11.90/6.02 => ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X2 @ Y4 ) @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4862_apfst__def,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType] :
% 11.90/6.02 ( ( product_apfst @ A @ C @ B )
% 11.90/6.02 = ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4863_apsnd__def,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType] :
% 11.90/6.02 ( ( product_apsnd @ B @ C @ A )
% 11.90/6.02 = ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4864_map__prod__def,axiom,
% 11.90/6.02 ! [B: $tType,D: $tType,C: $tType,A: $tType] :
% 11.90/6.02 ( ( product_map_prod @ A @ C @ B @ D )
% 11.90/6.02 = ( ^ [F2: A > C,G2: B > D] :
% 11.90/6.02 ( product_case_prod @ A @ B @ ( product_prod @ C @ D )
% 11.90/6.02 @ ^ [X3: A,Y3: B] : ( product_Pair @ C @ D @ ( F2 @ X3 ) @ ( G2 @ Y3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4865_map__prod__compose,axiom,
% 11.90/6.02 ! [D: $tType,C: $tType,A: $tType,E: $tType,F5: $tType,B: $tType,F1: E > C,F22: A > E,G1: F5 > D,G22: B > F5] :
% 11.90/6.02 ( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F5 @ D @ B @ G1 @ G22 ) )
% 11.90/6.02 = ( comp @ ( product_prod @ E @ F5 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F5 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F5 @ F22 @ G22 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4866_map__prod_Ocomp,axiom,
% 11.90/6.02 ! [A: $tType,C: $tType,E: $tType,F5: $tType,D: $tType,B: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D] :
% 11.90/6.02 ( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F5 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F5 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H3 @ I ) )
% 11.90/6.02 = ( product_map_prod @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4867_map__prod__surj__on,axiom,
% 11.90/6.02 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A4: set @ B,A18: set @ A,G: D > C,B6: set @ D,B14: set @ C] :
% 11.90/6.02 ( ( ( image2 @ B @ A @ F @ A4 )
% 11.90/6.02 = A18 )
% 11.90/6.02 => ( ( ( image2 @ D @ C @ G @ B6 )
% 11.90/6.02 = B14 )
% 11.90/6.02 => ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
% 11.90/6.02 @ ( product_Sigma @ B @ D @ A4
% 11.90/6.02 @ ^ [Uu: B] : B6 ) )
% 11.90/6.02 = ( product_Sigma @ A @ C @ A18
% 11.90/6.02 @ ^ [Uu: A] : B14 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4868_map__prod__o__convol,axiom,
% 11.90/6.02 ! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H22: E > C,F: A > D,G: A > E] :
% 11.90/6.02 ( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H22 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
% 11.90/6.02 = ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H22 @ G ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4869_map__prod__inj__on,axiom,
% 11.90/6.02 ! [D: $tType,B: $tType,C: $tType,A: $tType,F: A > B,A4: set @ A,G: C > D,B6: set @ C] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ A4 )
% 11.90/6.02 => ( ( inj_on @ C @ D @ G @ B6 )
% 11.90/6.02 => ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G )
% 11.90/6.02 @ ( product_Sigma @ A @ C @ A4
% 11.90/6.02 @ ^ [Uu: A] : B6 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4870_map__prod__surj,axiom,
% 11.90/6.02 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
% 11.90/6.02 ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( ( image2 @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
% 11.90/6.02 = ( top_top @ ( set @ D ) ) )
% 11.90/6.02 => ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
% 11.90/6.02 = ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4871_Zorns__po__lemma,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.02 => ( ! [C8: set @ A] :
% 11.90/6.02 ( ( member @ ( set @ A ) @ C8 @ ( chains @ A @ R ) )
% 11.90/6.02 => ? [X7: A] :
% 11.90/6.02 ( ( member @ A @ X7 @ ( field2 @ A @ R ) )
% 11.90/6.02 & ! [Xa3: A] :
% 11.90/6.02 ( ( member @ A @ Xa3 @ C8 )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X7 ) @ R ) ) ) )
% 11.90/6.02 => ? [X2: A] :
% 11.90/6.02 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.90/6.02 & ! [Xa2: A] :
% 11.90/6.02 ( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Xa2 ) @ R )
% 11.90/6.02 => ( Xa2 = X2 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4872_assert__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( heap_Time_assert @ A )
% 11.90/6.02 = ( ^ [P4: A > $o,X3: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ X3 ) @ ( heap_Time_return @ A @ X3 ) @ ( heap_Time_raise @ A @ ( cons @ char @ ( char2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ $false ) @ ( nil @ char ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4873_power__int__add__1,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [X: A,M2: int] :
% 11.90/6.02 ( ( ( X
% 11.90/6.02 != ( zero_zero @ A ) )
% 11.90/6.02 | ( M2
% 11.90/6.02 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 11.90/6.02 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
% 11.90/6.02 = ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4874_power__int__1__left,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [N: int] :
% 11.90/6.02 ( ( power_int @ A @ ( one_one @ A ) @ N )
% 11.90/6.02 = ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4875_power__int__0__right,axiom,
% 11.90/6.02 ! [B: $tType] :
% 11.90/6.02 ( ( ( inverse @ B )
% 11.90/6.02 & ( power @ B ) )
% 11.90/6.02 => ! [X: B] :
% 11.90/6.02 ( ( power_int @ B @ X @ ( zero_zero @ int ) )
% 11.90/6.02 = ( one_one @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4876_power__int__minus__one__mult__self,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [M2: int] :
% 11.90/6.02 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
% 11.90/6.02 = ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4877_power__int__minus__one__mult__self_H,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [M2: int,B2: A] :
% 11.90/6.02 ( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B2 ) )
% 11.90/6.02 = B2 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4878_power__int__add__numeral,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [X: A,M2: num,N: num] :
% 11.90/6.02 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
% 11.90/6.02 = ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4879_power__int__add__numeral2,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [X: A,M2: num,N: num,B2: A] :
% 11.90/6.02 ( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
% 11.90/6.02 = ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4880_power__int__one__over,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [X: A,N: int] :
% 11.90/6.02 ( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
% 11.90/6.02 = ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4881_power__int__0__left__If,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [M2: int] :
% 11.90/6.02 ( ( ( M2
% 11.90/6.02 = ( zero_zero @ int ) )
% 11.90/6.02 => ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/6.02 = ( one_one @ A ) ) )
% 11.90/6.02 & ( ( M2
% 11.90/6.02 != ( zero_zero @ int ) )
% 11.90/6.02 => ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
% 11.90/6.02 = ( zero_zero @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4882_power__int__increasing,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [N: int,N6: int,A3: A] :
% 11.90/6.02 ( ( ord_less_eq @ int @ N @ N6 )
% 11.90/6.02 => ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
% 11.90/6.02 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4883_power__int__strict__increasing,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [N: int,N6: int,A3: A] :
% 11.90/6.02 ( ( ord_less @ int @ N @ N6 )
% 11.90/6.02 => ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/6.02 => ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4884_power__int__minus__one__minus,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [N: int] :
% 11.90/6.02 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
% 11.90/6.02 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4885_power__int__minus__one__diff__commute,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [A3: int,B2: int] :
% 11.90/6.02 ( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
% 11.90/6.02 = ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4886_power__int__strict__decreasing,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [N: int,N6: int,A3: A] :
% 11.90/6.02 ( ( ord_less @ int @ N @ N6 )
% 11.90/6.02 => ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/6.02 => ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
% 11.90/6.02 => ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4887_one__le__power__int,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [X: A,N: int] :
% 11.90/6.02 ( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
% 11.90/6.02 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 11.90/6.02 => ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4888_one__less__power__int,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [A3: A,N: int] :
% 11.90/6.02 ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
% 11.90/6.02 => ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
% 11.90/6.02 => ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4889_power__int__add,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [X: A,M2: int,N: int] :
% 11.90/6.02 ( ( ( X
% 11.90/6.02 != ( zero_zero @ A ) )
% 11.90/6.02 | ( ( plus_plus @ int @ M2 @ N )
% 11.90/6.02 != ( zero_zero @ int ) ) )
% 11.90/6.02 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
% 11.90/6.02 = ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4890_power__int__minus__left__distrib,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType] :
% 11.90/6.02 ( ( ( division_ring @ A )
% 11.90/6.02 & ( one @ B )
% 11.90/6.02 & ( uminus @ B ) )
% 11.90/6.02 => ! [X: C,A3: A,N: int] :
% 11.90/6.02 ( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
% 11.90/6.02 => ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 11.90/6.02 = ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4891_power__int__decreasing,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [N: int,N6: int,A3: A] :
% 11.90/6.02 ( ( ord_less_eq @ int @ N @ N6 )
% 11.90/6.02 => ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
% 11.90/6.02 => ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
% 11.90/6.02 => ( ( ( A3
% 11.90/6.02 != ( zero_zero @ A ) )
% 11.90/6.02 | ( N6
% 11.90/6.02 != ( zero_zero @ int ) )
% 11.90/6.02 | ( N
% 11.90/6.02 = ( zero_zero @ int ) ) )
% 11.90/6.02 => ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4892_power__int__le__one,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [X: A,N: int] :
% 11.90/6.02 ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
% 11.90/6.02 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 11.90/6.02 => ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
% 11.90/6.02 => ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4893_power__int__le__imp__le__exp,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [X: A,M2: int,N: int] :
% 11.90/6.02 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 11.90/6.02 => ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
% 11.90/6.02 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 11.90/6.02 => ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4894_power__int__le__imp__less__exp,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( linordered_field @ A )
% 11.90/6.02 => ! [X: A,M2: int,N: int] :
% 11.90/6.02 ( ( ord_less @ A @ ( one_one @ A ) @ X )
% 11.90/6.02 => ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
% 11.90/6.02 => ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
% 11.90/6.02 => ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4895_power__int__add__1_H,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( division_ring @ A )
% 11.90/6.02 => ! [X: A,M2: int] :
% 11.90/6.02 ( ( ( X
% 11.90/6.02 != ( zero_zero @ A ) )
% 11.90/6.02 | ( M2
% 11.90/6.02 != ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
% 11.90/6.02 => ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
% 11.90/6.02 = ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4896_integer__of__char__code,axiom,
% 11.90/6.02 ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
% 11.90/6.02 ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
% 11.90/6.02 = ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4897_char__of__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( bit_un5681908812861735899ations @ A )
% 11.90/6.02 => ( ( unique5772411509450598832har_of @ A )
% 11.90/6.02 = ( ^ [N5: A] :
% 11.90/6.02 ( char2
% 11.90/6.02 @ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( one_one @ nat ) )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
% 11.90/6.02 @ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4898_numeral__unfold__funpow,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_1 @ A )
% 11.90/6.02 => ( ( numeral_numeral @ A )
% 11.90/6.02 = ( ^ [K4: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K4 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4899_relpowp__1,axiom,
% 11.90/6.02 ! [A: $tType,P: A > A > $o] :
% 11.90/6.02 ( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
% 11.90/6.02 = P ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4900_Suc__funpow,axiom,
% 11.90/6.02 ! [N: nat] :
% 11.90/6.02 ( ( compow @ ( nat > nat ) @ N @ suc )
% 11.90/6.02 = ( plus_plus @ nat @ N ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4901_comp__funpow,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,N: nat,F: A > A] :
% 11.90/6.02 ( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
% 11.90/6.02 = ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4902_funpow_Osimps_I2_J,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,F: A > A] :
% 11.90/6.02 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 11.90/6.02 = ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4903_funpow__Suc__right,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,F: A > A] :
% 11.90/6.02 ( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
% 11.90/6.02 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ F ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4904_funpow__add,axiom,
% 11.90/6.02 ! [A: $tType,M2: nat,N: nat,F: A > A] :
% 11.90/6.02 ( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N ) @ F )
% 11.90/6.02 = ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F ) @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4905_of__nat__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_1 @ A )
% 11.90/6.02 => ( ( semiring_1_of_nat @ A )
% 11.90/6.02 = ( ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4906_numeral__add__unfold__funpow,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( semiring_numeral @ A )
% 11.90/6.02 => ! [K: num,A3: A] :
% 11.90/6.02 ( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
% 11.90/6.02 = ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4907_inj__on__funpow__least,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,F: A > A,S: A] :
% 11.90/6.02 ( ( ( compow @ ( A > A ) @ N @ F @ S )
% 11.90/6.02 = S )
% 11.90/6.02 => ( ! [M5: nat] :
% 11.90/6.02 ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M5 )
% 11.90/6.02 => ( ( ord_less @ nat @ M5 @ N )
% 11.90/6.02 => ( ( compow @ ( A > A ) @ M5 @ F @ S )
% 11.90/6.02 != S ) ) )
% 11.90/6.02 => ( inj_on @ nat @ A
% 11.90/6.02 @ ^ [K4: nat] : ( compow @ ( A > A ) @ K4 @ F @ S )
% 11.90/6.02 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4908_card__Un__disjnt,axiom,
% 11.90/6.02 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.90/6.02 ( ( finite_finite2 @ A @ A4 )
% 11.90/6.02 => ( ( finite_finite2 @ A @ B6 )
% 11.90/6.02 => ( ( disjnt @ A @ A4 @ B6 )
% 11.90/6.02 => ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/6.02 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4909_plus__multiset__def,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( plus_plus @ ( multiset @ A ) )
% 11.90/6.02 = ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( count @ A ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
% 11.90/6.02 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4910_mset__subset__add__iff2,axiom,
% 11.90/6.02 ! [A: $tType,I: nat,J: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 11.90/6.02 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/6.02 => ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 11.90/6.02 = ( subset_mset @ A @ M2 @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4911_repeat__mset__distrib2,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,A4: multiset @ A,B6: multiset @ A] :
% 11.90/6.02 ( ( repeat_mset @ A @ N @ ( plus_plus @ ( multiset @ A ) @ A4 @ B6 ) )
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ N @ A4 ) @ ( repeat_mset @ A @ N @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4912_repeat__mset__Suc,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,M3: multiset @ A] :
% 11.90/6.02 ( ( repeat_mset @ A @ ( suc @ N ) @ M3 )
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ M3 @ ( repeat_mset @ A @ N @ M3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4913_disjnt__Times1__iff,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,C5: set @ A,A4: set @ B,B6: set @ B] :
% 11.90/6.02 ( ( disjnt @ ( product_prod @ A @ B )
% 11.90/6.02 @ ( product_Sigma @ A @ B @ C5
% 11.90/6.02 @ ^ [Uu: A] : A4 )
% 11.90/6.02 @ ( product_Sigma @ A @ B @ C5
% 11.90/6.02 @ ^ [Uu: A] : B6 ) )
% 11.90/6.02 = ( ( C5
% 11.90/6.02 = ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 | ( disjnt @ B @ A4 @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4914_disjnt__Times2__iff,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A4: set @ A,C5: set @ B,B6: set @ A] :
% 11.90/6.02 ( ( disjnt @ ( product_prod @ A @ B )
% 11.90/6.02 @ ( product_Sigma @ A @ B @ A4
% 11.90/6.02 @ ^ [Uu: A] : C5 )
% 11.90/6.02 @ ( product_Sigma @ A @ B @ B6
% 11.90/6.02 @ ^ [Uu: A] : C5 ) )
% 11.90/6.02 = ( ( C5
% 11.90/6.02 = ( bot_bot @ ( set @ B ) ) )
% 11.90/6.02 | ( disjnt @ A @ A4 @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4915_disjnt__Sigma__iff,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A4: set @ A,C5: A > ( set @ B ),B6: set @ A] :
% 11.90/6.02 ( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A4 @ C5 ) @ ( product_Sigma @ A @ B @ B6 @ C5 ) )
% 11.90/6.02 = ( ! [X3: A] :
% 11.90/6.02 ( ( member @ A @ X3 @ ( inf_inf @ ( set @ A ) @ A4 @ B6 ) )
% 11.90/6.02 => ( ( C5 @ X3 )
% 11.90/6.02 = ( bot_bot @ ( set @ B ) ) ) )
% 11.90/6.02 | ( disjnt @ A @ A4 @ B6 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4916_left__add__mult__distrib__mset,axiom,
% 11.90/6.02 ! [A: $tType,I: nat,U: multiset @ A,J: nat,K: multiset @ A] :
% 11.90/6.02 ( ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ K ) )
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4917_repeat__mset__distrib,axiom,
% 11.90/6.02 ! [A: $tType,M2: nat,N: nat,A4: multiset @ A] :
% 11.90/6.02 ( ( repeat_mset @ A @ ( plus_plus @ nat @ M2 @ N ) @ A4 )
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ M2 @ A4 ) @ ( repeat_mset @ A @ N @ A4 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4918_mset__subseteq__add__iff1,axiom,
% 11.90/6.02 ! [A: $tType,J: nat,I: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 11.90/6.02 ( ( ord_less_eq @ nat @ J @ I )
% 11.90/6.02 => ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 11.90/6.02 = ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4919_mset__subseteq__add__iff2,axiom,
% 11.90/6.02 ! [A: $tType,I: nat,J: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 11.90/6.02 ( ( ord_less_eq @ nat @ I @ J )
% 11.90/6.02 => ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 11.90/6.02 = ( subseteq_mset @ A @ M2 @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4920_mset__subset__add__iff1,axiom,
% 11.90/6.02 ! [A: $tType,J: nat,I: nat,U: multiset @ A,M2: multiset @ A,N: multiset @ A] :
% 11.90/6.02 ( ( ord_less_eq @ nat @ J @ I )
% 11.90/6.02 => ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M2 ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
% 11.90/6.02 = ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4921_plus__multiset_Oabs__eq,axiom,
% 11.90/6.02 ! [A: $tType,Xa: A > nat,X: A > nat] :
% 11.90/6.02 ( ( bNF_eq_onp @ ( A > nat )
% 11.90/6.02 @ ^ [F2: A > nat] :
% 11.90/6.02 ( finite_finite2 @ A
% 11.90/6.02 @ ( collect @ A
% 11.90/6.02 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
% 11.90/6.02 @ Xa
% 11.90/6.02 @ Xa )
% 11.90/6.02 => ( ( bNF_eq_onp @ ( A > nat )
% 11.90/6.02 @ ^ [F2: A > nat] :
% 11.90/6.02 ( finite_finite2 @ A
% 11.90/6.02 @ ( collect @ A
% 11.90/6.02 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) )
% 11.90/6.02 @ X
% 11.90/6.02 @ X )
% 11.90/6.02 => ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X ) )
% 11.90/6.02 = ( abs_multiset @ A
% 11.90/6.02 @ ^ [A7: A] : ( plus_plus @ nat @ ( Xa @ A7 ) @ ( X @ A7 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4922_repeat__mset__distrib__add__mset,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,A3: A,A4: multiset @ A] :
% 11.90/6.02 ( ( repeat_mset @ A @ N @ ( add_mset @ A @ A3 @ A4 ) )
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ ( replicate_mset @ A @ N @ A3 ) @ ( repeat_mset @ A @ N @ A4 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4923_o__prs_I2_J,axiom,
% 11.90/6.02 ! [F5: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H8: $tType,G3: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F5,Rep32: F5 > E] :
% 11.90/6.02 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 11.90/6.02 => ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
% 11.90/6.02 => ( ( quotient3 @ E @ F5 @ R32 @ Abs32 @ Rep32 )
% 11.90/6.02 => ( ( map_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( A > G3 ) > A > H8 ) @ ( ( B > G3 ) > B > H8 ) @ ( id @ ( G3 > H8 ) ) @ ( map_fun @ ( B > G3 ) @ ( A > G3 ) @ ( A > H8 ) @ ( B > H8 ) @ ( map_fun @ A @ B @ G3 @ G3 @ Abs1 @ ( id @ G3 ) ) @ ( map_fun @ B @ A @ H8 @ H8 @ Rep1 @ ( id @ H8 ) ) ) @ ( comp @ G3 @ H8 @ A ) )
% 11.90/6.02 = ( comp @ G3 @ H8 @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4924_abs__o__rep,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A] :
% 11.90/6.02 ( ( quotient3 @ A @ B @ R5 @ Abs @ Rep )
% 11.90/6.02 => ( ( comp @ A @ B @ B @ Abs @ Rep )
% 11.90/6.02 = ( id @ B ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4925_o__prs_I1_J,axiom,
% 11.90/6.02 ! [C: $tType,E: $tType,A: $tType,B: $tType,F5: $tType,D: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F5,Rep32: F5 > E] :
% 11.90/6.02 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 11.90/6.02 => ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
% 11.90/6.02 => ( ( quotient3 @ E @ F5 @ R32 @ Abs32 @ Rep32 )
% 11.90/6.02 => ( ( map_fun @ ( D > F5 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F5 ) @ ( map_fun @ C @ D @ F5 @ E @ Abs22 @ Rep32 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F5 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) @ ( map_fun @ B @ A @ E @ F5 @ Rep1 @ Abs32 ) ) @ ( comp @ C @ E @ A ) )
% 11.90/6.02 = ( comp @ D @ F5 @ B ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4926_plus__multiset_Orsp,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
% 11.90/6.02 @ ( bNF_eq_onp @ ( A > nat )
% 11.90/6.02 @ ^ [F2: A > nat] :
% 11.90/6.02 ( finite_finite2 @ A
% 11.90/6.02 @ ( collect @ A
% 11.90/6.02 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) )
% 11.90/6.02 @ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
% 11.90/6.02 @ ( bNF_eq_onp @ ( A > nat )
% 11.90/6.02 @ ^ [F2: A > nat] :
% 11.90/6.02 ( finite_finite2 @ A
% 11.90/6.02 @ ( collect @ A
% 11.90/6.02 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) )
% 11.90/6.02 @ ( bNF_eq_onp @ ( A > nat )
% 11.90/6.02 @ ^ [F2: A > nat] :
% 11.90/6.02 ( finite_finite2 @ A
% 11.90/6.02 @ ( collect @ A
% 11.90/6.02 @ ^ [X3: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ) )
% 11.90/6.02 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) )
% 11.90/6.02 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4927_or__not__num__neg_Opelims,axiom,
% 11.90/6.02 ! [X: num,Xa: num,Y: num] :
% 11.90/6.02 ( ( ( bit_or_not_num_neg @ X @ Xa )
% 11.90/6.02 = Y )
% 11.90/6.02 => ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 11.90/6.02 => ( ( ( X = one2 )
% 11.90/6.02 => ( ( Xa = one2 )
% 11.90/6.02 => ( ( Y = one2 )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 11.90/6.02 => ( ( ( X = one2 )
% 11.90/6.02 => ! [M5: num] :
% 11.90/6.02 ( ( Xa
% 11.90/6.02 = ( bit0 @ M5 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bit1 @ M5 ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M5 ) ) ) ) ) )
% 11.90/6.02 => ( ( ( X = one2 )
% 11.90/6.02 => ! [M5: num] :
% 11.90/6.02 ( ( Xa
% 11.90/6.02 = ( bit1 @ M5 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bit1 @ M5 ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M5 ) ) ) ) ) )
% 11.90/6.02 => ( ! [N3: num] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( bit0 @ N3 ) )
% 11.90/6.02 => ( ( Xa = one2 )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bit0 @ one2 ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ one2 ) ) ) ) )
% 11.90/6.02 => ( ! [N3: num] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( bit0 @ N3 ) )
% 11.90/6.02 => ! [M5: num] :
% 11.90/6.02 ( ( Xa
% 11.90/6.02 = ( bit0 @ M5 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 11.90/6.02 => ( ! [N3: num] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( bit0 @ N3 ) )
% 11.90/6.02 => ! [M5: num] :
% 11.90/6.02 ( ( Xa
% 11.90/6.02 = ( bit1 @ M5 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) )
% 11.90/6.02 => ( ! [N3: num] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( bit1 @ N3 ) )
% 11.90/6.02 => ( ( Xa = one2 )
% 11.90/6.02 => ( ( Y = one2 )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ one2 ) ) ) ) )
% 11.90/6.02 => ( ! [N3: num] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( bit1 @ N3 ) )
% 11.90/6.02 => ! [M5: num] :
% 11.90/6.02 ( ( Xa
% 11.90/6.02 = ( bit0 @ M5 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
% 11.90/6.02 => ~ ! [N3: num] :
% 11.90/6.02 ( ( X
% 11.90/6.02 = ( bit1 @ N3 ) )
% 11.90/6.02 => ! [M5: num] :
% 11.90/6.02 ( ( Xa
% 11.90/6.02 = ( bit1 @ M5 ) )
% 11.90/6.02 => ( ( Y
% 11.90/6.02 = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
% 11.90/6.02 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4928_cr__natural__def,axiom,
% 11.90/6.02 ( code_cr_natural
% 11.90/6.02 = ( ^ [X3: nat,Y3: code_natural] :
% 11.90/6.02 ( X3
% 11.90/6.02 = ( code_nat_of_natural @ Y3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4929_inv__o__cancel,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 11.90/6.02 = ( id @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4930_o__inv__o__cancel,axiom,
% 11.90/6.02 ! [B: $tType,C: $tType,A: $tType,F: A > B,G: A > C] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) @ F )
% 11.90/6.02 = G ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4931_inj__map__inv__f,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B,L: list @ A] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( map @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( map @ A @ B @ F @ L ) )
% 11.90/6.02 = L ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4932_inv__into__comp,axiom,
% 11.90/6.02 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A4: set @ C,X: B] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) )
% 11.90/6.02 => ( ( inj_on @ C @ A @ G @ A4 )
% 11.90/6.02 => ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A4 ) ) )
% 11.90/6.02 => ( ( hilbert_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F @ G ) @ X )
% 11.90/6.02 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A4 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A4 ) @ F ) @ X ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4933_inv__unique__comp,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: B > A,G: A > B] :
% 11.90/6.02 ( ( ( comp @ B @ A @ A @ F @ G )
% 11.90/6.02 = ( id @ A ) )
% 11.90/6.02 => ( ( ( comp @ A @ B @ B @ G @ F )
% 11.90/6.02 = ( id @ B ) )
% 11.90/6.02 => ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
% 11.90/6.02 = G ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4934_o__inv__distrib,axiom,
% 11.90/6.02 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > A] :
% 11.90/6.02 ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 => ( ( bij_betw @ C @ A @ G @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F @ G ) )
% 11.90/6.02 = ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4935_surj__iff,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: B > A] :
% 11.90/6.02 ( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 11.90/6.02 = ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( ( comp @ B @ A @ A @ F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) )
% 11.90/6.02 = ( id @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4936_inj__iff,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,F: A > B] :
% 11.90/6.02 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 = ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 11.90/6.02 = ( id @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4937_fn__o__inv__fn__is__id,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A,N: nat] :
% 11.90/6.02 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
% 11.90/6.02 = ( ^ [X3: A] : X3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4938_inv__fn__o__fn__is__id,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A,N: nat] :
% 11.90/6.02 ( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
% 11.90/6.02 => ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N @ F ) )
% 11.90/6.02 = ( ^ [X3: A] : X3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4939_bijection_Oinv__comp__right,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A] :
% 11.90/6.02 ( ( hilbert_bijection @ A @ F )
% 11.90/6.02 => ( ( comp @ A @ A @ A @ F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
% 11.90/6.02 = ( id @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4940_bijection_Oinv__comp__left,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A] :
% 11.90/6.02 ( ( hilbert_bijection @ A @ F )
% 11.90/6.02 => ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ F )
% 11.90/6.02 = ( id @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4941_finite__reachable__restrictedI,axiom,
% 11.90/6.02 ! [A: $tType,Q: set @ A,I4: set @ A,E4: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( finite_finite2 @ A @ Q )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ I4 @ Q )
% 11.90/6.02 => ( ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ E4 ) @ Q )
% 11.90/6.02 => ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E4 ) @ I4 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4942_Range__insert,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A3: B,B2: A,R: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) )
% 11.90/6.02 = ( insert2 @ A @ B2 @ ( range2 @ B @ A @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4943_Range__iff,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( ( member @ A @ A3 @ ( range2 @ B @ A @ R ) )
% 11.90/6.02 = ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A3 ) @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4944_RangeE,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
% 11.90/6.02 ( ( member @ A @ B2 @ ( range2 @ B @ A @ R ) )
% 11.90/6.02 => ~ ! [A5: B] :
% 11.90/6.02 ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A5 @ B2 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4945_Range_Ointros,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 11.90/6.02 => ( member @ B @ B2 @ ( range2 @ A @ B @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4946_Range_Osimps,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
% 11.90/6.02 = ( ? [A7: A,B5: B] :
% 11.90/6.02 ( ( A3 = B5 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4947_Range_Ocases,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
% 11.90/6.02 => ~ ! [A5: A] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ A3 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4948_trancl__Image__in__Range,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),V6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_trancl @ A @ R5 ) @ V6 ) @ ( range2 @ A @ A @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4949_Range__rel__restrict,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( range2 @ A @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R5 ) @ A4 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4950_dom__ran__disj__comp,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) )
% 11.90/6.02 = ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( ( relcomp @ A @ A @ A @ R5 @ R5 )
% 11.90/6.02 = ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4951_wf__max,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( wf @ A @ ( converse @ A @ A @ R5 ) )
% 11.90/6.02 => ( ( R5
% 11.90/6.02 != ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 11.90/6.02 => ~ ! [M5: A] :
% 11.90/6.02 ~ ( member @ A @ M5 @ ( minus_minus @ ( set @ A ) @ ( range2 @ A @ A @ R5 ) @ ( domain @ A @ A @ R5 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4952_Rangep__Range__eq,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( rangep @ A @ B
% 11.90/6.02 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 11.90/6.02 = ( ^ [X3: B] : ( member @ B @ X3 @ ( range2 @ A @ B @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4953_Domain__insert,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R ) )
% 11.90/6.02 = ( insert2 @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4954_Domain_Ocases,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 11.90/6.02 => ~ ! [B3: B] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4955_Domain_Osimps,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 11.90/6.02 = ( ? [A7: A,B5: B] :
% 11.90/6.02 ( ( A3 = A7 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4956_Domain_ODomainI,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
% 11.90/6.02 => ( member @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4957_DomainE,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 11.90/6.02 => ~ ! [B3: B] :
% 11.90/6.02 ~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4958_Domain__iff,axiom,
% 11.90/6.02 ! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
% 11.90/6.02 = ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4959_Not__Domain__rtrancl,axiom,
% 11.90/6.02 ! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A ),Y: A] :
% 11.90/6.02 ( ~ ( member @ A @ X @ ( domain @ A @ A @ R5 ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R5 ) )
% 11.90/6.02 = ( X = Y ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4960_Domain__unfold,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( domain @ A @ B )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( collect @ A
% 11.90/6.02 @ ^ [X3: A] :
% 11.90/6.02 ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4961_for__in__RI,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType,X: A,R5: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( ( member @ A @ X @ ( domain @ A @ B @ R5 ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ ( fun_of_rel @ A @ B @ R5 @ X ) ) @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4962_Domain__rel__restrict,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ A @ ( rel_restrict @ A @ R5 @ A4 ) ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ A4 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4963_wf__no__path,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( ( inf_inf @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) )
% 11.90/6.02 = ( bot_bot @ ( set @ A ) ) )
% 11.90/6.02 => ( wf @ A @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4964_wf__min,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( wf @ A @ R5 )
% 11.90/6.02 => ( ( R5
% 11.90/6.02 != ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
% 11.90/6.02 => ~ ! [M5: A] :
% 11.90/6.02 ~ ( member @ A @ M5 @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ A @ R5 ) @ ( range2 @ A @ A @ R5 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4965_Range__def,axiom,
% 11.90/6.02 ! [B: $tType,A: $tType] :
% 11.90/6.02 ( ( range2 @ A @ B )
% 11.90/6.02 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 11.90/6.02 ( collect @ B
% 11.90/6.02 @ ( rangep @ A @ B
% 11.90/6.02 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4966_and_Omonoid__axioms,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/6.02 => ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4967_cr__integer__def,axiom,
% 11.90/6.02 ( code_cr_integer
% 11.90/6.02 = ( ^ [X3: int,Y3: code_integer] :
% 11.90/6.02 ( X3
% 11.90/6.02 = ( code_int_of_integer @ Y3 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4968_wo__rel_OisMinim__def,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ A,B2: A] :
% 11.90/6.02 ( ( bNF_Wellorder_wo_rel @ A @ R )
% 11.90/6.02 => ( ( bNF_We4791949203932849705sMinim @ A @ R @ A4 @ B2 )
% 11.90/6.02 = ( ( member @ A @ B2 @ A4 )
% 11.90/6.02 & ! [X3: A] :
% 11.90/6.02 ( ( member @ A @ X3 @ A4 )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X3 ) @ R ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4969_monoid_Oleft__neutral,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A > A,Z2: A,A3: A] :
% 11.90/6.02 ( ( monoid @ A @ F @ Z2 )
% 11.90/6.02 => ( ( F @ Z2 @ A3 )
% 11.90/6.02 = A3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4970_monoid_Oright__neutral,axiom,
% 11.90/6.02 ! [A: $tType,F: A > A > A,Z2: A,A3: A] :
% 11.90/6.02 ( ( monoid @ A @ F @ Z2 )
% 11.90/6.02 => ( ( F @ A3 @ Z2 )
% 11.90/6.02 = A3 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4971_add_Omonoid__axioms,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_add @ A )
% 11.90/6.02 => ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4972_mult_Omonoid__axioms,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( monoid_mult @ A )
% 11.90/6.02 => ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4973_integer_Opcr__cr__eq,axiom,
% 11.90/6.02 code_pcr_integer = code_cr_integer ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4974_multp__implies__one__step,axiom,
% 11.90/6.02 ! [A: $tType,R: A > A > $o,M3: multiset @ A,N6: multiset @ A] :
% 11.90/6.02 ( ( transp @ A @ R )
% 11.90/6.02 => ( ( multp @ A @ R @ M3 @ N6 )
% 11.90/6.02 => ? [I8: multiset @ A,J5: multiset @ A] :
% 11.90/6.02 ( ( N6
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ I8 @ J5 ) )
% 11.90/6.02 & ? [K10: multiset @ A] :
% 11.90/6.02 ( ( M3
% 11.90/6.02 = ( plus_plus @ ( multiset @ A ) @ I8 @ K10 ) )
% 11.90/6.02 & ( J5
% 11.90/6.02 != ( zero_zero @ ( multiset @ A ) ) )
% 11.90/6.02 & ! [X7: A] :
% 11.90/6.02 ( ( member @ A @ X7 @ ( set_mset @ A @ K10 ) )
% 11.90/6.02 => ? [Xa3: A] :
% 11.90/6.02 ( ( member @ A @ Xa3 @ ( set_mset @ A @ J5 ) )
% 11.90/6.02 & ( R @ X7 @ Xa3 ) ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4975_relpow__fun__conv,axiom,
% 11.90/6.02 ! [A: $tType,A3: A,B2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 = ( ? [F2: nat > A] :
% 11.90/6.02 ( ( ( F2 @ ( zero_zero @ nat ) )
% 11.90/6.02 = A3 )
% 11.90/6.02 & ( ( F2 @ N )
% 11.90/6.02 = B2 )
% 11.90/6.02 & ! [I3: nat] :
% 11.90/6.02 ( ( ord_less @ nat @ I3 @ N )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I3 ) @ ( F2 @ ( suc @ I3 ) ) ) @ R5 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4976_irreflp__irrefl__eq,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( irreflp @ A
% 11.90/6.02 @ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) )
% 11.90/6.02 = ( irrefl @ A @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4977_relpow__1,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R5 )
% 11.90/6.02 = R5 ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4978_multp__cancel,axiom,
% 11.90/6.02 ! [A: $tType,R: A > A > $o,X10: multiset @ A,Z9: multiset @ A,Y9: multiset @ A] :
% 11.90/6.02 ( ( transp @ A @ R )
% 11.90/6.02 => ( ( irreflp @ A @ R )
% 11.90/6.02 => ( ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ X10 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y9 @ Z9 ) )
% 11.90/6.02 = ( multp @ A @ R @ X10 @ Y9 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4979_relpow__Suc__D2_H,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A ),X7: A,Y6: A,Z6: A] :
% 11.90/6.02 ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ Z6 ) @ R5 ) )
% 11.90/6.02 => ? [W4: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X7 @ W4 ) @ R5 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W4 @ Z6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4980_relpow__0__E,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) )
% 11.90/6.02 => ( X = Y ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4981_relpow__0__I,axiom,
% 11.90/6.02 ! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4982_relpow__Suc__E,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
% 11.90/6.02 => ~ ! [Y4: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4983_relpow__Suc__I,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Y: A,N: nat,R5: set @ ( product_prod @ A @ A ),Z2: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R5 )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4984_relpow__Suc__D2,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
% 11.90/6.02 => ? [Y4: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
% 11.90/6.02 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4985_relpow__Suc__E2,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
% 11.90/6.02 => ~ ! [Y4: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4986_relpow__Suc__I2,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Y: A,R5: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4987_relpow__add,axiom,
% 11.90/6.02 ! [A: $tType,M2: nat,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M2 @ N ) @ R5 )
% 11.90/6.02 = ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M2 @ R5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4988_transp__trans__eq,axiom,
% 11.90/6.02 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( transp @ A
% 11.90/6.02 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.90/6.02 = ( trans @ A @ R ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4989_relpowp__relpow__eq,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( compow @ ( A > A > $o ) @ N
% 11.90/6.02 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R5 ) )
% 11.90/6.02 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4990_relpow__E2,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 => ( ( ( N
% 11.90/6.02 = ( zero_zero @ nat ) )
% 11.90/6.02 => ( X != Z2 ) )
% 11.90/6.02 => ~ ! [Y4: A,M5: nat] :
% 11.90/6.02 ( ( N
% 11.90/6.02 = ( suc @ M5 ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R5 ) ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4991_relpow__E,axiom,
% 11.90/6.02 ! [A: $tType,X: A,Z2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
% 11.90/6.02 => ( ( ( N
% 11.90/6.02 = ( zero_zero @ nat ) )
% 11.90/6.02 => ( X != Z2 ) )
% 11.90/6.02 => ~ ! [Y4: A,M5: nat] :
% 11.90/6.02 ( ( N
% 11.90/6.02 = ( suc @ M5 ) )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M5 @ R5 ) )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z2 ) @ R5 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4992_nths__drop,axiom,
% 11.90/6.02 ! [A: $tType,N: nat,Xs: list @ A,I4: set @ nat] :
% 11.90/6.02 ( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I4 )
% 11.90/6.02 = ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I4 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4993_nths__append,axiom,
% 11.90/6.02 ! [A: $tType,L: list @ A,L4: list @ A,A4: set @ nat] :
% 11.90/6.02 ( ( nths @ A @ ( append @ A @ L @ L4 ) @ A4 )
% 11.90/6.02 = ( append @ A @ ( nths @ A @ L @ A4 )
% 11.90/6.02 @ ( nths @ A @ L4
% 11.90/6.02 @ ( collect @ nat
% 11.90/6.02 @ ^ [J2: nat] : ( member @ nat @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ L ) ) @ A4 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4994_asymp__asym__eq,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( asymp @ A
% 11.90/6.02 @ ^ [A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R5 ) )
% 11.90/6.02 = ( asym @ A @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4995_asym_Ocases,axiom,
% 11.90/6.02 ! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( asym @ A @ A3 )
% 11.90/6.02 => ! [A17: A,B8: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A17 @ B8 ) @ A3 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B8 @ A17 ) @ A3 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4996_asym_Osimps,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( asym @ A )
% 11.90/6.02 = ( ^ [A7: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ? [R6: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ( A7 = R6 )
% 11.90/6.02 & ! [X3: A,Y3: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) ) ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4997_asym_Ointros,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ( ! [A5: A,B3: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ R5 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A5 ) @ R5 ) )
% 11.90/6.02 => ( asym @ A @ R5 ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4998_asymD,axiom,
% 11.90/6.02 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.90/6.02 ( ( asym @ A @ R5 )
% 11.90/6.02 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R5 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R5 ) ) ) ).
% 11.90/6.02
% 11.90/6.02 thf(fact_4999_asym__iff,axiom,
% 11.90/6.02 ! [A: $tType] :
% 11.90/6.02 ( ( asym @ A )
% 11.90/6.02 = ( ^ [R6: set @ ( product_prod @ A @ A )] :
% 11.90/6.02 ! [X3: A,Y3: A] :
% 11.90/6.02 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R6 )
% 11.90/6.02 => ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5000_lexord__asymmetric,axiom,
% 11.90/6.03 ! [A: $tType,R5: set @ ( product_prod @ A @ A ),A3: list @ A,B2: list @ A] :
% 11.90/6.03 ( ( asym @ A @ R5 )
% 11.90/6.03 => ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A3 @ B2 ) @ ( lexord @ A @ R5 ) )
% 11.90/6.03 => ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A3 ) @ ( lexord @ A @ R5 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5001_ordLess__iff,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.03 = ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.90/6.03 & ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.90/6.03 & ~ ? [X8: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R @ X8 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5002_option_Oin__rel,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType] :
% 11.90/6.03 ( ( rel_option @ A @ B )
% 11.90/6.03 = ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
% 11.90/6.03 ? [Z5: option @ ( product_prod @ A @ B )] :
% 11.90/6.03 ( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z5
% 11.90/6.03 @ ( collect @ ( option @ ( product_prod @ A @ B ) )
% 11.90/6.03 @ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
% 11.90/6.03 & ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
% 11.90/6.03 = A7 )
% 11.90/6.03 & ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
% 11.90/6.03 = B5 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5003_bit__double__iff,axiom,
% 11.90/6.03 ! [A: $tType] :
% 11.90/6.03 ( ( bit_semiring_bits @ A )
% 11.90/6.03 => ! [A3: A,N: nat] :
% 11.90/6.03 ( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
% 11.90/6.03 = ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
% 11.90/6.03 & ( N
% 11.90/6.03 != ( zero_zero @ nat ) )
% 11.90/6.03 & ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5004_rel__option__None2,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,P: A > B > $o,X: option @ A] :
% 11.90/6.03 ( ( rel_option @ A @ B @ P @ X @ ( none @ B ) )
% 11.90/6.03 = ( X
% 11.90/6.03 = ( none @ A ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5005_rel__option__None1,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,P: A > B > $o,X: option @ B] :
% 11.90/6.03 ( ( rel_option @ A @ B @ P @ ( none @ A ) @ X )
% 11.90/6.03 = ( X
% 11.90/6.03 = ( none @ B ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5006_bit__minus__1__iff,axiom,
% 11.90/6.03 ! [A: $tType] :
% 11.90/6.03 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/6.03 => ! [N: nat] :
% 11.90/6.03 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
% 11.90/6.03 = ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5007_rel__option__iff,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType] :
% 11.90/6.03 ( ( rel_option @ A @ B )
% 11.90/6.03 = ( ^ [R6: A > B > $o,X3: option @ A,Y3: option @ B] :
% 11.90/6.03 ( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
% 11.90/6.03 @ ^ [A7: option @ A,B5: option @ B] :
% 11.90/6.03 ( case_option @ $o @ A
% 11.90/6.03 @ ( case_option @ $o @ B @ $true
% 11.90/6.03 @ ^ [C3: B] : $false
% 11.90/6.03 @ B5 )
% 11.90/6.03 @ ^ [Z5: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z5 ) @ B5 )
% 11.90/6.03 @ A7 )
% 11.90/6.03 @ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X3 @ Y3 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5008_option_Orel__mono,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
% 11.90/6.03 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
% 11.90/6.03 => ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ A @ B @ Ra2 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5009_option_Orel__inject_I2_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
% 11.90/6.03 ( ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) )
% 11.90/6.03 = ( R5 @ X22 @ Y2 ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5010_option_Orel__intros_I2_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
% 11.90/6.03 ( ( R5 @ X22 @ Y2 )
% 11.90/6.03 => ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5011_option__rel__Some1,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: option @ B] :
% 11.90/6.03 ( ( rel_option @ A @ B @ A4 @ ( some @ A @ X ) @ Y )
% 11.90/6.03 = ( ? [Y8: B] :
% 11.90/6.03 ( ( Y
% 11.90/6.03 = ( some @ B @ Y8 ) )
% 11.90/6.03 & ( A4 @ X @ Y8 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5012_option__rel__Some2,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,A4: A > B > $o,X: option @ A,Y: B] :
% 11.90/6.03 ( ( rel_option @ A @ B @ A4 @ X @ ( some @ B @ Y ) )
% 11.90/6.03 = ( ? [X9: A] :
% 11.90/6.03 ( ( X
% 11.90/6.03 = ( some @ A @ X9 ) )
% 11.90/6.03 & ( A4 @ X9 @ Y ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5013_option_Octr__transfer_I1_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o] : ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( none @ B ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5014_rel__option__inf,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,A4: A > B > $o,B6: A > B > $o] :
% 11.90/6.03 ( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A4 ) @ ( rel_option @ A @ B @ B6 ) )
% 11.90/6.03 = ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A4 @ B6 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5015_option_Orel__refl,axiom,
% 11.90/6.03 ! [B: $tType,Ra2: B > B > $o,X: option @ B] :
% 11.90/6.03 ( ! [X2: B] : ( Ra2 @ X2 @ X2 )
% 11.90/6.03 => ( rel_option @ B @ B @ Ra2 @ X @ X ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5016_option_Orel__eq,axiom,
% 11.90/6.03 ! [A: $tType] :
% 11.90/6.03 ( ( rel_option @ A @ A
% 11.90/6.03 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 11.90/6.03 = ( ^ [Y5: option @ A,Z4: option @ A] : ( Y5 = Z4 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5017_option_Orel__cong,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,X: option @ A,Ya: option @ A,Y: option @ B,Xa: option @ B,R5: A > B > $o,Ra2: A > B > $o] :
% 11.90/6.03 ( ( X = Ya )
% 11.90/6.03 => ( ( Y = Xa )
% 11.90/6.03 => ( ! [Z3: A,Yb: B] :
% 11.90/6.03 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 11.90/6.03 => ( ( member @ B @ Yb @ ( set_option @ B @ Xa ) )
% 11.90/6.03 => ( ( R5 @ Z3 @ Yb )
% 11.90/6.03 = ( Ra2 @ Z3 @ Yb ) ) ) )
% 11.90/6.03 => ( ( rel_option @ A @ B @ R5 @ X @ Y )
% 11.90/6.03 = ( rel_option @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5018_option_Orel__mono__strong,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y: option @ B,Ra2: A > B > $o] :
% 11.90/6.03 ( ( rel_option @ A @ B @ R5 @ X @ Y )
% 11.90/6.03 => ( ! [Z3: A,Yb: B] :
% 11.90/6.03 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 11.90/6.03 => ( ( member @ B @ Yb @ ( set_option @ B @ Y ) )
% 11.90/6.03 => ( ( R5 @ Z3 @ Yb )
% 11.90/6.03 => ( Ra2 @ Z3 @ Yb ) ) ) )
% 11.90/6.03 => ( rel_option @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5019_option_Orel__refl__strong,axiom,
% 11.90/6.03 ! [A: $tType,X: option @ A,Ra2: A > A > $o] :
% 11.90/6.03 ( ! [Z3: A] :
% 11.90/6.03 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 11.90/6.03 => ( Ra2 @ Z3 @ Z3 ) )
% 11.90/6.03 => ( rel_option @ A @ A @ Ra2 @ X @ X ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5020_rel__option__reflI,axiom,
% 11.90/6.03 ! [A: $tType,Y: option @ A,P: A > A > $o] :
% 11.90/6.03 ( ! [X2: A] :
% 11.90/6.03 ( ( member @ A @ X2 @ ( set_option @ A @ Y ) )
% 11.90/6.03 => ( P @ X2 @ X2 ) )
% 11.90/6.03 => ( rel_option @ A @ A @ P @ Y @ Y ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5021_option_Orel__map_I1_J,axiom,
% 11.90/6.03 ! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: option @ A,Y: option @ B] :
% 11.90/6.03 ( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X ) @ Y )
% 11.90/6.03 = ( rel_option @ A @ B
% 11.90/6.03 @ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
% 11.90/6.03 @ X
% 11.90/6.03 @ Y ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5022_option_Orel__map_I2_J,axiom,
% 11.90/6.03 ! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G: B > C,Y: option @ B] :
% 11.90/6.03 ( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G @ Y ) )
% 11.90/6.03 = ( rel_option @ A @ B
% 11.90/6.03 @ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
% 11.90/6.03 @ X
% 11.90/6.03 @ Y ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5023_option_Orel__flip,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ B,B2: option @ A] :
% 11.90/6.03 ( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
% 11.90/6.03 = ( rel_option @ A @ B @ R5 @ B2 @ A3 ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5024_option_Orel__conversep,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 11.90/6.03 ( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) )
% 11.90/6.03 = ( conversep @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5025_option_Octr__transfer_I2_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R5 @ ( rel_option @ A @ B @ R5 ) @ ( some @ A ) @ ( some @ B ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5026_option_Odisc__transfer_I2_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 11.90/6.03 ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
% 11.90/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 11.90/6.03 @ ^ [Option3: option @ A] :
% 11.90/6.03 ( Option3
% 11.90/6.03 != ( none @ A ) )
% 11.90/6.03 @ ^ [Option3: option @ B] :
% 11.90/6.03 ( Option3
% 11.90/6.03 != ( none @ B ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5027_option_Odisc__transfer_I1_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 11.90/6.03 ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
% 11.90/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 )
% 11.90/6.03 @ ^ [Option3: option @ A] :
% 11.90/6.03 ( Option3
% 11.90/6.03 = ( none @ A ) )
% 11.90/6.03 @ ^ [Option3: option @ B] :
% 11.90/6.03 ( Option3
% 11.90/6.03 = ( none @ B ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5028_option__bind__transfer,axiom,
% 11.90/6.03 ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A4 @ ( rel_option @ C @ D @ B6 ) ) @ ( rel_option @ C @ D @ B6 ) ) @ ( bind @ A @ C ) @ ( bind @ B @ D ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5029_option_Orel__transp,axiom,
% 11.90/6.03 ! [A: $tType,R5: A > A > $o] :
% 11.90/6.03 ( ( transp @ A @ R5 )
% 11.90/6.03 => ( transp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5030_option_Orel__distinct_I2_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,Y2: A] :
% 11.90/6.03 ~ ( rel_option @ A @ B @ R5 @ ( some @ A @ Y2 ) @ ( none @ B ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5031_option_Orel__distinct_I1_J,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,Y2: B] :
% 11.90/6.03 ~ ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( some @ B @ Y2 ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5032_option_Orel__cases,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ A,B2: option @ B] :
% 11.90/6.03 ( ( rel_option @ A @ B @ R5 @ A3 @ B2 )
% 11.90/6.03 => ( ( ( A3
% 11.90/6.03 = ( none @ A ) )
% 11.90/6.03 => ( B2
% 11.90/6.03 != ( none @ B ) ) )
% 11.90/6.03 => ~ ! [X2: A] :
% 11.90/6.03 ( ( A3
% 11.90/6.03 = ( some @ A @ X2 ) )
% 11.90/6.03 => ! [Y4: B] :
% 11.90/6.03 ( ( B2
% 11.90/6.03 = ( some @ B @ Y4 ) )
% 11.90/6.03 => ~ ( R5 @ X2 @ Y4 ) ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5033_option_Orel__induct,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
% 11.90/6.03 ( ( rel_option @ A @ B @ R5 @ X @ Y )
% 11.90/6.03 => ( ( Q @ ( none @ A ) @ ( none @ B ) )
% 11.90/6.03 => ( ! [A24: A,B24: B] :
% 11.90/6.03 ( ( R5 @ A24 @ B24 )
% 11.90/6.03 => ( Q @ ( some @ A @ A24 ) @ ( some @ B @ B24 ) ) )
% 11.90/6.03 => ( Q @ X @ Y ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5034_option_Omap__transfer,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,F5: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F5 ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E ) > ( option @ F5 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F5 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E ) @ ( option @ B ) @ ( option @ F5 ) @ ( rel_option @ A @ E @ Rb2 ) @ ( rel_option @ B @ F5 @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E @ F5 ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5035_option_Ocase__transfer,axiom,
% 11.90/6.03 ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S2 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5036_option_Orel__transfer,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 11.90/6.03 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
% 11.90/6.03 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 11.90/6.03 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 11.90/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 11.90/6.03 @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
% 11.90/6.03 @ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
% 11.90/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 11.90/6.03 @ ( rel_option @ A @ B )
% 11.90/6.03 @ ( rel_option @ C @ D ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5037_bit__minus__iff,axiom,
% 11.90/6.03 ! [A: $tType] :
% 11.90/6.03 ( ( bit_ri3973907225187159222ations @ A )
% 11.90/6.03 => ! [A3: A,N: nat] :
% 11.90/6.03 ( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
% 11.90/6.03 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 11.90/6.03 & ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5038_option_Orel__sel,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType] :
% 11.90/6.03 ( ( rel_option @ A @ B )
% 11.90/6.03 = ( ^ [R6: A > B > $o,A7: option @ A,B5: option @ B] :
% 11.90/6.03 ( ( ( A7
% 11.90/6.03 = ( none @ A ) )
% 11.90/6.03 = ( B5
% 11.90/6.03 = ( none @ B ) ) )
% 11.90/6.03 & ( ( A7
% 11.90/6.03 != ( none @ A ) )
% 11.90/6.03 => ( ( B5
% 11.90/6.03 != ( none @ B ) )
% 11.90/6.03 => ( R6 @ ( the2 @ A @ A7 ) @ ( the2 @ B @ B5 ) ) ) ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5039_ordLess__not__embed,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.90/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.03 => ~ ? [X_1: B > A] : ( bNF_Wellorder_embed @ B @ A @ R4 @ R @ X_1 ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5040_option_Orec__transfer,axiom,
% 11.90/6.03 ! [C: $tType,A: $tType,B: $tType,D: $tType,S2: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S2 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S2 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S2 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5041_bit__2__iff,axiom,
% 11.90/6.03 ! [A: $tType] :
% 11.90/6.03 ( ( bit_semiring_bits @ A )
% 11.90/6.03 => ! [N: nat] :
% 11.90/6.03 ( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
% 11.90/6.03 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
% 11.90/6.03 & ( N
% 11.90/6.03 = ( one_one @ nat ) ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5042_BNF__Wellorder__Constructions_OordLess__Field,axiom,
% 11.90/6.03 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B] :
% 11.90/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.03 => ( ( bNF_Wellorder_embed @ A @ B @ R1 @ R22 @ F )
% 11.90/6.03 => ( ( image2 @ A @ B @ F @ ( field2 @ A @ R1 ) )
% 11.90/6.03 != ( field2 @ B @ R22 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5043_bit__mask__sub__iff,axiom,
% 11.90/6.03 ! [A: $tType] :
% 11.90/6.03 ( ( bit_semiring_bits @ A )
% 11.90/6.03 => ! [M2: nat,N: nat] :
% 11.90/6.03 ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
% 11.90/6.03 = ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
% 11.90/6.03 & ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5044_embed__ordLess__ofilterIncl,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),F13: A > C,F232: B > C] :
% 11.90/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.90/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R22 @ R33 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
% 11.90/6.03 => ( ( bNF_Wellorder_embed @ A @ C @ R1 @ R33 @ F13 )
% 11.90/6.03 => ( ( bNF_Wellorder_embed @ B @ C @ R22 @ R33 @ F232 )
% 11.90/6.03 => ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F13 @ ( field2 @ A @ R1 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R22 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R33 ) ) ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5045_comp__embed,axiom,
% 11.90/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.90/6.03 ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 11.90/6.03 => ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
% 11.90/6.03 => ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5046_override__on__insert,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X10: set @ A] :
% 11.90/6.03 ( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X10 ) )
% 11.90/6.03 = ( fun_upd @ A @ B @ ( override_on @ A @ B @ F @ G @ X10 ) @ X @ ( G @ X ) ) ) ).
% 11.90/6.03
% 11.90/6.03 thf(fact_5047_override__on__apply__in,axiom,
% 11.90/6.03 ! [B: $tType,A: $tType,A3: A,A4: set @ A,F: A > B,G: A > B] :
% 11.93/6.03 ( ( member @ A @ A3 @ A4 )
% 11.93/6.03 => ( ( override_on @ A @ B @ F @ G @ A4 @ A3 )
% 11.93/6.03 = ( G @ A3 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5048_override__on__apply__notin,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A3: A,A4: set @ A,F: A > B,G: A > B] :
% 11.93/6.03 ( ~ ( member @ A @ A3 @ A4 )
% 11.93/6.03 => ( ( override_on @ A @ B @ F @ G @ A4 @ A3 )
% 11.93/6.03 = ( F @ A3 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5049_override__on__emptyset,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 11.93/6.03 ( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 = F ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5050_override__on__def,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( override_on @ A @ B )
% 11.93/6.03 = ( ^ [F2: A > B,G2: A > B,A9: set @ A,A7: A] : ( if @ B @ ( member @ A @ A7 @ A9 ) @ ( G2 @ A7 ) @ ( F2 @ A7 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5051_override__on__insert_H,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X10: set @ A] :
% 11.93/6.03 ( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X10 ) )
% 11.93/6.03 = ( override_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ ( G @ X ) ) @ G @ X10 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5052_embedS__comp__embed,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5053_embed__comp__embedS,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_embedS @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5054_pred__option__parametric,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: A > B > $o] :
% 11.93/6.03 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
% 11.93/6.03 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.03 @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A4 )
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.03 @ ( pred_option @ A )
% 11.93/6.03 @ ( pred_option @ B ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5055_option_Opred__inject_I2_J,axiom,
% 11.93/6.03 ! [A: $tType,P: A > $o,A3: A] :
% 11.93/6.03 ( ( pred_option @ A @ P @ ( some @ A @ A3 ) )
% 11.93/6.03 = ( P @ A3 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5056_option_Opred__mono,axiom,
% 11.93/6.03 ! [A: $tType,P: A > $o,Pa: A > $o] :
% 11.93/6.03 ( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
% 11.93/6.03 => ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P ) @ ( pred_option @ A @ Pa ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5057_option_Opred__inject_I1_J,axiom,
% 11.93/6.03 ! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5058_option_Opred__True,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( pred_option @ A
% 11.93/6.03 @ ^ [Uu: A] : $true )
% 11.93/6.03 = ( ^ [Uu: option @ A] : $true ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5059_option_Opred__mono__strong,axiom,
% 11.93/6.03 ! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
% 11.93/6.03 ( ( pred_option @ A @ P @ X )
% 11.93/6.03 => ( ! [Z3: A] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( set_option @ A @ X ) )
% 11.93/6.03 => ( ( P @ Z3 )
% 11.93/6.03 => ( Pa @ Z3 ) ) )
% 11.93/6.03 => ( pred_option @ A @ Pa @ X ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5060_option_Opred__cong,axiom,
% 11.93/6.03 ! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
% 11.93/6.03 ( ( X = Ya )
% 11.93/6.03 => ( ! [Z3: A] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( set_option @ A @ Ya ) )
% 11.93/6.03 => ( ( P @ Z3 )
% 11.93/6.03 = ( Pa @ Z3 ) ) )
% 11.93/6.03 => ( ( pred_option @ A @ P @ X )
% 11.93/6.03 = ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5061_option_Omap__cong__pred,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
% 11.93/6.03 ( ( X = Ya )
% 11.93/6.03 => ( ( pred_option @ A
% 11.93/6.03 @ ^ [Z5: A] :
% 11.93/6.03 ( ( F @ Z5 )
% 11.93/6.03 = ( G @ Z5 ) )
% 11.93/6.03 @ Ya )
% 11.93/6.03 => ( ( map_option @ A @ B @ F @ X )
% 11.93/6.03 = ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5062_option_Opred__set,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( pred_option @ A )
% 11.93/6.03 = ( ^ [P4: A > $o,X3: option @ A] :
% 11.93/6.03 ! [Y3: A] :
% 11.93/6.03 ( ( member @ A @ Y3 @ ( set_option @ A @ X3 ) )
% 11.93/6.03 => ( P4 @ Y3 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5063_option_Opred__rel,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( pred_option @ A )
% 11.93/6.03 = ( ^ [P4: A > $o,X3: option @ A] : ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X3 @ X3 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5064_option_Orel__eq__onp,axiom,
% 11.93/6.03 ! [A: $tType,P: A > $o] :
% 11.93/6.03 ( ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P ) )
% 11.93/6.03 = ( bNF_eq_onp @ ( option @ A ) @ ( pred_option @ A @ P ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5065_option_Opred__map,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: option @ A] :
% 11.93/6.03 ( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F @ X ) )
% 11.93/6.03 = ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5066_option_Opred__transfer,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 11.93/6.03 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
% 11.93/6.03 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.03 @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.03 @ ( pred_option @ A )
% 11.93/6.03 @ ( pred_option @ B ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5067_iso__comp__embedS,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( order_well_order_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_embedS @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5068_cut__def,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( cut @ A @ B )
% 11.93/6.03 = ( ^ [F2: A > B,R6: set @ ( product_prod @ A @ A ),X3: A,Y3: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R6 ) @ ( F2 @ Y3 ) @ ( undefined @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5069_above__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( order_above @ A )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 11.93/6.03 ( collect @ A
% 11.93/6.03 @ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5070_comp__iso,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_iso @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5071_iso__forward,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.93/6.03 => ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y ) ) @ R4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5072_cut__apply,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,X: A,A3: A,R5: set @ ( product_prod @ A @ A ),F: A > B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R5 )
% 11.93/6.03 => ( ( cut @ A @ B @ F @ R5 @ A3 @ X )
% 11.93/6.03 = ( F @ X ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5073_cuts__eq,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,R5: set @ ( product_prod @ A @ A ),X: A,G: A > B] :
% 11.93/6.03 ( ( ( cut @ A @ B @ F @ R5 @ X )
% 11.93/6.03 = ( cut @ A @ B @ G @ R5 @ X ) )
% 11.93/6.03 = ( ! [Y3: A] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R5 )
% 11.93/6.03 => ( ( F @ Y3 )
% 11.93/6.03 = ( G @ Y3 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5074_iso__comp__embed,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( bNF_Wellorder_iso @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_embed @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5075_embed__comp__iso,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_embed @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5076_embedS__comp__iso,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B,R9: set @ ( product_prod @ C @ C ),F3: B > C] :
% 11.93/6.03 ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
% 11.93/6.03 => ( ( bNF_Wellorder_iso @ B @ C @ R4 @ R9 @ F3 )
% 11.93/6.03 => ( bNF_Wellorder_embedS @ A @ C @ R @ R9 @ ( comp @ B @ C @ A @ F3 @ F ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5077_iso__iff2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( bNF_Wellorder_iso @ A @ B )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ B @ B ),F2: A > B] :
% 11.93/6.03 ( ( bij_betw @ A @ B @ F2 @ ( field2 @ A @ R3 ) @ ( field2 @ B @ R11 ) )
% 11.93/6.03 & ! [X3: A] :
% 11.93/6.03 ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
% 11.93/6.03 => ! [Y3: A] :
% 11.93/6.03 ( ( member @ A @ Y3 @ ( field2 @ A @ R3 ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 )
% 11.93/6.03 = ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) @ R11 ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5078_iso__backward,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,X: A,Y: A,R4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),F: B > A] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R4 )
% 11.93/6.03 => ( ( bNF_Wellorder_iso @ B @ A @ R @ R4 @ F )
% 11.93/6.03 => ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ X ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ Y ) ) @ R ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5079_lists__of__len__fin1,axiom,
% 11.93/6.03 ! [A: $tType,P: set @ A,N: nat] :
% 11.93/6.03 ( ( finite_finite2 @ A @ P )
% 11.93/6.03 => ( finite_finite2 @ ( list @ A )
% 11.93/6.03 @ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
% 11.93/6.03 @ ( collect @ ( list @ A )
% 11.93/6.03 @ ^ [L3: list @ A] :
% 11.93/6.03 ( ( size_size @ ( list @ A ) @ L3 )
% 11.93/6.03 = N ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5080_lists__of__len__fin2,axiom,
% 11.93/6.03 ! [A: $tType,P: set @ A,N: nat] :
% 11.93/6.03 ( ( finite_finite2 @ A @ P )
% 11.93/6.03 => ( finite_finite2 @ ( list @ A )
% 11.93/6.03 @ ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ P )
% 11.93/6.03 @ ( collect @ ( list @ A )
% 11.93/6.03 @ ^ [L3: list @ A] :
% 11.93/6.03 ( N
% 11.93/6.03 = ( size_size @ ( list @ A ) @ L3 ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5081_Random__Pred_Ounion__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( random_union @ A )
% 11.93/6.03 = ( ^ [R13: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R24: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S4: product_prod @ code_natural @ code_natural] :
% 11.93/6.03 ( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
% 11.93/6.03 @ ^ [P12: pred @ A,S8: product_prod @ code_natural @ code_natural] :
% 11.93/6.03 ( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
% 11.93/6.03 @ ^ [P23: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P12 @ P23 ) )
% 11.93/6.03 @ ( R24 @ S8 ) )
% 11.93/6.03 @ ( R13 @ S4 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5082_lists__image__witness,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,X: list @ A,F: B > A,Q: set @ B] :
% 11.93/6.03 ( ( member @ ( list @ A ) @ X @ ( lists @ A @ ( image2 @ B @ A @ F @ Q ) ) )
% 11.93/6.03 => ~ ! [Xo2: list @ B] :
% 11.93/6.03 ( ( member @ ( list @ B ) @ Xo2 @ ( lists @ B @ Q ) )
% 11.93/6.03 => ( X
% 11.93/6.03 != ( map @ B @ A @ F @ Xo2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5083_Random__Pred_Oempty__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( random_empty @ A )
% 11.93/6.03 = ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5084_Random__Pred_Obind__def,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( random_bind @ A @ B )
% 11.93/6.03 = ( ^ [R6: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),F2: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S4: product_prod @ code_natural @ code_natural] :
% 11.93/6.03 ( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
% 11.93/6.03 @ ^ [P4: pred @ A,S8: product_prod @ code_natural @ code_natural] :
% 11.93/6.03 ( product_case_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
% 11.93/6.03 @ ^ [S12: product_prod @ code_natural @ code_natural] :
% 11.93/6.03 ( product_Pair @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural )
% 11.93/6.03 @ ( bind2 @ A @ B @ P4
% 11.93/6.03 @ ^ [A7: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F2 @ A7 @ S12 ) ) ) )
% 11.93/6.03 @ ( split_seed @ S8 ) )
% 11.93/6.03 @ ( R6 @ S4 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5085_card__Plus__conv__if,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( ( ( finite_finite2 @ A @ A4 )
% 11.93/6.03 & ( finite_finite2 @ B @ B6 ) )
% 11.93/6.03 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 11.93/6.03 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) )
% 11.93/6.03 & ( ~ ( ( finite_finite2 @ A @ A4 )
% 11.93/6.03 & ( finite_finite2 @ B @ B6 ) )
% 11.93/6.03 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 11.93/6.03 = ( zero_zero @ nat ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5086_card__Plus,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( finite_finite2 @ B @ B6 )
% 11.93/6.03 => ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 11.93/6.03 = ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5087_option_Orel__compp__Grp,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( rel_option @ A @ B )
% 11.93/6.03 = ( ^ [R6: A > B > $o] :
% 11.93/6.03 ( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
% 11.93/6.03 @ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
% 11.93/6.03 @ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
% 11.93/6.03 @ ( collect @ ( option @ ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 11.93/6.03 @ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
% 11.93/6.03 @ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
% 11.93/6.03 @ ( collect @ ( option @ ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 11.93/6.03 @ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5088_mult__one__div__unit__factor,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.03 => ! [A3: A,B2: A] :
% 11.93/6.03 ( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
% 11.93/6.03 = ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5089_unit__factor__mult__unit__left,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semido2269285787275462019factor @ A )
% 11.93/6.03 => ! [A3: A,B2: A] :
% 11.93/6.03 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
% 11.93/6.03 = ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5090_unit__factor__simps_I2_J,axiom,
% 11.93/6.03 ! [N: nat] :
% 11.93/6.03 ( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
% 11.93/6.03 = ( one_one @ nat ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5091_unit__factor__1,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
% 11.93/6.03 = ( one_one @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5092_inv__unit__factor__eq__0__iff,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.03 => ! [A3: A] :
% 11.93/6.03 ( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 11.93/6.03 = ( zero_zero @ A ) )
% 11.93/6.03 = ( A3
% 11.93/6.03 = ( zero_zero @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5093_unit__factor__mult__unit__right,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semido2269285787275462019factor @ A )
% 11.93/6.03 => ! [A3: A,B2: A] :
% 11.93/6.03 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
% 11.93/6.03 = ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5094_option_Orel__compp,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S2: B > C > $o] :
% 11.93/6.03 ( ( rel_option @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S2 ) )
% 11.93/6.03 = ( relcompp @ ( option @ A ) @ ( option @ B ) @ ( option @ C ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ B @ C @ S2 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5095_OOO__quotient3,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: B > B > $o,Abs22: B > C,Rep22: C > B,R25: A > A > $o] :
% 11.93/6.03 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 11.93/6.03 => ( ( quotient3 @ B @ C @ R23 @ Abs22 @ Rep22 )
% 11.93/6.03 => ( ! [X2: A,Y4: A] :
% 11.93/6.03 ( ( R25 @ X2 @ Y4 )
% 11.93/6.03 => ( ( R12 @ X2 @ X2 )
% 11.93/6.03 => ( ( R12 @ Y4 @ Y4 )
% 11.93/6.03 => ( R23 @ ( Abs1 @ X2 ) @ ( Abs1 @ Y4 ) ) ) ) )
% 11.93/6.03 => ( ! [X2: B,Y4: B] :
% 11.93/6.03 ( ( R23 @ X2 @ Y4 )
% 11.93/6.03 => ( R25 @ ( Rep1 @ X2 ) @ ( Rep1 @ Y4 ) ) )
% 11.93/6.03 => ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R12 @ ( relcompp @ A @ A @ A @ R25 @ R12 ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5096_OOO__eq__quotient3,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,Abs22: B > C,Rep22: C > B] :
% 11.93/6.03 ( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
% 11.93/6.03 => ( ( quotient3 @ B @ C
% 11.93/6.03 @ ^ [Y5: B,Z4: B] : ( Y5 = Z4 )
% 11.93/6.03 @ Abs22
% 11.93/6.03 @ Rep22 )
% 11.93/6.03 => ( quotient3 @ A @ C
% 11.93/6.03 @ ( relcompp @ A @ A @ A @ R12
% 11.93/6.03 @ ( relcompp @ A @ A @ A
% 11.93/6.03 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.93/6.03 @ R12 ) )
% 11.93/6.03 @ ( comp @ B @ C @ A @ Abs22 @ Abs1 )
% 11.93/6.03 @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5097_relpowp__add,axiom,
% 11.93/6.03 ! [A: $tType,M2: nat,N: nat,P: A > A > $o] :
% 11.93/6.03 ( ( compow @ ( A > A > $o ) @ ( plus_plus @ nat @ M2 @ N ) @ P )
% 11.93/6.03 = ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ M2 @ P ) @ ( compow @ ( A > A > $o ) @ N @ P ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5098_relcompp__relcomp__eq,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
% 11.93/6.03 ( ( relcompp @ A @ B @ C
% 11.93/6.03 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R )
% 11.93/6.03 @ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S ) )
% 11.93/6.03 = ( ^ [X3: A,Y3: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5099_is__unit__unit__factor,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semido2269285787275462019factor @ A )
% 11.93/6.03 => ! [A3: A] :
% 11.93/6.03 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ A3 )
% 11.93/6.03 = A3 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5100_unit__factor__nat__def,axiom,
% 11.93/6.03 ( ( unit_f5069060285200089521factor @ nat )
% 11.93/6.03 = ( ^ [N5: nat] :
% 11.93/6.03 ( if @ nat
% 11.93/6.03 @ ( N5
% 11.93/6.03 = ( zero_zero @ nat ) )
% 11.93/6.03 @ ( zero_zero @ nat )
% 11.93/6.03 @ ( one_one @ nat ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5101_option_Orel__Grp,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 11.93/6.03 ( ( rel_option @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F ) )
% 11.93/6.03 = ( bNF_Grp @ ( option @ A ) @ ( option @ B )
% 11.93/6.03 @ ( collect @ ( option @ A )
% 11.93/6.03 @ ^ [X3: option @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_option @ A @ X3 ) @ A4 ) )
% 11.93/6.03 @ ( map_option @ A @ B @ F ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5102_fun_Orel__Grp,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 11.93/6.03 ( ( bNF_rel_fun @ D @ D @ A @ B
% 11.93/6.03 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_Grp @ A @ B @ A4 @ F ) )
% 11.93/6.03 = ( bNF_Grp @ ( D > A ) @ ( D > B )
% 11.93/6.03 @ ( collect @ ( D > A )
% 11.93/6.03 @ ^ [X3: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) @ A4 ) )
% 11.93/6.03 @ ( comp @ A @ B @ D @ F ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5103_relcomp__def,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType] :
% 11.93/6.03 ( ( relcomp @ A @ B @ C )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ B ),S4: set @ ( product_prod @ B @ C )] :
% 11.93/6.03 ( collect @ ( product_prod @ A @ C )
% 11.93/6.03 @ ( product_case_prod @ A @ C @ $o
% 11.93/6.03 @ ( relcompp @ A @ B @ C
% 11.93/6.03 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 11.93/6.03 @ ^ [X3: B,Y3: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X3 @ Y3 ) @ S4 ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5104_unit__factor__is__unit,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semido2269285787275462019factor @ A )
% 11.93/6.03 => ! [A3: A] :
% 11.93/6.03 ( ( A3
% 11.93/6.03 != ( zero_zero @ A ) )
% 11.93/6.03 => ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5105_unit__factor__gcd,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_gcd @ A )
% 11.93/6.03 => ! [A3: A,B2: A] :
% 11.93/6.03 ( ( ( ( A3
% 11.93/6.03 = ( zero_zero @ A ) )
% 11.93/6.03 & ( B2
% 11.93/6.03 = ( zero_zero @ A ) ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 11.93/6.03 = ( zero_zero @ A ) ) )
% 11.93/6.03 & ( ~ ( ( A3
% 11.93/6.03 = ( zero_zero @ A ) )
% 11.93/6.03 & ( B2
% 11.93/6.03 = ( zero_zero @ A ) ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
% 11.93/6.03 = ( one_one @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5106_type__copy__vimage2p__Grp__Abs,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: A > B,Abs: B > A,G: D > C,P: C > $o,H3: C > A] :
% 11.93/6.03 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.93/6.03 => ( ( bNF_vimage2p @ D @ C @ B @ A @ $o @ G @ Abs @ ( bNF_Grp @ C @ A @ ( collect @ C @ P ) @ H3 ) )
% 11.93/6.03 = ( bNF_Grp @ D @ B
% 11.93/6.03 @ ( collect @ D
% 11.93/6.03 @ ^ [X3: D] : ( P @ ( G @ X3 ) ) )
% 11.93/6.03 @ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H3 ) @ G ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5107_type__copy__vimage2p__Grp__Rep,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: A > B,Abs: B > A,F: C > D,P: D > $o,H3: D > B] :
% 11.93/6.03 ( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
% 11.93/6.03 => ( ( bNF_vimage2p @ C @ D @ A @ B @ $o @ F @ Rep @ ( bNF_Grp @ D @ B @ ( collect @ D @ P ) @ H3 ) )
% 11.93/6.03 = ( bNF_Grp @ C @ A
% 11.93/6.03 @ ( collect @ C
% 11.93/6.03 @ ^ [X3: C] : ( P @ ( F @ X3 ) ) )
% 11.93/6.03 @ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H3 ) @ F ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5108_Nat_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5109_pcr__natural__def,axiom,
% 11.93/6.03 ( code_pcr_natural
% 11.93/6.03 = ( relcompp @ nat @ nat @ code_natural
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ code_cr_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5110_natural_Orep__transfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ^ [X3: nat] : X3
% 11.93/6.03 @ code_nat_of_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5111_natural_Oid__abs__transfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ code_pcr_natural
% 11.93/6.03 @ ^ [X3: nat] : X3
% 11.93/6.03 @ code_natural_of_nat ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5112_natural_Opcr__cr__eq,axiom,
% 11.93/6.03 code_pcr_natural = code_cr_natural ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5113_zero__natural_Otransfer,axiom,
% 11.93/6.03 code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5114_one__natural_Otransfer,axiom,
% 11.93/6.03 code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5115_less__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.03 @ ( ord_less @ nat )
% 11.93/6.03 @ ( ord_less @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5116_vimage2p__comp,axiom,
% 11.93/6.03 ! [E: $tType,D: $tType,F5: $tType,A: $tType,C: $tType,B: $tType,G3: $tType,F1: F5 > A,F22: D > F5,G1: G3 > B,G22: E > G3] :
% 11.93/6.03 ( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F5 @ A @ D @ F1 @ F22 ) @ ( comp @ G3 @ B @ E @ G1 @ G22 ) )
% 11.93/6.03 = ( comp @ ( F5 > G3 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F5 @ E @ G3 @ C @ F22 @ G22 ) @ ( bNF_vimage2p @ F5 @ A @ G3 @ B @ C @ F1 @ G1 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5117_plus__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( plus_plus @ nat ) @ ( plus_plus @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5118_Suc_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5119_minus__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( minus_minus @ nat ) @ ( minus_minus @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5120_pcr__integer__def,axiom,
% 11.93/6.03 ( code_pcr_integer
% 11.93/6.03 = ( relcompp @ int @ int @ code_integer
% 11.93/6.03 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.93/6.03 @ code_cr_integer ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5121_less__eq__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.03 @ ( ord_less_eq @ nat )
% 11.93/6.03 @ ( ord_less_eq @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5122_times__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( times_times @ nat ) @ ( times_times @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5123_divide__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( divide_divide @ nat ) @ ( divide_divide @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5124_modulo__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( modulo_modulo @ nat ) @ ( modulo_modulo @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5125_take__bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 11.93/6.03 @ ( bit_se2584673776208193580ke_bit @ nat )
% 11.93/6.03 @ ( bit_se2584673776208193580ke_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5126_xor__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se5824344971392196577ns_xor @ nat ) @ ( bit_se5824344971392196577ns_xor @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5127_or__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se1065995026697491101ons_or @ nat ) @ ( bit_se1065995026697491101ons_or @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5128_and__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se5824344872417868541ns_and @ nat ) @ ( bit_se5824344872417868541ns_and @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5129_bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_natural
% 11.93/6.03 @ ^ [Y5: nat > $o,Z4: nat > $o] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bit_se5641148757651400278ts_bit @ nat )
% 11.93/6.03 @ ( bit_se5641148757651400278ts_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5130_push__bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 11.93/6.03 @ ( bit_se4730199178511100633sh_bit @ nat )
% 11.93/6.03 @ ( bit_se4730199178511100633sh_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5131_mask__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ nat @ code_natural
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ code_pcr_natural
% 11.93/6.03 @ ( bit_se2239418461657761734s_mask @ nat )
% 11.93/6.03 @ ( bit_se2239418461657761734s_mask @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5132_drop__bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 11.93/6.03 @ ( bit_se4197421643247451524op_bit @ nat )
% 11.93/6.03 @ ( bit_se4197421643247451524op_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5133_division__segment__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ ( euclid7384307370059645450egment @ nat ) @ ( euclid7384307370059645450egment @ code_natural ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5134_flip__bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 11.93/6.03 @ ( bit_se8732182000553998342ip_bit @ nat )
% 11.93/6.03 @ ( bit_se8732182000553998342ip_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5135_set__bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 11.93/6.03 @ ( bit_se5668285175392031749et_bit @ nat )
% 11.93/6.03 @ ( bit_se5668285175392031749et_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5136_unset__bit__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
% 11.93/6.03 @ ( bit_se2638667681897837118et_bit @ nat )
% 11.93/6.03 @ ( bit_se2638667681897837118et_bit @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5137_euclidean__size__natural_Otransfer,axiom,
% 11.93/6.03 ( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
% 11.93/6.03 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.03 @ ( euclid6346220572633701492n_size @ nat )
% 11.93/6.03 @ ( euclid6346220572633701492n_size @ code_natural ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5138_integer__of__natural_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ nat @ code_natural @ int @ code_integer @ code_pcr_natural @ code_pcr_integer @ ( semiring_1_of_nat @ int ) @ code_i5400310926305786745atural ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5139_natural__of__integer_Otransfer,axiom,
% 11.93/6.03 bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5140_fun_Orel__compp__Grp,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,A: $tType,R5: A > B > $o] :
% 11.93/6.03 ( ( bNF_rel_fun @ D @ D @ A @ B
% 11.93/6.03 @ ^ [Y5: D,Z4: D] : ( Y5 = Z4 )
% 11.93/6.03 @ R5 )
% 11.93/6.03 = ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
% 11.93/6.03 @ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
% 11.93/6.03 @ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
% 11.93/6.03 @ ( collect @ ( D > ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
% 11.93/6.03 @ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
% 11.93/6.03 @ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
% 11.93/6.03 @ ( collect @ ( D > ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X3 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
% 11.93/6.03 @ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5141_plus__num__def,axiom,
% 11.93/6.03 ( ( plus_plus @ num )
% 11.93/6.03 = ( ^ [M: num,N5: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5142_Lcm__no__units,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ( ( gcd_Lcm @ A )
% 11.93/6.03 = ( ^ [A9: set @ A] :
% 11.93/6.03 ( gcd_Lcm @ A
% 11.93/6.03 @ ( minus_minus @ ( set @ A ) @ A9
% 11.93/6.03 @ ( collect @ A
% 11.93/6.03 @ ^ [A7: A] : ( dvd_dvd @ A @ A7 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5143_remdups__adj__altdef,axiom,
% 11.93/6.03 ! [A: $tType,Xs: list @ A,Ys2: list @ A] :
% 11.93/6.03 ( ( ( remdups_adj @ A @ Xs )
% 11.93/6.03 = Ys2 )
% 11.93/6.03 = ( ? [F2: nat > nat] :
% 11.93/6.03 ( ( order_mono @ nat @ nat @ F2 )
% 11.93/6.03 & ( ( image2 @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
% 11.93/6.03 = ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
% 11.93/6.03 & ! [I3: nat] :
% 11.93/6.03 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.93/6.03 => ( ( nth @ A @ Xs @ I3 )
% 11.93/6.03 = ( nth @ A @ Ys2 @ ( F2 @ I3 ) ) ) )
% 11.93/6.03 & ! [I3: nat] :
% 11.93/6.03 ( ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
% 11.93/6.03 => ( ( ( nth @ A @ Xs @ I3 )
% 11.93/6.03 = ( nth @ A @ Xs @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) )
% 11.93/6.03 = ( ( F2 @ I3 )
% 11.93/6.03 = ( F2 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5144_Lcm__empty,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 = ( one_one @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5145_Lcm__1__iff,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ! [A4: set @ A] :
% 11.93/6.03 ( ( ( gcd_Lcm @ A @ A4 )
% 11.93/6.03 = ( one_one @ A ) )
% 11.93/6.03 = ( ! [X3: A] :
% 11.93/6.03 ( ( member @ A @ X3 @ A4 )
% 11.93/6.03 => ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5146_mono__add,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( ordere6658533253407199908up_add @ A )
% 11.93/6.03 => ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5147_Lcm__nat__empty,axiom,
% 11.93/6.03 ( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.03 = ( one_one @ nat ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5148_nat__of__num_Osimps_I2_J,axiom,
% 11.93/6.03 ! [X: num] :
% 11.93/6.03 ( ( nat_of_num @ ( bit0 @ X ) )
% 11.93/6.03 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5149_nat__of__num__code_I1_J,axiom,
% 11.93/6.03 ( ( nat_of_num @ one2 )
% 11.93/6.03 = ( one_one @ nat ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5150_nat__of__num__add,axiom,
% 11.93/6.03 ! [X: num,Y: num] :
% 11.93/6.03 ( ( nat_of_num @ ( plus_plus @ num @ X @ Y ) )
% 11.93/6.03 = ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5151_nat__of__num__code_I2_J,axiom,
% 11.93/6.03 ! [N: num] :
% 11.93/6.03 ( ( nat_of_num @ ( bit0 @ N ) )
% 11.93/6.03 = ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5152_mono__image__least,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType] :
% 11.93/6.03 ( ( ( order @ B )
% 11.93/6.03 & ( order @ A ) )
% 11.93/6.03 => ! [F: A > B,M2: A,N: A,M7: B,N2: B] :
% 11.93/6.03 ( ( order_mono @ A @ B @ F )
% 11.93/6.03 => ( ( ( image2 @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
% 11.93/6.03 = ( set_or7035219750837199246ssThan @ B @ M7 @ N2 ) )
% 11.93/6.03 => ( ( ord_less @ A @ M2 @ N )
% 11.93/6.03 => ( ( F @ M2 )
% 11.93/6.03 = M7 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5153_nat__of__num_Osimps_I3_J,axiom,
% 11.93/6.03 ! [X: num] :
% 11.93/6.03 ( ( nat_of_num @ ( bit1 @ X ) )
% 11.93/6.03 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5154_unit__factor__Lcm,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ! [A4: set @ A] :
% 11.93/6.03 ( ( ( ( gcd_Lcm @ A @ A4 )
% 11.93/6.03 = ( zero_zero @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
% 11.93/6.03 = ( zero_zero @ A ) ) )
% 11.93/6.03 & ( ( ( gcd_Lcm @ A @ A4 )
% 11.93/6.03 != ( zero_zero @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
% 11.93/6.03 = ( one_one @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5155_nat__of__num__code_I3_J,axiom,
% 11.93/6.03 ! [N: num] :
% 11.93/6.03 ( ( nat_of_num @ ( bit1 @ N ) )
% 11.93/6.03 = ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5156_remdups__adj_Opelims,axiom,
% 11.93/6.03 ! [A: $tType,X: list @ A,Y: list @ A] :
% 11.93/6.03 ( ( ( remdups_adj @ A @ X )
% 11.93/6.03 = Y )
% 11.93/6.03 => ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
% 11.93/6.03 => ( ( ( X
% 11.93/6.03 = ( nil @ A ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( nil @ A ) )
% 11.93/6.03 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
% 11.93/6.03 => ( ! [X2: A] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.93/6.03 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 11.93/6.03 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 11.93/6.03 => ( ( ( ( X2 = Y4 )
% 11.93/6.03 => ( Y
% 11.93/6.03 = ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs2 ) ) ) )
% 11.93/6.03 & ( ( X2 != Y4 )
% 11.93/6.03 => ( Y
% 11.93/6.03 = ( cons @ A @ X2 @ ( remdups_adj @ A @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) )
% 11.93/6.03 => ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5157_relChain__def,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( ord @ B )
% 11.93/6.03 => ( ( bNF_Ca3754400796208372196lChain @ A @ B )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ A ),As3: A > B] :
% 11.93/6.03 ! [I3: A,J2: A] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I3 @ J2 ) @ R3 )
% 11.93/6.03 => ( ord_less_eq @ B @ ( As3 @ I3 ) @ ( As3 @ J2 ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5158_unit__factor__Gcd,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ! [A4: set @ A] :
% 11.93/6.03 ( ( ( ( gcd_Gcd @ A @ A4 )
% 11.93/6.03 = ( zero_zero @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
% 11.93/6.03 = ( zero_zero @ A ) ) )
% 11.93/6.03 & ( ( ( gcd_Gcd @ A @ A4 )
% 11.93/6.03 != ( zero_zero @ A ) )
% 11.93/6.03 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
% 11.93/6.03 = ( one_one @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5159_Gcd__UNIV,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
% 11.93/6.03 = ( one_one @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5160_Gcd__1,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ! [A4: set @ A] :
% 11.93/6.03 ( ( member @ A @ ( one_one @ A ) @ A4 )
% 11.93/6.03 => ( ( gcd_Gcd @ A @ A4 )
% 11.93/6.03 = ( one_one @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5161_Gcd__nat__eq__one,axiom,
% 11.93/6.03 ! [N6: set @ nat] :
% 11.93/6.03 ( ( member @ nat @ ( one_one @ nat ) @ N6 )
% 11.93/6.03 => ( ( gcd_Gcd @ nat @ N6 )
% 11.93/6.03 = ( one_one @ nat ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5162_Gcd__eq__1__I,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( semiring_Gcd @ A )
% 11.93/6.03 => ! [A3: A,A4: set @ A] :
% 11.93/6.03 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.03 => ( ( member @ A @ A3 @ A4 )
% 11.93/6.03 => ( ( gcd_Gcd @ A @ A4 )
% 11.93/6.03 = ( one_one @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5163_cofinal__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( bNF_Ca7293521722713021262ofinal @ A )
% 11.93/6.03 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ! [X3: A] :
% 11.93/6.03 ( ( member @ A @ X3 @ ( field2 @ A @ R3 ) )
% 11.93/6.03 => ? [Y3: A] :
% 11.93/6.03 ( ( member @ A @ Y3 @ A9 )
% 11.93/6.03 & ( X3 != Y3 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5164_card__of__UNION__ordLeq__infinite,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,B6: set @ A,I4: set @ B,A4: B > ( set @ C )] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ B6 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ! [X2: B] :
% 11.93/6.03 ( ( member @ B @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5165_card__of__ordLess,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( ~ ? [F2: A > B] :
% 11.93/6.03 ( ( inj_on @ A @ B @ F2 @ A4 )
% 11.93/6.03 & ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B6 ) ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5166_card__of__Times3,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
% 11.93/6.03 @ ( product_Sigma @ A @ A @ A4
% 11.93/6.03 @ ^ [Uu: A] : A4 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5167_card__of__Sigma__mono1,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: A > ( set @ C )] :
% 11.93/6.03 ( ! [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B6 @ X2 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I4 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5168_card__of__Times__mono1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
% 11.93/6.03 @ ( product_Sigma @ A @ C @ A4
% 11.93/6.03 @ ^ [Uu: A] : C5 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
% 11.93/6.03 @ ( product_Sigma @ B @ C @ B6
% 11.93/6.03 @ ^ [Uu: B] : C5 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5169_card__of__Times__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
% 11.93/6.03 @ ( product_Sigma @ C @ A @ C5
% 11.93/6.03 @ ^ [Uu: C] : A4 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
% 11.93/6.03 @ ( product_Sigma @ C @ B @ C5
% 11.93/6.03 @ ^ [Uu: C] : B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5170_infinite__iff__card__of__nat,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] :
% 11.93/6.03 ( ( ~ ( finite_finite2 @ A @ A4 ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5171_card__of__UNION__Sigma,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: B > ( set @ A ),I4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A4 @ I4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I4 @ A4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5172_card__of__Times__commute,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ B6
% 11.93/6.03 @ ^ [Uu: B] : A4 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5173_card__of__Times__same__infinite,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
% 11.93/6.03 @ ( product_Sigma @ A @ A @ A4
% 11.93/6.03 @ ^ [Uu: A] : A4 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5174_card__of__Times1,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( A4
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ B6
% 11.93/6.03 @ ^ [Uu: B] : A4 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5175_card__of__Times2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( A4
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5176_card__of__bool,axiom,
% 11.93/6.03 ! [A: $tType,A1: A,A22: A] :
% 11.93/6.03 ( ( A1 != A22 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5177_card__of__Plus__empty1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5178_card__of__Plus__empty2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5179_card__of__Plus__Times__bool,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ A4 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
% 11.93/6.03 @ ( product_Sigma @ A @ $o @ A4
% 11.93/6.03 @ ^ [Uu: A] : ( top_top @ ( set @ $o ) ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5180_card__of__Plus__commute,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5181_card__of__Plus__cong2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5182_card__of__Plus__cong1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5183_card__of__Plus__assoc,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_Plus @ ( sum_sum @ A @ B ) @ C @ ( sum_Plus @ A @ B @ A4 @ B6 ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A4 @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5184_card__of__Plus__cong,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C,D4: set @ D] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5185_card__of__Times__Plus__distrib,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ ( sum_sum @ B @ C ) )
% 11.93/6.03 @ ( product_Sigma @ A @ ( sum_sum @ B @ C ) @ A4
% 11.93/6.03 @ ^ [Uu: A] : ( sum_Plus @ B @ C @ B6 @ C5 ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) )
% 11.93/6.03 @ ( sum_Plus @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 )
% 11.93/6.03 @ ( product_Sigma @ A @ C @ A4
% 11.93/6.03 @ ^ [Uu: A] : C5 ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5186_card__of__Plus__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C5 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C5 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5187_card__of__Plus__mono1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B6 @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5188_card__of__Plus__mono,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C,D4: set @ D] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C5 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A4 @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B6 @ D4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5189_card__of__Plus2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,B6: set @ A,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5190_card__of__Plus1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5191_card__of__Un__Plus__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,B6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A4 @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5192_card__of__Plus__infinite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5193_card__of__Plus__infinite1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5194_card__of__Plus__infinite2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B6 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5195_card__of__Plus__ordLess__infinite,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,C5: set @ A,A4: set @ B,B6: set @ C] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ C5 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C5 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5196_card__of__Sigma__ordLeq__infinite,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,B6: set @ A,I4: set @ B,A4: B > ( set @ C )] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ B6 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ! [X2: B] :
% 11.93/6.03 ( ( member @ B @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I4 @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5197_card__of__refl,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5198_card__of__Plus__Times__aux,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( ( A1 != A22 )
% 11.93/6.03 & ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5199_card__of__Times__infinite__simps_I4_J,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ B6
% 11.93/6.03 @ ^ [Uu: B] : A4 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5200_card__of__Times__infinite__simps_I3_J,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ B6
% 11.93/6.03 @ ^ [Uu: B] : A4 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5201_card__of__Times__infinite__simps_I2_J,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5202_card__of__Times__infinite__simps_I1_J,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5203_card__of__Times__infinite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ B6
% 11.93/6.03 @ ^ [Uu: B] : A4 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5204_card__of__Plus__Times,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A1: A,A22: A,A4: set @ A,B1: B,B22: B,B6: set @ B] :
% 11.93/6.03 ( ( ( A1 != A22 )
% 11.93/6.03 & ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 ) )
% 11.93/6.03 => ( ( ( B1 != B22 )
% 11.93/6.03 & ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B1 @ ( insert2 @ B @ B22 @ ( bot_bot @ ( set @ B ) ) ) ) @ B6 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5205_card__of__image,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: B > A,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( image2 @ B @ A @ F @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5206_card__of__empty,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5207_card__of__empty3,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( A4
% 11.93/6.03 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5208_card__of__ordLeq__finite,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( finite_finite2 @ B @ B6 )
% 11.93/6.03 => ( finite_finite2 @ A @ A4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5209_card__of__ordLeq__infinite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.03 => ~ ( finite_finite2 @ B @ B6 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5210_card__of__mono1,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.93/6.03 ( ( ord_less_eq @ ( set @ A ) @ A4 @ B6 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5211_card__of__empty__ordIso,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5212_card__of__empty2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( A4
% 11.93/6.03 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5213_card__of__ordIso__finite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( finite_finite2 @ A @ A4 )
% 11.93/6.03 = ( finite_finite2 @ B @ B6 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5214_card__of__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5215_card__of__cong,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R4 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5216_card__of__ordLeqI,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( inj_on @ A @ B @ F @ A4 )
% 11.93/6.03 => ( ! [A5: A] :
% 11.93/6.03 ( ( member @ A @ A5 @ A4 )
% 11.93/6.03 => ( member @ B @ ( F @ A5 ) @ B6 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5217_ex__bij__betw,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ? [F4: B > A,B10: set @ B] : ( bij_betw @ B @ A @ F4 @ B10 @ A4 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5218_card__of__least,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( order_well_order_on @ A @ A4 @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5219_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5220_card__of__ordIsoI,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( bij_betw @ A @ B @ F @ A4 @ B6 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5221_card__of__ordIso,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A4 @ B6 ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5222_type__copy__set__bd,axiom,
% 11.93/6.03 ! [A: $tType,D: $tType,C: $tType,B: $tType,S2: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X: D] :
% 11.93/6.03 ( ! [Y4: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S2 @ Y4 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S2 @ Rep @ X ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5223_card__of__ordLeq2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( A4
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( ( ? [G2: B > A] :
% 11.93/6.03 ( ( image2 @ B @ A @ G2 @ B6 )
% 11.93/6.03 = A4 ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5224_surj__imp__ordLeq,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,B6: set @ A,F: B > A,A4: set @ B] :
% 11.93/6.03 ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F @ A4 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5225_card__of__singl__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,B2: B] :
% 11.93/6.03 ( ( A4
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5226_card__of__ordLess2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,B6: set @ A,A4: set @ B] :
% 11.93/6.03 ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( ( ~ ? [F2: B > A] :
% 11.93/6.03 ( ( image2 @ B @ A @ F2 @ A4 )
% 11.93/6.03 = B6 ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5227_internalize__card__of__ordLeq2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,C5: set @ B] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 = ( ? [B9: set @ B] :
% 11.93/6.03 ( ( ord_less_eq @ ( set @ B ) @ B9 @ C5 )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5228_card__of__Field__ordLess,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5229_ordLeq__Plus__mono,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P5 @ P8 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5230_ordLeq__Plus__mono1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5231_ordLeq__Plus__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A4 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5232_ordIso__Plus__cong,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),P5: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P5 @ P8 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R4 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5233_ordIso__Plus__cong1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R4 ) @ C5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5234_ordIso__Plus__cong2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A4 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A4 @ ( field2 @ B @ R4 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5235_ordLeq__Times__mono1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),C5: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
% 11.93/6.03 @ ( product_Sigma @ A @ C @ ( field2 @ A @ R )
% 11.93/6.03 @ ^ [Uu: A] : C5 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
% 11.93/6.03 @ ( product_Sigma @ B @ C @ ( field2 @ B @ R4 )
% 11.93/6.03 @ ^ [Uu: B] : C5 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5236_ordLeq__Times__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),A4: set @ C] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
% 11.93/6.03 @ ( product_Sigma @ C @ A @ A4
% 11.93/6.03 @ ^ [Uu: C] : ( field2 @ A @ R ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
% 11.93/6.03 @ ( product_Sigma @ C @ B @ A4
% 11.93/6.03 @ ^ [Uu: C] : ( field2 @ B @ R4 ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5237_card__of__ordLeq,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] :
% 11.93/6.03 ( ( ? [F2: A > B] :
% 11.93/6.03 ( ( inj_on @ A @ B @ F2 @ A4 )
% 11.93/6.03 & ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A4 ) @ B6 ) ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5238_internalize__card__of__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 = ( ? [B9: set @ B] :
% 11.93/6.03 ( ( ord_less_eq @ ( set @ B ) @ B9 @ ( field2 @ B @ R ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B9 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5239_regularCard__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( bNF_Ca7133664381575040944arCard @ A )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ! [K3: set @ A] :
% 11.93/6.03 ( ( ( ord_less_eq @ ( set @ A ) @ K3 @ ( field2 @ A @ R3 ) )
% 11.93/6.03 & ( bNF_Ca7293521722713021262ofinal @ A @ K3 @ R3 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K3 ) @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5240_card__of__ordIso__subst,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,B6: set @ A] :
% 11.93/6.03 ( ( A4 = B6 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5241_comp__set__bd__Union__o__collect,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,X: C,X10: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A
% 11.93/6.03 @ ( complete_Sup_Sup @ ( set @ A )
% 11.93/6.03 @ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
% 11.93/6.03 @ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
% 11.93/6.03 @ ^ [F2: C > ( set @ ( set @ A ) )] : ( F2 @ X )
% 11.93/6.03 @ X10 ) ) ) )
% 11.93/6.03 @ Hbd )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X10 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5242_card__of__Plus__ordLess__infinite__Field,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5243_card__of__Times__ordLeq__infinite__Field,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
% 11.93/6.03 @ ( product_Sigma @ B @ C @ A4
% 11.93/6.03 @ ^ [Uu: B] : B6 ) )
% 11.93/6.03 @ R )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5244_card__of__Sigma__ordLeq__infinite__Field,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ! [X2: B] :
% 11.93/6.03 ( ( member @ B @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I4 @ A4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5245_Card__order__trans,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( X != Y )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.93/6.03 => ( ( Y != Z2 )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 11.93/6.03 => ( ( X != Z2 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ R ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5246_card__order__on__ordIso,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5247_infinite__Card__order__limit,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.93/6.03 => ? [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.93/6.03 & ( A3 != X2 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5248_ordLeq__refl,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5249_Card__order__ordIso,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
% 11.93/6.03 => ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5250_Card__order__ordIso2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5251_ordIso__refl,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5252_card__of__unique,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ A4 @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5253_card__order__on__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A )
% 11.93/6.03 = ( ^ [A9: set @ A,R3: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( order_well_order_on @ A @ A9 @ R3 )
% 11.93/6.03 & ! [R11: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( order_well_order_on @ A @ A9 @ R11 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R11 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5254_Card__order__iff__ordLeq__card__of,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5255_ordIso__card__of__imp__Card__order,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5256_Card__order__iff__ordIso__card__of,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5257_card__of__Field__ordIso,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5258_dir__image,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ! [X2: A,Y4: A] :
% 11.93/6.03 ( ( ( F @ X2 )
% 11.93/6.03 = ( F @ Y4 ) )
% 11.93/6.03 = ( X2 = Y4 ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5259_exists__minim__Card__order,axiom,
% 11.93/6.03 ! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
% 11.93/6.03 ( ( R5
% 11.93/6.03 != ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
% 11.93/6.03 => ( ! [X2: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 11.93/6.03 => ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X2 ) @ X2 ) )
% 11.93/6.03 => ? [X2: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ R5 )
% 11.93/6.03 & ! [Xa2: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X2 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5260_Card__order__empty,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5261_card__of__ordIso__finite__Field,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 = ( finite_finite2 @ B @ A4 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5262_card__of__underS,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R @ A3 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5263_Card__order__Plus1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5264_Card__order__Plus2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A4 @ ( field2 @ A @ R ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5265_Card__order__singl__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),B2: B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( ( field2 @ A @ R )
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5266_card__of__Un__ordLeq__infinite__Field,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ B] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5267_card__of__empty1,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 | ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5268_Card__order__Times1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B6: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( B6
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
% 11.93/6.03 @ ^ [Uu: A] : B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5269_Card__order__Times2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( A4
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ A4
% 11.93/6.03 @ ^ [Uu: B] : ( field2 @ A @ R ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5270_Card__order__Times__same__infinite,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
% 11.93/6.03 @ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
% 11.93/6.03 @ ^ [Uu: A] : ( field2 @ A @ R ) ) )
% 11.93/6.03 @ R )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5271_card__of__UNION__ordLeq__infinite__Field,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ! [X2: B] :
% 11.93/6.03 ( ( member @ B @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A4 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5272_Card__order__Plus__infinite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ ( field2 @ B @ P5 ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( field2 @ B @ P5 ) @ ( field2 @ A @ R ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5273_Card__order__iff__Restr__underS,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 = ( ! [X3: A] :
% 11.93/6.03 ( ( member @ A @ X3 @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
% 11.93/6.03 @ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X3 )
% 11.93/6.03 @ ^ [Uu: A] : ( order_underS @ A @ R @ X3 ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) )
% 11.93/6.03 @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5274_regularCard__UNION,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: A > ( set @ B ),B6: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca7133664381575040944arCard @ A @ R )
% 11.93/6.03 => ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R @ As4 )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As4 @ ( field2 @ A @ R ) ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
% 11.93/6.03 => ? [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.93/6.03 & ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5275_card__of__Plus__ordLeq__infinite__Field,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A4: set @ B,B6: set @ C] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5276_Card__order__Times__infinite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( ( field2 @ B @ P5 )
% 11.93/6.03 != ( bot_bot @ ( set @ B ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
% 11.93/6.03 @ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
% 11.93/6.03 @ ^ [Uu: A] : ( field2 @ B @ P5 ) ) )
% 11.93/6.03 @ R )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
% 11.93/6.03 @ ( product_Sigma @ B @ A @ ( field2 @ B @ P5 )
% 11.93/6.03 @ ^ [Uu: B] : ( field2 @ A @ R ) ) )
% 11.93/6.03 @ R )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5277_ex__toCard__pred,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 11.93/6.03 => ? [X_12: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R @ X_12 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5278_cardSuc__UNION,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B6: set @ B] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ? [X2: set @ A] :
% 11.93/6.03 ( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
% 11.93/6.03 & ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5279_toCard__inj,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),X: A,Y: A] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 11.93/6.03 => ( ( member @ A @ X @ A4 )
% 11.93/6.03 => ( ( member @ A @ Y @ A4 )
% 11.93/6.03 => ( ( ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ X )
% 11.93/6.03 = ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ Y ) )
% 11.93/6.03 = ( X = Y ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5280_cardSuc__greater,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5281_cardSuc__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5282_toCard__pred__toCard,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 11.93/6.03 => ( bNF_Gr1419584066657907630d_pred @ A @ B @ A4 @ R @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5283_cardSuc__least__aux,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R4 ) @ R4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5284_cardSuc__ordLeq__ordLess,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5285_cardSuc__ordLess__ordLeq,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5286_cardSuc__least,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R4 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5287_cardSuc__mono__ordLeq,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ B ) ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5288_cardSuc__invar__ordIso,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R4 ) @ R4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R4 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( set @ B ) ) )
% 11.93/6.03 = ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5289_fromCard__toCard,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),B2: A] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 11.93/6.03 => ( ( member @ A @ B2 @ A4 )
% 11.93/6.03 => ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A4 @ R @ ( bNF_Greatest_toCard @ A @ B @ A4 @ R @ B2 ) )
% 11.93/6.03 = B2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5290_cardSuc__UNION__Cinfinite,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B6: set @ B] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ? [X2: set @ A] :
% 11.93/6.03 ( ( member @ ( set @ A ) @ X2 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
% 11.93/6.03 & ( ord_less_eq @ ( set @ B ) @ B6 @ ( As4 @ X2 ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5291_isCardSuc__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( bNF_Ca6246979054910435723ardSuc @ A )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R11 ) @ R11 )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
% 11.93/6.03 & ! [R14: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
% 11.93/6.03 ( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R14 ) @ R14 )
% 11.93/6.03 & ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ R14 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R11 @ R14 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5292_cinfinite__mono,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
% 11.93/6.03 => ( bNF_Ca4139267488887388095finite @ B @ R22 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5293_Cinfinite__limit2,axiom,
% 11.93/6.03 ! [A: $tType,X1: A,R: set @ ( product_prod @ A @ A ),X22: A] :
% 11.93/6.03 ( ( member @ A @ X1 @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( member @ A @ X22 @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ? [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.93/6.03 & ( X1 != X2 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X2 ) @ R )
% 11.93/6.03 & ( X22 != X2 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X2 ) @ R ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5294_Cinfinite__limit,axiom,
% 11.93/6.03 ! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( member @ A @ X @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ? [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.93/6.03 & ( X != X2 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X2 ) @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5295_Cinfinite__cong,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 11.93/6.03 => ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5296_Cinfinite__limit__finite,axiom,
% 11.93/6.03 ! [A: $tType,X10: set @ A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( finite_finite2 @ A @ X10 )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ A ) @ X10 @ ( field2 @ A @ R ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ? [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ ( field2 @ A @ R ) )
% 11.93/6.03 & ! [Xa2: A] :
% 11.93/6.03 ( ( member @ A @ Xa2 @ X10 )
% 11.93/6.03 => ( ( Xa2 != X2 )
% 11.93/6.03 & ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X2 ) @ R ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5297_Un__Cinfinite__bound,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ B @ B ),B6: set @ A] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5298_UNION__Cinfinite__bound,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,C: $tType,I4: set @ A,R: set @ ( product_prod @ B @ B ),A4: A > ( set @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ! [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A4 @ I4 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5299_Cfinite__ordLess__Cinfinite,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ( bNF_Cardinal_cfinite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ S ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5300_comp__single__set__bd,axiom,
% 11.93/6.03 ! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X: D] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
% 11.93/6.03 => ( ! [X2: B] : ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( Fset @ X2 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
% 11.93/6.03 => ( ! [X2: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X2 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5301_cprod__dup,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P5 @ P8 ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P5 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5302_cprod__com,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5303_Cfinite__cprod__Cinfinite,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ( bNF_Cardinal_cfinite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ S ) @ S ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5304_cprod__mono,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5305_cprod__mono1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5306_cprod__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5307_cprod__cong,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5308_cprod__cong1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ P24 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5309_cprod__cong2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5310_cprod__infinite,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5311_cprod__cinfinite__bound,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P5: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P5 ) @ P5 )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P5 @ Q3 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5312_card__of__Csum__Times_H,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I4: set @ B,A4: B > ( set @ C )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ! [X2: B] :
% 11.93/6.03 ( ( member @ B @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ X2 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
% 11.93/6.03 @ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 )
% 11.93/6.03 @ ^ [I3: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A4 @ I3 ) ) )
% 11.93/6.03 @ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I4 ) @ R ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5313_card__of__Csum__Times,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,I4: set @ A,A4: A > ( set @ B ),B6: set @ C] :
% 11.93/6.03 ( ! [X2: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ I4 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ X2 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) )
% 11.93/6.03 @ ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 )
% 11.93/6.03 @ ^ [I3: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A4 @ I3 ) ) )
% 11.93/6.03 @ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I4 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B6 ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5314_cprod__infinite1_H,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P5 ) @ P5 ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ P5 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5315_czero__ordIso,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_czero @ A ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5316_cinfinite__not__czero,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 => ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5317_czeroE,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( field2 @ A @ R )
% 11.93/6.03 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5318_card__of__ordIso__czero__iff__empty,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 = ( A4
% 11.93/6.03 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5319_czeroI,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( ( field2 @ A @ R )
% 11.93/6.03 = ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5320_Cnotzero__imp__not__empty,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( field2 @ A @ R )
% 11.93/6.03 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5321_Cnotzero__mono,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ Q3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5322_Cinfinite__Cnotzero,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5323_Cnotzero__UNIV,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5324_cinfinite__cprod2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
% 11.93/6.03 => ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5325_Cinfinite__cprod2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
% 11.93/6.03 => ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5326_ordLeq__cprod2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P13 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P24 ) @ P24 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ P24 @ ( bNF_Cardinal_cprod @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5327_cone__ordLeq__Cnotzero,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5328_csum__dup,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P5: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Cardinal_csum @ B @ C @ P5 @ P8 ) @ ( bNF_Cardinal_csum @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ ( sum_sum @ A @ A ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ C @ P5 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5329_csum__Cnotzero1,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
% 11.93/6.03 => ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5330_csum__csum,axiom,
% 11.93/6.03 ! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_csum @ C @ D @ R33 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R1 @ R33 ) @ ( bNF_Cardinal_csum @ B @ D @ R22 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5331_csum__assoc,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B ),P32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ P32 ) @ ( bNF_Cardinal_csum @ A @ ( sum_sum @ B @ C ) @ P13 @ ( bNF_Cardinal_csum @ B @ C @ P24 @ P32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5332_cprod__csum__distrib1,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cprod @ A @ C @ R1 @ R33 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R1 @ ( bNF_Cardinal_csum @ B @ C @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5333_cone__not__czero,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ~ ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ product_unit @ A ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5334_csum__com,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ A @ P24 @ P13 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5335_Un__csum,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,B6: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B6 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B6 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5336_ordLeq__csum1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),P24: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P13 ) @ P13 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ P13 @ ( bNF_Cardinal_csum @ A @ B @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5337_ordLeq__csum2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),P13: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ P24 @ ( bNF_Cardinal_csum @ B @ A @ P13 @ P24 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5338_csum__cong,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5339_csum__cong1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5340_csum__cong2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5341_csum__mono,axiom,
% 11.93/6.03 ! [D: $tType,B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5342_csum__mono1,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5343_csum__mono2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_csum @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5344_Plus__csum,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,B6: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B6 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5345_csum__absorb2_H,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
% 11.93/6.03 | ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ A @ R1 @ R22 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5346_csum__absorb1_H,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
% 11.93/6.03 | ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5347_csum__absorb1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R22 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5348_csum__cinfinite__bound,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P5: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q3 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P5 ) @ P5 )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_csum @ A @ C @ P5 @ Q3 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5349_natLeq__ordLeq__cinfinite,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ R ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5350_cexp__mono2__Cnotzero,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P24 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5351_cexp__mono2,axiom,
% 11.93/6.03 ! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ ( bNF_Cardinal_czero @ D ) ) @ ( bNF_Wellorder_ordIso @ A @ D ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5352_cprod__cexp,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C ),T2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5353_cexp__cprod,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ C @ C ),R33: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R1 @ ( bNF_Cardinal_cprod @ C @ B @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5354_cprod__cexp__csum__cexp__Cinfinite,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R @ S ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5355_card__of__Field__natLeq,axiom,
% 11.93/6.03 member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5356_cexp__cone,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R @ bNF_Cardinal_cone ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5357_finite__iff__ordLess__natLeq,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( finite_finite2 @ A )
% 11.93/6.03 = ( ^ [A9: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A9 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5358_cexp__cprod__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
% 11.93/6.03 => ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R33 @ ( bNF_Cardinal_czero @ C ) ) @ ( bNF_Wellorder_ordIso @ C @ C ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ R33 ) @ R33 ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R33 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5359_cexp__mono_H,axiom,
% 11.93/6.03 ! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 11.93/6.03 => ( ( ( ( field2 @ C @ P24 )
% 11.93/6.03 = ( bot_bot @ ( set @ C ) ) )
% 11.93/6.03 => ( ( field2 @ D @ R22 )
% 11.93/6.03 = ( bot_bot @ ( set @ D ) ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5360_cexp__mono1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5361_cexp__cong2,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P24 ) @ P24 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5362_cexp__cong1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ Q3 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q3 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5363_cexp__cong,axiom,
% 11.93/6.03 ! [B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R22 ) @ R22 )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5364_card__of__nat,axiom,
% 11.93/6.03 member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5365_cexp__mono2_H,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,P24: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q3: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q3 ) @ Q3 )
% 11.93/6.03 => ( ( ( ( field2 @ A @ P24 )
% 11.93/6.03 = ( bot_bot @ ( set @ A ) ) )
% 11.93/6.03 => ( ( field2 @ B @ R22 )
% 11.93/6.03 = ( bot_bot @ ( set @ B ) ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q3 @ P24 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q3 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5366_ordLeq__cexp1,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q3: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q3 ) @ Q3 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q3 @ ( bNF_Cardinal_cexp @ B @ A @ Q3 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5367_infinite__iff__natLeq__ordLeq,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] :
% 11.93/6.03 ( ( finite_finite2 @ A @ A4 )
% 11.93/6.03 != ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5368_cexp__mono,axiom,
% 11.93/6.03 ! [E: $tType,F5: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P13: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P24: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P13 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P24 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
% 11.93/6.03 => ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) @ P24 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F5 @ F5 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F5 @ F5 ) ) @ R22 @ ( bNF_Cardinal_czero @ F5 ) ) @ ( bNF_Wellorder_ordIso @ D @ F5 ) ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P24 ) @ P24 )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P13 @ P24 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5369_ordLeq__cexp2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
% 11.93/6.03 => ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5370_Cfinite__cexp__Cinfinite,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( ( bNF_Cardinal_cfinite @ A @ S )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S ) @ S ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5371_cinfinite__cexp,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 11.93/6.03 => ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5372_ctwo__Cnotzero,axiom,
% 11.93/6.03 ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ $o ) ) @ ( bNF_Wellorder_ordIso @ $o @ $o ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ $o @ ( field2 @ $o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5373_ctwo__ordLess__natLeq,axiom,
% 11.93/6.03 member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ $o @ nat ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5374_ordLess__ctwo__cexp,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5375_cprod__csum__cexp,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5376_ctwo__not__czero,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5377_ctwo__ordLeq__Cinfinite,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5378_ctwo__ordLess__Cinfinite,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5379_csum__Cfinite__cexp__Cinfinite,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( ( ( bNF_Cardinal_cfinite @ B @ S )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5380_Cinfinite__cexp,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,Q3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
% 11.93/6.03 ( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q3 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
% 11.93/6.03 => ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
% 11.93/6.03 => ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) )
% 11.93/6.03 & ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q3 @ R ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5381_card__of__Func__UNIV,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,B6: set @ B] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B6 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
% 11.93/6.03 @ ( collect @ ( A > B )
% 11.93/6.03 @ ^ [F2: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) @ B6 ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5382_Card__order__Pow,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
% 11.93/6.03 => ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ ( field2 @ A @ R ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5383_Func__Times__Range,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
% 11.93/6.03 @ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A4
% 11.93/6.03 @ ( product_Sigma @ B @ C @ B6
% 11.93/6.03 @ ^ [Uu: B] : C5 ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
% 11.93/6.03 @ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A4 @ B6 )
% 11.93/6.03 @ ^ [Uu: A > B] : ( bNF_Wellorder_Func @ A @ C @ A4 @ C5 ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5384_card__of__Func__Times,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,A4: set @ A,B6: set @ B,C5: set @ C] :
% 11.93/6.03 ( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
% 11.93/6.03 @ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
% 11.93/6.03 @ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
% 11.93/6.03 @ ( product_Sigma @ A @ B @ A4
% 11.93/6.03 @ ^ [Uu: A] : B6 )
% 11.93/6.03 @ C5 ) )
% 11.93/6.03 @ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A4 @ ( bNF_Wellorder_Func @ B @ C @ B6 @ C5 ) ) ) )
% 11.93/6.03 @ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5385_card__of__Pow__Func,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A4 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5386_Cantors__paradox,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] :
% 11.93/6.03 ~ ? [F8: A > ( set @ A )] :
% 11.93/6.03 ( ( image2 @ A @ ( set @ A ) @ F8 @ A4 )
% 11.93/6.03 = ( pow @ A @ A4 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5387_inj__on__image__Pow,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,A4: set @ A] :
% 11.93/6.03 ( ( inj_on @ A @ B @ F @ A4 )
% 11.93/6.03 => ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( pow @ A @ A4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5388_card__of__Pow,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow @ A @ A4 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5389_lfp__induct2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,A3: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) )
% 11.93/6.03 => ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F )
% 11.93/6.03 => ( ! [A5: A,B3: B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( F @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
% 11.93/6.03 => ( P @ A5 @ B3 ) )
% 11.93/6.03 => ( P @ A3 @ B2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5390_sort__quicksort__by__rel,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( linorder @ A )
% 11.93/6.03 => ( ( linorder_sort_key @ A @ A
% 11.93/6.03 @ ^ [X3: A] : X3 )
% 11.93/6.03 = ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5391_Gr__def,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( bNF_Gr @ A @ B )
% 11.93/6.03 = ( ^ [A9: set @ A,F2: A > B] :
% 11.93/6.03 ( collect @ ( product_prod @ A @ B )
% 11.93/6.03 @ ^ [Uu: product_prod @ A @ B] :
% 11.93/6.03 ? [A7: A] :
% 11.93/6.03 ( ( Uu
% 11.93/6.03 = ( product_Pair @ A @ B @ A7 @ ( F2 @ A7 ) ) )
% 11.93/6.03 & ( member @ A @ A7 @ A9 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5392_GrD2,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,X: A,Fx: B,A4: set @ A,F: A > B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F ) )
% 11.93/6.03 => ( ( F @ X )
% 11.93/6.03 = Fx ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5393_GrD1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,X: A,Fx: B,A4: set @ A,F: A > B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A4 @ F ) )
% 11.93/6.03 => ( member @ A @ X @ A4 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5394_sort__mergesort,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( linorder @ A )
% 11.93/6.03 => ( ( linorder_sort_key @ A @ A
% 11.93/6.03 @ ^ [X3: A] : X3 )
% 11.93/6.03 = ( mergesort @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5395_sort__mergesort__by__rel,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( linorder @ A )
% 11.93/6.03 => ( ( linorder_sort_key @ A @ A
% 11.93/6.03 @ ^ [X3: A] : X3 )
% 11.93/6.03 = ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5396_remdup__sort__mergesort__remdups,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( linorder @ A )
% 11.93/6.03 => ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
% 11.93/6.03 @ ( linorder_sort_key @ A @ A
% 11.93/6.03 @ ^ [X3: A] : X3 ) )
% 11.93/6.03 = ( mergesort_remdups @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5397_prod_H__def,axiom,
% 11.93/6.03 ! [C: $tType,A: $tType] :
% 11.93/6.03 ( ( comm_monoid_mult @ A )
% 11.93/6.03 => ( ( groups1962203154675924110t_prod @ C @ A )
% 11.93/6.03 = ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5398_prod_Ocomm__monoid__list__set__axioms,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( comm_monoid_mult @ A )
% 11.93/6.03 => ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5399_length__remdups__card,axiom,
% 11.93/6.03 ! [A: $tType,L: list @ A] :
% 11.93/6.03 ( ( size_size @ ( list @ A ) @ ( remdups @ A @ L ) )
% 11.93/6.03 = ( finite_card @ A @ ( set2 @ A @ L ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5400_sum_H__def,axiom,
% 11.93/6.03 ! [C: $tType,A: $tType] :
% 11.93/6.03 ( ( comm_monoid_add @ A )
% 11.93/6.03 => ( ( groups1027152243600224163dd_sum @ C @ A )
% 11.93/6.03 = ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5401_sum_Ocomm__monoid__list__set__axioms,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( comm_monoid_add @ A )
% 11.93/6.03 => ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5402_and_Ocomm__monoid__axioms,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( bit_ri3973907225187159222ations @ A )
% 11.93/6.03 => ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5403_bijective__alt,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( bijective @ A @ B )
% 11.93/6.03 = ( ^ [R6: set @ ( product_prod @ A @ B )] :
% 11.93/6.03 ( ( single_valued @ A @ B @ R6 )
% 11.93/6.03 & ( single_valued @ B @ A @ ( converse @ A @ B @ R6 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5404_Heap_Osize__gen__o__map,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: B > nat,G: A > B] :
% 11.93/6.03 ( ( comp @ ( heap_Time_Heap @ B ) @ nat @ ( heap_Time_Heap @ A ) @ ( heap_Time_size_Heap @ B @ F ) @ ( heap_Time_map_Heap @ A @ B @ G ) )
% 11.93/6.03 = ( heap_Time_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5405_single__valued__inter1,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 11.93/6.03 ( ( single_valued @ A @ B @ R5 )
% 11.93/6.03 => ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S2 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5406_single__valued__inter2,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 11.93/6.03 ( ( single_valued @ A @ B @ R5 )
% 11.93/6.03 => ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 @ R5 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5407_comm__monoid_Ocomm__neutral,axiom,
% 11.93/6.03 ! [A: $tType,F: A > A > A,Z2: A,A3: A] :
% 11.93/6.03 ( ( comm_monoid @ A @ F @ Z2 )
% 11.93/6.03 => ( ( F @ A3 @ Z2 )
% 11.93/6.03 = A3 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5408_Heap_Omap__ident,axiom,
% 11.93/6.03 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 11.93/6.03 ( ( heap_Time_map_Heap @ A @ A
% 11.93/6.03 @ ^ [X3: A] : X3
% 11.93/6.03 @ T2 )
% 11.93/6.03 = T2 ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5409_Heap_Omap__id0,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
% 11.93/6.03 = ( id @ ( heap_Time_Heap @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5410_Heap_Omap__id,axiom,
% 11.93/6.03 ! [A: $tType,T2: heap_Time_Heap @ A] :
% 11.93/6.03 ( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) @ T2 )
% 11.93/6.03 = T2 ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5411_Heap_Omap__comp,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Time_Heap @ A] :
% 11.93/6.03 ( ( heap_Time_map_Heap @ B @ C @ G @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 11.93/6.03 = ( heap_Time_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5412_Heap_Omap,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.03 ( ( heap_Time_map_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 11.93/6.03 = ( heap_Time_Heap2 @ B @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5413_single__valuedD,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X: A,Y: B,Z2: B] :
% 11.93/6.03 ( ( single_valued @ A @ B @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z2 ) @ R )
% 11.93/6.03 => ( Y = Z2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5414_single__valuedI,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 11.93/6.03 ( ! [X2: A,Y4: B,Z3: B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R )
% 11.93/6.03 => ( Y4 = Z3 ) ) )
% 11.93/6.03 => ( single_valued @ A @ B @ R ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5415_single__valued__def,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( single_valued @ A @ B )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 11.93/6.03 ! [X3: A,Y3: B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 )
% 11.93/6.03 => ! [Z5: B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Z5 ) @ R3 )
% 11.93/6.03 => ( Y3 = Z5 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5416_add_Ocomm__monoid__axioms,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( comm_monoid_add @ A )
% 11.93/6.03 => ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5417_mult_Ocomm__monoid__axioms,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( comm_monoid_mult @ A )
% 11.93/6.03 => ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5418_Heap_Oinj__map,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B] :
% 11.93/6.03 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.93/6.03 => ( inj_on @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_map_Heap @ A @ B @ F ) @ ( top_top @ ( set @ ( heap_Time_Heap @ A ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5419_single__valued__confluent,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y: A,Z2: A] :
% 11.93/6.03 ( ( single_valued @ A @ A @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.93/6.03 | ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z2 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5420_single__valued__below__Id,axiom,
% 11.93/6.03 ! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) )
% 11.93/6.03 => ( single_valued @ A @ A @ R5 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5421_Heap_Orec__o__map,axiom,
% 11.93/6.03 ! [B: $tType,C: $tType,A: $tType,G: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,F: A > B] :
% 11.93/6.03 ( ( comp @ ( heap_Time_Heap @ B ) @ C @ ( heap_Time_Heap @ A ) @ ( heap_Time_rec_Heap @ B @ C @ G ) @ ( heap_Time_map_Heap @ A @ B @ F ) )
% 11.93/6.03 = ( heap_Time_rec_Heap @ A @ C
% 11.93/6.03 @ ^ [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( G @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X3 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5422_single__valuedp__single__valued__eq,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 11.93/6.03 ( ( single_valuedp @ A @ B
% 11.93/6.03 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 11.93/6.03 = ( single_valued @ A @ B @ R ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5423_disjnt__equiv__class,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( disjnt @ A @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.93/6.03 = ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5424_equiv__class__eq__iff,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.93/6.03 = ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 11.93/6.03 = ( image @ A @ A @ R @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.93/6.03 & ( member @ A @ X @ A4 )
% 11.93/6.03 & ( member @ A @ Y @ A4 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5425_eq__equiv__class__iff,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ A @ X @ A4 )
% 11.93/6.03 => ( ( member @ A @ Y @ A4 )
% 11.93/6.03 => ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 11.93/6.03 = ( image @ A @ A @ R @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.93/6.03 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5426_equiv__class__eq,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.93/6.03 => ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 11.93/6.03 = ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5427_eq__equiv__class,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,A4: set @ A] :
% 11.93/6.03 ( ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
% 11.93/6.03 = ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.93/6.03 => ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ A @ B2 @ A4 )
% 11.93/6.03 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5428_Heap_Orec,axiom,
% 11.93/6.03 ! [C: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.03 ( ( heap_Time_rec_Heap @ A @ C @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 11.93/6.03 = ( F @ X ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5429_subset__equiv__class,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 11.93/6.03 => ( ( member @ A @ B2 @ A4 )
% 11.93/6.03 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5430_equiv__class__subset,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.93/6.03 => ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5431_equiv__class__nondisjoint,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 11.93/6.03 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5432_proj__iff,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A4 )
% 11.93/6.03 => ( ( ( equiv_proj @ A @ A @ R @ X )
% 11.93/6.03 = ( equiv_proj @ A @ A @ R @ Y ) )
% 11.93/6.03 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5433_in__quotient__imp__in__rel,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X10 )
% 11.93/6.03 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5434_congruent2__commuteI,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > A > B] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ! [Y4: A,Z3: A] :
% 11.93/6.03 ( ( member @ A @ Y4 @ A4 )
% 11.93/6.03 => ( ( member @ A @ Z3 @ A4 )
% 11.93/6.03 => ( ( F @ Y4 @ Z3 )
% 11.93/6.03 = ( F @ Z3 @ Y4 ) ) ) )
% 11.93/6.03 => ( ! [Y4: A,Z3: A,W4: A] :
% 11.93/6.03 ( ( member @ A @ W4 @ A4 )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.93/6.03 => ( ( F @ W4 @ Y4 )
% 11.93/6.03 = ( F @ W4 @ Z3 ) ) ) )
% 11.93/6.03 => ( equiv_congruent2 @ A @ A @ B @ R @ R @ F ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5435_quotient__eqI,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,Y9: set @ A,X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( member @ A @ X @ X10 )
% 11.93/6.03 => ( ( member @ A @ Y @ Y9 )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.93/6.03 => ( X10 = Y9 ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5436_quotient__eq__iff,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,Y9: set @ A,X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( member @ A @ X @ X10 )
% 11.93/6.03 => ( ( member @ A @ Y @ Y9 )
% 11.93/6.03 => ( ( X10 = Y9 )
% 11.93/6.03 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5437_in__quotient__imp__closed,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X10: set @ A,X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( member @ A @ X @ X10 )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
% 11.93/6.03 => ( member @ A @ Y @ X10 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5438_congruent2D,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z1: A,Y2: B,Z22: B] :
% 11.93/6.03 ( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
% 11.93/6.03 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R22 )
% 11.93/6.03 => ( ( F @ Y1 @ Y2 )
% 11.93/6.03 = ( F @ Z1 @ Z22 ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5439_congruent2I_H,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 11.93/6.03 ( ! [Y12: A,Z12: A,Y22: B,Z23: B] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z12 ) @ R1 )
% 11.93/6.03 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z23 ) @ R22 )
% 11.93/6.03 => ( ( F @ Y12 @ Y22 )
% 11.93/6.03 = ( F @ Z12 @ Z23 ) ) ) )
% 11.93/6.03 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5440_eq__equiv__class__iff2,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y: A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( member @ A @ X @ A4 )
% 11.93/6.03 => ( ( member @ A @ Y @ A4 )
% 11.93/6.03 => ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R )
% 11.93/6.03 = ( equiv_quotient @ A @ ( insert2 @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R ) )
% 11.93/6.03 = ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5441_congruent2I,axiom,
% 11.93/6.03 ! [C: $tType,B: $tType,A: $tType,A110: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A110 @ R1 )
% 11.93/6.03 => ( ( equiv_equiv @ B @ A26 @ R22 )
% 11.93/6.03 => ( ! [Y4: A,Z3: A,W4: B] :
% 11.93/6.03 ( ( member @ B @ W4 @ A26 )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R1 )
% 11.93/6.03 => ( ( F @ Y4 @ W4 )
% 11.93/6.03 = ( F @ Z3 @ W4 ) ) ) )
% 11.93/6.03 => ( ! [Y4: B,Z3: B,W4: A] :
% 11.93/6.03 ( ( member @ A @ W4 @ A110 )
% 11.93/6.03 => ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Z3 ) @ R22 )
% 11.93/6.03 => ( ( F @ W4 @ Y4 )
% 11.93/6.03 = ( F @ W4 @ Z3 ) ) ) )
% 11.93/6.03 => ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5442_UN__equiv__class__inject,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X10: set @ A,Y9: set @ A] :
% 11.93/6.03 ( ( equiv_equiv @ A @ A4 @ R )
% 11.93/6.03 => ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
% 11.93/6.03 => ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X10 ) )
% 11.93/6.03 = ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ Y9 ) ) )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ X10 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A4 @ R ) )
% 11.93/6.03 => ( ! [X2: A,Y4: A] :
% 11.93/6.03 ( ( member @ A @ X2 @ A4 )
% 11.93/6.03 => ( ( member @ A @ Y4 @ A4 )
% 11.93/6.03 => ( ( ( F @ X2 )
% 11.93/6.03 = ( F @ Y4 ) )
% 11.93/6.03 => ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y4 ) @ R ) ) ) )
% 11.93/6.03 => ( X10 = Y9 ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5443_equivp__equiv,axiom,
% 11.93/6.03 ! [A: $tType,A4: set @ ( product_prod @ A @ A )] :
% 11.93/6.03 ( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A4 )
% 11.93/6.03 = ( equiv_equivp @ A
% 11.93/6.03 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ A4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5444_prod__mset__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( comm_monoid_mult @ A )
% 11.93/6.03 => ( ( comm_m9189036328036947845d_mset @ A )
% 11.93/6.03 = ( comm_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5445_congruentI,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
% 11.93/6.03 ( ! [Y4: A,Z3: A] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
% 11.93/6.03 => ( ( F @ Y4 )
% 11.93/6.03 = ( F @ Z3 ) ) )
% 11.93/6.03 => ( equiv_congruent @ A @ B @ R @ F ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5446_congruentD,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B,Y: A,Z2: A] :
% 11.93/6.03 ( ( equiv_congruent @ A @ B @ R @ F )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z2 ) @ R )
% 11.93/6.03 => ( ( F @ Y )
% 11.93/6.03 = ( F @ Z2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5447_sum__mset__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( comm_monoid_add @ A )
% 11.93/6.03 => ( ( comm_m7189776963980413722m_mset @ A )
% 11.93/6.03 = ( comm_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5448_or__num_Opelims,axiom,
% 11.93/6.03 ! [X: num,Xa: num,Y: num] :
% 11.93/6.03 ( ( ( bit_un6697907153464112080or_num @ X @ Xa )
% 11.93/6.03 = Y )
% 11.93/6.03 => ( ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
% 11.93/6.03 => ( ( ( X = one2 )
% 11.93/6.03 => ( ( Xa = one2 )
% 11.93/6.03 => ( ( Y = one2 )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
% 11.93/6.03 => ( ( ( X = one2 )
% 11.93/6.03 => ! [N3: num] :
% 11.93/6.03 ( ( Xa
% 11.93/6.03 = ( bit0 @ N3 ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ N3 ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) ) ) ) )
% 11.93/6.03 => ( ( ( X = one2 )
% 11.93/6.03 => ! [N3: num] :
% 11.93/6.03 ( ( Xa
% 11.93/6.03 = ( bit1 @ N3 ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ N3 ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) ) ) ) )
% 11.93/6.03 => ( ! [M5: num] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( bit0 @ M5 ) )
% 11.93/6.03 => ( ( Xa = one2 )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ M5 ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ one2 ) ) ) ) )
% 11.93/6.03 => ( ! [M5: num] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( bit0 @ M5 ) )
% 11.93/6.03 => ! [N3: num] :
% 11.93/6.03 ( ( Xa
% 11.93/6.03 = ( bit0 @ N3 ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit0 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.93/6.03 => ( ! [M5: num] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( bit0 @ M5 ) )
% 11.93/6.03 => ! [N3: num] :
% 11.93/6.03 ( ( Xa
% 11.93/6.03 = ( bit1 @ N3 ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
% 11.93/6.03 => ( ! [M5: num] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( bit1 @ M5 ) )
% 11.93/6.03 => ( ( Xa = one2 )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ M5 ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ one2 ) ) ) ) )
% 11.93/6.03 => ( ! [M5: num] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( bit1 @ M5 ) )
% 11.93/6.03 => ! [N3: num] :
% 11.93/6.03 ( ( Xa
% 11.93/6.03 = ( bit0 @ N3 ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
% 11.93/6.03 => ~ ! [M5: num] :
% 11.93/6.03 ( ( X
% 11.93/6.03 = ( bit1 @ M5 ) )
% 11.93/6.03 => ! [N3: num] :
% 11.93/6.03 ( ( Xa
% 11.93/6.03 = ( bit1 @ N3 ) )
% 11.93/6.03 => ( ( Y
% 11.93/6.03 = ( bit1 @ ( bit_un6697907153464112080or_num @ M5 @ N3 ) ) )
% 11.93/6.03 => ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4773296044027857193um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5449_Heap_Oset,axiom,
% 11.93/6.03 ! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.03 ( ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) )
% 11.93/6.03 = ( complete_Sup_Sup @ ( set @ A )
% 11.93/6.03 @ ( image2 @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( set @ A )
% 11.93/6.03 @ ^ [Uu3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set @ A ) @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Uu3 ) ) )
% 11.93/6.03 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5450_under__incr,axiom,
% 11.93/6.03 ! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
% 11.93/6.03 ( ( trans @ A @ R )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.93/6.03 => ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A3 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5451_Heap_Oset__intros,axiom,
% 11.93/6.03 ! [A: $tType,Y: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ),Yb2: A] :
% 11.93/6.03 ( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Y @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
% 11.93/6.03 => ( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Ya @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Y ) )
% 11.93/6.03 => ( ( member @ A @ Yb2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Ya ) )
% 11.93/6.03 => ( member @ A @ Yb2 @ ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5452_Heap_Oset__cases,axiom,
% 11.93/6.03 ! [A: $tType,E2: A,A3: heap_Time_Heap @ A] :
% 11.93/6.03 ( ( member @ A @ E2 @ ( heap_Time_set_Heap @ A @ A3 ) )
% 11.93/6.03 => ~ ! [Z3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.03 ( ( A3
% 11.93/6.03 = ( heap_Time_Heap2 @ A @ Z3 ) )
% 11.93/6.03 => ! [Xa3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
% 11.93/6.03 ( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Xa3 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Z3 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
% 11.93/6.03 => ! [Xb3: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat )] :
% 11.93/6.03 ( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xb3 @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xa3 ) )
% 11.93/6.03 => ~ ( member @ A @ E2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Xb3 ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5453_under__def,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( order_under @ A )
% 11.93/6.03 = ( ^ [R3: set @ ( product_prod @ A @ A ),A7: A] :
% 11.93/6.03 ( collect @ A
% 11.93/6.03 @ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A7 ) @ R3 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5454_Heap_Oinj__map__strong,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
% 11.93/6.03 ( ! [Z3: A,Za: A] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 11.93/6.03 => ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa ) )
% 11.93/6.03 => ( ( ( F @ Z3 )
% 11.93/6.03 = ( Fa @ Za ) )
% 11.93/6.03 => ( Z3 = Za ) ) ) )
% 11.93/6.03 => ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 11.93/6.03 = ( heap_Time_map_Heap @ A @ B @ Fa @ Xa ) )
% 11.93/6.03 => ( X = Xa ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5455_Heap_Omap__cong0,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 11.93/6.03 ( ! [Z3: A] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 11.93/6.03 => ( ( F @ Z3 )
% 11.93/6.03 = ( G @ Z3 ) ) )
% 11.93/6.03 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 11.93/6.03 = ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5456_Heap_Omap__cong,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 11.93/6.03 ( ( X = Ya )
% 11.93/6.03 => ( ! [Z3: A] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 11.93/6.03 => ( ( F @ Z3 )
% 11.93/6.03 = ( G @ Z3 ) ) )
% 11.93/6.03 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 11.93/6.03 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5457_Heap_Oset__map,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,F: A > B,V: heap_Time_Heap @ A] :
% 11.93/6.03 ( ( heap_Time_set_Heap @ B @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
% 11.93/6.03 = ( image2 @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5458_Heap_Orel__Grp,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,A4: set @ A,F: A > B] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ B @ ( bNF_Grp @ A @ B @ A4 @ F ) )
% 11.93/6.03 = ( bNF_Grp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
% 11.93/6.03 @ ( collect @ ( heap_Time_Heap @ A )
% 11.93/6.03 @ ^ [X3: heap_Time_Heap @ A] : ( ord_less_eq @ ( set @ A ) @ ( heap_Time_set_Heap @ A @ X3 ) @ A4 ) )
% 11.93/6.03 @ ( heap_Time_map_Heap @ A @ B @ F ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5459_Heap_Orel__compp__Grp,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ B )
% 11.93/6.03 = ( ^ [R6: A > B > $o] :
% 11.93/6.03 ( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
% 11.93/6.03 @ ( conversep @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
% 11.93/6.03 @ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
% 11.93/6.03 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 11.93/6.03 @ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
% 11.93/6.03 @ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
% 11.93/6.03 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
% 11.93/6.03 @ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5460_Heap_Oin__rel,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ B )
% 11.93/6.03 = ( ^ [R6: A > B > $o,A7: heap_Time_Heap @ A,B5: heap_Time_Heap @ B] :
% 11.93/6.03 ? [Z5: heap_Time_Heap @ ( product_prod @ A @ B )] :
% 11.93/6.03 ( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z5
% 11.93/6.03 @ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
% 11.93/6.03 @ ^ [X3: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
% 11.93/6.03 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
% 11.93/6.03 = A7 )
% 11.93/6.03 & ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
% 11.93/6.03 = B5 ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5461_Heap_Orel__cong,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Xa: heap_Time_Heap @ B,R5: A > B > $o,Ra2: A > B > $o] :
% 11.93/6.03 ( ( X = Ya )
% 11.93/6.03 => ( ( Y = Xa )
% 11.93/6.03 => ( ! [Z3: A,Yb: B] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 11.93/6.03 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa ) )
% 11.93/6.03 => ( ( R5 @ Z3 @ Yb )
% 11.93/6.03 = ( Ra2 @ Z3 @ Yb ) ) ) )
% 11.93/6.03 => ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
% 11.93/6.03 = ( heap_Time_rel_Heap @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5462_Heap_Orel__mono__strong,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Ra2: A > B > $o] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
% 11.93/6.03 => ( ! [Z3: A,Yb: B] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 11.93/6.03 => ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y ) )
% 11.93/6.03 => ( ( R5 @ Z3 @ Yb )
% 11.93/6.03 => ( Ra2 @ Z3 @ Yb ) ) ) )
% 11.93/6.03 => ( heap_Time_rel_Heap @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5463_Heap_Orel__refl__strong,axiom,
% 11.93/6.03 ! [A: $tType,X: heap_Time_Heap @ A,Ra2: A > A > $o] :
% 11.93/6.03 ( ! [Z3: A] :
% 11.93/6.03 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 11.93/6.03 => ( Ra2 @ Z3 @ Z3 ) )
% 11.93/6.03 => ( heap_Time_rel_Heap @ A @ A @ Ra2 @ X @ X ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5464_Heap_Orel__map_I2_J,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: heap_Time_Heap @ A,G: B > C,Y: heap_Time_Heap @ B] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ C @ Sa @ X @ ( heap_Time_map_Heap @ B @ C @ G @ Y ) )
% 11.93/6.03 = ( heap_Time_rel_Heap @ A @ B
% 11.93/6.03 @ ^ [X3: A,Y3: B] : ( Sa @ X3 @ ( G @ Y3 ) )
% 11.93/6.03 @ X
% 11.93/6.03 @ Y ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5465_Heap_Orel__map_I1_J,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ C @ B @ Sb @ ( heap_Time_map_Heap @ A @ C @ I @ X ) @ Y )
% 11.93/6.03 = ( heap_Time_rel_Heap @ A @ B
% 11.93/6.03 @ ^ [X3: A] : ( Sb @ ( I @ X3 ) )
% 11.93/6.03 @ X
% 11.93/6.03 @ Y ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5466_Heap_Orel__mono,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
% 11.93/6.03 ( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
% 11.93/6.03 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra2 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5467_Heap_Orel__transp,axiom,
% 11.93/6.03 ! [A: $tType,R5: A > A > $o] :
% 11.93/6.03 ( ( transp @ A @ R5 )
% 11.93/6.03 => ( transp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5468_Heap_Orel__refl,axiom,
% 11.93/6.03 ! [B: $tType,Ra2: B > B > $o,X: heap_Time_Heap @ B] :
% 11.93/6.03 ( ! [X2: B] : ( Ra2 @ X2 @ X2 )
% 11.93/6.03 => ( heap_Time_rel_Heap @ B @ B @ Ra2 @ X @ X ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5469_Heap_Orel__eq,axiom,
% 11.93/6.03 ! [A: $tType] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ A
% 11.93/6.03 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 11.93/6.03 = ( ^ [Y5: heap_Time_Heap @ A,Z4: heap_Time_Heap @ A] : ( Y5 = Z4 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5470_Heap_Orel__conversep,axiom,
% 11.93/6.03 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) )
% 11.93/6.03 = ( conversep @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5471_Heap_Orel__flip,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ B,B2: heap_Time_Heap @ A] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
% 11.93/6.03 = ( heap_Time_rel_Heap @ A @ B @ R5 @ B2 @ A3 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5472_Heap_Orel__compp,axiom,
% 11.93/6.03 ! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S2: B > C > $o] :
% 11.93/6.03 ( ( heap_Time_rel_Heap @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S2 ) )
% 11.93/6.03 = ( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ C ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ B @ C @ S2 ) ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5473_Heap_Orel__transfer,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
% 11.93/6.03 ( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ C ) > ( heap_Time_Heap @ D ) > $o )
% 11.93/6.03 @ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
% 11.93/6.03 @ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 11.93/6.03 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ C ) @ ( ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ D ) > $o ) @ ( heap_Time_rel_Heap @ A @ C @ Sa )
% 11.93/6.03 @ ( bNF_rel_fun @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ D ) @ $o @ $o @ ( heap_Time_rel_Heap @ B @ D @ Sc )
% 11.93/6.03 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) ) )
% 11.93/6.03 @ ( heap_Time_rel_Heap @ A @ B )
% 11.93/6.03 @ ( heap_Time_rel_Heap @ C @ D ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5474_Heap_Omap__transfer,axiom,
% 11.93/6.03 ! [A: $tType,B: $tType,F5: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F5 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F5 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E ) > ( heap_Time_Heap @ F5 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F5 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F5 ) @ ( heap_Time_rel_Heap @ A @ E @ Rb2 ) @ ( heap_Time_rel_Heap @ B @ F5 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E @ F5 ) ) ).
% 11.93/6.03
% 11.93/6.03 thf(fact_5475_those_Osimps_I2_J,axiom,
% 11.93/6.03 ! [A: $tType,X: option @ A,Xs: list @ ( option @ A )] :
% 11.93/6.03 ( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs ) )
% 11.93/6.03 = ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
% 11.93/6.03 @ ^ [Y3: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( those @ A @ Xs ) )
% 11.93/6.04 @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5476_Quotient__natural,axiom,
% 11.93/6.04 ( quotient @ nat @ code_natural
% 11.93/6.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 )
% 11.93/6.04 @ code_natural_of_nat
% 11.93/6.04 @ code_nat_of_natural
% 11.93/6.04 @ code_cr_natural ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5477_Quotient__integer,axiom,
% 11.93/6.04 ( quotient @ int @ code_integer
% 11.93/6.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 )
% 11.93/6.04 @ code_integer_of_int
% 11.93/6.04 @ code_int_of_integer
% 11.93/6.04 @ code_cr_integer ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5478_Heap_OQuotient,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
% 11.93/6.04 ( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T6 )
% 11.93/6.04 => ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T6 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5479_option_OQuotient,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
% 11.93/6.04 ( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T6 )
% 11.93/6.04 => ( quotient @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ A @ R5 ) @ ( map_option @ A @ B @ Abs ) @ ( map_option @ B @ A @ Rep ) @ ( rel_option @ A @ B @ T6 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5480_Quotient__compose,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,B: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,T1: A > B > $o,R23: B > B > $o,Abs22: B > C,Rep22: C > B,T22: B > C > $o] :
% 11.93/6.04 ( ( quotient @ A @ B @ R12 @ Abs1 @ Rep1 @ T1 )
% 11.93/6.04 => ( ( quotient @ B @ C @ R23 @ Abs22 @ Rep22 @ T22 )
% 11.93/6.04 => ( quotient @ A @ C @ ( relcompp @ A @ B @ A @ T1 @ ( relcompp @ B @ B @ A @ R23 @ ( conversep @ A @ B @ T1 ) ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) @ ( relcompp @ A @ B @ C @ T1 @ T22 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5481_Heap_Opred__transfer,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 11.93/6.04 ( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Time_Heap @ A ) > $o ) @ ( ( heap_Time_Heap @ B ) > $o )
% 11.93/6.04 @ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
% 11.93/6.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.04 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R5 )
% 11.93/6.04 @ ^ [Y5: $o,Z4: $o] : ( Y5 = Z4 ) )
% 11.93/6.04 @ ( heap_Time_pred_Heap @ A )
% 11.93/6.04 @ ( heap_Time_pred_Heap @ B ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5482_Heap_Orec__transfer,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S2: C > D > $o] :
% 11.93/6.04 ( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
% 11.93/6.04 @ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
% 11.93/6.04 @ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) ) )
% 11.93/6.04 @ S2 )
% 11.93/6.04 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S2 )
% 11.93/6.04 @ ( heap_Time_rec_Heap @ A @ C )
% 11.93/6.04 @ ( heap_Time_rec_Heap @ B @ D ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5483_rel__prod__inject,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A3: A,B2: C,C2: B,D3: D] :
% 11.93/6.04 ( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
% 11.93/6.04 = ( ( R12 @ A3 @ C2 )
% 11.93/6.04 & ( R23 @ B2 @ D3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5484_Heap_Opred__mono,axiom,
% 11.93/6.04 ! [A: $tType,P: A > $o,Pa: A > $o] :
% 11.93/6.04 ( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
% 11.93/6.04 => ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > $o ) @ ( heap_Time_pred_Heap @ A @ P ) @ ( heap_Time_pred_Heap @ A @ Pa ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5485_rel__prod_Ointros,axiom,
% 11.93/6.04 ! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A3: A,B2: B,R23: C > D > $o,C2: C,D3: D] :
% 11.93/6.04 ( ( R12 @ A3 @ B2 )
% 11.93/6.04 => ( ( R23 @ C2 @ D3 )
% 11.93/6.04 => ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( product_Pair @ B @ D @ B2 @ D3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5486_rel__prod_Osimps,axiom,
% 11.93/6.04 ! [C: $tType,D: $tType,B: $tType,A: $tType] :
% 11.93/6.04 ( ( basic_rel_prod @ A @ B @ C @ D )
% 11.93/6.04 = ( ^ [R13: A > B > $o,R24: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
% 11.93/6.04 ? [A7: A,B5: B,C3: C,D5: D] :
% 11.93/6.04 ( ( A12
% 11.93/6.04 = ( product_Pair @ A @ C @ A7 @ C3 ) )
% 11.93/6.04 & ( A23
% 11.93/6.04 = ( product_Pair @ B @ D @ B5 @ D5 ) )
% 11.93/6.04 & ( R13 @ A7 @ B5 )
% 11.93/6.04 & ( R24 @ C3 @ D5 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5487_rel__prod_Ocases,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
% 11.93/6.04 ( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ A1 @ A22 )
% 11.93/6.04 => ~ ! [A5: A,B3: B,C4: C] :
% 11.93/6.04 ( ( A1
% 11.93/6.04 = ( product_Pair @ A @ C @ A5 @ C4 ) )
% 11.93/6.04 => ! [D2: D] :
% 11.93/6.04 ( ( A22
% 11.93/6.04 = ( product_Pair @ B @ D @ B3 @ D2 ) )
% 11.93/6.04 => ( ( R12 @ A5 @ B3 )
% 11.93/6.04 => ~ ( R23 @ C4 @ D2 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5488_Heap_Opred__True,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap_Time_pred_Heap @ A
% 11.93/6.04 @ ^ [Uu: A] : $true )
% 11.93/6.04 = ( ^ [Uu: heap_Time_Heap @ A] : $true ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5489_Heap_Opred__mono__strong,axiom,
% 11.93/6.04 ! [A: $tType,P: A > $o,X: heap_Time_Heap @ A,Pa: A > $o] :
% 11.93/6.04 ( ( heap_Time_pred_Heap @ A @ P @ X )
% 11.93/6.04 => ( ! [Z3: A] :
% 11.93/6.04 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ X ) )
% 11.93/6.04 => ( ( P @ Z3 )
% 11.93/6.04 => ( Pa @ Z3 ) ) )
% 11.93/6.04 => ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5490_Heap_Opred__cong,axiom,
% 11.93/6.04 ! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
% 11.93/6.04 ( ( X = Ya )
% 11.93/6.04 => ( ! [Z3: A] :
% 11.93/6.04 ( ( member @ A @ Z3 @ ( heap_Time_set_Heap @ A @ Ya ) )
% 11.93/6.04 => ( ( P @ Z3 )
% 11.93/6.04 = ( Pa @ Z3 ) ) )
% 11.93/6.04 => ( ( heap_Time_pred_Heap @ A @ P @ X )
% 11.93/6.04 = ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5491_Heap_Octr__transfer,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R5: A > B > $o] :
% 11.93/6.04 ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
% 11.93/6.04 @ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) ) )
% 11.93/6.04 @ ( heap_Time_rel_Heap @ A @ B @ R5 )
% 11.93/6.04 @ ( heap_Time_Heap2 @ A )
% 11.93/6.04 @ ( heap_Time_Heap2 @ B ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5492_Heap_Orel__intros,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 11.93/6.04 @ X
% 11.93/6.04 @ Y )
% 11.93/6.04 => ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5493_Heap_Orel__inject,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y ) )
% 11.93/6.04 = ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 11.93/6.04 @ X
% 11.93/6.04 @ Y ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5494_Heap_Orel__induct,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y: heap_Time_Heap @ B,Q: ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o] :
% 11.93/6.04 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y )
% 11.93/6.04 => ( ! [A5: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),B3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 11.93/6.04 @ A5
% 11.93/6.04 @ B3 )
% 11.93/6.04 => ( Q @ ( heap_Time_Heap2 @ A @ A5 ) @ ( heap_Time_Heap2 @ B @ B3 ) ) )
% 11.93/6.04 => ( Q @ X @ Y ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5495_Heap_Orel__cases,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ A,B2: heap_Time_Heap @ B] :
% 11.93/6.04 ( ( heap_Time_rel_Heap @ A @ B @ R5 @ A3 @ B2 )
% 11.93/6.04 => ~ ! [Xa3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( A3
% 11.93/6.04 = ( heap_Time_Heap2 @ A @ Xa3 ) )
% 11.93/6.04 => ! [Ya2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( B2
% 11.93/6.04 = ( heap_Time_Heap2 @ B @ Ya2 ) )
% 11.93/6.04 => ~ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) )
% 11.93/6.04 @ Xa3
% 11.93/6.04 @ Ya2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5496_Heap_Omap__cong__pred,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
% 11.93/6.04 ( ( X = Ya )
% 11.93/6.04 => ( ( heap_Time_pred_Heap @ A
% 11.93/6.04 @ ^ [Z5: A] :
% 11.93/6.04 ( ( F @ Z5 )
% 11.93/6.04 = ( G @ Z5 ) )
% 11.93/6.04 @ Ya )
% 11.93/6.04 => ( ( heap_Time_map_Heap @ A @ B @ F @ X )
% 11.93/6.04 = ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5497_Heap_Orel__eq__onp,axiom,
% 11.93/6.04 ! [A: $tType,P: A > $o] :
% 11.93/6.04 ( ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P ) )
% 11.93/6.04 = ( bNF_eq_onp @ ( heap_Time_Heap @ A ) @ ( heap_Time_pred_Heap @ A @ P ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5498_Heap_Opred__rel,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap_Time_pred_Heap @ A )
% 11.93/6.04 = ( ^ [P4: A > $o,X3: heap_Time_Heap @ A] : ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P4 ) @ X3 @ X3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5499_Heap_Opred__set,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap_Time_pred_Heap @ A )
% 11.93/6.04 = ( ^ [P4: A > $o,X3: heap_Time_Heap @ A] :
% 11.93/6.04 ! [Y3: A] :
% 11.93/6.04 ( ( member @ A @ Y3 @ ( heap_Time_set_Heap @ A @ X3 ) )
% 11.93/6.04 => ( P4 @ Y3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5500_Pair__transfer,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B6: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A4 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B6 @ ( basic_rel_prod @ A @ B @ C @ D @ A4 @ B6 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5501_Heap_Opred__map,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: heap_Time_Heap @ A] :
% 11.93/6.04 ( ( heap_Time_pred_Heap @ B @ Q @ ( heap_Time_map_Heap @ A @ B @ F @ X ) )
% 11.93/6.04 = ( heap_Time_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5502_Heap_Ocase__transfer,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S2: C > D > $o] :
% 11.93/6.04 ( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
% 11.93/6.04 @ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
% 11.93/6.04 @ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Y5: heap_ext @ product_unit,Z4: heap_ext @ product_unit] : ( Y5 = Z4 )
% 11.93/6.04 @ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
% 11.93/6.04 @ ^ [Y5: product_prod @ ( heap_ext @ product_unit ) @ nat,Z4: product_prod @ ( heap_ext @ product_unit ) @ nat] : ( Y5 = Z4 ) ) ) )
% 11.93/6.04 @ S2 )
% 11.93/6.04 @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S2 )
% 11.93/6.04 @ ( heap_Time_case_Heap @ A @ C )
% 11.93/6.04 @ ( heap_Time_case_Heap @ B @ D ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5503_sndOp__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,C: $tType] :
% 11.93/6.04 ( ( bNF_sndOp @ C @ A @ B )
% 11.93/6.04 = ( ^ [P4: C > A > $o,Q6: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P4 @ Q6 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5504_fstOp__def,axiom,
% 11.93/6.04 ! [C: $tType,B: $tType,A: $tType] :
% 11.93/6.04 ( ( bNF_fstOp @ A @ B @ C )
% 11.93/6.04 = ( ^ [P4: A > B > $o,Q6: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P4 @ Q6 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5505_Heap_Ocase__distrib,axiom,
% 11.93/6.04 ! [C: $tType,B: $tType,A: $tType,H3: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
% 11.93/6.04 ( ( H3 @ ( heap_Time_case_Heap @ A @ B @ F @ Heap ) )
% 11.93/6.04 = ( heap_Time_case_Heap @ A @ C
% 11.93/6.04 @ ^ [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( H3 @ ( F @ X3 ) )
% 11.93/6.04 @ Heap ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5506_Heap_Ocase,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( heap_Time_case_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
% 11.93/6.04 = ( F @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5507_trancl__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( transitive_trancl @ A )
% 11.93/6.04 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( collect @ ( product_prod @ A @ A )
% 11.93/6.04 @ ( product_case_prod @ A @ A @ $o
% 11.93/6.04 @ ( transitive_tranclp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5508_Nitpick_Otranclp__unfold,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( transitive_tranclp @ A )
% 11.93/6.04 = ( ^ [R3: A > A > $o,A7: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5509_reflp__refl__eq,axiom,
% 11.93/6.04 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( reflp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.93/6.04 = ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5510_tranclp__induct2,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
% 11.93/6.04 ( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 11.93/6.04 => ( ! [A5: A,B3: B] :
% 11.93/6.04 ( ( R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.93/6.04 => ( P @ A5 @ B3 ) )
% 11.93/6.04 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 11.93/6.04 ( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.93/6.04 => ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
% 11.93/6.04 => ( ( P @ A5 @ B3 )
% 11.93/6.04 => ( P @ Aa3 @ Ba ) ) ) )
% 11.93/6.04 => ( P @ Bx @ By ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5511_Heap_Orel__reflp,axiom,
% 11.93/6.04 ! [A: $tType,R5: A > A > $o] :
% 11.93/6.04 ( ( reflp @ A @ R5 )
% 11.93/6.04 => ( reflp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5512_option_Orel__reflp,axiom,
% 11.93/6.04 ! [A: $tType,R5: A > A > $o] :
% 11.93/6.04 ( ( reflp @ A @ R5 )
% 11.93/6.04 => ( reflp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5513_tranclp__trancl__eq,axiom,
% 11.93/6.04 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( transitive_tranclp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.93/6.04 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5514_Lcm__fin__1__iff,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A4: set @ A] :
% 11.93/6.04 ( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
% 11.93/6.04 = ( one_one @ A ) )
% 11.93/6.04 = ( ! [X3: A] :
% 11.93/6.04 ( ( member @ A @ X3 @ A4 )
% 11.93/6.04 => ( dvd_dvd @ A @ X3 @ ( one_one @ A ) ) )
% 11.93/6.04 & ( finite_finite2 @ A @ A4 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5515_heap__interpretation,axiom,
% 11.93/6.04 ! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ ( heap_Time_Heap_lub @ A ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5516_old_Orec__bool__def,axiom,
% 11.93/6.04 ! [T: $tType] :
% 11.93/6.04 ( ( product_rec_bool @ T )
% 11.93/6.04 = ( ^ [F12: T,F23: T,X3: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F23 @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5517_old_Obool_Osimps_I6_J,axiom,
% 11.93/6.04 ! [T: $tType,F1: T,F22: T] :
% 11.93/6.04 ( ( product_rec_bool @ T @ F1 @ F22 @ $false )
% 11.93/6.04 = F22 ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5518_old_Obool_Osimps_I5_J,axiom,
% 11.93/6.04 ! [T: $tType,F1: T,F22: T] :
% 11.93/6.04 ( ( product_rec_bool @ T @ F1 @ F22 @ $true )
% 11.93/6.04 = F1 ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5519_Lcm__fin_Oempty,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
% 11.93/6.04 = ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5520_is__unit__Lcm__fin__iff,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A4: set @ A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) @ ( one_one @ A ) )
% 11.93/6.04 = ( ( semiring_gcd_Lcm_fin @ A @ A4 )
% 11.93/6.04 = ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5521_Lcm__fin_Oeq__fold,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( ( semiring_gcd_Lcm_fin @ A )
% 11.93/6.04 = ( ^ [A9: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A9 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A9 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5522_Lcm__fin__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( ( semiring_gcd_Lcm_fin @ A )
% 11.93/6.04 = ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5523_fun_Opred__map,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
% 11.93/6.04 ( ( basic_pred_fun @ D @ B
% 11.93/6.04 @ ^ [Uu: D] : $true
% 11.93/6.04 @ Q
% 11.93/6.04 @ ( comp @ A @ B @ D @ F @ X ) )
% 11.93/6.04 = ( basic_pred_fun @ D @ A
% 11.93/6.04 @ ^ [Uu: D] : $true
% 11.93/6.04 @ ( comp @ B @ $o @ A @ Q @ F )
% 11.93/6.04 @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5524_lcm__eq__1__iff,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A,B2: A] :
% 11.93/6.04 ( ( ( gcd_lcm @ A @ A3 @ B2 )
% 11.93/6.04 = ( one_one @ A ) )
% 11.93/6.04 = ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 & ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5525_unit__factor__lcm,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A,B2: A] :
% 11.93/6.04 ( ( ( ( A3
% 11.93/6.04 = ( zero_zero @ A ) )
% 11.93/6.04 | ( B2
% 11.93/6.04 = ( zero_zero @ A ) ) )
% 11.93/6.04 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 11.93/6.04 = ( zero_zero @ A ) ) )
% 11.93/6.04 & ( ~ ( ( A3
% 11.93/6.04 = ( zero_zero @ A ) )
% 11.93/6.04 | ( B2
% 11.93/6.04 = ( zero_zero @ A ) ) )
% 11.93/6.04 => ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
% 11.93/6.04 = ( one_one @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5526_fun_Omap__cong__pred,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
% 11.93/6.04 ( ( X = Ya )
% 11.93/6.04 => ( ( basic_pred_fun @ D @ A
% 11.93/6.04 @ ^ [Uu: D] : $true
% 11.93/6.04 @ ^ [Z5: A] :
% 11.93/6.04 ( ( F @ Z5 )
% 11.93/6.04 = ( G @ Z5 ) )
% 11.93/6.04 @ Ya )
% 11.93/6.04 => ( ( comp @ A @ B @ D @ F @ X )
% 11.93/6.04 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5527_lcm__mult__unit2,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A,B2: A,C2: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
% 11.93/6.04 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5528_lcm__mult__unit1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A,B2: A,C2: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
% 11.93/6.04 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5529_lcm__div__unit1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A,B2: A,C2: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
% 11.93/6.04 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5530_lcm__div__unit2,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A,B2: A,C2: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
% 11.93/6.04 = ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5531_Lcm__nat__def,axiom,
% 11.93/6.04 ( ( gcd_Lcm @ nat )
% 11.93/6.04 = ( ^ [M8: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M8 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M8 ) @ ( zero_zero @ nat ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5532_Heap_Opred__inject,axiom,
% 11.93/6.04 ! [A: $tType,P: A > $o,A3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
% 11.93/6.04 ( ( heap_Time_pred_Heap @ A @ P @ ( heap_Time_Heap2 @ A @ A3 ) )
% 11.93/6.04 = ( basic_pred_fun @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
% 11.93/6.04 @ ^ [Uu: heap_ext @ product_unit] : $true
% 11.93/6.04 @ ( pred_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( basic_pred_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ P @ ( top_top @ ( ( product_prod @ ( heap_ext @ product_unit ) @ nat ) > $o ) ) ) )
% 11.93/6.04 @ A3 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5533_Lcm__fin_Oset__eq__fold,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [Xs: list @ A] :
% 11.93/6.04 ( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
% 11.93/6.04 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5534_pred__prod__inject,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P14: A > $o,P25: B > $o,A3: A,B2: B] :
% 11.93/6.04 ( ( basic_pred_prod @ A @ B @ P14 @ P25 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
% 11.93/6.04 = ( ( P14 @ A3 )
% 11.93/6.04 & ( P25 @ B2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5535_fold__append,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,F: B > A > A,Xs: list @ B,Ys2: list @ B] :
% 11.93/6.04 ( ( fold @ B @ A @ F @ ( append @ B @ Xs @ Ys2 ) )
% 11.93/6.04 = ( comp @ A @ A @ A @ ( fold @ B @ A @ F @ Ys2 ) @ ( fold @ B @ A @ F @ Xs ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5536_pred__prod__split,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,P: $o > $o,Q: A > $o,R5: B > $o,Xy2: product_prod @ A @ B] :
% 11.93/6.04 ( ( P @ ( basic_pred_prod @ A @ B @ Q @ R5 @ Xy2 ) )
% 11.93/6.04 = ( ! [X3: A,Y3: B] :
% 11.93/6.04 ( ( Xy2
% 11.93/6.04 = ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 11.93/6.04 => ( P
% 11.93/6.04 @ ( ( Q @ X3 )
% 11.93/6.04 & ( R5 @ Y3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5537_pred__prod_Ocases,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P14: A > $o,P25: B > $o,A3: product_prod @ A @ B] :
% 11.93/6.04 ( ( basic_pred_prod @ A @ B @ P14 @ P25 @ A3 )
% 11.93/6.04 => ~ ! [A5: A,B3: B] :
% 11.93/6.04 ( ( A3
% 11.93/6.04 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.93/6.04 => ( ( P14 @ A5 )
% 11.93/6.04 => ~ ( P25 @ B3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5538_pred__prod_Osimps,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( basic_pred_prod @ A @ B )
% 11.93/6.04 = ( ^ [P12: A > $o,P23: B > $o,A7: product_prod @ A @ B] :
% 11.93/6.04 ? [B5: A,C3: B] :
% 11.93/6.04 ( ( A7
% 11.93/6.04 = ( product_Pair @ A @ B @ B5 @ C3 ) )
% 11.93/6.04 & ( P12 @ B5 )
% 11.93/6.04 & ( P23 @ C3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5539_pred__prod_Ointros,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P14: A > $o,A3: A,P25: B > $o,B2: B] :
% 11.93/6.04 ( ( P14 @ A3 )
% 11.93/6.04 => ( ( P25 @ B2 )
% 11.93/6.04 => ( basic_pred_prod @ A @ B @ P14 @ P25 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5540_fold__Cons,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,F: A > B > B,X: A,Xs: list @ A] :
% 11.93/6.04 ( ( fold @ A @ B @ F @ ( cons @ A @ X @ Xs ) )
% 11.93/6.04 = ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ Xs ) @ ( F @ X ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5541_fold__commute__apply,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H3: B > C,G: A > B > B,F: A > C > C,S: B] :
% 11.93/6.04 ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( comp @ B @ C @ B @ H3 @ ( G @ X2 ) )
% 11.93/6.04 = ( comp @ C @ C @ B @ ( F @ X2 ) @ H3 ) ) )
% 11.93/6.04 => ( ( H3 @ ( fold @ A @ B @ G @ Xs @ S ) )
% 11.93/6.04 = ( fold @ A @ C @ F @ Xs @ ( H3 @ S ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5542_fold__commute,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H3: B > C,G: A > B > B,F: A > C > C] :
% 11.93/6.04 ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( comp @ B @ C @ B @ H3 @ ( G @ X2 ) )
% 11.93/6.04 = ( comp @ C @ C @ B @ ( F @ X2 ) @ H3 ) ) )
% 11.93/6.04 => ( ( comp @ B @ C @ B @ H3 @ ( fold @ A @ B @ G @ Xs ) )
% 11.93/6.04 = ( comp @ C @ C @ B @ ( fold @ A @ C @ F @ Xs ) @ H3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5543_Lcm__nat__set__eq__fold,axiom,
% 11.93/6.04 ! [Xs: list @ nat] :
% 11.93/6.04 ( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
% 11.93/6.04 = ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5544_fold__multiset__equiv,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,Ys2: list @ A] :
% 11.93/6.04 ( ! [X2: A,Y4: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) )
% 11.93/6.04 = ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) )
% 11.93/6.04 => ( ( ( mset @ A @ Xs )
% 11.93/6.04 = ( mset @ A @ Ys2 ) )
% 11.93/6.04 => ( ( fold @ A @ B @ F @ Xs )
% 11.93/6.04 = ( fold @ A @ B @ F @ Ys2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5545_fold__rev,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
% 11.93/6.04 ( ! [X2: A,Y4: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 11.93/6.04 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) )
% 11.93/6.04 => ( ( fold @ A @ B @ F @ ( rev @ A @ Xs ) )
% 11.93/6.04 = ( fold @ A @ B @ F @ Xs ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5546_foldr__fold,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B] :
% 11.93/6.04 ( ! [X2: A,Y4: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 11.93/6.04 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) ) ) )
% 11.93/6.04 => ( ( foldr @ A @ B @ F @ Xs )
% 11.93/6.04 = ( fold @ A @ B @ F @ Xs ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5547_fold__plus__sum__list__rev,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( monoid_add @ A )
% 11.93/6.04 => ! [Xs: list @ A] :
% 11.93/6.04 ( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
% 11.93/6.04 = ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5548_fold__remove1__split,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,Xs: list @ A,F: A > B > B,X: A] :
% 11.93/6.04 ( ! [X2: A,Y4: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( member @ A @ Y4 @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) )
% 11.93/6.04 = ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) ) ) )
% 11.93/6.04 => ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
% 11.93/6.04 => ( ( fold @ A @ B @ F @ Xs )
% 11.93/6.04 = ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ ( remove1 @ A @ X @ Xs ) ) @ ( F @ X ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5549_Lcm__set__eq__fold,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_Gcd @ A )
% 11.93/6.04 => ! [Xs: list @ A] :
% 11.93/6.04 ( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
% 11.93/6.04 = ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5550_subset__mset_Omax__add__distrib__left,axiom,
% 11.93/6.04 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 11.93/6.04 ( ( plus_plus @ ( multiset @ A ) @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X @ Y ) @ Z2 )
% 11.93/6.04 = ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5551_subset__mset_Omax__add__distrib__right,axiom,
% 11.93/6.04 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 11.93/6.04 ( ( plus_plus @ ( multiset @ A ) @ X @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
% 11.93/6.04 = ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5552_prod__list_Ocomm__monoid__list__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_mult @ A )
% 11.93/6.04 => ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5553_sum__list_Ocomm__monoid__list__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_add @ A )
% 11.93/6.04 => ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5554_lazI,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
% 11.93/6.04 ( ( ( size_size @ ( list @ A ) @ A3 )
% 11.93/6.04 = ( size_size @ ( list @ B ) @ B2 ) )
% 11.93/6.04 => ( ! [I2: nat] :
% 11.93/6.04 ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ B2 ) )
% 11.93/6.04 => ( P @ ( nth @ A @ A3 @ I2 ) @ ( nth @ B @ B2 @ I2 ) ) )
% 11.93/6.04 => ( list_all_zip @ A @ B @ P @ A3 @ B2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5555_list__all__zip__alt,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( list_all_zip @ A @ B )
% 11.93/6.04 = ( ^ [P4: A > B > $o,As5: list @ A,Bs3: list @ B] :
% 11.93/6.04 ( ( ( size_size @ ( list @ A ) @ As5 )
% 11.93/6.04 = ( size_size @ ( list @ B ) @ Bs3 ) )
% 11.93/6.04 & ! [I3: nat] :
% 11.93/6.04 ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ As5 ) )
% 11.93/6.04 => ( P4 @ ( nth @ A @ As5 @ I3 ) @ ( nth @ B @ Bs3 @ I3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5556_is__none__bind,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A )] :
% 11.93/6.04 ( ( is_none @ A @ ( bind @ B @ A @ F @ G ) )
% 11.93/6.04 = ( ( is_none @ B @ F )
% 11.93/6.04 | ( is_none @ A @ ( G @ ( the2 @ B @ F ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5557_laz__conj,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,A3: list @ A,B2: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B
% 11.93/6.04 @ ^ [X3: A,Y3: B] :
% 11.93/6.04 ( ( P @ X3 @ Y3 )
% 11.93/6.04 & ( Q @ X3 @ Y3 ) )
% 11.93/6.04 @ A3
% 11.93/6.04 @ B2 )
% 11.93/6.04 = ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ Q @ A3 @ B2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5558_is__none__code_I2_J,axiom,
% 11.93/6.04 ! [B: $tType,X: B] :
% 11.93/6.04 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5559_is__none__code_I1_J,axiom,
% 11.93/6.04 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5560_is__none__map__option,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,F: B > A,X: option @ B] :
% 11.93/6.04 ( ( is_none @ A @ ( map_option @ B @ A @ F @ X ) )
% 11.93/6.04 = ( is_none @ B @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5561_laz__weak__Pb,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: B > $o,A4: list @ A,B6: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B
% 11.93/6.04 @ ^ [A7: A] : P
% 11.93/6.04 @ A4
% 11.93/6.04 @ B6 )
% 11.93/6.04 = ( ( ( size_size @ ( list @ A ) @ A4 )
% 11.93/6.04 = ( size_size @ ( list @ B ) @ B6 ) )
% 11.93/6.04 & ! [X3: B] :
% 11.93/6.04 ( ( member @ B @ X3 @ ( set2 @ B @ B6 ) )
% 11.93/6.04 => ( P @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5562_laz__weak__Pa,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,P: A > $o,A4: list @ A,B6: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B
% 11.93/6.04 @ ^ [A7: A,B5: B] : ( P @ A7 )
% 11.93/6.04 @ A4
% 11.93/6.04 @ B6 )
% 11.93/6.04 = ( ( ( size_size @ ( list @ A ) @ A4 )
% 11.93/6.04 = ( size_size @ ( list @ B ) @ B6 ) )
% 11.93/6.04 & ! [X3: A] :
% 11.93/6.04 ( ( member @ A @ X3 @ ( set2 @ A @ A4 ) )
% 11.93/6.04 => ( P @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5563_list__all__zip_Osimps_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: A > B > $o] : ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( nil @ B ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5564_laz__eq,axiom,
% 11.93/6.04 ! [A: $tType,A3: list @ A,B2: list @ A] :
% 11.93/6.04 ( ( list_all_zip @ A @ A
% 11.93/6.04 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 )
% 11.93/6.04 @ A3
% 11.93/6.04 @ B2 )
% 11.93/6.04 = ( A3 = B2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5565_laz__len,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: A > B > $o,A3: list @ A,B2: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B @ P @ A3 @ B2 )
% 11.93/6.04 => ( ( size_size @ ( list @ A ) @ A3 )
% 11.93/6.04 = ( size_size @ ( list @ B ) @ B2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5566_laz__swap__ex,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,C: $tType,P: A > B > C > $o,A4: list @ A,B6: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B
% 11.93/6.04 @ ^ [A7: A,B5: B] :
% 11.93/6.04 ? [X8: C] : ( P @ A7 @ B5 @ X8 )
% 11.93/6.04 @ A4
% 11.93/6.04 @ B6 )
% 11.93/6.04 => ~ ! [C8: list @ C] :
% 11.93/6.04 ( ( list_all_zip @ A @ C
% 11.93/6.04 @ ^ [A7: A,C3: C] :
% 11.93/6.04 ? [B5: B] : ( P @ A7 @ B5 @ C3 )
% 11.93/6.04 @ A4
% 11.93/6.04 @ C8 )
% 11.93/6.04 => ~ ( list_all_zip @ B @ C
% 11.93/6.04 @ ^ [B5: B,C3: C] :
% 11.93/6.04 ? [A7: A] : ( P @ A7 @ B5 @ C3 )
% 11.93/6.04 @ B6
% 11.93/6.04 @ C8 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5567_is__none__simps_I2_J,axiom,
% 11.93/6.04 ! [B: $tType,X: B] :
% 11.93/6.04 ~ ( is_none @ B @ ( some @ B @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5568_list__all__zip__map2,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As2: list @ A,F: C > B,Bs2: list @ C] :
% 11.93/6.04 ( ( list_all_zip @ A @ B @ P @ As2 @ ( map @ C @ B @ F @ Bs2 ) )
% 11.93/6.04 = ( list_all_zip @ A @ C
% 11.93/6.04 @ ^ [A7: A,B5: C] : ( P @ A7 @ ( F @ B5 ) )
% 11.93/6.04 @ As2
% 11.93/6.04 @ Bs2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5569_list__all__zip__map1,axiom,
% 11.93/6.04 ! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As2: list @ C,Bs2: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B @ P @ ( map @ C @ A @ F @ As2 ) @ Bs2 )
% 11.93/6.04 = ( list_all_zip @ C @ B
% 11.93/6.04 @ ^ [A7: C] : ( P @ ( F @ A7 ) )
% 11.93/6.04 @ As2
% 11.93/6.04 @ Bs2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5570_is__none__simps_I1_J,axiom,
% 11.93/6.04 ! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5571_Option_Ois__none__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( is_none @ A )
% 11.93/6.04 = ( ^ [X3: option @ A] :
% 11.93/6.04 ( X3
% 11.93/6.04 = ( none @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5572_list__all__zip_Osimps_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: A > B > $o,A3: A,As2: list @ A,B2: B,Bs2: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B @ P @ ( cons @ A @ A3 @ As2 ) @ ( cons @ B @ B2 @ Bs2 ) )
% 11.93/6.04 = ( ( P @ A3 @ B2 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ P @ As2 @ Bs2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5573_list__all__zip_Oelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 11.93/6.04 ( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 11.93/6.04 => ( ! [A5: A,As: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ A5 @ As ) )
% 11.93/6.04 => ! [B3: B,Bs: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ B3 @ Bs ) )
% 11.93/6.04 => ( ( X @ A5 @ B3 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) )
% 11.93/6.04 => ( ( ? [V4: A,Va: list @ A] :
% 11.93/6.04 ( Xa
% 11.93/6.04 = ( cons @ A @ V4 @ Va ) )
% 11.93/6.04 => ( Xb
% 11.93/6.04 != ( nil @ B ) ) )
% 11.93/6.04 => ~ ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ! [V4: B,Va: list @ B] :
% 11.93/6.04 ( Xb
% 11.93/6.04 != ( cons @ B @ V4 @ Va ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5574_list__all__zip_Oelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ( Xb
% 11.93/6.04 != ( nil @ B ) ) )
% 11.93/6.04 => ~ ! [A5: A,As: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ A5 @ As ) )
% 11.93/6.04 => ! [B3: B,Bs: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ B3 @ Bs ) )
% 11.93/6.04 => ~ ( ( X @ A5 @ B3 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5575_list__all__zip_Oelims_I1_J,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
% 11.93/6.04 ( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ( ( Xb
% 11.93/6.04 = ( nil @ B ) )
% 11.93/6.04 => ~ Y ) )
% 11.93/6.04 => ( ! [A5: A,As: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ A5 @ As ) )
% 11.93/6.04 => ! [B3: B,Bs: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ B3 @ Bs ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ( ( X @ A5 @ B3 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) )
% 11.93/6.04 => ( ( ? [V4: A,Va: list @ A] :
% 11.93/6.04 ( Xa
% 11.93/6.04 = ( cons @ A @ V4 @ Va ) )
% 11.93/6.04 => ( ( Xb
% 11.93/6.04 = ( nil @ B ) )
% 11.93/6.04 => Y ) )
% 11.93/6.04 => ~ ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ( ? [V4: B,Va: list @ B] :
% 11.93/6.04 ( Xb
% 11.93/6.04 = ( cons @ B @ V4 @ Va ) )
% 11.93/6.04 => Y ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5576_list__all__zip_Osimps_I4_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: A > B > $o,V: B,Va2: list @ B] :
% 11.93/6.04 ~ ( list_all_zip @ A @ B @ P @ ( nil @ A ) @ ( cons @ B @ V @ Va2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5577_list__all__zip_Osimps_I3_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: A > B > $o,V: A,Va2: list @ A] :
% 11.93/6.04 ~ ( list_all_zip @ A @ B @ P @ ( cons @ A @ V @ Va2 ) @ ( nil @ B ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5578_the__map__option,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: option @ A,F: A > B] :
% 11.93/6.04 ( ~ ( is_none @ A @ X )
% 11.93/6.04 => ( ( the2 @ B @ ( map_option @ A @ B @ F @ X ) )
% 11.93/6.04 = ( F @ ( the2 @ A @ X ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5579_rel__optionI,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: option @ A,Y: option @ B,P: A > B > $o] :
% 11.93/6.04 ( ( ( is_none @ A @ X )
% 11.93/6.04 = ( is_none @ B @ Y ) )
% 11.93/6.04 => ( ( ~ ( is_none @ A @ X )
% 11.93/6.04 => ( ~ ( is_none @ B @ Y )
% 11.93/6.04 => ( P @ ( the2 @ A @ X ) @ ( the2 @ B @ Y ) ) ) )
% 11.93/6.04 => ( rel_option @ A @ B @ P @ X @ Y ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5580_rel__option__unfold,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( rel_option @ A @ B )
% 11.93/6.04 = ( ^ [R6: A > B > $o,X3: option @ A,Y3: option @ B] :
% 11.93/6.04 ( ( ( is_none @ A @ X3 )
% 11.93/6.04 = ( is_none @ B @ Y3 ) )
% 11.93/6.04 & ( ~ ( is_none @ A @ X3 )
% 11.93/6.04 => ( ~ ( is_none @ B @ Y3 )
% 11.93/6.04 => ( R6 @ ( the2 @ A @ X3 ) @ ( the2 @ B @ Y3 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5581_list__all__zip_Opelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B,Y: $o] :
% 11.93/6.04 ( ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ( ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ( ( Xb
% 11.93/6.04 = ( nil @ B ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
% 11.93/6.04 => ( ! [A5: A,As: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ A5 @ As ) )
% 11.93/6.04 => ! [B3: B,Bs: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ B3 @ Bs ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ( X @ A5 @ B3 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) ) ) ) )
% 11.93/6.04 => ( ! [V4: A,Va: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ V4 @ Va ) )
% 11.93/6.04 => ( ( Xb
% 11.93/6.04 = ( nil @ B ) )
% 11.93/6.04 => ( ~ Y
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
% 11.93/6.04 => ~ ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ! [V4: B,Va: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ V4 @ Va ) )
% 11.93/6.04 => ( ~ Y
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5582_list__all__zip_Opelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 11.93/6.04 ( ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ( ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ( ( Xb
% 11.93/6.04 = ( nil @ B ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) )
% 11.93/6.04 => ~ ! [A5: A,As: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ A5 @ As ) )
% 11.93/6.04 => ! [B3: B,Bs: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ B3 @ Bs ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 11.93/6.04 => ~ ( ( X @ A5 @ B3 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5583_list__all__zip_Opelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: A > B > $o,Xa: list @ A,Xb: list @ B] :
% 11.93/6.04 ( ~ ( list_all_zip @ A @ B @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ( ! [A5: A,As: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ A5 @ As ) )
% 11.93/6.04 => ! [B3: B,Bs: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ B3 @ Bs ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A5 @ As ) @ ( cons @ B @ B3 @ Bs ) ) ) )
% 11.93/6.04 => ( ( X @ A5 @ B3 )
% 11.93/6.04 & ( list_all_zip @ A @ B @ X @ As @ Bs ) ) ) ) )
% 11.93/6.04 => ( ! [V4: A,Va: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ V4 @ Va ) )
% 11.93/6.04 => ( ( Xb
% 11.93/6.04 = ( nil @ B ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V4 @ Va ) @ ( nil @ B ) ) ) ) ) )
% 11.93/6.04 => ~ ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ! [V4: B,Va: list @ B] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( cons @ B @ V4 @ Va ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( list_all_zip_rel @ A @ B ) @ ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V4 @ Va ) ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5584_plus__multiset_Otransfer,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
% 11.93/6.04 @ ( pcr_multiset @ A @ A
% 11.93/6.04 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 11.93/6.04 @ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
% 11.93/6.04 @ ( pcr_multiset @ A @ A
% 11.93/6.04 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) )
% 11.93/6.04 @ ( pcr_multiset @ A @ A
% 11.93/6.04 @ ^ [Y5: A,Z4: A] : ( Y5 = Z4 ) ) )
% 11.93/6.04 @ ^ [M8: A > nat,N9: A > nat,A7: A] : ( plus_plus @ nat @ ( M8 @ A7 ) @ ( N9 @ A7 ) )
% 11.93/6.04 @ ( plus_plus @ ( multiset @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5585_map__option__o__case__sum,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,D: $tType,B: $tType,F: D > C,G: A > ( option @ D ),H3: B > ( option @ D )] :
% 11.93/6.04 ( ( comp @ ( option @ D ) @ ( option @ C ) @ ( sum_sum @ A @ B ) @ ( map_option @ D @ C @ F ) @ ( sum_case_sum @ A @ ( option @ D ) @ B @ G @ H3 ) )
% 11.93/6.04 = ( sum_case_sum @ A @ ( option @ C ) @ B @ ( comp @ ( option @ D ) @ ( option @ C ) @ A @ ( map_option @ D @ C @ F ) @ G ) @ ( comp @ ( option @ D ) @ ( option @ C ) @ B @ ( map_option @ D @ C @ F ) @ H3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5586_semilattice__set_Oeq__fold_H,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,A4: set @ A] :
% 11.93/6.04 ( ( lattic149705377957585745ce_set @ A @ F )
% 11.93/6.04 => ( ( lattic1715443433743089157tice_F @ A @ F @ A4 )
% 11.93/6.04 = ( the2 @ A
% 11.93/6.04 @ ( finite_fold @ A @ ( option @ A )
% 11.93/6.04 @ ^ [X3: A,Y3: option @ A] : ( some @ A @ ( case_option @ A @ A @ X3 @ ( F @ X3 ) @ Y3 ) )
% 11.93/6.04 @ ( none @ A )
% 11.93/6.04 @ A4 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5587_o__case__sum,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,D: $tType,B: $tType,H3: D > C,F: A > D,G: B > D] :
% 11.93/6.04 ( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H3 @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
% 11.93/6.04 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H3 @ F ) @ ( comp @ D @ C @ B @ H3 @ G ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5588_semilattice__set_Oinfinite,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,A4: set @ A] :
% 11.93/6.04 ( ( lattic149705377957585745ce_set @ A @ F )
% 11.93/6.04 => ( ~ ( finite_finite2 @ A @ A4 )
% 11.93/6.04 => ( ( lattic1715443433743089157tice_F @ A @ F @ A4 )
% 11.93/6.04 = ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5589_group_Oaxioms_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 11.93/6.04 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.93/6.04 => ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5590_normalize__div,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
% 11.93/6.04 = ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5591_strict__mono__inv,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType] :
% 11.93/6.04 ( ( ( linorder @ B )
% 11.93/6.04 & ( linorder @ A ) )
% 11.93/6.04 => ! [F: A > B,G: B > A] :
% 11.93/6.04 ( ( order_strict_mono @ A @ B @ F )
% 11.93/6.04 => ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 11.93/6.04 = ( top_top @ ( set @ B ) ) )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( G @ ( F @ X2 ) )
% 11.93/6.04 = X2 )
% 11.93/6.04 => ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5592_gcd_Onormalize__bottom,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 11.93/6.04 = ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5593_normalize__1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
% 11.93/6.04 = ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5594_lcm_Otop__left__normalize,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
% 11.93/6.04 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5595_lcm_Otop__right__normalize,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5596_normalize__mult__unit__right,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [B2: A,A3: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 11.93/6.04 = ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5597_normalize__mult__unit__left,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A,B2: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
% 11.93/6.04 = ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5598_unit__factor__normalize,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( A3
% 11.93/6.04 != ( zero_zero @ A ) )
% 11.93/6.04 => ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
% 11.93/6.04 = ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5599_normalize__unit__factor,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( A3
% 11.93/6.04 != ( zero_zero @ A ) )
% 11.93/6.04 => ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
% 11.93/6.04 = ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5600_associated__unit,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A,B2: A] :
% 11.93/6.04 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 11.93/6.04 = ( normal6383669964737779283malize @ A @ B2 ) )
% 11.93/6.04 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5601_normalize__1__iff,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 11.93/6.04 = ( one_one @ A ) )
% 11.93/6.04 = ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5602_is__unit__normalize,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 => ( ( normal6383669964737779283malize @ A @ A3 )
% 11.93/6.04 = ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5603_normalize__idem__imp__is__unit__iff,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( normal8620421768224518004emidom @ A )
% 11.93/6.04 => ! [A3: A] :
% 11.93/6.04 ( ( ( normal6383669964737779283malize @ A @ A3 )
% 11.93/6.04 = A3 )
% 11.93/6.04 => ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
% 11.93/6.04 = ( A3
% 11.93/6.04 = ( one_one @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5604_group__axioms__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( group_axioms @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A,Z5: A,Inverse2: A > A] :
% 11.93/6.04 ( ! [A7: A] :
% 11.93/6.04 ( ( F2 @ Z5 @ A7 )
% 11.93/6.04 = A7 )
% 11.93/6.04 & ! [A7: A] :
% 11.93/6.04 ( ( F2 @ ( Inverse2 @ A7 ) @ A7 )
% 11.93/6.04 = Z5 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5605_group__axioms_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 11.93/6.04 ( ! [A5: A] :
% 11.93/6.04 ( ( F @ Z2 @ A5 )
% 11.93/6.04 = A5 )
% 11.93/6.04 => ( ! [A5: A] :
% 11.93/6.04 ( ( F @ ( Inverse @ A5 ) @ A5 )
% 11.93/6.04 = Z2 )
% 11.93/6.04 => ( group_axioms @ A @ F @ Z2 @ Inverse ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5606_strict__mono__imp__inj__on,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( ( linorder @ A )
% 11.93/6.04 & ( order @ B ) )
% 11.93/6.04 => ! [F: A > B,A4: set @ A] :
% 11.93/6.04 ( ( order_strict_mono @ A @ B @ F )
% 11.93/6.04 => ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5607_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5608_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5609_lcm_Obounded__quasi__semilattice__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5610_gcd_Obounded__quasi__semilattice__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semiring_gcd @ A )
% 11.93/6.04 => ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5611_group__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( group @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A,Z5: A,Inverse2: A > A] :
% 11.93/6.04 ( ( semigroup @ A @ F2 )
% 11.93/6.04 & ( group_axioms @ A @ F2 @ Z5 @ Inverse2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5612_group_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 11.93/6.04 ( ( semigroup @ A @ F )
% 11.93/6.04 => ( ( group_axioms @ A @ F @ Z2 @ Inverse )
% 11.93/6.04 => ( group @ A @ F @ Z2 @ Inverse ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5613_Predicate_Oiterate__upto_Opsimps,axiom,
% 11.93/6.04 ! [A: $tType,F: code_natural > A,N: code_natural,M2: code_natural] :
% 11.93/6.04 ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F @ ( product_Pair @ code_natural @ code_natural @ N @ M2 ) ) )
% 11.93/6.04 => ( ( iterate_upto @ A @ F @ N @ M2 )
% 11.93/6.04 = ( seq2 @ A
% 11.93/6.04 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M2 @ N ) @ ( empty @ A ) @ ( insert @ A @ ( F @ N ) @ ( iterate_upto @ A @ F @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5614_group_Oaxioms_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,Inverse: A > A] :
% 11.93/6.04 ( ( group @ A @ F @ Z2 @ Inverse )
% 11.93/6.04 => ( semigroup @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5615_monoid_Oaxioms_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ( monoid @ A @ F @ Z2 )
% 11.93/6.04 => ( semigroup @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5616_semigroup__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semigroup @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A] :
% 11.93/6.04 ! [A7: A,B5: A,C3: A] :
% 11.93/6.04 ( ( F2 @ ( F2 @ A7 @ B5 ) @ C3 )
% 11.93/6.04 = ( F2 @ A7 @ ( F2 @ B5 @ C3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5617_semigroup_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A] :
% 11.93/6.04 ( ! [A5: A,B3: A,C4: A] :
% 11.93/6.04 ( ( F @ ( F @ A5 @ B3 ) @ C4 )
% 11.93/6.04 = ( F @ A5 @ ( F @ B3 @ C4 ) ) )
% 11.93/6.04 => ( semigroup @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5618_semigroup_Oassoc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,A3: A,B2: A,C2: A] :
% 11.93/6.04 ( ( semigroup @ A @ F )
% 11.93/6.04 => ( ( F @ ( F @ A3 @ B2 ) @ C2 )
% 11.93/6.04 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5619_mult_Osemigroup__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semigroup_mult @ A )
% 11.93/6.04 => ( semigroup @ A @ ( times_times @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5620_Predicate_Oiterate__upto_Osimps,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( iterate_upto @ A )
% 11.93/6.04 = ( ^ [F2: code_natural > A,N5: code_natural,M: code_natural] :
% 11.93/6.04 ( seq2 @ A
% 11.93/6.04 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M @ N5 ) @ ( empty @ A ) @ ( insert @ A @ ( F2 @ N5 ) @ ( iterate_upto @ A @ F2 @ ( plus_plus @ code_natural @ N5 @ ( one_one @ code_natural ) ) @ M ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5621_Predicate_Oiterate__upto_Oelims,axiom,
% 11.93/6.04 ! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
% 11.93/6.04 ( ( ( iterate_upto @ A @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( seq2 @ A
% 11.93/6.04 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5622_add_Osemigroup__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( semigroup_add @ A )
% 11.93/6.04 => ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5623_Predicate_Oiterate__upto_Opelims,axiom,
% 11.93/6.04 ! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y: pred @ A] :
% 11.93/6.04 ( ( ( iterate_upto @ A @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ( ( Y
% 11.93/6.04 = ( seq2 @ A
% 11.93/6.04 @ ^ [U2: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5624_Random__Pred_Oiterate__upto__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( random_iterate_upto @ A )
% 11.93/6.04 = ( ^ [F2: code_natural > A,N5: code_natural,M: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F2 @ N5 @ M ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5625_monoid_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ( semigroup @ A @ F )
% 11.93/6.04 => ( ( monoid_axioms @ A @ F @ Z2 )
% 11.93/6.04 => ( monoid @ A @ F @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5626_monoid__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( monoid @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A,Z5: A] :
% 11.93/6.04 ( ( semigroup @ A @ F2 )
% 11.93/6.04 & ( monoid_axioms @ A @ F2 @ Z5 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5627_monoid__axioms_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ! [A5: A] :
% 11.93/6.04 ( ( F @ Z2 @ A5 )
% 11.93/6.04 = A5 )
% 11.93/6.04 => ( ! [A5: A] :
% 11.93/6.04 ( ( F @ A5 @ Z2 )
% 11.93/6.04 = A5 )
% 11.93/6.04 => ( monoid_axioms @ A @ F @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5628_monoid__axioms__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( monoid_axioms @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A,Z5: A] :
% 11.93/6.04 ( ! [A7: A] :
% 11.93/6.04 ( ( F2 @ Z5 @ A7 )
% 11.93/6.04 = A7 )
% 11.93/6.04 & ! [A7: A] :
% 11.93/6.04 ( ( F2 @ A7 @ Z5 )
% 11.93/6.04 = A7 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5629_monoid_Oaxioms_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ( monoid @ A @ F @ Z2 )
% 11.93/6.04 => ( monoid_axioms @ A @ F @ Z2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5630_mod__h__bot__normalize,axiom,
% 11.93/6.04 ! [A: $tType,H3: heap_ext @ product_unit,P: assn] :
% 11.93/6.04 ( ( syntax7388354845996824322omatch @ A @ ( heap_ext @ product_unit ) @ ( undefined @ A ) @ H3 )
% 11.93/6.04 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ ( undefined @ ( heap_ext @ product_unit ) ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5631_image__bind,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,C: $tType,F: B > A,A4: set @ C,G: C > ( set @ B )] :
% 11.93/6.04 ( ( image2 @ B @ A @ F @ ( bind3 @ C @ B @ A4 @ G ) )
% 11.93/6.04 = ( bind3 @ C @ A @ A4 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image2 @ B @ A @ F ) @ G ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5632_pure__assn__raw_Ocases,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: product_prod @ $o @ ( product_prod @ A @ ( set @ B ) )] :
% 11.93/6.04 ~ ! [B3: $o,H2: A,As: set @ B] :
% 11.93/6.04 ( X
% 11.93/6.04 != ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ B3 @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5633_mod__h__bot__iff_I7_J,axiom,
% 11.93/6.04 ! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( rep_assn @ ( sup_sup @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 | ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5634_mod__h__bot__iff_I5_J,axiom,
% 11.93/6.04 ! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5635_mod__h__bot__iff_I6_J,axiom,
% 11.93/6.04 ! [P: assn,Q: assn,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( rep_assn @ ( inf_inf @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5636_mod__h__bot__iff_I2_J,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit] : ( rep_assn @ ( top_top @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5637_times__assn__raw_Ocases,axiom,
% 11.93/6.04 ! [X: product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
% 11.93/6.04 ~ ! [P7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q7: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( X
% 11.93/6.04 != ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ P7 @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Q7 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5638_sngr__assn__raw_Ocases,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
% 11.93/6.04 ~ ! [R2: ref @ A,X2: A,H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( X
% 11.93/6.04 != ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R2 @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5639_snga__assn__raw_Ocases,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) )] :
% 11.93/6.04 ~ ! [R2: array @ A,X2: list @ A,H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( X
% 11.93/6.04 != ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ R2 @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5640_one__assn__raw_Ocases,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( X
% 11.93/6.04 != ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5641_mod__star__conv,axiom,
% 11.93/6.04 ! [A4: assn,B6: assn,H3: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( rep_assn @ ( times_times @ assn @ A4 @ B6 ) @ H3 )
% 11.93/6.04 = ( ? [Hr: heap_ext @ product_unit,As1: set @ nat,As22: set @ nat] :
% 11.93/6.04 ( ( H3
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( rep_assn @ A4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As1 ) )
% 11.93/6.04 & ( rep_assn @ B6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ Hr @ As22 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5642_star__assnI,axiom,
% 11.93/6.04 ! [P: assn,H3: heap_ext @ product_unit,As2: set @ nat,Q: assn,As6: set @ nat] :
% 11.93/6.04 ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) )
% 11.93/6.04 => ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As6 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 => ( rep_assn @ ( times_times @ assn @ P @ Q ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5643_mod__emp__simp,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit] : ( rep_assn @ ( one_one @ assn ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5644_mod__h__bot__indep,axiom,
% 11.93/6.04 ! [P: assn,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 11.93/6.04 ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5645_bind__image,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,C: $tType,F: C > B,A4: set @ C,G: B > ( set @ A )] :
% 11.93/6.04 ( ( bind3 @ B @ A @ ( image2 @ C @ B @ F @ A4 ) @ G )
% 11.93/6.04 = ( bind3 @ C @ A @ A4 @ ( comp @ B @ ( set @ A ) @ C @ G @ F ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5646_mod__h__bot__iff_I3_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [P5: ref @ A,X: A,H3: heap_ext @ product_unit] :
% 11.93/6.04 ~ ( rep_assn @ ( sngr_assn @ A @ P5 @ X ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5647_mod__h__bot__iff_I4_J,axiom,
% 11.93/6.04 ! [B: $tType] :
% 11.93/6.04 ( ( heap @ B )
% 11.93/6.04 => ! [Q3: array @ B,Y: list @ B,H3: heap_ext @ product_unit] :
% 11.93/6.04 ~ ( rep_assn @ ( snga_assn @ B @ Q3 @ Y ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5648_times__assn__raw_Osimps,axiom,
% 11.93/6.04 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( times_assn_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( ? [As1: set @ nat,As22: set @ nat] :
% 11.93/6.04 ( ( As2
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As1 ) )
% 11.93/6.04 & ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As22 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5649_times__assn__raw_Oelims_I3_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ? [As12: set @ nat,As23: set @ nat] :
% 11.93/6.04 ( ( As
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As12 ) )
% 11.93/6.04 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As23 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5650_times__assn__raw_Oelims_I2_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( times_assn_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ~ ? [As13: set @ nat,As24: set @ nat] :
% 11.93/6.04 ( ( As
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As13 ) )
% 11.93/6.04 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As24 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5651_times__assn__raw_Oelims_I1_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ? [As1: set @ nat,As22: set @ nat] :
% 11.93/6.04 ( ( As
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
% 11.93/6.04 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5652_times__assn__raw_Opelims_I1_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( times_assn_raw @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ? [As1: set @ nat,As22: set @ nat] :
% 11.93/6.04 ( ( As
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As1 @ As22 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As1 @ As22 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As1 ) )
% 11.93/6.04 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As22 ) ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5653_times__assn__raw_Opelims_I2_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( times_assn_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ~ ? [As13: set @ nat,As24: set @ nat] :
% 11.93/6.04 ( ( As
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As13 @ As24 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As13 @ As24 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As13 ) )
% 11.93/6.04 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As24 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5654_times__assn__raw_Opelims_I3_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( times_assn_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ times_assn_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ? [As12: set @ nat,As23: set @ nat] :
% 11.93/6.04 ( ( As
% 11.93/6.04 = ( sup_sup @ ( set @ nat ) @ As12 @ As23 ) )
% 11.93/6.04 & ( ( inf_inf @ ( set @ nat ) @ As12 @ As23 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As12 ) )
% 11.93/6.04 & ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As23 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5655_mod__h__bot__iff_I8_J,axiom,
% 11.93/6.04 ! [C: $tType,R5: C > assn,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( rep_assn @ ( ex_assn @ C @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = ( ? [X3: C] : ( rep_assn @ ( R5 @ X3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5656_make__rule,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [N: nat,F: nat > A] :
% 11.93/6.04 ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_make @ A @ N @ F )
% 11.93/6.04 @ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5657_mod__h__bot__iff_I1_J,axiom,
% 11.93/6.04 ! [B2: $o,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( rep_assn @ ( pure_assn @ B2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 = B2 ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5658_bind__rule,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap @ A,R5: A > assn,G: A > ( heap_Time_Heap @ B ),Q: B > assn] :
% 11.93/6.04 ( ( hoare_hoare_triple @ A @ P @ F @ R5 )
% 11.93/6.04 => ( ! [X2: A] : ( hoare_hoare_triple @ B @ ( R5 @ X2 ) @ ( G @ X2 ) @ Q )
% 11.93/6.04 => ( hoare_hoare_triple @ B @ P @ ( heap_Time_bind @ A @ B @ F @ G ) @ Q ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5659_raise__iff,axiom,
% 11.93/6.04 ! [A: $tType,P: assn,S: list @ char,Q: A > assn] :
% 11.93/6.04 ( ( hoare_hoare_triple @ A @ P @ ( heap_Time_raise @ A @ S ) @ Q )
% 11.93/6.04 = ( P
% 11.93/6.04 = ( bot_bot @ assn ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5660_raise__rule,axiom,
% 11.93/6.04 ! [A: $tType,S: list @ char,Q: A > assn] : ( hoare_hoare_triple @ A @ ( bot_bot @ assn ) @ ( heap_Time_raise @ A @ S ) @ Q ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5661_bind__rule_H,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap @ B,R5: B > assn,G: B > ( heap_Time_Heap @ C ),Q: C > assn] :
% 11.93/6.04 ( ! [R2: A] :
% 11.93/6.04 ( hoare_hoare_triple @ B @ P @ F
% 11.93/6.04 @ ^ [S4: B] : ( times_times @ assn @ P @ ( R5 @ S4 ) ) )
% 11.93/6.04 => ( ! [X2: B] : ( hoare_hoare_triple @ C @ ( times_times @ assn @ ( R5 @ X2 ) @ P ) @ ( G @ X2 ) @ Q )
% 11.93/6.04 => ( hoare_hoare_triple @ C @ P @ ( heap_Time_bind @ B @ C @ F @ G ) @ Q ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5662_case__prod__rule,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,C: $tType,X: product_prod @ A @ B,P: assn,F: A > B > ( heap_Time_Heap @ C ),Q: C > assn] :
% 11.93/6.04 ( ! [A5: A,B3: B] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.93/6.04 => ( hoare_hoare_triple @ C @ P @ ( F @ A5 @ B3 ) @ Q ) )
% 11.93/6.04 => ( hoare_hoare_triple @ C @ P @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ X ) @ Q ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5663_hoare__triple__success,axiom,
% 11.93/6.04 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
% 11.93/6.04 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( heap_Time_success @ A @ C2 @ H3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5664_case__option__rule,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,V: option @ A,P: assn,Fn: heap_Time_Heap @ B,Q: B > assn,Fs: A > ( heap_Time_Heap @ B )] :
% 11.93/6.04 ( ( ( V
% 11.93/6.04 = ( none @ A ) )
% 11.93/6.04 => ( hoare_hoare_triple @ B @ P @ Fn @ Q ) )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( V
% 11.93/6.04 = ( some @ A @ X2 ) )
% 11.93/6.04 => ( hoare_hoare_triple @ B @ P @ ( Fs @ X2 ) @ Q ) )
% 11.93/6.04 => ( hoare_hoare_triple @ B @ P @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ Fn @ Fs @ V ) @ Q ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5665_of__list__rule,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [Xs: list @ A] :
% 11.93/6.04 ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_of_list @ A @ Xs )
% 11.93/6.04 @ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ Xs ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5666_new__rule,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [N: nat,X: A] :
% 11.93/6.04 ( hoare_hoare_triple @ ( array @ A ) @ ( one_one @ assn ) @ ( array_new @ A @ N @ X )
% 11.93/6.04 @ ^ [R3: array @ A] : ( snga_assn @ A @ R3 @ ( replicate @ A @ N @ X ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5667_hoare__triple__effect,axiom,
% 11.93/6.04 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
% 11.93/6.04 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ? [H6: heap_ext @ product_unit,R2: A,T7: nat] :
% 11.93/6.04 ( ( heap_Time_effect @ A @ C2 @ H3 @ H6 @ R2 @ T7 )
% 11.93/6.04 & ( rep_assn @ ( Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( hoare_new_addrs @ H3 @ As2 @ H6 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5668_hoare__triple__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( hoare_hoare_triple @ A )
% 11.93/6.04 = ( ^ [P4: assn,C3: heap_Time_Heap @ A,Q6: A > assn] :
% 11.93/6.04 ! [H: heap_ext @ product_unit,As5: set @ nat] :
% 11.93/6.04 ( ( rep_assn @ P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 11.93/6.04 => ? [R3: A,H7: heap_ext @ product_unit] :
% 11.93/6.04 ( ? [T3: nat] :
% 11.93/6.04 ( ( heap_Time_execute @ A @ C3 @ H )
% 11.93/6.04 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ T3 ) ) ) )
% 11.93/6.04 & ( rep_assn @ ( Q6 @ R3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( hoare_new_addrs @ H @ As5 @ H7 ) ) )
% 11.93/6.04 & ( relH
% 11.93/6.04 @ ( collect @ nat
% 11.93/6.04 @ ^ [A7: nat] :
% 11.93/6.04 ( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H ) )
% 11.93/6.04 & ~ ( member @ nat @ A7 @ As5 ) ) )
% 11.93/6.04 @ H
% 11.93/6.04 @ H7 )
% 11.93/6.04 & ( ord_less_eq @ nat @ ( lim @ product_unit @ H ) @ ( lim @ product_unit @ H7 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5669_hoare__tripleI,axiom,
% 11.93/6.04 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn] :
% 11.93/6.04 ( ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ? [R15: A,H9: heap_ext @ product_unit,T8: nat] :
% 11.93/6.04 ( ( ( heap_Time_execute @ A @ C2 @ H2 )
% 11.93/6.04 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R15 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H9 @ T8 ) ) ) )
% 11.93/6.04 & ( rep_assn @ ( Q @ R15 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( hoare_new_addrs @ H2 @ As @ H9 ) ) )
% 11.93/6.04 & ( relH
% 11.93/6.04 @ ( collect @ nat
% 11.93/6.04 @ ^ [A7: nat] :
% 11.93/6.04 ( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H2 ) )
% 11.93/6.04 & ~ ( member @ nat @ A7 @ As ) ) )
% 11.93/6.04 @ H2
% 11.93/6.04 @ H9 )
% 11.93/6.04 & ( ord_less_eq @ nat @ ( lim @ product_unit @ H2 ) @ ( lim @ product_unit @ H9 ) ) ) )
% 11.93/6.04 => ( hoare_hoare_triple @ A @ P @ C2 @ Q ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5670_mod__relH,axiom,
% 11.93/6.04 ! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,P: assn] :
% 11.93/6.04 ( ( relH @ As2 @ H3 @ H4 )
% 11.93/6.04 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5671_hoare__tripleE,axiom,
% 11.93/6.04 ! [A: $tType,P: assn,C2: heap_Time_Heap @ A,Q: A > assn,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( hoare_hoare_triple @ A @ P @ C2 @ Q )
% 11.93/6.04 => ( ( rep_assn @ P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ~ ! [R2: A,H6: heap_ext @ product_unit] :
% 11.93/6.04 ( ? [T7: nat] :
% 11.93/6.04 ( ( heap_Time_execute @ A @ C2 @ H3 )
% 11.93/6.04 = ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ T7 ) ) ) )
% 11.93/6.04 => ( ( rep_assn @ ( Q @ R2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( hoare_new_addrs @ H3 @ As2 @ H6 ) ) )
% 11.93/6.04 => ( ( relH
% 11.93/6.04 @ ( collect @ nat
% 11.93/6.04 @ ^ [A7: nat] :
% 11.93/6.04 ( ( ord_less @ nat @ A7 @ ( lim @ product_unit @ H3 ) )
% 11.93/6.04 & ~ ( member @ nat @ A7 @ As2 ) ) )
% 11.93/6.04 @ H3
% 11.93/6.04 @ H6 )
% 11.93/6.04 => ~ ( ord_less_eq @ nat @ ( lim @ product_unit @ H3 ) @ ( lim @ product_unit @ H6 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5672_Array__Time_Oalloc__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ( ( array_alloc @ A )
% 11.93/6.04 = ( ^ [Xs3: list @ A,H: heap_ext @ product_unit] :
% 11.93/6.04 ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H ) )
% 11.93/6.04 @ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H ) ) @ Xs3
% 11.93/6.04 @ ( lim_update @ product_unit
% 11.93/6.04 @ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
% 11.93/6.04 @ H ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5673_snga__assn__raw_Opelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ( ( array_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5674_snga__assn__raw_Opelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ~ ( ( ( array_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5675_Array__Time_Opresent__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ( ( array_present @ A )
% 11.93/6.04 = ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A7 ) @ ( lim @ product_unit @ H ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5676_snga__assn__raw_Osimps,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: array @ A,X: list @ A,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( snga_assn_raw @ A @ R @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( ( ( array_get @ A @ H3 @ R )
% 11.93/6.04 = X )
% 11.93/6.04 & ( As2
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ R ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ R ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5677_snga__assn__raw_Oelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ( ( ( array_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5678_snga__assn__raw_Oelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ~ ( ( ( array_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5679_snga__assn__raw_Oelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( ( array_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5680_snga__assn__raw_Opelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: array @ A,Xa: list @ A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( snga_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( snga_assn_raw_rel @ A ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ( ( ( array_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_array @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_array @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5681_relH__set__array,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: array @ A,As2: set @ nat,H3: heap_ext @ product_unit,X: list @ A] :
% 11.93/6.04 ( ~ ( member @ nat @ ( addr_of_array @ A @ R ) @ As2 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( relH @ As2 @ H3 @ ( array_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5682_sngr__assn__raw_Opelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ( ( ref_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5683_sngr__assn__raw_Opelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ~ ( ( ( ref_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5684_in__range__empty,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit] : ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5685_in__range__dist__union,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit,As2: set @ nat,As6: set @ nat] :
% 11.93/6.04 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ ( sup_sup @ ( set @ nat ) @ As2 @ As6 ) ) )
% 11.93/6.04 = ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5686_relH__set__ref,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: ref @ A,As2: set @ nat,H3: heap_ext @ product_unit,X: A] :
% 11.93/6.04 ( ~ ( member @ nat @ ( addr_of_ref @ A @ R ) @ As2 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( relH @ As2 @ H3 @ ( ref_set @ A @ R @ X @ H3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5687_relH__refl,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( relH @ As2 @ H3 @ H3 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5688_relH__in__rangeI_I1_J,axiom,
% 11.93/6.04 ! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 11.93/6.04 ( ( relH @ As2 @ H3 @ H4 )
% 11.93/6.04 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5689_relH__in__rangeI_I2_J,axiom,
% 11.93/6.04 ! [As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 11.93/6.04 ( ( relH @ As2 @ H3 @ H4 )
% 11.93/6.04 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5690_in__range__subset,axiom,
% 11.93/6.04 ! [As2: set @ nat,As6: set @ nat,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( ord_less_eq @ ( set @ nat ) @ As2 @ As6 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As6 ) )
% 11.93/6.04 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5691_in__range_Oelims_I3_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( in_range @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ! [X2: nat] :
% 11.93/6.04 ( ( member @ nat @ X2 @ As )
% 11.93/6.04 => ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5692_in__range_Oelims_I2_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( in_range @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ~ ! [X7: nat] :
% 11.93/6.04 ( ( member @ nat @ X7 @ As )
% 11.93/6.04 => ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H2 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5693_in__range_Oelims_I1_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( in_range @ X )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ! [X3: nat] :
% 11.93/6.04 ( ( member @ nat @ X3 @ As )
% 11.93/6.04 => ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5694_in__range_Osimps,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( ! [X3: nat] :
% 11.93/6.04 ( ( member @ nat @ X3 @ As2 )
% 11.93/6.04 => ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5695_sngr__assn__raw_Oelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( ( ref_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5696_sngr__assn__raw_Oelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ~ ( ( ( ref_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5697_sngr__assn__raw_Oelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ( ( ( ref_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5698_sngr__assn__raw_Osimps,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: ref @ A,X: A,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( sngr_assn_raw @ A @ R @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( ( ( ref_get @ A @ H3 @ R )
% 11.93/6.04 = X )
% 11.93/6.04 & ( As2
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ R ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ R ) @ ( lim @ product_unit @ H3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5699_sngr__assn__raw_Opelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [X: ref @ A,Xa: A,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( sngr_assn_raw @ A @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ ( sngr_assn_raw_rel @ A ) @ ( product_Pair @ ( ref @ A ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ( ( ( ref_get @ A @ H2 @ X )
% 11.93/6.04 = Xa )
% 11.93/6.04 & ( As
% 11.93/6.04 = ( insert2 @ nat @ ( addr_of_ref @ A @ X ) @ ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 & ( ord_less @ nat @ ( addr_of_ref @ A @ X ) @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5700_wand__assnI,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit,As2: set @ nat,Q: assn,R5: assn] :
% 11.93/6.04 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( ! [H6: heap_ext @ product_unit,As7: set @ nat] :
% 11.93/6.04 ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As7 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 => ( ( relH @ As2 @ H3 @ H6 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As2 ) )
% 11.93/6.04 => ( ( rep_assn @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) )
% 11.93/6.04 => ( rep_assn @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As2 @ As7 ) ) ) ) ) ) )
% 11.93/6.04 => ( rep_assn @ ( wand_assn @ Q @ R5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5701_wand__raw_Osimps,axiom,
% 11.93/6.04 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Q: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( wand_raw @ P @ Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 & ! [H7: heap_ext @ product_unit,As8: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As2 @ As8 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As2 @ H3 @ H7 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As2 ) )
% 11.93/6.04 & ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
% 11.93/6.04 => ( Q @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As2 @ As8 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5702_wand__raw_Oelims_I1_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( wand_raw @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 & ! [H7: heap_ext @ product_unit,As8: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As @ H2 @ H7 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
% 11.93/6.04 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5703_wand__raw_Oelims_I3_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 & ! [H6: heap_ext @ product_unit,As7: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As @ H2 @ H6 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) ) )
% 11.93/6.04 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5704_wand__raw_Oelims_I2_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( wand_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 & ! [H9: heap_ext @ product_unit,As9: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As @ H2 @ H9 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
% 11.93/6.04 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5705_wand__raw_Opelims_I3_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( wand_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 & ! [H6: heap_ext @ product_unit,As7: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As7 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As @ H2 @ H6 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As7 ) ) )
% 11.93/6.04 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ ( sup_sup @ ( set @ nat ) @ As @ As7 ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5706_wand__raw_Opelims_I2_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( wand_raw @ X @ Xa @ Xb )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) )
% 11.93/6.04 => ~ ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 & ! [H9: heap_ext @ product_unit,As9: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As9 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As @ H2 @ H9 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ As9 ) ) )
% 11.93/6.04 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H9 @ ( sup_sup @ ( set @ nat ) @ As @ As9 ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5707_wand__raw_Opelims_I1_J,axiom,
% 11.93/6.04 ! [X: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xa: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,Xb: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( wand_raw @ X @ Xa @ Xb )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ Xb ) ) )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( Xb
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 & ! [H7: heap_ext @ product_unit,As8: set @ nat] :
% 11.93/6.04 ( ( ( ( inf_inf @ ( set @ nat ) @ As @ As8 )
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) )
% 11.93/6.04 & ( relH @ As @ H2 @ H7 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As ) )
% 11.93/6.04 & ( X @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As8 ) ) )
% 11.93/6.04 => ( Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ ( sup_sup @ ( set @ nat ) @ As @ As8 ) ) ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) ) @ wand_raw_rel @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) ) @ X @ ( product_Pair @ ( ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o ) @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ Xa @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5708_in__range_Opelims_I3_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( in_range @ X )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ! [X2: nat] :
% 11.93/6.04 ( ( member @ nat @ X2 @ As )
% 11.93/6.04 => ( ord_less @ nat @ X2 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5709_in__range_Opelims_I2_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( in_range @ X )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ~ ! [X7: nat] :
% 11.93/6.04 ( ( member @ nat @ X7 @ As )
% 11.93/6.04 => ( ord_less @ nat @ X7 @ ( lim @ product_unit @ H2 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5710_in__range_Opelims_I1_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( in_range @ X )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ! [X3: nat] :
% 11.93/6.04 ( ( member @ nat @ X3 @ As )
% 11.93/6.04 => ( ord_less @ nat @ X3 @ ( lim @ product_unit @ H2 ) ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ in_range_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5711_Ref__Time_Oalloc__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ( ( ref_alloc @ A )
% 11.93/6.04 = ( ^ [X3: A,H: heap_ext @ product_unit] :
% 11.93/6.04 ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( ref2 @ A @ ( lim @ product_unit @ H ) )
% 11.93/6.04 @ ( ref_set @ A @ ( ref2 @ A @ ( lim @ product_unit @ H ) ) @ X3
% 11.93/6.04 @ ( lim_update @ product_unit
% 11.93/6.04 @ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H ) @ ( one_one @ nat ) )
% 11.93/6.04 @ H ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5712_pure__assn__raw_Osimps,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,B2: $o,H3: A,As2: set @ B] :
% 11.93/6.04 ( ( pure_assn_raw @ A @ B @ B2 @ ( product_Pair @ A @ ( set @ B ) @ H3 @ As2 ) )
% 11.93/6.04 = ( ( As2
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & B2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5713_pure__assn__raw_Oelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
% 11.93/6.04 ( ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: A,As: set @ B] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( ~ ( ( As
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & X ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5714_pure__assn__raw_Oelims_I3_J,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 11.93/6.04 ( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
% 11.93/6.04 => ~ ! [H2: A,As: set @ B] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 11.93/6.04 => ( ( As
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & X ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5715_pure__assn__raw_Oelims_I2_J,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 11.93/6.04 ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 11.93/6.04 => ~ ! [H2: A,As: set @ B] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 11.93/6.04 => ~ ( ( As
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & X ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5716_pure__assn__raw_Opelims_I3_J,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 11.93/6.04 ( ~ ( pure_assn_raw @ A @ B @ X @ Xa )
% 11.93/6.04 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
% 11.93/6.04 => ~ ! [H2: A,As: set @ B] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) )
% 11.93/6.04 => ( ( As
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & X ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5717_pure__assn__raw_Opelims_I2_J,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,X: $o,Xa: product_prod @ A @ ( set @ B )] :
% 11.93/6.04 ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 11.93/6.04 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
% 11.93/6.04 => ~ ! [H2: A,As: set @ B] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) )
% 11.93/6.04 => ~ ( ( As
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & X ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5718_pure__assn__raw_Opelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,X: $o,Xa: product_prod @ A @ ( set @ B ),Y: $o] :
% 11.93/6.04 ( ( ( pure_assn_raw @ A @ B @ X @ Xa )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ Xa ) )
% 11.93/6.04 => ~ ! [H2: A,As: set @ B] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ( As
% 11.93/6.04 = ( bot_bot @ ( set @ B ) ) )
% 11.93/6.04 & X ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ $o @ ( product_prod @ A @ ( set @ B ) ) ) @ ( pure_assn_raw_rel @ A @ B ) @ ( product_Pair @ $o @ ( product_prod @ A @ ( set @ B ) ) @ X @ ( product_Pair @ A @ ( set @ B ) @ H2 @ As ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5719_effect__refI,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: ref @ A,H4: heap_ext @ product_unit,V: A,H3: heap_ext @ product_unit,N: nat] :
% 11.93/6.04 ( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
% 11.93/6.04 = ( ref_alloc @ A @ V @ H3 ) )
% 11.93/6.04 => ( ( N
% 11.93/6.04 = ( one_one @ nat ) )
% 11.93/6.04 => ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 @ H4 @ R @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5720_ref__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ( ( ref_ref @ A )
% 11.93/6.04 = ( ^ [V2: A] :
% 11.93/6.04 ( heap_Time_heap @ ( ref @ A )
% 11.93/6.04 @ ^ [H: heap_ext @ product_unit] :
% 11.93/6.04 ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ^ [R3: ref @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( one_one @ nat ) ) )
% 11.93/6.04 @ ( ref_alloc @ A @ V2 @ H ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5721_execute__ref,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [V: A,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( heap_Time_execute @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 )
% 11.93/6.04 = ( some @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ( product_case_prod @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
% 11.93/6.04 @ ^ [R3: ref @ A,H7: heap_ext @ product_unit] : ( product_Pair @ ( ref @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ ( one_one @ nat ) ) )
% 11.93/6.04 @ ( ref_alloc @ A @ V @ H3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5722_effect__refE,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [V: A,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: ref @ A,N: nat] :
% 11.93/6.04 ( ( heap_Time_effect @ ( ref @ A ) @ ( ref_ref @ A @ V ) @ H3 @ H4 @ R @ N )
% 11.93/6.04 => ~ ( ( ( ref_get @ A @ H4 @ R )
% 11.93/6.04 = V )
% 11.93/6.04 => ( ( ref_present @ A @ H4 @ R )
% 11.93/6.04 => ( ~ ( ref_present @ A @ H3 @ R )
% 11.93/6.04 => ( N
% 11.93/6.04 != ( one_one @ nat ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5723_next__fresh,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: ref @ A,H4: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
% 11.93/6.04 = ( ref_alloc @ A @ X @ H3 ) )
% 11.93/6.04 => ~ ( ref_present @ A @ H3 @ R ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5724_next__present,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ! [R: ref @ A,H4: heap_ext @ product_unit,X: A,H3: heap_ext @ product_unit] :
% 11.93/6.04 ( ( ( product_Pair @ ( ref @ A ) @ ( heap_ext @ product_unit ) @ R @ H4 )
% 11.93/6.04 = ( ref_alloc @ A @ X @ H3 ) )
% 11.93/6.04 => ( ref_present @ A @ H4 @ R ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5725_one__assn__raw_Oelims_I3_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( one_assn_raw @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( As
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5726_one__assn__raw_Oelims_I2_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( one_assn_raw @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( As
% 11.93/6.04 != ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5727_one__assn__raw_Oelims_I1_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( one_assn_raw @ X )
% 11.93/6.04 = Y )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 = ( As
% 11.93/6.04 != ( bot_bot @ ( set @ nat ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5728_one__assn__raw_Osimps,axiom,
% 11.93/6.04 ! [H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( one_assn_raw @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( As2
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5729_one__assn__raw_Opelims_I3_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ~ ( one_assn_raw @ X )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( As
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5730_one__assn__raw_Opelims_I2_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat )] :
% 11.93/6.04 ( ( one_assn_raw @ X )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( As
% 11.93/6.04 != ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5731_one__assn__raw_Opelims_I1_J,axiom,
% 11.93/6.04 ! [X: product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ),Y: $o] :
% 11.93/6.04 ( ( ( one_assn_raw @ X )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ X )
% 11.93/6.04 => ~ ! [H2: heap_ext @ product_unit,As: set @ nat] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( As
% 11.93/6.04 = ( bot_bot @ ( set @ nat ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) @ one_assn_raw_rel @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5732_proper__iff,axiom,
% 11.93/6.04 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,As2: set @ nat,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit] :
% 11.93/6.04 ( ( proper @ P )
% 11.93/6.04 => ( ( relH @ As2 @ H3 @ H4 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
% 11.93/6.04 => ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 = ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5733_proper__def,axiom,
% 11.93/6.04 ( proper
% 11.93/6.04 = ( ^ [P4: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
% 11.93/6.04 ! [H: heap_ext @ product_unit,H7: heap_ext @ product_unit,As5: set @ nat] :
% 11.93/6.04 ( ( ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 11.93/6.04 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) ) )
% 11.93/6.04 & ( ( ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 11.93/6.04 & ( relH @ As5 @ H @ H7 )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) ) )
% 11.93/6.04 => ( P4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5734_properD2,axiom,
% 11.93/6.04 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat,H4: heap_ext @ product_unit] :
% 11.93/6.04 ( ( proper @ P )
% 11.93/6.04 => ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( ( relH @ As2 @ H3 @ H4 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) )
% 11.93/6.04 => ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H4 @ As2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5735_properD1,axiom,
% 11.93/6.04 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o,H3: heap_ext @ product_unit,As2: set @ nat] :
% 11.93/6.04 ( ( proper @ P )
% 11.93/6.04 => ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) )
% 11.93/6.04 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H3 @ As2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5736_properI,axiom,
% 11.93/6.04 ! [P: ( product_prod @ ( heap_ext @ product_unit ) @ ( set @ nat ) ) > $o] :
% 11.93/6.04 ( ! [As: set @ nat,H2: heap_ext @ product_unit] :
% 11.93/6.04 ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) ) )
% 11.93/6.04 => ( ! [As: set @ nat,H2: heap_ext @ product_unit,H6: heap_ext @ product_unit] :
% 11.93/6.04 ( ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H2 @ As ) )
% 11.93/6.04 => ( ( relH @ As @ H2 @ H6 )
% 11.93/6.04 => ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) )
% 11.93/6.04 => ( P @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H6 @ As ) ) ) ) )
% 11.93/6.04 => ( proper @ P ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5737_Array__Time_Onoteq__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( ( heap @ A )
% 11.93/6.04 & ( heap @ B ) )
% 11.93/6.04 => ( ( array_noteq @ A @ B )
% 11.93/6.04 = ( ^ [R3: array @ A,S4: array @ B] :
% 11.93/6.04 ( ( ( typerep_typerep @ A @ ( type2 @ A ) )
% 11.93/6.04 != ( typerep_typerep @ B @ ( type2 @ B ) ) )
% 11.93/6.04 | ( ( addr_of_array @ A @ R3 )
% 11.93/6.04 != ( addr_of_array @ B @ S4 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5738_comm__monoid__mset_Ounion,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M3: multiset @ A,N6: multiset @ A] :
% 11.93/6.04 ( ( comm_monoid_mset @ A @ F @ Z2 )
% 11.93/6.04 => ( ( comm_monoid_F @ A @ F @ Z2 @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.93/6.04 = ( F @ ( comm_monoid_F @ A @ F @ Z2 @ M3 ) @ ( comm_monoid_F @ A @ F @ Z2 @ N6 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5739_prod__list_Omonoid__list__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( monoid_mult @ A )
% 11.93/6.04 => ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5740_sum__mset_Ocomm__monoid__mset__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_add @ A )
% 11.93/6.04 => ( comm_monoid_mset @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5741_prod__mset_Ocomm__monoid__mset__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_mult @ A )
% 11.93/6.04 => ( comm_monoid_mset @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5742_sum__list_Omonoid__list__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( monoid_add @ A )
% 11.93/6.04 => ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5743_Array__Time_Oset__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ( ( array_set @ A )
% 11.93/6.04 = ( ^ [A7: array @ A,X3: list @ A] :
% 11.93/6.04 ( arrays_update @ product_unit
% 11.93/6.04 @ ^ [H: typerep > nat > ( list @ nat )] : ( fun_upd @ typerep @ ( nat > ( list @ nat ) ) @ H @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( fun_upd @ nat @ ( list @ nat ) @ ( H @ ( typerep_typerep @ A @ ( type2 @ A ) ) ) @ ( addr_of_array @ A @ A7 ) @ ( map @ A @ nat @ ( to_nat @ A ) @ X3 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5744_Array__Time_Oget__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( heap @ A )
% 11.93/6.04 => ( ( array_get @ A )
% 11.93/6.04 = ( ^ [H: heap_ext @ product_unit,A7: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A7 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5745_Int_Osub__code_I8_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
% 11.93/6.04 = ( plus_plus @ int @ ( dup @ ( sub @ M2 @ N ) ) @ ( one_one @ int ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5746_Int_Odup__def,axiom,
% 11.93/6.04 ( dup
% 11.93/6.04 = ( ^ [K4: int] : ( plus_plus @ int @ K4 @ K4 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5747_relH__def,axiom,
% 11.93/6.04 ( relH
% 11.93/6.04 = ( ^ [As5: set @ nat,H: heap_ext @ product_unit,H7: heap_ext @ product_unit] :
% 11.93/6.04 ( ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H @ As5 ) )
% 11.93/6.04 & ( in_range @ ( product_Pair @ ( heap_ext @ product_unit ) @ ( set @ nat ) @ H7 @ As5 ) )
% 11.93/6.04 & ! [T3: typerep,X3: nat] :
% 11.93/6.04 ( ( member @ nat @ X3 @ As5 )
% 11.93/6.04 => ( ( ( refs @ product_unit @ H @ T3 @ X3 )
% 11.93/6.04 = ( refs @ product_unit @ H7 @ T3 @ X3 ) )
% 11.93/6.04 & ( ( arrays @ product_unit @ H @ T3 @ X3 )
% 11.93/6.04 = ( arrays @ product_unit @ H7 @ T3 @ X3 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5748_case__sum__o__map__sum__id,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,C: $tType,G: B > A,F: C > A,X: sum_sum @ C @ B] :
% 11.93/6.04 ( ( comp @ ( sum_sum @ A @ B ) @ A @ ( sum_sum @ C @ B ) @ ( sum_case_sum @ A @ A @ B @ ( id @ A ) @ G ) @ ( sum_map_sum @ C @ A @ B @ B @ F @ ( id @ B ) ) @ X )
% 11.93/6.04 = ( sum_case_sum @ C @ A @ B @ ( comp @ C @ A @ C @ F @ ( id @ C ) ) @ G @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5749_comm__monoid_Oaxioms_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ( comm_monoid @ A @ F @ Z2 )
% 11.93/6.04 => ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5750_sum_Omap__comp,axiom,
% 11.93/6.04 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F5,F1: A > C,F22: B > D,V: sum_sum @ A @ B] :
% 11.93/6.04 ( ( sum_map_sum @ C @ E @ D @ F5 @ G1 @ G22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F22 @ V ) )
% 11.93/6.04 = ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F5 @ B @ G22 @ F22 ) @ V ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5751_comm__monoid__axioms_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ! [A5: A] :
% 11.93/6.04 ( ( F @ A5 @ Z2 )
% 11.93/6.04 = A5 )
% 11.93/6.04 => ( comm_monoid_axioms @ A @ F @ Z2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5752_comm__monoid__axioms__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_axioms @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A,Z5: A] :
% 11.93/6.04 ! [A7: A] :
% 11.93/6.04 ( ( F2 @ A7 @ Z5 )
% 11.93/6.04 = A7 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5753_case__sum__map__sum,axiom,
% 11.93/6.04 ! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: B > A,R: C > A,F: D > B,G: E > C,X: sum_sum @ D @ E] :
% 11.93/6.04 ( ( sum_case_sum @ B @ A @ C @ L @ R @ ( sum_map_sum @ D @ B @ E @ C @ F @ G @ X ) )
% 11.93/6.04 = ( sum_case_sum @ D @ A @ E @ ( comp @ B @ A @ D @ L @ F ) @ ( comp @ C @ A @ E @ R @ G ) @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5754_case__sum__o__map__sum,axiom,
% 11.93/6.04 ! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: D > C,G: E > C,H1: A > D,H22: B > E] :
% 11.93/6.04 ( ( comp @ ( sum_sum @ D @ E ) @ C @ ( sum_sum @ A @ B ) @ ( sum_case_sum @ D @ C @ E @ F @ G ) @ ( sum_map_sum @ A @ D @ B @ E @ H1 @ H22 ) )
% 11.93/6.04 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ F @ H1 ) @ ( comp @ E @ C @ B @ G @ H22 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5755_map__sum_Ocomp,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,E: $tType,F5: $tType,D: $tType,B: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D] :
% 11.93/6.04 ( ( comp @ ( sum_sum @ C @ D ) @ ( sum_sum @ E @ F5 ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ C @ E @ D @ F5 @ F @ G ) @ ( sum_map_sum @ A @ C @ B @ D @ H3 @ I ) )
% 11.93/6.04 = ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5756_map__sum_Ocompositionality,axiom,
% 11.93/6.04 ! [D: $tType,F5: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F5,H3: A > C,I: B > D,Sum: sum_sum @ A @ B] :
% 11.93/6.04 ( ( sum_map_sum @ C @ E @ D @ F5 @ F @ G @ ( sum_map_sum @ A @ C @ B @ D @ H3 @ I @ Sum ) )
% 11.93/6.04 = ( sum_map_sum @ A @ E @ B @ F5 @ ( comp @ C @ E @ A @ F @ H3 ) @ ( comp @ D @ F5 @ B @ G @ I ) @ Sum ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5757_comm__monoid__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A,Z5: A] :
% 11.93/6.04 ( ( abel_semigroup @ A @ F2 )
% 11.93/6.04 & ( comm_monoid_axioms @ A @ F2 @ Z5 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5758_mult_Oabel__semigroup__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ab_semigroup_mult @ A )
% 11.93/6.04 => ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5759_comm__monoid_Oaxioms_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ( comm_monoid @ A @ F @ Z2 )
% 11.93/6.04 => ( abel_semigroup @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5760_abel__semigroup_Oleft__commute,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,B2: A,A3: A,C2: A] :
% 11.93/6.04 ( ( abel_semigroup @ A @ F )
% 11.93/6.04 => ( ( F @ B2 @ ( F @ A3 @ C2 ) )
% 11.93/6.04 = ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5761_abel__semigroup_Ocommute,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,A3: A,B2: A] :
% 11.93/6.04 ( ( abel_semigroup @ A @ F )
% 11.93/6.04 => ( ( F @ A3 @ B2 )
% 11.93/6.04 = ( F @ B2 @ A3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5762_add_Oabel__semigroup__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ab_semigroup_add @ A )
% 11.93/6.04 => ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5763_abel__semigroup_Oaxioms_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A] :
% 11.93/6.04 ( ( abel_semigroup @ A @ F )
% 11.93/6.04 => ( semigroup @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5764_comm__monoid_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A] :
% 11.93/6.04 ( ( abel_semigroup @ A @ F )
% 11.93/6.04 => ( ( comm_monoid_axioms @ A @ F @ Z2 )
% 11.93/6.04 => ( comm_monoid @ A @ F @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5765_abel__semigroup_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A] :
% 11.93/6.04 ( ( semigroup @ A @ F )
% 11.93/6.04 => ( ( abel_s757365448890700780axioms @ A @ F )
% 11.93/6.04 => ( abel_semigroup @ A @ F ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5766_abel__semigroup__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( abel_semigroup @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A] :
% 11.93/6.04 ( ( semigroup @ A @ F2 )
% 11.93/6.04 & ( abel_s757365448890700780axioms @ A @ F2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5767_curr__def,axiom,
% 11.93/6.04 ! [B: $tType,C: $tType,A: $tType] :
% 11.93/6.04 ( ( bNF_Wellorder_curr @ A @ B @ C )
% 11.93/6.04 = ( ^ [A9: set @ A,F2: ( product_prod @ A @ B ) > C,A7: A] :
% 11.93/6.04 ( if @ ( B > C ) @ ( member @ A @ A7 @ A9 )
% 11.93/6.04 @ ^ [B5: B] : ( F2 @ ( product_Pair @ A @ B @ A7 @ B5 ) )
% 11.93/6.04 @ ( undefined @ ( B > C ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5768_abel__semigroup__axioms_Ointro,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A] :
% 11.93/6.04 ( ! [A5: A,B3: A] :
% 11.93/6.04 ( ( F @ A5 @ B3 )
% 11.93/6.04 = ( F @ B3 @ A5 ) )
% 11.93/6.04 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5769_abel__semigroup__axioms__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( abel_s757365448890700780axioms @ A )
% 11.93/6.04 = ( ^ [F2: A > A > A] :
% 11.93/6.04 ! [A7: A,B5: A] :
% 11.93/6.04 ( ( F2 @ A7 @ B5 )
% 11.93/6.04 = ( F2 @ B5 @ A7 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5770_abel__semigroup_Oaxioms_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A] :
% 11.93/6.04 ( ( abel_semigroup @ A @ F )
% 11.93/6.04 => ( abel_s757365448890700780axioms @ A @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5771_comm__monoid__set_Ozero__middle,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,P5: nat,K: nat,G: nat > A,H3: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ K @ P5 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( if @ A @ ( J2 = K ) @ Z2 @ ( H3 @ ( minus_minus @ nat @ J2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ P5 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [J2: nat] : ( if @ A @ ( ord_less @ nat @ J2 @ K ) @ ( G @ J2 ) @ ( H3 @ J2 ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5772_iterate__add__diff__add__eq2,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ I @ J )
% 11.93/6.04 => ( ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( minus_minus @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5773_iterate__add__add__eq1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ! [J: nat,I: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ J @ I )
% 11.93/6.04 => ( ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( minus_minus @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5774_iterate__add__simps_I2_J,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ! [N: nat,A3: A] :
% 11.93/6.04 ( ( iterate_add @ A @ ( suc @ N ) @ A3 )
% 11.93/6.04 = ( plus_plus @ A @ A3 @ ( iterate_add @ A @ N @ A3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5775_iterate__add__distrib,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ! [M2: nat,N: nat,A3: A] :
% 11.93/6.04 ( ( iterate_add @ A @ ( plus_plus @ nat @ M2 @ N ) @ A3 )
% 11.93/6.04 = ( plus_plus @ A @ ( iterate_add @ A @ M2 @ A3 ) @ ( iterate_add @ A @ N @ A3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5776_iterate__add__1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 & ( semiring_1 @ A ) )
% 11.93/6.04 => ! [N: nat] :
% 11.93/6.04 ( ( iterate_add @ A @ N @ ( one_one @ A ) )
% 11.93/6.04 = ( semiring_1_of_nat @ A @ N ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5777_prod_Ocomm__monoid__set__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_mult @ A )
% 11.93/6.04 => ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5778_comm__monoid__set_Onat__group,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,K: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [M: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ K ) ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5779_comm__monoid__set_Oub__add__nat,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A,P5: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P5 ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5780_comm__monoid__set_Oivl__cong,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType] :
% 11.93/6.04 ( ( ord @ B )
% 11.93/6.04 => ! [F: A > A > A,Z2: A,A3: B,C2: B,B2: B,D3: B,G: B > A,H3: B > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( A3 = C2 )
% 11.93/6.04 => ( ( B2 = D3 )
% 11.93/6.04 => ( ! [X2: B] :
% 11.93/6.04 ( ( ord_less_eq @ B @ C2 @ X2 )
% 11.93/6.04 => ( ( ord_less @ B @ X2 @ D3 )
% 11.93/6.04 => ( ( G @ X2 )
% 11.93/6.04 = ( H3 @ X2 ) ) ) )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ H3 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D3 ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5781_comm__monoid__set_Ohead,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.93/6.04 = ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5782_comm__monoid__set_Olast__plus,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.93/6.04 = ( F @ ( G @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5783_comm__monoid__set_OatLeast__Suc__atMost,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.93/6.04 = ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5784_comm__monoid__set_Onat__ivl__Suc_H,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( G @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5785_comm__monoid__set_OatLeastLessThan__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,A3: nat,B2: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ A3 @ B2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5786_comm__monoid__set_OatLeastLessThan__concat,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,P5: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ N @ P5 )
% 11.93/6.04 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P5 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5787_comm__monoid__set_OSuc__reindex__ivl,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.93/6.04 => ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( G @ M2 )
% 11.93/6.04 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5788_comm__monoid__set_OatLeast1__atMost__eq,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [K4: nat] : ( G @ ( suc @ K4 ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5789_comm__monoid__set_OatLeastLessThan__rev,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I3 ) ) )
% 11.93/6.04 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5790_comm__monoid__set_Onested__swap,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,A3: nat > nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( A3 @ I3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I3 ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [J2: nat] :
% 11.93/6.04 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 11.93/6.04 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5791_comm__monoid__set_Onat__diff__reindex,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I3 ) ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5792_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5793_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.93/6.04 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5794_comm__monoid__set_OlessThan__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5795_comm__monoid__set_OatMost__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5796_comm__monoid__set_Ohead__if,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ( ord_less @ nat @ N @ M2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.93/6.04 = Z2 ) )
% 11.93/6.04 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5797_comm__monoid__set_Ocl__ivl__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.93/6.04 = Z2 ) )
% 11.93/6.04 & ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5798_comm__monoid__set_Oop__ivl__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,N: nat,M2: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ( ord_less @ nat @ N @ M2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 11.93/6.04 = Z2 ) )
% 11.93/6.04 & ( ~ ( ord_less @ nat @ N @ M2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5799_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less @ nat @ M2 @ N )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.93/6.04 = ( F @ ( G @ M2 ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5800_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5801_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5802_comm__monoid__set_OatMost__Suc__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( G @ ( zero_zero @ nat ) )
% 11.93/6.04 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5803_comm__monoid__set_OatLeastAtMost__rev,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5804_comm__monoid__set_OlessThan__Suc__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( G @ ( zero_zero @ nat ) )
% 11.93/6.04 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5805_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5806_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( plus_plus @ nat @ I3 @ K ) )
% 11.93/6.04 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5807_sum_Ocomm__monoid__set__axioms,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comm_monoid_add @ A )
% 11.93/6.04 => ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5808_comm__monoid__set_OatMost__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ N ) )
% 11.93/6.04 = ( F @ ( G @ ( zero_zero @ nat ) )
% 11.93/6.04 @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5809_comm__monoid__set_Onested__swap_H,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,A3: nat > nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( A3 @ I3 ) @ ( set_ord_lessThan @ nat @ I3 ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [J2: nat] :
% 11.93/6.04 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( A3 @ I3 @ J2 )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J2 ) @ N ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5810_comm__monoid__set_Oreindex__nontrivial,axiom,
% 11.93/6.04 ! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z2: A,A4: set @ B,H3: B > C,G: C > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( finite_finite2 @ B @ A4 )
% 11.93/6.04 => ( ! [X2: B,Y4: B] :
% 11.93/6.04 ( ( member @ B @ X2 @ A4 )
% 11.93/6.04 => ( ( member @ B @ Y4 @ A4 )
% 11.93/6.04 => ( ( X2 != Y4 )
% 11.93/6.04 => ( ( ( H3 @ X2 )
% 11.93/6.04 = ( H3 @ Y4 ) )
% 11.93/6.04 => ( ( G @ ( H3 @ X2 ) )
% 11.93/6.04 = Z2 ) ) ) ) )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5811_comm__monoid__set_Oreindex,axiom,
% 11.93/6.04 ! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z2: A,H3: B > C,A4: set @ B,G: C > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( inj_on @ B @ C @ H3 @ A4 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ C @ F @ Z2 @ G @ ( image2 @ B @ C @ H3 @ A4 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ ( comp @ C @ A @ B @ G @ H3 ) @ A4 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5812_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5813_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5814_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5815_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,K: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5816_comm__monoid__set_OatLeastLessThan__reindex,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( ord @ B )
% 11.93/6.04 => ! [F: A > A > A,Z2: A,H3: nat > B,M2: nat,N: nat,G: B > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( bij_betw @ nat @ B @ H3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5817_comm__monoid__set_OatLeastAtMost__reindex,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( ord @ B )
% 11.93/6.04 => ! [F: A > A > A,Z2: A,H3: nat > B,M2: nat,N: nat,G: B > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( bij_betw @ nat @ B @ H3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ B @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ B @ ( H3 @ M2 ) @ ( H3 @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ B @ A @ nat @ G @ H3 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5818_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5819_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5820_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5821_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat,M2: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I3 ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5822_iterate__add__eq__add__iff1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ I @ J )
% 11.93/6.04 => ( ( ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ M2 )
% 11.93/6.04 = ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ N ) )
% 11.93/6.04 = ( ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ M2 )
% 11.93/6.04 = N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5823_iterate__add__eq__add__iff2,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ I @ J )
% 11.93/6.04 => ( ( ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 )
% 11.93/6.04 = ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( M2
% 11.93/6.04 = ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5824_iterate__add__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 => ( ( iterate_add @ A )
% 11.93/6.04 = ( ^ [N5: nat,A7: A] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5825_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ( comp @ nat @ A @ nat @ G
% 11.93/6.04 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5826_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ( comp @ nat @ A @ nat @ G
% 11.93/6.04 @ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
% 11.93/6.04 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5827_comm__monoid__add__class_Osum__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( comm_monoid_add @ A )
% 11.93/6.04 => ( ( groups7311177749621191930dd_sum @ B @ A )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5828_comm__monoid__mult__class_Oprod__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( comm_monoid_mult @ A )
% 11.93/6.04 => ( ( groups7121269368397514597t_prod @ B @ A )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5829_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,M2: nat,N: nat,G: nat > A] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( ord_less_eq @ nat @ M2 @ N )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5830_comm__monoid__set_Oin__pairs,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 11.93/6.04 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5831_comm__monoid__set_Oin__pairs__0,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I3 ) ) ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5832_iterate__add__less__eq__iff2,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 & ( ordere2412721322843649153imp_le @ A ) )
% 11.93/6.04 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ I @ J )
% 11.93/6.04 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( ord_less_eq @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5833_iterate__add__less__eq__iff1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 & ( ordere2412721322843649153imp_le @ A ) )
% 11.93/6.04 => ! [J: nat,I: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ J @ I )
% 11.93/6.04 => ( ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( ord_less_eq @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5834_iterate__add__less__iff1,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 & ( ordere2412721322843649153imp_le @ A ) )
% 11.93/6.04 => ! [J: nat,I: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ J @ I )
% 11.93/6.04 => ( ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5835_iterate__add__less__iff2,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( ( cancel1802427076303600483id_add @ A )
% 11.93/6.04 & ( ordere2412721322843649153imp_le @ A ) )
% 11.93/6.04 => ! [I: nat,J: nat,U: A,M2: A,N: A] :
% 11.93/6.04 ( ( ord_less_eq @ nat @ I @ J )
% 11.93/6.04 => ( ( ord_less @ A @ ( plus_plus @ A @ ( iterate_add @ A @ I @ U ) @ M2 ) @ ( plus_plus @ A @ ( iterate_add @ A @ J @ U ) @ N ) )
% 11.93/6.04 = ( ord_less @ A @ M2 @ ( plus_plus @ A @ ( iterate_add @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5836_comm__monoid__set_Otriangle__reindex,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z2 @ ( product_case_prod @ nat @ nat @ A @ G )
% 11.93/6.04 @ ( collect @ ( product_prod @ nat @ nat )
% 11.93/6.04 @ ( product_case_prod @ nat @ nat @ $o
% 11.93/6.04 @ ^ [I3: nat,J2: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [K4: nat] :
% 11.93/6.04 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ K4 ) )
% 11.93/6.04 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5837_comm__monoid__set_Otriangle__reindex__eq,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: nat > nat > A,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z2 @ ( product_case_prod @ nat @ nat @ A @ G )
% 11.93/6.04 @ ( collect @ ( product_prod @ nat @ nat )
% 11.93/6.04 @ ( product_case_prod @ nat @ nat @ $o
% 11.93/6.04 @ ^ [I3: nat,J2: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I3 @ J2 ) @ N ) ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [K4: nat] :
% 11.93/6.04 ( groups_comm_monoid_F @ A @ nat @ F @ Z2
% 11.93/6.04 @ ^ [I3: nat] : ( G @ I3 @ ( minus_minus @ nat @ K4 @ I3 ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ K4 ) )
% 11.93/6.04 @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5838_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: int > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z2 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5839_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
% 11.93/6.04 ! [A: $tType,F: A > A > A,Z2: A,G: int > A,M2: nat,N: nat] :
% 11.93/6.04 ( ( groups778175481326437816id_set @ A @ F @ Z2 )
% 11.93/6.04 => ( ( groups_comm_monoid_F @ A @ int @ F @ Z2 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
% 11.93/6.04 = ( groups_comm_monoid_F @ A @ nat @ F @ Z2 @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5840_old_Ounit_Ocase,axiom,
% 11.93/6.04 ! [A: $tType,F: A] :
% 11.93/6.04 ( ( product_case_unit @ A @ F @ product_Unity )
% 11.93/6.04 = F ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5841_wfP__wf__eq,axiom,
% 11.93/6.04 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( wfP @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.93/6.04 = ( wf @ A @ R ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5842_subset__mset_Omin__add__distrib__left,axiom,
% 11.93/6.04 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 11.93/6.04 ( ( plus_plus @ ( multiset @ A ) @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X @ Y ) @ Z2 )
% 11.93/6.04 = ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5843_subset__mset_Omin__arg__le_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,N: multiset @ A,M2: multiset @ A] :
% 11.93/6.04 ( ( subseteq_mset @ A @ N @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N ) )
% 11.93/6.04 = ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N )
% 11.93/6.04 = N ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5844_subset__mset_Omin__arg__le_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,M2: multiset @ A,N: multiset @ A] :
% 11.93/6.04 ( ( subseteq_mset @ A @ M2 @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N ) )
% 11.93/6.04 = ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M2 @ N )
% 11.93/6.04 = M2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5845_wfP__accp__iff,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( wfP @ A )
% 11.93/6.04 = ( ^ [R3: A > A > $o] :
% 11.93/6.04 ! [X8: A] : ( accp @ A @ R3 @ X8 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5846_accp__wfPI,axiom,
% 11.93/6.04 ! [A: $tType,R: A > A > $o] :
% 11.93/6.04 ( ! [X_12: A] : ( accp @ A @ R @ X_12 )
% 11.93/6.04 => ( wfP @ A @ R ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5847_accp__wfPD,axiom,
% 11.93/6.04 ! [A: $tType,R: A > A > $o,X: A] :
% 11.93/6.04 ( ( wfP @ A @ R )
% 11.93/6.04 => ( accp @ A @ R @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5848_unit_Ocase__distrib,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,H3: A > B,F: A,Unit: product_unit] :
% 11.93/6.04 ( ( H3 @ ( product_case_unit @ A @ F @ Unit ) )
% 11.93/6.04 = ( product_case_unit @ B @ ( H3 @ F ) @ Unit ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5849_subset__mset_Omin__add__distrib__right,axiom,
% 11.93/6.04 ! [A: $tType,X: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
% 11.93/6.04 ( ( plus_plus @ ( multiset @ A ) @ X @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
% 11.93/6.04 = ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5850_accp__eq__acc,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( accp @ A )
% 11.93/6.04 = ( ^ [R3: A > A > $o,X3: A] : ( member @ A @ X3 @ ( acc @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5851_rtrancl__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( transitive_rtrancl @ A )
% 11.93/6.04 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( collect @ ( product_prod @ A @ A )
% 11.93/6.04 @ ( product_case_prod @ A @ A @ $o
% 11.93/6.04 @ ( transitive_rtranclp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5852_Enum_Ortranclp__rtrancl__eq,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( transitive_rtranclp @ A )
% 11.93/6.04 = ( ^ [R3: A > A > $o,X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5853_acc_Ocases,axiom,
% 11.93/6.04 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 11.93/6.04 => ! [Y6: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ A3 ) @ R )
% 11.93/6.04 => ( member @ A @ Y6 @ ( acc @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5854_acc_Osimps,axiom,
% 11.93/6.04 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 11.93/6.04 = ( ? [X3: A] :
% 11.93/6.04 ( ( A3 = X3 )
% 11.93/6.04 & ! [Y3: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R )
% 11.93/6.04 => ( member @ A @ Y3 @ ( acc @ A @ R ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5855_acc_Ointros,axiom,
% 11.93/6.04 ! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ! [Y4: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R )
% 11.93/6.04 => ( member @ A @ Y4 @ ( acc @ A @ R ) ) )
% 11.93/6.04 => ( member @ A @ X @ ( acc @ A @ R ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5856_acc__induct,axiom,
% 11.93/6.04 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 11.93/6.04 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( ! [Y6: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 11.93/6.04 => ( P @ Y6 ) )
% 11.93/6.04 => ( P @ X2 ) ) )
% 11.93/6.04 => ( P @ A3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5857_acc__downward,axiom,
% 11.93/6.04 ! [A: $tType,B2: A,R: set @ ( product_prod @ A @ A ),A3: A] :
% 11.93/6.04 ( ( member @ A @ B2 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
% 11.93/6.04 => ( member @ A @ A3 @ ( acc @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5858_not__acc__down,axiom,
% 11.93/6.04 ! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ~ ( member @ A @ X @ ( acc @ A @ R5 ) )
% 11.93/6.04 => ~ ! [Z3: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X ) @ R5 )
% 11.93/6.04 => ( member @ A @ Z3 @ ( acc @ A @ R5 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5859_acc__induct__rule,axiom,
% 11.93/6.04 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
% 11.93/6.04 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( ! [Y6: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X2 ) @ R )
% 11.93/6.04 => ( P @ Y6 ) )
% 11.93/6.04 => ( P @ X2 ) ) )
% 11.93/6.04 => ( P @ A3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5860_rtranclp__induct2,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
% 11.93/6.04 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 11.93/6.04 => ( ( P @ Ax @ Ay )
% 11.93/6.04 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 11.93/6.04 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.93/6.04 => ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
% 11.93/6.04 => ( ( P @ A5 @ B3 )
% 11.93/6.04 => ( P @ Aa3 @ Ba ) ) ) )
% 11.93/6.04 => ( P @ Bx @ By ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5861_converse__rtranclpE2,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za2: A,Zb: B] :
% 11.93/6.04 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) )
% 11.93/6.04 => ( ( ( product_Pair @ A @ B @ Xa @ Xb )
% 11.93/6.04 != ( product_Pair @ A @ B @ Za2 @ Zb ) )
% 11.93/6.04 => ~ ! [A5: A,B3: B] :
% 11.93/6.04 ( ( R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A5 @ B3 ) )
% 11.93/6.04 => ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5862_converse__rtranclp__induct2,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
% 11.93/6.04 ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 11.93/6.04 => ( ( P @ Bx @ By )
% 11.93/6.04 => ( ! [A5: A,B3: B,Aa3: A,Ba: B] :
% 11.93/6.04 ( ( R @ ( product_Pair @ A @ B @ A5 @ B3 ) @ ( product_Pair @ A @ B @ Aa3 @ Ba ) )
% 11.93/6.04 => ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Aa3 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
% 11.93/6.04 => ( ( P @ Aa3 @ Ba )
% 11.93/6.04 => ( P @ A5 @ B3 ) ) ) )
% 11.93/6.04 => ( P @ Ax @ Ay ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5863_accp__downwards__aux,axiom,
% 11.93/6.04 ! [A: $tType,R: A > A > $o,B2: A,A3: A] :
% 11.93/6.04 ( ( transitive_rtranclp @ A @ R @ B2 @ A3 )
% 11.93/6.04 => ( ( accp @ A @ R @ A3 )
% 11.93/6.04 => ( accp @ A @ R @ B2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5864_accp__downwards,axiom,
% 11.93/6.04 ! [A: $tType,R: A > A > $o,A3: A,B2: A] :
% 11.93/6.04 ( ( accp @ A @ R @ A3 )
% 11.93/6.04 => ( ( transitive_rtranclp @ A @ R @ B2 @ A3 )
% 11.93/6.04 => ( accp @ A @ R @ B2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5865_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
% 11.93/6.04 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( transitive_rtranclp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.93/6.04 = ( ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5866_acc__subset__induct,axiom,
% 11.93/6.04 ! [A: $tType,D4: set @ A,R5: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
% 11.93/6.04 ( ( ord_less_eq @ ( set @ A ) @ D4 @ ( acc @ A @ R5 ) )
% 11.93/6.04 => ( ! [X2: A,Z3: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ D4 )
% 11.93/6.04 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X2 ) @ R5 )
% 11.93/6.04 => ( member @ A @ Z3 @ D4 ) ) )
% 11.93/6.04 => ( ( member @ A @ X @ D4 )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ D4 )
% 11.93/6.04 => ( ! [Z6: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X2 ) @ R5 )
% 11.93/6.04 => ( P @ Z6 ) )
% 11.93/6.04 => ( P @ X2 ) ) )
% 11.93/6.04 => ( P @ X ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5867_acc__downwards,axiom,
% 11.93/6.04 ! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),B2: A] :
% 11.93/6.04 ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.93/6.04 => ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5868_acc__downwards__aux,axiom,
% 11.93/6.04 ! [A: $tType,B2: A,A3: A,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
% 11.93/6.04 => ( ( member @ A @ A3 @ ( acc @ A @ R ) )
% 11.93/6.04 => ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5869_accp__acc__eq,axiom,
% 11.93/6.04 ! [A: $tType,R: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( ( accp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R ) )
% 11.93/6.04 = ( ^ [X3: A] : ( member @ A @ X3 @ ( acc @ A @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5870_acc__def,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( acc @ A )
% 11.93/6.04 = ( ^ [R3: set @ ( product_prod @ A @ A )] :
% 11.93/6.04 ( collect @ A
% 11.93/6.04 @ ( accp @ A
% 11.93/6.04 @ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5871_scomp__fcomp,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,B: $tType,E: $tType,D: $tType,F: A > ( product_prod @ D @ E ),G: D > E > C,H3: C > B] :
% 11.93/6.04 ( ( fcomp @ A @ C @ B @ ( product_scomp @ A @ D @ E @ C @ F @ G ) @ H3 )
% 11.93/6.04 = ( product_scomp @ A @ D @ E @ B @ F
% 11.93/6.04 @ ^ [X3: D] : ( fcomp @ E @ C @ B @ ( G @ X3 ) @ H3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5872_successively_Opelims_I2_J,axiom,
% 11.93/6.04 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 11.93/6.04 ( ( successively @ A @ X @ Xa )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.93/6.04 => ( ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
% 11.93/6.04 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
% 11.93/6.04 => ~ ( ( X @ X2 @ Y4 )
% 11.93/6.04 & ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5873_successively_Opelims_I1_J,axiom,
% 11.93/6.04 ! [A: $tType,X: A > A > $o,Xa: list @ A,Y: $o] :
% 11.93/6.04 ( ( ( successively @ A @ X @ Xa )
% 11.93/6.04 = Y )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.93/6.04 => ( ( ( Xa
% 11.93/6.04 = ( nil @ A ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ X2 @ ( nil @ A ) ) )
% 11.93/6.04 => ( Y
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
% 11.93/6.04 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 11.93/6.04 => ( ( Y
% 11.93/6.04 = ( ( X @ X2 @ Y4 )
% 11.93/6.04 & ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
% 11.93/6.04 => ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5874_fcomp__apply,axiom,
% 11.93/6.04 ! [A: $tType,C: $tType,B: $tType] :
% 11.93/6.04 ( ( fcomp @ B @ C @ A )
% 11.93/6.04 = ( ^ [F2: B > C,G2: C > A,X3: B] : ( G2 @ ( F2 @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5875_id__fcomp,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,G: A > B] :
% 11.93/6.04 ( ( fcomp @ A @ A @ B @ ( id @ A ) @ G )
% 11.93/6.04 = G ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5876_fcomp__id,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,F: A > B] :
% 11.93/6.04 ( ( fcomp @ A @ B @ B @ F @ ( id @ B ) )
% 11.93/6.04 = F ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5877_fcomp__scomp,axiom,
% 11.93/6.04 ! [A: $tType,E: $tType,B: $tType,D: $tType,C: $tType,F: A > E,G: E > ( product_prod @ C @ D ),H3: C > D > B] :
% 11.93/6.04 ( ( product_scomp @ A @ C @ D @ B @ ( fcomp @ A @ E @ ( product_prod @ C @ D ) @ F @ G ) @ H3 )
% 11.93/6.04 = ( fcomp @ A @ E @ B @ F @ ( product_scomp @ E @ C @ D @ B @ G @ H3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5878_fcomp__assoc,axiom,
% 11.93/6.04 ! [A: $tType,D: $tType,B: $tType,C: $tType,F: A > D,G: D > C,H3: C > B] :
% 11.93/6.04 ( ( fcomp @ A @ C @ B @ ( fcomp @ A @ D @ C @ F @ G ) @ H3 )
% 11.93/6.04 = ( fcomp @ A @ D @ B @ F @ ( fcomp @ D @ C @ B @ G @ H3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5879_fcomp__def,axiom,
% 11.93/6.04 ! [C: $tType,B: $tType,A: $tType] :
% 11.93/6.04 ( ( fcomp @ A @ B @ C )
% 11.93/6.04 = ( ^ [F2: A > B,G2: B > C,X3: A] : ( G2 @ ( F2 @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5880_fcomp__comp,axiom,
% 11.93/6.04 ! [B: $tType,C: $tType,A: $tType] :
% 11.93/6.04 ( ( fcomp @ A @ C @ B )
% 11.93/6.04 = ( ^ [F2: A > C,G2: C > B] : ( comp @ C @ B @ A @ G2 @ F2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5881_successively_Opelims_I3_J,axiom,
% 11.93/6.04 ! [A: $tType,X: A > A > $o,Xa: list @ A] :
% 11.93/6.04 ( ~ ( successively @ A @ X @ Xa )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
% 11.93/6.04 => ~ ! [X2: A,Y4: A,Xs2: list @ A] :
% 11.93/6.04 ( ( Xa
% 11.93/6.04 = ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) )
% 11.93/6.04 => ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( successively_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
% 11.93/6.04 => ( ( X @ X2 @ Y4 )
% 11.93/6.04 & ( successively @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5882_folding__idem__axioms_Ointro,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,F: A > B > B] :
% 11.93/6.04 ( ! [X2: A] :
% 11.93/6.04 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 11.93/6.04 = ( F @ X2 ) )
% 11.93/6.04 => ( finite7837460588564673216axioms @ A @ B @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5883_comp__fun__idem__axioms_Ointro,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,F: A > B > B] :
% 11.93/6.04 ( ! [X2: A] :
% 11.93/6.04 ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 11.93/6.04 = ( F @ X2 ) )
% 11.93/6.04 => ( finite1981172056906289455axioms @ A @ B @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5884_folding__idem__on__axioms_Ointro,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,S2: set @ A,F: A > B > B] :
% 11.93/6.04 ( ! [X2: A,Y4: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ S2 )
% 11.93/6.04 => ( ( member @ A @ Y4 @ S2 )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 11.93/6.04 = ( F @ X2 ) ) ) )
% 11.93/6.04 => ( finite6916993218817215295axioms @ A @ B @ S2 @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5885_folding__idem__on__axioms__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( finite6916993218817215295axioms @ A @ B )
% 11.93/6.04 = ( ^ [S5: set @ A,F2: A > B > B] :
% 11.93/6.04 ! [X3: A,Y3: A] :
% 11.93/6.04 ( ( member @ A @ X3 @ S5 )
% 11.93/6.04 => ( ( member @ A @ Y3 @ S5 )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 11.93/6.04 = ( F2 @ X3 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5886_comp__fun__idem__axioms__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( finite1981172056906289455axioms @ A @ B )
% 11.93/6.04 = ( ^ [F2: A > B > B] :
% 11.93/6.04 ! [X3: A] :
% 11.93/6.04 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 11.93/6.04 = ( F2 @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5887_folding__idem__axioms__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( finite7837460588564673216axioms @ A @ B )
% 11.93/6.04 = ( ^ [F2: A > B > B] :
% 11.93/6.04 ! [X3: A] :
% 11.93/6.04 ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 11.93/6.04 = ( F2 @ X3 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5888_comp__fun__idem__on__axioms__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( finite4980608107308702382axioms @ A @ B )
% 11.93/6.04 = ( ^ [S5: set @ A,F2: A > B > B] :
% 11.93/6.04 ! [X3: A] :
% 11.93/6.04 ( ( member @ A @ X3 @ S5 )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ X3 ) )
% 11.93/6.04 = ( F2 @ X3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5889_comp__fun__idem__on__axioms_Ointro,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,S2: set @ A,F: A > B > B] :
% 11.93/6.04 ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ S2 )
% 11.93/6.04 => ( ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ X2 ) )
% 11.93/6.04 = ( F @ X2 ) ) )
% 11.93/6.04 => ( finite4980608107308702382axioms @ A @ B @ S2 @ F ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5890_image__mset__If,axiom,
% 11.93/6.04 ! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,A4: multiset @ B] :
% 11.93/6.04 ( ( image_mset @ B @ A
% 11.93/6.04 @ ^ [X3: B] : ( if @ A @ ( P @ X3 ) @ ( F @ X3 ) @ ( G @ X3 ) )
% 11.93/6.04 @ A4 )
% 11.93/6.04 = ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ ( filter_mset @ B @ P @ A4 ) )
% 11.93/6.04 @ ( image_mset @ B @ A @ G
% 11.93/6.04 @ ( filter_mset @ B
% 11.93/6.04 @ ^ [X3: B] :
% 11.93/6.04 ~ ( P @ X3 )
% 11.93/6.04 @ A4 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5891_union__filter__mset__complement,axiom,
% 11.93/6.04 ! [A: $tType,P: A > $o,Q: A > $o,M3: multiset @ A] :
% 11.93/6.04 ( ! [X2: A] :
% 11.93/6.04 ( ( P @ X2 )
% 11.93/6.04 = ( ~ ( Q @ X2 ) ) )
% 11.93/6.04 => ( ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ Q @ M3 ) )
% 11.93/6.04 = M3 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5892_filter__union__mset,axiom,
% 11.93/6.04 ! [A: $tType,P: A > $o,M3: multiset @ A,N6: multiset @ A] :
% 11.93/6.04 ( ( filter_mset @ A @ P @ ( plus_plus @ ( multiset @ A ) @ M3 @ N6 ) )
% 11.93/6.04 = ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ P @ N6 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5893_multiset__partition,axiom,
% 11.93/6.04 ! [A: $tType,M3: multiset @ A,P: A > $o] :
% 11.93/6.04 ( M3
% 11.93/6.04 = ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 )
% 11.93/6.04 @ ( filter_mset @ A
% 11.93/6.04 @ ^ [X3: A] :
% 11.93/6.04 ~ ( P @ X3 )
% 11.93/6.04 @ M3 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5894_heap_Olub__upper,axiom,
% 11.93/6.04 ! [A: $tType,A4: set @ ( heap_Time_Heap @ A ),X: heap_Time_Heap @ A] :
% 11.93/6.04 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A4 )
% 11.93/6.04 => ( ( member @ ( heap_Time_Heap @ A ) @ X @ A4 )
% 11.93/6.04 => ( heap_Time_Heap_ord @ A @ X @ ( heap_Time_Heap_lub @ A @ A4 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5895_heap_Olub__least,axiom,
% 11.93/6.04 ! [A: $tType,A4: set @ ( heap_Time_Heap @ A ),Z2: heap_Time_Heap @ A] :
% 11.93/6.04 ( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A4 )
% 11.93/6.04 => ( ! [X2: heap_Time_Heap @ A] :
% 11.93/6.04 ( ( member @ ( heap_Time_Heap @ A ) @ X2 @ A4 )
% 11.93/6.04 => ( heap_Time_Heap_ord @ A @ X2 @ Z2 ) )
% 11.93/6.04 => ( heap_Time_Heap_ord @ A @ ( heap_Time_Heap_lub @ A @ A4 ) @ Z2 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5896_plus__int__code_I6_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( plus_plus @ int @ ( neg @ M2 ) @ ( neg @ N ) )
% 11.93/6.04 = ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5897_ccpo__Sup__mono,axiom,
% 11.93/6.04 ! [A: $tType] :
% 11.93/6.04 ( ( comple9053668089753744459l_ccpo @ A )
% 11.93/6.04 => ! [A4: set @ A,B6: set @ A] :
% 11.93/6.04 ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A4 )
% 11.93/6.04 => ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B6 )
% 11.93/6.04 => ( ! [X2: A] :
% 11.93/6.04 ( ( member @ A @ X2 @ A4 )
% 11.93/6.04 => ? [Xa2: A] :
% 11.93/6.04 ( ( member @ A @ Xa2 @ B6 )
% 11.93/6.04 & ( ord_less_eq @ A @ X2 @ Xa2 ) ) )
% 11.93/6.04 => ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A4 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5898_plus__int__code_I5_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( plus_plus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
% 11.93/6.04 = ( sub @ N @ M2 ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5899_plus__int__code_I4_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( plus_plus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
% 11.93/6.04 = ( sub @ M2 @ N ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5900_minus__int__code_I4_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( minus_minus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
% 11.93/6.04 = ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5901_plus__int__code_I3_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( plus_plus @ int @ ( pos @ M2 ) @ ( pos @ N ) )
% 11.93/6.04 = ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5902_minus__int__code_I5_J,axiom,
% 11.93/6.04 ! [M2: num,N: num] :
% 11.93/6.04 ( ( minus_minus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
% 11.93/6.04 = ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5903_Unity__def,axiom,
% 11.93/6.04 ( product_Unity
% 11.93/6.04 = ( product_Abs_unit @ $true ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5904_Domainp__Domain__eq,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
% 11.93/6.04 ( ( domainp @ A @ B
% 11.93/6.04 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) )
% 11.93/6.04 = ( ^ [X3: A] : ( member @ A @ X3 @ ( domain @ A @ B @ R ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5905_Domain__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( domain @ A @ B )
% 11.93/6.04 = ( ^ [R3: set @ ( product_prod @ A @ B )] :
% 11.93/6.04 ( collect @ A
% 11.93/6.04 @ ( domainp @ A @ B
% 11.93/6.04 @ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5906_Abs__unit__inject,axiom,
% 11.93/6.04 ! [X: $o,Y: $o] :
% 11.93/6.04 ( ( member @ $o @ X @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 11.93/6.04 => ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 11.93/6.04 => ( ( ( product_Abs_unit @ X )
% 11.93/6.04 = ( product_Abs_unit @ Y ) )
% 11.93/6.04 = ( X = Y ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5907_Abs__unit__induct,axiom,
% 11.93/6.04 ! [P: product_unit > $o,X: product_unit] :
% 11.93/6.04 ( ! [Y4: $o] :
% 11.93/6.04 ( ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 11.93/6.04 => ( P @ ( product_Abs_unit @ Y4 ) ) )
% 11.93/6.04 => ( P @ X ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5908_Abs__unit__cases,axiom,
% 11.93/6.04 ! [X: product_unit] :
% 11.93/6.04 ~ ! [Y4: $o] :
% 11.93/6.04 ( ( X
% 11.93/6.04 = ( product_Abs_unit @ Y4 ) )
% 11.93/6.04 => ~ ( member @ $o @ Y4 @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5909_Heap_ODomainp__rel,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 11.93/6.04 ( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) )
% 11.93/6.04 = ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5910_integer_Odomain,axiom,
% 11.93/6.04 ! [P: int > $o] :
% 11.93/6.04 ( ( ( domainp @ int @ int
% 11.93/6.04 @ ^ [Y5: int,Z4: int] : ( Y5 = Z4 ) )
% 11.93/6.04 = P )
% 11.93/6.04 => ( ( domainp @ int @ code_integer @ code_pcr_integer )
% 11.93/6.04 = P ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5911_natural_Odomain,axiom,
% 11.93/6.04 ! [P: nat > $o] :
% 11.93/6.04 ( ( ( domainp @ nat @ nat
% 11.93/6.04 @ ^ [Y5: nat,Z4: nat] : ( Y5 = Z4 ) )
% 11.93/6.04 = P )
% 11.93/6.04 => ( ( domainp @ nat @ code_natural @ code_pcr_natural )
% 11.93/6.04 = P ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5912_option_ODomainp__rel,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType,R5: A > B > $o] :
% 11.93/6.04 ( ( domainp @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) )
% 11.93/6.04 = ( pred_option @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5913_Abs__unit__inverse,axiom,
% 11.93/6.04 ! [Y: $o] :
% 11.93/6.04 ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 11.93/6.04 => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
% 11.93/6.04 = Y ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5914_type__definition__unit,axiom,
% 11.93/6.04 type_definition @ product_unit @ $o @ product_Rep_unit @ product_Abs_unit @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5915_Rep__unit__inverse,axiom,
% 11.93/6.04 ! [X: product_unit] :
% 11.93/6.04 ( ( product_Abs_unit @ ( product_Rep_unit @ X ) )
% 11.93/6.04 = X ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5916_Rep__unit__inject,axiom,
% 11.93/6.04 ! [X: product_unit,Y: product_unit] :
% 11.93/6.04 ( ( ( product_Rep_unit @ X )
% 11.93/6.04 = ( product_Rep_unit @ Y ) )
% 11.93/6.04 = ( X = Y ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5917_Rep__unit,axiom,
% 11.93/6.04 ! [X: product_unit] : ( member @ $o @ ( product_Rep_unit @ X ) @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5918_Rep__unit__cases,axiom,
% 11.93/6.04 ! [Y: $o] :
% 11.93/6.04 ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 11.93/6.04 => ~ ! [X2: product_unit] :
% 11.93/6.04 ( Y
% 11.93/6.04 = ( ~ ( product_Rep_unit @ X2 ) ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5919_Rep__unit__induct,axiom,
% 11.93/6.04 ! [Y: $o,P: $o > $o] :
% 11.93/6.04 ( ( member @ $o @ Y @ ( insert2 @ $o @ $true @ ( bot_bot @ ( set @ $o ) ) ) )
% 11.93/6.04 => ( ! [X2: product_unit] : ( P @ ( product_Rep_unit @ X2 ) )
% 11.93/6.04 => ( P @ Y ) ) ) ).
% 11.93/6.04
% 11.93/6.04 thf(fact_5920_adm__wf__def,axiom,
% 11.93/6.04 ! [B: $tType,A: $tType] :
% 11.93/6.04 ( ( adm_wf @ A @ B )
% 11.93/6.04 = ( ^ [R6: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
% 11.93/6.04 ! [F2: A > B,G2: A > B,X3: A] :
% 11.93/6.04 ( ! [Z5: A] :
% 11.93/6.04 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ X3 ) @ R6 )
% 11.93/6.05 => ( ( F2 @ Z5 )
% 11.93/6.05 = ( G2 @ Z5 ) ) )
% 11.93/6.05 => ( ( F7 @ F2 @ X3 )
% 11.93/6.05 = ( F7 @ G2 @ X3 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5921_folding__idem_Ocomp__fun__idem,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,F: A > B > B,X: A] :
% 11.93/6.05 ( ( finite_folding_idem @ A @ B @ F )
% 11.93/6.05 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 11.93/6.05 = ( F @ X ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5922_compat__def,axiom,
% 11.93/6.05 ! [A2: $tType,A: $tType] :
% 11.93/6.05 ( ( bNF_Wellorder_compat @ A @ A2 )
% 11.93/6.05 = ( ^ [R3: set @ ( product_prod @ A @ A ),R11: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
% 11.93/6.05 ! [A7: A,B5: A] :
% 11.93/6.05 ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A7 @ B5 ) @ R3 )
% 11.93/6.05 => ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F2 @ A7 ) @ ( F2 @ B5 ) ) @ R11 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5923_positive__add,axiom,
% 11.93/6.05 ! [X: rat,Y: rat] :
% 11.93/6.05 ( ( positive @ X )
% 11.93/6.05 => ( ( positive @ Y )
% 11.93/6.05 => ( positive @ ( plus_plus @ rat @ X @ Y ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5924_inj__on__iff__Uniq,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( inj_on @ A @ B )
% 11.93/6.05 = ( ^ [F2: A > B,A9: set @ A] :
% 11.93/6.05 ! [X3: A] :
% 11.93/6.05 ( ( member @ A @ X3 @ A9 )
% 11.93/6.05 => ( uniq @ A
% 11.93/6.05 @ ^ [Y3: A] :
% 11.93/6.05 ( ( member @ A @ Y3 @ A9 )
% 11.93/6.05 & ( ( F2 @ X3 )
% 11.93/6.05 = ( F2 @ Y3 ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5925_mono__imp__mono__on,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( order @ A )
% 11.93/6.05 & ( order @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A] :
% 11.93/6.05 ( ( order_mono @ A @ B @ F )
% 11.93/6.05 => ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5926_mono__on__subset,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A,B6: set @ A] :
% 11.93/6.05 ( ( mono_on @ A @ B @ F @ A4 )
% 11.93/6.05 => ( ( ord_less_eq @ ( set @ A ) @ B6 @ A4 )
% 11.93/6.05 => ( mono_on @ A @ B @ F @ B6 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5927_mono__on__greaterD,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( linorder @ A )
% 11.93/6.05 & ( linorder @ B ) )
% 11.93/6.05 => ! [G: A > B,A4: set @ A,X: A,Y: A] :
% 11.93/6.05 ( ( mono_on @ A @ B @ G @ A4 )
% 11.93/6.05 => ( ( member @ A @ X @ A4 )
% 11.93/6.05 => ( ( member @ A @ Y @ A4 )
% 11.93/6.05 => ( ( ord_less @ B @ ( G @ Y ) @ ( G @ X ) )
% 11.93/6.05 => ( ord_less @ A @ Y @ X ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5928_mono__onD,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A,R: A,S: A] :
% 11.93/6.05 ( ( mono_on @ A @ B @ F @ A4 )
% 11.93/6.05 => ( ( member @ A @ R @ A4 )
% 11.93/6.05 => ( ( member @ A @ S @ A4 )
% 11.93/6.05 => ( ( ord_less_eq @ A @ R @ S )
% 11.93/6.05 => ( ord_less_eq @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5929_mono__onI,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ! [A4: set @ A,F: A > B] :
% 11.93/6.05 ( ! [R2: A,S6: A] :
% 11.93/6.05 ( ( member @ A @ R2 @ A4 )
% 11.93/6.05 => ( ( member @ A @ S6 @ A4 )
% 11.93/6.05 => ( ( ord_less_eq @ A @ R2 @ S6 )
% 11.93/6.05 => ( ord_less_eq @ B @ ( F @ R2 ) @ ( F @ S6 ) ) ) ) )
% 11.93/6.05 => ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5930_mono__on__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ( ( mono_on @ A @ B )
% 11.93/6.05 = ( ^ [F2: A > B,A9: set @ A] :
% 11.93/6.05 ! [R3: A,S4: A] :
% 11.93/6.05 ( ( ( member @ A @ R3 @ A9 )
% 11.93/6.05 & ( member @ A @ S4 @ A9 )
% 11.93/6.05 & ( ord_less_eq @ A @ R3 @ S4 ) )
% 11.93/6.05 => ( ord_less_eq @ B @ ( F2 @ R3 ) @ ( F2 @ S4 ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5931_strict__mono__on__imp__mono__on,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( linorder @ A )
% 11.93/6.05 & ( preorder @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A] :
% 11.93/6.05 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 11.93/6.05 => ( mono_on @ A @ B @ F @ A4 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5932_folding_Ointro,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,F: A > B > B] :
% 11.93/6.05 ( ! [Y4: A,X2: A] :
% 11.93/6.05 ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X2 ) )
% 11.93/6.05 = ( comp @ B @ B @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 11.93/6.05 => ( finite_folding @ A @ B @ F ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5933_folding_Ocomp__fun__commute,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,F: A > B > B,Y: A,X: A] :
% 11.93/6.05 ( ( finite_folding @ A @ B @ F )
% 11.93/6.05 => ( ( comp @ B @ B @ B @ ( F @ Y ) @ ( F @ X ) )
% 11.93/6.05 = ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5934_strict__mono__on__eqD,axiom,
% 11.93/6.05 ! [D: $tType,C: $tType] :
% 11.93/6.05 ( ( ( linorder @ C )
% 11.93/6.05 & ( preorder @ D ) )
% 11.93/6.05 => ! [F: C > D,A4: set @ C,X: C,Y: C] :
% 11.93/6.05 ( ( strict_mono_on @ C @ D @ F @ A4 )
% 11.93/6.05 => ( ( ( F @ X )
% 11.93/6.05 = ( F @ Y ) )
% 11.93/6.05 => ( ( member @ C @ X @ A4 )
% 11.93/6.05 => ( ( member @ C @ Y @ A4 )
% 11.93/6.05 => ( Y = X ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5935_strict__mono__on__leD,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( linorder @ A )
% 11.93/6.05 & ( preorder @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A,X: A,Y: A] :
% 11.93/6.05 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 11.93/6.05 => ( ( member @ A @ X @ A4 )
% 11.93/6.05 => ( ( member @ A @ Y @ A4 )
% 11.93/6.05 => ( ( ord_less_eq @ A @ X @ Y )
% 11.93/6.05 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5936_strict__mono__on__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ( ( strict_mono_on @ A @ B )
% 11.93/6.05 = ( ^ [F2: A > B,A9: set @ A] :
% 11.93/6.05 ! [R3: A,S4: A] :
% 11.93/6.05 ( ( ( member @ A @ R3 @ A9 )
% 11.93/6.05 & ( member @ A @ S4 @ A9 )
% 11.93/6.05 & ( ord_less @ A @ R3 @ S4 ) )
% 11.93/6.05 => ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S4 ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5937_strict__mono__onI,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ! [A4: set @ A,F: A > B] :
% 11.93/6.05 ( ! [R2: A,S6: A] :
% 11.93/6.05 ( ( member @ A @ R2 @ A4 )
% 11.93/6.05 => ( ( member @ A @ S6 @ A4 )
% 11.93/6.05 => ( ( ord_less @ A @ R2 @ S6 )
% 11.93/6.05 => ( ord_less @ B @ ( F @ R2 ) @ ( F @ S6 ) ) ) ) )
% 11.93/6.05 => ( strict_mono_on @ A @ B @ F @ A4 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5938_strict__mono__onD,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( ord @ A )
% 11.93/6.05 & ( ord @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A,R: A,S: A] :
% 11.93/6.05 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 11.93/6.05 => ( ( member @ A @ R @ A4 )
% 11.93/6.05 => ( ( member @ A @ S @ A4 )
% 11.93/6.05 => ( ( ord_less @ A @ R @ S )
% 11.93/6.05 => ( ord_less @ B @ ( F @ R ) @ ( F @ S ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5939_strict__mono__on__imp__inj__on,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( ( linorder @ A )
% 11.93/6.05 & ( preorder @ B ) )
% 11.93/6.05 => ! [F: A > B,A4: set @ A] :
% 11.93/6.05 ( ( strict_mono_on @ A @ B @ F @ A4 )
% 11.93/6.05 => ( inj_on @ A @ B @ F @ A4 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5940_folding__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( finite_folding @ A @ B )
% 11.93/6.05 = ( ^ [F2: A > B > B] :
% 11.93/6.05 ! [Y3: A,X3: A] :
% 11.93/6.05 ( ( comp @ B @ B @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) )
% 11.93/6.05 = ( comp @ B @ B @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5941_folding__idem__on_Ocomp__fun__idem__on,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,S2: set @ A,F: A > B > B,X: A,Y: A] :
% 11.93/6.05 ( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F )
% 11.93/6.05 => ( ( member @ A @ X @ S2 )
% 11.93/6.05 => ( ( member @ A @ Y @ S2 )
% 11.93/6.05 => ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
% 11.93/6.05 = ( F @ X ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5942_sum_OPlus,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,C: $tType] :
% 11.93/6.05 ( ( comm_monoid_add @ A )
% 11.93/6.05 => ! [A4: set @ B,B6: set @ C,G: ( sum_sum @ B @ C ) > A] :
% 11.93/6.05 ( ( finite_finite2 @ B @ A4 )
% 11.93/6.05 => ( ( finite_finite2 @ C @ B6 )
% 11.93/6.05 => ( ( groups7311177749621191930dd_sum @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A4 @ B6 ) )
% 11.93/6.05 = ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A4 ) @ ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B6 ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5943_map__sum__Inl__conv,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,D: $tType,C: $tType,Fl: C > A,Fr: D > B,S: sum_sum @ C @ D,Y: A] :
% 11.93/6.05 ( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S )
% 11.93/6.05 = ( sum_Inl @ A @ B @ Y ) )
% 11.93/6.05 = ( ? [X3: C] :
% 11.93/6.05 ( ( S
% 11.93/6.05 = ( sum_Inl @ C @ D @ X3 ) )
% 11.93/6.05 & ( Y
% 11.93/6.05 = ( Fl @ X3 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5944_map__sum__Inr__conv,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,C: $tType,D: $tType,Fl: C > A,Fr: D > B,S: sum_sum @ C @ D,Y: B] :
% 11.93/6.05 ( ( ( sum_map_sum @ C @ A @ D @ B @ Fl @ Fr @ S )
% 11.93/6.05 = ( sum_Inr @ B @ A @ Y ) )
% 11.93/6.05 = ( ? [X3: D] :
% 11.93/6.05 ( ( S
% 11.93/6.05 = ( sum_Inr @ D @ C @ X3 ) )
% 11.93/6.05 & ( Y
% 11.93/6.05 = ( Fr @ X3 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5945_Union__plus,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,C: $tType,F: ( sum_sum @ B @ C ) > ( set @ A ),A4: set @ B,B6: set @ C] :
% 11.93/6.05 ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( sum_sum @ B @ C ) @ ( set @ A ) @ F @ ( sum_Plus @ B @ C @ A4 @ B6 ) ) )
% 11.93/6.05 = ( sup_sup @ ( set @ A )
% 11.93/6.05 @ ( complete_Sup_Sup @ ( set @ A )
% 11.93/6.05 @ ( image2 @ B @ ( set @ A )
% 11.93/6.05 @ ^ [A7: B] : ( F @ ( sum_Inl @ B @ C @ A7 ) )
% 11.93/6.05 @ A4 ) )
% 11.93/6.05 @ ( complete_Sup_Sup @ ( set @ A )
% 11.93/6.05 @ ( image2 @ C @ ( set @ A )
% 11.93/6.05 @ ^ [B5: C] : ( F @ ( sum_Inr @ C @ B @ B5 ) )
% 11.93/6.05 @ B6 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5946_Union__sum,axiom,
% 11.93/6.05 ! [C: $tType,A: $tType,B: $tType,F: ( sum_sum @ A @ B ) > ( set @ C )] :
% 11.93/6.05 ( ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ ( sum_sum @ A @ B ) @ ( set @ C ) @ F @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) ) )
% 11.93/6.05 = ( sup_sup @ ( set @ C )
% 11.93/6.05 @ ( complete_Sup_Sup @ ( set @ C )
% 11.93/6.05 @ ( image2 @ A @ ( set @ C )
% 11.93/6.05 @ ^ [L3: A] : ( F @ ( sum_Inl @ A @ B @ L3 ) )
% 11.93/6.05 @ ( top_top @ ( set @ A ) ) ) )
% 11.93/6.05 @ ( complete_Sup_Sup @ ( set @ C )
% 11.93/6.05 @ ( image2 @ B @ ( set @ C )
% 11.93/6.05 @ ^ [R3: B] : ( F @ ( sum_Inr @ B @ A @ R3 ) )
% 11.93/6.05 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5947_sum_Osize__gen_I1_J,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,Xa: A > nat,X: B > nat,X1: A] :
% 11.93/6.05 ( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inl @ A @ B @ X1 ) )
% 11.93/6.05 = ( plus_plus @ nat @ ( Xa @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5948_sum_Osize__gen_I2_J,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,Xa: A > nat,X: B > nat,X22: B] :
% 11.93/6.05 ( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inr @ B @ A @ X22 ) )
% 11.93/6.05 = ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5949_Node__def,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType] :
% 11.93/6.05 ( ( old_Node @ B @ A )
% 11.93/6.05 = ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
% 11.93/6.05 @ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
% 11.93/6.05 ? [F2: nat > ( sum_sum @ B @ nat ),X3: sum_sum @ A @ nat,K4: nat] :
% 11.93/6.05 ( ( P6
% 11.93/6.05 = ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F2 @ X3 ) )
% 11.93/6.05 & ( ( F2 @ K4 )
% 11.93/6.05 = ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5950_Node__K0__I,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
% 11.93/6.05 ( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
% 11.93/6.05 @ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
% 11.93/6.05 @ ^ [K4: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
% 11.93/6.05 @ A3 )
% 11.93/6.05 @ ( old_Node @ A @ B ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5951_nth__item_Opinduct,axiom,
% 11.93/6.05 ! [A0: nat,P: nat > $o] :
% 11.93/6.05 ( ( accp @ nat @ nth_item_rel @ A0 )
% 11.93/6.05 => ( ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
% 11.93/6.05 => ( P @ ( zero_zero @ nat ) ) )
% 11.93/6.05 => ( ! [N3: nat] :
% 11.93/6.05 ( ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) )
% 11.93/6.05 => ( ! [A17: nat,Aa4: nat] :
% 11.93/6.05 ( ( ( nat_sum_decode @ N3 )
% 11.93/6.05 = ( sum_Inl @ nat @ nat @ A17 ) )
% 11.93/6.05 => ( ( ( nat_sum_decode @ A17 )
% 11.93/6.05 = ( sum_Inl @ nat @ nat @ Aa4 ) )
% 11.93/6.05 => ( P @ Aa4 ) ) )
% 11.93/6.05 => ( ! [A17: nat,B8: nat] :
% 11.93/6.05 ( ( ( nat_sum_decode @ N3 )
% 11.93/6.05 = ( sum_Inl @ nat @ nat @ A17 ) )
% 11.93/6.05 => ( ( ( nat_sum_decode @ A17 )
% 11.93/6.05 = ( sum_Inr @ nat @ nat @ B8 ) )
% 11.93/6.05 => ( P @ B8 ) ) )
% 11.93/6.05 => ( ! [B8: nat,Ba2: nat,X7: nat,Y6: nat] :
% 11.93/6.05 ( ( ( nat_sum_decode @ N3 )
% 11.93/6.05 = ( sum_Inr @ nat @ nat @ B8 ) )
% 11.93/6.05 => ( ( ( nat_sum_decode @ B8 )
% 11.93/6.05 = ( sum_Inr @ nat @ nat @ Ba2 ) )
% 11.93/6.05 => ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
% 11.93/6.05 = ( nat_prod_decode @ Ba2 ) )
% 11.93/6.05 => ( P @ X7 ) ) ) )
% 11.93/6.05 => ( ! [B8: nat,Ba2: nat,X7: nat,Y6: nat] :
% 11.93/6.05 ( ( ( nat_sum_decode @ N3 )
% 11.93/6.05 = ( sum_Inr @ nat @ nat @ B8 ) )
% 11.93/6.05 => ( ( ( nat_sum_decode @ B8 )
% 11.93/6.05 = ( sum_Inr @ nat @ nat @ Ba2 ) )
% 11.93/6.05 => ( ( ( product_Pair @ nat @ nat @ X7 @ Y6 )
% 11.93/6.05 = ( nat_prod_decode @ Ba2 ) )
% 11.93/6.05 => ( P @ Y6 ) ) ) )
% 11.93/6.05 => ( P @ ( suc @ N3 ) ) ) ) ) ) )
% 11.93/6.05 => ( P @ A0 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5952_ndepth__K0,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
% 11.93/6.05 ( ( old_ndepth @ A @ B
% 11.93/6.05 @ ( old_Abs_Node @ B @ A
% 11.93/6.05 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 11.93/6.05 @ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 11.93/6.05 @ X ) ) )
% 11.93/6.05 = ( zero_zero @ nat ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5953_Atom__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( old_Atom @ A @ B )
% 11.93/6.05 = ( ^ [X3: sum_sum @ A @ nat] :
% 11.93/6.05 ( insert2 @ ( old_node @ A @ B )
% 11.93/6.05 @ ( old_Abs_Node @ B @ A
% 11.93/6.05 @ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
% 11.93/6.05 @ ^ [K4: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
% 11.93/6.05 @ X3 ) )
% 11.93/6.05 @ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5954_nth__item_Opsimps_I1_J,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( countable @ A )
% 11.93/6.05 => ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
% 11.93/6.05 => ( ( nth_item @ A @ ( zero_zero @ nat ) )
% 11.93/6.05 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5955_nth__item_Opelims,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( countable @ A )
% 11.93/6.05 => ! [X: nat,Y: set @ ( old_node @ A @ product_unit )] :
% 11.93/6.05 ( ( ( nth_item @ A @ X )
% 11.93/6.05 = Y )
% 11.93/6.05 => ( ( accp @ nat @ nth_item_rel @ X )
% 11.93/6.05 => ( ( ( X
% 11.93/6.05 = ( zero_zero @ nat ) )
% 11.93/6.05 => ( ( Y
% 11.93/6.05 = ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
% 11.93/6.05 => ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
% 11.93/6.05 => ~ ! [N3: nat] :
% 11.93/6.05 ( ( X
% 11.93/6.05 = ( suc @ N3 ) )
% 11.93/6.05 => ( ( Y
% 11.93/6.05 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 11.93/6.05 @ ^ [I3: nat] :
% 11.93/6.05 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 11.93/6.05 @ ^ [J2: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 11.93/6.05 @ ^ [J2: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 11.93/6.05 @ ( nat_sum_decode @ I3 ) )
% 11.93/6.05 @ ^ [I3: nat] :
% 11.93/6.05 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 11.93/6.05 @ ^ [J2: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J2 ) )
% 11.93/6.05 @ ^ [J2: nat] :
% 11.93/6.05 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 11.93/6.05 @ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
% 11.93/6.05 @ ( nat_prod_decode @ J2 ) )
% 11.93/6.05 @ ( nat_sum_decode @ I3 ) )
% 11.93/6.05 @ ( nat_sum_decode @ N3 ) ) )
% 11.93/6.05 => ~ ( accp @ nat @ nth_item_rel @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5956_nth__item_Opsimps_I2_J,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( countable @ A )
% 11.93/6.05 => ! [N: nat] :
% 11.93/6.05 ( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
% 11.93/6.05 => ( ( nth_item @ A @ ( suc @ N ) )
% 11.93/6.05 = ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 11.93/6.05 @ ^ [I3: nat] :
% 11.93/6.05 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 11.93/6.05 @ ^ [J2: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 11.93/6.05 @ ^ [J2: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J2 ) )
% 11.93/6.05 @ ( nat_sum_decode @ I3 ) )
% 11.93/6.05 @ ^ [I3: nat] :
% 11.93/6.05 ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
% 11.93/6.05 @ ^ [J2: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J2 ) )
% 11.93/6.05 @ ^ [J2: nat] :
% 11.93/6.05 ( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
% 11.93/6.05 @ ^ [A7: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A7 ) @ ( nth_item @ A @ B5 ) )
% 11.93/6.05 @ ( nat_prod_decode @ J2 ) )
% 11.93/6.05 @ ( nat_sum_decode @ I3 ) )
% 11.93/6.05 @ ( nat_sum_decode @ N ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5957_Scons__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( old_Scons @ A @ B )
% 11.93/6.05 = ( ^ [M8: set @ ( old_node @ A @ B ),N9: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M8 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N9 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5958_In1__def,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType] :
% 11.93/6.05 ( ( old_In1 @ A @ B )
% 11.93/6.05 = ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5959_dsum__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( old_dsum @ A @ B )
% 11.93/6.05 = ( ^ [R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( sup_sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X3 ) @ ( old_In0 @ A @ B @ X9 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
% 11.93/6.05 @ R3 ) )
% 11.93/6.05 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y3 ) @ ( old_In1 @ A @ B @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
% 11.93/6.05 @ S4 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5960_dsumE,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,W: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ W @ ( old_dsum @ A @ B @ R @ S ) )
% 11.93/6.05 => ( ! [X2: set @ ( old_node @ A @ B ),X5: set @ ( old_node @ A @ B )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X5 ) @ R )
% 11.93/6.05 => ( W
% 11.93/6.05 != ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X2 ) @ ( old_In0 @ A @ B @ X5 ) ) ) )
% 11.93/6.05 => ~ ! [Y4: set @ ( old_node @ A @ B ),Y14: set @ ( old_node @ A @ B )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y14 ) @ S )
% 11.93/6.05 => ( W
% 11.93/6.05 != ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y4 ) @ ( old_In1 @ A @ B @ Y14 ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5961_dsum__In0I,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,M3: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M3 @ M10 ) @ R )
% 11.93/6.05 => ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ M3 ) @ ( old_In0 @ A @ B @ M10 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5962_dsum__In1I,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,N6: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N11 ) @ S )
% 11.93/6.05 => ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ N6 ) @ ( old_In1 @ A @ B @ N11 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5963_dprod__def,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType] :
% 11.93/6.05 ( ( old_dprod @ A @ B )
% 11.93/6.05 = ( ^ [R3: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ^ [X3: set @ ( old_node @ A @ B ),X9: set @ ( old_node @ A @ B )] :
% 11.93/6.05 ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
% 11.93/6.05 @ ^ [Y3: set @ ( old_node @ A @ B ),Y8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X3 @ Y3 ) @ ( old_Scons @ A @ B @ X9 @ Y8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
% 11.93/6.05 @ S4 ) ) )
% 11.93/6.05 @ R3 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5964_dprodI,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,M3: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),N6: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M3 @ M10 ) @ R )
% 11.93/6.05 => ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N11 ) @ S )
% 11.93/6.05 => ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ M3 @ N6 ) @ ( old_Scons @ A @ B @ M10 @ N11 ) ) @ ( old_dprod @ A @ B @ R @ S ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5965_dprodE,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,C2: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ C2 @ ( old_dprod @ A @ B @ R @ S ) )
% 11.93/6.05 => ~ ! [X2: set @ ( old_node @ A @ B ),Y4: set @ ( old_node @ A @ B ),X5: set @ ( old_node @ A @ B )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X2 @ X5 ) @ R )
% 11.93/6.05 => ! [Y14: set @ ( old_node @ A @ B )] :
% 11.93/6.05 ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y14 ) @ S )
% 11.93/6.05 => ( C2
% 11.93/6.05 != ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y4 ) @ ( old_Scons @ A @ B @ X5 @ Y14 ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5966_prop__matchD,axiom,
% 11.93/6.05 ! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,E6: A,Bl2: list @ A,P: A > $o] :
% 11.93/6.05 ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
% 11.93/6.05 = ( append @ A @ Al2 @ ( cons @ A @ E6 @ Bl2 ) ) )
% 11.93/6.05 => ( ( list_all @ A @ P @ Al )
% 11.93/6.05 => ( ~ ( P @ E2 )
% 11.93/6.05 => ( ~ ( P @ E6 )
% 11.93/6.05 => ( ( list_all @ A @ P @ Bl )
% 11.93/6.05 => ( ( Al = Al2 )
% 11.93/6.05 & ( E2 = E6 )
% 11.93/6.05 & ( Bl = Bl2 ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5967_prop__match,axiom,
% 11.93/6.05 ! [A: $tType,P: A > $o,Al: list @ A,E2: A,E6: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
% 11.93/6.05 ( ( list_all @ A @ P @ Al )
% 11.93/6.05 => ( ~ ( P @ E2 )
% 11.93/6.05 => ( ~ ( P @ E6 )
% 11.93/6.05 => ( ( list_all @ A @ P @ Bl )
% 11.93/6.05 => ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
% 11.93/6.05 = ( append @ A @ Al2 @ ( cons @ A @ E6 @ Bl2 ) ) )
% 11.93/6.05 = ( ( Al = Al2 )
% 11.93/6.05 & ( E2 = E6 )
% 11.93/6.05 & ( Bl = Bl2 ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5968_Heap__lub__def,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( heap_Time_Heap_lub @ A )
% 11.93/6.05 = ( partial_img_lub @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_execute @ A ) @ ( heap_Time_Heap2 @ A ) @ ( partial_fun_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_lub @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5969_heap__step__admissible,axiom,
% 11.93/6.05 ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: E > A > C > B > D > $o,X: E] :
% 11.93/6.05 ( comple1908693960933563346ssible @ ( A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) @ ( partial_fun_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_lub @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) ) @ ( partial_fun_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ A @ ( partial_flat_ord @ ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) ) )
% 11.93/6.05 @ ^ [F2: A > ( option @ ( product_prod @ B @ ( product_prod @ C @ D ) ) )] :
% 11.93/6.05 ! [H: A,H7: C,R3: B,N5: D] :
% 11.93/6.05 ( ( ( F2 @ H )
% 11.93/6.05 = ( some @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ R3 @ ( product_Pair @ C @ D @ H7 @ N5 ) ) ) )
% 11.93/6.05 => ( P @ X @ H @ H7 @ R3 @ N5 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5970_option__admissible,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,P: A > B > $o] :
% 11.93/6.05 ( comple1908693960933563346ssible @ ( A > ( option @ B ) ) @ ( partial_fun_lub @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_lub @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) )
% 11.93/6.05 @ ^ [F2: A > ( option @ B )] :
% 11.93/6.05 ! [X3: A,Y3: B] :
% 11.93/6.05 ( ( ( F2 @ X3 )
% 11.93/6.05 = ( some @ B @ Y3 ) )
% 11.93/6.05 => ( P @ X3 @ Y3 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5971_option_Olub__upper,axiom,
% 11.93/6.05 ! [A: $tType,A4: set @ ( option @ A ),X: option @ A] :
% 11.93/6.05 ( ( comple1602240252501008431_chain @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ A4 )
% 11.93/6.05 => ( ( member @ ( option @ A ) @ X @ A4 )
% 11.93/6.05 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) @ A4 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5972_option_Oleq__antisym,axiom,
% 11.93/6.05 ! [A: $tType,X: option @ A,Y: option @ A] :
% 11.93/6.05 ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Y )
% 11.93/6.05 => ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y @ X )
% 11.93/6.05 => ( X = Y ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5973_option_Oleq__trans,axiom,
% 11.93/6.05 ! [A: $tType,X: option @ A,Y: option @ A,Z2: option @ A] :
% 11.93/6.05 ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Y )
% 11.93/6.05 => ( ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ Y @ Z2 )
% 11.93/6.05 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ Z2 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5974_option_Oleq__refl,axiom,
% 11.93/6.05 ! [A: $tType,X: option @ A] : ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X @ X ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5975_option_Olub__least,axiom,
% 11.93/6.05 ! [A: $tType,A4: set @ ( option @ A ),Z2: option @ A] :
% 11.93/6.05 ( ( comple1602240252501008431_chain @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ A4 )
% 11.93/6.05 => ( ! [X2: option @ A] :
% 11.93/6.05 ( ( member @ ( option @ A ) @ X2 @ A4 )
% 11.93/6.05 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ X2 @ Z2 ) )
% 11.93/6.05 => ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) @ A4 ) @ Z2 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5976_Heap__ord__def,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( heap_Time_Heap_ord @ A )
% 11.93/6.05 = ( partial_img_ord @ ( heap_Time_Heap @ A ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ $o @ ( heap_Time_execute @ A ) @ ( partial_fun_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( partial_flat_ord @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5977_Partial__Function_Obind__mono,axiom,
% 11.93/6.05 ! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( option @ B ) ) > ( option @ C ),C5: C > ( A > ( option @ B ) ) > ( option @ D )] :
% 11.93/6.05 ( ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ C ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ C ) @ ( none @ C ) ) @ B6 )
% 11.93/6.05 => ( ! [Y4: C] : ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ D ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ D ) @ ( none @ D ) ) @ ( C5 @ Y4 ) )
% 11.93/6.05 => ( comple7038119648293358887notone @ ( A > ( option @ B ) ) @ ( option @ D ) @ ( partial_fun_ord @ ( option @ B ) @ ( option @ B ) @ A @ ( partial_flat_ord @ ( option @ B ) @ ( none @ B ) ) ) @ ( partial_flat_ord @ ( option @ D ) @ ( none @ D ) )
% 11.93/6.05 @ ^ [F2: A > ( option @ B )] :
% 11.93/6.05 ( bind @ C @ D @ ( B6 @ F2 )
% 11.93/6.05 @ ^ [Y3: C] : ( C5 @ Y3 @ F2 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5978_option_Oconst__mono,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,Ord: B > B > $o,C2: option @ A] :
% 11.93/6.05 ( comple7038119648293358887notone @ B @ ( option @ A ) @ Ord @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 11.93/6.05 @ ^ [F2: B] : C2 ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5979_option_Ofixp__induct__uc,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: ( B > ( option @ A ) ) > $o] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 11.93/6.05 => ( ( F
% 11.93/6.05 = ( C5
% 11.93/6.05 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 11.93/6.05 => ( ! [F4: B > ( option @ A )] :
% 11.93/6.05 ( ( U4 @ ( C5 @ F4 ) )
% 11.93/6.05 = F4 )
% 11.93/6.05 => ( ( comple1908693960933563346ssible @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ P )
% 11.93/6.05 => ( ( P
% 11.93/6.05 @ ^ [Uu: B] : ( none @ A ) )
% 11.93/6.05 => ( ! [F4: C] :
% 11.93/6.05 ( ( P @ ( U4 @ F4 ) )
% 11.93/6.05 => ( P @ ( U4 @ ( F6 @ F4 ) ) ) )
% 11.93/6.05 => ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5980_fixp__induct__option,axiom,
% 11.93/6.05 ! [C: $tType,B: $tType,A: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C,P: B > A > $o,X: B,Y: A] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 11.93/6.05 => ( ( F
% 11.93/6.05 = ( C5
% 11.93/6.05 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 11.93/6.05 => ( ! [F4: B > ( option @ A )] :
% 11.93/6.05 ( ( U4 @ ( C5 @ F4 ) )
% 11.93/6.05 = F4 )
% 11.93/6.05 => ( ! [F4: C,X2: B,Y4: A] :
% 11.93/6.05 ( ! [Xa2: B,Ya3: A] :
% 11.93/6.05 ( ( ( U4 @ F4 @ Xa2 )
% 11.93/6.05 = ( some @ A @ Ya3 ) )
% 11.93/6.05 => ( P @ Xa2 @ Ya3 ) )
% 11.93/6.05 => ( ( ( U4 @ ( F6 @ F4 ) @ X2 )
% 11.93/6.05 = ( some @ A @ Y4 ) )
% 11.93/6.05 => ( P @ X2 @ Y4 ) ) )
% 11.93/6.05 => ( ( ( U4 @ F @ X )
% 11.93/6.05 = ( some @ A @ Y ) )
% 11.93/6.05 => ( P @ X @ Y ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5981_option_Ofixp__rule__uc,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( option @ A ),F6: C > C,C5: ( B > ( option @ A ) ) > C,F: C] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 11.93/6.05 => ( ( F
% 11.93/6.05 = ( C5
% 11.93/6.05 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 11.93/6.05 => ( ! [F4: C] :
% 11.93/6.05 ( ( C5 @ ( U4 @ F4 ) )
% 11.93/6.05 = F4 )
% 11.93/6.05 => ( F
% 11.93/6.05 = ( F6 @ F ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5982_option_Omono__body__fixp,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,F6: ( B > ( option @ A ) ) > B > ( option @ A )] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( option @ A ) ) @ ( option @ A ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( option @ A )] : ( F6 @ F2 @ X2 ) )
% 11.93/6.05 => ( ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F6 )
% 11.93/6.05 = ( F6 @ ( comple187402453842119260l_fixp @ ( B > ( option @ A ) ) @ ( partial_fun_lub @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) @ ( partial_fun_ord @ ( option @ A ) @ ( option @ A ) @ B @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) ) @ F6 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5983_option_Opartial__function__definitions__axioms,axiom,
% 11.93/6.05 ! [A: $tType] : ( partia7178651479351089652itions @ ( option @ A ) @ ( partial_flat_ord @ ( option @ A ) @ ( none @ A ) ) @ ( partial_flat_lub @ ( option @ A ) @ ( none @ A ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5984_fixp__mono,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( comple9053668089753744459l_ccpo @ A )
% 11.93/6.05 => ! [F: A > A,G: A > A] :
% 11.93/6.05 ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
% 11.93/6.05 => ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ G )
% 11.93/6.05 => ( ! [Z11: A] : ( ord_less_eq @ A @ ( F @ Z11 ) @ ( G @ Z11 ) )
% 11.93/6.05 => ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple115746919287870866o_fixp @ A @ G ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5985_heap_Oconst__mono,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
% 11.93/6.05 ( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
% 11.93/6.05 @ ^ [F2: B] : C2 ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5986_Heap__Time__Monad_Obind__mono,axiom,
% 11.93/6.05 ! [C: $tType,D: $tType,B: $tType,A: $tType,B6: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C5: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
% 11.93/6.05 ( ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ C ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ C ) @ B6 )
% 11.93/6.05 => ( ! [Y4: C] : ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D ) @ ( C5 @ Y4 ) )
% 11.93/6.05 => ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D )
% 11.93/6.05 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 11.93/6.05 ( heap_Time_bind @ C @ D @ ( B6 @ F2 )
% 11.93/6.05 @ ^ [Y3: C] : ( C5 @ Y3 @ F2 ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5987_fixp__induct__heap,axiom,
% 11.93/6.05 ! [C: $tType,A: $tType,B: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: B > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o,X: B,H3: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 11.93/6.05 => ( ( F
% 11.93/6.05 = ( C5
% 11.93/6.05 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 11.93/6.05 => ( ! [F4: B > ( heap_Time_Heap @ A )] :
% 11.93/6.05 ( ( U4 @ ( C5 @ F4 ) )
% 11.93/6.05 = F4 )
% 11.93/6.05 => ( ! [F4: C,X2: B,H2: heap_ext @ product_unit,H6: heap_ext @ product_unit,R2: A,N3: nat] :
% 11.93/6.05 ( ! [Xa2: B,Ha: heap_ext @ product_unit,H_a: heap_ext @ product_unit,Ra3: A,Na: nat] :
% 11.93/6.05 ( ( heap_Time_effect @ A @ ( U4 @ F4 @ Xa2 ) @ Ha @ H_a @ Ra3 @ Na )
% 11.93/6.05 => ( P @ Xa2 @ Ha @ H_a @ Ra3 @ Na ) )
% 11.93/6.05 => ( ( heap_Time_effect @ A @ ( U4 @ ( F6 @ F4 ) @ X2 ) @ H2 @ H6 @ R2 @ N3 )
% 11.93/6.05 => ( P @ X2 @ H2 @ H6 @ R2 @ N3 ) ) )
% 11.93/6.05 => ( ( heap_Time_effect @ A @ ( U4 @ F @ X ) @ H3 @ H4 @ R @ N )
% 11.93/6.05 => ( P @ X @ H3 @ H4 @ R @ N ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5988_refines__adm,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,T2: A > ( heap_Time_Heap @ B )] :
% 11.93/6.05 ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
% 11.93/6.05 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 11.93/6.05 ! [X3: A] : ( refine_Imp_refines @ B @ ( T2 @ X3 ) @ ( F2 @ X3 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5989_heap_Ofixp__induct__uc,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 11.93/6.05 => ( ( F
% 11.93/6.05 = ( C5
% 11.93/6.05 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 11.93/6.05 => ( ! [F4: B > ( heap_Time_Heap @ A )] :
% 11.93/6.05 ( ( U4 @ ( C5 @ F4 ) )
% 11.93/6.05 = F4 )
% 11.93/6.05 => ( ( comple1908693960933563346ssible @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ P )
% 11.93/6.05 => ( ( P
% 11.93/6.05 @ ^ [Uu: B] :
% 11.93/6.05 ( heap_Time_Heap2 @ A
% 11.93/6.05 @ ^ [X3: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
% 11.93/6.05 => ( ! [F4: C] :
% 11.93/6.05 ( ( P @ ( U4 @ F4 ) )
% 11.93/6.05 => ( P @ ( U4 @ ( F6 @ F4 ) ) ) )
% 11.93/6.05 => ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5990_heap_Omono__body__fixp,axiom,
% 11.93/6.05 ! [A: $tType,B: $tType,F6: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( F6 @ F2 @ X2 ) )
% 11.93/6.05 => ( ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F6 )
% 11.93/6.05 = ( F6 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F6 ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5991_heap_Ofixp__rule__uc,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F6: C > C,C5: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
% 11.93/6.05 ( ! [X2: B] :
% 11.93/6.05 ( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) @ X2 ) )
% 11.93/6.05 => ( ( F
% 11.93/6.05 = ( C5
% 11.93/6.05 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
% 11.93/6.05 @ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F6 @ ( C5 @ F2 ) ) ) ) ) )
% 11.93/6.05 => ( ! [F4: C] :
% 11.93/6.05 ( ( C5 @ ( U4 @ F4 ) )
% 11.93/6.05 = F4 )
% 11.93/6.05 => ( F
% 11.93/6.05 = ( F6 @ F ) ) ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(fact_5992_admissible__heap,axiom,
% 11.93/6.05 ! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
% 11.93/6.05 ( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
% 11.93/6.05 @ ^ [F2: A > ( heap_Time_Heap @ B )] :
% 11.93/6.05 ! [X3: A,H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R3: B,N5: nat] :
% 11.93/6.05 ( ( heap_Time_effect @ B @ ( F2 @ X3 ) @ H @ H7 @ R3 @ N5 )
% 11.93/6.05 => ( P @ X3 @ H @ H7 @ R3 @ N5 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( heap @ A )
% 11.93/6.05 => ( typerep2 @ A ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(subcl_Heap_Oheap___HOL_Otype,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( heap @ A )
% 11.93/6.05 => ( type @ A ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
% 11.93/6.05 ! [A: $tType] :
% 11.93/6.05 ( ( heap @ A )
% 11.93/6.05 => ( countable @ A ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( heap_Time_Heap @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( heap_Time_Heap @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of_1,axiom,
% 11.93/6.05 code_term_of @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_2,axiom,
% 11.93/6.05 typerep2 @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_3,axiom,
% 11.93/6.05 code_term_of @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_4,axiom,
% 11.93/6.05 typerep2 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_5,axiom,
% 11.93/6.05 code_term_of @ code_term ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_6,axiom,
% 11.93/6.05 typerep2 @ code_term ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_7,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( heap_ext @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_8,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( heap_ext @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_9,axiom,
% 11.93/6.05 code_term_of @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
% 11.93/6.05 enum @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Typerep_Otyperep_10,axiom,
% 11.93/6.05 typerep2 @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
% 11.93/6.05 bounded_lattice_top @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_11,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( code_term_of @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Enum_Oenum_12,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( enum @ A20 )
% 11.93/6.05 & ( enum @ A21 ) )
% 11.93/6.05 => ( enum @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Typerep_Otyperep_13,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( typerep2 @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_14,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( typerep2 @ ( old_node @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_15,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Typerep_Otyperep_16,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Typerep_Otyperep___Code__Evaluation_Oterm__of_17,axiom,
% 11.93/6.05 code_term_of @ typerep ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Typerep_Otyperep___Typerep_Otyperep_18,axiom,
% 11.93/6.05 typerep2 @ typerep ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
% 11.93/6.05 typerep2 @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice__top_20,axiom,
% 11.93/6.05 bounded_lattice_top @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Code__Evaluation_Oterm__of_21,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( pred @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Typerep_Otyperep_22,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( pred @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Lattices_Obounded__lattice__top_23,axiom,
% 11.93/6.05 ! [A20: $tType] : ( bounded_lattice_top @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Oseq___Code__Evaluation_Oterm__of_24,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( seq @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Oseq___Typerep_Otyperep_25,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( seq @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Code__Evaluation_Oterm__of_26,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Enum_Oenum_27,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( enum @ A20 )
% 11.93/6.05 => ( enum @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Typerep_Otyperep_28,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_29,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( bounded_lattice_top @ A20 )
% 11.93/6.05 => ( bounded_lattice_top @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_30,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( code_term_of @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Enum_Oenum_31,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( enum @ A20 )
% 11.93/6.05 & ( enum @ A21 ) )
% 11.93/6.05 => ( enum @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Typerep_Otyperep_32,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( typerep2 @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Code__Evaluation_Oterm__of_33,axiom,
% 11.93/6.05 code_term_of @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Enum_Oenum_34,axiom,
% 11.93/6.05 enum @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Typerep_Otyperep_35,axiom,
% 11.93/6.05 typerep2 @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_36,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( array @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___Typerep_Otyperep_37,axiom,
% 11.93/6.05 ! [A20: $tType] : ( typerep2 @ ( array @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___Code__Evaluation_Oterm__of_38,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( list @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___Typerep_Otyperep_39,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( list @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___Code__Evaluation_Oterm__of_40,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( ref @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___Typerep_Otyperep_41,axiom,
% 11.93/6.05 ! [A20: $tType] : ( typerep2 @ ( ref @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_42,axiom,
% 11.93/6.05 code_term_of @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Enum_Oenum_43,axiom,
% 11.93/6.05 enum @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Typerep_Otyperep_44,axiom,
% 11.93/6.05 typerep2 @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_45,axiom,
% 11.93/6.05 bounded_lattice_top @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_46,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Enum_Oenum_47,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( enum @ A20 )
% 11.93/6.05 => ( enum @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Typerep_Otyperep_48,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_49,axiom,
% 11.93/6.05 ! [A20: $tType] : ( bounded_lattice_top @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Code__Evaluation_Oterm__of_50,axiom,
% 11.93/6.05 code_term_of @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Typerep_Otyperep_51,axiom,
% 11.93/6.05 typerep2 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Code__Evaluation_Oterm__of_52,axiom,
% 11.93/6.05 code_term_of @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Typerep_Otyperep_53,axiom,
% 11.93/6.05 typerep2 @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_54,axiom,
% 11.93/6.05 code_term_of @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Typerep_Otyperep_55,axiom,
% 11.93/6.05 typerep2 @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Code__Evaluation_Oterm__of_56,axiom,
% 11.93/6.05 code_term_of @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Typerep_Otyperep_57,axiom,
% 11.93/6.05 typerep2 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_itself___Code__Evaluation_Oterm__of_58,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( code_term_of @ ( itself @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_itself___Typerep_Otyperep_59,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( typerep2 @ ( itself @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Code__Evaluation_Oterm__of_60,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( code_term_of @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Enum_Oenum_61,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( enum @ A20 )
% 11.93/6.05 & ( enum @ A21 ) )
% 11.93/6.05 => ( enum @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Typerep_Otyperep_62,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( typerep2 @ A20 )
% 11.93/6.05 & ( typerep2 @ A21 ) )
% 11.93/6.05 => ( typerep2 @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Lattices_Obounded__lattice__top_63,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( bounded_lattice @ A21 )
% 11.93/6.05 => ( bounded_lattice_top @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( cl_HOL_Oequal @ A20 )
% 11.93/6.05 & ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 & ( quickc3360725361186068524ustive @ A21 ) )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( comple6319245703460814977attice @ A21 )
% 11.93/6.05 => ( comple6319245703460814977attice @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( cl_HOL_Oequal @ A20 )
% 11.93/6.05 & ( quickc658316121487927005ustive @ A20 )
% 11.93/6.05 & ( quickc658316121487927005ustive @ A21 ) )
% 11.93/6.05 => ( quickc658316121487927005ustive @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( boolea8198339166811842893lgebra @ A21 )
% 11.93/6.05 => ( boolea8198339166811842893lgebra @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( comple6319245703460814977attice @ A21 )
% 11.93/6.05 => ( comple9053668089753744459l_ccpo @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( code_term_of @ A20 )
% 11.93/6.05 & ( cl_HOL_Oequal @ A20 )
% 11.93/6.05 & ( quickcheck_random @ A21 ) )
% 11.93/6.05 => ( quickcheck_random @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( semilattice_sup @ A21 )
% 11.93/6.05 => ( semilattice_sup @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( semilattice_inf @ A21 )
% 11.93/6.05 => ( semilattice_inf @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Lattices_Obounded__lattice,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( bounded_lattice @ A21 )
% 11.93/6.05 => ( bounded_lattice @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Complete__Lattices_OSup,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( complete_Sup @ A21 )
% 11.93/6.05 => ( complete_Sup @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Complete__Lattices_OInf,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( complete_Inf @ A21 )
% 11.93/6.05 => ( complete_Inf @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( order_top @ A21 )
% 11.93/6.05 => ( order_top @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( order_bot @ A21 )
% 11.93/6.05 => ( order_bot @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Countable_Ocountable,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( finite_finite @ A20 )
% 11.93/6.05 & ( countable @ A21 ) )
% 11.93/6.05 => ( countable @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Orderings_Opreorder,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( preorder @ A21 )
% 11.93/6.05 => ( preorder @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( finite_finite @ A20 )
% 11.93/6.05 & ( finite_finite @ A21 ) )
% 11.93/6.05 => ( finite_finite @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Orderings_Oorder,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( order @ A21 )
% 11.93/6.05 => ( order @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Orderings_Oord,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ord @ A21 )
% 11.93/6.05 => ( ord @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Groups_Ouminus,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( uminus @ A21 )
% 11.93/6.05 => ( uminus @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Lattices_Osup,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( semilattice_sup @ A21 )
% 11.93/6.05 => ( sup @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___Lattices_Oinf,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( semilattice_inf @ A21 )
% 11.93/6.05 => ( inf @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_fun___HOL_Oequal,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( enum @ A20 )
% 11.93/6.05 & ( cl_HOL_Oequal @ A21 ) )
% 11.93/6.05 => ( cl_HOL_Oequal @ ( A20 > A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_itself___Quickcheck__Random_Orandom_64,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( itself @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_itself___HOL_Oequal_65,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( itself @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
% 11.93/6.05 bit_un5681908812861735899ations @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
% 11.93/6.05 semiri1453513574482234551roduct @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
% 11.93/6.05 euclid5411537665997757685th_nat @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
% 11.93/6.05 euclid8789492081693882211th_nat @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
% 11.93/6.05 ordere1937475149494474687imp_le @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
% 11.93/6.05 euclid3128863361964157862miring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
% 11.93/6.05 euclid4440199948858584721cancel @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
% 11.93/6.05 unique1627219031080169319umeral @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
% 11.93/6.05 euclid8851590272496341667cancel @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
% 11.93/6.05 strict9044650504122735259up_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
% 11.93/6.05 ordere580206878836729694up_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
% 11.93/6.05 ordere2412721322843649153imp_le @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
% 11.93/6.05 bit_se359711467146920520ations @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_66,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
% 11.93/6.05 strict7427464778891057005id_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
% 11.93/6.05 ordere8940638589300402666id_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
% 11.93/6.05 euclid3725896446679973847miring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
% 11.93/6.05 linord715952674999750819strict @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
% 11.93/6.05 linord4140545234300271783up_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
% 11.93/6.05 bit_ri3973907225187159222ations @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
% 11.93/6.05 linord181362715937106298miring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
% 11.93/6.05 semido2269285787275462019factor @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_67,axiom,
% 11.93/6.05 quickc658316121487927005ustive @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
% 11.93/6.05 semiri3467727345109120633visors @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
% 11.93/6.05 ordere6658533253407199908up_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
% 11.93/6.05 ordere166539214618696060dd_abs @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
% 11.93/6.05 ordere6911136660526730532id_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
% 11.93/6.05 linord5086331880401160121up_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
% 11.93/6.05 cancel2418104881723323429up_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
% 11.93/6.05 ring_15535105094025558882visors @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
% 11.93/6.05 cancel1802427076303600483id_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
% 11.93/6.05 linord4710134922213307826strict @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
% 11.93/6.05 comm_s4317794764714335236cancel @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
% 11.93/6.05 bit_semiring_bits @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
% 11.93/6.05 normal8620421768224518004emidom @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
% 11.93/6.05 linord6961819062388156250ring_1 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
% 11.93/6.05 ordered_ab_group_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
% 11.93/6.05 cancel_semigroup_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
% 11.93/6.05 linordered_semidom @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Quickcheck__Random_Orandom_68,axiom,
% 11.93/6.05 quickcheck_random @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Lattices_Osemilattice__sup_69,axiom,
% 11.93/6.05 semilattice_sup @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Lattices_Osemilattice__inf_70,axiom,
% 11.93/6.05 semilattice_inf @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
% 11.93/6.05 ab_semigroup_mult @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
% 11.93/6.05 algebraic_semidom @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
% 11.93/6.05 comm_monoid_mult @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
% 11.93/6.05 ab_semigroup_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
% 11.93/6.05 semiring_parity @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
% 11.93/6.05 comm_monoid_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
% 11.93/6.05 semiring_modulo @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
% 11.93/6.05 linordered_ring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
% 11.93/6.05 linordered_idom @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
% 11.93/6.05 comm_semiring_1 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
% 11.93/6.05 comm_semiring_0 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
% 11.93/6.05 semigroup_mult @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Complete__Lattices_OSup_71,axiom,
% 11.93/6.05 complete_Sup @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Complete__Lattices_OInf_72,axiom,
% 11.93/6.05 complete_Inf @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
% 11.93/6.05 semidom_modulo @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
% 11.93/6.05 semidom_divide @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
% 11.93/6.05 semiring_numeral @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
% 11.93/6.05 semigroup_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
% 11.93/6.05 zero_less_one @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
% 11.93/6.05 comm_semiring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
% 11.93/6.05 semiring_char_0 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
% 11.93/6.05 ab_group_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Countable_Ocountable_73,axiom,
% 11.93/6.05 countable @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
% 11.93/6.05 zero_neq_one @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
% 11.93/6.05 ordered_ring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
% 11.93/6.05 idom_abs_sgn @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
% 11.93/6.05 ring_parity @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Orderings_Opreorder_74,axiom,
% 11.93/6.05 preorder @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
% 11.93/6.05 linorder @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
% 11.93/6.05 monoid_mult @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
% 11.93/6.05 comm_ring_1 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
% 11.93/6.05 monoid_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
% 11.93/6.05 semiring_1 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
% 11.93/6.05 semiring_0 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
% 11.93/6.05 no_top @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
% 11.93/6.05 no_bot @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
% 11.93/6.05 group_add @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
% 11.93/6.05 semiring_gcd @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
% 11.93/6.05 semiring_Gcd @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
% 11.93/6.05 comm_ring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Orderings_Oorder_75,axiom,
% 11.93/6.05 order @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
% 11.93/6.05 neg_numeral @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
% 11.93/6.05 ring_char_0 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Osemiring,axiom,
% 11.93/6.05 semiring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Orderings_Oord_76,axiom,
% 11.93/6.05 ord @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ouminus_77,axiom,
% 11.93/6.05 uminus @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Oring__1,axiom,
% 11.93/6.05 ring_1 @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Lattices_Osup_78,axiom,
% 11.93/6.05 sup @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Lattices_Oinf_79,axiom,
% 11.93/6.05 inf @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
% 11.93/6.05 ring_gcd @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Power_Opower,axiom,
% 11.93/6.05 power @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Num_Onumeral,axiom,
% 11.93/6.05 numeral @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Ozero,axiom,
% 11.93/6.05 zero @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oplus,axiom,
% 11.93/6.05 plus @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Oring,axiom,
% 11.93/6.05 ring @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Groups_Oone,axiom,
% 11.93/6.05 one @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Rings_Odvd,axiom,
% 11.93/6.05 dvd @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___Heap_Oheap,axiom,
% 11.93/6.05 heap @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Int_Oint___HOL_Oequal_80,axiom,
% 11.93/6.05 cl_HOL_Oequal @ int ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_81,axiom,
% 11.93/6.05 bit_un5681908812861735899ations @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_82,axiom,
% 11.93/6.05 semiri1453513574482234551roduct @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_83,axiom,
% 11.93/6.05 euclid5411537665997757685th_nat @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_84,axiom,
% 11.93/6.05 ordere1937475149494474687imp_le @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_85,axiom,
% 11.93/6.05 euclid3128863361964157862miring @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_86,axiom,
% 11.93/6.05 euclid4440199948858584721cancel @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_87,axiom,
% 11.93/6.05 unique1627219031080169319umeral @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_88,axiom,
% 11.93/6.05 strict9044650504122735259up_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
% 11.93/6.05 ordere1170586879665033532d_diff @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_89,axiom,
% 11.93/6.05 ordere580206878836729694up_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_90,axiom,
% 11.93/6.05 ordere2412721322843649153imp_le @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_91,axiom,
% 11.93/6.05 bit_se359711467146920520ations @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_92,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_93,axiom,
% 11.93/6.05 strict7427464778891057005id_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_94,axiom,
% 11.93/6.05 ordere8940638589300402666id_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
% 11.93/6.05 canoni5634975068530333245id_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_95,axiom,
% 11.93/6.05 euclid3725896446679973847miring @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_96,axiom,
% 11.93/6.05 linord4140545234300271783up_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_97,axiom,
% 11.93/6.05 linord181362715937106298miring @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_98,axiom,
% 11.93/6.05 semido2269285787275462019factor @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_99,axiom,
% 11.93/6.05 quickc658316121487927005ustive @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_100,axiom,
% 11.93/6.05 semiri3467727345109120633visors @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_101,axiom,
% 11.93/6.05 ordere6658533253407199908up_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_102,axiom,
% 11.93/6.05 ordere6911136660526730532id_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_103,axiom,
% 11.93/6.05 cancel2418104881723323429up_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_104,axiom,
% 11.93/6.05 cancel1802427076303600483id_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_105,axiom,
% 11.93/6.05 comm_s4317794764714335236cancel @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_106,axiom,
% 11.93/6.05 bit_semiring_bits @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Onormalization__semidom_107,axiom,
% 11.93/6.05 normal8620421768224518004emidom @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_108,axiom,
% 11.93/6.05 cancel_semigroup_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Olinordered__semidom_109,axiom,
% 11.93/6.05 linordered_semidom @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_110,axiom,
% 11.93/6.05 quickcheck_random @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_111,axiom,
% 11.93/6.05 semilattice_sup @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_112,axiom,
% 11.93/6.05 semilattice_inf @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_113,axiom,
% 11.93/6.05 ab_semigroup_mult @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_114,axiom,
% 11.93/6.05 algebraic_semidom @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_115,axiom,
% 11.93/6.05 comm_monoid_mult @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
% 11.93/6.05 comm_monoid_diff @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_116,axiom,
% 11.93/6.05 ab_semigroup_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Parity_Osemiring__parity_117,axiom,
% 11.93/6.05 semiring_parity @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_118,axiom,
% 11.93/6.05 comm_monoid_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemiring__modulo_119,axiom,
% 11.93/6.05 semiring_modulo @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_120,axiom,
% 11.93/6.05 comm_semiring_1 @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_121,axiom,
% 11.93/6.05 comm_semiring_0 @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Osemigroup__mult_122,axiom,
% 11.93/6.05 semigroup_mult @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Complete__Lattices_OSup_123,axiom,
% 11.93/6.05 complete_Sup @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Complete__Lattices_OInf_124,axiom,
% 11.93/6.05 complete_Inf @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemidom__modulo_125,axiom,
% 11.93/6.05 semidom_modulo @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemidom__divide_126,axiom,
% 11.93/6.05 semidom_divide @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Num_Osemiring__numeral_127,axiom,
% 11.93/6.05 semiring_numeral @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Osemigroup__add_128,axiom,
% 11.93/6.05 semigroup_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Ozero__less__one_129,axiom,
% 11.93/6.05 zero_less_one @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Ocomm__semiring_130,axiom,
% 11.93/6.05 comm_semiring @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Orderings_Oorder__bot_131,axiom,
% 11.93/6.05 order_bot @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Nat_Osemiring__char__0_132,axiom,
% 11.93/6.05 semiring_char_0 @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Countable_Ocountable_133,axiom,
% 11.93/6.05 countable @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Ozero__neq__one_134,axiom,
% 11.93/6.05 zero_neq_one @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Orderings_Opreorder_135,axiom,
% 11.93/6.05 preorder @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Orderings_Olinorder_136,axiom,
% 11.93/6.05 linorder @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Omonoid__mult_137,axiom,
% 11.93/6.05 monoid_mult @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Omonoid__add_138,axiom,
% 11.93/6.05 monoid_add @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemiring__1_139,axiom,
% 11.93/6.05 semiring_1 @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemiring__0_140,axiom,
% 11.93/6.05 semiring_0 @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Orderings_Ono__top_141,axiom,
% 11.93/6.05 no_top @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___GCD_Osemiring__gcd_142,axiom,
% 11.93/6.05 semiring_gcd @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_143,axiom,
% 11.93/6.05 semiring_Gcd @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Orderings_Oorder_144,axiom,
% 11.93/6.05 order @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Osemiring_145,axiom,
% 11.93/6.05 semiring @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Orderings_Oord_146,axiom,
% 11.93/6.05 ord @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Lattices_Osup_147,axiom,
% 11.93/6.05 sup @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Lattices_Oinf_148,axiom,
% 11.93/6.05 inf @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Power_Opower_149,axiom,
% 11.93/6.05 power @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Num_Onumeral_150,axiom,
% 11.93/6.05 numeral @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Ozero_151,axiom,
% 11.93/6.05 zero @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oplus_152,axiom,
% 11.93/6.05 plus @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Groups_Oone_153,axiom,
% 11.93/6.05 one @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Rings_Odvd_154,axiom,
% 11.93/6.05 dvd @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___Heap_Oheap_155,axiom,
% 11.93/6.05 heap @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Nat_Onat___HOL_Oequal_156,axiom,
% 11.93/6.05 cl_HOL_Oequal @ nat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_157,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Quickcheck__Random_Orandom_158,axiom,
% 11.93/6.05 quickcheck_random @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Orderings_Opreorder_159,axiom,
% 11.93/6.05 preorder @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Orderings_Olinorder_160,axiom,
% 11.93/6.05 linorder @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Orderings_Oorder_161,axiom,
% 11.93/6.05 order @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Orderings_Oord_162,axiom,
% 11.93/6.05 ord @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___Groups_Oplus_163,axiom,
% 11.93/6.05 plus @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Num_Onum___HOL_Oequal_164,axiom,
% 11.93/6.05 cl_HOL_Oequal @ num ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_165,axiom,
% 11.93/6.05 semiri1453513574482234551roduct @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_166,axiom,
% 11.93/6.05 ordere1937475149494474687imp_le @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167,axiom,
% 11.93/6.05 strict9044650504122735259up_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168,axiom,
% 11.93/6.05 ordere580206878836729694up_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169,axiom,
% 11.93/6.05 ordere2412721322843649153imp_le @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_170,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171,axiom,
% 11.93/6.05 strict7427464778891057005id_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172,axiom,
% 11.93/6.05 ordere8940638589300402666id_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
% 11.93/6.05 archim462609752435547400_field @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_173,axiom,
% 11.93/6.05 linord715952674999750819strict @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_174,axiom,
% 11.93/6.05 linord4140545234300271783up_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_175,axiom,
% 11.93/6.05 linord181362715937106298miring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_176,axiom,
% 11.93/6.05 quickc658316121487927005ustive @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_177,axiom,
% 11.93/6.05 semiri3467727345109120633visors @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_178,axiom,
% 11.93/6.05 ordere6658533253407199908up_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_179,axiom,
% 11.93/6.05 ordere166539214618696060dd_abs @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
% 11.93/6.05 archim2362893244070406136eiling @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_180,axiom,
% 11.93/6.05 ordere6911136660526730532id_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_181,axiom,
% 11.93/6.05 linord5086331880401160121up_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_182,axiom,
% 11.93/6.05 cancel2418104881723323429up_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_183,axiom,
% 11.93/6.05 ring_15535105094025558882visors @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_184,axiom,
% 11.93/6.05 cancel1802427076303600483id_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_185,axiom,
% 11.93/6.05 linord4710134922213307826strict @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_186,axiom,
% 11.93/6.05 comm_s4317794764714335236cancel @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_187,axiom,
% 11.93/6.05 linord6961819062388156250ring_1 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_188,axiom,
% 11.93/6.05 ordered_ab_group_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_189,axiom,
% 11.93/6.05 cancel_semigroup_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__semidom_190,axiom,
% 11.93/6.05 linordered_semidom @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Quickcheck__Random_Orandom_191,axiom,
% 11.93/6.05 quickcheck_random @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
% 11.93/6.05 dense_linorder @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_192,axiom,
% 11.93/6.05 semilattice_sup @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_193,axiom,
% 11.93/6.05 semilattice_inf @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_194,axiom,
% 11.93/6.05 ab_semigroup_mult @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_195,axiom,
% 11.93/6.05 comm_monoid_mult @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_196,axiom,
% 11.93/6.05 ab_semigroup_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
% 11.93/6.05 linordered_field @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_197,axiom,
% 11.93/6.05 comm_monoid_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__ring_198,axiom,
% 11.93/6.05 linordered_ring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Olinordered__idom_199,axiom,
% 11.93/6.05 linordered_idom @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_200,axiom,
% 11.93/6.05 comm_semiring_1 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_201,axiom,
% 11.93/6.05 comm_semiring_0 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Osemigroup__mult_202,axiom,
% 11.93/6.05 semigroup_mult @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Osemidom__divide_203,axiom,
% 11.93/6.05 semidom_divide @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Num_Osemiring__numeral_204,axiom,
% 11.93/6.05 semiring_numeral @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Osemigroup__add_205,axiom,
% 11.93/6.05 semigroup_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
% 11.93/6.05 division_ring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ozero__less__one_206,axiom,
% 11.93/6.05 zero_less_one @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ocomm__semiring_207,axiom,
% 11.93/6.05 comm_semiring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Nat_Osemiring__char__0_208,axiom,
% 11.93/6.05 semiring_char_0 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oab__group__add_209,axiom,
% 11.93/6.05 ab_group_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
% 11.93/6.05 field_char_0 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Countable_Ocountable_210,axiom,
% 11.93/6.05 countable @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ozero__neq__one_211,axiom,
% 11.93/6.05 zero_neq_one @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Oordered__ring_212,axiom,
% 11.93/6.05 ordered_ring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_213,axiom,
% 11.93/6.05 idom_abs_sgn @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Opreorder_214,axiom,
% 11.93/6.05 preorder @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Olinorder_215,axiom,
% 11.93/6.05 linorder @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Omonoid__mult_216,axiom,
% 11.93/6.05 monoid_mult @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_217,axiom,
% 11.93/6.05 comm_ring_1 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Omonoid__add_218,axiom,
% 11.93/6.05 monoid_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Osemiring__1_219,axiom,
% 11.93/6.05 semiring_1 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Osemiring__0_220,axiom,
% 11.93/6.05 semiring_0 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Ono__top_221,axiom,
% 11.93/6.05 no_top @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Ono__bot_222,axiom,
% 11.93/6.05 no_bot @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ogroup__add_223,axiom,
% 11.93/6.05 group_add @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Ocomm__ring_224,axiom,
% 11.93/6.05 comm_ring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Oorder_225,axiom,
% 11.93/6.05 order @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Num_Oneg__numeral_226,axiom,
% 11.93/6.05 neg_numeral @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Nat_Oring__char__0_227,axiom,
% 11.93/6.05 ring_char_0 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Osemiring_228,axiom,
% 11.93/6.05 semiring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
% 11.93/6.05 inverse @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Orderings_Oord_229,axiom,
% 11.93/6.05 ord @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ouminus_230,axiom,
% 11.93/6.05 uminus @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Oring__1_231,axiom,
% 11.93/6.05 ring_1 @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Lattices_Osup_232,axiom,
% 11.93/6.05 sup @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Lattices_Oinf_233,axiom,
% 11.93/6.05 inf @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Fields_Ofield,axiom,
% 11.93/6.05 field @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Power_Opower_234,axiom,
% 11.93/6.05 power @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Num_Onumeral_235,axiom,
% 11.93/6.05 numeral @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Ozero_236,axiom,
% 11.93/6.05 zero @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oplus_237,axiom,
% 11.93/6.05 plus @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Oring_238,axiom,
% 11.93/6.05 ring @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Groups_Oone_239,axiom,
% 11.93/6.05 one @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___Rings_Odvd_240,axiom,
% 11.93/6.05 dvd @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Rat_Orat___HOL_Oequal_241,axiom,
% 11.93/6.05 cl_HOL_Oequal @ rat ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_242,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_243,axiom,
% 11.93/6.05 ! [A20: $tType] : ( comple6319245703460814977attice @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_244,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickc658316121487927005ustive @ A20 )
% 11.93/6.05 => ( quickc658316121487927005ustive @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_245,axiom,
% 11.93/6.05 ! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_246,axiom,
% 11.93/6.05 ! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Quickcheck__Random_Orandom_247,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickcheck_random @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_248,axiom,
% 11.93/6.05 ! [A20: $tType] : ( semilattice_sup @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Lattices_Osemilattice__inf_249,axiom,
% 11.93/6.05 ! [A20: $tType] : ( semilattice_inf @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Lattices_Obounded__lattice_250,axiom,
% 11.93/6.05 ! [A20: $tType] : ( bounded_lattice @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Complete__Lattices_OSup_251,axiom,
% 11.93/6.05 ! [A20: $tType] : ( complete_Sup @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Complete__Lattices_OInf_252,axiom,
% 11.93/6.05 ! [A20: $tType] : ( complete_Inf @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Orderings_Oorder__top_253,axiom,
% 11.93/6.05 ! [A20: $tType] : ( order_top @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Orderings_Oorder__bot_254,axiom,
% 11.93/6.05 ! [A20: $tType] : ( order_bot @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Countable_Ocountable_255,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( finite_finite @ A20 )
% 11.93/6.05 => ( countable @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Orderings_Opreorder_256,axiom,
% 11.93/6.05 ! [A20: $tType] : ( preorder @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Finite__Set_Ofinite_257,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( finite_finite @ A20 )
% 11.93/6.05 => ( finite_finite @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Orderings_Oorder_258,axiom,
% 11.93/6.05 ! [A20: $tType] : ( order @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Orderings_Oord_259,axiom,
% 11.93/6.05 ! [A20: $tType] : ( ord @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Groups_Ouminus_260,axiom,
% 11.93/6.05 ! [A20: $tType] : ( uminus @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Lattices_Osup_261,axiom,
% 11.93/6.05 ! [A20: $tType] : ( sup @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___Lattices_Oinf_262,axiom,
% 11.93/6.05 ! [A20: $tType] : ( inf @ ( set @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Set_Oset___HOL_Oequal_263,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( cl_HOL_Oequal @ A20 )
% 11.93/6.05 => ( cl_HOL_Oequal @ ( set @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_265,axiom,
% 11.93/6.05 comple6319245703460814977attice @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_266,axiom,
% 11.93/6.05 boolea8198339166811842893lgebra @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_267,axiom,
% 11.93/6.05 comple9053668089753744459l_ccpo @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_268,axiom,
% 11.93/6.05 quickcheck_random @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_269,axiom,
% 11.93/6.05 semilattice_sup @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_270,axiom,
% 11.93/6.05 semilattice_inf @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Lattices_Obounded__lattice_271,axiom,
% 11.93/6.05 bounded_lattice @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Complete__Lattices_OSup_272,axiom,
% 11.93/6.05 complete_Sup @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Complete__Lattices_OInf_273,axiom,
% 11.93/6.05 complete_Inf @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Orderings_Oorder__top_274,axiom,
% 11.93/6.05 order_top @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Orderings_Oorder__bot_275,axiom,
% 11.93/6.05 order_bot @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Countable_Ocountable_276,axiom,
% 11.93/6.05 countable @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Orderings_Opreorder_277,axiom,
% 11.93/6.05 preorder @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Orderings_Olinorder_278,axiom,
% 11.93/6.05 linorder @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Finite__Set_Ofinite_279,axiom,
% 11.93/6.05 finite_finite @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Orderings_Oorder_280,axiom,
% 11.93/6.05 order @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Orderings_Oord_281,axiom,
% 11.93/6.05 ord @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Groups_Ouminus_282,axiom,
% 11.93/6.05 uminus @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Lattices_Osup_283,axiom,
% 11.93/6.05 sup @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Lattices_Oinf_284,axiom,
% 11.93/6.05 inf @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___Heap_Oheap_285,axiom,
% 11.93/6.05 heap @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_HOL_Obool___HOL_Oequal_286,axiom,
% 11.93/6.05 cl_HOL_Oequal @ $o ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___Quickcheck__Exhaustive_Ofull__exhaustive_287,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( ref @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___Quickcheck__Random_Orandom_288,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( ref @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___Countable_Ocountable_289,axiom,
% 11.93/6.05 ! [A20: $tType] : ( countable @ ( ref @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___Heap_Oheap_290,axiom,
% 11.93/6.05 ! [A20: $tType] : ( heap @ ( ref @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oref___HOL_Oequal_291,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( ref @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_292,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( list @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___Quickcheck__Random_Orandom_293,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickcheck_random @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( list @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___Countable_Ocountable_294,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( countable @ A20 )
% 11.93/6.05 => ( countable @ ( list @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___Heap_Oheap_295,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( heap @ A20 )
% 11.93/6.05 => ( heap @ ( list @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_List_Olist___HOL_Oequal_296,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( list @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___Quickcheck__Exhaustive_Ofull__exhaustive_297,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( array @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___Quickcheck__Random_Orandom_298,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( typerep2 @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( array @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___Countable_Ocountable_299,axiom,
% 11.93/6.05 ! [A20: $tType] : ( countable @ ( array @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___Heap_Oheap_300,axiom,
% 11.93/6.05 ! [A20: $tType] : ( heap @ ( array @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oarray___HOL_Oequal_301,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( array @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_302,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Quickcheck__Random_Orandom_303,axiom,
% 11.93/6.05 quickcheck_random @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Countable_Ocountable_304,axiom,
% 11.93/6.05 countable @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Finite__Set_Ofinite_305,axiom,
% 11.93/6.05 finite_finite @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___Heap_Oheap_306,axiom,
% 11.93/6.05 heap @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_String_Ochar___HOL_Oequal_307,axiom,
% 11.93/6.05 cl_HOL_Oequal @ char ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_308,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 & ( quickc3360725361186068524ustive @ A21 ) )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_309,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( quickcheck_random @ A20 )
% 11.93/6.05 & ( quickcheck_random @ A21 ) )
% 11.93/6.05 => ( quickcheck_random @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Countable_Ocountable_310,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( countable @ A20 )
% 11.93/6.05 & ( countable @ A21 ) )
% 11.93/6.05 => ( countable @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_311,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( finite_finite @ A20 )
% 11.93/6.05 & ( finite_finite @ A21 ) )
% 11.93/6.05 => ( finite_finite @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___Heap_Oheap_312,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( heap @ A20 )
% 11.93/6.05 & ( heap @ A21 ) )
% 11.93/6.05 => ( heap @ ( sum_sum @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Sum__Type_Osum___HOL_Oequal_313,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A20 @ A21 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_314,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_315,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( comple6319245703460814977attice @ A20 )
% 11.93/6.05 => ( comple6319245703460814977attice @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_316,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( comple6319245703460814977attice @ A20 )
% 11.93/6.05 => ( comple9053668089753744459l_ccpo @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Quickcheck__Random_Orandom_317,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickcheck_random @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_318,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( semilattice_sup @ A20 )
% 11.93/6.05 => ( semilattice_sup @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_319,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( semilattice_inf @ A20 )
% 11.93/6.05 => ( semilattice_inf @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Lattices_Obounded__lattice_320,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( bounded_lattice_top @ A20 )
% 11.93/6.05 => ( bounded_lattice @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Complete__Lattices_OSup_321,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( comple6319245703460814977attice @ A20 )
% 11.93/6.05 => ( complete_Sup @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Complete__Lattices_OInf_322,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( comple6319245703460814977attice @ A20 )
% 11.93/6.05 => ( complete_Inf @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Orderings_Oorder__top_323,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( order_top @ A20 )
% 11.93/6.05 => ( order_top @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Orderings_Oorder__bot_324,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( order @ A20 )
% 11.93/6.05 => ( order_bot @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Countable_Ocountable_325,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( countable @ A20 )
% 11.93/6.05 => ( countable @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Orderings_Opreorder_326,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( preorder @ A20 )
% 11.93/6.05 => ( preorder @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Orderings_Olinorder_327,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( linorder @ A20 )
% 11.93/6.05 => ( linorder @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Finite__Set_Ofinite_328,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( finite_finite @ A20 )
% 11.93/6.05 => ( finite_finite @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Orderings_Oorder_329,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( order @ A20 )
% 11.93/6.05 => ( order @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Orderings_Oord_330,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( preorder @ A20 )
% 11.93/6.05 => ( ord @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Lattices_Osup_331,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( sup @ A20 )
% 11.93/6.05 => ( sup @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Lattices_Oinf_332,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( inf @ A20 )
% 11.93/6.05 => ( inf @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___Heap_Oheap_333,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( heap @ A20 )
% 11.93/6.05 => ( heap @ ( option @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Option_Ooption___HOL_Oequal_334,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( option @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Oseq___HOL_Oequal_335,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( seq @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_336,axiom,
% 11.93/6.05 ! [A20: $tType] : ( comple6319245703460814977attice @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_337,axiom,
% 11.93/6.05 ! [A20: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_338,axiom,
% 11.93/6.05 ! [A20: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_339,axiom,
% 11.93/6.05 ! [A20: $tType] : ( semilattice_sup @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_340,axiom,
% 11.93/6.05 ! [A20: $tType] : ( semilattice_inf @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_341,axiom,
% 11.93/6.05 ! [A20: $tType] : ( bounded_lattice @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Complete__Lattices_OSup_342,axiom,
% 11.93/6.05 ! [A20: $tType] : ( complete_Sup @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Complete__Lattices_OInf_343,axiom,
% 11.93/6.05 ! [A20: $tType] : ( complete_Inf @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Orderings_Oorder__top_344,axiom,
% 11.93/6.05 ! [A20: $tType] : ( order_top @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Orderings_Oorder__bot_345,axiom,
% 11.93/6.05 ! [A20: $tType] : ( order_bot @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Orderings_Opreorder_346,axiom,
% 11.93/6.05 ! [A20: $tType] : ( preorder @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Orderings_Oorder_347,axiom,
% 11.93/6.05 ! [A20: $tType] : ( order @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Orderings_Oord_348,axiom,
% 11.93/6.05 ! [A20: $tType] : ( ord @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Groups_Ouminus_349,axiom,
% 11.93/6.05 ! [A20: $tType] : ( uminus @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Lattices_Osup_350,axiom,
% 11.93/6.05 ! [A20: $tType] : ( sup @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___Lattices_Oinf_351,axiom,
% 11.93/6.05 ! [A20: $tType] : ( inf @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Predicate_Opred___HOL_Oequal_352,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( pred @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_353,axiom,
% 11.93/6.05 boolea8198339166811842893lgebra @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Lattices_Osemilattice__sup_354,axiom,
% 11.93/6.05 semilattice_sup @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Lattices_Osemilattice__inf_355,axiom,
% 11.93/6.05 semilattice_inf @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Lattices_Obounded__lattice_356,axiom,
% 11.93/6.05 bounded_lattice @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_357,axiom,
% 11.93/6.05 ab_semigroup_mult @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_358,axiom,
% 11.93/6.05 comm_monoid_mult @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Groups_Osemigroup__mult_359,axiom,
% 11.93/6.05 semigroup_mult @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Orderings_Oorder__top_360,axiom,
% 11.93/6.05 order_top @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Orderings_Oorder__bot_361,axiom,
% 11.93/6.05 order_bot @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Orderings_Opreorder_362,axiom,
% 11.93/6.05 preorder @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Groups_Omonoid__mult_363,axiom,
% 11.93/6.05 monoid_mult @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Orderings_Oorder_364,axiom,
% 11.93/6.05 order @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Orderings_Oord_365,axiom,
% 11.93/6.05 ord @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Groups_Ouminus_366,axiom,
% 11.93/6.05 uminus @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Lattices_Osup_367,axiom,
% 11.93/6.05 sup @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Lattices_Oinf_368,axiom,
% 11.93/6.05 inf @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Power_Opower_369,axiom,
% 11.93/6.05 power @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Groups_Oone_370,axiom,
% 11.93/6.05 one @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Assertions_Oassn___Rings_Odvd_371,axiom,
% 11.93/6.05 dvd @ assn ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Typerep_Otyperep___Countable_Ocountable_372,axiom,
% 11.93/6.05 countable @ typerep ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Typerep_Otyperep___Heap_Oheap_373,axiom,
% 11.93/6.05 heap @ typerep ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Typerep_Otyperep___HOL_Oequal_374,axiom,
% 11.93/6.05 cl_HOL_Oequal @ typerep ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Quickcheck__Exhaustive_Ofull__exhaustive_375,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_376,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( preorder @ A20 )
% 11.93/6.05 => ( ordere6658533253407199908up_add @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_377,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_378,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_379,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cancel_semigroup_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Quickcheck__Random_Orandom_380,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickcheck_random @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_381,axiom,
% 11.93/6.05 ! [A20: $tType] : ( comm_monoid_diff @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_382,axiom,
% 11.93/6.05 ! [A20: $tType] : ( ab_semigroup_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_383,axiom,
% 11.93/6.05 ! [A20: $tType] : ( comm_monoid_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_384,axiom,
% 11.93/6.05 ! [A20: $tType] : ( complete_Sup @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_385,axiom,
% 11.93/6.05 ! [A20: $tType] : ( complete_Inf @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_386,axiom,
% 11.93/6.05 ! [A20: $tType] : ( semigroup_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Orderings_Opreorder_387,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( preorder @ A20 )
% 11.93/6.05 => ( preorder @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_388,axiom,
% 11.93/6.05 ! [A20: $tType] : ( monoid_add @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Orderings_Oorder_389,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( preorder @ A20 )
% 11.93/6.05 => ( order @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Orderings_Oord_390,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( preorder @ A20 )
% 11.93/6.05 => ( ord @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Ozero_391,axiom,
% 11.93/6.05 ! [A20: $tType] : ( zero @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___Groups_Oplus_392,axiom,
% 11.93/6.05 ! [A20: $tType] : ( plus @ ( multiset @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Multiset_Omultiset___HOL_Oequal_393,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( cl_HOL_Oequal @ A20 )
% 11.93/6.05 => ( cl_HOL_Oequal @ ( multiset @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_394,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( quickc3360725361186068524ustive @ A20 )
% 11.93/6.05 & ( quickc3360725361186068524ustive @ A21 ) )
% 11.93/6.05 => ( quickc3360725361186068524ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_395,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( quickc658316121487927005ustive @ A20 )
% 11.93/6.05 & ( quickc658316121487927005ustive @ A21 ) )
% 11.93/6.05 => ( quickc658316121487927005ustive @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_396,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( quickcheck_random @ A20 )
% 11.93/6.05 & ( quickcheck_random @ A21 ) )
% 11.93/6.05 => ( quickcheck_random @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Countable_Ocountable_397,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( countable @ A20 )
% 11.93/6.05 & ( countable @ A21 ) )
% 11.93/6.05 => ( countable @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_398,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( finite_finite @ A20 )
% 11.93/6.05 & ( finite_finite @ A21 ) )
% 11.93/6.05 => ( finite_finite @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___Heap_Oheap_399,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] :
% 11.93/6.05 ( ( ( heap @ A20 )
% 11.93/6.05 & ( heap @ A21 ) )
% 11.93/6.05 => ( heap @ ( product_prod @ A20 @ A21 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Oprod___HOL_Oequal_400,axiom,
% 11.93/6.05 ! [A20: $tType,A21: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A20 @ A21 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_401,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_402,axiom,
% 11.93/6.05 comple6319245703460814977attice @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_403,axiom,
% 11.93/6.05 boolea8198339166811842893lgebra @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_404,axiom,
% 11.93/6.05 comple9053668089753744459l_ccpo @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_405,axiom,
% 11.93/6.05 quickcheck_random @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_406,axiom,
% 11.93/6.05 semilattice_sup @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_407,axiom,
% 11.93/6.05 semilattice_inf @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_408,axiom,
% 11.93/6.05 bounded_lattice @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_409,axiom,
% 11.93/6.05 complete_Sup @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_410,axiom,
% 11.93/6.05 complete_Inf @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_411,axiom,
% 11.93/6.05 order_top @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_412,axiom,
% 11.93/6.05 order_bot @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Countable_Ocountable_413,axiom,
% 11.93/6.05 countable @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Orderings_Opreorder_414,axiom,
% 11.93/6.05 preorder @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Orderings_Olinorder_415,axiom,
% 11.93/6.05 linorder @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_416,axiom,
% 11.93/6.05 finite_finite @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Orderings_Oorder_417,axiom,
% 11.93/6.05 order @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Orderings_Oord_418,axiom,
% 11.93/6.05 ord @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Groups_Ouminus_419,axiom,
% 11.93/6.05 uminus @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Lattices_Osup_420,axiom,
% 11.93/6.05 sup @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Lattices_Oinf_421,axiom,
% 11.93/6.05 inf @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___Heap_Oheap_422,axiom,
% 11.93/6.05 heap @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Product__Type_Ounit___HOL_Oequal_423,axiom,
% 11.93/6.05 cl_HOL_Oequal @ product_unit ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Random_Orandom_424,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickcheck_random @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( heap_ext @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_425,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_426,axiom,
% 11.93/6.05 bit_un5681908812861735899ations @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_427,axiom,
% 11.93/6.05 semiri1453513574482234551roduct @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_428,axiom,
% 11.93/6.05 euclid5411537665997757685th_nat @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_429,axiom,
% 11.93/6.05 euclid8789492081693882211th_nat @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_430,axiom,
% 11.93/6.05 ordere1937475149494474687imp_le @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_431,axiom,
% 11.93/6.05 euclid3128863361964157862miring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_432,axiom,
% 11.93/6.05 euclid4440199948858584721cancel @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_433,axiom,
% 11.93/6.05 unique1627219031080169319umeral @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_434,axiom,
% 11.93/6.05 euclid8851590272496341667cancel @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_435,axiom,
% 11.93/6.05 strict9044650504122735259up_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_436,axiom,
% 11.93/6.05 ordere580206878836729694up_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_437,axiom,
% 11.93/6.05 ordere2412721322843649153imp_le @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_438,axiom,
% 11.93/6.05 bit_se359711467146920520ations @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_439,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_440,axiom,
% 11.93/6.05 strict7427464778891057005id_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_441,axiom,
% 11.93/6.05 ordere8940638589300402666id_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_442,axiom,
% 11.93/6.05 euclid3725896446679973847miring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_443,axiom,
% 11.93/6.05 linord715952674999750819strict @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_444,axiom,
% 11.93/6.05 linord4140545234300271783up_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_445,axiom,
% 11.93/6.05 bit_ri3973907225187159222ations @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_446,axiom,
% 11.93/6.05 linord181362715937106298miring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_447,axiom,
% 11.93/6.05 quickc658316121487927005ustive @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_448,axiom,
% 11.93/6.05 semiri3467727345109120633visors @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_449,axiom,
% 11.93/6.05 ordere6658533253407199908up_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_450,axiom,
% 11.93/6.05 ordere166539214618696060dd_abs @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_451,axiom,
% 11.93/6.05 ordere6911136660526730532id_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_452,axiom,
% 11.93/6.05 linord5086331880401160121up_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_453,axiom,
% 11.93/6.05 cancel2418104881723323429up_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_454,axiom,
% 11.93/6.05 ring_15535105094025558882visors @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_455,axiom,
% 11.93/6.05 cancel1802427076303600483id_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_456,axiom,
% 11.93/6.05 linord4710134922213307826strict @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_457,axiom,
% 11.93/6.05 comm_s4317794764714335236cancel @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_458,axiom,
% 11.93/6.05 bit_semiring_bits @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_459,axiom,
% 11.93/6.05 linord6961819062388156250ring_1 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_460,axiom,
% 11.93/6.05 ordered_ab_group_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_461,axiom,
% 11.93/6.05 cancel_semigroup_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_462,axiom,
% 11.93/6.05 linordered_semidom @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Quickcheck__Random_Orandom_463,axiom,
% 11.93/6.05 quickcheck_random @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_464,axiom,
% 11.93/6.05 ab_semigroup_mult @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_465,axiom,
% 11.93/6.05 algebraic_semidom @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_466,axiom,
% 11.93/6.05 comm_monoid_mult @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_467,axiom,
% 11.93/6.05 ab_semigroup_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_468,axiom,
% 11.93/6.05 semiring_parity @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_469,axiom,
% 11.93/6.05 comm_monoid_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_470,axiom,
% 11.93/6.05 semiring_modulo @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_471,axiom,
% 11.93/6.05 linordered_ring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_472,axiom,
% 11.93/6.05 linordered_idom @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_473,axiom,
% 11.93/6.05 comm_semiring_1 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_474,axiom,
% 11.93/6.05 comm_semiring_0 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_475,axiom,
% 11.93/6.05 semigroup_mult @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_476,axiom,
% 11.93/6.05 semidom_modulo @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_477,axiom,
% 11.93/6.05 semidom_divide @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_478,axiom,
% 11.93/6.05 semiring_numeral @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_479,axiom,
% 11.93/6.05 semigroup_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_480,axiom,
% 11.93/6.05 zero_less_one @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_481,axiom,
% 11.93/6.05 comm_semiring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_482,axiom,
% 11.93/6.05 semiring_char_0 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_483,axiom,
% 11.93/6.05 ab_group_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_484,axiom,
% 11.93/6.05 zero_neq_one @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_485,axiom,
% 11.93/6.05 ordered_ring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_486,axiom,
% 11.93/6.05 idom_abs_sgn @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_487,axiom,
% 11.93/6.05 ring_parity @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_488,axiom,
% 11.93/6.05 preorder @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_489,axiom,
% 11.93/6.05 linorder @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_490,axiom,
% 11.93/6.05 monoid_mult @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_491,axiom,
% 11.93/6.05 comm_ring_1 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_492,axiom,
% 11.93/6.05 monoid_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_493,axiom,
% 11.93/6.05 semiring_1 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_494,axiom,
% 11.93/6.05 semiring_0 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_495,axiom,
% 11.93/6.05 group_add @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_496,axiom,
% 11.93/6.05 comm_ring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_497,axiom,
% 11.93/6.05 order @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_498,axiom,
% 11.93/6.05 neg_numeral @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_499,axiom,
% 11.93/6.05 ring_char_0 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_500,axiom,
% 11.93/6.05 semiring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_501,axiom,
% 11.93/6.05 ord @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_502,axiom,
% 11.93/6.05 uminus @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_503,axiom,
% 11.93/6.05 ring_1 @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Power_Opower_504,axiom,
% 11.93/6.05 power @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_505,axiom,
% 11.93/6.05 numeral @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_506,axiom,
% 11.93/6.05 zero @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_507,axiom,
% 11.93/6.05 plus @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Oring_508,axiom,
% 11.93/6.05 ring @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Groups_Oone_509,axiom,
% 11.93/6.05 one @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_510,axiom,
% 11.93/6.05 dvd @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_511,axiom,
% 11.93/6.05 cl_HOL_Oequal @ code_integer ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_512,axiom,
% 11.93/6.05 bit_un5681908812861735899ations @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_513,axiom,
% 11.93/6.05 euclid5411537665997757685th_nat @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_514,axiom,
% 11.93/6.05 ordere1937475149494474687imp_le @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_515,axiom,
% 11.93/6.05 euclid3128863361964157862miring @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_516,axiom,
% 11.93/6.05 euclid4440199948858584721cancel @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_517,axiom,
% 11.93/6.05 strict9044650504122735259up_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_518,axiom,
% 11.93/6.05 ordere580206878836729694up_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_519,axiom,
% 11.93/6.05 ordere2412721322843649153imp_le @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_520,axiom,
% 11.93/6.05 bit_se359711467146920520ations @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_521,axiom,
% 11.93/6.05 quickc3360725361186068524ustive @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_522,axiom,
% 11.93/6.05 strict7427464778891057005id_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_523,axiom,
% 11.93/6.05 ordere8940638589300402666id_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_524,axiom,
% 11.93/6.05 euclid3725896446679973847miring @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_525,axiom,
% 11.93/6.05 linord4140545234300271783up_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_526,axiom,
% 11.93/6.05 linord181362715937106298miring @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_527,axiom,
% 11.93/6.05 quickc658316121487927005ustive @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_528,axiom,
% 11.93/6.05 semiri3467727345109120633visors @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_529,axiom,
% 11.93/6.05 ordere6658533253407199908up_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_530,axiom,
% 11.93/6.05 ordere6911136660526730532id_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_531,axiom,
% 11.93/6.05 cancel2418104881723323429up_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_532,axiom,
% 11.93/6.05 cancel1802427076303600483id_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_533,axiom,
% 11.93/6.05 comm_s4317794764714335236cancel @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_534,axiom,
% 11.93/6.05 bit_semiring_bits @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_535,axiom,
% 11.93/6.05 cancel_semigroup_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_536,axiom,
% 11.93/6.05 linordered_semidom @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_537,axiom,
% 11.93/6.05 quickcheck_random @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_538,axiom,
% 11.93/6.05 ab_semigroup_mult @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_539,axiom,
% 11.93/6.05 algebraic_semidom @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_540,axiom,
% 11.93/6.05 comm_monoid_mult @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_541,axiom,
% 11.93/6.05 comm_monoid_diff @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_542,axiom,
% 11.93/6.05 ab_semigroup_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_543,axiom,
% 11.93/6.05 semiring_parity @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_544,axiom,
% 11.93/6.05 comm_monoid_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_545,axiom,
% 11.93/6.05 semiring_modulo @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_546,axiom,
% 11.93/6.05 comm_semiring_1 @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_547,axiom,
% 11.93/6.05 comm_semiring_0 @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_548,axiom,
% 11.93/6.05 semigroup_mult @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_549,axiom,
% 11.93/6.05 semidom_modulo @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_550,axiom,
% 11.93/6.05 semidom_divide @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_551,axiom,
% 11.93/6.05 semiring_numeral @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_552,axiom,
% 11.93/6.05 semigroup_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_553,axiom,
% 11.93/6.05 zero_less_one @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_554,axiom,
% 11.93/6.05 comm_semiring @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_555,axiom,
% 11.93/6.05 semiring_char_0 @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_556,axiom,
% 11.93/6.05 zero_neq_one @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_557,axiom,
% 11.93/6.05 preorder @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_558,axiom,
% 11.93/6.05 linorder @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_559,axiom,
% 11.93/6.05 monoid_mult @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_560,axiom,
% 11.93/6.05 monoid_add @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_561,axiom,
% 11.93/6.05 semiring_1 @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_562,axiom,
% 11.93/6.05 semiring_0 @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_563,axiom,
% 11.93/6.05 order @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_564,axiom,
% 11.93/6.05 semiring @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Orderings_Oord_565,axiom,
% 11.93/6.05 ord @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Power_Opower_566,axiom,
% 11.93/6.05 power @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Num_Onumeral_567,axiom,
% 11.93/6.05 numeral @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Ozero_568,axiom,
% 11.93/6.05 zero @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oplus_569,axiom,
% 11.93/6.05 plus @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Groups_Oone_570,axiom,
% 11.93/6.05 one @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___Rings_Odvd_571,axiom,
% 11.93/6.05 dvd @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Code__Numeral_Onatural___HOL_Oequal_572,axiom,
% 11.93/6.05 cl_HOL_Oequal @ code_natural ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap__Time__Monad_OHeap___Quickcheck__Random_Orandom_573,axiom,
% 11.93/6.05 ! [A20: $tType] :
% 11.93/6.05 ( ( quickcheck_random @ A20 )
% 11.93/6.05 => ( quickcheck_random @ ( heap_Time_Heap @ A20 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_574,axiom,
% 11.93/6.05 ! [A20: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A20 ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(help_If_3_1_T,axiom,
% 11.93/6.05 ! [P: $o] :
% 11.93/6.05 ( ( P = $true )
% 11.93/6.05 | ( P = $false ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(help_If_2_1_T,axiom,
% 11.93/6.05 ! [A: $tType,X: A,Y: A] :
% 11.93/6.05 ( ( if @ A @ $false @ X @ Y )
% 11.93/6.05 = Y ) ).
% 11.93/6.05
% 11.93/6.05 thf(help_If_1_1_T,axiom,
% 11.93/6.05 ! [A: $tType,X: A,Y: A] :
% 11.93/6.05 ( ( if @ A @ $true @ X @ Y )
% 11.93/6.05 = X ) ).
% 11.93/6.05
% 11.93/6.05 thf(help_fChoice_1_1_T,axiom,
% 11.93/6.05 ! [A: $tType,P: A > $o] :
% 11.93/6.05 ( ( P @ ( fChoice @ A @ P ) )
% 11.93/6.05 = ( ? [X8: A] : ( P @ X8 ) ) ) ).
% 11.93/6.05
% 11.93/6.05 thf(tfree_0,hypothesis,
% 11.93/6.05 heap @ a ).
% 11.93/6.05
% 11.93/6.05 thf(conj_0,conjecture,
% 11.93/6.05 ( ( heap_Time_execute @ ( array @ a ) @ ( array_of_list @ a @ xs ) @ h )
% 11.93/6.05 != ( none @ ( product_prod @ ( array @ a ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% 11.93/6.05 ------------------------
% 11.93/6.48 % (24867)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/183Mi)
% 11.93/6.48 % (24872)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/275Mi)
% 11.93/6.48 % (24869)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/27Mi)
% 11.93/6.48 % (24873)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/18Mi)
% 11.93/6.48 % (24868)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/4Mi)
% 11.93/6.48 % (24870)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/2Mi)
% 11.93/6.48 % (24871)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/2Mi)
% 11.93/6.49 % (24870)Instruction limit reached!
% 11.93/6.49 % (24870)------------------------------
% 11.93/6.49 % (24870)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.49 % (24870)Termination reason: Unknown
% 11.93/6.49 % (24870)Termination phase: shuffling
% 11.93/6.49
% 11.93/6.49 % (24870)Memory used [KB]: 11769
% 11.93/6.49 % (24870)Time elapsed: 0.004 s
% 11.93/6.49 % (24870)Instructions burned: 2 (million)
% 11.93/6.49 % (24870)------------------------------
% 11.93/6.49 % (24870)------------------------------
% 11.93/6.49 % (24871)Instruction limit reached!
% 11.93/6.49 % (24871)------------------------------
% 11.93/6.49 % (24871)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.49 % (24871)Termination reason: Unknown
% 11.93/6.49 % (24871)Termination phase: shuffling
% 11.93/6.49
% 11.93/6.49 % (24871)Memory used [KB]: 11769
% 11.93/6.49 % (24871)Time elapsed: 0.004 s
% 11.93/6.49 % (24871)Instructions burned: 2 (million)
% 11.93/6.49 % (24871)------------------------------
% 11.93/6.49 % (24871)------------------------------
% 11.93/6.49 % (24868)Instruction limit reached!
% 11.93/6.49 % (24868)------------------------------
% 11.93/6.49 % (24868)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.49 % (24868)Termination reason: Unknown
% 11.93/6.49 % (24868)Termination phase: shuffling
% 11.93/6.49
% 11.93/6.49 % (24868)Memory used [KB]: 11897
% 11.93/6.49 % (24868)Time elapsed: 0.008 s
% 11.93/6.49 % (24868)Instructions burned: 4 (million)
% 11.93/6.49 % (24868)------------------------------
% 11.93/6.49 % (24868)------------------------------
% 11.93/6.50 % (24873)Instruction limit reached!
% 11.93/6.50 % (24873)------------------------------
% 11.93/6.50 % (24873)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.50 % (24873)Termination reason: Unknown
% 11.93/6.50 % (24873)Termination phase: shuffling
% 11.93/6.50
% 11.93/6.50 % (24873)Memory used [KB]: 12281
% 11.93/6.50 % (24873)Time elapsed: 0.017 s
% 11.93/6.50 % (24873)Instructions burned: 19 (million)
% 11.93/6.50 % (24873)------------------------------
% 11.93/6.50 % (24873)------------------------------
% 11.93/6.50 % (24869)Instruction limit reached!
% 11.93/6.50 % (24869)------------------------------
% 11.93/6.50 % (24869)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.50 % (24869)Termination reason: Unknown
% 11.93/6.50 % (24869)Termination phase: shuffling
% 11.93/6.50
% 11.93/6.50 % (24869)Memory used [KB]: 12537
% 11.93/6.50 % (24869)Time elapsed: 0.025 s
% 11.93/6.50 % (24869)Instructions burned: 28 (million)
% 11.93/6.50 % (24869)------------------------------
% 11.93/6.50 % (24869)------------------------------
% 11.93/6.51 % (24875)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/37Mi)
% 11.93/6.51 % (24874)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/3Mi)
% 11.93/6.51 % (24876)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/15Mi)
% 11.93/6.51 % (24874)Instruction limit reached!
% 11.93/6.51 % (24874)------------------------------
% 11.93/6.51 % (24874)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.51 % (24874)Termination reason: Unknown
% 11.93/6.51 % (24874)Termination phase: shuffling
% 11.93/6.51
% 11.93/6.51 % (24874)Memory used [KB]: 11769
% 11.93/6.51 % (24874)Time elapsed: 0.007 s
% 11.93/6.51 % (24874)Instructions burned: 3 (million)
% 11.93/6.51 % (24874)------------------------------
% 11.93/6.51 % (24874)------------------------------
% 11.93/6.52 % (24878)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/1041Mi)
% 11.93/6.52 % (24877)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/3Mi)
% 11.93/6.52 % (24876)Instruction limit reached!
% 11.93/6.52 % (24876)------------------------------
% 11.93/6.52 % (24876)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.52 % (24876)Termination reason: Unknown
% 11.93/6.52 % (24876)Termination phase: shuffling
% 11.93/6.52
% 11.93/6.52 % (24876)Memory used [KB]: 12281
% 11.93/6.52 % (24876)Time elapsed: 0.012 s
% 11.93/6.52 % (24876)Instructions burned: 16 (million)
% 11.93/6.52 % (24876)------------------------------
% 11.93/6.52 % (24876)------------------------------
% 11.93/6.52 % (24877)Instruction limit reached!
% 11.93/6.52 % (24877)------------------------------
% 11.93/6.52 % (24877)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.52 % (24877)Termination reason: Unknown
% 11.93/6.52 % (24877)Termination phase: shuffling
% 11.93/6.52
% 11.93/6.52 % (24877)Memory used [KB]: 11769
% 11.93/6.52 % (24877)Time elapsed: 0.006 s
% 11.93/6.52 % (24877)Instructions burned: 3 (million)
% 11.93/6.52 % (24877)------------------------------
% 11.93/6.52 % (24877)------------------------------
% 11.93/6.53 % (24879)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/7Mi)
% 11.93/6.53 % (24880)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/16Mi)
% 11.93/6.53 % (24881)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/3Mi)
% 11.93/6.54 % (24875)Instruction limit reached!
% 11.93/6.54 % (24875)------------------------------
% 11.93/6.54 % (24875)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.54 % (24875)Termination reason: Unknown
% 11.93/6.54 % (24875)Termination phase: shuffling
% 11.93/6.54
% 11.93/6.54 % (24875)Memory used [KB]: 12665
% 11.93/6.54 % (24875)Time elapsed: 0.031 s
% 11.93/6.54 % (24875)Instructions burned: 39 (million)
% 11.93/6.54 % (24875)------------------------------
% 11.93/6.54 % (24875)------------------------------
% 11.93/6.54 % (24879)Instruction limit reached!
% 11.93/6.54 % (24879)------------------------------
% 11.93/6.54 % (24879)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.54 % (24879)Termination reason: Unknown
% 11.93/6.54 % (24879)Termination phase: shuffling
% 11.93/6.54
% 11.93/6.54 % (24879)Memory used [KB]: 12025
% 11.93/6.54 % (24879)Time elapsed: 0.010 s
% 11.93/6.54 % (24879)Instructions burned: 8 (million)
% 11.93/6.54 % (24879)------------------------------
% 11.93/6.54 % (24879)------------------------------
% 11.93/6.54 % (24881)Instruction limit reached!
% 11.93/6.54 % (24881)------------------------------
% 11.93/6.54 % (24881)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.54 % (24881)Termination reason: Unknown
% 11.93/6.54 % (24881)Termination phase: shuffling
% 11.93/6.54
% 11.93/6.54 % (24881)Memory used [KB]: 11769
% 11.93/6.54 % (24881)Time elapsed: 0.005 s
% 11.93/6.54 % (24881)Instructions burned: 3 (million)
% 11.93/6.54 % (24881)------------------------------
% 11.93/6.54 % (24881)------------------------------
% 11.93/6.54 % (24880)Instruction limit reached!
% 11.93/6.54 % (24880)------------------------------
% 11.93/6.54 % (24880)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.54 % (24880)Termination reason: Unknown
% 11.93/6.54 % (24880)Termination phase: shuffling
% 11.93/6.54
% 11.93/6.54 % (24880)Memory used [KB]: 12281
% 11.93/6.54 % (24880)Time elapsed: 0.012 s
% 11.93/6.54 % (24880)Instructions burned: 17 (million)
% 11.93/6.54 % (24880)------------------------------
% 11.93/6.54 % (24880)------------------------------
% 11.93/6.54 % (24867)Instruction limit reached!
% 11.93/6.54 % (24867)------------------------------
% 11.93/6.54 % (24867)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.54 % (24867)Termination reason: Unknown
% 11.93/6.54 % (24867)Termination phase: shuffling
% 11.93/6.54
% 11.93/6.54 % (24867)Memory used [KB]: 14967
% 11.93/6.54 % (24867)Time elapsed: 0.069 s
% 11.93/6.54 % (24867)Instructions burned: 184 (million)
% 11.93/6.54 % (24867)------------------------------
% 11.93/6.54 % (24867)------------------------------
% 11.93/6.55 % (24883)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/7Mi)
% 11.93/6.55 % (24882)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/3Mi)
% 11.93/6.55 % (24884)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/3Mi)
% 11.93/6.55 % (24884)Instruction limit reached!
% 11.93/6.55 % (24884)------------------------------
% 11.93/6.55 % (24884)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.55 % (24884)Termination reason: Unknown
% 11.93/6.55 % (24884)Termination phase: shuffling
% 11.93/6.55
% 11.93/6.55 % (24884)Memory used [KB]: 11769
% 11.93/6.55 % (24884)Time elapsed: 0.004 s
% 11.93/6.55 % (24884)Instructions burned: 3 (million)
% 11.93/6.55 % (24884)------------------------------
% 11.93/6.55 % (24884)------------------------------
% 11.93/6.55 % (24882)Instruction limit reached!
% 11.93/6.55 % (24882)------------------------------
% 11.93/6.55 % (24882)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.55 % (24882)Termination reason: Unknown
% 11.93/6.55 % (24882)Termination phase: shuffling
% 11.93/6.55
% 11.93/6.55 % (24882)Memory used [KB]: 11769
% 11.93/6.55 % (24882)Time elapsed: 0.004 s
% 11.93/6.55 % (24882)Instructions burned: 3 (million)
% 11.93/6.55 % (24882)------------------------------
% 11.93/6.55 % (24882)------------------------------
% 11.93/6.55 % (24883)Instruction limit reached!
% 11.93/6.55 % (24883)------------------------------
% 11.93/6.55 % (24883)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.55 % (24883)Termination reason: Unknown
% 11.93/6.55 % (24883)Termination phase: shuffling
% 11.93/6.55
% 11.93/6.55 % (24883)Memory used [KB]: 12153
% 11.93/6.55 % (24883)Time elapsed: 0.009 s
% 11.93/6.55 % (24883)Instructions burned: 9 (million)
% 11.93/6.55 % (24883)------------------------------
% 11.93/6.55 % (24883)------------------------------
% 11.93/6.56 % (24886)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/18Mi)
% 11.93/6.56 % (24885)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/4Mi)
% 11.93/6.56 % (24887)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/710Mi)
% 11.93/6.56 % (24888)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/6Mi)
% 11.93/6.56 % (24885)Instruction limit reached!
% 11.93/6.56 % (24885)------------------------------
% 11.93/6.56 % (24885)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.56 % (24885)Termination reason: Unknown
% 11.93/6.56 % (24885)Termination phase: shuffling
% 11.93/6.56
% 11.93/6.56 % (24885)Memory used [KB]: 12025
% 11.93/6.56 % (24885)Time elapsed: 0.007 s
% 11.93/6.56 % (24885)Instructions burned: 5 (million)
% 11.93/6.56 % (24885)------------------------------
% 11.93/6.56 % (24885)------------------------------
% 11.93/6.57 % (24886)Instruction limit reached!
% 11.93/6.57 % (24886)------------------------------
% 11.93/6.57 % (24886)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.57 % (24886)Termination reason: Unknown
% 11.93/6.57 % (24886)Termination phase: shuffling
% 11.93/6.57
% 11.93/6.57 % (24886)Memory used [KB]: 12281
% 11.93/6.57 % (24886)Time elapsed: 0.013 s
% 11.93/6.57 % (24886)Instructions burned: 19 (million)
% 11.93/6.57 % (24886)------------------------------
% 11.93/6.57 % (24886)------------------------------
% 11.93/6.57 % (24889)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_24760 for (2995ds/902Mi)
% 11.93/6.57 % (24888)Instruction limit reached!
% 11.93/6.57 % (24888)------------------------------
% 11.93/6.57 % (24888)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.57 % (24888)Termination reason: Unknown
% 11.93/6.57 % (24888)Termination phase: shuffling
% 11.93/6.57
% 11.93/6.57 % (24888)Memory used [KB]: 12025
% 11.93/6.57 % (24888)Time elapsed: 0.008 s
% 11.93/6.57 % (24888)Instructions burned: 7 (million)
% 11.93/6.57 % (24888)------------------------------
% 11.93/6.57 % (24888)------------------------------
% 11.93/6.57 % (24890)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/21Mi)
% 11.93/6.58 % (24891)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/5Mi)
% 11.93/6.58 % (24892)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/6Mi)
% 11.93/6.58 % (24872)Instruction limit reached!
% 11.93/6.58 % (24872)------------------------------
% 11.93/6.58 % (24872)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.58 % (24872)Termination reason: Unknown
% 11.93/6.58 % (24872)Termination phase: shuffling
% 11.93/6.58
% 11.93/6.58 % (24872)Memory used [KB]: 16247
% 11.93/6.58 % (24872)Time elapsed: 0.107 s
% 11.93/6.58 % (24872)Instructions burned: 275 (million)
% 11.93/6.58 % (24872)------------------------------
% 11.93/6.58 % (24872)------------------------------
% 11.93/6.58 % (24891)Instruction limit reached!
% 11.93/6.58 % (24891)------------------------------
% 11.93/6.58 % (24891)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.58 % (24891)Termination reason: Unknown
% 11.93/6.58 % (24891)Termination phase: shuffling
% 11.93/6.58
% 11.93/6.58 % (24891)Memory used [KB]: 12025
% 11.93/6.58 % (24891)Time elapsed: 0.007 s
% 11.93/6.58 % (24891)Instructions burned: 5 (million)
% 11.93/6.58 % (24891)------------------------------
% 11.93/6.58 % (24891)------------------------------
% 11.93/6.58 % (24892)Instruction limit reached!
% 11.93/6.58 % (24892)------------------------------
% 11.93/6.58 % (24892)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.58 % (24892)Termination reason: Unknown
% 11.93/6.58 % (24892)Termination phase: shuffling
% 11.93/6.58
% 11.93/6.58 % (24892)Memory used [KB]: 12025
% 11.93/6.58 % (24892)Time elapsed: 0.008 s
% 11.93/6.58 % (24892)Instructions burned: 7 (million)
% 11.93/6.58 % (24892)------------------------------
% 11.93/6.58 % (24892)------------------------------
% 11.93/6.59 % (24890)Instruction limit reached!
% 11.93/6.59 % (24890)------------------------------
% 11.93/6.59 % (24890)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.59 % (24890)Termination reason: Unknown
% 11.93/6.59 % (24890)Termination phase: shuffling
% 11.93/6.59
% 11.93/6.59 % (24890)Memory used [KB]: 12281
% 11.93/6.59 % (24890)Time elapsed: 0.017 s
% 11.93/6.59 % (24890)Instructions burned: 21 (million)
% 11.93/6.59 % (24890)------------------------------
% 11.93/6.59 % (24890)------------------------------
% 11.93/6.59 % (24893)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/377Mi)
% 11.93/6.59 % (24894)dis+1010_1:4_atotf=0.2:c=on:cbe=off:cnfonf=lazy_simp:fe=off:ins=2:ntd=on:s2a=on:s2at=5.0:sgt=5:ss=axioms:st=1.5:i=779:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/779Mi)
% 11.93/6.59 % (24895)lrs+10_1:1_cnfonf=lazy_not_be_gen:ntd=on:sp=const_min:ss=axioms:sup=off:i=19:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/19Mi)
% 11.93/6.60 % (24896)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/879Mi)
% 11.93/6.61 % (24895)Instruction limit reached!
% 11.93/6.61 % (24895)------------------------------
% 11.93/6.61 % (24895)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.61 % (24895)Termination reason: Unknown
% 11.93/6.61 % (24895)Termination phase: shuffling
% 11.93/6.61
% 11.93/6.61 % (24895)Memory used [KB]: 12281
% 11.93/6.61 % (24895)Time elapsed: 0.012 s
% 11.93/6.61 % (24895)Instructions burned: 21 (million)
% 11.93/6.61 % (24895)------------------------------
% 11.93/6.61 % (24895)------------------------------
% 11.93/6.62 % (24897)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/17Mi)
% 11.93/6.63 % (24897)Instruction limit reached!
% 11.93/6.63 % (24897)------------------------------
% 11.93/6.63 % (24897)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.63 % (24897)Termination reason: Unknown
% 11.93/6.63 % (24897)Termination phase: shuffling
% 11.93/6.63
% 11.93/6.63 % (24897)Memory used [KB]: 12281
% 11.93/6.63 % (24897)Time elapsed: 0.013 s
% 11.93/6.63 % (24897)Instructions burned: 19 (million)
% 11.93/6.63 % (24897)------------------------------
% 11.93/6.63 % (24897)------------------------------
% 11.93/6.64 % (24898)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/3Mi)
% 11.93/6.64 % (24898)Instruction limit reached!
% 11.93/6.64 % (24898)------------------------------
% 11.93/6.64 % (24898)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.64 % (24898)Termination reason: Unknown
% 11.93/6.64 % (24898)Termination phase: shuffling
% 11.93/6.64
% 11.93/6.64 % (24898)Memory used [KB]: 11769
% 11.93/6.64 % (24898)Time elapsed: 0.005 s
% 11.93/6.64 % (24898)Instructions burned: 3 (million)
% 11.93/6.64 % (24898)------------------------------
% 11.93/6.64 % (24898)------------------------------
% 11.93/6.65 % (24899)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_24760 for (2994ds/30Mi)
% 11.93/6.67 % (24899)Instruction limit reached!
% 11.93/6.67 % (24899)------------------------------
% 11.93/6.67 % (24899)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.67 % (24899)Termination reason: Unknown
% 11.93/6.67 % (24899)Termination phase: shuffling
% 11.93/6.67
% 11.93/6.67 % (24899)Memory used [KB]: 12409
% 11.93/6.67 % (24899)Time elapsed: 0.016 s
% 11.93/6.67 % (24899)Instructions burned: 31 (million)
% 11.93/6.67 % (24899)------------------------------
% 11.93/6.67 % (24899)------------------------------
% 11.93/6.68 % (24900)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_24760 for (2993ds/127Mi)
% 11.93/6.73 % (24900)Instruction limit reached!
% 11.93/6.73 % (24900)------------------------------
% 11.93/6.73 % (24900)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.73 % (24900)Termination reason: Unknown
% 11.93/6.73 % (24900)Termination phase: shuffling
% 11.93/6.73
% 11.93/6.73 % (24900)Memory used [KB]: 14200
% 11.93/6.73 % (24900)Time elapsed: 0.050 s
% 11.93/6.73 % (24900)Instructions burned: 129 (million)
% 11.93/6.73 % (24900)------------------------------
% 11.93/6.73 % (24900)------------------------------
% 11.93/6.73 % (24893)Instruction limit reached!
% 11.93/6.73 % (24893)------------------------------
% 11.93/6.73 % (24893)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.73 % (24893)Termination reason: Unknown
% 11.93/6.73 % (24893)Termination phase: shuffling
% 11.93/6.73
% 11.93/6.73 % (24893)Memory used [KB]: 17654
% 11.93/6.73 % (24893)Time elapsed: 0.139 s
% 11.93/6.73 % (24893)Instructions burned: 379 (million)
% 11.93/6.73 % (24893)------------------------------
% 11.93/6.73 % (24893)------------------------------
% 11.93/6.74 % (24901)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_24760 for (2993ds/100Mi)
% 11.93/6.74 % (24902)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=3:si=on:rtra=on_0 on DTF2THF_24760 for (2993ds/3Mi)
% 11.93/6.74 % (24902)Instruction limit reached!
% 11.93/6.74 % (24902)------------------------------
% 11.93/6.74 % (24902)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.74 % (24902)Termination reason: Unknown
% 11.93/6.74 % (24902)Termination phase: shuffling
% 11.93/6.74
% 11.93/6.74 % (24902)Memory used [KB]: 11769
% 11.93/6.74 % (24902)Time elapsed: 0.004 s
% 11.93/6.74 % (24902)Instructions burned: 3 (million)
% 11.93/6.74 % (24902)------------------------------
% 11.93/6.74 % (24902)------------------------------
% 11.93/6.76 % (24903)lrs+10_8:1_au=on:avsq=on:e2e=on:ins=3:s2a=on:s2at=3.0:ss=axioms:i=20:si=on:rtra=on_0 on DTF2THF_24760 for (2993ds/20Mi)
% 11.93/6.77 % (24903)Instruction limit reached!
% 11.93/6.77 % (24903)------------------------------
% 11.93/6.77 % (24903)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.77 % (24903)Termination reason: Unknown
% 11.93/6.77 % (24903)Termination phase: shuffling
% 11.93/6.77
% 11.93/6.77 % (24903)Memory used [KB]: 12281
% 11.93/6.77 % (24903)Time elapsed: 0.013 s
% 11.93/6.77 % (24903)Instructions burned: 20 (million)
% 11.93/6.77 % (24903)------------------------------
% 11.93/6.77 % (24903)------------------------------
% 11.93/6.77 % (24901)Instruction limit reached!
% 11.93/6.77 % (24901)------------------------------
% 11.93/6.77 % (24901)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.77 % (24901)Termination reason: Unknown
% 11.93/6.77 % (24901)Termination phase: shuffling
% 11.93/6.77
% 11.93/6.77 % (24901)Memory used [KB]: 13688
% 11.93/6.77 % (24901)Time elapsed: 0.036 s
% 11.93/6.77 % (24901)Instructions burned: 102 (million)
% 11.93/6.77 % (24901)------------------------------
% 11.93/6.77 % (24901)------------------------------
% 11.93/6.78 % (24904)dis+1002_1:1_cbe=off:hud=5:nm=4:plsq=on:plsqr=7,1:prag=on:sp=const_max:tnu=1:i=86:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/86Mi)
% 11.93/6.79 % (24905)lrs+1010_1:1_au=on:cbe=off:nm=2:ntd=on:sd=2:ss=axioms:st=5.0:i=107:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/107Mi)
% 11.93/6.81 % (24904)Instruction limit reached!
% 11.93/6.81 % (24904)------------------------------
% 11.93/6.81 % (24904)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.81 % (24904)Termination reason: Unknown
% 11.93/6.81 % (24904)Termination phase: shuffling
% 11.93/6.81
% 11.93/6.81 % (24904)Memory used [KB]: 13560
% 11.93/6.81 % (24904)Time elapsed: 0.036 s
% 11.93/6.81 % (24904)Instructions burned: 88 (million)
% 11.93/6.81 % (24904)------------------------------
% 11.93/6.81 % (24904)------------------------------
% 11.93/6.81 % (24887)Instruction limit reached!
% 11.93/6.81 % (24887)------------------------------
% 11.93/6.81 % (24887)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.81 % (24887)Termination reason: Unknown
% 11.93/6.81 % (24887)Termination phase: shuffling
% 11.93/6.81
% 11.93/6.81 % (24887)Memory used [KB]: 22643
% 11.93/6.81 % (24887)Time elapsed: 0.255 s
% 11.93/6.81 % (24887)Instructions burned: 712 (million)
% 11.93/6.81 % (24887)------------------------------
% 11.93/6.81 % (24887)------------------------------
% 11.93/6.82 % (24906)lrs+2_1:1024_cnfonf=lazy_gen:fe=off:hud=15:plsq=on:plsqc=1:plsqr=32,1:i=39:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/39Mi)
% 11.93/6.83 % (24907)dis+10_1:1_cnfonf=lazy_not_gen:fsr=off:kws=precedence:nwc=5.0:s2a=on:ss=axioms:st=1.5:i=448:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/448Mi)
% 11.93/6.83 % (24905)Instruction limit reached!
% 11.93/6.83 % (24905)------------------------------
% 11.93/6.83 % (24905)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.83 % (24905)Termination reason: Unknown
% 11.93/6.83 % (24905)Termination phase: shuffling
% 11.93/6.83
% 11.93/6.83 % (24905)Memory used [KB]: 13816
% 11.93/6.83 % (24905)Time elapsed: 0.043 s
% 11.93/6.83 % (24905)Instructions burned: 109 (million)
% 11.93/6.83 % (24905)------------------------------
% 11.93/6.83 % (24905)------------------------------
% 11.93/6.84 % (24908)lrs+10_1:512_au=on:fde=unused:lma=on:nm=32:plsq=on:plsqc=1:plsqr=16121663,131072:sfv=off:sp=const_max:ss=axioms:st=3.0:tgt=full:i=46:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/46Mi)
% 11.93/6.84 % (24878)Instruction limit reached!
% 11.93/6.84 % (24878)------------------------------
% 11.93/6.84 % (24878)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.84 % (24878)Termination reason: Unknown
% 11.93/6.84 % (24878)Termination phase: shuffling
% 11.93/6.84
% 11.93/6.84 % (24878)Memory used [KB]: 32494
% 11.93/6.84 % (24878)Time elapsed: 0.324 s
% 11.93/6.84 % (24878)Instructions burned: 1042 (million)
% 11.93/6.84 % (24878)------------------------------
% 11.93/6.84 % (24878)------------------------------
% 11.93/6.84 % (24894)Instruction limit reached!
% 11.93/6.84 % (24894)------------------------------
% 11.93/6.84 % (24894)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.84 % (24894)Termination reason: Unknown
% 11.93/6.84 % (24894)Termination phase: shuffling
% 11.93/6.84
% 11.93/6.84 % (24894)Memory used [KB]: 25074
% 11.93/6.84 % (24894)Time elapsed: 0.247 s
% 11.93/6.84 % (24894)Instructions burned: 781 (million)
% 11.93/6.84 % (24894)------------------------------
% 11.93/6.84 % (24894)------------------------------
% 11.93/6.84 % (24906)Instruction limit reached!
% 11.93/6.84 % (24906)------------------------------
% 11.93/6.84 % (24906)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.84 % (24906)Termination reason: Unknown
% 11.93/6.84 % (24906)Termination phase: shuffling
% 11.93/6.84
% 11.93/6.84 % (24906)Memory used [KB]: 12665
% 11.93/6.84 % (24906)Time elapsed: 0.018 s
% 11.93/6.84 % (24906)Instructions burned: 40 (million)
% 11.93/6.84 % (24906)------------------------------
% 11.93/6.84 % (24906)------------------------------
% 11.93/6.85 % (24909)lrs+10_1:10_au=on:av=off:cbe=off:cnfonf=lazy_pi_sigma_gen:ntd=on:plsq=on:plsqc=1:plsqr=32,1:i=98:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/98Mi)
% 11.93/6.85 % (24910)ott+1002_1:1_apa=on:au=on:bd=off:cnfonf=off:fd=off:sos=on:sp=weighted_frequency:i=507:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/507Mi)
% 11.93/6.85 % (24911)dis+1010_2:3_amm=off:fd=preordered:ixr=off:nm=0:pe=on:piset=equals:prag=on:sac=on:tgt=ground:i=149:si=on:rtra=on_0 on DTF2THF_24760 for (2992ds/149Mi)
% 11.93/6.86 % (24908)Instruction limit reached!
% 11.93/6.86 % (24908)------------------------------
% 11.93/6.86 % (24908)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.86 % (24908)Termination reason: Unknown
% 11.93/6.86 % (24908)Termination phase: shuffling
% 11.93/6.86
% 11.93/6.86 % (24908)Memory used [KB]: 12792
% 11.93/6.86 % (24908)Time elapsed: 0.023 s
% 11.93/6.86 % (24908)Instructions burned: 48 (million)
% 11.93/6.86 % (24908)------------------------------
% 11.93/6.86 % (24908)------------------------------
% 11.93/6.87 % (24912)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/39Mi)
% 11.93/6.88 % (24909)Instruction limit reached!
% 11.93/6.88 % (24909)------------------------------
% 11.93/6.88 % (24909)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.88 % (24909)Termination reason: Unknown
% 11.93/6.88 % (24909)Termination phase: shuffling
% 11.93/6.88
% 11.93/6.88 % (24909)Memory used [KB]: 13560
% 11.93/6.88 % (24909)Time elapsed: 0.035 s
% 11.93/6.88 % (24909)Instructions burned: 99 (million)
% 11.93/6.88 % (24909)------------------------------
% 11.93/6.88 % (24909)------------------------------
% 11.93/6.89 % (24912)Instruction limit reached!
% 11.93/6.89 % (24912)------------------------------
% 11.93/6.89 % (24912)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.89 % (24912)Termination reason: Unknown
% 11.93/6.89 % (24912)Termination phase: shuffling
% 11.93/6.89
% 11.93/6.89 % (24912)Memory used [KB]: 12665
% 11.93/6.89 % (24912)Time elapsed: 0.020 s
% 11.93/6.89 % (24912)Instructions burned: 41 (million)
% 11.93/6.89 % (24912)------------------------------
% 11.93/6.89 % (24912)------------------------------
% 11.93/6.90 % (24913)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/6Mi)
% 11.93/6.90 % (24914)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=1087:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/1087Mi)
% 11.93/6.90 % (24889)Instruction limit reached!
% 11.93/6.90 % (24889)------------------------------
% 11.93/6.90 % (24889)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.90 % (24889)Termination reason: Unknown
% 11.93/6.90 % (24889)Termination phase: Property scanning
% 11.93/6.90
% 11.93/6.90 % (24889)Memory used [KB]: 27249
% 11.93/6.90 % (24889)Time elapsed: 0.334 s
% 11.93/6.90 % (24889)Instructions burned: 904 (million)
% 11.93/6.90 % (24889)------------------------------
% 11.93/6.90 % (24889)------------------------------
% 11.93/6.90 % (24913)Instruction limit reached!
% 11.93/6.90 % (24913)------------------------------
% 11.93/6.90 % (24913)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.90 % (24913)Termination reason: Unknown
% 11.93/6.90 % (24913)Termination phase: shuffling
% 11.93/6.90
% 11.93/6.90 % (24913)Memory used [KB]: 12025
% 11.93/6.90 % (24913)Time elapsed: 0.008 s
% 11.93/6.90 % (24913)Instructions burned: 6 (million)
% 11.93/6.90 % (24913)------------------------------
% 11.93/6.90 % (24913)------------------------------
% 11.93/6.91 % (24911)Instruction limit reached!
% 11.93/6.91 % (24911)------------------------------
% 11.93/6.91 % (24911)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.91 % (24911)Termination reason: Unknown
% 11.93/6.91 % (24911)Termination phase: shuffling
% 11.93/6.91
% 11.93/6.91 % (24911)Memory used [KB]: 14456
% 11.93/6.91 % (24911)Time elapsed: 0.056 s
% 11.93/6.91 % (24911)Instructions burned: 149 (million)
% 11.93/6.91 % (24911)------------------------------
% 11.93/6.91 % (24911)------------------------------
% 11.93/6.91 % (24896)Instruction limit reached!
% 11.93/6.91 % (24896)------------------------------
% 11.93/6.91 % (24896)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.91 % (24896)Termination reason: Unknown
% 11.93/6.91 % (24896)Termination phase: shuffling
% 11.93/6.91
% 11.93/6.91 % (24896)Memory used [KB]: 26865
% 11.93/6.91 % (24896)Time elapsed: 0.311 s
% 11.93/6.91 % (24896)Instructions burned: 880 (million)
% 11.93/6.91 % (24896)------------------------------
% 11.93/6.91 % (24896)------------------------------
% 11.93/6.91 % (24915)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/47Mi)
% 11.93/6.91 % (24916)lrs+1010_1:128_av=off:cbe=off:cnfonf=lazy_simp:fsr=off:ntd=on:ss=axioms:st=1.5:i=44:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/44Mi)
% 11.93/6.92 % (24917)dis+10_1:1_av=off:bd=off:cnfonf=lazy_gen:fe=off:hud=10:plsq=on:plsqc=1:plsqr=32,1:prag=on:tnu=1:i=432:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/432Mi)
% 11.93/6.92 % (24918)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/62Mi)
% 11.93/6.93 % (24916)Instruction limit reached!
% 11.93/6.93 % (24916)------------------------------
% 11.93/6.93 % (24916)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.93 % (24916)Termination reason: Unknown
% 11.93/6.93 % (24916)Termination phase: shuffling
% 11.93/6.93
% 11.93/6.93 % (24916)Memory used [KB]: 12665
% 11.93/6.93 % (24916)Time elapsed: 0.019 s
% 11.93/6.93 % (24916)Instructions burned: 44 (million)
% 11.93/6.93 % (24916)------------------------------
% 11.93/6.93 % (24916)------------------------------
% 11.93/6.93 % (24915)Instruction limit reached!
% 11.93/6.93 % (24915)------------------------------
% 11.93/6.93 % (24915)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.93 % (24915)Termination reason: Unknown
% 11.93/6.93 % (24915)Termination phase: shuffling
% 11.93/6.93
% 11.93/6.93 % (24915)Memory used [KB]: 12792
% 11.93/6.93 % (24915)Time elapsed: 0.023 s
% 11.93/6.93 % (24915)Instructions burned: 49 (million)
% 11.93/6.93 % (24915)------------------------------
% 11.93/6.93 % (24915)------------------------------
% 11.93/6.94 % (24919)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/31Mi)
% 11.93/6.94 % (24920)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/96Mi)
% 11.93/6.95 % (24918)Instruction limit reached!
% 11.93/6.95 % (24918)------------------------------
% 11.93/6.95 % (24918)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.95 % (24918)Termination reason: Unknown
% 11.93/6.95 % (24918)Termination phase: shuffling
% 11.93/6.95
% 11.93/6.95 % (24918)Memory used [KB]: 13048
% 11.93/6.95 % (24918)Time elapsed: 0.025 s
% 11.93/6.95 % (24918)Instructions burned: 64 (million)
% 11.93/6.95 % (24918)------------------------------
% 11.93/6.95 % (24918)------------------------------
% 11.93/6.96 % (24919)Instruction limit reached!
% 11.93/6.96 % (24919)------------------------------
% 11.93/6.96 % (24919)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.96 % (24919)Termination reason: Unknown
% 11.93/6.96 % (24919)Termination phase: shuffling
% 11.93/6.96
% 11.93/6.96 % (24919)Memory used [KB]: 12537
% 11.93/6.96 % (24919)Time elapsed: 0.017 s
% 11.93/6.96 % (24919)Instructions burned: 31 (million)
% 11.93/6.96 % (24919)------------------------------
% 11.93/6.96 % (24919)------------------------------
% 11.93/6.96 % (24921)lrs+1002_1:4_au=on:cbe=off:cnfonf=lazy_pi_sigma_gen:hud=15:inj=on:nwc=4.0:s2a=on:sac=on:sp=reverse_arity:i=453:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/453Mi)
% 11.93/6.96 % (24907)Instruction limit reached!
% 11.93/6.96 % (24907)------------------------------
% 11.93/6.96 % (24907)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.96 % (24907)Termination reason: Unknown
% 11.93/6.96 % (24907)Termination phase: shuffling
% 11.93/6.96
% 11.93/6.96 % (24907)Memory used [KB]: 20468
% 11.93/6.96 % (24907)Time elapsed: 0.139 s
% 11.93/6.96 % (24907)Instructions burned: 448 (million)
% 11.93/6.96 % (24907)------------------------------
% 11.93/6.96 % (24907)------------------------------
% 11.93/6.97 % (24922)lrs-1004_1:1_au=on:bsd=on:cbe=off:cnfonf=off:fe=axiom:fsd=on:i=71:si=on:rtra=on_0 on DTF2THF_24760 for (2991ds/71Mi)
% 11.93/6.98 % (24923)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/83Mi)
% 11.93/6.98 % (24920)Instruction limit reached!
% 11.93/6.98 % (24920)------------------------------
% 11.93/6.98 % (24920)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/6.98 % (24920)Termination reason: Unknown
% 11.93/6.98 % (24920)Termination phase: shuffling
% 11.93/6.98
% 11.93/6.98 % (24920)Memory used [KB]: 13688
% 11.93/6.98 % (24920)Time elapsed: 0.039 s
% 11.93/6.98 % (24920)Instructions burned: 98 (million)
% 11.93/6.98 % (24920)------------------------------
% 11.93/6.98 % (24920)------------------------------
% 11.93/6.99 % (24924)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/11Mi)
% 11.93/7.00 % (24924)Instruction limit reached!
% 11.93/7.00 % (24924)------------------------------
% 11.93/7.00 % (24924)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/7.00 % (24924)Termination reason: Unknown
% 11.93/7.00 % (24924)Termination phase: shuffling
% 11.93/7.00
% 11.93/7.00 % (24924)Memory used [KB]: 12153
% 11.93/7.00 % (24924)Time elapsed: 0.010 s
% 11.93/7.00 % (24924)Instructions burned: 11 (million)
% 11.93/7.00 % (24924)------------------------------
% 11.93/7.00 % (24924)------------------------------
% 11.93/7.00 % (24922)Instruction limit reached!
% 11.93/7.00 % (24922)------------------------------
% 11.93/7.00 % (24922)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/7.00 % (24922)Termination reason: Unknown
% 11.93/7.00 % (24922)Termination phase: shuffling
% 11.93/7.00
% 11.93/7.00 % (24922)Memory used [KB]: 13304
% 11.93/7.00 % (24922)Time elapsed: 0.038 s
% 11.93/7.00 % (24922)Instructions burned: 73 (million)
% 11.93/7.00 % (24922)------------------------------
% 11.93/7.00 % (24922)------------------------------
% 11.93/7.01 % (24923)Instruction limit reached!
% 11.93/7.01 % (24923)------------------------------
% 11.93/7.01 % (24923)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.93/7.01 % (24923)Termination reason: Unknown
% 11.93/7.01 % (24923)Termination phase: shuffling
% 11.93/7.01
% 11.93/7.01 % (24923)Memory used [KB]: 13432
% 11.93/7.01 % (24923)Time elapsed: 0.035 s
% 11.93/7.01 % (24923)Instructions burned: 84 (million)
% 11.93/7.01 % (24923)------------------------------
% 11.93/7.01 % (24923)------------------------------
% 16.08/7.01 % (24925)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_gen_be_off:fsr=off:spb=non_intro:i=41:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/41Mi)
% 16.08/7.02 % (24927)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/150Mi)
% 16.08/7.02 % (24926)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/6Mi)
% 16.08/7.03 % (24926)Instruction limit reached!
% 16.08/7.03 % (24926)------------------------------
% 16.08/7.03 % (24926)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.03 % (24926)Termination reason: Unknown
% 16.08/7.03 % (24926)Termination phase: shuffling
% 16.08/7.03
% 16.08/7.03 % (24926)Memory used [KB]: 12025
% 16.08/7.03 % (24926)Time elapsed: 0.008 s
% 16.08/7.03 % (24926)Instructions burned: 7 (million)
% 16.08/7.03 % (24926)------------------------------
% 16.08/7.03 % (24926)------------------------------
% 16.08/7.03 % (24925)Instruction limit reached!
% 16.08/7.03 % (24925)------------------------------
% 16.08/7.03 % (24925)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.03 % (24925)Termination reason: Unknown
% 16.08/7.03 % (24925)Termination phase: shuffling
% 16.08/7.03
% 16.08/7.03 % (24925)Memory used [KB]: 12665
% 16.08/7.03 % (24925)Time elapsed: 0.019 s
% 16.08/7.03 % (24925)Instructions burned: 42 (million)
% 16.08/7.03 % (24925)------------------------------
% 16.08/7.03 % (24925)------------------------------
% 16.08/7.03 % (24910)Instruction limit reached!
% 16.08/7.03 % (24910)------------------------------
% 16.08/7.03 % (24910)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.03 % (24910)Termination reason: Unknown
% 16.08/7.03 % (24910)Termination phase: shuffling
% 16.08/7.03
% 16.08/7.03 % (24910)Memory used [KB]: 19573
% 16.08/7.03 % (24910)Time elapsed: 0.181 s
% 16.08/7.03 % (24910)Instructions burned: 508 (million)
% 16.08/7.03 % (24910)------------------------------
% 16.08/7.03 % (24910)------------------------------
% 16.08/7.04 % (24928)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/343Mi)
% 16.08/7.04 % (24929)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=109:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/109Mi)
% 16.08/7.04 % (24930)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=253:si=on:rtra=on_0 on DTF2THF_24760 for (2990ds/253Mi)
% 16.08/7.05 % (24917)Instruction limit reached!
% 16.08/7.05 % (24917)------------------------------
% 16.08/7.05 % (24917)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.05 % (24917)Termination reason: Unknown
% 16.08/7.05 % (24917)Termination phase: shuffling
% 16.08/7.05
% 16.08/7.05 % (24917)Memory used [KB]: 20212
% 16.08/7.05 % (24917)Time elapsed: 0.135 s
% 16.08/7.05 % (24917)Instructions burned: 433 (million)
% 16.08/7.05 % (24917)------------------------------
% 16.08/7.05 % (24917)------------------------------
% 16.08/7.06 % (24931)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_24760 for (2989ds/140Mi)
% 16.08/7.08 % (24927)Instruction limit reached!
% 16.08/7.08 % (24927)------------------------------
% 16.08/7.08 % (24927)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.08 % (24927)Termination reason: Unknown
% 16.08/7.08 % (24927)Termination phase: shuffling
% 16.08/7.08
% 16.08/7.08 % (24927)Memory used [KB]: 14456
% 16.08/7.08 % (24927)Time elapsed: 0.058 s
% 16.08/7.08 % (24927)Instructions burned: 151 (million)
% 16.08/7.08 % (24927)------------------------------
% 16.08/7.08 % (24927)------------------------------
% 16.08/7.08 % (24929)Instruction limit reached!
% 16.08/7.08 % (24929)------------------------------
% 16.08/7.08 % (24929)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.08 % (24929)Termination reason: Unknown
% 16.08/7.08 % (24929)Termination phase: shuffling
% 16.08/7.08
% 16.08/7.08 % (24929)Memory used [KB]: 13816
% 16.08/7.08 % (24929)Time elapsed: 0.043 s
% 16.08/7.08 % (24929)Instructions burned: 109 (million)
% 16.08/7.08 % (24929)------------------------------
% 16.08/7.08 % (24929)------------------------------
% 16.08/7.09 % (24932)dis+1010_16:1_acc=model:au=on:avsq=on:cond=fast:plsq=on:plsqc=2:plsqr=9267,262144:i=614:si=on:rtra=on_0 on DTF2THF_24760 for (2989ds/614Mi)
% 16.08/7.09 % (24933)lrs+1002_1:1_e2e=on:fde=unused:fe=axiom:prag=on:sd=1:sos=on:ss=axioms:st=2.0:i=202:si=on:rtra=on_0 on DTF2THF_24760 for (2989ds/202Mi)
% 16.08/7.10 % (24921)Instruction limit reached!
% 16.08/7.10 % (24921)------------------------------
% 16.08/7.10 % (24921)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.08/7.10 % (24921)Termination reason: Unknown
% 16.08/7.10 % (24921)Termination phase: shuffling
% 16.08/7.10
% 16.08/7.10 % (24921)Memory used [KB]: 20468
% 16.08/7.10 % (24921)Time elapsed: 0.142 s
% 16.08/7.10 % (24921)Instructions burned: 454 (million)
% 16.08/7.10 % (24921)------------------------------
% 16.08/7.10 % (24921)------------------------------
% 16.08/7.11 % (24934)dis+1002_1:1_au=on:cs=on:ins=3:kws=inv_frequency:nwc=10.0:ss=axioms:st=3.0:i=339:si=on:rtra=on_0 on DTF2THF_24760 for (2989ds/339Mi)
% 16.53/7.12 % (24931)Instruction limit reached!
% 16.53/7.12 % (24931)------------------------------
% 16.53/7.12 % (24931)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.53/7.12 % (24931)Termination reason: Unknown
% 16.53/7.12 % (24931)Termination phase: shuffling
% 16.53/7.12
% 16.53/7.12 % (24931)Memory used [KB]: 14328
% 16.53/7.12 % (24931)Time elapsed: 0.055 s
% 16.53/7.12 % (24931)Instructions burned: 142 (million)
% 16.53/7.12 % (24931)------------------------------
% 16.53/7.12 % (24931)------------------------------
% 16.53/7.13 % (24935)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=283:si=on:rtra=on_0 on DTF2THF_24760 for (2989ds/283Mi)
% 16.53/7.13 % (24930)Instruction limit reached!
% 16.53/7.13 % (24930)------------------------------
% 16.53/7.13 % (24930)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.53/7.13 % (24930)Termination reason: Unknown
% 16.53/7.13 % (24930)Termination phase: shuffling
% 16.53/7.13
% 16.53/7.13 % (24930)Memory used [KB]: 16119
% 16.53/7.13 % (24930)Time elapsed: 0.094 s
% 16.53/7.13 % (24930)Instructions burned: 255 (million)
% 16.53/7.13 % (24930)------------------------------
% 16.53/7.13 % (24930)------------------------------
% 16.53/7.15 % (24936)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_24760 for (2989ds/134Mi)
% 16.53/7.17 % (24933)Instruction limit reached!
% 16.53/7.17 % (24933)------------------------------
% 16.53/7.17 % (24933)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.53/7.17 % (24933)Termination reason: Unknown
% 16.53/7.17 % (24933)Termination phase: shuffling
% 16.53/7.17
% 16.53/7.17 % (24933)Memory used [KB]: 15223
% 16.53/7.17 % (24933)Time elapsed: 0.075 s
% 16.53/7.17 % (24933)Instructions burned: 202 (million)
% 16.53/7.17 % (24933)------------------------------
% 16.53/7.17 % (24933)------------------------------
% 16.53/7.17 % (24928)Instruction limit reached!
% 16.53/7.17 % (24928)------------------------------
% 16.53/7.17 % (24928)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.53/7.17 % (24928)Termination reason: Unknown
% 16.53/7.17 % (24928)Termination phase: shuffling
% 16.53/7.17
% 16.53/7.17 % (24928)Memory used [KB]: 17142
% 16.53/7.17 % (24928)Time elapsed: 0.134 s
% 16.53/7.17 % (24928)Instructions burned: 344 (million)
% 16.53/7.17 % (24928)------------------------------
% 16.53/7.17 % (24928)------------------------------
% 16.53/7.18 % (24937)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_24760 for (2988ds/298Mi)
% 16.53/7.18 % (24938)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_24760 for (2988ds/192Mi)
% 17.02/7.20 % (24936)Instruction limit reached!
% 17.02/7.20 % (24936)------------------------------
% 17.02/7.20 % (24936)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.20 % (24936)Termination reason: Unknown
% 17.02/7.20 % (24936)Termination phase: shuffling
% 17.02/7.20
% 17.02/7.20 % (24936)Memory used [KB]: 14200
% 17.02/7.20 % (24936)Time elapsed: 0.053 s
% 17.02/7.20 % (24936)Instructions burned: 135 (million)
% 17.02/7.20 % (24936)------------------------------
% 17.02/7.20 % (24936)------------------------------
% 17.02/7.21 % (24939)lrs+10_1:1_bd=off:hud=1:nm=0:prag=on:sd=2:sgt=32:sos=all:ss=axioms:i=258:si=on:rtra=on_0 on DTF2THF_24760 for (2988ds/258Mi)
% 17.02/7.22 % (24935)Instruction limit reached!
% 17.02/7.22 % (24935)------------------------------
% 17.02/7.22 % (24935)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.22 % (24935)Termination reason: Unknown
% 17.02/7.22 % (24935)Termination phase: shuffling
% 17.02/7.22
% 17.02/7.22 % (24935)Memory used [KB]: 18166
% 17.02/7.22 % (24935)Time elapsed: 0.091 s
% 17.02/7.22 % (24935)Instructions burned: 284 (million)
% 17.02/7.22 % (24935)------------------------------
% 17.02/7.22 % (24935)------------------------------
% 17.02/7.23 % (24940)lrs+1002_1:1024_au=on:cond=fast:ntd=on:piset=and:plsq=on:plsqc=1:plsqr=183491,1048576:sac=on:tgt=ground:i=90:si=on:rtra=on_0 on DTF2THF_24760 for (2988ds/90Mi)
% 17.02/7.24 % (24934)Instruction limit reached!
% 17.02/7.24 % (24934)------------------------------
% 17.02/7.24 % (24934)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.24 % (24934)Termination reason: Unknown
% 17.02/7.24 % (24934)Termination phase: shuffling
% 17.02/7.24
% 17.02/7.24 % (24934)Memory used [KB]: 17270
% 17.02/7.24 % (24934)Time elapsed: 0.127 s
% 17.02/7.24 % (24934)Instructions burned: 339 (million)
% 17.02/7.24 % (24934)------------------------------
% 17.02/7.24 % (24934)------------------------------
% 17.02/7.25 % (24941)dis+1004_16:1_avsq=on:avsqr=1,16:cnfonf=lazy_gen:cs=on:fe=off:tgt=ground:tnu=1:i=231:si=on:rtra=on_0 on DTF2THF_24760 for (2988ds/231Mi)
% 17.02/7.25 % (24938)Instruction limit reached!
% 17.02/7.25 % (24938)------------------------------
% 17.02/7.25 % (24938)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.25 % (24938)Termination reason: Unknown
% 17.02/7.25 % (24938)Termination phase: shuffling
% 17.02/7.25
% 17.02/7.25 % (24938)Memory used [KB]: 15095
% 17.02/7.25 % (24938)Time elapsed: 0.072 s
% 17.02/7.25 % (24938)Instructions burned: 192 (million)
% 17.02/7.25 % (24938)------------------------------
% 17.02/7.25 % (24938)------------------------------
% 17.02/7.26 % (24940)Instruction limit reached!
% 17.02/7.26 % (24940)------------------------------
% 17.02/7.26 % (24940)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.26 % (24940)Termination reason: Unknown
% 17.02/7.26 % (24940)Termination phase: shuffling
% 17.02/7.26
% 17.02/7.26 % (24940)Memory used [KB]: 13560
% 17.02/7.26 % (24940)Time elapsed: 0.037 s
% 17.02/7.26 % (24940)Instructions burned: 90 (million)
% 17.02/7.26 % (24940)------------------------------
% 17.02/7.26 % (24940)------------------------------
% 17.02/7.26 % (24942)lrs+1002_1:1_au=on:s2a=on:sgt=32:slsq=on:slsqc=1:slsql=off:ss=axioms:i=458:si=on:rtra=on_0 on DTF2THF_24760 for (2988ds/458Mi)
% 17.02/7.27 % (24937)Instruction limit reached!
% 17.02/7.27 % (24937)------------------------------
% 17.02/7.27 % (24937)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.27 % (24937)Termination reason: Unknown
% 17.02/7.27 % (24937)Termination phase: shuffling
% 17.02/7.27
% 17.02/7.27 % (24937)Memory used [KB]: 18421
% 17.02/7.27 % (24937)Time elapsed: 0.095 s
% 17.02/7.27 % (24937)Instructions burned: 298 (million)
% 17.02/7.27 % (24937)------------------------------
% 17.02/7.27 % (24937)------------------------------
% 17.02/7.27 % (24943)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/31Mi)
% 17.02/7.28 % (24944)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:s2a=on:sp=weighted_frequency:ss=axioms:i=297:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/297Mi)
% 17.02/7.29 % (24943)Instruction limit reached!
% 17.02/7.29 % (24943)------------------------------
% 17.02/7.29 % (24943)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.02/7.29 % (24943)Termination reason: Unknown
% 17.02/7.29 % (24943)Termination phase: shuffling
% 17.02/7.29
% 17.02/7.29 % (24943)Memory used [KB]: 12537
% 17.02/7.29 % (24943)Time elapsed: 0.017 s
% 17.02/7.29 % (24943)Instructions burned: 32 (million)
% 17.02/7.29 % (24943)------------------------------
% 17.02/7.29 % (24943)------------------------------
% 17.68/7.30 % (24945)dis+2_1:4_amm=off:au=on:cbe=off:cnfonf=lazy_gen:hfaw=0:hfsq=on:hfsql=off:plsq=on:plsqc=1:plsqr=35,4:sac=on:i=987:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/987Mi)
% 17.68/7.30 % (24939)Instruction limit reached!
% 17.68/7.30 % (24939)------------------------------
% 17.68/7.30 % (24939)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.30 % (24939)Termination reason: Unknown
% 17.68/7.30 % (24939)Termination phase: shuffling
% 17.68/7.30
% 17.68/7.30 % (24939)Memory used [KB]: 15991
% 17.68/7.30 % (24939)Time elapsed: 0.095 s
% 17.68/7.30 % (24939)Instructions burned: 259 (million)
% 17.68/7.30 % (24939)------------------------------
% 17.68/7.30 % (24939)------------------------------
% 17.68/7.30 % (24932)Instruction limit reached!
% 17.68/7.30 % (24932)------------------------------
% 17.68/7.30 % (24932)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.30 % (24932)Termination reason: Unknown
% 17.68/7.30 % (24932)Termination phase: shuffling
% 17.68/7.30
% 17.68/7.30 % (24932)Memory used [KB]: 21108
% 17.68/7.30 % (24932)Time elapsed: 0.217 s
% 17.68/7.30 % (24932)Instructions burned: 615 (million)
% 17.68/7.30 % (24932)------------------------------
% 17.68/7.30 % (24932)------------------------------
% 17.68/7.31 % (24946)lrs+10_1:1024_avsq=on:avsqc=4:avsqr=584411,1048576:bd=off:bet=on:br=off:chr=on:fsd=on:fsdmm=1:hud=10:nm=0:ntd=on:nwc=9.0:piset=or:plsq=on:plsqc=2:plsql=on:plsqr=852219,1048576:rawr=on:sp=const_frequency:spb=units:i=182:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/182Mi)
% 17.68/7.31 % (24947)dis+10_1:1_au=on:e2e=on:fde=none:nwc=10.0:s2a=on:s2at=2.0:i=182:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/182Mi)
% 17.68/7.32 % (24914)Instruction limit reached!
% 17.68/7.32 % (24914)------------------------------
% 17.68/7.32 % (24914)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.32 % (24914)Termination reason: Unknown
% 17.68/7.32 % (24914)Termination phase: Property scanning
% 17.68/7.32
% 17.68/7.32 % (24914)Memory used [KB]: 27249
% 17.68/7.32 % (24914)Time elapsed: 0.423 s
% 17.68/7.32 % (24914)Instructions burned: 1088 (million)
% 17.68/7.32 % (24914)------------------------------
% 17.68/7.32 % (24914)------------------------------
% 17.68/7.32 % (24941)Instruction limit reached!
% 17.68/7.32 % (24941)------------------------------
% 17.68/7.32 % (24941)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.32 % (24941)Termination reason: Unknown
% 17.68/7.32 % (24941)Termination phase: shuffling
% 17.68/7.32
% 17.68/7.32 % (24941)Memory used [KB]: 15607
% 17.68/7.32 % (24941)Time elapsed: 0.074 s
% 17.68/7.32 % (24941)Instructions burned: 232 (million)
% 17.68/7.32 % (24941)------------------------------
% 17.68/7.32 % (24941)------------------------------
% 17.68/7.33 % (24948)dis+10_1:8_cha=on:cnfonf=lazy_pi_sigma_gen:fe=axiom:hud=5:plsq=on:plsqr=32,1:prag=on:sos=all:i=154:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/154Mi)
% 17.68/7.33 % (24949)dis+1003_1:64_atotf=0.4:au=on:fsr=off:hflw=5:hfsq=on:hfsqc=20:hfsql=off:plsq=on:plsqc=1:plsqr=32,1:i=79:si=on:rtra=on_0 on DTF2THF_24760 for (2987ds/79Mi)
% 17.68/7.36 % (24949)Instruction limit reached!
% 17.68/7.36 % (24949)------------------------------
% 17.68/7.36 % (24949)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.36 % (24949)Termination reason: Unknown
% 17.68/7.36 % (24949)Termination phase: shuffling
% 17.68/7.36
% 17.68/7.36 % (24949)Memory used [KB]: 13304
% 17.68/7.36 % (24949)Time elapsed: 0.032 s
% 17.68/7.36 % (24949)Instructions burned: 79 (million)
% 17.68/7.36 % (24949)------------------------------
% 17.68/7.36 % (24949)------------------------------
% 17.68/7.38 % (24950)dis+1010_5:2_anc=all_dependent:awrs=converge:bd=off:bet=on:c=on:cha=on:cnfonf=lazy_not_gen_be_off:fe=axiom:hud=5:kws=precedence:piset=and:prag=on:s2a=on:sac=on:slsq=on:slsqc=1:sp=reverse_arity:ss=axioms:st=1.5:i=216:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/216Mi)
% 17.68/7.38 % (24944)Instruction limit reached!
% 17.68/7.38 % (24944)------------------------------
% 17.68/7.38 % (24944)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.38 % (24944)Termination reason: Unknown
% 17.68/7.38 % (24944)Termination phase: shuffling
% 17.68/7.38
% 17.68/7.38 % (24944)Memory used [KB]: 18421
% 17.68/7.38 % (24944)Time elapsed: 0.094 s
% 17.68/7.38 % (24944)Instructions burned: 300 (million)
% 17.68/7.38 % (24944)------------------------------
% 17.68/7.38 % (24944)------------------------------
% 17.68/7.38 % (24946)Instruction limit reached!
% 17.68/7.38 % (24946)------------------------------
% 17.68/7.38 % (24946)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.38 % (24946)Termination reason: Unknown
% 17.68/7.38 % (24946)Termination phase: shuffling
% 17.68/7.38
% 17.68/7.38 % (24946)Memory used [KB]: 14967
% 17.68/7.38 % (24946)Time elapsed: 0.069 s
% 17.68/7.38 % (24946)Instructions burned: 184 (million)
% 17.68/7.38 % (24946)------------------------------
% 17.68/7.38 % (24946)------------------------------
% 17.68/7.38 % (24947)Instruction limit reached!
% 17.68/7.38 % (24947)------------------------------
% 17.68/7.38 % (24947)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.38 % (24947)Termination reason: Unknown
% 17.68/7.38 % (24947)Termination phase: shuffling
% 17.68/7.38
% 17.68/7.38 % (24947)Memory used [KB]: 14967
% 17.68/7.38 % (24947)Time elapsed: 0.068 s
% 17.68/7.38 % (24947)Instructions burned: 184 (million)
% 17.68/7.38 % (24947)------------------------------
% 17.68/7.38 % (24947)------------------------------
% 17.68/7.38 % (24948)Instruction limit reached!
% 17.68/7.38 % (24948)------------------------------
% 17.68/7.38 % (24948)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.68/7.38 % (24948)Termination reason: Unknown
% 17.68/7.38 % (24948)Termination phase: shuffling
% 17.68/7.38
% 17.68/7.38 % (24948)Memory used [KB]: 14456
% 17.68/7.38 % (24948)Time elapsed: 0.051 s
% 17.68/7.38 % (24948)Instructions burned: 155 (million)
% 17.68/7.38 % (24948)------------------------------
% 17.68/7.38 % (24948)------------------------------
% 17.68/7.39 % (24951)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/114Mi)
% 17.68/7.39 % (24953)lrs+10_1:128_au=on:bd=off:cnfonf=lazy_not_gen_be_off:nwc=10.0:piset=and:i=182:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/182Mi)
% 17.68/7.39 % (24952)dis+1002_1:1_acc=on:au=on:bd=off:e2e=on:sgt=8:ss=axioms:i=922:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/922Mi)
% 17.68/7.39 % (24954)ott+1010_1:1024_au=on:av=off:c=on:cnfonf=conj_eager:piset=equals:s2a=on:sp=occurrence:spb=units:ss=axioms:i=164:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/164Mi)
% 18.50/7.43 % (24942)Instruction limit reached!
% 18.50/7.43 % (24942)------------------------------
% 18.50/7.43 % (24942)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.43 % (24942)Termination reason: Unknown
% 18.50/7.43 % (24942)Termination phase: shuffling
% 18.50/7.43
% 18.50/7.43 % (24942)Memory used [KB]: 18805
% 18.50/7.43 % (24942)Time elapsed: 0.164 s
% 18.50/7.43 % (24942)Instructions burned: 459 (million)
% 18.50/7.43 % (24942)------------------------------
% 18.50/7.43 % (24942)------------------------------
% 18.50/7.43 % (24951)Instruction limit reached!
% 18.50/7.43 % (24951)------------------------------
% 18.50/7.43 % (24951)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.43 % (24951)Termination reason: Unknown
% 18.50/7.43 % (24951)Termination phase: shuffling
% 18.50/7.43
% 18.50/7.43 % (24951)Memory used [KB]: 13944
% 18.50/7.43 % (24951)Time elapsed: 0.045 s
% 18.50/7.43 % (24951)Instructions burned: 114 (million)
% 18.50/7.43 % (24951)------------------------------
% 18.50/7.43 % (24951)------------------------------
% 18.50/7.44 % (24955)lrs+1004_1:1_cnfonf=lazy_pi_sigma_gen:fe=off:gs=on:hud=10:prag=on:tnu=1:i=50:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/50Mi)
% 18.50/7.44 % (24956)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/368Mi)
% 18.50/7.44 % (24950)Instruction limit reached!
% 18.50/7.44 % (24950)------------------------------
% 18.50/7.44 % (24950)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.44 % (24950)Termination reason: Unknown
% 18.50/7.44 % (24950)Termination phase: shuffling
% 18.50/7.44
% 18.50/7.44 % (24950)Memory used [KB]: 15351
% 18.50/7.44 % (24950)Time elapsed: 0.070 s
% 18.50/7.44 % (24950)Instructions burned: 218 (million)
% 18.50/7.44 % (24950)------------------------------
% 18.50/7.44 % (24950)------------------------------
% 18.50/7.44 % (24954)Instruction limit reached!
% 18.50/7.44 % (24954)------------------------------
% 18.50/7.44 % (24954)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.44 % (24954)Termination reason: Unknown
% 18.50/7.44 % (24954)Termination phase: shuffling
% 18.50/7.44
% 18.50/7.44 % (24954)Memory used [KB]: 14583
% 18.50/7.44 % (24954)Time elapsed: 0.053 s
% 18.50/7.44 % (24954)Instructions burned: 164 (million)
% 18.50/7.44 % (24954)------------------------------
% 18.50/7.44 % (24954)------------------------------
% 18.50/7.45 % (24953)Instruction limit reached!
% 18.50/7.45 % (24953)------------------------------
% 18.50/7.45 % (24953)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.45 % (24953)Termination reason: Unknown
% 18.50/7.45 % (24953)Termination phase: shuffling
% 18.50/7.45
% 18.50/7.45 % (24953)Memory used [KB]: 14839
% 18.50/7.45 % (24953)Time elapsed: 0.058 s
% 18.50/7.45 % (24953)Instructions burned: 182 (million)
% 18.50/7.45 % (24953)------------------------------
% 18.50/7.45 % (24953)------------------------------
% 18.50/7.45 % (24957)dis+2_1:1_bd=off:e2e=on:hud=1:nm=3:piset=or:prag=on:s2a=on:sp=const_max:i=1021:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/1021Mi)
% 18.50/7.46 % (24958)dis+10_1:1_cnfonf=lazy_gen:fe=off:plsq=on:plsqc=1:plsqr=32,1:prag=on:i=267:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/267Mi)
% 18.50/7.46 % (24955)Instruction limit reached!
% 18.50/7.46 % (24955)------------------------------
% 18.50/7.46 % (24955)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.46 % (24955)Termination reason: Unknown
% 18.50/7.46 % (24955)Termination phase: shuffling
% 18.50/7.46
% 18.50/7.46 % (24955)Memory used [KB]: 12920
% 18.50/7.46 % (24955)Time elapsed: 0.022 s
% 18.50/7.46 % (24955)Instructions burned: 53 (million)
% 18.50/7.46 % (24955)------------------------------
% 18.50/7.46 % (24955)------------------------------
% 18.50/7.46 % (24959)dis+21_1:5_au=on:ntd=on:nwc=5.0:s2a=on:ss=axioms:st=5.0:i=204:si=on:rtra=on_0 on DTF2THF_24760 for (2986ds/204Mi)
% 18.50/7.47 % (24960)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_24760 for (2985ds/26Mi)
% 18.50/7.48 % (24960)Instruction limit reached!
% 18.50/7.48 % (24960)------------------------------
% 18.50/7.48 % (24960)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.50/7.48 % (24960)Termination reason: Unknown
% 18.50/7.48 % (24960)Termination phase: shuffling
% 18.50/7.48
% 18.50/7.48 % (24960)Memory used [KB]: 12409
% 18.50/7.48 % (24960)Time elapsed: 0.014 s
% 18.50/7.48 % (24960)Instructions burned: 26 (million)
% 18.50/7.48 % (24960)------------------------------
% 18.50/7.48 % (24960)------------------------------
% 18.50/7.49 % (24961)dis+1002_1:1_bd=preordered:c=on:cbe=off:cnfonf=lazy_not_gen_be_off:hud=10:ixr=off:prag=on:i=53:si=on:rtra=on_0 on DTF2THF_24760 for (2985ds/53Mi)
% 18.99/7.51 % (24961)Instruction limit reached!
% 18.99/7.51 % (24961)------------------------------
% 18.99/7.51 % (24961)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.99/7.51 % (24961)Termination reason: Unknown
% 18.99/7.51 % (24961)Termination phase: shuffling
% 18.99/7.51
% 18.99/7.51 % (24961)Memory used [KB]: 12920
% 18.99/7.51 % (24961)Time elapsed: 0.022 s
% 18.99/7.51 % (24961)Instructions burned: 53 (million)
% 18.99/7.51 % (24961)------------------------------
% 18.99/7.51 % (24961)------------------------------
% 18.99/7.53 % (24962)dis+1002_1:1_aac=none:au=on:e2e=on:nm=2:nwc=10.0:sd=1:ss=axioms:st=3.0:i=4436:si=on:rtra=on_0 on DTF2THF_24760 for (2985ds/4436Mi)
% 18.99/7.53 % (24959)Instruction limit reached!
% 18.99/7.53 % (24959)------------------------------
% 18.99/7.53 % (24959)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.99/7.53 % (24959)Termination reason: Unknown
% 18.99/7.53 % (24959)Termination phase: shuffling
% 18.99/7.53
% 18.99/7.53 % (24959)Memory used [KB]: 15223
% 18.99/7.53 % (24959)Time elapsed: 0.075 s
% 18.99/7.53 % (24959)Instructions burned: 204 (million)
% 18.99/7.53 % (24959)------------------------------
% 18.99/7.53 % (24959)------------------------------
% 18.99/7.54 % (24958)Instruction limit reached!
% 18.99/7.54 % (24958)------------------------------
% 18.99/7.54 % (24958)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.99/7.54 % (24958)Termination reason: Unknown
% 18.99/7.54 % (24958)Termination phase: shuffling
% 18.99/7.54
% 18.99/7.54 % (24958)Memory used [KB]: 17910
% 18.99/7.54 % (24958)Time elapsed: 0.084 s
% 18.99/7.54 % (24958)Instructions burned: 268 (million)
% 18.99/7.54 % (24958)------------------------------
% 18.99/7.54 % (24958)------------------------------
% 18.99/7.55 % (24963)dis+1010_1:1_acc=model:avsq=on:ixr=off:prag=on:s2a=on:sac=on:i=1539:si=on:rtra=on_0 on DTF2THF_24760 for (2985ds/1539Mi)
% 18.99/7.55 % (24964)lrs+2_1:1_bd=off:e2e=on:fde=none:kws=inv_precedence:plsq=on:plsqr=32,1:sos=on:i=408:si=on:rtra=on_0 on DTF2THF_24760 for (2985ds/408Mi)
% 18.99/7.57 % (24956)Instruction limit reached!
% 18.99/7.57 % (24956)------------------------------
% 18.99/7.57 % (24956)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.99/7.57 % (24956)Termination reason: Unknown
% 18.99/7.57 % (24956)Termination phase: shuffling
% 18.99/7.57
% 18.99/7.57 % (24956)Memory used [KB]: 17654
% 18.99/7.57 % (24956)Time elapsed: 0.131 s
% 18.99/7.57 % (24956)Instructions burned: 369 (million)
% 18.99/7.57 % (24956)------------------------------
% 18.99/7.57 % (24956)------------------------------
% 18.99/7.58 % (24965)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=5205:si=on:rtra=on_0 on DTF2THF_24760 for (2984ds/5205Mi)
% 19.33/7.60 % (24945)Instruction limit reached!
% 19.33/7.60 % (24945)------------------------------
% 19.33/7.60 % (24945)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.33/7.60 % (24945)Termination reason: Unknown
% 19.33/7.60 % (24945)Termination phase: shuffling
% 19.33/7.60
% 19.33/7.60 % (24945)Memory used [KB]: 31854
% 19.33/7.60 % (24945)Time elapsed: 0.300 s
% 19.33/7.60 % (24945)Instructions burned: 988 (million)
% 19.33/7.60 % (24945)------------------------------
% 19.33/7.60 % (24945)------------------------------
% 19.33/7.61 % (24966)dis+1010_1:4_au=on:av=off:cbe=off:nwc=10.0:s2a=on:s2at=3.0:sd=2:ss=axioms:st=5.0:i=1913:si=on:rtra=on_0 on DTF2THF_24760 for (2984ds/1913Mi)
% 19.33/7.69 % (24964)Instruction limit reached!
% 19.33/7.69 % (24964)------------------------------
% 19.33/7.69 % (24964)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.33/7.69 % (24964)Termination reason: Unknown
% 19.33/7.69 % (24964)Termination phase: shuffling
% 19.33/7.69
% 19.33/7.69 % (24964)Memory used [KB]: 18166
% 19.33/7.69 % (24964)Time elapsed: 0.145 s
% 19.33/7.69 % (24964)Instructions burned: 408 (million)
% 19.33/7.69 % (24964)------------------------------
% 19.33/7.69 % (24964)------------------------------
% 19.81/7.70 % (24967)lrs+2_1:1_bd=off:cbe=off:piset=equals:sd=1:sgt=20:sos=all:ss=axioms:i=4350:si=on:rtra=on_0 on DTF2THF_24760 for (2983ds/4350Mi)
% 19.81/7.72 % (24952)Instruction limit reached!
% 19.81/7.72 % (24952)------------------------------
% 19.81/7.72 % (24952)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.81/7.72 % (24952)Termination reason: Unknown
% 19.81/7.72 % (24952)Termination phase: Property scanning
% 19.81/7.72
% 19.81/7.72 % (24952)Memory used [KB]: 27376
% 19.81/7.72 % (24952)Time elapsed: 0.326 s
% 19.81/7.72 % (24952)Instructions burned: 925 (million)
% 19.81/7.72 % (24952)------------------------------
% 19.81/7.72 % (24952)------------------------------
% 19.81/7.73 % (24968)ott+1010_1:4_cbe=off:chr=on:hfsq=on:hfsqr=1,16:hud=3:sac=on:sp=unary_frequency:ss=axioms:st=5.0:tnu=2:i=68:si=on:rtra=on_0 on DTF2THF_24760 for (2983ds/68Mi)
% 19.81/7.76 % (24968)Instruction limit reached!
% 19.81/7.76 % (24968)------------------------------
% 19.81/7.76 % (24968)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.81/7.76 % (24968)Termination reason: Unknown
% 19.81/7.76 % (24968)Termination phase: shuffling
% 19.81/7.76
% 19.81/7.76 % (24968)Memory used [KB]: 13176
% 19.81/7.76 % (24968)Time elapsed: 0.030 s
% 19.81/7.76 % (24968)Instructions burned: 69 (million)
% 19.81/7.76 % (24968)------------------------------
% 19.81/7.76 % (24968)------------------------------
% 19.81/7.77 % (24969)dis+10_1:1_au=on:nwc=5.0:s2a=on:s2at=2.0:sac=on:sd=7:ss=axioms:st=2.0:i=358:si=on:rtra=on_0 on DTF2THF_24760 for (2982ds/358Mi)
% 20.19/7.81 % (24957)Instruction limit reached!
% 20.19/7.81 % (24957)------------------------------
% 20.19/7.81 % (24957)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.19/7.81 % (24957)Termination reason: Unknown
% 20.19/7.81 % (24957)Termination phase: Property scanning
% 20.19/7.81
% 20.19/7.81 % (24957)Memory used [KB]: 27249
% 20.19/7.81 % (24957)Time elapsed: 0.352 s
% 20.19/7.81 % (24957)Instructions burned: 1021 (million)
% 20.19/7.81 % (24957)------------------------------
% 20.19/7.81 % (24957)------------------------------
% 20.19/7.82 % (24970)lrs+1010_1:16_add=large:amm=off:anc=all_dependent:chr=on:fd=off:ins=2:sp=const_max:tnu=5:uhcvi=on:i=31:si=on:rtra=on_0 on DTF2THF_24760 for (2982ds/31Mi)
% 20.19/7.83 % (24970)Instruction limit reached!
% 20.19/7.83 % (24970)------------------------------
% 20.19/7.83 % (24970)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.19/7.83 % (24970)Termination reason: Unknown
% 20.19/7.83 % (24970)Termination phase: shuffling
% 20.19/7.83
% 20.19/7.83 % (24970)Memory used [KB]: 12537
% 20.19/7.83 % (24970)Time elapsed: 0.018 s
% 20.19/7.83 % (24970)Instructions burned: 33 (million)
% 20.19/7.83 % (24970)------------------------------
% 20.19/7.83 % (24970)------------------------------
% 20.19/7.85 % (24971)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_24760 for (2982ds/279Mi)
% 20.58/7.90 % (24969)Instruction limit reached!
% 20.58/7.90 % (24969)------------------------------
% 20.58/7.90 % (24969)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/7.90 % (24969)Termination reason: Unknown
% 20.58/7.90 % (24969)Termination phase: shuffling
% 20.58/7.90
% 20.58/7.90 % (24969)Memory used [KB]: 17398
% 20.58/7.90 % (24969)Time elapsed: 0.132 s
% 20.58/7.90 % (24969)Instructions burned: 358 (million)
% 20.58/7.90 % (24969)------------------------------
% 20.58/7.90 % (24969)------------------------------
% 20.58/7.91 % (24972)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_24760 for (2981ds/258Mi)
% 20.58/7.93 % (24971)Instruction limit reached!
% 20.58/7.93 % (24971)------------------------------
% 20.58/7.93 % (24971)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/7.93 % (24971)Termination reason: Unknown
% 20.58/7.93 % (24971)Termination phase: shuffling
% 20.58/7.93
% 20.58/7.93 % (24971)Memory used [KB]: 18038
% 20.58/7.93 % (24971)Time elapsed: 0.090 s
% 20.58/7.93 % (24971)Instructions burned: 280 (million)
% 20.58/7.93 % (24971)------------------------------
% 20.58/7.93 % (24971)------------------------------
% 20.58/7.95 % (24982)dis+2_1:1024_atotf=0.0802881:au=on:bs=unit_only:cnfonf=lazy_not_gen_be_off:fde=none:nwc=10.0:prag=on:i=26:si=on:rtra=on_0 on DTF2THF_24760 for (2981ds/26Mi)
% 20.58/7.96 % (24982)Instruction limit reached!
% 20.58/7.96 % (24982)------------------------------
% 20.58/7.96 % (24982)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/7.96 % (24982)Termination reason: Unknown
% 20.58/7.96 % (24982)Termination phase: shuffling
% 20.58/7.96
% 20.58/7.96 % (24982)Memory used [KB]: 12409
% 20.58/7.96 % (24982)Time elapsed: 0.014 s
% 20.58/7.96 % (24982)Instructions burned: 26 (million)
% 20.58/7.96 % (24982)------------------------------
% 20.58/7.96 % (24982)------------------------------
% 20.58/7.97 % (24993)lrs+1010_1:1_au=on:cbe=off:cnfonf=lazy_not_be_gen:ntd=on:pe=on:i=995:si=on:rtra=on_0 on DTF2THF_24760 for (2980ds/995Mi)
% 20.58/8.01 % (24972)Instruction limit reached!
% 20.58/8.01 % (24972)------------------------------
% 20.58/8.01 % (24972)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.58/8.01 % (24972)Termination reason: Unknown
% 20.58/8.01 % (24972)Termination phase: shuffling
% 20.58/8.01
% 20.58/8.01 % (24972)Memory used [KB]: 16119
% 20.58/8.01 % (24972)Time elapsed: 0.096 s
% 20.58/8.01 % (24972)Instructions burned: 258 (million)
% 20.58/8.01 % (24972)------------------------------
% 20.58/8.01 % (24972)------------------------------
% 23.05/8.02 % (25013)dis+10_1:8_amm=sco:cbe=off:hflw=5:hfsq=on:hfsqc=14:hfsql=off:hfsqr=4,1:hud=5:ile=on:lma=on:nm=2:ntd=on:prag=on:sac=on:sp=frequency:i=373:si=on:rtra=on_0 on DTF2THF_24760 for (2980ds/373Mi)
% 23.05/8.07 % (24963)Instruction limit reached!
% 23.05/8.07 % (24963)------------------------------
% 23.05/8.07 % (24963)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.05/8.07 % (24963)Termination reason: Unknown
% 23.05/8.07 % (24963)Termination phase: Preprocessing 1
% 23.05/8.07
% 23.05/8.07 % (24963)Memory used [KB]: 27888
% 23.05/8.07 % (24963)Time elapsed: 0.523 s
% 23.05/8.07 % (24963)Instructions burned: 1540 (million)
% 23.05/8.07 % (24963)------------------------------
% 23.05/8.07 % (24963)------------------------------
% 23.05/8.08 % (25033)lrs+1010_1:128_au=on:cnfonf=lazy_pi_sigma_gen:hflw=4:hfsq=on:hfsqc=20:hfsql=off:i=253:si=on:rtra=on_0 on DTF2THF_24760 for (2979ds/253Mi)
% 23.62/8.15 % (25013)Instruction limit reached!
% 23.62/8.15 % (25013)------------------------------
% 23.62/8.15 % (25013)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.62/8.15 % (25013)Termination reason: Unknown
% 23.62/8.15 % (25013)Termination phase: shuffling
% 23.62/8.15
% 23.62/8.15 % (25013)Memory used [KB]: 17654
% 23.62/8.15 % (25013)Time elapsed: 0.134 s
% 23.62/8.15 % (25013)Instructions burned: 373 (million)
% 23.62/8.15 % (25013)------------------------------
% 23.62/8.15 % (25013)------------------------------
% 23.62/8.16 % (25033)Instruction limit reached!
% 23.62/8.16 % (25033)------------------------------
% 23.62/8.16 % (25033)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.62/8.16 % (25033)Termination reason: Unknown
% 23.62/8.16 % (25033)Termination phase: shuffling
% 23.62/8.16
% 23.62/8.16 % (25033)Memory used [KB]: 17782
% 23.62/8.16 % (25033)Time elapsed: 0.081 s
% 23.62/8.16 % (25033)Instructions burned: 255 (million)
% 23.62/8.16 % (25033)------------------------------
% 23.62/8.16 % (25033)------------------------------
% 23.62/8.16 % (25073)lrs+1002_1:14_au=on:bd=off:e2e=on:fde=unused:sac=on:sos=on:i=832:si=on:rtra=on_0 on DTF2THF_24760 for (2978ds/832Mi)
% 23.62/8.17 % (25079)lrs+1010_1:1_chr=on:cnfonf=lazy_not_be_gen:gs=on:hud=5:plsq=on:plsqc=1:plsqr=5,1:prag=on:rawr=on:i=668:si=on:rtra=on_0 on DTF2THF_24760 for (2978ds/668Mi)
% 24.29/8.26 % (24966)Instruction limit reached!
% 24.29/8.26 % (24966)------------------------------
% 24.29/8.26 % (24966)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.29/8.26 % (24966)Termination reason: Unknown
% 24.29/8.26 % (24966)Termination phase: Preprocessing 3
% 24.29/8.26
% 24.29/8.26 % (24966)Memory used [KB]: 28912
% 24.29/8.26 % (24966)Time elapsed: 0.650 s
% 24.29/8.26 % (24966)Instructions burned: 1914 (million)
% 24.29/8.26 % (24966)------------------------------
% 24.29/8.26 % (24966)------------------------------
% 24.29/8.27 % (25124)dis+10_1:1_anc=none:cnfonf=lazy_gen:fd=preordered:fe=off:hud=10:ins=3:ixr=off:nwc=5.0:plsq=on:plsqc=1:plsqr=32,1:sp=const_frequency:uhcvi=on:i=40:si=on:rtra=on_0 on DTF2THF_24760 for (2977ds/40Mi)
% 24.29/8.28 % (24993)Instruction limit reached!
% 24.29/8.28 % (24993)------------------------------
% 24.29/8.28 % (24993)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.29/8.28 % (24993)Termination reason: Unknown
% 24.29/8.28 % (24993)Termination phase: shuffling
% 24.29/8.28
% 24.29/8.28 % (24993)Memory used [KB]: 31854
% 24.29/8.28 % (24993)Time elapsed: 0.306 s
% 24.29/8.28 % (24993)Instructions burned: 997 (million)
% 24.29/8.28 % (24993)------------------------------
% 24.29/8.28 % (24993)------------------------------
% 24.29/8.29 % (25124)Instruction limit reached!
% 24.29/8.29 % (25124)------------------------------
% 24.29/8.29 % (25124)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.29/8.29 % (25124)Termination reason: Unknown
% 24.29/8.29 % (25124)Termination phase: shuffling
% 24.29/8.29
% 24.29/8.29 % (25124)Memory used [KB]: 12665
% 24.29/8.29 % (25124)Time elapsed: 0.019 s
% 24.29/8.29 % (25124)Instructions burned: 42 (million)
% 24.29/8.29 % (25124)------------------------------
% 24.29/8.29 % (25124)------------------------------
% 24.29/8.29 % (25130)lrs+1010_1:1_cnfonf=lazy_pi_sigma_gen:fd=off:ntd=on:sd=1:sgt=32:sos=on:ss=axioms:i=294:si=on:rtra=on_0 on DTF2THF_24760 for (2977ds/294Mi)
% 24.80/8.30 % (25134)dis+1010_1:1_avsq=on:fe=off:fsr=off:hud=10:plsq=on:plsqr=32,1:prag=on:i=187:si=on:rtra=on_0 on DTF2THF_24760 for (2977ds/187Mi)
% 24.80/8.33 % (24967)First to succeed.
% 24.80/8.34 % (24967)Refutation found. Thanks to Tanya!
% 24.80/8.34 % SZS status Theorem for DTF2THF_24760
% 24.80/8.34 % SZS output start Proof for DTF2THF_24760
% 24.80/8.34 thf(type_def_5, type, int: $tType).
% 24.80/8.34 thf(type_def_7, type, list: ($tType) > $tType).
% 24.80/8.34 thf(type_def_8, type, product_unit: $tType).
% 24.80/8.34 thf(type_def_9, type, heap_ext: ($tType) > $tType).
% 24.80/8.34 thf(type_def_10, type, array: ($tType) > $tType).
% 24.80/8.34 thf(type_def_11, type, product_prod: ($tType * $tType) > $tType).
% 24.80/8.34 thf(type_def_12, type, heap_Time_Heap: ($tType) > $tType).
% 24.80/8.34 thf(type_def_13, type, nat: $tType).
% 24.80/8.34 thf(type_def_14, type, code_integer: $tType).
% 24.80/8.34 thf(type_def_15, type, assn: $tType).
% 24.80/8.34 thf(type_def_16, type, set: ($tType) > $tType).
% 24.80/8.34 thf(type_def_17, type, ref: ($tType) > $tType).
% 24.80/8.34 thf(type_def_18, type, sum_sum: ($tType * $tType) > $tType).
% 24.80/8.34 thf(type_def_19, type, num: $tType).
% 24.80/8.34 thf(type_def_20, type, option: ($tType) > $tType).
% 24.80/8.34 thf(type_def_21, type, itself: ($tType) > $tType).
% 24.80/8.34 thf(type_def_22, type, code_natural: $tType).
% 24.80/8.34 thf(type_def_23, type, old_node: ($tType * $tType) > $tType).
% 24.80/8.34 thf(type_def_24, type, typerep: $tType).
% 24.80/8.34 thf(type_def_25, type, char: $tType).
% 24.80/8.34 thf(type_def_26, type, multiset: ($tType) > $tType).
% 24.80/8.34 thf(type_def_27, type, seq: ($tType) > $tType).
% 24.80/8.34 thf(type_def_28, type, pred: ($tType) > $tType).
% 24.80/8.34 thf(type_def_29, type, rat: $tType).
% 24.80/8.34 thf(type_def_30, type, a: $tType).
% 24.80/8.34 thf(type_def_31, type, code_term: $tType).
% 24.80/8.34 thf(func_def_0, type, heap_Time_Heap: ($tType) > $tType).
% 24.80/8.34 thf(func_def_1, type, code_natural: $tType).
% 24.80/8.34 thf(func_def_2, type, code_integer: $tType).
% 24.80/8.34 thf(func_def_3, type, code_term: $tType).
% 24.80/8.34 thf(func_def_4, type, heap_ext: ($tType) > $tType).
% 24.80/8.34 thf(func_def_5, type, product_unit: $tType).
% 24.80/8.34 thf(func_def_6, type, product_prod: ($tType * $tType) > $tType).
% 24.80/8.34 thf(func_def_7, type, old_node: ($tType * $tType) > $tType).
% 24.80/8.34 thf(func_def_8, type, multiset: ($tType) > $tType).
% 24.80/8.34 thf(func_def_9, type, typerep: $tType).
% 24.80/8.34 thf(func_def_10, type, assn: $tType).
% 24.80/8.34 thf(func_def_11, type, pred: ($tType) > $tType).
% 24.80/8.34 thf(func_def_12, type, sum_sum: ($tType * $tType) > $tType).
% 24.80/8.34 thf(func_def_13, type, seq: ($tType) > $tType).
% 24.80/8.34 thf(func_def_14, type, option: ($tType) > $tType).
% 24.80/8.34 thf(func_def_15, type, char: $tType).
% 24.80/8.34 thf(func_def_16, type, array: ($tType) > $tType).
% 24.80/8.34 thf(func_def_17, type, list: ($tType) > $tType).
% 24.80/8.34 thf(func_def_18, type, ref: ($tType) > $tType).
% 24.80/8.34 thf(func_def_19, type, set: ($tType) > $tType).
% 24.80/8.34 thf(func_def_20, type, rat: $tType).
% 24.80/8.34 thf(func_def_21, type, num: $tType).
% 24.80/8.34 thf(func_def_22, type, nat: $tType).
% 24.80/8.34 thf(func_def_23, type, int: $tType).
% 24.80/8.34 thf(func_def_24, type, itself: ($tType) > $tType).
% 24.80/8.34 thf(func_def_25, type, a: $tType).
% 24.80/8.34 thf(func_def_26, type, bounded_lattice_top: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_27, type, typerep2: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_28, type, enum: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_29, type, code_term_of: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_30, type, type: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_31, type, cl_HOL_Oequal: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_32, type, heap: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_33, type, dvd: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_34, type, one: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_35, type, ring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_36, type, plus: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_37, type, zero: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_38, type, numeral: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_39, type, power: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_40, type, field: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_41, type, ring_gcd: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_42, type, inf: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_43, type, sup: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_44, type, ring_1: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_45, type, uminus: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_46, type, ord: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_47, type, inverse: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_48, type, semiring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_49, type, ring_char_0: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_50, type, neg_numeral: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_51, type, order: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_52, type, comm_ring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_53, type, semiring_Gcd: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_54, type, semiring_gcd: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_55, type, group_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_56, type, no_bot: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_57, type, no_top: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_58, type, semiring_0: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_59, type, semiring_1: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_60, type, finite_finite: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_61, type, monoid_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_62, type, comm_ring_1: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_63, type, monoid_mult: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_64, type, linorder: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_65, type, preorder: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_66, type, ring_parity: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_67, type, idom_abs_sgn: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_68, type, ordered_ring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_69, type, zero_neq_one: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_70, type, countable: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_71, type, field_char_0: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_72, type, ab_group_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_73, type, semiring_char_0: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_74, type, order_bot: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_75, type, order_top: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_76, type, comm_semiring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_77, type, zero_less_one: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_78, type, division_ring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_79, type, semigroup_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_80, type, semiring_numeral: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_81, type, semidom_divide: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_82, type, semidom_modulo: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_83, type, complete_Inf: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_84, type, complete_Sup: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_85, type, semigroup_mult: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_86, type, comm_semiring_0: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_87, type, comm_semiring_1: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_88, type, linordered_idom: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_89, type, linordered_ring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_90, type, semiring_modulo: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_91, type, comm_monoid_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_92, type, semiring_parity: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_93, type, linordered_field: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_94, type, ab_semigroup_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_95, type, comm_monoid_diff: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_96, type, comm_monoid_mult: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_97, type, algebraic_semidom: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_98, type, ab_semigroup_mult: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_99, type, bounded_lattice: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_100, type, semilattice_inf: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_101, type, semilattice_sup: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_102, type, dense_linorder: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_103, type, quickcheck_random: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_104, type, linordered_semidom: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_105, type, comple9053668089753744459l_ccpo: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_106, type, cancel_semigroup_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_107, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_108, type, linord6961819062388156250ring_1: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_109, type, normal8620421768224518004emidom: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_110, type, bit_semiring_bits: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_111, type, comm_s4317794764714335236cancel: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_112, type, linord4710134922213307826strict: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_113, type, cancel1802427076303600483id_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_114, type, ring_15535105094025558882visors: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_115, type, cancel2418104881723323429up_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_116, type, linord5086331880401160121up_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_117, type, ordere6911136660526730532id_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_118, type, archim2362893244070406136eiling: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_119, type, ordere166539214618696060dd_abs: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_120, type, ordere6658533253407199908up_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_121, type, semiri3467727345109120633visors: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_122, type, boolea8198339166811842893lgebra: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_123, type, quickc658316121487927005ustive: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_124, type, semido2269285787275462019factor: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_125, type, linord181362715937106298miring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_126, type, bit_ri3973907225187159222ations: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_127, type, comple6319245703460814977attice: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_128, type, linord4140545234300271783up_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_129, type, linord715952674999750819strict: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_130, type, archim462609752435547400_field: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_131, type, euclid3725896446679973847miring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_132, type, canoni5634975068530333245id_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_133, type, ordere8940638589300402666id_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_134, type, strict7427464778891057005id_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_135, type, quickc3360725361186068524ustive: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_136, type, bit_se359711467146920520ations: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_137, type, ordere2412721322843649153imp_le: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_138, type, ordere580206878836729694up_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_139, type, ordere1170586879665033532d_diff: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_140, type, strict9044650504122735259up_add: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_141, type, euclid8851590272496341667cancel: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_142, type, unique1627219031080169319umeral: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_143, type, euclid4440199948858584721cancel: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_144, type, euclid3128863361964157862miring: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_145, type, ordere1937475149494474687imp_le: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_146, type, euclid8789492081693882211th_nat: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_147, type, euclid5411537665997757685th_nat: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_148, type, semiri1453513574482234551roduct: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_149, type, bit_un5681908812861735899ations: !>[X0: $tType]:($o)).
% 24.80/8.34 thf(func_def_150, type, archimedean_ceiling: !>[X0: $tType]:(X0 > int)).
% 24.80/8.34 thf(func_def_151, type, archim6421214686448440834_floor: !>[X0: $tType]:(X0 > int)).
% 24.80/8.34 thf(func_def_152, type, archimedean_frac: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_153, type, archimedean_round: !>[X0: $tType]:(X0 > int)).
% 24.80/8.34 thf(func_def_154, type, array_alloc: !>[X0: $tType]:(list @ X0 > heap_ext @ product_unit > product_prod @ array @ X0 @ heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_155, type, array_freeze: !>[X0: $tType]:(array @ X0 > heap_Time_Heap @ list @ X0)).
% 24.80/8.34 thf(func_def_156, type, array_get: !>[X0: $tType]:(heap_ext @ product_unit > array @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_157, type, array_len: !>[X0: $tType]:(array @ X0 > heap_Time_Heap @ nat)).
% 24.80/8.34 thf(func_def_158, type, array_len2: !>[X0: $tType, X1: $tType]:(array @ X0 > heap_Time_Heap @ X1)).
% 24.80/8.34 thf(func_def_159, type, array_length: !>[X0: $tType]:(heap_ext @ product_unit > array @ X0 > nat)).
% 24.80/8.34 thf(func_def_160, type, array_make: !>[X0: $tType]:(nat > (nat > X0) > heap_Time_Heap @ array @ X0)).
% 24.80/8.34 thf(func_def_161, type, array_make2: !>[X0: $tType, X1: $tType]:(code_integer > (X0 > X1) > heap_Time_Heap @ array @ X1)).
% 24.80/8.34 thf(func_def_162, type, array_map_entry: !>[X0: $tType]:(nat > (X0 > X0) > array @ X0 > heap_Time_Heap @ array @ X0)).
% 24.80/8.34 thf(func_def_163, type, array_new: !>[X0: $tType]:(nat > X0 > heap_Time_Heap @ array @ X0)).
% 24.80/8.34 thf(func_def_164, type, array_new2: !>[X0: $tType]:(code_integer > X0 > heap_Time_Heap @ array @ X0)).
% 24.80/8.34 thf(func_def_165, type, array_noteq: !>[X0: $tType, X1: $tType]:(array @ X0 > array @ X1 > $o)).
% 24.80/8.34 thf(func_def_166, type, array_nth: !>[X0: $tType]:(array @ X0 > nat > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_167, type, array_nth2: !>[X0: $tType]:(array @ X0 > code_integer > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_168, type, array_of_list: !>[X0: $tType]:(list @ X0 > heap_Time_Heap @ array @ X0)).
% 24.80/8.34 thf(func_def_169, type, array_present: !>[X0: $tType]:(heap_ext @ product_unit > array @ X0 > $o)).
% 24.80/8.34 thf(func_def_170, type, array_set: !>[X0: $tType]:(array @ X0 > list @ X0 > heap_ext @ product_unit > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_171, type, array_swap: !>[X0: $tType]:(nat > X0 > array @ X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_172, type, array_upd: !>[X0: $tType]:(nat > X0 > array @ X0 > heap_Time_Heap @ array @ X0)).
% 24.80/8.34 thf(func_def_173, type, array_upd2: !>[X0: $tType]:(array @ X0 > code_integer > X0 > heap_Time_Heap @ product_unit)).
% 24.80/8.34 thf(func_def_174, type, array_update: !>[X0: $tType]:(array @ X0 > nat > X0 > heap_ext @ product_unit > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_175, type, rep_assn: assn > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_176, type, ex_assn: !>[X0: $tType]:((X0 > assn) > assn)).
% 24.80/8.34 thf(func_def_177, type, in_range: product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_178, type, in_range_rel: product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_179, type, one_assn_raw: product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_180, type, one_assn_raw_rel: product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_181, type, proper: (product_prod @ heap_ext @ product_unit @ set @ nat > $o) > $o).
% 24.80/8.34 thf(func_def_182, type, pure_assn: $o > assn).
% 24.80/8.34 thf(func_def_183, type, pure_assn_raw: !>[X0: $tType, X1: $tType]:($o > product_prod @ X0 @ set @ X1 > $o)).
% 24.80/8.34 thf(func_def_184, type, pure_assn_raw_rel: !>[X0: $tType, X1: $tType]:(product_prod @ $o @ product_prod @ X0 @ set @ X1 > product_prod @ $o @ product_prod @ X0 @ set @ X1 > $o)).
% 24.80/8.34 thf(func_def_185, type, relH: set @ nat > heap_ext @ product_unit > heap_ext @ product_unit > $o).
% 24.80/8.34 thf(func_def_186, type, snga_assn: !>[X0: $tType]:(array @ X0 > list @ X0 > assn)).
% 24.80/8.34 thf(func_def_187, type, snga_assn_raw: !>[X0: $tType]:(array @ X0 > list @ X0 > product_prod @ heap_ext @ product_unit @ set @ nat > $o)).
% 24.80/8.34 thf(func_def_188, type, snga_assn_raw_rel: !>[X0: $tType]:(product_prod @ array @ X0 @ product_prod @ list @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ array @ X0 @ product_prod @ list @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > $o)).
% 24.80/8.34 thf(func_def_189, type, sngr_assn: !>[X0: $tType]:(ref @ X0 > X0 > assn)).
% 24.80/8.34 thf(func_def_190, type, sngr_assn_raw: !>[X0: $tType]:(ref @ X0 > X0 > product_prod @ heap_ext @ product_unit @ set @ nat > $o)).
% 24.80/8.34 thf(func_def_191, type, sngr_assn_raw_rel: !>[X0: $tType]:(product_prod @ ref @ X0 @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ ref @ X0 @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > $o)).
% 24.80/8.34 thf(func_def_192, type, times_assn_raw: (product_prod @ heap_ext @ product_unit @ set @ nat > $o) > (product_prod @ heap_ext @ product_unit @ set @ nat > $o) > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_193, type, times_assn_raw_rel: product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_194, type, wand_assn: assn > assn > assn).
% 24.80/8.34 thf(func_def_195, type, wand_raw: (product_prod @ heap_ext @ product_unit @ set @ nat > $o) > (product_prod @ heap_ext @ product_unit @ set @ nat > $o) > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_196, type, wand_raw_rel: product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_197, type, bNF_Cardinal_Csum: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_198, type, bNF_Cardinal_cexp: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ (X1 > X0) @ (X1 > X0))).
% 24.80/8.34 thf(func_def_199, type, bNF_Cardinal_cfinite: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_200, type, bNF_Ca4139267488887388095finite: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_201, type, bNF_Cardinal_cone: set @ product_prod @ product_unit @ product_unit).
% 24.80/8.34 thf(func_def_202, type, bNF_Cardinal_cprod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_203, type, bNF_Cardinal_csum: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ sum_sum @ X0 @ X1 @ sum_sum @ X0 @ X1)).
% 24.80/8.34 thf(func_def_204, type, bNF_Cardinal_ctwo: set @ product_prod @ $o @ $o).
% 24.80/8.34 thf(func_def_205, type, bNF_Cardinal_czero: !>[X0: $tType]:(set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_206, type, bNF_Ca8387033319878233205ardSuc: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 24.80/8.34 thf(func_def_207, type, bNF_Ca6860139660246222851ard_of: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_208, type, bNF_Ca8970107618336181345der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_209, type, bNF_Ca7293521722713021262ofinal: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_210, type, bNF_Ca6246979054910435723ardSuc: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0 > $o)).
% 24.80/8.34 thf(func_def_211, type, bNF_Ca8665028551170535155natLeq: set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_212, type, bNF_Ca7133664381575040944arCard: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_213, type, bNF_Ca3754400796208372196lChain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_214, type, bNF_Gr: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > set @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_215, type, bNF_Grp: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X0 > X1 > $o)).
% 24.80/8.34 thf(func_def_216, type, bNF_collect: !>[X0: $tType, X1: $tType]:(set @ (X0 > set @ X1) > X0 > set @ X1)).
% 24.80/8.34 thf(func_def_217, type, bNF_convol: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X0 > X2) > X0 > product_prod @ X1 @ X2)).
% 24.80/8.34 thf(func_def_218, type, bNF_eq_onp: !>[X0: $tType]:((X0 > $o) > X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_219, type, bNF_fstOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_220, type, bNF_pick_middlep: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > X1)).
% 24.80/8.34 thf(func_def_221, type, bNF_rel_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > (X0 > X2) > (X1 > X3) > $o)).
% 24.80/8.34 thf(func_def_222, type, bNF_sndOp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 24.80/8.34 thf(func_def_223, type, bNF_vimage2p: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((X0 > X1) > (X2 > X3) > (X1 > X3 > X4) > X0 > X2 > X4)).
% 24.80/8.34 thf(func_def_224, type, bNF_Gr5436034075474128252omCard: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > X1 > X0)).
% 24.80/8.34 thf(func_def_225, type, bNF_Greatest_image2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (X0 > X1) > (X0 > X2) > set @ product_prod @ X1 @ X2)).
% 24.80/8.34 thf(func_def_226, type, bNF_Gr4221423524335903396lImage: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_227, type, bNF_Gr7122648621184425601vImage: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_228, type, bNF_Greatest_toCard: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > X0 > X1)).
% 24.80/8.34 thf(func_def_229, type, bNF_Gr1419584066657907630d_pred: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_230, type, bNF_Wellorder_Func: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ (X0 > X1))).
% 24.80/8.34 thf(func_def_231, type, bNF_Wellorder_bsqr: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ product_prod @ X0 @ X0 @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_232, type, bNF_Wellorder_curr: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ X0 > (product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 24.80/8.34 thf(func_def_233, type, bNF_We2720479622203943262_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_234, type, bNF_We413866401316099525erIncl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 24.80/8.34 thf(func_def_235, type, bNF_Wellorder_ordIso: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_236, type, bNF_Wellorder_ordLeq: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_237, type, bNF_We4044943003108391690rdLess: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_238, type, bNF_Wellorder_compat: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_239, type, bNF_Wellorder_embed: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_240, type, bNF_Wellorder_embedS: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_241, type, bNF_Wellorder_iso: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_242, type, bNF_Wellorder_wo_rel: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_243, type, bNF_We4791949203932849705sMinim: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_244, type, bNF_We1388413361240627857o_max2: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_245, type, bNF_We6954850376910717587_minim: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0)).
% 24.80/8.34 thf(func_def_246, type, bNF_Wellorder_wo_suc: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > X0)).
% 24.80/8.34 thf(func_def_247, type, basic_BNF_size_prod: !>[X0: $tType, X1: $tType]:((X0 > nat) > (X1 > nat) > product_prod @ X0 @ X1 > nat)).
% 24.80/8.34 thf(func_def_248, type, basic_BNF_size_sum: !>[X0: $tType, X1: $tType]:((X0 > nat) > (X1 > nat) > sum_sum @ X0 @ X1 > nat)).
% 24.80/8.34 thf(func_def_249, type, basic_fsts: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > set @ X0)).
% 24.80/8.34 thf(func_def_250, type, basic_pred_fun: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > $o) > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_251, type, basic_pred_prod: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > $o) > product_prod @ X0 @ X1 > $o)).
% 24.80/8.34 thf(func_def_252, type, basic_rel_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1 > $o) > (X2 > X3 > $o) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3 > $o)).
% 24.80/8.34 thf(func_def_253, type, basic_snds: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > set @ X1)).
% 24.80/8.34 thf(func_def_254, type, binomial: nat > nat > nat).
% 24.80/8.34 thf(func_def_255, type, gbinomial: !>[X0: $tType]:(X0 > nat > X0)).
% 24.80/8.34 thf(func_def_256, type, bit_and_int_rel: product_prod @ int @ int > product_prod @ int @ int > $o).
% 24.80/8.34 thf(func_def_257, type, bit_and_not_num: num > num > option @ num).
% 24.80/8.34 thf(func_def_258, type, bit_and_not_num_rel: product_prod @ num @ num > product_prod @ num @ num > $o).
% 24.80/8.34 thf(func_def_259, type, bit_concat_bit: nat > int > int > int).
% 24.80/8.34 thf(func_def_260, type, bit_or_not_num_neg: num > num > num).
% 24.80/8.34 thf(func_def_261, type, bit_or3848514188828904588eg_rel: product_prod @ num @ num > product_prod @ num @ num > $o).
% 24.80/8.34 thf(func_def_262, type, bit_ri4277139882892585799ns_not: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_263, type, bit_ri4674362597316999326ke_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_264, type, bit_se5824344872417868541ns_and: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_265, type, bit_se4197421643247451524op_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_266, type, bit_se8732182000553998342ip_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_267, type, bit_se2239418461657761734s_mask: !>[X0: $tType]:(nat > X0)).
% 24.80/8.34 thf(func_def_268, type, bit_se1065995026697491101ons_or: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_269, type, bit_se4730199178511100633sh_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_270, type, bit_se5668285175392031749et_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_271, type, bit_se2584673776208193580ke_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_272, type, bit_se2638667681897837118et_bit: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_273, type, bit_se5824344971392196577ns_xor: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_274, type, bit_se5641148757651400278ts_bit: !>[X0: $tType]:(X0 > nat > $o)).
% 24.80/8.34 thf(func_def_275, type, bit_se6407376104438227557le_bit: !>[X0: $tType]:(itself @ X0 > nat > $o)).
% 24.80/8.34 thf(func_def_276, type, bit_take_bit_num: nat > num > option @ num).
% 24.80/8.34 thf(func_def_277, type, bit_un7362597486090784418nd_num: num > num > option @ num).
% 24.80/8.34 thf(func_def_278, type, bit_un4731106466462545111um_rel: product_prod @ num @ num > product_prod @ num @ num > $o).
% 24.80/8.34 thf(func_def_279, type, bit_un6697907153464112080or_num: num > num > num).
% 24.80/8.34 thf(func_def_280, type, bit_un4773296044027857193um_rel: product_prod @ num @ num > product_prod @ num @ num > $o).
% 24.80/8.34 thf(func_def_281, type, bit_un2480387367778600638or_num: num > num > option @ num).
% 24.80/8.34 thf(func_def_282, type, bit_un2901131394128224187um_rel: product_prod @ num @ num > product_prod @ num @ num > $o).
% 24.80/8.34 thf(func_def_283, type, boolea2506097494486148201lgebra: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > (X0 > X0) > X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_284, type, boolea3799213064322606851m_diff: !>[X0: $tType]:((X0 > X0 > X0) > (X0 > X0 > X0) > (X0 > X0) > X0 > X0 > (X0 > X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_285, type, iterate_add: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_286, type, code_Nat: code_integer > code_natural).
% 24.80/8.34 thf(func_def_287, type, code_Neg: num > code_integer).
% 24.80/8.34 thf(func_def_288, type, code_Pos: num > code_integer).
% 24.80/8.34 thf(func_def_289, type, code_Suc: code_natural > code_natural).
% 24.80/8.34 thf(func_def_290, type, code_bit_cut_integer: code_integer > product_prod @ code_integer @ $o).
% 24.80/8.34 thf(func_def_291, type, code_cr_integer: int > code_integer > $o).
% 24.80/8.34 thf(func_def_292, type, code_cr_natural: nat > code_natural > $o).
% 24.80/8.34 thf(func_def_293, type, code_divmod_abs: code_integer > code_integer > product_prod @ code_integer @ code_integer).
% 24.80/8.34 thf(func_def_294, type, code_divmod_integer: code_integer > code_integer > product_prod @ code_integer @ code_integer).
% 24.80/8.34 thf(func_def_295, type, code_dup: code_integer > code_integer).
% 24.80/8.34 thf(func_def_296, type, code_int_of_integer: code_integer > int).
% 24.80/8.34 thf(func_def_297, type, code_integer_of_int: int > code_integer).
% 24.80/8.34 thf(func_def_298, type, code_integer_of_nat: nat > code_integer).
% 24.80/8.34 thf(func_def_299, type, code_i5400310926305786745atural: code_natural > code_integer).
% 24.80/8.34 thf(func_def_300, type, code_integer_of_num: num > code_integer).
% 24.80/8.34 thf(func_def_301, type, code_nat_of_integer: code_integer > nat).
% 24.80/8.34 thf(func_def_302, type, code_case_natural: !>[X0: $tType]:(X0 > (code_natural > X0) > code_natural > X0)).
% 24.80/8.34 thf(func_def_303, type, code_nat_of_natural: code_natural > nat).
% 24.80/8.34 thf(func_def_304, type, code_natural_of_nat: nat > code_natural).
% 24.80/8.34 thf(func_def_305, type, code_rec_natural: !>[X0: $tType]:(X0 > (code_natural > X0 > X0) > code_natural > X0)).
% 24.80/8.34 thf(func_def_306, type, code_rec_set_natural: !>[X0: $tType]:(X0 > (code_natural > X0 > X0) > code_natural > X0 > $o)).
% 24.80/8.34 thf(func_def_307, type, code_n4118661773612635043nteger: code_integer > code_natural).
% 24.80/8.34 thf(func_def_308, type, code_negative: num > code_integer).
% 24.80/8.34 thf(func_def_309, type, code_num_of_integer: code_integer > num).
% 24.80/8.34 thf(func_def_310, type, code_pcr_integer: int > code_integer > $o).
% 24.80/8.34 thf(func_def_311, type, code_pcr_natural: nat > code_natural > $o).
% 24.80/8.34 thf(func_def_312, type, code_positive: num > code_integer).
% 24.80/8.34 thf(func_def_313, type, code_size_nat: code_natural > nat).
% 24.80/8.34 thf(func_def_314, type, code_sub: num > num > code_integer).
% 24.80/8.34 thf(func_def_315, type, complete_Inf_Inf: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_316, type, complete_Sup_Sup: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_317, type, comple1908693960933563346ssible: !>[X0: $tType]:((set @ X0 > X0) > (X0 > X0 > $o) > (X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_318, type, comple187402453842119260l_fixp: !>[X0: $tType]:((set @ X0 > X0) > (X0 > X0 > $o) > (X0 > X0) > X0)).
% 24.80/8.34 thf(func_def_319, type, comple115746919287870866o_fixp: !>[X0: $tType]:((X0 > X0) > X0)).
% 24.80/8.34 thf(func_def_320, type, comple1602240252501008431_chain: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_321, type, comple7038119648293358887notone: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X1 > X1 > $o) > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_322, type, from_nat: !>[X0: $tType]:(nat > X0)).
% 24.80/8.34 thf(func_def_323, type, nth_item: !>[X0: $tType]:(nat > set @ old_node @ X0 @ product_unit)).
% 24.80/8.34 thf(func_def_324, type, nth_item_rel: nat > nat > $o).
% 24.80/8.34 thf(func_def_325, type, to_nat: !>[X0: $tType]:(X0 > nat)).
% 24.80/8.34 thf(func_def_326, type, adjust_div: product_prod @ int @ int > int).
% 24.80/8.34 thf(func_def_327, type, divmod_nat: nat > nat > product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_328, type, eucl_rel_int: int > int > product_prod @ int @ int > $o).
% 24.80/8.34 thf(func_def_329, type, unique5940410009612947441es_aux: !>[X0: $tType]:(product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_330, type, unique8689654367752047608divmod: !>[X0: $tType]:(num > num > product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_331, type, unique1321980374590559556d_step: !>[X0: $tType]:(num > product_prod @ X0 @ X0 > product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_332, type, equiv_congruent: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_333, type, equiv_congruent2: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > (X0 > X1 > X2) > $o)).
% 24.80/8.34 thf(func_def_334, type, equiv_equiv: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_335, type, equiv_equivp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_336, type, equiv_proj: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > set @ X1)).
% 24.80/8.34 thf(func_def_337, type, equiv_quotient: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > set @ set @ X0)).
% 24.80/8.34 thf(func_def_338, type, euclid6346220572633701492n_size: !>[X0: $tType]:(X0 > nat)).
% 24.80/8.34 thf(func_def_339, type, euclid7384307370059645450egment: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_340, type, comm_s3205402744901411588hammer: !>[X0: $tType]:(X0 > nat > X0)).
% 24.80/8.34 thf(func_def_341, type, semiring_char_0_fact: !>[X0: $tType]:(nat > X0)).
% 24.80/8.34 thf(func_def_342, type, inverse_inverse: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_343, type, finite_card: !>[X0: $tType]:(set @ X0 > nat)).
% 24.80/8.34 thf(func_def_344, type, finite6289374366891150609ommute: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_345, type, finite1981172056906289455axioms: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_346, type, finite4980608107308702382axioms: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_347, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 24.80/8.34 thf(func_def_348, type, finite_fold: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > set @ X0 > X1)).
% 24.80/8.34 thf(func_def_349, type, finite_folding: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_350, type, finite_folding_idem: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_351, type, finite7837460588564673216axioms: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_352, type, finite1890593828518410140dem_on: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_353, type, finite6916993218817215295axioms: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1 > X1) > $o)).
% 24.80/8.34 thf(func_def_354, type, bij_betw: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1 > $o)).
% 24.80/8.34 thf(func_def_355, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 24.80/8.34 thf(func_def_356, type, fcomp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X1 > X2) > X0 > X2)).
% 24.80/8.34 thf(func_def_357, type, fun_upd: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > X1 > X0 > X1)).
% 24.80/8.34 thf(func_def_358, type, id: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_359, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_360, type, map_fun: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (X1 > X2) > X0 > X3)).
% 24.80/8.34 thf(func_def_361, type, mono_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_362, type, override_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X0 > X1) > set @ X0 > X0 > X1)).
% 24.80/8.34 thf(func_def_363, type, strict_mono_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_364, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 24.80/8.34 thf(func_def_365, type, fun_max_strict: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_366, type, fun_max_weak: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_367, type, fun_min_strict: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_368, type, fun_min_weak: set @ product_prod @ set @ product_prod @ nat @ nat @ set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_369, type, fun_pair_leq: set @ product_prod @ product_prod @ nat @ nat @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_370, type, fun_pair_less: set @ product_prod @ product_prod @ nat @ nat @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_371, type, fun_reduction_pair: !>[X0: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_372, type, fun_rp_inv_image: !>[X0: $tType, X1: $tType]:(product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0 > (X1 > X0) > product_prod @ set @ product_prod @ X1 @ X1 @ set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_373, type, gcd_Gcd: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_374, type, gcd_Lcm: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_375, type, bezw: nat > nat > product_prod @ int @ int).
% 24.80/8.34 thf(func_def_376, type, bezw_rel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 24.80/8.34 thf(func_def_377, type, bounde8507323023520639062attice: !>[X0: $tType]:((X0 > X0 > X0) > X0 > X0 > (X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_378, type, bounde6485984586167503788ce_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > X0 > (X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_379, type, bounde2362111253966948842tice_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > X0 > set @ X0 > X0)).
% 24.80/8.34 thf(func_def_380, type, gcd_gcd: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_381, type, gcd_lcm: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_382, type, gcd_nat_rel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 24.80/8.34 thf(func_def_383, type, semiring_gcd_Gcd_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_384, type, semiring_gcd_Lcm_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_385, type, abel_semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_386, type, abel_s757365448890700780axioms: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_387, type, abs_abs: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_388, type, comm_monoid: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_389, type, comm_monoid_axioms: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_390, type, group: !>[X0: $tType]:((X0 > X0 > X0) > X0 > (X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_391, type, group_axioms: !>[X0: $tType]:((X0 > X0 > X0) > X0 > (X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_392, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_393, type, monoid: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_394, type, monoid_axioms: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_395, type, one_one: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(func_def_396, type, plus_plus: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_397, type, semigroup: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_398, type, sgn_sgn: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_399, type, times_times: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_400, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_401, type, zero_zero: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(func_def_402, type, groups3894954378712506084id_sum: !>[X0: $tType, X1: $tType]:((X0 > X0 > X0) > X0 > (X1 > X0) > set @ X1 > X0)).
% 24.80/8.34 thf(func_def_403, type, groups7311177749621191930dd_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 24.80/8.34 thf(func_def_404, type, groups1027152243600224163dd_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 24.80/8.34 thf(func_def_405, type, groups7121269368397514597t_prod: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 24.80/8.34 thf(func_def_406, type, groups1962203154675924110t_prod: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1)).
% 24.80/8.34 thf(func_def_407, type, groups778175481326437816id_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_408, type, groups_comm_monoid_F: !>[X0: $tType, X1: $tType]:((X0 > X0 > X0) > X0 > (X1 > X0) > set @ X1 > X0)).
% 24.80/8.34 thf(func_def_409, type, groups_comm_monoid_G: !>[X0: $tType, X1: $tType]:((X0 > X0 > X0) > X0 > (X1 > X0) > set @ X1 > X0)).
% 24.80/8.34 thf(func_def_410, type, groups1828464146339083142d_list: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_411, type, groups4802862169904069756st_set: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_412, type, groups4207007520872428315er_sum: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1 > list @ X0 > X1)).
% 24.80/8.34 thf(func_def_413, type, groups4543113879258116180m_list: !>[X0: $tType]:((X0 > X0 > X0) > X0 > list @ X0 > X0)).
% 24.80/8.34 thf(func_def_414, type, groups8242544230860333062m_list: !>[X0: $tType]:(list @ X0 > X0)).
% 24.80/8.34 thf(func_def_415, type, groups_monoid_list: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_416, type, groups_monoid_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > list @ X0 > X0)).
% 24.80/8.34 thf(func_def_417, type, groups5270119922927024881d_list: !>[X0: $tType]:(list @ X0 > X0)).
% 24.80/8.34 thf(func_def_418, type, nO_MATCH: !>[X0: $tType, X1: $tType]:(X0 > X1 > $o)).
% 24.80/8.34 thf(func_def_419, type, the: !>[X0: $tType]:((X0 > $o) > X0)).
% 24.80/8.34 thf(func_def_420, type, uniq: !>[X0: $tType]:((X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_421, type, default_default: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(func_def_422, type, undefined: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(func_def_423, type, addr_of_array: !>[X0: $tType]:(array @ X0 > nat)).
% 24.80/8.34 thf(func_def_424, type, addr_of_ref: !>[X0: $tType]:(ref @ X0 > nat)).
% 24.80/8.34 thf(func_def_425, type, array2: !>[X0: $tType]:(nat > array @ X0)).
% 24.80/8.34 thf(func_def_426, type, arrays: !>[X0: $tType]:(heap_ext @ X0 > typerep > nat > list @ nat)).
% 24.80/8.34 thf(func_def_427, type, arrays_update: !>[X0: $tType]:(((typerep > nat > list @ nat) > typerep > nat > list @ nat) > heap_ext @ X0 > heap_ext @ X0)).
% 24.80/8.34 thf(func_def_428, type, lim: !>[X0: $tType]:(heap_ext @ X0 > nat)).
% 24.80/8.34 thf(func_def_429, type, lim_update: !>[X0: $tType]:((nat > nat) > heap_ext @ X0 > heap_ext @ X0)).
% 24.80/8.34 thf(func_def_430, type, refs: !>[X0: $tType]:(heap_ext @ X0 > typerep > nat > nat)).
% 24.80/8.34 thf(func_def_431, type, ref2: !>[X0: $tType]:(nat > ref @ X0)).
% 24.80/8.34 thf(func_def_432, type, heap_Time_Heap2: !>[X0: $tType]:((heap_ext @ product_unit > option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat) > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_433, type, heap_Time_case_Heap: !>[X0: $tType, X1: $tType]:(((heap_ext @ product_unit > option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat) > X1) > heap_Time_Heap @ X0 > X1)).
% 24.80/8.34 thf(func_def_434, type, heap_Time_map_Heap: !>[X0: $tType, X1: $tType]:((X0 > X1) > heap_Time_Heap @ X0 > heap_Time_Heap @ X1)).
% 24.80/8.34 thf(func_def_435, type, heap_Time_pred_Heap: !>[X0: $tType]:((X0 > $o) > heap_Time_Heap @ X0 > $o)).
% 24.80/8.34 thf(func_def_436, type, heap_Time_rec_Heap: !>[X0: $tType, X1: $tType]:(((heap_ext @ product_unit > option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat) > X1) > heap_Time_Heap @ X0 > X1)).
% 24.80/8.34 thf(func_def_437, type, heap_Time_rel_Heap: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > heap_Time_Heap @ X0 > heap_Time_Heap @ X1 > $o)).
% 24.80/8.34 thf(func_def_438, type, heap_Time_set_Heap: !>[X0: $tType]:(heap_Time_Heap @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_439, type, heap_Time_size_Heap: !>[X0: $tType]:((X0 > nat) > heap_Time_Heap @ X0 > nat)).
% 24.80/8.34 thf(func_def_440, type, heap_Time_Heap_lub: !>[X0: $tType]:(set @ heap_Time_Heap @ X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_441, type, heap_Time_Heap_ord: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_Time_Heap @ X0 > $o)).
% 24.80/8.34 thf(func_def_442, type, heap_Time_assert: !>[X0: $tType]:((X0 > $o) > X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_443, type, heap_Time_bind: !>[X0: $tType, X1: $tType]:(heap_Time_Heap @ X0 > (X0 > heap_Time_Heap @ X1) > heap_Time_Heap @ X1)).
% 24.80/8.34 thf(func_def_444, type, heap_Time_effect: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > heap_ext @ product_unit > X0 > nat > $o)).
% 24.80/8.34 thf(func_def_445, type, heap_Time_execute: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat)).
% 24.80/8.34 thf(func_def_446, type, heap_Time_guard: !>[X0: $tType]:((heap_ext @ product_unit > $o) > (heap_ext @ product_unit > product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat) > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_447, type, heap_Time_heap: !>[X0: $tType]:((heap_ext @ product_unit > product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat) > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_448, type, heap_Time_lift: !>[X0: $tType, X1: $tType]:((X0 > X1) > X0 > heap_Time_Heap @ X1)).
% 24.80/8.34 thf(func_def_449, type, heap_Time_raise: !>[X0: $tType]:(list @ char > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_450, type, heap_Time_return: !>[X0: $tType]:(X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_451, type, heap_Time_success: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > $o)).
% 24.80/8.34 thf(func_def_452, type, heap_Time_tap: !>[X0: $tType]:((heap_ext @ product_unit > X0) > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_453, type, heap_Time_timeFrame: !>[X0: $tType]:(nat > option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat)).
% 24.80/8.34 thf(func_def_454, type, heap_T5500966940807335491me_rel: !>[X0: $tType]:(product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > $o)).
% 24.80/8.34 thf(func_def_455, type, heap_Time_ureturn: !>[X0: $tType]:(X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_456, type, heap_Time_wait: nat > heap_Time_Heap @ product_unit).
% 24.80/8.34 thf(func_def_457, type, hilbert_bijection: !>[X0: $tType]:((X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_458, type, hilbert_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 24.80/8.34 thf(func_def_459, type, hoare_hoare_triple: !>[X0: $tType]:(assn > heap_Time_Heap @ X0 > (X0 > assn) > $o)).
% 24.80/8.34 thf(func_def_460, type, hoare_new_addrs: heap_ext @ product_unit > set @ nat > heap_ext @ product_unit > set @ nat).
% 24.80/8.34 thf(func_def_461, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_462, type, complete_lattice_lfp: !>[X0: $tType]:((X0 > X0) > X0)).
% 24.80/8.34 thf(func_def_463, type, abs_Integ: product_prod @ nat @ nat > int).
% 24.80/8.34 thf(func_def_464, type, neg: num > int).
% 24.80/8.34 thf(func_def_465, type, pos: num > int).
% 24.80/8.34 thf(func_def_466, type, rep_Integ: int > product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_467, type, dup: int > int).
% 24.80/8.34 thf(func_def_468, type, intrel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 24.80/8.34 thf(func_def_469, type, nat2: int > nat).
% 24.80/8.34 thf(func_def_470, type, pcr_int: product_prod @ nat @ nat > int > $o).
% 24.80/8.34 thf(func_def_471, type, power_int: !>[X0: $tType]:(X0 > int > X0)).
% 24.80/8.34 thf(func_def_472, type, ring_1_Ints: !>[X0: $tType]:(set @ X0)).
% 24.80/8.34 thf(func_def_473, type, ring_1_of_int: !>[X0: $tType]:(int > X0)).
% 24.80/8.34 thf(func_def_474, type, sub: num > num > int).
% 24.80/8.34 thf(func_def_475, type, inf_inf: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_476, type, semilattice_neutr: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_477, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_478, type, lattic643756798349783984er_Max: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_479, type, lattic643756798350308766er_Min: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_480, type, lattic7752659483105999362nf_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_481, type, lattic5214292709420241887eutr_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > set @ X0 > X0)).
% 24.80/8.34 thf(func_def_482, type, lattic149705377957585745ce_set: !>[X0: $tType]:((X0 > X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_483, type, lattic1715443433743089157tice_F: !>[X0: $tType]:((X0 > X0 > X0) > set @ X0 > X0)).
% 24.80/8.34 thf(func_def_484, type, lattic5882676163264333800up_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 24.80/8.34 thf(func_def_485, type, quotient: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > (X0 > X1 > $o) > $o)).
% 24.80/8.34 thf(func_def_486, type, append: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_487, type, arg_min_list: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > X0)).
% 24.80/8.34 thf(func_def_488, type, arg_min_list_rel: !>[X0: $tType, X1: $tType]:(product_prod @ (X0 > X1) @ list @ X0 > product_prod @ (X0 > X1) @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_489, type, butlast: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_490, type, concat: !>[X0: $tType]:(list @ list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_491, type, count_list: !>[X0: $tType]:(list @ X0 > X0 > nat)).
% 24.80/8.34 thf(func_def_492, type, distinct: !>[X0: $tType]:(list @ X0 > $o)).
% 24.80/8.34 thf(func_def_493, type, drop: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_494, type, dropWhile: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_495, type, enumerate: !>[X0: $tType]:(nat > list @ X0 > list @ product_prod @ nat @ X0)).
% 24.80/8.34 thf(func_def_496, type, extract: !>[X0: $tType]:((X0 > $o) > list @ X0 > option @ product_prod @ list @ X0 @ product_prod @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_497, type, filter: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_498, type, find: !>[X0: $tType]:((X0 > $o) > list @ X0 > option @ X0)).
% 24.80/8.34 thf(func_def_499, type, fold: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > list @ X0 > X1 > X1)).
% 24.80/8.34 thf(func_def_500, type, foldl: !>[X0: $tType, X1: $tType]:((X0 > X1 > X0) > X0 > list @ X1 > X0)).
% 24.80/8.34 thf(func_def_501, type, foldr: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > list @ X0 > X1 > X1)).
% 24.80/8.34 thf(func_def_502, type, gen_length: !>[X0: $tType]:(nat > list @ X0 > nat)).
% 24.80/8.34 thf(func_def_503, type, last: !>[X0: $tType]:(list @ X0 > X0)).
% 24.80/8.34 thf(func_def_504, type, lenlex: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_505, type, lex: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_506, type, lexn: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > nat > set @ product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_507, type, lexord: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_508, type, lexordp: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_509, type, linorder_sort_key: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_510, type, linord4507533701916653071of_set: !>[X0: $tType]:(set @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_511, type, cons: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_512, type, nil: !>[X0: $tType]:(list @ X0)).
% 24.80/8.34 thf(func_def_513, type, case_list: !>[X0: $tType, X1: $tType]:(X0 > (X1 > list @ X1 > X0) > list @ X1 > X0)).
% 24.80/8.34 thf(func_def_514, type, hd: !>[X0: $tType]:(list @ X0 > X0)).
% 24.80/8.34 thf(func_def_515, type, list_all: !>[X0: $tType]:((X0 > $o) > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_516, type, list_all2: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > list @ X0 > list @ X1 > $o)).
% 24.80/8.34 thf(func_def_517, type, map: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X1)).
% 24.80/8.34 thf(func_def_518, type, set2: !>[X0: $tType]:(list @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_519, type, size_list: !>[X0: $tType]:((X0 > nat) > list @ X0 > nat)).
% 24.80/8.34 thf(func_def_520, type, tl: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_521, type, list_update: !>[X0: $tType]:(list @ X0 > nat > X0 > list @ X0)).
% 24.80/8.34 thf(func_def_522, type, listrel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ list @ X0 @ list @ X1)).
% 24.80/8.34 thf(func_def_523, type, listrel1: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_524, type, listrel1p: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_525, type, listrelp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > list @ X0 > list @ X1 > $o)).
% 24.80/8.34 thf(func_def_526, type, lists: !>[X0: $tType]:(set @ X0 > set @ list @ X0)).
% 24.80/8.34 thf(func_def_527, type, map_filter: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > list @ X0 > list @ X1)).
% 24.80/8.34 thf(func_def_528, type, map_project: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ X0 > set @ X1)).
% 24.80/8.34 thf(func_def_529, type, map_tailrec_rev: !>[X0: $tType, X1: $tType]:((X0 > X1) > list @ X0 > list @ X1 > list @ X1)).
% 24.80/8.34 thf(func_def_530, type, map_tailrec_rev_rel: !>[X0: $tType, X1: $tType]:(product_prod @ (X0 > X1) @ product_prod @ list @ X0 @ list @ X1 > product_prod @ (X0 > X1) @ product_prod @ list @ X0 @ list @ X1 > $o)).
% 24.80/8.34 thf(func_def_531, type, measures: !>[X0: $tType]:(list @ (X0 > nat) > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_532, type, min_list: !>[X0: $tType]:(list @ X0 > X0)).
% 24.80/8.34 thf(func_def_533, type, min_list_rel: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_534, type, nth: !>[X0: $tType]:(list @ X0 > nat > X0)).
% 24.80/8.34 thf(func_def_535, type, nths: !>[X0: $tType]:(list @ X0 > set @ nat > list @ X0)).
% 24.80/8.34 thf(func_def_536, type, ord_lexordp: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_537, type, partition: !>[X0: $tType]:((X0 > $o) > list @ X0 > product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_538, type, product: !>[X0: $tType, X1: $tType]:(list @ X0 > list @ X1 > list @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_539, type, product_lists: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 24.80/8.34 thf(func_def_540, type, remdups: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_541, type, remdups_adj: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_542, type, remdups_adj_rel: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_543, type, remove1: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_544, type, replicate: !>[X0: $tType]:(nat > X0 > list @ X0)).
% 24.80/8.34 thf(func_def_545, type, rev: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_546, type, rotate: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_547, type, rotate1: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_548, type, shuffles: !>[X0: $tType]:(list @ X0 > list @ X0 > set @ list @ X0)).
% 24.80/8.34 thf(func_def_549, type, shuffles_rel: !>[X0: $tType]:(product_prod @ list @ X0 @ list @ X0 > product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_550, type, sorted_wrt: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_551, type, sorted_wrt_rel: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ list @ X0 > product_prod @ (X0 > X0 > $o) @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_552, type, splice: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_553, type, splice_rel: !>[X0: $tType]:(product_prod @ list @ X0 @ list @ X0 > product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_554, type, successively: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_555, type, successively_rel: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ list @ X0 > product_prod @ (X0 > X0 > $o) @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_556, type, take: !>[X0: $tType]:(nat > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_557, type, takeWhile: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_558, type, those: !>[X0: $tType]:(list @ option @ X0 > option @ list @ X0)).
% 24.80/8.34 thf(func_def_559, type, transpose: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0)).
% 24.80/8.34 thf(func_def_560, type, transpose_rel: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_561, type, upt: nat > nat > list @ nat).
% 24.80/8.34 thf(func_def_562, type, upto: int > int > list @ int).
% 24.80/8.34 thf(func_def_563, type, upto_rel: product_prod @ int @ int > product_prod @ int @ int > $o).
% 24.80/8.34 thf(func_def_564, type, zip: !>[X0: $tType, X1: $tType]:(list @ X0 > list @ X1 > list @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_565, type, dom: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ X0)).
% 24.80/8.34 thf(func_def_566, type, graph: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_567, type, map_add: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > (X0 > option @ X1) > X0 > option @ X1)).
% 24.80/8.34 thf(func_def_568, type, map_comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > option @ X1) > (X2 > option @ X0) > X2 > option @ X1)).
% 24.80/8.34 thf(func_def_569, type, map_le: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > (X0 > option @ X1) > $o)).
% 24.80/8.34 thf(func_def_570, type, map_of: !>[X0: $tType, X1: $tType]:(list @ product_prod @ X0 @ X1 > X0 > option @ X1)).
% 24.80/8.34 thf(func_def_571, type, map_upds: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > list @ X0 > list @ X1 > X0 > option @ X1)).
% 24.80/8.34 thf(func_def_572, type, ran: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ X1)).
% 24.80/8.34 thf(func_def_573, type, restrict_map: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ X0 > X0 > option @ X1)).
% 24.80/8.34 thf(func_def_574, type, cODE_ABORT: !>[X0: $tType]:((product_unit > X0) > X0)).
% 24.80/8.34 thf(func_def_575, type, eps_Opt: !>[X0: $tType]:((X0 > $o) > option @ X0)).
% 24.80/8.34 thf(func_def_576, type, bijective: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > $o)).
% 24.80/8.34 thf(func_def_577, type, brk_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ product_prod @ $o @ X0 @ product_prod @ $o @ X1)).
% 24.80/8.34 thf(func_def_578, type, dflt_None_set: !>[X0: $tType]:(set @ X0 > option @ set @ X0)).
% 24.80/8.34 thf(func_def_579, type, filter_rev: !>[X0: $tType]:((X0 > $o) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_580, type, filter_rev_aux: !>[X0: $tType]:(list @ X0 > (X0 > $o) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_581, type, fun_of_rel: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > X1)).
% 24.80/8.34 thf(func_def_582, type, inv_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > X1 > X0)).
% 24.80/8.34 thf(func_def_583, type, list_all_zip: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > list @ X0 > list @ X1 > $o)).
% 24.80/8.34 thf(func_def_584, type, list_all_zip_rel: !>[X0: $tType, X1: $tType]:(product_prod @ (X0 > X1 > $o) @ product_prod @ list @ X0 @ list @ X1 > product_prod @ (X0 > X1 > $o) @ product_prod @ list @ X0 @ list @ X1 > $o)).
% 24.80/8.34 thf(func_def_585, type, list_collect_set: !>[X0: $tType, X1: $tType]:((X0 > set @ X1) > list @ X0 > set @ X1)).
% 24.80/8.34 thf(func_def_586, type, map_mmupd: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ X0 > X1 > X0 > option @ X1)).
% 24.80/8.34 thf(func_def_587, type, map_to_set: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > set @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_588, type, merge: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_589, type, merge_list: !>[X0: $tType]:(list @ list @ X0 > list @ list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_590, type, merge_list_rel: !>[X0: $tType]:(product_prod @ list @ list @ X0 @ list @ list @ X0 > product_prod @ list @ list @ X0 @ list @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_591, type, merge_rel: !>[X0: $tType]:(product_prod @ list @ X0 @ list @ X0 > product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_592, type, mergesort: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_593, type, mergesort_by_rel: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_594, type, merges9089515139780605204_merge: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_595, type, merges2244889521215249637ge_rel: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ product_prod @ list @ X0 @ list @ X0 > product_prod @ (X0 > X0 > $o) @ product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_596, type, mergesort_by_rel_rel: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ list @ X0 > product_prod @ (X0 > X0 > $o) @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_597, type, merges295452479951948502_split: !>[X0: $tType]:(product_prod @ list @ X0 @ list @ X0 > list @ X0 > product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_598, type, merges7066485432131860899it_rel: !>[X0: $tType]:(product_prod @ product_prod @ list @ X0 @ list @ X0 @ list @ X0 > product_prod @ product_prod @ list @ X0 @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_599, type, mergesort_remdups: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_600, type, pairself: !>[X0: $tType, X1: $tType]:((X0 > X1) > product_prod @ X0 @ X0 > product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_601, type, pairself_rel: !>[X0: $tType, X1: $tType]:(product_prod @ (X0 > X1) @ product_prod @ X0 @ X0 > product_prod @ (X0 > X1) @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_602, type, partition_rev: !>[X0: $tType]:((X0 > $o) > product_prod @ list @ X0 @ list @ X0 > list @ X0 > product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_603, type, partition_rev_rel: !>[X0: $tType]:(product_prod @ (X0 > $o) @ product_prod @ product_prod @ list @ X0 @ list @ X0 @ list @ X0 > product_prod @ (X0 > $o) @ product_prod @ product_prod @ list @ X0 @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_604, type, quicksort_by_rel: !>[X0: $tType]:((X0 > X0 > $o) > list @ X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_605, type, quicksort_by_rel_rel: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ product_prod @ list @ X0 @ list @ X0 > product_prod @ (X0 > X0 > $o) @ product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_606, type, rel_of: !>[X0: $tType, X1: $tType]:((X0 > option @ X1) > (product_prod @ X0 @ X1 > $o) > set @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_607, type, rel_restrict: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_608, type, remove_rev: !>[X0: $tType]:(X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_609, type, revg: !>[X0: $tType]:(list @ X0 > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_610, type, revg_rel: !>[X0: $tType]:(product_prod @ list @ X0 @ list @ X0 > product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_611, type, set_to_map: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > X0 > option @ X1)).
% 24.80/8.34 thf(func_def_612, type, slice: !>[X0: $tType]:(nat > nat > list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_613, type, su_rel_fun: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > (X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_614, type, the_default: !>[X0: $tType]:(X0 > option @ X0 > X0)).
% 24.80/8.34 thf(func_def_615, type, uncurry: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 24.80/8.34 thf(func_def_616, type, zipf: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > list @ X0 > list @ X1 > list @ X2)).
% 24.80/8.34 thf(func_def_617, type, zipf_rel: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ (X0 > X1 > X2) @ product_prod @ list @ X0 @ list @ X1 > product_prod @ (X0 > X1 > X2) @ product_prod @ list @ X0 @ list @ X1 > $o)).
% 24.80/8.34 thf(func_def_618, type, add_mset: !>[X0: $tType]:(X0 > multiset @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_619, type, comm_monoid_sum_mset: !>[X0: $tType]:((X0 > X0 > X0) > X0 > multiset @ X0 > X0)).
% 24.80/8.34 thf(func_def_620, type, comm_m7189776963980413722m_mset: !>[X0: $tType]:(multiset @ X0 > X0)).
% 24.80/8.34 thf(func_def_621, type, comm_monoid_mset: !>[X0: $tType]:((X0 > X0 > X0) > X0 > $o)).
% 24.80/8.34 thf(func_def_622, type, comm_monoid_F: !>[X0: $tType]:((X0 > X0 > X0) > X0 > multiset @ X0 > X0)).
% 24.80/8.34 thf(func_def_623, type, comm_m9189036328036947845d_mset: !>[X0: $tType]:(multiset @ X0 > X0)).
% 24.80/8.34 thf(func_def_624, type, filter_mset: !>[X0: $tType]:((X0 > $o) > multiset @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_625, type, fold_mset: !>[X0: $tType, X1: $tType]:((X0 > X1 > X1) > X1 > multiset @ X0 > X1)).
% 24.80/8.34 thf(func_def_626, type, image_mset: !>[X0: $tType, X1: $tType]:((X0 > X1) > multiset @ X0 > multiset @ X1)).
% 24.80/8.34 thf(func_def_627, type, inter_mset: !>[X0: $tType]:(multiset @ X0 > multiset @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_628, type, linorder_part: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1 > list @ X0 > product_prod @ list @ X0 @ product_prod @ list @ X0 @ list @ X0)).
% 24.80/8.34 thf(func_def_629, type, ms_strict: set @ product_prod @ multiset @ product_prod @ nat @ nat @ multiset @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_630, type, ms_weak: set @ product_prod @ multiset @ product_prod @ nat @ nat @ multiset @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_631, type, mset: !>[X0: $tType]:(list @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_632, type, mset_set: !>[X0: $tType]:(set @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_633, type, mult: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ multiset @ X0 @ multiset @ X0)).
% 24.80/8.34 thf(func_def_634, type, mult1: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ multiset @ X0 @ multiset @ X0)).
% 24.80/8.34 thf(func_def_635, type, multeqp_code: !>[X0: $tType]:((X0 > X0 > $o) > multiset @ X0 > multiset @ X0 > $o)).
% 24.80/8.34 thf(func_def_636, type, abs_multiset: !>[X0: $tType]:((X0 > nat) > multiset @ X0)).
% 24.80/8.34 thf(func_def_637, type, count: !>[X0: $tType]:(multiset @ X0 > X0 > nat)).
% 24.80/8.34 thf(func_def_638, type, multp: !>[X0: $tType]:((X0 > X0 > $o) > multiset @ X0 > multiset @ X0 > $o)).
% 24.80/8.34 thf(func_def_639, type, multp_code: !>[X0: $tType]:((X0 > X0 > $o) > multiset @ X0 > multiset @ X0 > $o)).
% 24.80/8.34 thf(func_def_640, type, pcr_multiset: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > (X0 > nat) > multiset @ X1 > $o)).
% 24.80/8.34 thf(func_def_641, type, pw_leq: multiset @ product_prod @ nat @ nat > multiset @ product_prod @ nat @ nat > $o).
% 24.80/8.34 thf(func_def_642, type, repeat_mset: !>[X0: $tType]:(nat > multiset @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_643, type, replicate_mset: !>[X0: $tType]:(nat > X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_644, type, set_mset: !>[X0: $tType]:(multiset @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_645, type, size_multiset: !>[X0: $tType]:((X0 > nat) > multiset @ X0 > nat)).
% 24.80/8.34 thf(func_def_646, type, subset_eq_mset_impl: !>[X0: $tType]:(list @ X0 > list @ X0 > option @ $o)).
% 24.80/8.34 thf(func_def_647, type, subset751672762298770561pl_rel: !>[X0: $tType]:(product_prod @ list @ X0 @ list @ X0 > product_prod @ list @ X0 @ list @ X0 > $o)).
% 24.80/8.34 thf(func_def_648, type, subset_mset: !>[X0: $tType]:(multiset @ X0 > multiset @ X0 > $o)).
% 24.80/8.34 thf(func_def_649, type, subseteq_mset: !>[X0: $tType]:(multiset @ X0 > multiset @ X0 > $o)).
% 24.80/8.34 thf(func_def_650, type, union_mset: !>[X0: $tType]:(multiset @ X0 > multiset @ X0 > multiset @ X0)).
% 24.80/8.34 thf(func_def_651, type, wcount: !>[X0: $tType]:((X0 > nat) > multiset @ X0 > X0 > nat)).
% 24.80/8.34 thf(func_def_652, type, suc: nat > nat).
% 24.80/8.34 thf(func_def_653, type, compow: !>[X0: $tType]:(nat > X0 > X0)).
% 24.80/8.34 thf(func_def_654, type, case_nat: !>[X0: $tType]:(X0 > (nat > X0) > nat > X0)).
% 24.80/8.34 thf(func_def_655, type, rec_nat: !>[X0: $tType]:(X0 > (nat > X0 > X0) > nat > X0)).
% 24.80/8.34 thf(func_def_656, type, semiring_1_of_nat: !>[X0: $tType]:(nat > X0)).
% 24.80/8.34 thf(func_def_657, type, semiri8178284476397505188at_aux: !>[X0: $tType]:((X0 > X0) > nat > X0 > X0)).
% 24.80/8.34 thf(func_def_658, type, size_size: !>[X0: $tType]:(X0 > nat)).
% 24.80/8.34 thf(func_def_659, type, nat_list_decode: nat > list @ nat).
% 24.80/8.34 thf(func_def_660, type, nat_list_decode_rel: nat > nat > $o).
% 24.80/8.34 thf(func_def_661, type, nat_list_encode: list @ nat > nat).
% 24.80/8.34 thf(func_def_662, type, nat_list_encode_rel: list @ nat > list @ nat > $o).
% 24.80/8.34 thf(func_def_663, type, nat_prod_decode: nat > product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_664, type, nat_prod_decode_aux: nat > nat > product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_665, type, nat_pr5047031295181774490ux_rel: product_prod @ nat @ nat > product_prod @ nat @ nat > $o).
% 24.80/8.34 thf(func_def_666, type, nat_prod_encode: product_prod @ nat @ nat > nat).
% 24.80/8.34 thf(func_def_667, type, nat_set_decode: nat > set @ nat).
% 24.80/8.34 thf(func_def_668, type, nat_set_encode: set @ nat > nat).
% 24.80/8.34 thf(func_def_669, type, nat_sum_decode: nat > sum_sum @ nat @ nat).
% 24.80/8.34 thf(func_def_670, type, nat_triangle: nat > nat).
% 24.80/8.34 thf(func_def_671, type, bitM: num > num).
% 24.80/8.34 thf(func_def_672, type, inc: num > num).
% 24.80/8.34 thf(func_def_673, type, nat_of_num: num > nat).
% 24.80/8.34 thf(func_def_674, type, neg_numeral_dbl: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_675, type, neg_numeral_dbl_dec: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_676, type, neg_numeral_dbl_inc: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_677, type, neg_numeral_is_num: !>[X0: $tType]:(X0 > $o)).
% 24.80/8.34 thf(func_def_678, type, neg_numeral_sub: !>[X0: $tType]:(num > num > X0)).
% 24.80/8.34 thf(func_def_679, type, bit0: num > num).
% 24.80/8.34 thf(func_def_680, type, bit1: num > num).
% 24.80/8.34 thf(func_def_681, type, one2: num).
% 24.80/8.34 thf(func_def_682, type, case_num: !>[X0: $tType]:(X0 > (num > X0) > (num > X0) > num > X0)).
% 24.80/8.34 thf(func_def_683, type, size_num: num > nat).
% 24.80/8.34 thf(func_def_684, type, num_of_nat: nat > num).
% 24.80/8.34 thf(func_def_685, type, numeral_numeral: !>[X0: $tType]:(num > X0)).
% 24.80/8.34 thf(func_def_686, type, pred_numeral: num > nat).
% 24.80/8.34 thf(func_def_687, type, ring_1_iszero: !>[X0: $tType]:(X0 > $o)).
% 24.80/8.34 thf(func_def_688, type, sqr: num > num).
% 24.80/8.34 thf(func_def_689, type, old_Atom: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ nat > set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_690, type, old_In0: !>[X0: $tType, X1: $tType]:(set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_691, type, old_In1: !>[X0: $tType, X1: $tType]:(set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_692, type, old_Leaf: !>[X0: $tType, X1: $tType]:(X0 > set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_693, type, old_Node: !>[X0: $tType, X1: $tType]:(set @ product_prod @ (nat > sum_sum @ X0 @ nat) @ sum_sum @ X1 @ nat)).
% 24.80/8.34 thf(func_def_694, type, old_Numb: !>[X0: $tType, X1: $tType]:(nat > set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_695, type, old_Push_Node: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ nat > old_node @ X1 @ X0 > old_node @ X1 @ X0)).
% 24.80/8.34 thf(func_def_696, type, old_Scons: !>[X0: $tType, X1: $tType]:(set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1 > set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_697, type, old_dprod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ old_node @ X0 @ X1 @ set @ old_node @ X0 @ X1 > set @ product_prod @ set @ old_node @ X0 @ X1 @ set @ old_node @ X0 @ X1 > set @ product_prod @ set @ old_node @ X0 @ X1 @ set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_698, type, old_dsum: !>[X0: $tType, X1: $tType]:(set @ product_prod @ set @ old_node @ X0 @ X1 @ set @ old_node @ X0 @ X1 > set @ product_prod @ set @ old_node @ X0 @ X1 @ set @ old_node @ X0 @ X1 > set @ product_prod @ set @ old_node @ X0 @ X1 @ set @ old_node @ X0 @ X1)).
% 24.80/8.34 thf(func_def_699, type, old_ndepth: !>[X0: $tType, X1: $tType]:(old_node @ X0 @ X1 > nat)).
% 24.80/8.34 thf(func_def_700, type, old_Abs_Node: !>[X0: $tType, X1: $tType]:(product_prod @ (nat > sum_sum @ X0 @ nat) @ sum_sum @ X1 @ nat > old_node @ X1 @ X0)).
% 24.80/8.34 thf(func_def_701, type, bind: !>[X0: $tType, X1: $tType]:(option @ X0 > (X0 > option @ X1) > option @ X1)).
% 24.80/8.34 thf(func_def_702, type, combine_options: !>[X0: $tType]:((X0 > X0 > X0) > option @ X0 > option @ X0 > option @ X0)).
% 24.80/8.34 thf(func_def_703, type, is_none: !>[X0: $tType]:(option @ X0 > $o)).
% 24.80/8.34 thf(func_def_704, type, none: !>[X0: $tType]:(option @ X0)).
% 24.80/8.34 thf(func_def_705, type, some: !>[X0: $tType]:(X0 > option @ X0)).
% 24.80/8.34 thf(func_def_706, type, case_option: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > option @ X1 > X0)).
% 24.80/8.34 thf(func_def_707, type, map_option: !>[X0: $tType, X1: $tType]:((X0 > X1) > option @ X0 > option @ X1)).
% 24.80/8.34 thf(func_def_708, type, pred_option: !>[X0: $tType]:((X0 > $o) > option @ X0 > $o)).
% 24.80/8.34 thf(func_def_709, type, rec_option: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > option @ X1 > X0)).
% 24.80/8.34 thf(func_def_710, type, rel_option: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > option @ X0 > option @ X1 > $o)).
% 24.80/8.34 thf(func_def_711, type, set_option: !>[X0: $tType]:(option @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_712, type, size_option: !>[X0: $tType]:((X0 > nat) > option @ X0 > nat)).
% 24.80/8.34 thf(func_def_713, type, the2: !>[X0: $tType]:(option @ X0 > X0)).
% 24.80/8.34 thf(func_def_714, type, these: !>[X0: $tType]:(set @ option @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_715, type, order_Above: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_716, type, order_AboveS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_717, type, order_Under: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_718, type, order_UnderS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_719, type, order_above: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_720, type, order_aboveS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_721, type, order_679001287576687338der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_722, type, order_ofilter: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_723, type, order_7125193373082350890der_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_724, type, order_preorder_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_725, type, order_relation_of: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_726, type, order_under: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_727, type, order_underS: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_728, type, order_well_order_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_729, type, bot_bot: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(func_def_730, type, max: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_731, type, min: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_732, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_733, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_734, type, ord_max: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_735, type, ord_min: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_736, type, order_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_737, type, order_strict_mono: !>[X0: $tType, X1: $tType]:((X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_738, type, top_top: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(func_def_739, type, partial_flat_lub: !>[X0: $tType]:(X0 > set @ X0 > X0)).
% 24.80/8.34 thf(func_def_740, type, partial_flat_ord: !>[X0: $tType]:(X0 > X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_741, type, partial_fun_lub: !>[X0: $tType, X1: $tType, X2: $tType]:((set @ X0 > X1) > set @ (X2 > X0) > X2 > X1)).
% 24.80/8.34 thf(func_def_742, type, partial_fun_ord: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X2 > X0) > (X2 > X1) > $o)).
% 24.80/8.34 thf(func_def_743, type, partial_img_lub: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > (set @ X1 > X2) > set @ X0 > X3)).
% 24.80/8.34 thf(func_def_744, type, partial_img_ord: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X1 > X1 > X2) > X0 > X0 > X2)).
% 24.80/8.34 thf(func_def_745, type, partia7178651479351089652itions: !>[X0: $tType]:((X0 > X0 > $o) > (set @ X0 > X0) > $o)).
% 24.80/8.34 thf(func_def_746, type, power2: !>[X0: $tType]:(X0 > (X0 > X0 > X0) > X0 > nat > X0)).
% 24.80/8.34 thf(func_def_747, type, power_power: !>[X0: $tType]:(X0 > nat > X0)).
% 24.80/8.34 thf(func_def_748, type, seq2: !>[X0: $tType]:((product_unit > seq @ X0) > pred @ X0)).
% 24.80/8.34 thf(func_def_749, type, bind2: !>[X0: $tType, X1: $tType]:(pred @ X0 > (X0 > pred @ X1) > pred @ X1)).
% 24.80/8.34 thf(func_def_750, type, iterate_upto: !>[X0: $tType]:((code_natural > X0) > code_natural > code_natural > pred @ X0)).
% 24.80/8.34 thf(func_def_751, type, iterate_upto_rel: !>[X0: $tType]:(product_prod @ (code_natural > X0) @ product_prod @ code_natural @ code_natural > product_prod @ (code_natural > X0) @ product_prod @ code_natural @ code_natural > $o)).
% 24.80/8.34 thf(func_def_752, type, empty: !>[X0: $tType]:(seq @ X0)).
% 24.80/8.34 thf(func_def_753, type, insert: !>[X0: $tType]:(X0 > pred @ X0 > seq @ X0)).
% 24.80/8.34 thf(func_def_754, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_755, type, product_Sigma: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_756, type, product_Unity: product_unit).
% 24.80/8.34 thf(func_def_757, type, product_apfst: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X0 @ X2 > product_prod @ X1 @ X2)).
% 24.80/8.34 thf(func_def_758, type, product_apsnd: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > product_prod @ X2 @ X0 > product_prod @ X2 @ X1)).
% 24.80/8.34 thf(func_def_759, type, product_curry: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X0 @ X1 > X2) > X0 > X1 > X2)).
% 24.80/8.34 thf(func_def_760, type, produc5280177257484947105e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 24.80/8.34 thf(func_def_761, type, product_map_prod: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > product_prod @ X0 @ X2 > product_prod @ X1 @ X3)).
% 24.80/8.34 thf(func_def_762, type, product_rec_bool: !>[X0: $tType]:(X0 > X0 > $o > X0)).
% 24.80/8.34 thf(func_def_763, type, product_rec_set_bool: !>[X0: $tType]:(X0 > X0 > $o > X0 > $o)).
% 24.80/8.34 thf(func_def_764, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 24.80/8.34 thf(func_def_765, type, product_rec_set_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2 > $o)).
% 24.80/8.34 thf(func_def_766, type, product_rec_set_unit: !>[X0: $tType]:(X0 > product_unit > X0 > $o)).
% 24.80/8.34 thf(func_def_767, type, product_rec_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 24.80/8.34 thf(func_def_768, type, product_case_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 24.80/8.34 thf(func_def_769, type, product_fst: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 24.80/8.34 thf(func_def_770, type, product_snd: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 24.80/8.34 thf(func_def_771, type, product_swap: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > product_prod @ X1 @ X0)).
% 24.80/8.34 thf(func_def_772, type, product_product: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_773, type, product_scomp: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > product_prod @ X1 @ X2) > (X1 > X2 > X3) > X0 > X3)).
% 24.80/8.34 thf(func_def_774, type, product_Abs_unit: $o > product_unit).
% 24.80/8.34 thf(func_def_775, type, product_Rep_unit: product_unit > $o).
% 24.80/8.34 thf(func_def_776, type, product_case_unit: !>[X0: $tType]:(X0 > product_unit > X0)).
% 24.80/8.34 thf(func_def_777, type, type2: !>[X0: $tType]:(itself @ X0)).
% 24.80/8.34 thf(func_def_778, type, linorder_quicksort: !>[X0: $tType]:(list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_779, type, linord6200660962353139674rt_rel: !>[X0: $tType]:(list @ X0 > list @ X0 > $o)).
% 24.80/8.34 thf(func_def_780, type, quotient3: !>[X0: $tType, X1: $tType]:((X0 > X0 > $o) > (X0 > X1) > (X1 > X0) > $o)).
% 24.80/8.34 thf(func_def_781, type, inc_shift: code_natural > code_natural > code_natural).
% 24.80/8.34 thf(func_def_782, type, iterate: !>[X0: $tType, X1: $tType]:(code_natural > (X0 > X1 > product_prod @ X0 @ X1) > X0 > X1 > product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_783, type, iterate_rel: !>[X0: $tType, X1: $tType]:(product_prod @ code_natural @ product_prod @ (X0 > X1 > product_prod @ X0 @ X1) @ X0 > product_prod @ code_natural @ product_prod @ (X0 > X1 > product_prod @ X0 @ X1) @ X0 > $o)).
% 24.80/8.34 thf(func_def_784, type, log: code_natural > code_natural > code_natural).
% 24.80/8.34 thf(func_def_785, type, log_rel: product_prod @ code_natural @ code_natural > product_prod @ code_natural @ code_natural > $o).
% 24.80/8.34 thf(func_def_786, type, minus_shift: code_natural > code_natural > code_natural > code_natural).
% 24.80/8.34 thf(func_def_787, type, next: product_prod @ code_natural @ code_natural > product_prod @ code_natural @ product_prod @ code_natural @ code_natural).
% 24.80/8.34 thf(func_def_788, type, pick: !>[X0: $tType]:(list @ product_prod @ code_natural @ X0 > code_natural > X0)).
% 24.80/8.34 thf(func_def_789, type, range: code_natural > product_prod @ code_natural @ code_natural > product_prod @ code_natural @ product_prod @ code_natural @ code_natural).
% 24.80/8.34 thf(func_def_790, type, select: !>[X0: $tType]:(list @ X0 > product_prod @ code_natural @ code_natural > product_prod @ X0 @ product_prod @ code_natural @ code_natural)).
% 24.80/8.34 thf(func_def_791, type, select_weight: !>[X0: $tType]:(list @ product_prod @ code_natural @ X0 > product_prod @ code_natural @ code_natural > product_prod @ X0 @ product_prod @ code_natural @ code_natural)).
% 24.80/8.34 thf(func_def_792, type, split_seed: product_prod @ code_natural @ code_natural > product_prod @ product_prod @ code_natural @ code_natural @ product_prod @ code_natural @ code_natural).
% 24.80/8.34 thf(func_def_793, type, random_bind: !>[X0: $tType, X1: $tType]:((product_prod @ code_natural @ code_natural > product_prod @ pred @ X0 @ product_prod @ code_natural @ code_natural) > (X0 > product_prod @ code_natural @ code_natural > product_prod @ pred @ X1 @ product_prod @ code_natural @ code_natural) > product_prod @ code_natural @ code_natural > product_prod @ pred @ X1 @ product_prod @ code_natural @ code_natural)).
% 24.80/8.34 thf(func_def_794, type, random_empty: !>[X0: $tType]:(product_prod @ code_natural @ code_natural > product_prod @ pred @ X0 @ product_prod @ code_natural @ code_natural)).
% 24.80/8.34 thf(func_def_795, type, random_iterate_upto: !>[X0: $tType]:((code_natural > X0) > code_natural > code_natural > product_prod @ code_natural @ code_natural > product_prod @ pred @ X0 @ product_prod @ code_natural @ code_natural)).
% 24.80/8.34 thf(func_def_796, type, random_union: !>[X0: $tType]:((product_prod @ code_natural @ code_natural > product_prod @ pred @ X0 @ product_prod @ code_natural @ code_natural) > (product_prod @ code_natural @ code_natural > product_prod @ pred @ X0 @ product_prod @ code_natural @ code_natural) > product_prod @ code_natural @ code_natural > product_prod @ pred @ X0 @ product_prod @ code_natural @ code_natural)).
% 24.80/8.34 thf(func_def_797, type, abs_Rat: product_prod @ int @ int > rat).
% 24.80/8.34 thf(func_def_798, type, fract: int > int > rat).
% 24.80/8.34 thf(func_def_799, type, frct: product_prod @ int @ int > rat).
% 24.80/8.34 thf(func_def_800, type, rep_Rat: rat > product_prod @ int @ int).
% 24.80/8.34 thf(func_def_801, type, field_char_0_of_rat: !>[X0: $tType]:(rat > X0)).
% 24.80/8.34 thf(func_def_802, type, normalize: product_prod @ int @ int > product_prod @ int @ int).
% 24.80/8.34 thf(func_def_803, type, of_int: int > rat).
% 24.80/8.34 thf(func_def_804, type, pcr_rat: product_prod @ int @ int > rat > $o).
% 24.80/8.34 thf(func_def_805, type, positive: rat > $o).
% 24.80/8.34 thf(func_def_806, type, quotient_of: rat > product_prod @ int @ int).
% 24.80/8.34 thf(func_def_807, type, ratrel: product_prod @ int @ int > product_prod @ int @ int > $o).
% 24.80/8.34 thf(func_def_808, type, ref_alloc: !>[X0: $tType]:(X0 > heap_ext @ product_unit > product_prod @ ref @ X0 @ heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_809, type, ref_change: !>[X0: $tType]:((X0 > X0) > ref @ X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_810, type, ref_get: !>[X0: $tType]:(heap_ext @ product_unit > ref @ X0 > X0)).
% 24.80/8.34 thf(func_def_811, type, ref_lookup: !>[X0: $tType]:(ref @ X0 > heap_Time_Heap @ X0)).
% 24.80/8.34 thf(func_def_812, type, ref_present: !>[X0: $tType]:(heap_ext @ product_unit > ref @ X0 > $o)).
% 24.80/8.34 thf(func_def_813, type, ref_ref: !>[X0: $tType]:(X0 > heap_Time_Heap @ ref @ X0)).
% 24.80/8.34 thf(func_def_814, type, ref_set: !>[X0: $tType]:(ref @ X0 > X0 > heap_ext @ product_unit > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_815, type, ref_update: !>[X0: $tType]:(ref @ X0 > X0 > heap_Time_Heap @ product_unit)).
% 24.80/8.34 thf(func_def_816, type, refine_Imp_assert: $o > heap_Time_Heap @ product_unit).
% 24.80/8.34 thf(func_def_817, type, refine_Imp_refines: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_Time_Heap @ X0 > $o)).
% 24.80/8.34 thf(func_def_818, type, domain: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0)).
% 24.80/8.34 thf(func_def_819, type, domainp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X0 > $o)).
% 24.80/8.34 thf(func_def_820, type, field2: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_821, type, id2: !>[X0: $tType]:(set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_822, type, id_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_823, type, image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X0 > set @ X1)).
% 24.80/8.34 thf(func_def_824, type, range2: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ X1)).
% 24.80/8.34 thf(func_def_825, type, rangep: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1 > $o)).
% 24.80/8.34 thf(func_def_826, type, antisym: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_827, type, antisymp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_828, type, asym: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_829, type, asymp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_830, type, converse: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X0)).
% 24.80/8.34 thf(func_def_831, type, conversep: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > X1 > X0 > $o)).
% 24.80/8.34 thf(func_def_832, type, inv_image: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > (X1 > X0) > set @ product_prod @ X1 @ X1)).
% 24.80/8.34 thf(func_def_833, type, irrefl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_834, type, irreflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_835, type, refl_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_836, type, reflp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_837, type, relcomp: !>[X0: $tType, X1: $tType, X2: $tType]:(set @ product_prod @ X0 @ X1 > set @ product_prod @ X1 @ X2 > set @ product_prod @ X0 @ X2)).
% 24.80/8.34 thf(func_def_838, type, relcompp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > $o) > (X1 > X2 > $o) > X0 > X2 > $o)).
% 24.80/8.34 thf(func_def_839, type, single_valued: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X1 > $o)).
% 24.80/8.34 thf(func_def_840, type, single_valuedp: !>[X0: $tType, X1: $tType]:((X0 > X1 > $o) > $o)).
% 24.80/8.34 thf(func_def_841, type, total_on: !>[X0: $tType]:(set @ X0 > set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_842, type, trans: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_843, type, transp: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_844, type, algebr8660921524188924756oprime: !>[X0: $tType]:(X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_845, type, divide_divide: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_846, type, dvd_dvd: !>[X0: $tType]:(X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_847, type, modulo_modulo: !>[X0: $tType]:(X0 > X0 > X0)).
% 24.80/8.34 thf(func_def_848, type, normal6383669964737779283malize: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_849, type, unit_f5069060285200089521factor: !>[X0: $tType]:(X0 > X0)).
% 24.80/8.34 thf(func_def_850, type, zero_neq_one_of_bool: !>[X0: $tType]:($o > X0)).
% 24.80/8.34 thf(func_def_851, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 24.80/8.34 thf(func_def_852, type, pow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 24.80/8.34 thf(func_def_853, type, bind3: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > set @ X1) > set @ X1)).
% 24.80/8.34 thf(func_def_854, type, disjnt: !>[X0: $tType]:(set @ X0 > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_855, type, image2: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 24.80/8.34 thf(func_def_856, type, insert2: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_857, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 24.80/8.34 thf(func_def_858, type, vimage: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X1 > set @ X0)).
% 24.80/8.34 thf(func_def_859, type, set_fo6178422350223883121st_nat: !>[X0: $tType]:((nat > X0 > X0) > nat > nat > X0 > X0)).
% 24.80/8.34 thf(func_def_860, type, set_fo1817059534552279752at_rel: !>[X0: $tType]:(product_prod @ (nat > X0 > X0) @ product_prod @ nat @ product_prod @ nat @ X0 > product_prod @ (nat > X0 > X0) @ product_prod @ nat @ product_prod @ nat @ X0 > $o)).
% 24.80/8.34 thf(func_def_861, type, set_ord_atLeast: !>[X0: $tType]:(X0 > set @ X0)).
% 24.80/8.34 thf(func_def_862, type, set_or1337092689740270186AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_863, type, set_or7035219750837199246ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_864, type, set_ord_atMost: !>[X0: $tType]:(X0 > set @ X0)).
% 24.80/8.34 thf(func_def_865, type, set_ord_greaterThan: !>[X0: $tType]:(X0 > set @ X0)).
% 24.80/8.34 thf(func_def_866, type, set_or3652927894154168847AtMost: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_867, type, set_or5935395276787703475ssThan: !>[X0: $tType]:(X0 > X0 > set @ X0)).
% 24.80/8.34 thf(func_def_868, type, set_ord_lessThan: !>[X0: $tType]:(X0 > set @ X0)).
% 24.80/8.34 thf(func_def_869, type, char2: $o > $o > $o > $o > $o > $o > $o > $o > char).
% 24.80/8.34 thf(func_def_870, type, integer_of_char: char > code_integer).
% 24.80/8.34 thf(func_def_871, type, unique5772411509450598832har_of: !>[X0: $tType]:(X0 > char)).
% 24.80/8.34 thf(func_def_872, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 24.80/8.34 thf(func_def_873, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 24.80/8.34 thf(func_def_874, type, sum_Plus: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ sum_sum @ X0 @ X1)).
% 24.80/8.34 thf(func_def_875, type, sum_map_sum: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((X0 > X1) > (X2 > X3) > sum_sum @ X0 @ X2 > sum_sum @ X1 @ X3)).
% 24.80/8.34 thf(func_def_876, type, sum_case_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 24.80/8.34 thf(func_def_877, type, syntax7388354845996824322omatch: !>[X0: $tType, X1: $tType]:(X0 > X1 > $o)).
% 24.80/8.34 thf(func_def_878, type, time_fails: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > $o)).
% 24.80/8.34 thf(func_def_879, type, time_the_heap: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_880, type, time_the_res: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > X0)).
% 24.80/8.34 thf(func_def_881, type, time_time: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_ext @ product_unit > nat)).
% 24.80/8.34 thf(func_def_882, type, transitive_acyclic: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_883, type, transitive_ntrancl: !>[X0: $tType]:(nat > set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_884, type, transitive_rtrancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_885, type, transitive_rtranclp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_886, type, transitive_trancl: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_887, type, transitive_tranclp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_888, type, type_definition: !>[X0: $tType, X1: $tType]:((X0 > X1) > (X1 > X0) > set @ X1 > $o)).
% 24.80/8.34 thf(func_def_889, type, typerep_typerep: !>[X0: $tType]:(itself @ X0 > typerep)).
% 24.80/8.34 thf(func_def_890, type, acc: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ X0)).
% 24.80/8.34 thf(func_def_891, type, accp: !>[X0: $tType]:((X0 > X0 > $o) > X0 > $o)).
% 24.80/8.34 thf(func_def_892, type, finite_psubset: !>[X0: $tType]:(set @ product_prod @ set @ X0 @ set @ X0)).
% 24.80/8.34 thf(func_def_893, type, less_than: set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_894, type, lex_prod: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ X1 @ X1 > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_895, type, max_ext: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 24.80/8.34 thf(func_def_896, type, max_extp: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_897, type, measure: !>[X0: $tType]:((X0 > nat) > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_898, type, min_ext: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ product_prod @ set @ X0 @ set @ X0)).
% 24.80/8.34 thf(func_def_899, type, mlex_prod: !>[X0: $tType]:((X0 > nat) > set @ product_prod @ X0 @ X0 > set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_900, type, pred_nat: set @ product_prod @ nat @ nat).
% 24.80/8.34 thf(func_def_901, type, wf: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > $o)).
% 24.80/8.34 thf(func_def_902, type, wfP: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 24.80/8.34 thf(func_def_903, type, adm_wf: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X0 @ X0 > ((X0 > X1) > X0 > X1) > $o)).
% 24.80/8.34 thf(func_def_904, type, cut: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ product_prod @ X0 @ X0 > X0 > X0 > X1)).
% 24.80/8.34 thf(func_def_905, type, same_fst: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X0 > set @ product_prod @ X1 @ X1) > set @ product_prod @ product_prod @ X0 @ X1 @ product_prod @ X0 @ X1)).
% 24.80/8.34 thf(func_def_906, type, chains: !>[X0: $tType]:(set @ product_prod @ X0 @ X0 > set @ set @ X0)).
% 24.80/8.34 thf(func_def_907, type, init_seg_of: !>[X0: $tType]:(set @ product_prod @ set @ product_prod @ X0 @ X0 @ set @ product_prod @ X0 @ X0)).
% 24.80/8.34 thf(func_def_908, type, pred_chain: !>[X0: $tType]:(set @ X0 > (X0 > X0 > $o) > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_909, type, fChoice: !>[X0: $tType]:((X0 > $o) > X0)).
% 24.80/8.34 thf(func_def_910, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 24.80/8.34 thf(func_def_911, type, h: heap_ext @ product_unit).
% 24.80/8.34 thf(func_def_912, type, xs: list @ a).
% 24.80/8.34 thf(func_def_934, type, sP0: !>[X0: $tType, X1: $tType]:((option @ X1 > option @ X0 > $o) > option @ X1 > option @ X0 > $o)).
% 24.80/8.34 thf(func_def_935, type, sK1: product_prod @ heap_ext @ product_unit @ set @ nat > heap_ext @ product_unit).
% 24.80/8.34 thf(func_def_936, type, sK2: product_prod @ heap_ext @ product_unit @ set @ nat > set @ nat).
% 24.80/8.34 thf(func_def_937, type, sK3: !>[X0: $tType]:(heap_Time_Heap @ X0 > heap_Time_Heap @ X0 > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_938, type, sK4: !>[X0: $tType]:(product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > nat)).
% 24.80/8.34 thf(func_def_939, type, sK5: !>[X0: $tType]:(product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > nat)).
% 24.80/8.34 thf(func_def_940, type, sK6: !>[X0: $tType]:(product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > X0)).
% 24.80/8.34 thf(func_def_941, type, sK7: !>[X0: $tType]:(product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_942, type, sK8: !>[X0: $tType]:(product_prod @ nat @ option @ product_prod @ X0 @ product_prod @ heap_ext @ product_unit @ nat > nat)).
% 24.80/8.34 thf(func_def_943, type, sK9: !>[X0: $tType]:(product_prod @ array @ X0 @ product_prod @ list @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > array @ X0)).
% 24.80/8.34 thf(func_def_944, type, sK10: !>[X0: $tType]:(product_prod @ array @ X0 @ product_prod @ list @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > list @ X0)).
% 24.80/8.34 thf(func_def_945, type, sK11: !>[X0: $tType]:(product_prod @ array @ X0 @ product_prod @ list @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > heap_ext @ product_unit)).
% 24.80/8.34 thf(func_def_946, type, sK12: !>[X0: $tType]:(product_prod @ array @ X0 @ product_prod @ list @ X0 @ product_prod @ heap_ext @ product_unit @ set @ nat > set @ nat)).
% 24.80/8.34 thf(func_def_947, type, sK13: !>[X0: $tType, X1: $tType]:((option @ X1 > option @ X0 > $o) > option @ X1 > option @ X0 > X1)).
% 24.80/8.34 thf(func_def_948, type, sK14: !>[X0: $tType, X1: $tType]:((option @ X1 > option @ X0 > $o) > option @ X1 > option @ X0 > X0)).
% 24.80/8.34 thf(func_def_949, type, sK15: !>[X0: $tType]:(option @ X0 > X0)).
% 24.80/8.34 thf(func_def_950, type, sK16: product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_951, type, sK17: product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > product_prod @ heap_ext @ product_unit @ set @ nat > $o).
% 24.80/8.34 thf(func_def_952, type, sK18: product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > heap_ext @ product_unit).
% 24.80/8.34 thf(func_def_953, type, sK19: product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ (product_prod @ heap_ext @ product_unit @ set @ nat > $o) @ product_prod @ heap_ext @ product_unit @ set @ nat > set @ nat).
% 24.80/8.34 thf(func_def_954, type, sK20: !>[X0: $tType]:(heap_ext @ product_unit > heap_Time_Heap @ X0 > X0)).
% 24.80/8.34 thf(func_def_955, type, sK21: !>[X0: $tType]:(heap_ext @ product_unit > heap_Time_Heap @ X0 > product_prod @ heap_ext @ product_unit @ nat)).
% 24.80/8.34 thf(func_def_956, type, sK22: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 24.80/8.34 thf(func_def_957, type, sK23: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ list @ X0 > X0 > X0 > $o)).
% 24.80/8.34 thf(func_def_958, type, sK24: !>[X0: $tType]:(product_prod @ (X0 > X0 > $o) @ list @ X0 > list @ X0)).
% 24.80/8.34 thf(func_def_960, type, ph26: !>[X0: $tType]:(X0)).
% 24.80/8.34 thf(f19068,plain,(
% 24.80/8.34 $false),
% 24.80/8.34 inference(trivial_inequality_removal,[status(thm)],[f19066])).
% 24.80/8.34 thf(f19066,plain,(
% 24.80/8.34 (none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat != none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat)),
% 24.80/8.34 inference(superposition,[status(thm)],[f18787,f17917])).
% 24.80/8.34 thf(f17917,plain,(
% 24.80/8.34 ((heap_Time_execute @ array @ a @ (array_of_list @ a @ xs) @ h) = none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat)),
% 24.80/8.34 inference(cnf_transformation,[status(thm)],[f17825])).
% 24.80/8.34 thf(f17825,plain,(
% 24.80/8.34 ((heap_Time_execute @ array @ a @ (array_of_list @ a @ xs) @ h) = none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat)),
% 24.80/8.34 inference(flattening,[status(thm)],[f6700])).
% 24.80/8.34 thf(f6700,negated_conjecture,(
% 24.80/8.34 ~((heap_Time_execute @ array @ a @ (array_of_list @ a @ xs) @ h) != none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat)),
% 24.80/8.34 inference(negated_conjecture,[status(cth)],[f6699])).
% 24.80/8.34 thf(f6699,conjecture,(
% 24.80/8.34 ((heap_Time_execute @ array @ a @ (array_of_list @ a @ xs) @ h) != none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat)),
% 24.80/8.34 file('/export/starexec/sandbox2/tmp/tmp.EbIqTwV4kM/DTF2THF_24760.p',conj_0)).
% 24.80/8.34 thf(f18787,plain,(
% 24.80/8.34 ( ! [X0 : list @ a,X1 : heap_ext @ product_unit] : ((none @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat != (heap_Time_execute @ array @ a @ (array_of_list @ a @ X0) @ X1))) )),
% 24.80/8.34 inference(superposition,[status(thm)],[f17914,f17960])).
% 24.80/8.34 thf(f17960,plain,(
% 24.80/8.34 ( ! [X0 : list @ a,X1 : heap_ext @ product_unit] : (((heap_Time_execute @ array @ a @ (array_of_list @ a @ X0) @ X1) = (some @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ a @ heap_ext @ product_unit @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat @ (^[Y0 : array @ a]: ((^[Y1 : heap_ext @ product_unit]: (product_Pair @ array @ a @ product_prod @ heap_ext @ product_unit @ nat @ Y0 @ (product_Pair @ heap_ext @ product_unit @ nat @ Y1 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ a @ X0))))))) @ (array_alloc @ a @ X0 @ X1))))) )),
% 24.80/8.34 inference(trivial_inequality_removal,[status(thm)],[f17947])).
% 24.80/8.34 thf(f17947,plain,(
% 24.80/8.34 ( ! [X0 : list @ a,X1 : heap_ext @ product_unit] : (($true != $true) | ((heap_Time_execute @ array @ a @ (array_of_list @ a @ X0) @ X1) = (some @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ a @ heap_ext @ product_unit @ product_prod @ array @ a @ product_prod @ heap_ext @ product_unit @ nat @ (^[Y0 : array @ a]: ((^[Y1 : heap_ext @ product_unit]: (product_Pair @ array @ a @ product_prod @ heap_ext @ product_unit @ nat @ Y0 @ (product_Pair @ heap_ext @ product_unit @ nat @ Y1 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ a @ X0))))))) @ (array_alloc @ a @ X0 @ X1))))) )),
% 24.80/8.34 inference(superposition,[status(thm)],[f17928,f17926])).
% 24.80/8.34 thf(f17926,plain,(
% 24.80/8.34 (heap @ a = $true)),
% 24.80/8.34 inference(cnf_transformation,[status(thm)],[f11117])).
% 24.80/8.34 thf(f11117,plain,(
% 24.80/8.34 (heap @ a = $true)),
% 24.80/8.34 inference(fool_elimination,[status(thm)],[f11116])).
% 24.80/8.34 thf(f11116,plain,(
% 24.80/8.34 heap @ a),
% 24.80/8.34 inference(rectify,[status(thm)],[f6698])).
% 24.80/8.34 thf(f6698,axiom,(
% 24.80/8.34 heap @ a),
% 24.80/8.34 file('/export/starexec/sandbox2/tmp/tmp.EbIqTwV4kM/DTF2THF_24760.p',tfree_0)).
% 24.80/8.34 thf(f17928,plain,(
% 24.80/8.34 ( ! [X0 : $tType,X2 : heap_ext @ product_unit,X1 : list @ X0] : ((heap @ X0 != $true) | ((heap_Time_execute @ array @ X0 @ (array_of_list @ X0 @ X1) @ X2) = (some @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ X0 @ heap_ext @ product_unit @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (^[Y0 : array @ X0]: ((^[Y1 : heap_ext @ product_unit]: (product_Pair @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ Y0 @ (product_Pair @ heap_ext @ product_unit @ nat @ Y1 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ X0 @ X1))))))) @ (array_alloc @ X0 @ X1 @ X2))))) )),
% 24.80/8.34 inference(cnf_transformation,[status(thm)],[f17848])).
% 24.80/8.34 thf(f17848,plain,(
% 24.80/8.34 ! [X0 : $tType] : (! [X1 : list @ X0,X2 : heap_ext @ product_unit] : ((heap_Time_execute @ array @ X0 @ (array_of_list @ X0 @ X1) @ X2) = (some @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ X0 @ heap_ext @ product_unit @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (^[Y0 : array @ X0]: ((^[Y1 : heap_ext @ product_unit]: (product_Pair @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ Y0 @ (product_Pair @ heap_ext @ product_unit @ nat @ Y1 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ X0 @ X1))))))) @ (array_alloc @ X0 @ X1 @ X2)))) | (heap @ X0 != $true))),
% 24.80/8.34 inference(ennf_transformation,[status(thm)],[f16442])).
% 24.80/8.34 thf(f16442,plain,(
% 24.80/8.34 ! [X0 : $tType] : ((heap @ X0 = $true) => ! [X1 : list @ X0,X2 : heap_ext @ product_unit] : ((heap_Time_execute @ array @ X0 @ (array_of_list @ X0 @ X1) @ X2) = (some @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ X0 @ heap_ext @ product_unit @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (^[Y0 : array @ X0]: ((^[Y1 : heap_ext @ product_unit]: (product_Pair @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ Y0 @ (product_Pair @ heap_ext @ product_unit @ nat @ Y1 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ X0 @ X1))))))) @ (array_alloc @ X0 @ X1 @ X2)))))),
% 24.80/8.34 inference(fool_elimination,[status(thm)],[f16441])).
% 24.80/8.34 thf(f16441,plain,(
% 24.80/8.34 ! [X0 : $tType] : (heap @ X0 => ! [X1 : list @ X0,X2 : heap_ext @ product_unit] : ((some @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ X0 @ heap_ext @ product_unit @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (^[X3 : array @ X0, X4 : heap_ext @ product_unit] : (product_Pair @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ X3 @ (product_Pair @ heap_ext @ product_unit @ nat @ X4 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ X0 @ X1))))) @ (array_alloc @ X0 @ X1 @ X2))) = (heap_Time_execute @ array @ X0 @ (array_of_list @ X0 @ X1) @ X2)))),
% 24.80/8.34 inference(rectify,[status(thm)],[f3531])).
% 24.80/8.34 thf(f3531,axiom,(
% 24.80/8.34 ! [X0 : $tType] : (heap @ X0 => ! [X29 : list @ X0,X17 : heap_ext @ product_unit] : ((some @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (product_case_prod @ array @ X0 @ heap_ext @ product_unit @ product_prod @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ (^[X70 : array @ X0, X69 : heap_ext @ product_unit] : (product_Pair @ array @ X0 @ product_prod @ heap_ext @ product_unit @ nat @ X70 @ (product_Pair @ heap_ext @ product_unit @ nat @ X69 @ (plus_plus @ nat @ one_one @ nat @ (size_size @ list @ X0 @ X29))))) @ (array_alloc @ X0 @ X29 @ X17))) = (heap_Time_execute @ array @ X0 @ (array_of_list @ X0 @ X29) @ X17)))),
% 24.80/8.34 file('/export/starexec/sandbox2/tmp/tmp.EbIqTwV4kM/DTF2THF_24760.p',fact_3530_execute__of__list)).
% 24.80/8.34 thf(f17914,plain,(
% 24.80/8.34 ( ! [X0 : $tType,X1 : X0] : ((none @ X0 != (some @ X0 @ X1))) )),
% 24.80/8.34 inference(cnf_transformation,[status(thm)],[f17834])).
% 24.80/8.34 thf(f17834,plain,(
% 24.80/8.34 ! [X0 : $tType,X1 : X0] : (none @ X0 != (some @ X0 @ X1))),
% 24.80/8.34 inference(rectify,[status(thm)],[f49])).
% 24.80/8.34 thf(f49,axiom,(
% 24.80/8.34 ! [X0 : $tType,X34 : X0] : (none @ X0 != (some @ X0 @ X34))),
% 24.80/8.34 file('/export/starexec/sandbox2/tmp/tmp.EbIqTwV4kM/DTF2THF_24760.p',fact_48_option_Odistinct_I1_J)).
% 24.80/8.34 % SZS output end Proof for DTF2THF_24760
% 24.80/8.34 % (24967)------------------------------
% 24.80/8.34 % (24967)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 24.80/8.34 % (24967)Termination reason: Refutation
% 24.80/8.34
% 24.80/8.34 % (24967)Memory used [KB]: 33645
% 24.80/8.34 % (24967)Time elapsed: 0.632 s
% 24.80/8.34 % (24967)Instructions burned: 1870 (million)
% 24.80/8.34 % (24967)------------------------------
% 24.80/8.34 % (24967)------------------------------
% 24.80/8.34 % (24866)Success in time 2.296 s
%------------------------------------------------------------------------------